Robust Finite-Time H-Infinity Control with Transients for...

18
Research Article Robust Finite-Time H-Infinity Control with Transients for Dynamic Positioning Ship Subject to Input Delay Xiaogong Lin, Kun Liang , Heng Li, Yuzhao Jiao , and Jun Nie College of Automation, Harbin Engineering University, Harbin, China Correspondence should be addressed to Kun Liang; [email protected] Received 1 January 2018; Revised 4 April 2018; Accepted 16 May 2018; Published 26 June 2018 Academic Editor: Sabri Arik Copyright © 2018 Xiaogong Lin et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. is paper presents the problem of robust finite-time ∞ control with transients for ocean surface vessels equipped with dynamic positioning (DP) system in presence of input delay. e main objective of this work is to design a finite-time ∞ state feedback controller, which ensures that all states of ship do not exceed a given threshold over a fixed time interval, with better robustness and transient performance subject to time-varying disturbance. Based on a novel augmented Lyapunov-Krasovskii-like function (LKLF) with triple integral terms and a method combining the Wirtinger inequality and reciprocally convex approach, a less conservative result is derived. In particular, an ∞ performance index with nonzero initial condition is introduced to attenuate the overconservatism caused by the assumption of zero initial condition and enhance the transient performance of ship subject to external disturbance. More precisely, the controller gain matrix for the DP system can be achieved by solving the linear matrix inequalities (LMIs), which can be easily facilitated by using some standard numerical packages. Finally, a numerical simulation for a ship is proposed to verify the effectiveness and less conservatism of the controller we designed. 1. Introductions With the increasing development in the ocean exploitation, dynamic positioning (DP) systems, regulating the horizontal position and heading of the vessel exclusively by means of active thrusters, have been developed for various marine and offshore applications such as drilling, salvage, pipe- laying, and oil production [1]. To achieve expected tra- jectory tracking or positioning, various control strategies have been proposed, including robust adaptive control [2], sliding mode control [3], prescribed performance control [4], hybrid control [5], and neural network control [6]. However, in some applications, it is significant to maintain the vessel’s states under some bounds, particularly of which transient performance is emphasized, during a specific time interval, for example, when facing the matter of saturations or when the task of trajectory tracking should be fulfilled in a prescribed time interval. In [7], the proposed controller can maintain the bound of ship over an infinity time interval regardless of disturbance. However, it is inconvenient to analyze and to enhance the transient performance when the operation should be arrived in short time, such as rescuing works. On the other hand, time delay is encountered in many dynamical systems and oſten leads to performance deterioration which engenders strongly growing interest in this topic in recent decades. In the DP system, the main kind of time delay is encountered in actuators [8], while another obvious kind of delay is the one produced between the sensors and the activation of the control mechanism [9]. e effect caused by time delay will be more significant in finite-time interval and it is the first motivation of this paper. e classical ∞ control theory [10] defines the control law, under the assumption of zero initial condition, providing the minimal value to the performance measure that is the worst-case norm of the regulated output over all exogenous signals and is applied to DP system successfully [11]. However, there exist some situations, when the initial state of ship is possibly nonzero, such as when saturation is activated or controller is switched to another one. It will cause an additional unknown disturbance [12] and the promising robustness is achieved at the expense of degraded nominal performance [13]. In [14], researchers introduced a perfor- mance measure that is the induced norm of the regulated output over all exogenous signals and initial states for finite Hindawi Mathematical Problems in Engineering Volume 2018, Article ID 2838749, 17 pages https://doi.org/10.1155/2018/2838749

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Research ArticleRobust Finite-Time H-Infinity Control with Transients forDynamic Positioning Ship Subject to Input Delay

Xiaogong Lin Kun Liang Heng Li Yuzhao Jiao and Jun Nie

College of Automation Harbin Engineering University Harbin China

Correspondence should be addressed to Kun Liang drliangkun126com

Received 1 January 2018 Revised 4 April 2018 Accepted 16 May 2018 Published 26 June 2018

Academic Editor Sabri Arik

Copyright copy 2018 Xiaogong Lin et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper presents the problem of robust finite-time119867infin control with transients for ocean surface vessels equipped with dynamicpositioning (DP) system in presence of input delay The main objective of this work is to design a finite-time 119867infin state feedbackcontroller which ensures that all states of ship do not exceed a given threshold over a fixed time interval with better robustnessand transient performance subject to time-varying disturbance Based on a novel augmented Lyapunov-Krasovskii-like function(LKLF) with triple integral terms and a method combining the Wirtinger inequality and reciprocally convex approach a lessconservative result is derived In particular an 119867infin performance index with nonzero initial condition is introduced to attenuatethe overconservatism caused by the assumption of zero initial condition and enhance the transient performance of ship subjectto external disturbance More precisely the controller gain matrix for the DP system can be achieved by solving the linear matrixinequalities (LMIs) which can be easily facilitated by using some standard numerical packages Finally a numerical simulation fora ship is proposed to verify the effectiveness and less conservatism of the controller we designed

1 Introductions

With the increasing development in the ocean exploitationdynamic positioning (DP) systems regulating the horizontalposition and heading of the vessel exclusively by means ofactive thrusters have been developed for various marineand offshore applications such as drilling salvage pipe-laying and oil production [1] To achieve expected tra-jectory tracking or positioning various control strategieshave been proposed including robust adaptive control [2]sliding mode control [3] prescribed performance control[4] hybrid control [5] and neural network control [6]However in some applications it is significant to maintainthe vesselrsquos states under some bounds particularly of whichtransient performance is emphasized during a specific timeinterval for example when facing the matter of saturationsor when the task of trajectory tracking should be fulfilledin a prescribed time interval In [7] the proposed controllercan maintain the bound of ship over an infinity time intervalregardless of disturbance However it is inconvenient toanalyze and to enhance the transient performance when theoperation should be arrived in short time such as rescuing

works On the other hand time delay is encountered inmany dynamical systems and often leads to performancedeterioration which engenders strongly growing interest inthis topic in recent decades In the DP system the main kindof time delay is encountered in actuators [8] while anotherobvious kind of delay is the one produced between the sensorsand the activation of the control mechanism [9] The effectcaused by time delay will be more significant in finite-timeinterval and it is the first motivation of this paper

The classical 119867infin control theory [10] defines the controllaw under the assumption of zero initial condition providingthe minimal value to the performance measure that is theworst-case norm of the regulated output over all exogenoussignals and is applied toDP system successfully [11] Howeverthere exist some situations when the initial state of shipis possibly nonzero such as when saturation is activatedor controller is switched to another one It will cause anadditional unknown disturbance [12] and the promisingrobustness is achieved at the expense of degraded nominalperformance [13] In [14] researchers introduced a perfor-mance measure that is the induced norm of the regulatedoutput over all exogenous signals and initial states for finite

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 2838749 17 pageshttpsdoiorg10115520182838749

2 Mathematical Problems in Engineering

and infinite horizons Unfortunately to the best the authorsrsquoknowledge it has not been extended to the system with inputdelay On another research frontier various approaches inthe framework of finite-time boundedness for time-delaysystems have been developed to obtain the results with lessconservatism [15 16] usually indicated by the bound ofstate [17] For DP system engineers are willing to obtaincontrol strategies which maintain the state of ship varyingin a small region around the desired set-point or trackingpath instead of the acceptable minimum state bound Thusthe second motivate is to obtain a result which reduces theoverconservatism caused by the assumption of zero initialcondition and the loss information in the proof for time delaysystems in a practical view

Based on the discussion aforementioned the problemof robust finite-time 119867infin control for DP system with inputdelay is studied The main contribution of this paper liesin three aspects Firstly a finite-time 119867infin controller withtransients is designed for DP systems with input delay bysolving a couple of LMIs which can guarantee the stateof ship within a desired value over a fixed time intervalin the presence of time-varying disturbances Secondly theconcept of 119867infin control under nonzero initial conditionis introduced to time-delay system and its advantage onenhancing the transient performance will be shown in anumerical simulation compared to the119867infin with zero initialcondition In particular the results are established in forms ofLMI which can be easily facilitated by using some standardnumerical packages Thirdly a novel augmented LKLF withtriple integral terms which contains more information isconstructed meanwhile a method combining the Wirtingerinequality and reciprocally convex approach is applied toobtain a tight bound of the integral terms of quadratic func-tions which may lead to a less conservative result comparedto previous works Moreover the practical significance of thismethod in engineering will be demonstrated later Finallya numerical simulation for a ship is proposed to verify theeffectiveness and advantage of the controller we designed

The rest of this paper is organized as follows In Sec-tion 2 the problem formulation of finite-time 119867infin controlis detailed for vessels while some definitions and lemmas areintroduced as preparation In Section 3 themethod to designfinite-time 119867infin controller is proposed and the proposedcontrol schemes are simulated in Section 4 In Section 5 theconclusion is drawn

Notation Throughout this paper R119899 is the 119899-dimensionalEuclidean vector space and R119898times119899 denotes the set of all119898 times 119899 real matrices For symmetric matrices 119883 and 119884119883 gt 119884 (respectively 119883 ge 119884 ) means that 119883 minus 119884is positive definite (respectively positive semidefinite) Thesuperscript ldquo119879rdquo represents the transpose The symmetricterms in a symmetric matrix are denoted by ldquolowastrdquo Moreoverwe use 120582max(sdot)(120582min(sdot) ) to denote the maximum (minimum)eigenvalue of a symmetric matrix

2 Problem Formulation

21 Model of DP System At first DP system model withthree-DOP under low speed can be described as [18]

120578 (119905) = 119869 (120595) 120592 (119905) (1)119872 120592 (119905) + 119863120592 (119905) = 119906 (119905 minus ℎ (119905)) + 120596 (119905) (2)

where 120592 = (120583 V 119903) is a vector of velocities given in thebody-fixed coordinate system and 120578 = (119883 119884 120595) is theposition and orientation of the vessel with respect to aninertial reference coordinate system 119906(119905) is a control vector offorces andmoments provided by the propulsion systems ℎ(119905)is a time-varying function that expresses the actuator delayand satisfies ℎ(119905) lt ℎ ℎ(119905) lt 120583 120596(119905) is the disturbance inputof the system and satisfies the condition of intinfin

0120596(119905)119879120596(119905)119889119905 lt119889 119869(120595) is the transformationmatrix between the inertial and

body-fixed coordinate framesThe inertia matrix119872 includeshydrodynamic added inertia and 119863 is a strictly positivedampingmatrix due to linearwave drift damping and laminarfriction Then the structures of the matrices 119869(120595)119872 and 119863can be explicitly given as follows

119869 (120595) = (cos (120595) minussin (120595) 0sin (120595) cos (120595) 00 0 1)

119872 = (119898 minus 119883 0 00 119898 minus 119884V 119898119909119892 minus 119884 1199030 119898119909119892 minus 119884 119903 119868119911 minus 119873 119903

)119863 = minus(119883119906 0 00 119884V 1198841199030 119873V 119873119903

)(3)

where 119883119906 119884V 119884119903 119873119903 119883 119884V 119884 119903 and 119873 119903 are the hydro-dynamic parameters of the vessel 119898 is the mass of the shipand 119868119911 is the moment of inertia about the yaw rotation 119909119892is the vertical distance from coordinate origin to center ofgravity in body-fixed frame

To simplify the model some assumptions are introducedfirst

Assumption 1 All the parameters of state are available

Assumption 2 The roll and pitch angles are small enough forDP system it is a reasonable assumption

Under assumption 2 we can obtain the simplified equa-tion in form of state-space [19] (119905) = 119860119909 (119905) + 1198611119906 (119905 minus ℎ (119905)) + 1198612120596 (119905) (4)119911 (119905) = 119862119909 (119905) + 119863119906 (119905 minus ℎ (119905)) (5)

where 119909 (119905) = (120578 (119905) 120592 (119905))119860 = (0 1198680 minus119872minus1119863)

Mathematical Problems in Engineering 3

1198611 = 1198612 = ( 0119872minus1)

(6)

Under assumption 1 we design the full-state feedback as119906 (119905 minus ℎ (119905)) = 119870119909 (119905 minus ℎ (119905)) (7)

so the (1) and (2) can be rewritten as (119905) = 119860119909 (119905) + 1198611119870119909 (119905 minus ℎ (119905)) + 1198612120596 (119905) (8)119911 (119905) = 119862119909 (119905) + 119863119870119909 (119905 minus ℎ (119905)) (9)

Remark 3 In the situation of DP motion the motion inheave roll and pitch will be ignored since we focus onthe motion on the surface of sea Meanwhile velocity ofvessel is low enough so the Coriolis-Centripetal matrix andnonlinearities in damping matrix can be neglected [18]

Remark 4 In measurement subsystem of DP system varioussensors are installed to obtain the state of the vessel motionaccurately including global position system for the vesselrsquo sposition gyrocompass for the vesselrsquo s heading and attitudesensor for the pitch and roll Meanwhile data fusion tech-nology is applied to DP system to obtain more accurate stateinformation of vessel motion [18]

22 Preliminaries In the sequel some definitions and lem-mas are introduced to obtain the results

Definition 5 ((FTB) see [20]) Given a positive definitematrix 119877 and three positive constants 1198881 1198882 119879 with 1198881 lt1198882 the time-delay system (4) with 119906(119905) = 0 is said to befinite-time boundedness with respect to (1198881 1198882 119889 119879 120591 119877)if supminus120591lt120579lt0119909119879(120579)119877119909(120579) 119879(120579)119877(120579) lt 1198881 rArr 119909119879(119905)119877119909(119905) lt1198882 forall119905 isin [0 119879] forall120596(119905) int1198790 120596(119905)119879120596(119905)119889119905 lt 119889Definition 6 Given 120574 gt 0 time-delay systems (4) and (5) aresaid to be 119867infin- FTB under nonzero initial condition (119867infin-FTB) with (1198881 1198882 119889 120574 119879 120591 119877) if the following conditionsare satisfied

(1) time-delay system (4) with 119906(119905) = 0 is FTB(2) under the initial condition that 119909(0) = 0 output 119911(119905)

satisfiesΓ= sup

120596(119905)2[0119879]2+119909(0)119879119878119909(0) =0

[ 119911 (119905)2[0119879]2120596 (119905)2[0119879]2 + 119909 (0)119879 119878119909 (0)]12lt 120574(10)

where 120574 is a prescribed scalar and 119878 is a weighting diagonalmatrix that penalizes the effect caused by the initial state

Remark 7 It is necessary to distinguish between finite-timestability and finite-time attractiveness [21] The first conceptis to maintain system states within a given boundary in aspecified time interval [22 23] while the latter describes the

fact that system state reaches the equilibrium point of systemin a finite time [24]

Remark 8 Similar to [13] this definition of 119867infin perfor-mance which depends on the initial condition has beenextended to linear system with acceptable maximal delaybound The performance measure is parametrised by aweighting matrix 119878 reflecting the relative importance of theuncertainty in the initial state contrary to the uncertainty inthe exogenous disturbance When 119878 = 0 it will reduce into asort of119867infin control with zero initial condition [25ndash27]

Lemma 9 (see [28]) For anymatrix119875 gt 0 and a differentiablesignal 119909 in [minus119886 minus119887] rarr R119899 the following inequality holdsintminus119887

minus119886119879 (119906) 1198781 (119906) 119889119904 ge 1119886 minus 119887sdot Π1 (1198781 119909119879 (minus119887) 119909119879 (minus119886) 1119886 minus 119887 intminus119887minus119886

119909119879 (119906) 119889119906) (11)

whereΠ1 (1198781 120572 120573 120574)= ( (120572 minus 120573)119879(120572 + 120573 minus 2120574)119879)119879(1198781 0lowast 31198781)( (120572 minus 120573)119879(120572 + 120573 minus 2120574)119879) (12)

Lemma 10 (see [29]) For constant matrices 1198781 gt 0 1198782 gt 0and constant scalars 119886 gt 119887 the following inequalities hold forall continuously function 119909 in [minus119886 minus119887] rarr R119899intminus119887

minus119886int119905minus119887119905+120579119879 (119906) 1198782 (119906) 119889119906119889120579ge Π2 (1198782 119909119879 (119905 minus 119887) 1119886 minus 119887 int119905minus119887119905minus119886

119909119879 (119906) 119889119906 1(119886 minus 119887)2sdot intminus119887minus119886int119905minus119887119905+120579119909119879 (119906) 119889119906119889120579)

intminus119887minus119886int119905+120579119905minus119886119879 (119906) 1198783 (119906) 119889119906119889120579ge Π3 (1198783 119909119879 (119905 minus 119886) 1119886 minus 119887 int119905minus119887119905minus119886

119909119879 (119906) 119889119906 1(119886 minus 119887)2sdot intminus119887minus119886int119905minus119887119905+120579119909119879 (119906) 119889119906119889120579)

(13)

where

Π2 (1198782 120572 120573 120574) = ( (120572 minus 120573)119879(1205722 minus 120574)119879)119879

sdot (61198782 minus121198782lowast 361198782 )( (120572 minus 120573)119879(1205722 minus 120574)119879)

4 Mathematical Problems in Engineering

Π3 (1198783 120572 120573 120574) = ( (120572 minus 120573)119879(1205722 minus 120573 + 120574)119879)119879

sdot (61198783 minus121198783lowast 361198783 )( (120572 minus 120573)119879(1205722 minus 120573 + 120574)119879)(14)

Lemma 11 (see [30]) Let 1198911 1198912 119891119873 R119898 997891rarr R havepositive values in an open subset119863 ofR119898Then the reciprocallyconvex combination of 119891119894 over119863 satisfies

min120572119894|120572119894gt0sum119894 120572119894=1

sum119894

1120572119894119891119894 (119905) = sum119894 119891119894 (119905) +max119892119894119895(119905)sum119894 =119895

119892119894119895 (119905)subject to 119892119894119895 119877119898 997891997888rarr 119877 119892119895119894 (119905) = 119892119894119895 (119905) [ 119891119894 (119905) 119892119894119895 (119905)119892119895119894 (119905) 119891119895 (119905) ] ge 0 (15)

Lemma 12 (see [31]) For any matrix 1198784 gt 0 and a vectorfunction 119909 in [minus119886 minus119887] rarr R119899 if the integrals concerned arewell defined then the following inequality holdsintminus119887

minus119886119909119879 (119904) 1198784119909 (119904) 119889119904ge 1119886 minus 119887 (intminus119887minus119886

119909119879 (119904) 119889119904) 1198784 (intminus119887minus119886119909 (119904) 119889119904) (16)

23 Control Objective Finally the objective of this paper is toderive the control gain 119870 such that

(1) the closed-loop system (8) with 119906(119905) = 0 is FTB(2) under given nonzero initial condition the closed

system (8) and (9) guarantees that 119869 = int1198790119911119879(119904)119911(119904) minus1205742120596119879(119904)120596(119904) minus 1205742(1119879)119909119879(0)119878119909(0)119889119904 lt 0 for all

nonzero 120596(119905) and 119905 isin [0 119879]3 Main Results

Thedesigning of the robust finite-time119867infin controller for theDP system with input delay is divided into three steps

31 FTB Analysis of DP System Firstly the result guarantee-ing the FTB of DP system is established in this subsection

Theorem 13 Given five positive scalars 120572 120573 1198881 1198882 119879 andpositive define matrix 119877 the finite-time boundedness problemof system (4) is solvable if there exist positive scalars 120579119897 (119897 =1 2 sdot sdot sdot 8) and matrices 11987511 gt 0 11987522 gt 0 119882 gt 0 119876119894 gt0 (119894 = 1 2)119880119895 gt 0 (119895 = 1 2)119872 gt 0 1198851 1198852 with appropriatedimensions satisfying the following conditionsΩ = (Ω119894119895)8times8 lt 0 (17)(119882 1198851lowast 119882) gt 0 (18)

(3119882 1198852lowast 3119882) gt 0 (19)

11988811205792 + ℎ211988811205793 + 120588ℎ11988811205794 + 120588ℎ11988811205795 + 12120588ℎ311988811205796+ 16120588ℎ311988811205797 + 13120588ℎ311988811205798 + 1205731205799119889 lt 119890minus12057211987911988821205791 (20)

whereΩ11 = 11986011987911987511 + 11987511119860 minus 12057211987511 + 1198761 + 1198762+ 119860119879 12ℎ2 (1198801 + 1198802 + 2119882)119860 minus 4120588119882 minus 31205881198801minus 91205881198802Ω22 = minus1205881198762 minus 4120588119882 minus 61205881198802Ω33 = 1198701198791198611119879 12ℎ2 (2119882 + 1198801 + 1198802) 1198611119870 minus 8120588119882+ 1205881198851119879 + 1205881198851 minus 1205881198852119879 minus 1205881198852 minus (1 minus 120583)1198762Ω44 = 1198612119879 12ℎ2 (2119882 + 1198801 + 1198802) 1198612 minus 120573119872Ω66 = minus12120588119882Ω77 = minus12120588119882Ω55 = 120572ℎ211987522 minus 181205881198802 minus 61205881198801Ω88 = minus361205881198801 minus 361205881198802Ω12 = 1205881198851 minus 1205881198852 + 61205881198802Ω15 = ℎ11987522 + 121205881198802Ω16 = 6120588119882Ω17 = 21205881198852

Mathematical Problems in Engineering 5

Ω13 = 119875111198611119870 + 119860119879 12ℎ2 (2119882 + 1198801 + 1198802) 1198611119870 minus 2120588119882minus 1205881198851 minus 1205881198852Ω14 = 119875111198612 + 119860119879 12ℎ2 (2119882 + 1198801 + 1198802) 1198612Ω18 = minus181205881198802 + 61205881198801Ω23 = minus1205881198851119879 minus 1205881198852119879 minus 2120588119882Ω25 = minusℎ11987522 minus 61205881198802Ω26 = 21205881198852119879Ω27 = 6120588119882Ω28 = 121205881198802Ω34 = 1198701198791198611119879 12ℎ2 (2119882 + 1198801 + 1198802) 1198612Ω36 = 6120588119882 + 21205881198852119879Ω37 = 6120588119882 + 21205881198852Ω58 = 121205881198801 + 241205881198802Ω67 = minus41205881198852Ω24 = Ω35 = Ω38 = Ω45 = Ω46 = Ω47 = Ω48 = Ω56= Ω57 = Ω68 = Ω78 = 01205791 = 120582min (11)1205792 = 120582max (11) 1205793 = 120582max (22) 1205794 = 120582max (1198761) 1205795 = 120582max (1198762) 1205796 = 120582max () 1205797 = 120582max (1) 1205798 = 120582max (2) 1205799 = 120582max (119872) 120588 = 119890120572ℎ

11 = 119877minus1211987511119877minus1222 = 119877minus1211987522119877minus12119876119894 = 119877minus12119876119894119877minus12 (119894 = 1 2 3) = 119877minus12119882119877minus12119895 = 119877minus12119880119895119877minus12 (119895 = 1 2) 0 lt 1205791119868 lt 11987511 lt 12057921198680 lt 11987522 lt 12057931198680 lt 1198761 lt 12057941198680 lt 1198762 lt 12057951198680 lt 119882 lt 12057961198680 lt 1198801 lt 12057971198680 lt 1198802 lt 12057981198680 lt 119872 lt 1205799119868(21)

Proof Consider the candidate augment LKLF as follows119881 (119905) = 1198811 (119905) + 1198812 (119905) + 1198813 (119905) + 1198814 (119905) (22)1198811 (119905) = 120576 (119905)119879 119875120576 (119905) (23)1198812 (119905) = int119905119905minusℎ119890120572(119905minus119904)119909 (119904)1198791198761119909 (119904) 119889119904+ int119905119905minusℎ(119905)119890120572(119905minus119904)119909 (119904)1198791198762119909 (119904) 119889119904 (24)

1198813 (119905) = ℎint0minusℎint119905119905+120579119890120572(119905minus119904) (119904)119879119882 (119904) 119889119904119889120579 (25)

1198814 (119905) = int0minusℎint0120579int119905119905+120573119890120572(119905minus119904) (119904)1198791198801 (119904) 119889119904119889120573119889120579

+ int0minusℎint120579minusℎint119905119905+120573119890120572(119905minus119904) (119904)1198791198802 (119904) 119889119904119889120573119889120579 (26)

where 120576(119905) = (119909(119905) int119905119905minusℎ119909(119904)119889119904) 119875 = ( 11987511 0

lowast 11987522)

To deal with the formulas conveniently let us providesuch definitions

120585 (119905)= (119909 (119905) 119909 (119905 minus ℎ) 119909 (119905 minus ℎ (119905)) 120596 (119905) 1ℎ int119905119905minusℎ 119909 (119904) 119889119904 1ℎ (119905) int119905119905minusℎ(119905) 119909 (119904) 119889119904 1ℎ minus ℎ (119905) int119905minusℎ(119905)119905minusℎ119909 (119904) 119889119904 1ℎ2 int0minusℎ int119905119905+120579 119909 (119904) 119889119904119889120579)119897119896119879 = (0 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896minus1

1 0 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟8minus119896) (27)

6 Mathematical Problems in Engineering

The time-derivative of 119881(119905) along system (4) can bebounded as1 (119905) = 1205721198811 (119905) + 120576 (119905)119879 119875120576 (119905) + 120576 (119905)119879 119875 120576 (119905) minus 1205721198811 (119905) (28)2 (119905)= 1205721198812 (119905) + 119909 (119905)1198791198761119909 (119905)minus 119890120572ℎ119909 (119905 minus ℎ)1198791198761119909 (119905 minus ℎ) + 119909 (119905)1198791198762119909 (119905)minus (1 minus ℎ (119905)) 119890120572ℎ(119905)119909 (119905 minus ℎ (119905))1198791198762119909 (119905 minus ℎ (119905))le 1205721198812 (119905) + 119909 (119905)1198791198761119909 (119905)minus 119890120572ℎ119909 (119905 minus ℎ)1198791198761119909 (119905 minus ℎ) + 119909 (119905)1198791198762119909 (119905)minus (1 minus 120583) 119909 (119905 minus ℎ (119905))1198791198762119909 (119905 minus ℎ (119905))

(29)

3 (119905)= 1205721198813 (119905) + ℎ2 (119905)1198791198822 (119905)minus 119890120572ℎℎint119905119905minusℎ (119904)1198791198822 (119904) 119889119904 (30)

4 (119905)= 1205721198814 (119905) + 12ℎ2 (119905)1198791198801 (119905)minus 119890120572ℎ int0minusℎint119905119905+120573 (119904)1198791198801 (119904) 119889119904119889120573+ 12ℎ2 (119905)1198791198802 (119905)minus 119890120572ℎ int0

minusℎint119905+120573119905minusℎ (119904)1198791198802 (119904) 119889119904119889120573

(31)

Invoking Lemma 9 we can obtain3 (119905) = 1205721198813 (119905) + ℎ2 (119905)119879119882 (119905)minus 119890120572ℎℎint119905119905minusℎ(119905) (119904)119879119882 (119904) 119889119904

minus 119890120572ℎℎint119905minusℎ(119905)119905minusℎ

(119904)119879119882 (119904) 119889119904le 1205721198813 (119905) + ℎ2 (119905)119879119882 (119905)minus 119890120572ℎ ℎℎ (119905)1206031119879Λ 11206031 minus 119890120572ℎ ℎℎ minus ℎ (119905)1206032119879Λ 21206032(32)

where 1206031119879 = 120585119879 (119905)( (1198971119879 minus 1198973119879)119879(1198971119879 + 1198973119879 minus 21198976119879)119879)119879

1206032119879 = 120585119879 (119905)( (1198973119879 minus 1198972119879)119879(1198973119879 + 1198972119879 minus 21198977119879))119879

Λ 1 = Λ 2 = (119882 0lowast 3119882) (33)

In order to obtain a tighter bound of integral termLemma 11 is applied to (32) as follows3 (119905) le 1205721198813 (119905) + ℎ2 (119905)119879119882 (119905) minus 119890120572ℎ1206033119879Λ 31206033 (34)

where 1206033119879 = ( 12060311198791206032119879 ) Λ 3 = (119882 1198851 0 0

lowast 119882 0 0lowast lowast 3119882 1198852lowast lowast lowast 3119882

)By applying Lemma 10 to (31) we can obtain the inequal-

ity 4 (119905) le 1205721198814 (119905) + 12ℎ2 (119905)1198791198801 (119905)+ 12ℎ2 (119905)1198791198802 (119905) minus 119890120572ℎ1206034119879Λ 41206034 (35)

where

1206034119879 = 120585119879 (119905)(((((

(1198971119879 minus 1198972119879)119879(12 1198971119879 minus 1198978119879)119879(1198972119879 minus 1198975119879)119879(12 1198972119879 minus 1198975119879 + 1198978119879)119879)))))

119879

Λ 4 =(61198801 minus121198801 0 0lowast 361198801 0 0lowast lowast 61198802 minus121198802lowast lowast lowast 361198802)(36)

Combining (28) (29) (34) and (35) with the definitionof 120588 = 119890120572ℎ we can obtain minus 120572119881 minus 120573120596119879 (119905)119872120596 (119905) lt 120585119879 (119905) Ω120585 (119905) (37)

Assuming that Ω lt 0 and integrating the left part ofinequality (37)119881 (119905) lt 119890120572119905119881 (0) + 120573int119905

0119890120572(119905minus119904)120596119879 (119904)119872120596 (119904) 119889119904lt 119890120572119879 (119881 (0) + 120573120582max (119872) 119889) (38)

Invoking the Jensen inequality shown in Lemma 12 onehas 119881 (0) le 119909 (0)119879 11987511119909 (0) + ℎint0

minusℎ119909 (119904)119879 11987522119909 (119904) 119889119904+ 120588int0

minusℎ119909 (119904)1198791198761119909 (119904) 119889119904+ 120588int0

minusℎ(119905)119909 (119904)1198791198762119909 (119904) 119889119904

Mathematical Problems in Engineering 7

+ ℎ120588int0minusℎint0120579 (119904)119879119882 (119904) 119889119904119889120579

+ 120588int0minusℎint0120579int0120573 (119904)1198791198801 (119904) 119889119904119889120573119889120579

+ 120588int0minusℎint120579minusℎint0120573 (119904)1198791198802 (119904) 119889119904119889120573119889120579le 120582max (11) 1198881 + ℎ120582max (22) ℎ1198881+ 120588120582max (1198761) ℎ1198881 + 120588120582max (1198762) ℎ1198881+ 120588120582max () 12ℎ31198881 + 120588120582max (1) 16ℎ31198881+ 120588120582max (2) 13ℎ31198881

(39)

Based on inequalities (38) and (39) we have119881 (119905) gt 119909119879 (119905) 11987511119909 (119905) = 119909119879 (119905) 119877121111987712119909 (119905)gt 120582min (11) 119909119879 (119905) 119877119909 (119905) (40)

Therefore conditions (17) to (20) can guarantee the FTBof system (4) This completes the proof

Remark 14 Among the existing approaches there are twothreads one is to construct a novel LKLF that involves moreinformation of delay the other is to find a tighter estimationof upper bound for cross terms coming from the derivativeof the LKLF In this paper these two techniques are appliedto obtain the result In addition the less conservatism inpractical engineering will be shown in the simulations whichare always ignored in most literature sources

32 Controller Design In this subsection we focus on theproblem of finite-time 119867infin state feedback designing basedon Theorem 13 that is designing a state feedback controllerin the form of (7) such that the resulting DP system satisfiesthe control objective proposed in Section 1

Theorem 15 For given positive 120572 1198881 1198882 119879 120574 and matrices 119877 gt0 119878 ge 0 if there exist positive scalars 120579119897 (119897 = 1 2 sdot sdot sdot 8) andmatrices 11987511 gt 0 11987522 gt 0119882 gt 0 119876119894 gt 0 (119894 = 1 2) 119880119895 gt0 (119895 = 1 2) 119884 1198851 1198852 with appropriate dimensions satisfyingthe following conditions Π = (Π119894119895)12times12 lt 0 (41)Σ = (Σ119894119895)9times9 lt 0 (42)

(119882 1198851lowast 119882) gt 0 (43)

(3119882 1198852lowast 3119882) gt 0 (44)

119868 minus 120574211987811987511 lt 0 (45)0 lt 1198671 lt 11987511 lt 11986720 lt 11987522 lt 11986730 lt 1198761 lt 11986740 lt 1198762 lt 11986750 lt 119882 lt 11986760 lt 1198801 lt 11986770 lt 1198802 lt 1198678(46)

where Π11 = 11987511119860119879 + 11986011987511 minus 12057211987511 + 1198761 + 1198762 minus 4120588119882minus 31205881198801 minus 91205881198802Π44 = minus1205742119868Π22 = minus1205881198762 minus 4120588119882 minus 61205881198802Π33 = minus8120588119882 + 1205881198851119879 + 1205881198851 minus 1205881198852119879 minus 1205881198852minus (1 minus 120583)1198762Π13 = 1198611119884 minus 2120588119882 minus 1205881198851 minus 1205881198852Π14 = 1198612Π66 = minus12120588119882Π77 = minus12120588119882Π55 = 120572ℎ211987522 minus 181205881198802 minus 61205881198801Π88 = minus361205881198801 minus 361205881198802Π12 = 1205881198851 minus 1205881198852 + 61205881198802Π15 = ℎ11987522 + 121205881198802Π16 = 6120588119882Π17 = 21205881198852Π25 = minusℎ11987522 minus 61205881198802Π18 = minus181205881198802 + 61205881198801Π26 = 21205881198852119879Π23 = minus1205881198851119879 minus 1205881198852119879 minus 2120588119882Π27 = 6120588119882Π28 = 121205881198802

8 Mathematical Problems in Engineering

Π36 = 6120588119882 + 21205881198852119879Π37 = 6120588119882 + 21205881198852Π58 = 121205881198801 + 241205881198802Π67 = minus41205881198852Π19 = 12ℎ11987511119860119879Π39 = 12ℎ1198841198791198611119879Π49 = 12ℎ1198612119879Π110 = 12ℎ11987511119860119879Π310 = 12ℎ1198841198791198611119879Π410 = 12ℎ1198612119879Π111 = ℎ11987511119860119879Π311 = ℎ1198841198791198611119879Π411 = ℎ1198612119879Π112 = 11987511119862119879Π312 = 119884119879119863119879Π99 = 121198801 minus 11987511Π1010 = 121198802 minus 11987511Π1111 = 119882 minus 211987511Π1212 = minus119890minus120572119879119868(47)

others Π119894119895 = 0Σ11 = minus119890minus12057211987911988821198671Σ22 = minus1198672Σ33 = minus1198673Σ44 = minus1198674Σ55 = minus1198675Σ66 = minus1198676Σ77 = minus1198677Σ88 = minus1198678

Σ99 = minus1205742119889Σ12 = radic11988811198672Σ13 = radicℎ211988811198673Σ14 = radic120588ℎ11988811198674Σ15 = radic120588ℎ11988811198675Σ16 = radic12120588ℎ311988811198676Σ17 = radic16120588ℎ311988811198677Σ18 = radic13120588ℎ311988811198678Σ19 = 11987511(48)

others Σ119894119895 = 0 (49)

then a state feedback119867infin controller in form of (7) existssuch that

(1) the closed-loop system (8) with 119906(119905) = 0 is FTB(2) under given nonzero initial condition the closed

system (8) and (9) guarantees that 119869 = int1198790119911119879(119904)119911(119904) minus1205742120596119879(119904)120596(119904) minus 1205742(1119879)119909119879(0)119878119909(0)119889119904 lt 0 for all nonzero 120596(119905)

and 119905 isin [0 119879]The controller gain can be calculated by 119870 = 11988411987511minus1

Proof In preparation for designing we set119872 = 1205742119868 120573 = 1for (37) then (37) and (38) can be rewritten as follows minus 120572119881 minus 1205742120596119879 (119905) 120596 (119905) lt 120585119879 (119905) Ω120585 (119905) (50)119881 (119905) lt 119890120572119905119881 (0) + 1205742 int119905

0119890120572(119905minus119904)120596119879 (119904) 120596 (119904) 119889119904lt 119890120572119879 (119881 (0) + 1205742119889) (51)

where Ω44 = 1198612119879 12ℎ2 (2119882 + 1198801 + 1198802) 1198612 minus 1205742119868Ω119894119895 = Ω119894119895

(52)

Then two steps are provided to design the desiredcontroller

Step 1 Here we will give some conditions that can guaranteethe 119867infin performance over the finite-time interval undernonzero initial conditions firstly

Mathematical Problems in Engineering 9

Consider the following inequality minus 120572119881 + 119890120572119905119911119879 (119905) 119911 (119905) minus 1205742119890120572119905120596119879 (119905) 120596 (119905)lt minus 120572119881 + 119890120572119879119911119879 (119905) 119911 (119905) minus 1205742120596119879 (119905) 120596 (119905)lt 120585119879 (119905) (Ω minus Ψ119879 (minus119890minus120572119879119868)minus1Ψ) 120585 (119905) (53)

Noting that119882 1198801 and 1198802 are coupled with 119860 1198611119870 and1198612 in Ω to decouple these terms the Ω can be rewritten asΩ = Ω minus Ψ119879 (minus121198801)minus1 Ψ minus Ψ119879 (minus121198802)minus1Ψminus Ψ119879 (minus119882)minus1 Ψ (54)

whereΩ11 = 11986011987911987511 + 11987511119860 minus 12057211987511 + 1198761 + 1198762 minus 4120588119882minus 31205881198801 minus 91205881198802Ω33 = minus8120588119882 + 1205881198851119879 + 1205881198851 minus 1205881198852119879 minus 1205881198852minus (1 minus 120583)1198762Ω44 = minus1205742119868Ω13 = 119875111198611119870 minus 2120588119882 minus 1205881198851 minus 1205881198852Ω14 = 119875111198612Ω34 = 0(55)

othersΩ119894119895 = Ω119894119895Ψ = (119862 0 119863119870 0 0 0 0 0)Ψ= (ℎ121198801119879119860 0 ℎ1211988011198791198611119870 ℎ1211988011198791198612 0 0 0 0) Ψ= (ℎ121198802119879119860 0 ℎ1211988021198791198611119870 ℎ1211988021198791198612 0 0 0 0)Ψ = (ℎ119882119879119860 0 ℎ1198821198791198611119870 ℎ1198821198791198612 0 0 0 0) (56)

By Schur complementΩ lt 0 and Ω minusΨ119879(minus119890minus120572119879119868)minus1Ψ areequivalent to

Π =(((((

Ω Ψ119879 Ψ119879 Ψ119879 Ψ119879lowast minus121198801 0 0 0lowast lowast minus121198802 0 0lowast lowast lowast minus119882 0lowast lowast lowast lowast minus119890minus120572119879119868)))))lt 0 (57)

To ensure the proposed conditions satisfy the 119867infin per-formance we set the inequality like

11987511 minus 1205742119878 lt 0 (58)

from which we have

minus 120572119881 + 119890120572119905119911119879 (119905) 119911 (119905) minus 1205742119890120572119905120596119879 (119905) 120596 (119905)minus 1205742119890120572119905 1119879119909119879 (0) 119878119909 (0) lt minus 120572119881 + 119890120572119905119911119879 (119905) 119911 (119905)minus 1205742119890120572119905120596119879 (119905) 120596 (119905) minus 1205742119890120572119905 1119879119909119879 (0) 119878119909 (0)+ 119890120572119905 1119879119909119879 (0) 11987511119909 (0) lt minus 120572119881 + 119890120572119905119911119879 (119905) 119911 (119905)minus 1205742119890120572119905120596119879 (119905) 120596 (119905) + 119890120572119905 1119879119909119879 (0) (11987511 minus 1205742119878) 119909 (0)lt 0(59)

Integrating (58) from 0 to 119879 we can obtain

int1198790119911119879 (119904) 119911 (119904) 119889119904 minus int119879

01205742120596119879 (119904) 120596 (119904) 119889119904

minus 1205742119909119879 (0) 119878119909 (0) lt minusint1198790(119890minus120572119905 minus 120572119890minus120572119905119881)119889119904 lt 0 (60)

so conditions (43) (44) (57) and (58) with the inequali-ties as follows can ensure that systems (17) and (18) satisfy the119867infin performance

11988811205792 + ℎ211988811205793 + 120588ℎ11988811205794 + 120588ℎ11988811205795 + 12120588ℎ311988811205796+ 16120588ℎ311988811205797 + 13120588ℎ311988811205798 + 1205742119889 lt 119890minus120572119879119888212057910 lt 1205791119868 lt 11987511 lt 12057921198680 lt 11987522 lt 12057931198680 lt 1198761 lt 12057941198680 lt 1198762 lt 12057951198680 lt 119882 lt 12057961198680 lt 1198801 lt 12057971198680 lt 1198802 lt 1205798119868

(61)

10 Mathematical Problems in Engineering

Step 2 In this step the approach to design the gain matrix 119870of the controller is provided in forms of LMIs which can becalculated expediently

To linearize the nonlinear terms in (57) we make itmultiply by the followingmatrix fromboth left and right sidesfirstly

diag 11987511minus1 11987511minus1 11987511minus1 119868 11987511minus1 11987511minus1 11987511minus1 11987511minus1 1198801minus1 1198802minus1 119882minus1 119868 (62)

Let us do the definitions as follows11987511 = 11987511minus11198761 = 11987511minus1119876111987511minus11198762 = 11987511minus1119876211987511minus1119882 = 11987511minus111988211987511minus11198801 = 11987511minus1119880111987511minus11198802 = 11987511minus1119880211987511minus111987522 = 11987511minus11198752211987511minus11198851 = 11987511minus1119885111987511minus11198852 = 11987511minus1119885211987511minus1119884 = 11987011987511minus1

(63)

so the inequality Π lt 0 can guarantee Π lt 0 whereΠ99 = minus12119875111198801

minus111987511Π1010 = minus12119875111198802

minus111987511Π1111 = minus11987511119882minus111987511(64)

others Π119894119895 = Π119894119895 (65)

Then based on the inequalities minus(12)119875111198801

minus111987511 le(12)1198801 minus 11987511 minus(12)119875111198802

minus111987511 le (12)1198802 minus 11987511minus11987511119882minus111987511 le 119882 minus 211987511 condition (57) can be convertedinto (41) in form of LMIs

Accordingly similar to the above-mentioned procedures(61) can be transformed into (42) with the Schur complementand the definition of119867119894 = 11987511minus112057911989411987511minus1 (119894 = 1 2 sdot sdot sdot 8)

Therefore conditions (41) to (45) can obtain the con-troller in the form of (7) guaranteeing the desired perfor-mance This completes the proof

Remark 16 The119867infin performance with nonzero initial con-dition is studied in this literature In a practical view it is

hard to suffer the worst-case over a finite time for surfacevessel with nonzero initial condition and it will reducethe conservatism caused by the assumption of zero initialcondition Therefore it is reasonable to introduce such 119867infinperformance index to handle the problem of finite-time119867infincontrol for ship subject to exogenous signals

Remark 17 It is challenging to achieve the result satisfyingthe control objective while it is very different from thestandard 119867infin control because a weighting diagonal matrixis introduced especially to establish the form of LMIs Inthe proof of Theorem 15 inequality (58) is introduced toensure the 119867infin performance with nonzero initial conditionand make it convenient to construct the conditions in formof LMIs

Remark 18 By introducing some new additional matrixvariables Cone Complementarity Linearization algorithm(CCL) [32] can be utilized to linearize the nonlinear termsas an alternative method It will reduce the conservatism atthe expense of increasing the computational burden and thenumber of decision variables

4 Simulations

41 Simulation Settings To demonstrate the effectiveness andsuperiority of the robust finite-time 119867infin control schemesimulation studies are conducted on the parameters of theship CyberShip II [33] which is a 170 scale-replica of a supplyshipwhosemain parameters are shown inTable 1 In additionwe choose the time-varying function like ℎ(119905) = 3sin (1199056 minus1205876) to describe the time delay occurred in the input signal

Unknown external disturbances 120596(119905) with significantlylarge magnitude and high frequency which can be roughlyconsidered as complex environment including ocean windswaves and currents are governed by [34]

120596 (119905) = (9 sin (01120587119905 minus 1205875 )6 sin (03120587119905 + 1205876 )3 sin (02120587119905 + 1205873 )) (66)

with initial conditions 120578(0) = (1 1 01) and 120592(0) =(08 06 01)42 Numerical Simulations We choose 120574 = 13 119879 = 10 119878 =03lowast119868 1198881 = 302 1198882 = 10 120572 = 0001 By usingTheorem 15 thecontrol gain 119870 is calculated by LMIs (41) to (45)

Mathematical Problems in Engineering 11

Table 1 The main parameters of the CyberShip II119898 238000 119883 -20 119883119906 -07225119868119911 17600 119884V -100 119884V -08612119909119892 00460 119884 119903 -00 119884119903 01079119873 119903 -10 119873119903 05

0 20 40 60 80 100Time(s) Time(s)

minus2

0

2

4

6

8

10

12

0 2 4 6 8 100

2

4

6

8

10

12

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

xT(t)Rx(t)xT(t)Rx(t)

c=c=

c1=302

c1=302

Figure 1 State response 119909119879(119905)119877119909(119905) of DP system

119870 = (minus32743 0 0 minus313907 0 00 minus42784 minus01572 0 minus412136 minus129820 minus01389 minus03497 0 minus13362 minus33091) (67)

To demonstrate the superiority and validity three sce-narios as comparative cases are provided here It should bepointed out that the results within 100s are also provided toillustrate the steady-state performance

Case 1 (with different 119878) For comparison purposes we set119878 with different value to demonstrate the effect of 119878 on stateresponse Without loss of generality we set 119878 = 05 lowast 119868 and119878 = 01 lowast 119868 respectivelyCase 2 (119867infin with zero initial condition]) This scenario willshow that the control law proposed in this paper can achievea better transient performance and disturbance rejectioncapability over traditional119867infin control approach for dynamicpositioning system conducted in [11]

Case 3 (a control design based on existing methods) Basedon the approach in [17] applied for discrete-time system acontrol law for dynamic positioning ship with input delay isproposed to illustrate the less conservatism of our designingapproach in practical engineering subsequently

At first the effectiveness of the control law we designedis confirmed from Figures 1ndash6 Figure 1 shows the state

response of 119909119879(119905)119877119909(119905) Obviously from that curve we cansee that the value of 119909119879(119905)119877119909(119905) is far below the value of1198882 = 10 in the presence of disturbance as in Figure 2which means the finite-time full-state feedback controllerdesigned can guarantee the DP system (8) sim (9) FTB with(302 10 119889 10 3 119868) and have less conservatism where weset 119889 = int119879

0120596(119905)119879120596(119905)119889119905 Meanwhile the control law can

stabilize the dynamic positioning shipwithminimum steady-state error in 100s and the control input will be small enoughas shown in Figure 6 From Figures 3ndash5 we can find that allstates in DP system are within a small range As shown inFigures 2ndash4 the angle and the angular velocity in yaw havechanged within 01∘ which means that the heading of thevessel stays in a fixed direction Furthermore the velocitiesin surge and sway are low enough which illustrates thatassumption 2 is reasonable and it meets the requirement ofstabilization control for DP system

To demonstrate the effect of 119878 state responses of systemswith different 119878 are exhibited in Figures 7ndash9 As the valueof 119878 decreases the restriction for 11987511 is more exclusive andthe robustness is stronger during preset time interval as inFigures 7 and 8 while the required control input will increasedramatically as in Figure 9 When the time oversteps the

12 Mathematical Problems in Engineering

0 10 20 30 40 50 60 70 80 90 100Time(s)

minus10

minus5

0

5

10

Dist

urba

nces

(N)

XY

Figure 2 Disturbance in surge sway and yaw

0 50 100Time(s)

minus050

051

152

X(m

)

0 5 10Time(s)

005

115

2

X(m

)

0 50 100Time(s)

minus050

051

152

Y(m

)

0 5 10Time(s)

005

115

2Y(

m)

0 50 100Time(s)

minus01

0

01

02

0 5 10Time(s)

0

01

02

(∘

)

(∘

)

Figure 3 State responses of position and orientation for DP System

0 50 100Time(s)

minus1

0

1

2

0 5 10Time(s)

minus1

0

1

2

0 50 100Time(s)

minus1

0

1

2

v(m

s)

0 5 10Time(s)

minus1

0

1

2

v(m

s)

0 50 100Time(s)

minus01

0

01

02

0 5 10Time(s)

minus01

0

01

02

r(∘ s

)

r(∘ s

)

(ms

)

(m

s)

Figure 4 State responses of velocities for DP system

Mathematical Problems in Engineering 13

minus05 0 05 1 15 2X(m)

minus05

0

05

1

15

2

Y(m

)

Figure 5 Trajectory of the shiprsquos horizontal motion

0 50 100Time(s)

minus30minus20minus10

010

u1(N

)

0 5 10Time(s)

minus30minus20minus10

010

u1(N

)

0 50 100Time(s)

minus30minus20minus10

010

u2(N

)

0 5 10Time(s)

minus30minus20minus10

010

u2(N

)

0 50 100Time(s)

minus15minus1

minus050

05

u3(N

m)

0 5 10Time(s)

minus15minus1

minus050

05

u3(N

m)

Figure 6 Control inputs

0 20 40 60 80 100Time(s)

minus1

0

1

2

3

4

5

6

S=03S=05S=01

0 2 4 6 8 10Time(s)

0

2

4

6

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 7 State response 119909119879(119905)119877119909(119905) with different 119878

14 Mathematical Problems in Engineering

Figure 8 State responses of position and orientation with different 119878

0 50 100Time(s)

minus60minus30

03060

u1(N

)

0 5 10Time(s)

minus40minus20

02040

u1(N

)

0 50 100Time(s)

minus100

0

100

u2(N

)

0 5 10Time(s)

minus40minus20

02040

u2(N

)

0 50 100Time(s)

minus4minus2

024

u3(N

m)

0 5 10Time(s)

minus2

0

2

u3(N

m)

S=03S=05S=01

Figure 9 Control input with different 119878preset time interval all the states can be guaranteed withinthe given upper bound as before

As shown in Figures 10ndash12 the overconservatism oftraditional119867infin control approach applied inDP ship is solvedby the 119867infin control strategy with nonzero initial conditionproposed in this article As is well known the classical 119867infincontrol law is designed under the so-called worst-case normwhile the performance of antidisturbance and transient issacrificed to some degree From Figures 10 and 11 we cansee that the state response can be controlled in a smallerrange and stabilized in a shorter time with a lower overshootshown in Figure 12 Hence a conclusion can be drawn thatthe control law we present can achieve a better disturbance

rejection capability and transient performance while thematter of overconservatism can be resolved at the sametime

Simulation results of Theorem 15 and Case 3 are shownin Figures 13 and 14 to show the better performance ofthe method we proposed We can find that the controllerbased on [17] and 119867infin control approach present in thisliterature can make all states of system within a smallenough region However compared with the controller weconstructed the state response has higher overshoot whichwill affect the control precision and transient performance inpractical engineering Based on the above simulation resultsthe developed controller can achieve a higher accuracy and

Mathematical Problems in Engineering 15

0 20 40 60 80 100Time(s)

minus2

0

2

4

6

8

10

12

Theorem 15Case 2

0 2 4 6 8 10Time(s)

minus2

0

2

4

6

8

10

12

Theorem 15Case 2

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 10 State response 119909119879(119905)119877119909(119905) of Theorem 15 and Case 2

0 50 100Time(s)

minus10123

X(m

)

0 5 10Time(s)

0123

X(m

)

0 50 100Time(s)

minus10123

Y(m

)

0 5 10Time(s)

0123

Y(m

)

0 50 100Time(s)

minus020

0204

0 5 10Time(s)

minus020

0204

Theorem 15Case 2

(∘

)

(∘

)

Figure 11 Position and orientation of Theorem 15 and Case 2

minus05 0 05 1 15 2 25X(m)

minus050

051

152

25

Y(m

)

Theorem 15Case 2

Figure 12 Trajectory of the shiprsquos horizontal motion of Theorem 15 and Case 2

16 Mathematical Problems in Engineering

0 20 40 60 80 100Time(s)

0

1

2

3

4

5

Theorem 15Case 3

0 2 4 6 8 10Time(s)

0

1

2

3

4

5

Theorem 15Case 3

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 13 State response 119909119879(119905)119877119909(119905) of Theorem 15 and Case 3

0 5 10Time(s)

05

1

15

X(m

)

Theorem 15Case 3

0 5 10Time(s)

05

1

15

Y(m

)

0 5 10Time(s)

0

005

01

015

02

(∘

)

Figure 14 Position and orientation of Theorem 15 and Case 3

lower overshoot which means that it is less conservative incomparison with existing result

5 Conclusion

In this paper the robust finite-time 119867infin control with tran-sients for DP system with input delay maintaining all statesof ocean surface vessel into a given threshold over a fixed timeinterval in presence of time-varying disturbance is proposedBased on a novel augmented LKLF with triple integral termsand a method combining theWirtinger inequality and recip-rocally convex approach a less conservative result is derivedIn particular a119867infin performance index with nonzero initialcondition attenuating the overconservatism caused by theassumption of zero initial condition and enhancing thetransient performance of ship subject to external disturbanceis adoptedThe obtained result is formulated in terms of LMIswhich can be easily solved by the standard numerical soft-ware Finally a numerical simulation for a ship is proposed toverify the effectiveness and less conservatism of the controller

we designed In the future the output feedback control can bedeveloped for the ocean surface vessel to avoid the use of thevessel-frame surge sway and yaw velocities

Data Availability

All the data supporting the conclusions of the study have beenprovided in Simulations and readers can access these data in[33 34]

Conflicts of Interest

The authors declare that there are no conflicts of interestrelated to this paper

Acknowledgments

This research was partially supported by the National Sci-ence Technology Support Program of China (Project no51609046)

Mathematical Problems in Engineering 17

References

[1] M Chen B Jiang and R Cui ldquoActuator fault-tolerant controlof ocean surface vessels with input saturationrdquo InternationalJournal of Robust amp Nonlinear Control vol 26 no 3 pp 542ndash564 2016

[2] J Du X Hu and Y Sun ldquoRobust dynamic positioning of shipswith disturbances under input saturationrdquo Automatica vol 73pp 207ndash214 2016

[3] H Ashrafiuon K R Muske L C McNinch and R ASoltan ldquoSliding-mode tracking control of surface vesselsrdquo IEEETransactions on Industrial Electronics vol 55 no 11 pp 4004ndash4012 2008

[4] W Si and X Dong ldquoNeural prescribed performance controlfor uncertain marine surface vessels without accurate initialerrorsrdquoMathematical Problems in Engineering Art ID 261832311 pages 2017

[5] T D Nguyen A J Soslashrensen and S T Quek ldquoDesign of hybridcontroller for dynamic positioning from calm to extreme seaconditionsrdquo Automatica vol 43 no 5 pp 768ndash785 2007

[6] J Du X Hu H Liu and C L Chen ldquoAdaptive robust outputfeedback control for a marine dynamic positioning systembased on a high-gain observerrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 11 pp 2775ndash27862015

[7] H Katayama ldquoNonlinear sampled-data stabilization of dynam-ically positioned shipsrdquo IEEE Transactions on Control SystemsTechnology vol 18 no 2 pp 463ndash468 2010

[8] K J Astrom andCG Kallstrom ldquoIdentification of ship steeringdynamicsrdquo Automatica vol 12 no 1 pp 9ndash22 1976

[9] Z Lei and C Guo ldquoDisturbance rejection control solutionfor ship steering system with uncertain time delayrdquo OceanEngineering vol 95 pp 78ndash83 2015

[10] J C Doyle K Glover P P Khargonekar and B A FrancisldquoState-space solutions to standard H-2 and Hinfin control prob-lemsrdquo IEEE Transactions on Automatic Control vol 34 no 8pp 831ndash847 1989

[11] M Katebi M Grimble and Y Zhang ldquoHinfin robust controldesign for dynamic ship positioningrdquo IEE Proceedings - ControlTheory and Applications vol 144 no 2 pp 110ndash120 1997

[12] D V Balandin and M M Kogan ldquoLMI-based Hinfin-optimalcontrol with transientsrdquo International Journal of Control vol 83no 8 pp 1664ndash1673 2010

[13] Z Feng J Lam S Xu and S Zhou ldquoHinfin Control withTransients for Singular Systemsrdquo Asian Journal of Control vol18 no 3 pp 817ndash827 2016

[14] P P Khargonekar K M Nagpal and K R Poolla ldquoHinfin controlwith transientsrdquo SIAM Journal onControl andOptimization vol29 no 6 pp 1373ndash1393 1991

[15] Z Zhang Z Zhang and H Zhang ldquoFinite-time stabilityanalysis and stabilization for uncertain continuous-time systemwith time-varying delayrdquo Journal of The Franklin Institute vol352 no 3 pp 1296ndash1317 2015

[16] J Cheng S Chen Z Liu H Wang and J Li ldquoRobust finite-time sampled-data control of linear systems subject to randomoccurring delays and its application to Four-Tank systemrdquoAppliedMathematics andComputation vol 281 pp 55ndash76 2016

[17] W Kang S Zhong K Shi and J Cheng ldquoFinite-time stabilityfor discrete-time system with time-varying delay and nonlinearperturbationsrdquo ISA Transactions vol 60 pp 67ndash73 2016

[18] T I Fossen Handbook of Marine Craft Hydrodynamics andMotion Control John Wiley amp Sons New York USA 2011

[19] D Zhao F Ding L Zhou W Zhang and H Xu ldquoRobustH-infinity Control of Neutral System with Time-Delay forDynamic Positioning Shipsrdquo Mathematical Problems in Engi-neering vol 2015 Article ID 976925 11 pages 2015

[20] F Amato M Ariola and C Cosentino ldquoFinite-time stabilityof linear time-varying systems analysis and controller designrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 55 no 4 pp 1003ndash1008 2010

[21] L Sun andZ Zheng ldquoFinite-time slidingmode trajectory track-ing control of uncertain mechanical systemsrdquo Asian Journal ofControl vol 19 no 1 pp 399ndash404 2017

[22] P Shi Y Zhang and R Agarwal ldquoStochastic finite-time stateestimation for discrete time-delay neural networks withMarko-vian jumpsrdquo Neurocomputing vol 151 part 1 pp 168ndash174 2014

[23] L Zhang S Wang H R Karimi and A Jasra ldquoRobust Finite-Time Control of Switched Linear Systems and Application toa Class of Servomechanism Systemsrdquo IEEEASME Transactionson Mechatronics vol 20 no 5 pp 2476ndash2485 2015

[24] L Wu P Shi and H Gao ldquoState estimation and sliding-mode control of Markovian jump singular systemsrdquo Institute ofElectrical and Electronics Engineers Transactions on AutomaticControl vol 55 no 5 pp 1213ndash1219 2010

[25] Y Q Zhang P Shi S K Nguang and Y D Song ldquoRobust finite-time Hinfin control for uncertain discrete-time singular systemswithMarkovian jumpsrdquo IET ControlTheory ampApplications vol8 no 12 pp 1105ndash1111 2014

[26] S Wang T Shi L Zhang A Jasra and M Zeng ldquoExtendedfinite-time Hinfin control for uncertain switched linear neutralsystemswith time-varying delaysrdquoNeurocomputing vol 152 pp377ndash387 2015

[27] J Hu G Sui S Du and X Li ldquoFinite-time stability of uncer-tain nonlinear systems with time-varying delayrdquo MathematicalProblems in Engineering Article ID 2538904 9 pages 2017

[28] A Seuret and F Gouaisbaut ldquoWirtinger-based integral inequal-ity application to time-delay systemsrdquo Automatica vol 49 no9 pp 2860ndash2866 2013

[29] Z Li Y Bai C Huang and Y Cai ldquoNovel delay-partitioningstabilization approach for networked control system viaWirtinger-based inequalitiesrdquo ISA Transactions vol 61 pp75ndash86 2016

[30] P Park JWKo andC Jeong ldquoReciprocally convex approach tostability of systems with time-varying delaysrdquo Automatica vol47 no 1 pp 235ndash238 2011

[31] K Gu V L Kharitonov and J Chen Stability of Time DelaySystems Birkhauser Boston Mass USA 2003

[32] L El Ghaoui F Oustry and M AitRami ldquoA cone comple-mentarity linearization algorithm for static output-feedbackand related problemsrdquo Institute of Electrical and ElectronicsEngineers Transactions on Automatic Control vol 42 no 8 pp1171ndash1176 1997

[33] R Skjetne T I Fossen and P Kokotovic ldquoAdaptive maneu-vering with experiments for a model ship in a marine controllaboratoryrdquo Automatica vol 41 no 2 pp 289ndash298 2005

[34] N Wang C Qian J-C Sun and Y-C Liu ldquoAdaptive RobustFinite-Time Trajectory Tracking Control of Fully ActuatedMarine Surface Vehiclesrdquo IEEE Transactions on Control SystemsTechnology vol 24 no 4 pp 1454ndash1462 2016

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Page 2: Robust Finite-Time H-Infinity Control with Transients for ...downloads.hindawi.com/journals/mpe/2018/2838749.pdf · ResearchArticle Robust Finite-Time H-Infinity Control with Transients

2 Mathematical Problems in Engineering

and infinite horizons Unfortunately to the best the authorsrsquoknowledge it has not been extended to the system with inputdelay On another research frontier various approaches inthe framework of finite-time boundedness for time-delaysystems have been developed to obtain the results with lessconservatism [15 16] usually indicated by the bound ofstate [17] For DP system engineers are willing to obtaincontrol strategies which maintain the state of ship varyingin a small region around the desired set-point or trackingpath instead of the acceptable minimum state bound Thusthe second motivate is to obtain a result which reduces theoverconservatism caused by the assumption of zero initialcondition and the loss information in the proof for time delaysystems in a practical view

Based on the discussion aforementioned the problemof robust finite-time 119867infin control for DP system with inputdelay is studied The main contribution of this paper liesin three aspects Firstly a finite-time 119867infin controller withtransients is designed for DP systems with input delay bysolving a couple of LMIs which can guarantee the stateof ship within a desired value over a fixed time intervalin the presence of time-varying disturbances Secondly theconcept of 119867infin control under nonzero initial conditionis introduced to time-delay system and its advantage onenhancing the transient performance will be shown in anumerical simulation compared to the119867infin with zero initialcondition In particular the results are established in forms ofLMI which can be easily facilitated by using some standardnumerical packages Thirdly a novel augmented LKLF withtriple integral terms which contains more information isconstructed meanwhile a method combining the Wirtingerinequality and reciprocally convex approach is applied toobtain a tight bound of the integral terms of quadratic func-tions which may lead to a less conservative result comparedto previous works Moreover the practical significance of thismethod in engineering will be demonstrated later Finallya numerical simulation for a ship is proposed to verify theeffectiveness and advantage of the controller we designed

The rest of this paper is organized as follows In Sec-tion 2 the problem formulation of finite-time 119867infin controlis detailed for vessels while some definitions and lemmas areintroduced as preparation In Section 3 themethod to designfinite-time 119867infin controller is proposed and the proposedcontrol schemes are simulated in Section 4 In Section 5 theconclusion is drawn

Notation Throughout this paper R119899 is the 119899-dimensionalEuclidean vector space and R119898times119899 denotes the set of all119898 times 119899 real matrices For symmetric matrices 119883 and 119884119883 gt 119884 (respectively 119883 ge 119884 ) means that 119883 minus 119884is positive definite (respectively positive semidefinite) Thesuperscript ldquo119879rdquo represents the transpose The symmetricterms in a symmetric matrix are denoted by ldquolowastrdquo Moreoverwe use 120582max(sdot)(120582min(sdot) ) to denote the maximum (minimum)eigenvalue of a symmetric matrix

2 Problem Formulation

21 Model of DP System At first DP system model withthree-DOP under low speed can be described as [18]

120578 (119905) = 119869 (120595) 120592 (119905) (1)119872 120592 (119905) + 119863120592 (119905) = 119906 (119905 minus ℎ (119905)) + 120596 (119905) (2)

where 120592 = (120583 V 119903) is a vector of velocities given in thebody-fixed coordinate system and 120578 = (119883 119884 120595) is theposition and orientation of the vessel with respect to aninertial reference coordinate system 119906(119905) is a control vector offorces andmoments provided by the propulsion systems ℎ(119905)is a time-varying function that expresses the actuator delayand satisfies ℎ(119905) lt ℎ ℎ(119905) lt 120583 120596(119905) is the disturbance inputof the system and satisfies the condition of intinfin

0120596(119905)119879120596(119905)119889119905 lt119889 119869(120595) is the transformationmatrix between the inertial and

body-fixed coordinate framesThe inertia matrix119872 includeshydrodynamic added inertia and 119863 is a strictly positivedampingmatrix due to linearwave drift damping and laminarfriction Then the structures of the matrices 119869(120595)119872 and 119863can be explicitly given as follows

119869 (120595) = (cos (120595) minussin (120595) 0sin (120595) cos (120595) 00 0 1)

119872 = (119898 minus 119883 0 00 119898 minus 119884V 119898119909119892 minus 119884 1199030 119898119909119892 minus 119884 119903 119868119911 minus 119873 119903

)119863 = minus(119883119906 0 00 119884V 1198841199030 119873V 119873119903

)(3)

where 119883119906 119884V 119884119903 119873119903 119883 119884V 119884 119903 and 119873 119903 are the hydro-dynamic parameters of the vessel 119898 is the mass of the shipand 119868119911 is the moment of inertia about the yaw rotation 119909119892is the vertical distance from coordinate origin to center ofgravity in body-fixed frame

To simplify the model some assumptions are introducedfirst

Assumption 1 All the parameters of state are available

Assumption 2 The roll and pitch angles are small enough forDP system it is a reasonable assumption

Under assumption 2 we can obtain the simplified equa-tion in form of state-space [19] (119905) = 119860119909 (119905) + 1198611119906 (119905 minus ℎ (119905)) + 1198612120596 (119905) (4)119911 (119905) = 119862119909 (119905) + 119863119906 (119905 minus ℎ (119905)) (5)

where 119909 (119905) = (120578 (119905) 120592 (119905))119860 = (0 1198680 minus119872minus1119863)

Mathematical Problems in Engineering 3

1198611 = 1198612 = ( 0119872minus1)

(6)

Under assumption 1 we design the full-state feedback as119906 (119905 minus ℎ (119905)) = 119870119909 (119905 minus ℎ (119905)) (7)

so the (1) and (2) can be rewritten as (119905) = 119860119909 (119905) + 1198611119870119909 (119905 minus ℎ (119905)) + 1198612120596 (119905) (8)119911 (119905) = 119862119909 (119905) + 119863119870119909 (119905 minus ℎ (119905)) (9)

Remark 3 In the situation of DP motion the motion inheave roll and pitch will be ignored since we focus onthe motion on the surface of sea Meanwhile velocity ofvessel is low enough so the Coriolis-Centripetal matrix andnonlinearities in damping matrix can be neglected [18]

Remark 4 In measurement subsystem of DP system varioussensors are installed to obtain the state of the vessel motionaccurately including global position system for the vesselrsquo sposition gyrocompass for the vesselrsquo s heading and attitudesensor for the pitch and roll Meanwhile data fusion tech-nology is applied to DP system to obtain more accurate stateinformation of vessel motion [18]

22 Preliminaries In the sequel some definitions and lem-mas are introduced to obtain the results

Definition 5 ((FTB) see [20]) Given a positive definitematrix 119877 and three positive constants 1198881 1198882 119879 with 1198881 lt1198882 the time-delay system (4) with 119906(119905) = 0 is said to befinite-time boundedness with respect to (1198881 1198882 119889 119879 120591 119877)if supminus120591lt120579lt0119909119879(120579)119877119909(120579) 119879(120579)119877(120579) lt 1198881 rArr 119909119879(119905)119877119909(119905) lt1198882 forall119905 isin [0 119879] forall120596(119905) int1198790 120596(119905)119879120596(119905)119889119905 lt 119889Definition 6 Given 120574 gt 0 time-delay systems (4) and (5) aresaid to be 119867infin- FTB under nonzero initial condition (119867infin-FTB) with (1198881 1198882 119889 120574 119879 120591 119877) if the following conditionsare satisfied

(1) time-delay system (4) with 119906(119905) = 0 is FTB(2) under the initial condition that 119909(0) = 0 output 119911(119905)

satisfiesΓ= sup

120596(119905)2[0119879]2+119909(0)119879119878119909(0) =0

[ 119911 (119905)2[0119879]2120596 (119905)2[0119879]2 + 119909 (0)119879 119878119909 (0)]12lt 120574(10)

where 120574 is a prescribed scalar and 119878 is a weighting diagonalmatrix that penalizes the effect caused by the initial state

Remark 7 It is necessary to distinguish between finite-timestability and finite-time attractiveness [21] The first conceptis to maintain system states within a given boundary in aspecified time interval [22 23] while the latter describes the

fact that system state reaches the equilibrium point of systemin a finite time [24]

Remark 8 Similar to [13] this definition of 119867infin perfor-mance which depends on the initial condition has beenextended to linear system with acceptable maximal delaybound The performance measure is parametrised by aweighting matrix 119878 reflecting the relative importance of theuncertainty in the initial state contrary to the uncertainty inthe exogenous disturbance When 119878 = 0 it will reduce into asort of119867infin control with zero initial condition [25ndash27]

Lemma 9 (see [28]) For anymatrix119875 gt 0 and a differentiablesignal 119909 in [minus119886 minus119887] rarr R119899 the following inequality holdsintminus119887

minus119886119879 (119906) 1198781 (119906) 119889119904 ge 1119886 minus 119887sdot Π1 (1198781 119909119879 (minus119887) 119909119879 (minus119886) 1119886 minus 119887 intminus119887minus119886

119909119879 (119906) 119889119906) (11)

whereΠ1 (1198781 120572 120573 120574)= ( (120572 minus 120573)119879(120572 + 120573 minus 2120574)119879)119879(1198781 0lowast 31198781)( (120572 minus 120573)119879(120572 + 120573 minus 2120574)119879) (12)

Lemma 10 (see [29]) For constant matrices 1198781 gt 0 1198782 gt 0and constant scalars 119886 gt 119887 the following inequalities hold forall continuously function 119909 in [minus119886 minus119887] rarr R119899intminus119887

minus119886int119905minus119887119905+120579119879 (119906) 1198782 (119906) 119889119906119889120579ge Π2 (1198782 119909119879 (119905 minus 119887) 1119886 minus 119887 int119905minus119887119905minus119886

119909119879 (119906) 119889119906 1(119886 minus 119887)2sdot intminus119887minus119886int119905minus119887119905+120579119909119879 (119906) 119889119906119889120579)

intminus119887minus119886int119905+120579119905minus119886119879 (119906) 1198783 (119906) 119889119906119889120579ge Π3 (1198783 119909119879 (119905 minus 119886) 1119886 minus 119887 int119905minus119887119905minus119886

119909119879 (119906) 119889119906 1(119886 minus 119887)2sdot intminus119887minus119886int119905minus119887119905+120579119909119879 (119906) 119889119906119889120579)

(13)

where

Π2 (1198782 120572 120573 120574) = ( (120572 minus 120573)119879(1205722 minus 120574)119879)119879

sdot (61198782 minus121198782lowast 361198782 )( (120572 minus 120573)119879(1205722 minus 120574)119879)

4 Mathematical Problems in Engineering

Π3 (1198783 120572 120573 120574) = ( (120572 minus 120573)119879(1205722 minus 120573 + 120574)119879)119879

sdot (61198783 minus121198783lowast 361198783 )( (120572 minus 120573)119879(1205722 minus 120573 + 120574)119879)(14)

Lemma 11 (see [30]) Let 1198911 1198912 119891119873 R119898 997891rarr R havepositive values in an open subset119863 ofR119898Then the reciprocallyconvex combination of 119891119894 over119863 satisfies

min120572119894|120572119894gt0sum119894 120572119894=1

sum119894

1120572119894119891119894 (119905) = sum119894 119891119894 (119905) +max119892119894119895(119905)sum119894 =119895

119892119894119895 (119905)subject to 119892119894119895 119877119898 997891997888rarr 119877 119892119895119894 (119905) = 119892119894119895 (119905) [ 119891119894 (119905) 119892119894119895 (119905)119892119895119894 (119905) 119891119895 (119905) ] ge 0 (15)

Lemma 12 (see [31]) For any matrix 1198784 gt 0 and a vectorfunction 119909 in [minus119886 minus119887] rarr R119899 if the integrals concerned arewell defined then the following inequality holdsintminus119887

minus119886119909119879 (119904) 1198784119909 (119904) 119889119904ge 1119886 minus 119887 (intminus119887minus119886

119909119879 (119904) 119889119904) 1198784 (intminus119887minus119886119909 (119904) 119889119904) (16)

23 Control Objective Finally the objective of this paper is toderive the control gain 119870 such that

(1) the closed-loop system (8) with 119906(119905) = 0 is FTB(2) under given nonzero initial condition the closed

system (8) and (9) guarantees that 119869 = int1198790119911119879(119904)119911(119904) minus1205742120596119879(119904)120596(119904) minus 1205742(1119879)119909119879(0)119878119909(0)119889119904 lt 0 for all

nonzero 120596(119905) and 119905 isin [0 119879]3 Main Results

Thedesigning of the robust finite-time119867infin controller for theDP system with input delay is divided into three steps

31 FTB Analysis of DP System Firstly the result guarantee-ing the FTB of DP system is established in this subsection

Theorem 13 Given five positive scalars 120572 120573 1198881 1198882 119879 andpositive define matrix 119877 the finite-time boundedness problemof system (4) is solvable if there exist positive scalars 120579119897 (119897 =1 2 sdot sdot sdot 8) and matrices 11987511 gt 0 11987522 gt 0 119882 gt 0 119876119894 gt0 (119894 = 1 2)119880119895 gt 0 (119895 = 1 2)119872 gt 0 1198851 1198852 with appropriatedimensions satisfying the following conditionsΩ = (Ω119894119895)8times8 lt 0 (17)(119882 1198851lowast 119882) gt 0 (18)

(3119882 1198852lowast 3119882) gt 0 (19)

11988811205792 + ℎ211988811205793 + 120588ℎ11988811205794 + 120588ℎ11988811205795 + 12120588ℎ311988811205796+ 16120588ℎ311988811205797 + 13120588ℎ311988811205798 + 1205731205799119889 lt 119890minus12057211987911988821205791 (20)

whereΩ11 = 11986011987911987511 + 11987511119860 minus 12057211987511 + 1198761 + 1198762+ 119860119879 12ℎ2 (1198801 + 1198802 + 2119882)119860 minus 4120588119882 minus 31205881198801minus 91205881198802Ω22 = minus1205881198762 minus 4120588119882 minus 61205881198802Ω33 = 1198701198791198611119879 12ℎ2 (2119882 + 1198801 + 1198802) 1198611119870 minus 8120588119882+ 1205881198851119879 + 1205881198851 minus 1205881198852119879 minus 1205881198852 minus (1 minus 120583)1198762Ω44 = 1198612119879 12ℎ2 (2119882 + 1198801 + 1198802) 1198612 minus 120573119872Ω66 = minus12120588119882Ω77 = minus12120588119882Ω55 = 120572ℎ211987522 minus 181205881198802 minus 61205881198801Ω88 = minus361205881198801 minus 361205881198802Ω12 = 1205881198851 minus 1205881198852 + 61205881198802Ω15 = ℎ11987522 + 121205881198802Ω16 = 6120588119882Ω17 = 21205881198852

Mathematical Problems in Engineering 5

Ω13 = 119875111198611119870 + 119860119879 12ℎ2 (2119882 + 1198801 + 1198802) 1198611119870 minus 2120588119882minus 1205881198851 minus 1205881198852Ω14 = 119875111198612 + 119860119879 12ℎ2 (2119882 + 1198801 + 1198802) 1198612Ω18 = minus181205881198802 + 61205881198801Ω23 = minus1205881198851119879 minus 1205881198852119879 minus 2120588119882Ω25 = minusℎ11987522 minus 61205881198802Ω26 = 21205881198852119879Ω27 = 6120588119882Ω28 = 121205881198802Ω34 = 1198701198791198611119879 12ℎ2 (2119882 + 1198801 + 1198802) 1198612Ω36 = 6120588119882 + 21205881198852119879Ω37 = 6120588119882 + 21205881198852Ω58 = 121205881198801 + 241205881198802Ω67 = minus41205881198852Ω24 = Ω35 = Ω38 = Ω45 = Ω46 = Ω47 = Ω48 = Ω56= Ω57 = Ω68 = Ω78 = 01205791 = 120582min (11)1205792 = 120582max (11) 1205793 = 120582max (22) 1205794 = 120582max (1198761) 1205795 = 120582max (1198762) 1205796 = 120582max () 1205797 = 120582max (1) 1205798 = 120582max (2) 1205799 = 120582max (119872) 120588 = 119890120572ℎ

11 = 119877minus1211987511119877minus1222 = 119877minus1211987522119877minus12119876119894 = 119877minus12119876119894119877minus12 (119894 = 1 2 3) = 119877minus12119882119877minus12119895 = 119877minus12119880119895119877minus12 (119895 = 1 2) 0 lt 1205791119868 lt 11987511 lt 12057921198680 lt 11987522 lt 12057931198680 lt 1198761 lt 12057941198680 lt 1198762 lt 12057951198680 lt 119882 lt 12057961198680 lt 1198801 lt 12057971198680 lt 1198802 lt 12057981198680 lt 119872 lt 1205799119868(21)

Proof Consider the candidate augment LKLF as follows119881 (119905) = 1198811 (119905) + 1198812 (119905) + 1198813 (119905) + 1198814 (119905) (22)1198811 (119905) = 120576 (119905)119879 119875120576 (119905) (23)1198812 (119905) = int119905119905minusℎ119890120572(119905minus119904)119909 (119904)1198791198761119909 (119904) 119889119904+ int119905119905minusℎ(119905)119890120572(119905minus119904)119909 (119904)1198791198762119909 (119904) 119889119904 (24)

1198813 (119905) = ℎint0minusℎint119905119905+120579119890120572(119905minus119904) (119904)119879119882 (119904) 119889119904119889120579 (25)

1198814 (119905) = int0minusℎint0120579int119905119905+120573119890120572(119905minus119904) (119904)1198791198801 (119904) 119889119904119889120573119889120579

+ int0minusℎint120579minusℎint119905119905+120573119890120572(119905minus119904) (119904)1198791198802 (119904) 119889119904119889120573119889120579 (26)

where 120576(119905) = (119909(119905) int119905119905minusℎ119909(119904)119889119904) 119875 = ( 11987511 0

lowast 11987522)

To deal with the formulas conveniently let us providesuch definitions

120585 (119905)= (119909 (119905) 119909 (119905 minus ℎ) 119909 (119905 minus ℎ (119905)) 120596 (119905) 1ℎ int119905119905minusℎ 119909 (119904) 119889119904 1ℎ (119905) int119905119905minusℎ(119905) 119909 (119904) 119889119904 1ℎ minus ℎ (119905) int119905minusℎ(119905)119905minusℎ119909 (119904) 119889119904 1ℎ2 int0minusℎ int119905119905+120579 119909 (119904) 119889119904119889120579)119897119896119879 = (0 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896minus1

1 0 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟8minus119896) (27)

6 Mathematical Problems in Engineering

The time-derivative of 119881(119905) along system (4) can bebounded as1 (119905) = 1205721198811 (119905) + 120576 (119905)119879 119875120576 (119905) + 120576 (119905)119879 119875 120576 (119905) minus 1205721198811 (119905) (28)2 (119905)= 1205721198812 (119905) + 119909 (119905)1198791198761119909 (119905)minus 119890120572ℎ119909 (119905 minus ℎ)1198791198761119909 (119905 minus ℎ) + 119909 (119905)1198791198762119909 (119905)minus (1 minus ℎ (119905)) 119890120572ℎ(119905)119909 (119905 minus ℎ (119905))1198791198762119909 (119905 minus ℎ (119905))le 1205721198812 (119905) + 119909 (119905)1198791198761119909 (119905)minus 119890120572ℎ119909 (119905 minus ℎ)1198791198761119909 (119905 minus ℎ) + 119909 (119905)1198791198762119909 (119905)minus (1 minus 120583) 119909 (119905 minus ℎ (119905))1198791198762119909 (119905 minus ℎ (119905))

(29)

3 (119905)= 1205721198813 (119905) + ℎ2 (119905)1198791198822 (119905)minus 119890120572ℎℎint119905119905minusℎ (119904)1198791198822 (119904) 119889119904 (30)

4 (119905)= 1205721198814 (119905) + 12ℎ2 (119905)1198791198801 (119905)minus 119890120572ℎ int0minusℎint119905119905+120573 (119904)1198791198801 (119904) 119889119904119889120573+ 12ℎ2 (119905)1198791198802 (119905)minus 119890120572ℎ int0

minusℎint119905+120573119905minusℎ (119904)1198791198802 (119904) 119889119904119889120573

(31)

Invoking Lemma 9 we can obtain3 (119905) = 1205721198813 (119905) + ℎ2 (119905)119879119882 (119905)minus 119890120572ℎℎint119905119905minusℎ(119905) (119904)119879119882 (119904) 119889119904

minus 119890120572ℎℎint119905minusℎ(119905)119905minusℎ

(119904)119879119882 (119904) 119889119904le 1205721198813 (119905) + ℎ2 (119905)119879119882 (119905)minus 119890120572ℎ ℎℎ (119905)1206031119879Λ 11206031 minus 119890120572ℎ ℎℎ minus ℎ (119905)1206032119879Λ 21206032(32)

where 1206031119879 = 120585119879 (119905)( (1198971119879 minus 1198973119879)119879(1198971119879 + 1198973119879 minus 21198976119879)119879)119879

1206032119879 = 120585119879 (119905)( (1198973119879 minus 1198972119879)119879(1198973119879 + 1198972119879 minus 21198977119879))119879

Λ 1 = Λ 2 = (119882 0lowast 3119882) (33)

In order to obtain a tighter bound of integral termLemma 11 is applied to (32) as follows3 (119905) le 1205721198813 (119905) + ℎ2 (119905)119879119882 (119905) minus 119890120572ℎ1206033119879Λ 31206033 (34)

where 1206033119879 = ( 12060311198791206032119879 ) Λ 3 = (119882 1198851 0 0

lowast 119882 0 0lowast lowast 3119882 1198852lowast lowast lowast 3119882

)By applying Lemma 10 to (31) we can obtain the inequal-

ity 4 (119905) le 1205721198814 (119905) + 12ℎ2 (119905)1198791198801 (119905)+ 12ℎ2 (119905)1198791198802 (119905) minus 119890120572ℎ1206034119879Λ 41206034 (35)

where

1206034119879 = 120585119879 (119905)(((((

(1198971119879 minus 1198972119879)119879(12 1198971119879 minus 1198978119879)119879(1198972119879 minus 1198975119879)119879(12 1198972119879 minus 1198975119879 + 1198978119879)119879)))))

119879

Λ 4 =(61198801 minus121198801 0 0lowast 361198801 0 0lowast lowast 61198802 minus121198802lowast lowast lowast 361198802)(36)

Combining (28) (29) (34) and (35) with the definitionof 120588 = 119890120572ℎ we can obtain minus 120572119881 minus 120573120596119879 (119905)119872120596 (119905) lt 120585119879 (119905) Ω120585 (119905) (37)

Assuming that Ω lt 0 and integrating the left part ofinequality (37)119881 (119905) lt 119890120572119905119881 (0) + 120573int119905

0119890120572(119905minus119904)120596119879 (119904)119872120596 (119904) 119889119904lt 119890120572119879 (119881 (0) + 120573120582max (119872) 119889) (38)

Invoking the Jensen inequality shown in Lemma 12 onehas 119881 (0) le 119909 (0)119879 11987511119909 (0) + ℎint0

minusℎ119909 (119904)119879 11987522119909 (119904) 119889119904+ 120588int0

minusℎ119909 (119904)1198791198761119909 (119904) 119889119904+ 120588int0

minusℎ(119905)119909 (119904)1198791198762119909 (119904) 119889119904

Mathematical Problems in Engineering 7

+ ℎ120588int0minusℎint0120579 (119904)119879119882 (119904) 119889119904119889120579

+ 120588int0minusℎint0120579int0120573 (119904)1198791198801 (119904) 119889119904119889120573119889120579

+ 120588int0minusℎint120579minusℎint0120573 (119904)1198791198802 (119904) 119889119904119889120573119889120579le 120582max (11) 1198881 + ℎ120582max (22) ℎ1198881+ 120588120582max (1198761) ℎ1198881 + 120588120582max (1198762) ℎ1198881+ 120588120582max () 12ℎ31198881 + 120588120582max (1) 16ℎ31198881+ 120588120582max (2) 13ℎ31198881

(39)

Based on inequalities (38) and (39) we have119881 (119905) gt 119909119879 (119905) 11987511119909 (119905) = 119909119879 (119905) 119877121111987712119909 (119905)gt 120582min (11) 119909119879 (119905) 119877119909 (119905) (40)

Therefore conditions (17) to (20) can guarantee the FTBof system (4) This completes the proof

Remark 14 Among the existing approaches there are twothreads one is to construct a novel LKLF that involves moreinformation of delay the other is to find a tighter estimationof upper bound for cross terms coming from the derivativeof the LKLF In this paper these two techniques are appliedto obtain the result In addition the less conservatism inpractical engineering will be shown in the simulations whichare always ignored in most literature sources

32 Controller Design In this subsection we focus on theproblem of finite-time 119867infin state feedback designing basedon Theorem 13 that is designing a state feedback controllerin the form of (7) such that the resulting DP system satisfiesthe control objective proposed in Section 1

Theorem 15 For given positive 120572 1198881 1198882 119879 120574 and matrices 119877 gt0 119878 ge 0 if there exist positive scalars 120579119897 (119897 = 1 2 sdot sdot sdot 8) andmatrices 11987511 gt 0 11987522 gt 0119882 gt 0 119876119894 gt 0 (119894 = 1 2) 119880119895 gt0 (119895 = 1 2) 119884 1198851 1198852 with appropriate dimensions satisfyingthe following conditions Π = (Π119894119895)12times12 lt 0 (41)Σ = (Σ119894119895)9times9 lt 0 (42)

(119882 1198851lowast 119882) gt 0 (43)

(3119882 1198852lowast 3119882) gt 0 (44)

119868 minus 120574211987811987511 lt 0 (45)0 lt 1198671 lt 11987511 lt 11986720 lt 11987522 lt 11986730 lt 1198761 lt 11986740 lt 1198762 lt 11986750 lt 119882 lt 11986760 lt 1198801 lt 11986770 lt 1198802 lt 1198678(46)

where Π11 = 11987511119860119879 + 11986011987511 minus 12057211987511 + 1198761 + 1198762 minus 4120588119882minus 31205881198801 minus 91205881198802Π44 = minus1205742119868Π22 = minus1205881198762 minus 4120588119882 minus 61205881198802Π33 = minus8120588119882 + 1205881198851119879 + 1205881198851 minus 1205881198852119879 minus 1205881198852minus (1 minus 120583)1198762Π13 = 1198611119884 minus 2120588119882 minus 1205881198851 minus 1205881198852Π14 = 1198612Π66 = minus12120588119882Π77 = minus12120588119882Π55 = 120572ℎ211987522 minus 181205881198802 minus 61205881198801Π88 = minus361205881198801 minus 361205881198802Π12 = 1205881198851 minus 1205881198852 + 61205881198802Π15 = ℎ11987522 + 121205881198802Π16 = 6120588119882Π17 = 21205881198852Π25 = minusℎ11987522 minus 61205881198802Π18 = minus181205881198802 + 61205881198801Π26 = 21205881198852119879Π23 = minus1205881198851119879 minus 1205881198852119879 minus 2120588119882Π27 = 6120588119882Π28 = 121205881198802

8 Mathematical Problems in Engineering

Π36 = 6120588119882 + 21205881198852119879Π37 = 6120588119882 + 21205881198852Π58 = 121205881198801 + 241205881198802Π67 = minus41205881198852Π19 = 12ℎ11987511119860119879Π39 = 12ℎ1198841198791198611119879Π49 = 12ℎ1198612119879Π110 = 12ℎ11987511119860119879Π310 = 12ℎ1198841198791198611119879Π410 = 12ℎ1198612119879Π111 = ℎ11987511119860119879Π311 = ℎ1198841198791198611119879Π411 = ℎ1198612119879Π112 = 11987511119862119879Π312 = 119884119879119863119879Π99 = 121198801 minus 11987511Π1010 = 121198802 minus 11987511Π1111 = 119882 minus 211987511Π1212 = minus119890minus120572119879119868(47)

others Π119894119895 = 0Σ11 = minus119890minus12057211987911988821198671Σ22 = minus1198672Σ33 = minus1198673Σ44 = minus1198674Σ55 = minus1198675Σ66 = minus1198676Σ77 = minus1198677Σ88 = minus1198678

Σ99 = minus1205742119889Σ12 = radic11988811198672Σ13 = radicℎ211988811198673Σ14 = radic120588ℎ11988811198674Σ15 = radic120588ℎ11988811198675Σ16 = radic12120588ℎ311988811198676Σ17 = radic16120588ℎ311988811198677Σ18 = radic13120588ℎ311988811198678Σ19 = 11987511(48)

others Σ119894119895 = 0 (49)

then a state feedback119867infin controller in form of (7) existssuch that

(1) the closed-loop system (8) with 119906(119905) = 0 is FTB(2) under given nonzero initial condition the closed

system (8) and (9) guarantees that 119869 = int1198790119911119879(119904)119911(119904) minus1205742120596119879(119904)120596(119904) minus 1205742(1119879)119909119879(0)119878119909(0)119889119904 lt 0 for all nonzero 120596(119905)

and 119905 isin [0 119879]The controller gain can be calculated by 119870 = 11988411987511minus1

Proof In preparation for designing we set119872 = 1205742119868 120573 = 1for (37) then (37) and (38) can be rewritten as follows minus 120572119881 minus 1205742120596119879 (119905) 120596 (119905) lt 120585119879 (119905) Ω120585 (119905) (50)119881 (119905) lt 119890120572119905119881 (0) + 1205742 int119905

0119890120572(119905minus119904)120596119879 (119904) 120596 (119904) 119889119904lt 119890120572119879 (119881 (0) + 1205742119889) (51)

where Ω44 = 1198612119879 12ℎ2 (2119882 + 1198801 + 1198802) 1198612 minus 1205742119868Ω119894119895 = Ω119894119895

(52)

Then two steps are provided to design the desiredcontroller

Step 1 Here we will give some conditions that can guaranteethe 119867infin performance over the finite-time interval undernonzero initial conditions firstly

Mathematical Problems in Engineering 9

Consider the following inequality minus 120572119881 + 119890120572119905119911119879 (119905) 119911 (119905) minus 1205742119890120572119905120596119879 (119905) 120596 (119905)lt minus 120572119881 + 119890120572119879119911119879 (119905) 119911 (119905) minus 1205742120596119879 (119905) 120596 (119905)lt 120585119879 (119905) (Ω minus Ψ119879 (minus119890minus120572119879119868)minus1Ψ) 120585 (119905) (53)

Noting that119882 1198801 and 1198802 are coupled with 119860 1198611119870 and1198612 in Ω to decouple these terms the Ω can be rewritten asΩ = Ω minus Ψ119879 (minus121198801)minus1 Ψ minus Ψ119879 (minus121198802)minus1Ψminus Ψ119879 (minus119882)minus1 Ψ (54)

whereΩ11 = 11986011987911987511 + 11987511119860 minus 12057211987511 + 1198761 + 1198762 minus 4120588119882minus 31205881198801 minus 91205881198802Ω33 = minus8120588119882 + 1205881198851119879 + 1205881198851 minus 1205881198852119879 minus 1205881198852minus (1 minus 120583)1198762Ω44 = minus1205742119868Ω13 = 119875111198611119870 minus 2120588119882 minus 1205881198851 minus 1205881198852Ω14 = 119875111198612Ω34 = 0(55)

othersΩ119894119895 = Ω119894119895Ψ = (119862 0 119863119870 0 0 0 0 0)Ψ= (ℎ121198801119879119860 0 ℎ1211988011198791198611119870 ℎ1211988011198791198612 0 0 0 0) Ψ= (ℎ121198802119879119860 0 ℎ1211988021198791198611119870 ℎ1211988021198791198612 0 0 0 0)Ψ = (ℎ119882119879119860 0 ℎ1198821198791198611119870 ℎ1198821198791198612 0 0 0 0) (56)

By Schur complementΩ lt 0 and Ω minusΨ119879(minus119890minus120572119879119868)minus1Ψ areequivalent to

Π =(((((

Ω Ψ119879 Ψ119879 Ψ119879 Ψ119879lowast minus121198801 0 0 0lowast lowast minus121198802 0 0lowast lowast lowast minus119882 0lowast lowast lowast lowast minus119890minus120572119879119868)))))lt 0 (57)

To ensure the proposed conditions satisfy the 119867infin per-formance we set the inequality like

11987511 minus 1205742119878 lt 0 (58)

from which we have

minus 120572119881 + 119890120572119905119911119879 (119905) 119911 (119905) minus 1205742119890120572119905120596119879 (119905) 120596 (119905)minus 1205742119890120572119905 1119879119909119879 (0) 119878119909 (0) lt minus 120572119881 + 119890120572119905119911119879 (119905) 119911 (119905)minus 1205742119890120572119905120596119879 (119905) 120596 (119905) minus 1205742119890120572119905 1119879119909119879 (0) 119878119909 (0)+ 119890120572119905 1119879119909119879 (0) 11987511119909 (0) lt minus 120572119881 + 119890120572119905119911119879 (119905) 119911 (119905)minus 1205742119890120572119905120596119879 (119905) 120596 (119905) + 119890120572119905 1119879119909119879 (0) (11987511 minus 1205742119878) 119909 (0)lt 0(59)

Integrating (58) from 0 to 119879 we can obtain

int1198790119911119879 (119904) 119911 (119904) 119889119904 minus int119879

01205742120596119879 (119904) 120596 (119904) 119889119904

minus 1205742119909119879 (0) 119878119909 (0) lt minusint1198790(119890minus120572119905 minus 120572119890minus120572119905119881)119889119904 lt 0 (60)

so conditions (43) (44) (57) and (58) with the inequali-ties as follows can ensure that systems (17) and (18) satisfy the119867infin performance

11988811205792 + ℎ211988811205793 + 120588ℎ11988811205794 + 120588ℎ11988811205795 + 12120588ℎ311988811205796+ 16120588ℎ311988811205797 + 13120588ℎ311988811205798 + 1205742119889 lt 119890minus120572119879119888212057910 lt 1205791119868 lt 11987511 lt 12057921198680 lt 11987522 lt 12057931198680 lt 1198761 lt 12057941198680 lt 1198762 lt 12057951198680 lt 119882 lt 12057961198680 lt 1198801 lt 12057971198680 lt 1198802 lt 1205798119868

(61)

10 Mathematical Problems in Engineering

Step 2 In this step the approach to design the gain matrix 119870of the controller is provided in forms of LMIs which can becalculated expediently

To linearize the nonlinear terms in (57) we make itmultiply by the followingmatrix fromboth left and right sidesfirstly

diag 11987511minus1 11987511minus1 11987511minus1 119868 11987511minus1 11987511minus1 11987511minus1 11987511minus1 1198801minus1 1198802minus1 119882minus1 119868 (62)

Let us do the definitions as follows11987511 = 11987511minus11198761 = 11987511minus1119876111987511minus11198762 = 11987511minus1119876211987511minus1119882 = 11987511minus111988211987511minus11198801 = 11987511minus1119880111987511minus11198802 = 11987511minus1119880211987511minus111987522 = 11987511minus11198752211987511minus11198851 = 11987511minus1119885111987511minus11198852 = 11987511minus1119885211987511minus1119884 = 11987011987511minus1

(63)

so the inequality Π lt 0 can guarantee Π lt 0 whereΠ99 = minus12119875111198801

minus111987511Π1010 = minus12119875111198802

minus111987511Π1111 = minus11987511119882minus111987511(64)

others Π119894119895 = Π119894119895 (65)

Then based on the inequalities minus(12)119875111198801

minus111987511 le(12)1198801 minus 11987511 minus(12)119875111198802

minus111987511 le (12)1198802 minus 11987511minus11987511119882minus111987511 le 119882 minus 211987511 condition (57) can be convertedinto (41) in form of LMIs

Accordingly similar to the above-mentioned procedures(61) can be transformed into (42) with the Schur complementand the definition of119867119894 = 11987511minus112057911989411987511minus1 (119894 = 1 2 sdot sdot sdot 8)

Therefore conditions (41) to (45) can obtain the con-troller in the form of (7) guaranteeing the desired perfor-mance This completes the proof

Remark 16 The119867infin performance with nonzero initial con-dition is studied in this literature In a practical view it is

hard to suffer the worst-case over a finite time for surfacevessel with nonzero initial condition and it will reducethe conservatism caused by the assumption of zero initialcondition Therefore it is reasonable to introduce such 119867infinperformance index to handle the problem of finite-time119867infincontrol for ship subject to exogenous signals

Remark 17 It is challenging to achieve the result satisfyingthe control objective while it is very different from thestandard 119867infin control because a weighting diagonal matrixis introduced especially to establish the form of LMIs Inthe proof of Theorem 15 inequality (58) is introduced toensure the 119867infin performance with nonzero initial conditionand make it convenient to construct the conditions in formof LMIs

Remark 18 By introducing some new additional matrixvariables Cone Complementarity Linearization algorithm(CCL) [32] can be utilized to linearize the nonlinear termsas an alternative method It will reduce the conservatism atthe expense of increasing the computational burden and thenumber of decision variables

4 Simulations

41 Simulation Settings To demonstrate the effectiveness andsuperiority of the robust finite-time 119867infin control schemesimulation studies are conducted on the parameters of theship CyberShip II [33] which is a 170 scale-replica of a supplyshipwhosemain parameters are shown inTable 1 In additionwe choose the time-varying function like ℎ(119905) = 3sin (1199056 minus1205876) to describe the time delay occurred in the input signal

Unknown external disturbances 120596(119905) with significantlylarge magnitude and high frequency which can be roughlyconsidered as complex environment including ocean windswaves and currents are governed by [34]

120596 (119905) = (9 sin (01120587119905 minus 1205875 )6 sin (03120587119905 + 1205876 )3 sin (02120587119905 + 1205873 )) (66)

with initial conditions 120578(0) = (1 1 01) and 120592(0) =(08 06 01)42 Numerical Simulations We choose 120574 = 13 119879 = 10 119878 =03lowast119868 1198881 = 302 1198882 = 10 120572 = 0001 By usingTheorem 15 thecontrol gain 119870 is calculated by LMIs (41) to (45)

Mathematical Problems in Engineering 11

Table 1 The main parameters of the CyberShip II119898 238000 119883 -20 119883119906 -07225119868119911 17600 119884V -100 119884V -08612119909119892 00460 119884 119903 -00 119884119903 01079119873 119903 -10 119873119903 05

0 20 40 60 80 100Time(s) Time(s)

minus2

0

2

4

6

8

10

12

0 2 4 6 8 100

2

4

6

8

10

12

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

xT(t)Rx(t)xT(t)Rx(t)

c=c=

c1=302

c1=302

Figure 1 State response 119909119879(119905)119877119909(119905) of DP system

119870 = (minus32743 0 0 minus313907 0 00 minus42784 minus01572 0 minus412136 minus129820 minus01389 minus03497 0 minus13362 minus33091) (67)

To demonstrate the superiority and validity three sce-narios as comparative cases are provided here It should bepointed out that the results within 100s are also provided toillustrate the steady-state performance

Case 1 (with different 119878) For comparison purposes we set119878 with different value to demonstrate the effect of 119878 on stateresponse Without loss of generality we set 119878 = 05 lowast 119868 and119878 = 01 lowast 119868 respectivelyCase 2 (119867infin with zero initial condition]) This scenario willshow that the control law proposed in this paper can achievea better transient performance and disturbance rejectioncapability over traditional119867infin control approach for dynamicpositioning system conducted in [11]

Case 3 (a control design based on existing methods) Basedon the approach in [17] applied for discrete-time system acontrol law for dynamic positioning ship with input delay isproposed to illustrate the less conservatism of our designingapproach in practical engineering subsequently

At first the effectiveness of the control law we designedis confirmed from Figures 1ndash6 Figure 1 shows the state

response of 119909119879(119905)119877119909(119905) Obviously from that curve we cansee that the value of 119909119879(119905)119877119909(119905) is far below the value of1198882 = 10 in the presence of disturbance as in Figure 2which means the finite-time full-state feedback controllerdesigned can guarantee the DP system (8) sim (9) FTB with(302 10 119889 10 3 119868) and have less conservatism where weset 119889 = int119879

0120596(119905)119879120596(119905)119889119905 Meanwhile the control law can

stabilize the dynamic positioning shipwithminimum steady-state error in 100s and the control input will be small enoughas shown in Figure 6 From Figures 3ndash5 we can find that allstates in DP system are within a small range As shown inFigures 2ndash4 the angle and the angular velocity in yaw havechanged within 01∘ which means that the heading of thevessel stays in a fixed direction Furthermore the velocitiesin surge and sway are low enough which illustrates thatassumption 2 is reasonable and it meets the requirement ofstabilization control for DP system

To demonstrate the effect of 119878 state responses of systemswith different 119878 are exhibited in Figures 7ndash9 As the valueof 119878 decreases the restriction for 11987511 is more exclusive andthe robustness is stronger during preset time interval as inFigures 7 and 8 while the required control input will increasedramatically as in Figure 9 When the time oversteps the

12 Mathematical Problems in Engineering

0 10 20 30 40 50 60 70 80 90 100Time(s)

minus10

minus5

0

5

10

Dist

urba

nces

(N)

XY

Figure 2 Disturbance in surge sway and yaw

0 50 100Time(s)

minus050

051

152

X(m

)

0 5 10Time(s)

005

115

2

X(m

)

0 50 100Time(s)

minus050

051

152

Y(m

)

0 5 10Time(s)

005

115

2Y(

m)

0 50 100Time(s)

minus01

0

01

02

0 5 10Time(s)

0

01

02

(∘

)

(∘

)

Figure 3 State responses of position and orientation for DP System

0 50 100Time(s)

minus1

0

1

2

0 5 10Time(s)

minus1

0

1

2

0 50 100Time(s)

minus1

0

1

2

v(m

s)

0 5 10Time(s)

minus1

0

1

2

v(m

s)

0 50 100Time(s)

minus01

0

01

02

0 5 10Time(s)

minus01

0

01

02

r(∘ s

)

r(∘ s

)

(ms

)

(m

s)

Figure 4 State responses of velocities for DP system

Mathematical Problems in Engineering 13

minus05 0 05 1 15 2X(m)

minus05

0

05

1

15

2

Y(m

)

Figure 5 Trajectory of the shiprsquos horizontal motion

0 50 100Time(s)

minus30minus20minus10

010

u1(N

)

0 5 10Time(s)

minus30minus20minus10

010

u1(N

)

0 50 100Time(s)

minus30minus20minus10

010

u2(N

)

0 5 10Time(s)

minus30minus20minus10

010

u2(N

)

0 50 100Time(s)

minus15minus1

minus050

05

u3(N

m)

0 5 10Time(s)

minus15minus1

minus050

05

u3(N

m)

Figure 6 Control inputs

0 20 40 60 80 100Time(s)

minus1

0

1

2

3

4

5

6

S=03S=05S=01

0 2 4 6 8 10Time(s)

0

2

4

6

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 7 State response 119909119879(119905)119877119909(119905) with different 119878

14 Mathematical Problems in Engineering

Figure 8 State responses of position and orientation with different 119878

0 50 100Time(s)

minus60minus30

03060

u1(N

)

0 5 10Time(s)

minus40minus20

02040

u1(N

)

0 50 100Time(s)

minus100

0

100

u2(N

)

0 5 10Time(s)

minus40minus20

02040

u2(N

)

0 50 100Time(s)

minus4minus2

024

u3(N

m)

0 5 10Time(s)

minus2

0

2

u3(N

m)

S=03S=05S=01

Figure 9 Control input with different 119878preset time interval all the states can be guaranteed withinthe given upper bound as before

As shown in Figures 10ndash12 the overconservatism oftraditional119867infin control approach applied inDP ship is solvedby the 119867infin control strategy with nonzero initial conditionproposed in this article As is well known the classical 119867infincontrol law is designed under the so-called worst-case normwhile the performance of antidisturbance and transient issacrificed to some degree From Figures 10 and 11 we cansee that the state response can be controlled in a smallerrange and stabilized in a shorter time with a lower overshootshown in Figure 12 Hence a conclusion can be drawn thatthe control law we present can achieve a better disturbance

rejection capability and transient performance while thematter of overconservatism can be resolved at the sametime

Simulation results of Theorem 15 and Case 3 are shownin Figures 13 and 14 to show the better performance ofthe method we proposed We can find that the controllerbased on [17] and 119867infin control approach present in thisliterature can make all states of system within a smallenough region However compared with the controller weconstructed the state response has higher overshoot whichwill affect the control precision and transient performance inpractical engineering Based on the above simulation resultsthe developed controller can achieve a higher accuracy and

Mathematical Problems in Engineering 15

0 20 40 60 80 100Time(s)

minus2

0

2

4

6

8

10

12

Theorem 15Case 2

0 2 4 6 8 10Time(s)

minus2

0

2

4

6

8

10

12

Theorem 15Case 2

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 10 State response 119909119879(119905)119877119909(119905) of Theorem 15 and Case 2

0 50 100Time(s)

minus10123

X(m

)

0 5 10Time(s)

0123

X(m

)

0 50 100Time(s)

minus10123

Y(m

)

0 5 10Time(s)

0123

Y(m

)

0 50 100Time(s)

minus020

0204

0 5 10Time(s)

minus020

0204

Theorem 15Case 2

(∘

)

(∘

)

Figure 11 Position and orientation of Theorem 15 and Case 2

minus05 0 05 1 15 2 25X(m)

minus050

051

152

25

Y(m

)

Theorem 15Case 2

Figure 12 Trajectory of the shiprsquos horizontal motion of Theorem 15 and Case 2

16 Mathematical Problems in Engineering

0 20 40 60 80 100Time(s)

0

1

2

3

4

5

Theorem 15Case 3

0 2 4 6 8 10Time(s)

0

1

2

3

4

5

Theorem 15Case 3

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 13 State response 119909119879(119905)119877119909(119905) of Theorem 15 and Case 3

0 5 10Time(s)

05

1

15

X(m

)

Theorem 15Case 3

0 5 10Time(s)

05

1

15

Y(m

)

0 5 10Time(s)

0

005

01

015

02

(∘

)

Figure 14 Position and orientation of Theorem 15 and Case 3

lower overshoot which means that it is less conservative incomparison with existing result

5 Conclusion

In this paper the robust finite-time 119867infin control with tran-sients for DP system with input delay maintaining all statesof ocean surface vessel into a given threshold over a fixed timeinterval in presence of time-varying disturbance is proposedBased on a novel augmented LKLF with triple integral termsand a method combining theWirtinger inequality and recip-rocally convex approach a less conservative result is derivedIn particular a119867infin performance index with nonzero initialcondition attenuating the overconservatism caused by theassumption of zero initial condition and enhancing thetransient performance of ship subject to external disturbanceis adoptedThe obtained result is formulated in terms of LMIswhich can be easily solved by the standard numerical soft-ware Finally a numerical simulation for a ship is proposed toverify the effectiveness and less conservatism of the controller

we designed In the future the output feedback control can bedeveloped for the ocean surface vessel to avoid the use of thevessel-frame surge sway and yaw velocities

Data Availability

All the data supporting the conclusions of the study have beenprovided in Simulations and readers can access these data in[33 34]

Conflicts of Interest

The authors declare that there are no conflicts of interestrelated to this paper

Acknowledgments

This research was partially supported by the National Sci-ence Technology Support Program of China (Project no51609046)

Mathematical Problems in Engineering 17

References

[1] M Chen B Jiang and R Cui ldquoActuator fault-tolerant controlof ocean surface vessels with input saturationrdquo InternationalJournal of Robust amp Nonlinear Control vol 26 no 3 pp 542ndash564 2016

[2] J Du X Hu and Y Sun ldquoRobust dynamic positioning of shipswith disturbances under input saturationrdquo Automatica vol 73pp 207ndash214 2016

[3] H Ashrafiuon K R Muske L C McNinch and R ASoltan ldquoSliding-mode tracking control of surface vesselsrdquo IEEETransactions on Industrial Electronics vol 55 no 11 pp 4004ndash4012 2008

[4] W Si and X Dong ldquoNeural prescribed performance controlfor uncertain marine surface vessels without accurate initialerrorsrdquoMathematical Problems in Engineering Art ID 261832311 pages 2017

[5] T D Nguyen A J Soslashrensen and S T Quek ldquoDesign of hybridcontroller for dynamic positioning from calm to extreme seaconditionsrdquo Automatica vol 43 no 5 pp 768ndash785 2007

[6] J Du X Hu H Liu and C L Chen ldquoAdaptive robust outputfeedback control for a marine dynamic positioning systembased on a high-gain observerrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 11 pp 2775ndash27862015

[7] H Katayama ldquoNonlinear sampled-data stabilization of dynam-ically positioned shipsrdquo IEEE Transactions on Control SystemsTechnology vol 18 no 2 pp 463ndash468 2010

[8] K J Astrom andCG Kallstrom ldquoIdentification of ship steeringdynamicsrdquo Automatica vol 12 no 1 pp 9ndash22 1976

[9] Z Lei and C Guo ldquoDisturbance rejection control solutionfor ship steering system with uncertain time delayrdquo OceanEngineering vol 95 pp 78ndash83 2015

[10] J C Doyle K Glover P P Khargonekar and B A FrancisldquoState-space solutions to standard H-2 and Hinfin control prob-lemsrdquo IEEE Transactions on Automatic Control vol 34 no 8pp 831ndash847 1989

[11] M Katebi M Grimble and Y Zhang ldquoHinfin robust controldesign for dynamic ship positioningrdquo IEE Proceedings - ControlTheory and Applications vol 144 no 2 pp 110ndash120 1997

[12] D V Balandin and M M Kogan ldquoLMI-based Hinfin-optimalcontrol with transientsrdquo International Journal of Control vol 83no 8 pp 1664ndash1673 2010

[13] Z Feng J Lam S Xu and S Zhou ldquoHinfin Control withTransients for Singular Systemsrdquo Asian Journal of Control vol18 no 3 pp 817ndash827 2016

[14] P P Khargonekar K M Nagpal and K R Poolla ldquoHinfin controlwith transientsrdquo SIAM Journal onControl andOptimization vol29 no 6 pp 1373ndash1393 1991

[15] Z Zhang Z Zhang and H Zhang ldquoFinite-time stabilityanalysis and stabilization for uncertain continuous-time systemwith time-varying delayrdquo Journal of The Franklin Institute vol352 no 3 pp 1296ndash1317 2015

[16] J Cheng S Chen Z Liu H Wang and J Li ldquoRobust finite-time sampled-data control of linear systems subject to randomoccurring delays and its application to Four-Tank systemrdquoAppliedMathematics andComputation vol 281 pp 55ndash76 2016

[17] W Kang S Zhong K Shi and J Cheng ldquoFinite-time stabilityfor discrete-time system with time-varying delay and nonlinearperturbationsrdquo ISA Transactions vol 60 pp 67ndash73 2016

[18] T I Fossen Handbook of Marine Craft Hydrodynamics andMotion Control John Wiley amp Sons New York USA 2011

[19] D Zhao F Ding L Zhou W Zhang and H Xu ldquoRobustH-infinity Control of Neutral System with Time-Delay forDynamic Positioning Shipsrdquo Mathematical Problems in Engi-neering vol 2015 Article ID 976925 11 pages 2015

[20] F Amato M Ariola and C Cosentino ldquoFinite-time stabilityof linear time-varying systems analysis and controller designrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 55 no 4 pp 1003ndash1008 2010

[21] L Sun andZ Zheng ldquoFinite-time slidingmode trajectory track-ing control of uncertain mechanical systemsrdquo Asian Journal ofControl vol 19 no 1 pp 399ndash404 2017

[22] P Shi Y Zhang and R Agarwal ldquoStochastic finite-time stateestimation for discrete time-delay neural networks withMarko-vian jumpsrdquo Neurocomputing vol 151 part 1 pp 168ndash174 2014

[23] L Zhang S Wang H R Karimi and A Jasra ldquoRobust Finite-Time Control of Switched Linear Systems and Application toa Class of Servomechanism Systemsrdquo IEEEASME Transactionson Mechatronics vol 20 no 5 pp 2476ndash2485 2015

[24] L Wu P Shi and H Gao ldquoState estimation and sliding-mode control of Markovian jump singular systemsrdquo Institute ofElectrical and Electronics Engineers Transactions on AutomaticControl vol 55 no 5 pp 1213ndash1219 2010

[25] Y Q Zhang P Shi S K Nguang and Y D Song ldquoRobust finite-time Hinfin control for uncertain discrete-time singular systemswithMarkovian jumpsrdquo IET ControlTheory ampApplications vol8 no 12 pp 1105ndash1111 2014

[26] S Wang T Shi L Zhang A Jasra and M Zeng ldquoExtendedfinite-time Hinfin control for uncertain switched linear neutralsystemswith time-varying delaysrdquoNeurocomputing vol 152 pp377ndash387 2015

[27] J Hu G Sui S Du and X Li ldquoFinite-time stability of uncer-tain nonlinear systems with time-varying delayrdquo MathematicalProblems in Engineering Article ID 2538904 9 pages 2017

[28] A Seuret and F Gouaisbaut ldquoWirtinger-based integral inequal-ity application to time-delay systemsrdquo Automatica vol 49 no9 pp 2860ndash2866 2013

[29] Z Li Y Bai C Huang and Y Cai ldquoNovel delay-partitioningstabilization approach for networked control system viaWirtinger-based inequalitiesrdquo ISA Transactions vol 61 pp75ndash86 2016

[30] P Park JWKo andC Jeong ldquoReciprocally convex approach tostability of systems with time-varying delaysrdquo Automatica vol47 no 1 pp 235ndash238 2011

[31] K Gu V L Kharitonov and J Chen Stability of Time DelaySystems Birkhauser Boston Mass USA 2003

[32] L El Ghaoui F Oustry and M AitRami ldquoA cone comple-mentarity linearization algorithm for static output-feedbackand related problemsrdquo Institute of Electrical and ElectronicsEngineers Transactions on Automatic Control vol 42 no 8 pp1171ndash1176 1997

[33] R Skjetne T I Fossen and P Kokotovic ldquoAdaptive maneu-vering with experiments for a model ship in a marine controllaboratoryrdquo Automatica vol 41 no 2 pp 289ndash298 2005

[34] N Wang C Qian J-C Sun and Y-C Liu ldquoAdaptive RobustFinite-Time Trajectory Tracking Control of Fully ActuatedMarine Surface Vehiclesrdquo IEEE Transactions on Control SystemsTechnology vol 24 no 4 pp 1454ndash1462 2016

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Page 3: Robust Finite-Time H-Infinity Control with Transients for ...downloads.hindawi.com/journals/mpe/2018/2838749.pdf · ResearchArticle Robust Finite-Time H-Infinity Control with Transients

Mathematical Problems in Engineering 3

1198611 = 1198612 = ( 0119872minus1)

(6)

Under assumption 1 we design the full-state feedback as119906 (119905 minus ℎ (119905)) = 119870119909 (119905 minus ℎ (119905)) (7)

so the (1) and (2) can be rewritten as (119905) = 119860119909 (119905) + 1198611119870119909 (119905 minus ℎ (119905)) + 1198612120596 (119905) (8)119911 (119905) = 119862119909 (119905) + 119863119870119909 (119905 minus ℎ (119905)) (9)

Remark 3 In the situation of DP motion the motion inheave roll and pitch will be ignored since we focus onthe motion on the surface of sea Meanwhile velocity ofvessel is low enough so the Coriolis-Centripetal matrix andnonlinearities in damping matrix can be neglected [18]

Remark 4 In measurement subsystem of DP system varioussensors are installed to obtain the state of the vessel motionaccurately including global position system for the vesselrsquo sposition gyrocompass for the vesselrsquo s heading and attitudesensor for the pitch and roll Meanwhile data fusion tech-nology is applied to DP system to obtain more accurate stateinformation of vessel motion [18]

22 Preliminaries In the sequel some definitions and lem-mas are introduced to obtain the results

Definition 5 ((FTB) see [20]) Given a positive definitematrix 119877 and three positive constants 1198881 1198882 119879 with 1198881 lt1198882 the time-delay system (4) with 119906(119905) = 0 is said to befinite-time boundedness with respect to (1198881 1198882 119889 119879 120591 119877)if supminus120591lt120579lt0119909119879(120579)119877119909(120579) 119879(120579)119877(120579) lt 1198881 rArr 119909119879(119905)119877119909(119905) lt1198882 forall119905 isin [0 119879] forall120596(119905) int1198790 120596(119905)119879120596(119905)119889119905 lt 119889Definition 6 Given 120574 gt 0 time-delay systems (4) and (5) aresaid to be 119867infin- FTB under nonzero initial condition (119867infin-FTB) with (1198881 1198882 119889 120574 119879 120591 119877) if the following conditionsare satisfied

(1) time-delay system (4) with 119906(119905) = 0 is FTB(2) under the initial condition that 119909(0) = 0 output 119911(119905)

satisfiesΓ= sup

120596(119905)2[0119879]2+119909(0)119879119878119909(0) =0

[ 119911 (119905)2[0119879]2120596 (119905)2[0119879]2 + 119909 (0)119879 119878119909 (0)]12lt 120574(10)

where 120574 is a prescribed scalar and 119878 is a weighting diagonalmatrix that penalizes the effect caused by the initial state

Remark 7 It is necessary to distinguish between finite-timestability and finite-time attractiveness [21] The first conceptis to maintain system states within a given boundary in aspecified time interval [22 23] while the latter describes the

fact that system state reaches the equilibrium point of systemin a finite time [24]

Remark 8 Similar to [13] this definition of 119867infin perfor-mance which depends on the initial condition has beenextended to linear system with acceptable maximal delaybound The performance measure is parametrised by aweighting matrix 119878 reflecting the relative importance of theuncertainty in the initial state contrary to the uncertainty inthe exogenous disturbance When 119878 = 0 it will reduce into asort of119867infin control with zero initial condition [25ndash27]

Lemma 9 (see [28]) For anymatrix119875 gt 0 and a differentiablesignal 119909 in [minus119886 minus119887] rarr R119899 the following inequality holdsintminus119887

minus119886119879 (119906) 1198781 (119906) 119889119904 ge 1119886 minus 119887sdot Π1 (1198781 119909119879 (minus119887) 119909119879 (minus119886) 1119886 minus 119887 intminus119887minus119886

119909119879 (119906) 119889119906) (11)

whereΠ1 (1198781 120572 120573 120574)= ( (120572 minus 120573)119879(120572 + 120573 minus 2120574)119879)119879(1198781 0lowast 31198781)( (120572 minus 120573)119879(120572 + 120573 minus 2120574)119879) (12)

Lemma 10 (see [29]) For constant matrices 1198781 gt 0 1198782 gt 0and constant scalars 119886 gt 119887 the following inequalities hold forall continuously function 119909 in [minus119886 minus119887] rarr R119899intminus119887

minus119886int119905minus119887119905+120579119879 (119906) 1198782 (119906) 119889119906119889120579ge Π2 (1198782 119909119879 (119905 minus 119887) 1119886 minus 119887 int119905minus119887119905minus119886

119909119879 (119906) 119889119906 1(119886 minus 119887)2sdot intminus119887minus119886int119905minus119887119905+120579119909119879 (119906) 119889119906119889120579)

intminus119887minus119886int119905+120579119905minus119886119879 (119906) 1198783 (119906) 119889119906119889120579ge Π3 (1198783 119909119879 (119905 minus 119886) 1119886 minus 119887 int119905minus119887119905minus119886

119909119879 (119906) 119889119906 1(119886 minus 119887)2sdot intminus119887minus119886int119905minus119887119905+120579119909119879 (119906) 119889119906119889120579)

(13)

where

Π2 (1198782 120572 120573 120574) = ( (120572 minus 120573)119879(1205722 minus 120574)119879)119879

sdot (61198782 minus121198782lowast 361198782 )( (120572 minus 120573)119879(1205722 minus 120574)119879)

4 Mathematical Problems in Engineering

Π3 (1198783 120572 120573 120574) = ( (120572 minus 120573)119879(1205722 minus 120573 + 120574)119879)119879

sdot (61198783 minus121198783lowast 361198783 )( (120572 minus 120573)119879(1205722 minus 120573 + 120574)119879)(14)

Lemma 11 (see [30]) Let 1198911 1198912 119891119873 R119898 997891rarr R havepositive values in an open subset119863 ofR119898Then the reciprocallyconvex combination of 119891119894 over119863 satisfies

min120572119894|120572119894gt0sum119894 120572119894=1

sum119894

1120572119894119891119894 (119905) = sum119894 119891119894 (119905) +max119892119894119895(119905)sum119894 =119895

119892119894119895 (119905)subject to 119892119894119895 119877119898 997891997888rarr 119877 119892119895119894 (119905) = 119892119894119895 (119905) [ 119891119894 (119905) 119892119894119895 (119905)119892119895119894 (119905) 119891119895 (119905) ] ge 0 (15)

Lemma 12 (see [31]) For any matrix 1198784 gt 0 and a vectorfunction 119909 in [minus119886 minus119887] rarr R119899 if the integrals concerned arewell defined then the following inequality holdsintminus119887

minus119886119909119879 (119904) 1198784119909 (119904) 119889119904ge 1119886 minus 119887 (intminus119887minus119886

119909119879 (119904) 119889119904) 1198784 (intminus119887minus119886119909 (119904) 119889119904) (16)

23 Control Objective Finally the objective of this paper is toderive the control gain 119870 such that

(1) the closed-loop system (8) with 119906(119905) = 0 is FTB(2) under given nonzero initial condition the closed

system (8) and (9) guarantees that 119869 = int1198790119911119879(119904)119911(119904) minus1205742120596119879(119904)120596(119904) minus 1205742(1119879)119909119879(0)119878119909(0)119889119904 lt 0 for all

nonzero 120596(119905) and 119905 isin [0 119879]3 Main Results

Thedesigning of the robust finite-time119867infin controller for theDP system with input delay is divided into three steps

31 FTB Analysis of DP System Firstly the result guarantee-ing the FTB of DP system is established in this subsection

Theorem 13 Given five positive scalars 120572 120573 1198881 1198882 119879 andpositive define matrix 119877 the finite-time boundedness problemof system (4) is solvable if there exist positive scalars 120579119897 (119897 =1 2 sdot sdot sdot 8) and matrices 11987511 gt 0 11987522 gt 0 119882 gt 0 119876119894 gt0 (119894 = 1 2)119880119895 gt 0 (119895 = 1 2)119872 gt 0 1198851 1198852 with appropriatedimensions satisfying the following conditionsΩ = (Ω119894119895)8times8 lt 0 (17)(119882 1198851lowast 119882) gt 0 (18)

(3119882 1198852lowast 3119882) gt 0 (19)

11988811205792 + ℎ211988811205793 + 120588ℎ11988811205794 + 120588ℎ11988811205795 + 12120588ℎ311988811205796+ 16120588ℎ311988811205797 + 13120588ℎ311988811205798 + 1205731205799119889 lt 119890minus12057211987911988821205791 (20)

whereΩ11 = 11986011987911987511 + 11987511119860 minus 12057211987511 + 1198761 + 1198762+ 119860119879 12ℎ2 (1198801 + 1198802 + 2119882)119860 minus 4120588119882 minus 31205881198801minus 91205881198802Ω22 = minus1205881198762 minus 4120588119882 minus 61205881198802Ω33 = 1198701198791198611119879 12ℎ2 (2119882 + 1198801 + 1198802) 1198611119870 minus 8120588119882+ 1205881198851119879 + 1205881198851 minus 1205881198852119879 minus 1205881198852 minus (1 minus 120583)1198762Ω44 = 1198612119879 12ℎ2 (2119882 + 1198801 + 1198802) 1198612 minus 120573119872Ω66 = minus12120588119882Ω77 = minus12120588119882Ω55 = 120572ℎ211987522 minus 181205881198802 minus 61205881198801Ω88 = minus361205881198801 minus 361205881198802Ω12 = 1205881198851 minus 1205881198852 + 61205881198802Ω15 = ℎ11987522 + 121205881198802Ω16 = 6120588119882Ω17 = 21205881198852

Mathematical Problems in Engineering 5

Ω13 = 119875111198611119870 + 119860119879 12ℎ2 (2119882 + 1198801 + 1198802) 1198611119870 minus 2120588119882minus 1205881198851 minus 1205881198852Ω14 = 119875111198612 + 119860119879 12ℎ2 (2119882 + 1198801 + 1198802) 1198612Ω18 = minus181205881198802 + 61205881198801Ω23 = minus1205881198851119879 minus 1205881198852119879 minus 2120588119882Ω25 = minusℎ11987522 minus 61205881198802Ω26 = 21205881198852119879Ω27 = 6120588119882Ω28 = 121205881198802Ω34 = 1198701198791198611119879 12ℎ2 (2119882 + 1198801 + 1198802) 1198612Ω36 = 6120588119882 + 21205881198852119879Ω37 = 6120588119882 + 21205881198852Ω58 = 121205881198801 + 241205881198802Ω67 = minus41205881198852Ω24 = Ω35 = Ω38 = Ω45 = Ω46 = Ω47 = Ω48 = Ω56= Ω57 = Ω68 = Ω78 = 01205791 = 120582min (11)1205792 = 120582max (11) 1205793 = 120582max (22) 1205794 = 120582max (1198761) 1205795 = 120582max (1198762) 1205796 = 120582max () 1205797 = 120582max (1) 1205798 = 120582max (2) 1205799 = 120582max (119872) 120588 = 119890120572ℎ

11 = 119877minus1211987511119877minus1222 = 119877minus1211987522119877minus12119876119894 = 119877minus12119876119894119877minus12 (119894 = 1 2 3) = 119877minus12119882119877minus12119895 = 119877minus12119880119895119877minus12 (119895 = 1 2) 0 lt 1205791119868 lt 11987511 lt 12057921198680 lt 11987522 lt 12057931198680 lt 1198761 lt 12057941198680 lt 1198762 lt 12057951198680 lt 119882 lt 12057961198680 lt 1198801 lt 12057971198680 lt 1198802 lt 12057981198680 lt 119872 lt 1205799119868(21)

Proof Consider the candidate augment LKLF as follows119881 (119905) = 1198811 (119905) + 1198812 (119905) + 1198813 (119905) + 1198814 (119905) (22)1198811 (119905) = 120576 (119905)119879 119875120576 (119905) (23)1198812 (119905) = int119905119905minusℎ119890120572(119905minus119904)119909 (119904)1198791198761119909 (119904) 119889119904+ int119905119905minusℎ(119905)119890120572(119905minus119904)119909 (119904)1198791198762119909 (119904) 119889119904 (24)

1198813 (119905) = ℎint0minusℎint119905119905+120579119890120572(119905minus119904) (119904)119879119882 (119904) 119889119904119889120579 (25)

1198814 (119905) = int0minusℎint0120579int119905119905+120573119890120572(119905minus119904) (119904)1198791198801 (119904) 119889119904119889120573119889120579

+ int0minusℎint120579minusℎint119905119905+120573119890120572(119905minus119904) (119904)1198791198802 (119904) 119889119904119889120573119889120579 (26)

where 120576(119905) = (119909(119905) int119905119905minusℎ119909(119904)119889119904) 119875 = ( 11987511 0

lowast 11987522)

To deal with the formulas conveniently let us providesuch definitions

120585 (119905)= (119909 (119905) 119909 (119905 minus ℎ) 119909 (119905 minus ℎ (119905)) 120596 (119905) 1ℎ int119905119905minusℎ 119909 (119904) 119889119904 1ℎ (119905) int119905119905minusℎ(119905) 119909 (119904) 119889119904 1ℎ minus ℎ (119905) int119905minusℎ(119905)119905minusℎ119909 (119904) 119889119904 1ℎ2 int0minusℎ int119905119905+120579 119909 (119904) 119889119904119889120579)119897119896119879 = (0 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896minus1

1 0 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟8minus119896) (27)

6 Mathematical Problems in Engineering

The time-derivative of 119881(119905) along system (4) can bebounded as1 (119905) = 1205721198811 (119905) + 120576 (119905)119879 119875120576 (119905) + 120576 (119905)119879 119875 120576 (119905) minus 1205721198811 (119905) (28)2 (119905)= 1205721198812 (119905) + 119909 (119905)1198791198761119909 (119905)minus 119890120572ℎ119909 (119905 minus ℎ)1198791198761119909 (119905 minus ℎ) + 119909 (119905)1198791198762119909 (119905)minus (1 minus ℎ (119905)) 119890120572ℎ(119905)119909 (119905 minus ℎ (119905))1198791198762119909 (119905 minus ℎ (119905))le 1205721198812 (119905) + 119909 (119905)1198791198761119909 (119905)minus 119890120572ℎ119909 (119905 minus ℎ)1198791198761119909 (119905 minus ℎ) + 119909 (119905)1198791198762119909 (119905)minus (1 minus 120583) 119909 (119905 minus ℎ (119905))1198791198762119909 (119905 minus ℎ (119905))

(29)

3 (119905)= 1205721198813 (119905) + ℎ2 (119905)1198791198822 (119905)minus 119890120572ℎℎint119905119905minusℎ (119904)1198791198822 (119904) 119889119904 (30)

4 (119905)= 1205721198814 (119905) + 12ℎ2 (119905)1198791198801 (119905)minus 119890120572ℎ int0minusℎint119905119905+120573 (119904)1198791198801 (119904) 119889119904119889120573+ 12ℎ2 (119905)1198791198802 (119905)minus 119890120572ℎ int0

minusℎint119905+120573119905minusℎ (119904)1198791198802 (119904) 119889119904119889120573

(31)

Invoking Lemma 9 we can obtain3 (119905) = 1205721198813 (119905) + ℎ2 (119905)119879119882 (119905)minus 119890120572ℎℎint119905119905minusℎ(119905) (119904)119879119882 (119904) 119889119904

minus 119890120572ℎℎint119905minusℎ(119905)119905minusℎ

(119904)119879119882 (119904) 119889119904le 1205721198813 (119905) + ℎ2 (119905)119879119882 (119905)minus 119890120572ℎ ℎℎ (119905)1206031119879Λ 11206031 minus 119890120572ℎ ℎℎ minus ℎ (119905)1206032119879Λ 21206032(32)

where 1206031119879 = 120585119879 (119905)( (1198971119879 minus 1198973119879)119879(1198971119879 + 1198973119879 minus 21198976119879)119879)119879

1206032119879 = 120585119879 (119905)( (1198973119879 minus 1198972119879)119879(1198973119879 + 1198972119879 minus 21198977119879))119879

Λ 1 = Λ 2 = (119882 0lowast 3119882) (33)

In order to obtain a tighter bound of integral termLemma 11 is applied to (32) as follows3 (119905) le 1205721198813 (119905) + ℎ2 (119905)119879119882 (119905) minus 119890120572ℎ1206033119879Λ 31206033 (34)

where 1206033119879 = ( 12060311198791206032119879 ) Λ 3 = (119882 1198851 0 0

lowast 119882 0 0lowast lowast 3119882 1198852lowast lowast lowast 3119882

)By applying Lemma 10 to (31) we can obtain the inequal-

ity 4 (119905) le 1205721198814 (119905) + 12ℎ2 (119905)1198791198801 (119905)+ 12ℎ2 (119905)1198791198802 (119905) minus 119890120572ℎ1206034119879Λ 41206034 (35)

where

1206034119879 = 120585119879 (119905)(((((

(1198971119879 minus 1198972119879)119879(12 1198971119879 minus 1198978119879)119879(1198972119879 minus 1198975119879)119879(12 1198972119879 minus 1198975119879 + 1198978119879)119879)))))

119879

Λ 4 =(61198801 minus121198801 0 0lowast 361198801 0 0lowast lowast 61198802 minus121198802lowast lowast lowast 361198802)(36)

Combining (28) (29) (34) and (35) with the definitionof 120588 = 119890120572ℎ we can obtain minus 120572119881 minus 120573120596119879 (119905)119872120596 (119905) lt 120585119879 (119905) Ω120585 (119905) (37)

Assuming that Ω lt 0 and integrating the left part ofinequality (37)119881 (119905) lt 119890120572119905119881 (0) + 120573int119905

0119890120572(119905minus119904)120596119879 (119904)119872120596 (119904) 119889119904lt 119890120572119879 (119881 (0) + 120573120582max (119872) 119889) (38)

Invoking the Jensen inequality shown in Lemma 12 onehas 119881 (0) le 119909 (0)119879 11987511119909 (0) + ℎint0

minusℎ119909 (119904)119879 11987522119909 (119904) 119889119904+ 120588int0

minusℎ119909 (119904)1198791198761119909 (119904) 119889119904+ 120588int0

minusℎ(119905)119909 (119904)1198791198762119909 (119904) 119889119904

Mathematical Problems in Engineering 7

+ ℎ120588int0minusℎint0120579 (119904)119879119882 (119904) 119889119904119889120579

+ 120588int0minusℎint0120579int0120573 (119904)1198791198801 (119904) 119889119904119889120573119889120579

+ 120588int0minusℎint120579minusℎint0120573 (119904)1198791198802 (119904) 119889119904119889120573119889120579le 120582max (11) 1198881 + ℎ120582max (22) ℎ1198881+ 120588120582max (1198761) ℎ1198881 + 120588120582max (1198762) ℎ1198881+ 120588120582max () 12ℎ31198881 + 120588120582max (1) 16ℎ31198881+ 120588120582max (2) 13ℎ31198881

(39)

Based on inequalities (38) and (39) we have119881 (119905) gt 119909119879 (119905) 11987511119909 (119905) = 119909119879 (119905) 119877121111987712119909 (119905)gt 120582min (11) 119909119879 (119905) 119877119909 (119905) (40)

Therefore conditions (17) to (20) can guarantee the FTBof system (4) This completes the proof

Remark 14 Among the existing approaches there are twothreads one is to construct a novel LKLF that involves moreinformation of delay the other is to find a tighter estimationof upper bound for cross terms coming from the derivativeof the LKLF In this paper these two techniques are appliedto obtain the result In addition the less conservatism inpractical engineering will be shown in the simulations whichare always ignored in most literature sources

32 Controller Design In this subsection we focus on theproblem of finite-time 119867infin state feedback designing basedon Theorem 13 that is designing a state feedback controllerin the form of (7) such that the resulting DP system satisfiesthe control objective proposed in Section 1

Theorem 15 For given positive 120572 1198881 1198882 119879 120574 and matrices 119877 gt0 119878 ge 0 if there exist positive scalars 120579119897 (119897 = 1 2 sdot sdot sdot 8) andmatrices 11987511 gt 0 11987522 gt 0119882 gt 0 119876119894 gt 0 (119894 = 1 2) 119880119895 gt0 (119895 = 1 2) 119884 1198851 1198852 with appropriate dimensions satisfyingthe following conditions Π = (Π119894119895)12times12 lt 0 (41)Σ = (Σ119894119895)9times9 lt 0 (42)

(119882 1198851lowast 119882) gt 0 (43)

(3119882 1198852lowast 3119882) gt 0 (44)

119868 minus 120574211987811987511 lt 0 (45)0 lt 1198671 lt 11987511 lt 11986720 lt 11987522 lt 11986730 lt 1198761 lt 11986740 lt 1198762 lt 11986750 lt 119882 lt 11986760 lt 1198801 lt 11986770 lt 1198802 lt 1198678(46)

where Π11 = 11987511119860119879 + 11986011987511 minus 12057211987511 + 1198761 + 1198762 minus 4120588119882minus 31205881198801 minus 91205881198802Π44 = minus1205742119868Π22 = minus1205881198762 minus 4120588119882 minus 61205881198802Π33 = minus8120588119882 + 1205881198851119879 + 1205881198851 minus 1205881198852119879 minus 1205881198852minus (1 minus 120583)1198762Π13 = 1198611119884 minus 2120588119882 minus 1205881198851 minus 1205881198852Π14 = 1198612Π66 = minus12120588119882Π77 = minus12120588119882Π55 = 120572ℎ211987522 minus 181205881198802 minus 61205881198801Π88 = minus361205881198801 minus 361205881198802Π12 = 1205881198851 minus 1205881198852 + 61205881198802Π15 = ℎ11987522 + 121205881198802Π16 = 6120588119882Π17 = 21205881198852Π25 = minusℎ11987522 minus 61205881198802Π18 = minus181205881198802 + 61205881198801Π26 = 21205881198852119879Π23 = minus1205881198851119879 minus 1205881198852119879 minus 2120588119882Π27 = 6120588119882Π28 = 121205881198802

8 Mathematical Problems in Engineering

Π36 = 6120588119882 + 21205881198852119879Π37 = 6120588119882 + 21205881198852Π58 = 121205881198801 + 241205881198802Π67 = minus41205881198852Π19 = 12ℎ11987511119860119879Π39 = 12ℎ1198841198791198611119879Π49 = 12ℎ1198612119879Π110 = 12ℎ11987511119860119879Π310 = 12ℎ1198841198791198611119879Π410 = 12ℎ1198612119879Π111 = ℎ11987511119860119879Π311 = ℎ1198841198791198611119879Π411 = ℎ1198612119879Π112 = 11987511119862119879Π312 = 119884119879119863119879Π99 = 121198801 minus 11987511Π1010 = 121198802 minus 11987511Π1111 = 119882 minus 211987511Π1212 = minus119890minus120572119879119868(47)

others Π119894119895 = 0Σ11 = minus119890minus12057211987911988821198671Σ22 = minus1198672Σ33 = minus1198673Σ44 = minus1198674Σ55 = minus1198675Σ66 = minus1198676Σ77 = minus1198677Σ88 = minus1198678

Σ99 = minus1205742119889Σ12 = radic11988811198672Σ13 = radicℎ211988811198673Σ14 = radic120588ℎ11988811198674Σ15 = radic120588ℎ11988811198675Σ16 = radic12120588ℎ311988811198676Σ17 = radic16120588ℎ311988811198677Σ18 = radic13120588ℎ311988811198678Σ19 = 11987511(48)

others Σ119894119895 = 0 (49)

then a state feedback119867infin controller in form of (7) existssuch that

(1) the closed-loop system (8) with 119906(119905) = 0 is FTB(2) under given nonzero initial condition the closed

system (8) and (9) guarantees that 119869 = int1198790119911119879(119904)119911(119904) minus1205742120596119879(119904)120596(119904) minus 1205742(1119879)119909119879(0)119878119909(0)119889119904 lt 0 for all nonzero 120596(119905)

and 119905 isin [0 119879]The controller gain can be calculated by 119870 = 11988411987511minus1

Proof In preparation for designing we set119872 = 1205742119868 120573 = 1for (37) then (37) and (38) can be rewritten as follows minus 120572119881 minus 1205742120596119879 (119905) 120596 (119905) lt 120585119879 (119905) Ω120585 (119905) (50)119881 (119905) lt 119890120572119905119881 (0) + 1205742 int119905

0119890120572(119905minus119904)120596119879 (119904) 120596 (119904) 119889119904lt 119890120572119879 (119881 (0) + 1205742119889) (51)

where Ω44 = 1198612119879 12ℎ2 (2119882 + 1198801 + 1198802) 1198612 minus 1205742119868Ω119894119895 = Ω119894119895

(52)

Then two steps are provided to design the desiredcontroller

Step 1 Here we will give some conditions that can guaranteethe 119867infin performance over the finite-time interval undernonzero initial conditions firstly

Mathematical Problems in Engineering 9

Consider the following inequality minus 120572119881 + 119890120572119905119911119879 (119905) 119911 (119905) minus 1205742119890120572119905120596119879 (119905) 120596 (119905)lt minus 120572119881 + 119890120572119879119911119879 (119905) 119911 (119905) minus 1205742120596119879 (119905) 120596 (119905)lt 120585119879 (119905) (Ω minus Ψ119879 (minus119890minus120572119879119868)minus1Ψ) 120585 (119905) (53)

Noting that119882 1198801 and 1198802 are coupled with 119860 1198611119870 and1198612 in Ω to decouple these terms the Ω can be rewritten asΩ = Ω minus Ψ119879 (minus121198801)minus1 Ψ minus Ψ119879 (minus121198802)minus1Ψminus Ψ119879 (minus119882)minus1 Ψ (54)

whereΩ11 = 11986011987911987511 + 11987511119860 minus 12057211987511 + 1198761 + 1198762 minus 4120588119882minus 31205881198801 minus 91205881198802Ω33 = minus8120588119882 + 1205881198851119879 + 1205881198851 minus 1205881198852119879 minus 1205881198852minus (1 minus 120583)1198762Ω44 = minus1205742119868Ω13 = 119875111198611119870 minus 2120588119882 minus 1205881198851 minus 1205881198852Ω14 = 119875111198612Ω34 = 0(55)

othersΩ119894119895 = Ω119894119895Ψ = (119862 0 119863119870 0 0 0 0 0)Ψ= (ℎ121198801119879119860 0 ℎ1211988011198791198611119870 ℎ1211988011198791198612 0 0 0 0) Ψ= (ℎ121198802119879119860 0 ℎ1211988021198791198611119870 ℎ1211988021198791198612 0 0 0 0)Ψ = (ℎ119882119879119860 0 ℎ1198821198791198611119870 ℎ1198821198791198612 0 0 0 0) (56)

By Schur complementΩ lt 0 and Ω minusΨ119879(minus119890minus120572119879119868)minus1Ψ areequivalent to

Π =(((((

Ω Ψ119879 Ψ119879 Ψ119879 Ψ119879lowast minus121198801 0 0 0lowast lowast minus121198802 0 0lowast lowast lowast minus119882 0lowast lowast lowast lowast minus119890minus120572119879119868)))))lt 0 (57)

To ensure the proposed conditions satisfy the 119867infin per-formance we set the inequality like

11987511 minus 1205742119878 lt 0 (58)

from which we have

minus 120572119881 + 119890120572119905119911119879 (119905) 119911 (119905) minus 1205742119890120572119905120596119879 (119905) 120596 (119905)minus 1205742119890120572119905 1119879119909119879 (0) 119878119909 (0) lt minus 120572119881 + 119890120572119905119911119879 (119905) 119911 (119905)minus 1205742119890120572119905120596119879 (119905) 120596 (119905) minus 1205742119890120572119905 1119879119909119879 (0) 119878119909 (0)+ 119890120572119905 1119879119909119879 (0) 11987511119909 (0) lt minus 120572119881 + 119890120572119905119911119879 (119905) 119911 (119905)minus 1205742119890120572119905120596119879 (119905) 120596 (119905) + 119890120572119905 1119879119909119879 (0) (11987511 minus 1205742119878) 119909 (0)lt 0(59)

Integrating (58) from 0 to 119879 we can obtain

int1198790119911119879 (119904) 119911 (119904) 119889119904 minus int119879

01205742120596119879 (119904) 120596 (119904) 119889119904

minus 1205742119909119879 (0) 119878119909 (0) lt minusint1198790(119890minus120572119905 minus 120572119890minus120572119905119881)119889119904 lt 0 (60)

so conditions (43) (44) (57) and (58) with the inequali-ties as follows can ensure that systems (17) and (18) satisfy the119867infin performance

11988811205792 + ℎ211988811205793 + 120588ℎ11988811205794 + 120588ℎ11988811205795 + 12120588ℎ311988811205796+ 16120588ℎ311988811205797 + 13120588ℎ311988811205798 + 1205742119889 lt 119890minus120572119879119888212057910 lt 1205791119868 lt 11987511 lt 12057921198680 lt 11987522 lt 12057931198680 lt 1198761 lt 12057941198680 lt 1198762 lt 12057951198680 lt 119882 lt 12057961198680 lt 1198801 lt 12057971198680 lt 1198802 lt 1205798119868

(61)

10 Mathematical Problems in Engineering

Step 2 In this step the approach to design the gain matrix 119870of the controller is provided in forms of LMIs which can becalculated expediently

To linearize the nonlinear terms in (57) we make itmultiply by the followingmatrix fromboth left and right sidesfirstly

diag 11987511minus1 11987511minus1 11987511minus1 119868 11987511minus1 11987511minus1 11987511minus1 11987511minus1 1198801minus1 1198802minus1 119882minus1 119868 (62)

Let us do the definitions as follows11987511 = 11987511minus11198761 = 11987511minus1119876111987511minus11198762 = 11987511minus1119876211987511minus1119882 = 11987511minus111988211987511minus11198801 = 11987511minus1119880111987511minus11198802 = 11987511minus1119880211987511minus111987522 = 11987511minus11198752211987511minus11198851 = 11987511minus1119885111987511minus11198852 = 11987511minus1119885211987511minus1119884 = 11987011987511minus1

(63)

so the inequality Π lt 0 can guarantee Π lt 0 whereΠ99 = minus12119875111198801

minus111987511Π1010 = minus12119875111198802

minus111987511Π1111 = minus11987511119882minus111987511(64)

others Π119894119895 = Π119894119895 (65)

Then based on the inequalities minus(12)119875111198801

minus111987511 le(12)1198801 minus 11987511 minus(12)119875111198802

minus111987511 le (12)1198802 minus 11987511minus11987511119882minus111987511 le 119882 minus 211987511 condition (57) can be convertedinto (41) in form of LMIs

Accordingly similar to the above-mentioned procedures(61) can be transformed into (42) with the Schur complementand the definition of119867119894 = 11987511minus112057911989411987511minus1 (119894 = 1 2 sdot sdot sdot 8)

Therefore conditions (41) to (45) can obtain the con-troller in the form of (7) guaranteeing the desired perfor-mance This completes the proof

Remark 16 The119867infin performance with nonzero initial con-dition is studied in this literature In a practical view it is

hard to suffer the worst-case over a finite time for surfacevessel with nonzero initial condition and it will reducethe conservatism caused by the assumption of zero initialcondition Therefore it is reasonable to introduce such 119867infinperformance index to handle the problem of finite-time119867infincontrol for ship subject to exogenous signals

Remark 17 It is challenging to achieve the result satisfyingthe control objective while it is very different from thestandard 119867infin control because a weighting diagonal matrixis introduced especially to establish the form of LMIs Inthe proof of Theorem 15 inequality (58) is introduced toensure the 119867infin performance with nonzero initial conditionand make it convenient to construct the conditions in formof LMIs

Remark 18 By introducing some new additional matrixvariables Cone Complementarity Linearization algorithm(CCL) [32] can be utilized to linearize the nonlinear termsas an alternative method It will reduce the conservatism atthe expense of increasing the computational burden and thenumber of decision variables

4 Simulations

41 Simulation Settings To demonstrate the effectiveness andsuperiority of the robust finite-time 119867infin control schemesimulation studies are conducted on the parameters of theship CyberShip II [33] which is a 170 scale-replica of a supplyshipwhosemain parameters are shown inTable 1 In additionwe choose the time-varying function like ℎ(119905) = 3sin (1199056 minus1205876) to describe the time delay occurred in the input signal

Unknown external disturbances 120596(119905) with significantlylarge magnitude and high frequency which can be roughlyconsidered as complex environment including ocean windswaves and currents are governed by [34]

120596 (119905) = (9 sin (01120587119905 minus 1205875 )6 sin (03120587119905 + 1205876 )3 sin (02120587119905 + 1205873 )) (66)

with initial conditions 120578(0) = (1 1 01) and 120592(0) =(08 06 01)42 Numerical Simulations We choose 120574 = 13 119879 = 10 119878 =03lowast119868 1198881 = 302 1198882 = 10 120572 = 0001 By usingTheorem 15 thecontrol gain 119870 is calculated by LMIs (41) to (45)

Mathematical Problems in Engineering 11

Table 1 The main parameters of the CyberShip II119898 238000 119883 -20 119883119906 -07225119868119911 17600 119884V -100 119884V -08612119909119892 00460 119884 119903 -00 119884119903 01079119873 119903 -10 119873119903 05

0 20 40 60 80 100Time(s) Time(s)

minus2

0

2

4

6

8

10

12

0 2 4 6 8 100

2

4

6

8

10

12

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

xT(t)Rx(t)xT(t)Rx(t)

c=c=

c1=302

c1=302

Figure 1 State response 119909119879(119905)119877119909(119905) of DP system

119870 = (minus32743 0 0 minus313907 0 00 minus42784 minus01572 0 minus412136 minus129820 minus01389 minus03497 0 minus13362 minus33091) (67)

To demonstrate the superiority and validity three sce-narios as comparative cases are provided here It should bepointed out that the results within 100s are also provided toillustrate the steady-state performance

Case 1 (with different 119878) For comparison purposes we set119878 with different value to demonstrate the effect of 119878 on stateresponse Without loss of generality we set 119878 = 05 lowast 119868 and119878 = 01 lowast 119868 respectivelyCase 2 (119867infin with zero initial condition]) This scenario willshow that the control law proposed in this paper can achievea better transient performance and disturbance rejectioncapability over traditional119867infin control approach for dynamicpositioning system conducted in [11]

Case 3 (a control design based on existing methods) Basedon the approach in [17] applied for discrete-time system acontrol law for dynamic positioning ship with input delay isproposed to illustrate the less conservatism of our designingapproach in practical engineering subsequently

At first the effectiveness of the control law we designedis confirmed from Figures 1ndash6 Figure 1 shows the state

response of 119909119879(119905)119877119909(119905) Obviously from that curve we cansee that the value of 119909119879(119905)119877119909(119905) is far below the value of1198882 = 10 in the presence of disturbance as in Figure 2which means the finite-time full-state feedback controllerdesigned can guarantee the DP system (8) sim (9) FTB with(302 10 119889 10 3 119868) and have less conservatism where weset 119889 = int119879

0120596(119905)119879120596(119905)119889119905 Meanwhile the control law can

stabilize the dynamic positioning shipwithminimum steady-state error in 100s and the control input will be small enoughas shown in Figure 6 From Figures 3ndash5 we can find that allstates in DP system are within a small range As shown inFigures 2ndash4 the angle and the angular velocity in yaw havechanged within 01∘ which means that the heading of thevessel stays in a fixed direction Furthermore the velocitiesin surge and sway are low enough which illustrates thatassumption 2 is reasonable and it meets the requirement ofstabilization control for DP system

To demonstrate the effect of 119878 state responses of systemswith different 119878 are exhibited in Figures 7ndash9 As the valueof 119878 decreases the restriction for 11987511 is more exclusive andthe robustness is stronger during preset time interval as inFigures 7 and 8 while the required control input will increasedramatically as in Figure 9 When the time oversteps the

12 Mathematical Problems in Engineering

0 10 20 30 40 50 60 70 80 90 100Time(s)

minus10

minus5

0

5

10

Dist

urba

nces

(N)

XY

Figure 2 Disturbance in surge sway and yaw

0 50 100Time(s)

minus050

051

152

X(m

)

0 5 10Time(s)

005

115

2

X(m

)

0 50 100Time(s)

minus050

051

152

Y(m

)

0 5 10Time(s)

005

115

2Y(

m)

0 50 100Time(s)

minus01

0

01

02

0 5 10Time(s)

0

01

02

(∘

)

(∘

)

Figure 3 State responses of position and orientation for DP System

0 50 100Time(s)

minus1

0

1

2

0 5 10Time(s)

minus1

0

1

2

0 50 100Time(s)

minus1

0

1

2

v(m

s)

0 5 10Time(s)

minus1

0

1

2

v(m

s)

0 50 100Time(s)

minus01

0

01

02

0 5 10Time(s)

minus01

0

01

02

r(∘ s

)

r(∘ s

)

(ms

)

(m

s)

Figure 4 State responses of velocities for DP system

Mathematical Problems in Engineering 13

minus05 0 05 1 15 2X(m)

minus05

0

05

1

15

2

Y(m

)

Figure 5 Trajectory of the shiprsquos horizontal motion

0 50 100Time(s)

minus30minus20minus10

010

u1(N

)

0 5 10Time(s)

minus30minus20minus10

010

u1(N

)

0 50 100Time(s)

minus30minus20minus10

010

u2(N

)

0 5 10Time(s)

minus30minus20minus10

010

u2(N

)

0 50 100Time(s)

minus15minus1

minus050

05

u3(N

m)

0 5 10Time(s)

minus15minus1

minus050

05

u3(N

m)

Figure 6 Control inputs

0 20 40 60 80 100Time(s)

minus1

0

1

2

3

4

5

6

S=03S=05S=01

0 2 4 6 8 10Time(s)

0

2

4

6

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 7 State response 119909119879(119905)119877119909(119905) with different 119878

14 Mathematical Problems in Engineering

Figure 8 State responses of position and orientation with different 119878

0 50 100Time(s)

minus60minus30

03060

u1(N

)

0 5 10Time(s)

minus40minus20

02040

u1(N

)

0 50 100Time(s)

minus100

0

100

u2(N

)

0 5 10Time(s)

minus40minus20

02040

u2(N

)

0 50 100Time(s)

minus4minus2

024

u3(N

m)

0 5 10Time(s)

minus2

0

2

u3(N

m)

S=03S=05S=01

Figure 9 Control input with different 119878preset time interval all the states can be guaranteed withinthe given upper bound as before

As shown in Figures 10ndash12 the overconservatism oftraditional119867infin control approach applied inDP ship is solvedby the 119867infin control strategy with nonzero initial conditionproposed in this article As is well known the classical 119867infincontrol law is designed under the so-called worst-case normwhile the performance of antidisturbance and transient issacrificed to some degree From Figures 10 and 11 we cansee that the state response can be controlled in a smallerrange and stabilized in a shorter time with a lower overshootshown in Figure 12 Hence a conclusion can be drawn thatthe control law we present can achieve a better disturbance

rejection capability and transient performance while thematter of overconservatism can be resolved at the sametime

Simulation results of Theorem 15 and Case 3 are shownin Figures 13 and 14 to show the better performance ofthe method we proposed We can find that the controllerbased on [17] and 119867infin control approach present in thisliterature can make all states of system within a smallenough region However compared with the controller weconstructed the state response has higher overshoot whichwill affect the control precision and transient performance inpractical engineering Based on the above simulation resultsthe developed controller can achieve a higher accuracy and

Mathematical Problems in Engineering 15

0 20 40 60 80 100Time(s)

minus2

0

2

4

6

8

10

12

Theorem 15Case 2

0 2 4 6 8 10Time(s)

minus2

0

2

4

6

8

10

12

Theorem 15Case 2

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 10 State response 119909119879(119905)119877119909(119905) of Theorem 15 and Case 2

0 50 100Time(s)

minus10123

X(m

)

0 5 10Time(s)

0123

X(m

)

0 50 100Time(s)

minus10123

Y(m

)

0 5 10Time(s)

0123

Y(m

)

0 50 100Time(s)

minus020

0204

0 5 10Time(s)

minus020

0204

Theorem 15Case 2

(∘

)

(∘

)

Figure 11 Position and orientation of Theorem 15 and Case 2

minus05 0 05 1 15 2 25X(m)

minus050

051

152

25

Y(m

)

Theorem 15Case 2

Figure 12 Trajectory of the shiprsquos horizontal motion of Theorem 15 and Case 2

16 Mathematical Problems in Engineering

0 20 40 60 80 100Time(s)

0

1

2

3

4

5

Theorem 15Case 3

0 2 4 6 8 10Time(s)

0

1

2

3

4

5

Theorem 15Case 3

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 13 State response 119909119879(119905)119877119909(119905) of Theorem 15 and Case 3

0 5 10Time(s)

05

1

15

X(m

)

Theorem 15Case 3

0 5 10Time(s)

05

1

15

Y(m

)

0 5 10Time(s)

0

005

01

015

02

(∘

)

Figure 14 Position and orientation of Theorem 15 and Case 3

lower overshoot which means that it is less conservative incomparison with existing result

5 Conclusion

In this paper the robust finite-time 119867infin control with tran-sients for DP system with input delay maintaining all statesof ocean surface vessel into a given threshold over a fixed timeinterval in presence of time-varying disturbance is proposedBased on a novel augmented LKLF with triple integral termsand a method combining theWirtinger inequality and recip-rocally convex approach a less conservative result is derivedIn particular a119867infin performance index with nonzero initialcondition attenuating the overconservatism caused by theassumption of zero initial condition and enhancing thetransient performance of ship subject to external disturbanceis adoptedThe obtained result is formulated in terms of LMIswhich can be easily solved by the standard numerical soft-ware Finally a numerical simulation for a ship is proposed toverify the effectiveness and less conservatism of the controller

we designed In the future the output feedback control can bedeveloped for the ocean surface vessel to avoid the use of thevessel-frame surge sway and yaw velocities

Data Availability

All the data supporting the conclusions of the study have beenprovided in Simulations and readers can access these data in[33 34]

Conflicts of Interest

The authors declare that there are no conflicts of interestrelated to this paper

Acknowledgments

This research was partially supported by the National Sci-ence Technology Support Program of China (Project no51609046)

Mathematical Problems in Engineering 17

References

[1] M Chen B Jiang and R Cui ldquoActuator fault-tolerant controlof ocean surface vessels with input saturationrdquo InternationalJournal of Robust amp Nonlinear Control vol 26 no 3 pp 542ndash564 2016

[2] J Du X Hu and Y Sun ldquoRobust dynamic positioning of shipswith disturbances under input saturationrdquo Automatica vol 73pp 207ndash214 2016

[3] H Ashrafiuon K R Muske L C McNinch and R ASoltan ldquoSliding-mode tracking control of surface vesselsrdquo IEEETransactions on Industrial Electronics vol 55 no 11 pp 4004ndash4012 2008

[4] W Si and X Dong ldquoNeural prescribed performance controlfor uncertain marine surface vessels without accurate initialerrorsrdquoMathematical Problems in Engineering Art ID 261832311 pages 2017

[5] T D Nguyen A J Soslashrensen and S T Quek ldquoDesign of hybridcontroller for dynamic positioning from calm to extreme seaconditionsrdquo Automatica vol 43 no 5 pp 768ndash785 2007

[6] J Du X Hu H Liu and C L Chen ldquoAdaptive robust outputfeedback control for a marine dynamic positioning systembased on a high-gain observerrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 11 pp 2775ndash27862015

[7] H Katayama ldquoNonlinear sampled-data stabilization of dynam-ically positioned shipsrdquo IEEE Transactions on Control SystemsTechnology vol 18 no 2 pp 463ndash468 2010

[8] K J Astrom andCG Kallstrom ldquoIdentification of ship steeringdynamicsrdquo Automatica vol 12 no 1 pp 9ndash22 1976

[9] Z Lei and C Guo ldquoDisturbance rejection control solutionfor ship steering system with uncertain time delayrdquo OceanEngineering vol 95 pp 78ndash83 2015

[10] J C Doyle K Glover P P Khargonekar and B A FrancisldquoState-space solutions to standard H-2 and Hinfin control prob-lemsrdquo IEEE Transactions on Automatic Control vol 34 no 8pp 831ndash847 1989

[11] M Katebi M Grimble and Y Zhang ldquoHinfin robust controldesign for dynamic ship positioningrdquo IEE Proceedings - ControlTheory and Applications vol 144 no 2 pp 110ndash120 1997

[12] D V Balandin and M M Kogan ldquoLMI-based Hinfin-optimalcontrol with transientsrdquo International Journal of Control vol 83no 8 pp 1664ndash1673 2010

[13] Z Feng J Lam S Xu and S Zhou ldquoHinfin Control withTransients for Singular Systemsrdquo Asian Journal of Control vol18 no 3 pp 817ndash827 2016

[14] P P Khargonekar K M Nagpal and K R Poolla ldquoHinfin controlwith transientsrdquo SIAM Journal onControl andOptimization vol29 no 6 pp 1373ndash1393 1991

[15] Z Zhang Z Zhang and H Zhang ldquoFinite-time stabilityanalysis and stabilization for uncertain continuous-time systemwith time-varying delayrdquo Journal of The Franklin Institute vol352 no 3 pp 1296ndash1317 2015

[16] J Cheng S Chen Z Liu H Wang and J Li ldquoRobust finite-time sampled-data control of linear systems subject to randomoccurring delays and its application to Four-Tank systemrdquoAppliedMathematics andComputation vol 281 pp 55ndash76 2016

[17] W Kang S Zhong K Shi and J Cheng ldquoFinite-time stabilityfor discrete-time system with time-varying delay and nonlinearperturbationsrdquo ISA Transactions vol 60 pp 67ndash73 2016

[18] T I Fossen Handbook of Marine Craft Hydrodynamics andMotion Control John Wiley amp Sons New York USA 2011

[19] D Zhao F Ding L Zhou W Zhang and H Xu ldquoRobustH-infinity Control of Neutral System with Time-Delay forDynamic Positioning Shipsrdquo Mathematical Problems in Engi-neering vol 2015 Article ID 976925 11 pages 2015

[20] F Amato M Ariola and C Cosentino ldquoFinite-time stabilityof linear time-varying systems analysis and controller designrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 55 no 4 pp 1003ndash1008 2010

[21] L Sun andZ Zheng ldquoFinite-time slidingmode trajectory track-ing control of uncertain mechanical systemsrdquo Asian Journal ofControl vol 19 no 1 pp 399ndash404 2017

[22] P Shi Y Zhang and R Agarwal ldquoStochastic finite-time stateestimation for discrete time-delay neural networks withMarko-vian jumpsrdquo Neurocomputing vol 151 part 1 pp 168ndash174 2014

[23] L Zhang S Wang H R Karimi and A Jasra ldquoRobust Finite-Time Control of Switched Linear Systems and Application toa Class of Servomechanism Systemsrdquo IEEEASME Transactionson Mechatronics vol 20 no 5 pp 2476ndash2485 2015

[24] L Wu P Shi and H Gao ldquoState estimation and sliding-mode control of Markovian jump singular systemsrdquo Institute ofElectrical and Electronics Engineers Transactions on AutomaticControl vol 55 no 5 pp 1213ndash1219 2010

[25] Y Q Zhang P Shi S K Nguang and Y D Song ldquoRobust finite-time Hinfin control for uncertain discrete-time singular systemswithMarkovian jumpsrdquo IET ControlTheory ampApplications vol8 no 12 pp 1105ndash1111 2014

[26] S Wang T Shi L Zhang A Jasra and M Zeng ldquoExtendedfinite-time Hinfin control for uncertain switched linear neutralsystemswith time-varying delaysrdquoNeurocomputing vol 152 pp377ndash387 2015

[27] J Hu G Sui S Du and X Li ldquoFinite-time stability of uncer-tain nonlinear systems with time-varying delayrdquo MathematicalProblems in Engineering Article ID 2538904 9 pages 2017

[28] A Seuret and F Gouaisbaut ldquoWirtinger-based integral inequal-ity application to time-delay systemsrdquo Automatica vol 49 no9 pp 2860ndash2866 2013

[29] Z Li Y Bai C Huang and Y Cai ldquoNovel delay-partitioningstabilization approach for networked control system viaWirtinger-based inequalitiesrdquo ISA Transactions vol 61 pp75ndash86 2016

[30] P Park JWKo andC Jeong ldquoReciprocally convex approach tostability of systems with time-varying delaysrdquo Automatica vol47 no 1 pp 235ndash238 2011

[31] K Gu V L Kharitonov and J Chen Stability of Time DelaySystems Birkhauser Boston Mass USA 2003

[32] L El Ghaoui F Oustry and M AitRami ldquoA cone comple-mentarity linearization algorithm for static output-feedbackand related problemsrdquo Institute of Electrical and ElectronicsEngineers Transactions on Automatic Control vol 42 no 8 pp1171ndash1176 1997

[33] R Skjetne T I Fossen and P Kokotovic ldquoAdaptive maneu-vering with experiments for a model ship in a marine controllaboratoryrdquo Automatica vol 41 no 2 pp 289ndash298 2005

[34] N Wang C Qian J-C Sun and Y-C Liu ldquoAdaptive RobustFinite-Time Trajectory Tracking Control of Fully ActuatedMarine Surface Vehiclesrdquo IEEE Transactions on Control SystemsTechnology vol 24 no 4 pp 1454ndash1462 2016

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Page 4: Robust Finite-Time H-Infinity Control with Transients for ...downloads.hindawi.com/journals/mpe/2018/2838749.pdf · ResearchArticle Robust Finite-Time H-Infinity Control with Transients

4 Mathematical Problems in Engineering

Π3 (1198783 120572 120573 120574) = ( (120572 minus 120573)119879(1205722 minus 120573 + 120574)119879)119879

sdot (61198783 minus121198783lowast 361198783 )( (120572 minus 120573)119879(1205722 minus 120573 + 120574)119879)(14)

Lemma 11 (see [30]) Let 1198911 1198912 119891119873 R119898 997891rarr R havepositive values in an open subset119863 ofR119898Then the reciprocallyconvex combination of 119891119894 over119863 satisfies

min120572119894|120572119894gt0sum119894 120572119894=1

sum119894

1120572119894119891119894 (119905) = sum119894 119891119894 (119905) +max119892119894119895(119905)sum119894 =119895

119892119894119895 (119905)subject to 119892119894119895 119877119898 997891997888rarr 119877 119892119895119894 (119905) = 119892119894119895 (119905) [ 119891119894 (119905) 119892119894119895 (119905)119892119895119894 (119905) 119891119895 (119905) ] ge 0 (15)

Lemma 12 (see [31]) For any matrix 1198784 gt 0 and a vectorfunction 119909 in [minus119886 minus119887] rarr R119899 if the integrals concerned arewell defined then the following inequality holdsintminus119887

minus119886119909119879 (119904) 1198784119909 (119904) 119889119904ge 1119886 minus 119887 (intminus119887minus119886

119909119879 (119904) 119889119904) 1198784 (intminus119887minus119886119909 (119904) 119889119904) (16)

23 Control Objective Finally the objective of this paper is toderive the control gain 119870 such that

(1) the closed-loop system (8) with 119906(119905) = 0 is FTB(2) under given nonzero initial condition the closed

system (8) and (9) guarantees that 119869 = int1198790119911119879(119904)119911(119904) minus1205742120596119879(119904)120596(119904) minus 1205742(1119879)119909119879(0)119878119909(0)119889119904 lt 0 for all

nonzero 120596(119905) and 119905 isin [0 119879]3 Main Results

Thedesigning of the robust finite-time119867infin controller for theDP system with input delay is divided into three steps

31 FTB Analysis of DP System Firstly the result guarantee-ing the FTB of DP system is established in this subsection

Theorem 13 Given five positive scalars 120572 120573 1198881 1198882 119879 andpositive define matrix 119877 the finite-time boundedness problemof system (4) is solvable if there exist positive scalars 120579119897 (119897 =1 2 sdot sdot sdot 8) and matrices 11987511 gt 0 11987522 gt 0 119882 gt 0 119876119894 gt0 (119894 = 1 2)119880119895 gt 0 (119895 = 1 2)119872 gt 0 1198851 1198852 with appropriatedimensions satisfying the following conditionsΩ = (Ω119894119895)8times8 lt 0 (17)(119882 1198851lowast 119882) gt 0 (18)

(3119882 1198852lowast 3119882) gt 0 (19)

11988811205792 + ℎ211988811205793 + 120588ℎ11988811205794 + 120588ℎ11988811205795 + 12120588ℎ311988811205796+ 16120588ℎ311988811205797 + 13120588ℎ311988811205798 + 1205731205799119889 lt 119890minus12057211987911988821205791 (20)

whereΩ11 = 11986011987911987511 + 11987511119860 minus 12057211987511 + 1198761 + 1198762+ 119860119879 12ℎ2 (1198801 + 1198802 + 2119882)119860 minus 4120588119882 minus 31205881198801minus 91205881198802Ω22 = minus1205881198762 minus 4120588119882 minus 61205881198802Ω33 = 1198701198791198611119879 12ℎ2 (2119882 + 1198801 + 1198802) 1198611119870 minus 8120588119882+ 1205881198851119879 + 1205881198851 minus 1205881198852119879 minus 1205881198852 minus (1 minus 120583)1198762Ω44 = 1198612119879 12ℎ2 (2119882 + 1198801 + 1198802) 1198612 minus 120573119872Ω66 = minus12120588119882Ω77 = minus12120588119882Ω55 = 120572ℎ211987522 minus 181205881198802 minus 61205881198801Ω88 = minus361205881198801 minus 361205881198802Ω12 = 1205881198851 minus 1205881198852 + 61205881198802Ω15 = ℎ11987522 + 121205881198802Ω16 = 6120588119882Ω17 = 21205881198852

Mathematical Problems in Engineering 5

Ω13 = 119875111198611119870 + 119860119879 12ℎ2 (2119882 + 1198801 + 1198802) 1198611119870 minus 2120588119882minus 1205881198851 minus 1205881198852Ω14 = 119875111198612 + 119860119879 12ℎ2 (2119882 + 1198801 + 1198802) 1198612Ω18 = minus181205881198802 + 61205881198801Ω23 = minus1205881198851119879 minus 1205881198852119879 minus 2120588119882Ω25 = minusℎ11987522 minus 61205881198802Ω26 = 21205881198852119879Ω27 = 6120588119882Ω28 = 121205881198802Ω34 = 1198701198791198611119879 12ℎ2 (2119882 + 1198801 + 1198802) 1198612Ω36 = 6120588119882 + 21205881198852119879Ω37 = 6120588119882 + 21205881198852Ω58 = 121205881198801 + 241205881198802Ω67 = minus41205881198852Ω24 = Ω35 = Ω38 = Ω45 = Ω46 = Ω47 = Ω48 = Ω56= Ω57 = Ω68 = Ω78 = 01205791 = 120582min (11)1205792 = 120582max (11) 1205793 = 120582max (22) 1205794 = 120582max (1198761) 1205795 = 120582max (1198762) 1205796 = 120582max () 1205797 = 120582max (1) 1205798 = 120582max (2) 1205799 = 120582max (119872) 120588 = 119890120572ℎ

11 = 119877minus1211987511119877minus1222 = 119877minus1211987522119877minus12119876119894 = 119877minus12119876119894119877minus12 (119894 = 1 2 3) = 119877minus12119882119877minus12119895 = 119877minus12119880119895119877minus12 (119895 = 1 2) 0 lt 1205791119868 lt 11987511 lt 12057921198680 lt 11987522 lt 12057931198680 lt 1198761 lt 12057941198680 lt 1198762 lt 12057951198680 lt 119882 lt 12057961198680 lt 1198801 lt 12057971198680 lt 1198802 lt 12057981198680 lt 119872 lt 1205799119868(21)

Proof Consider the candidate augment LKLF as follows119881 (119905) = 1198811 (119905) + 1198812 (119905) + 1198813 (119905) + 1198814 (119905) (22)1198811 (119905) = 120576 (119905)119879 119875120576 (119905) (23)1198812 (119905) = int119905119905minusℎ119890120572(119905minus119904)119909 (119904)1198791198761119909 (119904) 119889119904+ int119905119905minusℎ(119905)119890120572(119905minus119904)119909 (119904)1198791198762119909 (119904) 119889119904 (24)

1198813 (119905) = ℎint0minusℎint119905119905+120579119890120572(119905minus119904) (119904)119879119882 (119904) 119889119904119889120579 (25)

1198814 (119905) = int0minusℎint0120579int119905119905+120573119890120572(119905minus119904) (119904)1198791198801 (119904) 119889119904119889120573119889120579

+ int0minusℎint120579minusℎint119905119905+120573119890120572(119905minus119904) (119904)1198791198802 (119904) 119889119904119889120573119889120579 (26)

where 120576(119905) = (119909(119905) int119905119905minusℎ119909(119904)119889119904) 119875 = ( 11987511 0

lowast 11987522)

To deal with the formulas conveniently let us providesuch definitions

120585 (119905)= (119909 (119905) 119909 (119905 minus ℎ) 119909 (119905 minus ℎ (119905)) 120596 (119905) 1ℎ int119905119905minusℎ 119909 (119904) 119889119904 1ℎ (119905) int119905119905minusℎ(119905) 119909 (119904) 119889119904 1ℎ minus ℎ (119905) int119905minusℎ(119905)119905minusℎ119909 (119904) 119889119904 1ℎ2 int0minusℎ int119905119905+120579 119909 (119904) 119889119904119889120579)119897119896119879 = (0 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896minus1

1 0 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟8minus119896) (27)

6 Mathematical Problems in Engineering

The time-derivative of 119881(119905) along system (4) can bebounded as1 (119905) = 1205721198811 (119905) + 120576 (119905)119879 119875120576 (119905) + 120576 (119905)119879 119875 120576 (119905) minus 1205721198811 (119905) (28)2 (119905)= 1205721198812 (119905) + 119909 (119905)1198791198761119909 (119905)minus 119890120572ℎ119909 (119905 minus ℎ)1198791198761119909 (119905 minus ℎ) + 119909 (119905)1198791198762119909 (119905)minus (1 minus ℎ (119905)) 119890120572ℎ(119905)119909 (119905 minus ℎ (119905))1198791198762119909 (119905 minus ℎ (119905))le 1205721198812 (119905) + 119909 (119905)1198791198761119909 (119905)minus 119890120572ℎ119909 (119905 minus ℎ)1198791198761119909 (119905 minus ℎ) + 119909 (119905)1198791198762119909 (119905)minus (1 minus 120583) 119909 (119905 minus ℎ (119905))1198791198762119909 (119905 minus ℎ (119905))

(29)

3 (119905)= 1205721198813 (119905) + ℎ2 (119905)1198791198822 (119905)minus 119890120572ℎℎint119905119905minusℎ (119904)1198791198822 (119904) 119889119904 (30)

4 (119905)= 1205721198814 (119905) + 12ℎ2 (119905)1198791198801 (119905)minus 119890120572ℎ int0minusℎint119905119905+120573 (119904)1198791198801 (119904) 119889119904119889120573+ 12ℎ2 (119905)1198791198802 (119905)minus 119890120572ℎ int0

minusℎint119905+120573119905minusℎ (119904)1198791198802 (119904) 119889119904119889120573

(31)

Invoking Lemma 9 we can obtain3 (119905) = 1205721198813 (119905) + ℎ2 (119905)119879119882 (119905)minus 119890120572ℎℎint119905119905minusℎ(119905) (119904)119879119882 (119904) 119889119904

minus 119890120572ℎℎint119905minusℎ(119905)119905minusℎ

(119904)119879119882 (119904) 119889119904le 1205721198813 (119905) + ℎ2 (119905)119879119882 (119905)minus 119890120572ℎ ℎℎ (119905)1206031119879Λ 11206031 minus 119890120572ℎ ℎℎ minus ℎ (119905)1206032119879Λ 21206032(32)

where 1206031119879 = 120585119879 (119905)( (1198971119879 minus 1198973119879)119879(1198971119879 + 1198973119879 minus 21198976119879)119879)119879

1206032119879 = 120585119879 (119905)( (1198973119879 minus 1198972119879)119879(1198973119879 + 1198972119879 minus 21198977119879))119879

Λ 1 = Λ 2 = (119882 0lowast 3119882) (33)

In order to obtain a tighter bound of integral termLemma 11 is applied to (32) as follows3 (119905) le 1205721198813 (119905) + ℎ2 (119905)119879119882 (119905) minus 119890120572ℎ1206033119879Λ 31206033 (34)

where 1206033119879 = ( 12060311198791206032119879 ) Λ 3 = (119882 1198851 0 0

lowast 119882 0 0lowast lowast 3119882 1198852lowast lowast lowast 3119882

)By applying Lemma 10 to (31) we can obtain the inequal-

ity 4 (119905) le 1205721198814 (119905) + 12ℎ2 (119905)1198791198801 (119905)+ 12ℎ2 (119905)1198791198802 (119905) minus 119890120572ℎ1206034119879Λ 41206034 (35)

where

1206034119879 = 120585119879 (119905)(((((

(1198971119879 minus 1198972119879)119879(12 1198971119879 minus 1198978119879)119879(1198972119879 minus 1198975119879)119879(12 1198972119879 minus 1198975119879 + 1198978119879)119879)))))

119879

Λ 4 =(61198801 minus121198801 0 0lowast 361198801 0 0lowast lowast 61198802 minus121198802lowast lowast lowast 361198802)(36)

Combining (28) (29) (34) and (35) with the definitionof 120588 = 119890120572ℎ we can obtain minus 120572119881 minus 120573120596119879 (119905)119872120596 (119905) lt 120585119879 (119905) Ω120585 (119905) (37)

Assuming that Ω lt 0 and integrating the left part ofinequality (37)119881 (119905) lt 119890120572119905119881 (0) + 120573int119905

0119890120572(119905minus119904)120596119879 (119904)119872120596 (119904) 119889119904lt 119890120572119879 (119881 (0) + 120573120582max (119872) 119889) (38)

Invoking the Jensen inequality shown in Lemma 12 onehas 119881 (0) le 119909 (0)119879 11987511119909 (0) + ℎint0

minusℎ119909 (119904)119879 11987522119909 (119904) 119889119904+ 120588int0

minusℎ119909 (119904)1198791198761119909 (119904) 119889119904+ 120588int0

minusℎ(119905)119909 (119904)1198791198762119909 (119904) 119889119904

Mathematical Problems in Engineering 7

+ ℎ120588int0minusℎint0120579 (119904)119879119882 (119904) 119889119904119889120579

+ 120588int0minusℎint0120579int0120573 (119904)1198791198801 (119904) 119889119904119889120573119889120579

+ 120588int0minusℎint120579minusℎint0120573 (119904)1198791198802 (119904) 119889119904119889120573119889120579le 120582max (11) 1198881 + ℎ120582max (22) ℎ1198881+ 120588120582max (1198761) ℎ1198881 + 120588120582max (1198762) ℎ1198881+ 120588120582max () 12ℎ31198881 + 120588120582max (1) 16ℎ31198881+ 120588120582max (2) 13ℎ31198881

(39)

Based on inequalities (38) and (39) we have119881 (119905) gt 119909119879 (119905) 11987511119909 (119905) = 119909119879 (119905) 119877121111987712119909 (119905)gt 120582min (11) 119909119879 (119905) 119877119909 (119905) (40)

Therefore conditions (17) to (20) can guarantee the FTBof system (4) This completes the proof

Remark 14 Among the existing approaches there are twothreads one is to construct a novel LKLF that involves moreinformation of delay the other is to find a tighter estimationof upper bound for cross terms coming from the derivativeof the LKLF In this paper these two techniques are appliedto obtain the result In addition the less conservatism inpractical engineering will be shown in the simulations whichare always ignored in most literature sources

32 Controller Design In this subsection we focus on theproblem of finite-time 119867infin state feedback designing basedon Theorem 13 that is designing a state feedback controllerin the form of (7) such that the resulting DP system satisfiesthe control objective proposed in Section 1

Theorem 15 For given positive 120572 1198881 1198882 119879 120574 and matrices 119877 gt0 119878 ge 0 if there exist positive scalars 120579119897 (119897 = 1 2 sdot sdot sdot 8) andmatrices 11987511 gt 0 11987522 gt 0119882 gt 0 119876119894 gt 0 (119894 = 1 2) 119880119895 gt0 (119895 = 1 2) 119884 1198851 1198852 with appropriate dimensions satisfyingthe following conditions Π = (Π119894119895)12times12 lt 0 (41)Σ = (Σ119894119895)9times9 lt 0 (42)

(119882 1198851lowast 119882) gt 0 (43)

(3119882 1198852lowast 3119882) gt 0 (44)

119868 minus 120574211987811987511 lt 0 (45)0 lt 1198671 lt 11987511 lt 11986720 lt 11987522 lt 11986730 lt 1198761 lt 11986740 lt 1198762 lt 11986750 lt 119882 lt 11986760 lt 1198801 lt 11986770 lt 1198802 lt 1198678(46)

where Π11 = 11987511119860119879 + 11986011987511 minus 12057211987511 + 1198761 + 1198762 minus 4120588119882minus 31205881198801 minus 91205881198802Π44 = minus1205742119868Π22 = minus1205881198762 minus 4120588119882 minus 61205881198802Π33 = minus8120588119882 + 1205881198851119879 + 1205881198851 minus 1205881198852119879 minus 1205881198852minus (1 minus 120583)1198762Π13 = 1198611119884 minus 2120588119882 minus 1205881198851 minus 1205881198852Π14 = 1198612Π66 = minus12120588119882Π77 = minus12120588119882Π55 = 120572ℎ211987522 minus 181205881198802 minus 61205881198801Π88 = minus361205881198801 minus 361205881198802Π12 = 1205881198851 minus 1205881198852 + 61205881198802Π15 = ℎ11987522 + 121205881198802Π16 = 6120588119882Π17 = 21205881198852Π25 = minusℎ11987522 minus 61205881198802Π18 = minus181205881198802 + 61205881198801Π26 = 21205881198852119879Π23 = minus1205881198851119879 minus 1205881198852119879 minus 2120588119882Π27 = 6120588119882Π28 = 121205881198802

8 Mathematical Problems in Engineering

Π36 = 6120588119882 + 21205881198852119879Π37 = 6120588119882 + 21205881198852Π58 = 121205881198801 + 241205881198802Π67 = minus41205881198852Π19 = 12ℎ11987511119860119879Π39 = 12ℎ1198841198791198611119879Π49 = 12ℎ1198612119879Π110 = 12ℎ11987511119860119879Π310 = 12ℎ1198841198791198611119879Π410 = 12ℎ1198612119879Π111 = ℎ11987511119860119879Π311 = ℎ1198841198791198611119879Π411 = ℎ1198612119879Π112 = 11987511119862119879Π312 = 119884119879119863119879Π99 = 121198801 minus 11987511Π1010 = 121198802 minus 11987511Π1111 = 119882 minus 211987511Π1212 = minus119890minus120572119879119868(47)

others Π119894119895 = 0Σ11 = minus119890minus12057211987911988821198671Σ22 = minus1198672Σ33 = minus1198673Σ44 = minus1198674Σ55 = minus1198675Σ66 = minus1198676Σ77 = minus1198677Σ88 = minus1198678

Σ99 = minus1205742119889Σ12 = radic11988811198672Σ13 = radicℎ211988811198673Σ14 = radic120588ℎ11988811198674Σ15 = radic120588ℎ11988811198675Σ16 = radic12120588ℎ311988811198676Σ17 = radic16120588ℎ311988811198677Σ18 = radic13120588ℎ311988811198678Σ19 = 11987511(48)

others Σ119894119895 = 0 (49)

then a state feedback119867infin controller in form of (7) existssuch that

(1) the closed-loop system (8) with 119906(119905) = 0 is FTB(2) under given nonzero initial condition the closed

system (8) and (9) guarantees that 119869 = int1198790119911119879(119904)119911(119904) minus1205742120596119879(119904)120596(119904) minus 1205742(1119879)119909119879(0)119878119909(0)119889119904 lt 0 for all nonzero 120596(119905)

and 119905 isin [0 119879]The controller gain can be calculated by 119870 = 11988411987511minus1

Proof In preparation for designing we set119872 = 1205742119868 120573 = 1for (37) then (37) and (38) can be rewritten as follows minus 120572119881 minus 1205742120596119879 (119905) 120596 (119905) lt 120585119879 (119905) Ω120585 (119905) (50)119881 (119905) lt 119890120572119905119881 (0) + 1205742 int119905

0119890120572(119905minus119904)120596119879 (119904) 120596 (119904) 119889119904lt 119890120572119879 (119881 (0) + 1205742119889) (51)

where Ω44 = 1198612119879 12ℎ2 (2119882 + 1198801 + 1198802) 1198612 minus 1205742119868Ω119894119895 = Ω119894119895

(52)

Then two steps are provided to design the desiredcontroller

Step 1 Here we will give some conditions that can guaranteethe 119867infin performance over the finite-time interval undernonzero initial conditions firstly

Mathematical Problems in Engineering 9

Consider the following inequality minus 120572119881 + 119890120572119905119911119879 (119905) 119911 (119905) minus 1205742119890120572119905120596119879 (119905) 120596 (119905)lt minus 120572119881 + 119890120572119879119911119879 (119905) 119911 (119905) minus 1205742120596119879 (119905) 120596 (119905)lt 120585119879 (119905) (Ω minus Ψ119879 (minus119890minus120572119879119868)minus1Ψ) 120585 (119905) (53)

Noting that119882 1198801 and 1198802 are coupled with 119860 1198611119870 and1198612 in Ω to decouple these terms the Ω can be rewritten asΩ = Ω minus Ψ119879 (minus121198801)minus1 Ψ minus Ψ119879 (minus121198802)minus1Ψminus Ψ119879 (minus119882)minus1 Ψ (54)

whereΩ11 = 11986011987911987511 + 11987511119860 minus 12057211987511 + 1198761 + 1198762 minus 4120588119882minus 31205881198801 minus 91205881198802Ω33 = minus8120588119882 + 1205881198851119879 + 1205881198851 minus 1205881198852119879 minus 1205881198852minus (1 minus 120583)1198762Ω44 = minus1205742119868Ω13 = 119875111198611119870 minus 2120588119882 minus 1205881198851 minus 1205881198852Ω14 = 119875111198612Ω34 = 0(55)

othersΩ119894119895 = Ω119894119895Ψ = (119862 0 119863119870 0 0 0 0 0)Ψ= (ℎ121198801119879119860 0 ℎ1211988011198791198611119870 ℎ1211988011198791198612 0 0 0 0) Ψ= (ℎ121198802119879119860 0 ℎ1211988021198791198611119870 ℎ1211988021198791198612 0 0 0 0)Ψ = (ℎ119882119879119860 0 ℎ1198821198791198611119870 ℎ1198821198791198612 0 0 0 0) (56)

By Schur complementΩ lt 0 and Ω minusΨ119879(minus119890minus120572119879119868)minus1Ψ areequivalent to

Π =(((((

Ω Ψ119879 Ψ119879 Ψ119879 Ψ119879lowast minus121198801 0 0 0lowast lowast minus121198802 0 0lowast lowast lowast minus119882 0lowast lowast lowast lowast minus119890minus120572119879119868)))))lt 0 (57)

To ensure the proposed conditions satisfy the 119867infin per-formance we set the inequality like

11987511 minus 1205742119878 lt 0 (58)

from which we have

minus 120572119881 + 119890120572119905119911119879 (119905) 119911 (119905) minus 1205742119890120572119905120596119879 (119905) 120596 (119905)minus 1205742119890120572119905 1119879119909119879 (0) 119878119909 (0) lt minus 120572119881 + 119890120572119905119911119879 (119905) 119911 (119905)minus 1205742119890120572119905120596119879 (119905) 120596 (119905) minus 1205742119890120572119905 1119879119909119879 (0) 119878119909 (0)+ 119890120572119905 1119879119909119879 (0) 11987511119909 (0) lt minus 120572119881 + 119890120572119905119911119879 (119905) 119911 (119905)minus 1205742119890120572119905120596119879 (119905) 120596 (119905) + 119890120572119905 1119879119909119879 (0) (11987511 minus 1205742119878) 119909 (0)lt 0(59)

Integrating (58) from 0 to 119879 we can obtain

int1198790119911119879 (119904) 119911 (119904) 119889119904 minus int119879

01205742120596119879 (119904) 120596 (119904) 119889119904

minus 1205742119909119879 (0) 119878119909 (0) lt minusint1198790(119890minus120572119905 minus 120572119890minus120572119905119881)119889119904 lt 0 (60)

so conditions (43) (44) (57) and (58) with the inequali-ties as follows can ensure that systems (17) and (18) satisfy the119867infin performance

11988811205792 + ℎ211988811205793 + 120588ℎ11988811205794 + 120588ℎ11988811205795 + 12120588ℎ311988811205796+ 16120588ℎ311988811205797 + 13120588ℎ311988811205798 + 1205742119889 lt 119890minus120572119879119888212057910 lt 1205791119868 lt 11987511 lt 12057921198680 lt 11987522 lt 12057931198680 lt 1198761 lt 12057941198680 lt 1198762 lt 12057951198680 lt 119882 lt 12057961198680 lt 1198801 lt 12057971198680 lt 1198802 lt 1205798119868

(61)

10 Mathematical Problems in Engineering

Step 2 In this step the approach to design the gain matrix 119870of the controller is provided in forms of LMIs which can becalculated expediently

To linearize the nonlinear terms in (57) we make itmultiply by the followingmatrix fromboth left and right sidesfirstly

diag 11987511minus1 11987511minus1 11987511minus1 119868 11987511minus1 11987511minus1 11987511minus1 11987511minus1 1198801minus1 1198802minus1 119882minus1 119868 (62)

Let us do the definitions as follows11987511 = 11987511minus11198761 = 11987511minus1119876111987511minus11198762 = 11987511minus1119876211987511minus1119882 = 11987511minus111988211987511minus11198801 = 11987511minus1119880111987511minus11198802 = 11987511minus1119880211987511minus111987522 = 11987511minus11198752211987511minus11198851 = 11987511minus1119885111987511minus11198852 = 11987511minus1119885211987511minus1119884 = 11987011987511minus1

(63)

so the inequality Π lt 0 can guarantee Π lt 0 whereΠ99 = minus12119875111198801

minus111987511Π1010 = minus12119875111198802

minus111987511Π1111 = minus11987511119882minus111987511(64)

others Π119894119895 = Π119894119895 (65)

Then based on the inequalities minus(12)119875111198801

minus111987511 le(12)1198801 minus 11987511 minus(12)119875111198802

minus111987511 le (12)1198802 minus 11987511minus11987511119882minus111987511 le 119882 minus 211987511 condition (57) can be convertedinto (41) in form of LMIs

Accordingly similar to the above-mentioned procedures(61) can be transformed into (42) with the Schur complementand the definition of119867119894 = 11987511minus112057911989411987511minus1 (119894 = 1 2 sdot sdot sdot 8)

Therefore conditions (41) to (45) can obtain the con-troller in the form of (7) guaranteeing the desired perfor-mance This completes the proof

Remark 16 The119867infin performance with nonzero initial con-dition is studied in this literature In a practical view it is

hard to suffer the worst-case over a finite time for surfacevessel with nonzero initial condition and it will reducethe conservatism caused by the assumption of zero initialcondition Therefore it is reasonable to introduce such 119867infinperformance index to handle the problem of finite-time119867infincontrol for ship subject to exogenous signals

Remark 17 It is challenging to achieve the result satisfyingthe control objective while it is very different from thestandard 119867infin control because a weighting diagonal matrixis introduced especially to establish the form of LMIs Inthe proof of Theorem 15 inequality (58) is introduced toensure the 119867infin performance with nonzero initial conditionand make it convenient to construct the conditions in formof LMIs

Remark 18 By introducing some new additional matrixvariables Cone Complementarity Linearization algorithm(CCL) [32] can be utilized to linearize the nonlinear termsas an alternative method It will reduce the conservatism atthe expense of increasing the computational burden and thenumber of decision variables

4 Simulations

41 Simulation Settings To demonstrate the effectiveness andsuperiority of the robust finite-time 119867infin control schemesimulation studies are conducted on the parameters of theship CyberShip II [33] which is a 170 scale-replica of a supplyshipwhosemain parameters are shown inTable 1 In additionwe choose the time-varying function like ℎ(119905) = 3sin (1199056 minus1205876) to describe the time delay occurred in the input signal

Unknown external disturbances 120596(119905) with significantlylarge magnitude and high frequency which can be roughlyconsidered as complex environment including ocean windswaves and currents are governed by [34]

120596 (119905) = (9 sin (01120587119905 minus 1205875 )6 sin (03120587119905 + 1205876 )3 sin (02120587119905 + 1205873 )) (66)

with initial conditions 120578(0) = (1 1 01) and 120592(0) =(08 06 01)42 Numerical Simulations We choose 120574 = 13 119879 = 10 119878 =03lowast119868 1198881 = 302 1198882 = 10 120572 = 0001 By usingTheorem 15 thecontrol gain 119870 is calculated by LMIs (41) to (45)

Mathematical Problems in Engineering 11

Table 1 The main parameters of the CyberShip II119898 238000 119883 -20 119883119906 -07225119868119911 17600 119884V -100 119884V -08612119909119892 00460 119884 119903 -00 119884119903 01079119873 119903 -10 119873119903 05

0 20 40 60 80 100Time(s) Time(s)

minus2

0

2

4

6

8

10

12

0 2 4 6 8 100

2

4

6

8

10

12

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

xT(t)Rx(t)xT(t)Rx(t)

c=c=

c1=302

c1=302

Figure 1 State response 119909119879(119905)119877119909(119905) of DP system

119870 = (minus32743 0 0 minus313907 0 00 minus42784 minus01572 0 minus412136 minus129820 minus01389 minus03497 0 minus13362 minus33091) (67)

To demonstrate the superiority and validity three sce-narios as comparative cases are provided here It should bepointed out that the results within 100s are also provided toillustrate the steady-state performance

Case 1 (with different 119878) For comparison purposes we set119878 with different value to demonstrate the effect of 119878 on stateresponse Without loss of generality we set 119878 = 05 lowast 119868 and119878 = 01 lowast 119868 respectivelyCase 2 (119867infin with zero initial condition]) This scenario willshow that the control law proposed in this paper can achievea better transient performance and disturbance rejectioncapability over traditional119867infin control approach for dynamicpositioning system conducted in [11]

Case 3 (a control design based on existing methods) Basedon the approach in [17] applied for discrete-time system acontrol law for dynamic positioning ship with input delay isproposed to illustrate the less conservatism of our designingapproach in practical engineering subsequently

At first the effectiveness of the control law we designedis confirmed from Figures 1ndash6 Figure 1 shows the state

response of 119909119879(119905)119877119909(119905) Obviously from that curve we cansee that the value of 119909119879(119905)119877119909(119905) is far below the value of1198882 = 10 in the presence of disturbance as in Figure 2which means the finite-time full-state feedback controllerdesigned can guarantee the DP system (8) sim (9) FTB with(302 10 119889 10 3 119868) and have less conservatism where weset 119889 = int119879

0120596(119905)119879120596(119905)119889119905 Meanwhile the control law can

stabilize the dynamic positioning shipwithminimum steady-state error in 100s and the control input will be small enoughas shown in Figure 6 From Figures 3ndash5 we can find that allstates in DP system are within a small range As shown inFigures 2ndash4 the angle and the angular velocity in yaw havechanged within 01∘ which means that the heading of thevessel stays in a fixed direction Furthermore the velocitiesin surge and sway are low enough which illustrates thatassumption 2 is reasonable and it meets the requirement ofstabilization control for DP system

To demonstrate the effect of 119878 state responses of systemswith different 119878 are exhibited in Figures 7ndash9 As the valueof 119878 decreases the restriction for 11987511 is more exclusive andthe robustness is stronger during preset time interval as inFigures 7 and 8 while the required control input will increasedramatically as in Figure 9 When the time oversteps the

12 Mathematical Problems in Engineering

0 10 20 30 40 50 60 70 80 90 100Time(s)

minus10

minus5

0

5

10

Dist

urba

nces

(N)

XY

Figure 2 Disturbance in surge sway and yaw

0 50 100Time(s)

minus050

051

152

X(m

)

0 5 10Time(s)

005

115

2

X(m

)

0 50 100Time(s)

minus050

051

152

Y(m

)

0 5 10Time(s)

005

115

2Y(

m)

0 50 100Time(s)

minus01

0

01

02

0 5 10Time(s)

0

01

02

(∘

)

(∘

)

Figure 3 State responses of position and orientation for DP System

0 50 100Time(s)

minus1

0

1

2

0 5 10Time(s)

minus1

0

1

2

0 50 100Time(s)

minus1

0

1

2

v(m

s)

0 5 10Time(s)

minus1

0

1

2

v(m

s)

0 50 100Time(s)

minus01

0

01

02

0 5 10Time(s)

minus01

0

01

02

r(∘ s

)

r(∘ s

)

(ms

)

(m

s)

Figure 4 State responses of velocities for DP system

Mathematical Problems in Engineering 13

minus05 0 05 1 15 2X(m)

minus05

0

05

1

15

2

Y(m

)

Figure 5 Trajectory of the shiprsquos horizontal motion

0 50 100Time(s)

minus30minus20minus10

010

u1(N

)

0 5 10Time(s)

minus30minus20minus10

010

u1(N

)

0 50 100Time(s)

minus30minus20minus10

010

u2(N

)

0 5 10Time(s)

minus30minus20minus10

010

u2(N

)

0 50 100Time(s)

minus15minus1

minus050

05

u3(N

m)

0 5 10Time(s)

minus15minus1

minus050

05

u3(N

m)

Figure 6 Control inputs

0 20 40 60 80 100Time(s)

minus1

0

1

2

3

4

5

6

S=03S=05S=01

0 2 4 6 8 10Time(s)

0

2

4

6

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 7 State response 119909119879(119905)119877119909(119905) with different 119878

14 Mathematical Problems in Engineering

Figure 8 State responses of position and orientation with different 119878

0 50 100Time(s)

minus60minus30

03060

u1(N

)

0 5 10Time(s)

minus40minus20

02040

u1(N

)

0 50 100Time(s)

minus100

0

100

u2(N

)

0 5 10Time(s)

minus40minus20

02040

u2(N

)

0 50 100Time(s)

minus4minus2

024

u3(N

m)

0 5 10Time(s)

minus2

0

2

u3(N

m)

S=03S=05S=01

Figure 9 Control input with different 119878preset time interval all the states can be guaranteed withinthe given upper bound as before

As shown in Figures 10ndash12 the overconservatism oftraditional119867infin control approach applied inDP ship is solvedby the 119867infin control strategy with nonzero initial conditionproposed in this article As is well known the classical 119867infincontrol law is designed under the so-called worst-case normwhile the performance of antidisturbance and transient issacrificed to some degree From Figures 10 and 11 we cansee that the state response can be controlled in a smallerrange and stabilized in a shorter time with a lower overshootshown in Figure 12 Hence a conclusion can be drawn thatthe control law we present can achieve a better disturbance

rejection capability and transient performance while thematter of overconservatism can be resolved at the sametime

Simulation results of Theorem 15 and Case 3 are shownin Figures 13 and 14 to show the better performance ofthe method we proposed We can find that the controllerbased on [17] and 119867infin control approach present in thisliterature can make all states of system within a smallenough region However compared with the controller weconstructed the state response has higher overshoot whichwill affect the control precision and transient performance inpractical engineering Based on the above simulation resultsthe developed controller can achieve a higher accuracy and

Mathematical Problems in Engineering 15

0 20 40 60 80 100Time(s)

minus2

0

2

4

6

8

10

12

Theorem 15Case 2

0 2 4 6 8 10Time(s)

minus2

0

2

4

6

8

10

12

Theorem 15Case 2

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 10 State response 119909119879(119905)119877119909(119905) of Theorem 15 and Case 2

0 50 100Time(s)

minus10123

X(m

)

0 5 10Time(s)

0123

X(m

)

0 50 100Time(s)

minus10123

Y(m

)

0 5 10Time(s)

0123

Y(m

)

0 50 100Time(s)

minus020

0204

0 5 10Time(s)

minus020

0204

Theorem 15Case 2

(∘

)

(∘

)

Figure 11 Position and orientation of Theorem 15 and Case 2

minus05 0 05 1 15 2 25X(m)

minus050

051

152

25

Y(m

)

Theorem 15Case 2

Figure 12 Trajectory of the shiprsquos horizontal motion of Theorem 15 and Case 2

16 Mathematical Problems in Engineering

0 20 40 60 80 100Time(s)

0

1

2

3

4

5

Theorem 15Case 3

0 2 4 6 8 10Time(s)

0

1

2

3

4

5

Theorem 15Case 3

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 13 State response 119909119879(119905)119877119909(119905) of Theorem 15 and Case 3

0 5 10Time(s)

05

1

15

X(m

)

Theorem 15Case 3

0 5 10Time(s)

05

1

15

Y(m

)

0 5 10Time(s)

0

005

01

015

02

(∘

)

Figure 14 Position and orientation of Theorem 15 and Case 3

lower overshoot which means that it is less conservative incomparison with existing result

5 Conclusion

In this paper the robust finite-time 119867infin control with tran-sients for DP system with input delay maintaining all statesof ocean surface vessel into a given threshold over a fixed timeinterval in presence of time-varying disturbance is proposedBased on a novel augmented LKLF with triple integral termsand a method combining theWirtinger inequality and recip-rocally convex approach a less conservative result is derivedIn particular a119867infin performance index with nonzero initialcondition attenuating the overconservatism caused by theassumption of zero initial condition and enhancing thetransient performance of ship subject to external disturbanceis adoptedThe obtained result is formulated in terms of LMIswhich can be easily solved by the standard numerical soft-ware Finally a numerical simulation for a ship is proposed toverify the effectiveness and less conservatism of the controller

we designed In the future the output feedback control can bedeveloped for the ocean surface vessel to avoid the use of thevessel-frame surge sway and yaw velocities

Data Availability

All the data supporting the conclusions of the study have beenprovided in Simulations and readers can access these data in[33 34]

Conflicts of Interest

The authors declare that there are no conflicts of interestrelated to this paper

Acknowledgments

This research was partially supported by the National Sci-ence Technology Support Program of China (Project no51609046)

Mathematical Problems in Engineering 17

References

[1] M Chen B Jiang and R Cui ldquoActuator fault-tolerant controlof ocean surface vessels with input saturationrdquo InternationalJournal of Robust amp Nonlinear Control vol 26 no 3 pp 542ndash564 2016

[2] J Du X Hu and Y Sun ldquoRobust dynamic positioning of shipswith disturbances under input saturationrdquo Automatica vol 73pp 207ndash214 2016

[3] H Ashrafiuon K R Muske L C McNinch and R ASoltan ldquoSliding-mode tracking control of surface vesselsrdquo IEEETransactions on Industrial Electronics vol 55 no 11 pp 4004ndash4012 2008

[4] W Si and X Dong ldquoNeural prescribed performance controlfor uncertain marine surface vessels without accurate initialerrorsrdquoMathematical Problems in Engineering Art ID 261832311 pages 2017

[5] T D Nguyen A J Soslashrensen and S T Quek ldquoDesign of hybridcontroller for dynamic positioning from calm to extreme seaconditionsrdquo Automatica vol 43 no 5 pp 768ndash785 2007

[6] J Du X Hu H Liu and C L Chen ldquoAdaptive robust outputfeedback control for a marine dynamic positioning systembased on a high-gain observerrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 11 pp 2775ndash27862015

[7] H Katayama ldquoNonlinear sampled-data stabilization of dynam-ically positioned shipsrdquo IEEE Transactions on Control SystemsTechnology vol 18 no 2 pp 463ndash468 2010

[8] K J Astrom andCG Kallstrom ldquoIdentification of ship steeringdynamicsrdquo Automatica vol 12 no 1 pp 9ndash22 1976

[9] Z Lei and C Guo ldquoDisturbance rejection control solutionfor ship steering system with uncertain time delayrdquo OceanEngineering vol 95 pp 78ndash83 2015

[10] J C Doyle K Glover P P Khargonekar and B A FrancisldquoState-space solutions to standard H-2 and Hinfin control prob-lemsrdquo IEEE Transactions on Automatic Control vol 34 no 8pp 831ndash847 1989

[11] M Katebi M Grimble and Y Zhang ldquoHinfin robust controldesign for dynamic ship positioningrdquo IEE Proceedings - ControlTheory and Applications vol 144 no 2 pp 110ndash120 1997

[12] D V Balandin and M M Kogan ldquoLMI-based Hinfin-optimalcontrol with transientsrdquo International Journal of Control vol 83no 8 pp 1664ndash1673 2010

[13] Z Feng J Lam S Xu and S Zhou ldquoHinfin Control withTransients for Singular Systemsrdquo Asian Journal of Control vol18 no 3 pp 817ndash827 2016

[14] P P Khargonekar K M Nagpal and K R Poolla ldquoHinfin controlwith transientsrdquo SIAM Journal onControl andOptimization vol29 no 6 pp 1373ndash1393 1991

[15] Z Zhang Z Zhang and H Zhang ldquoFinite-time stabilityanalysis and stabilization for uncertain continuous-time systemwith time-varying delayrdquo Journal of The Franklin Institute vol352 no 3 pp 1296ndash1317 2015

[16] J Cheng S Chen Z Liu H Wang and J Li ldquoRobust finite-time sampled-data control of linear systems subject to randomoccurring delays and its application to Four-Tank systemrdquoAppliedMathematics andComputation vol 281 pp 55ndash76 2016

[17] W Kang S Zhong K Shi and J Cheng ldquoFinite-time stabilityfor discrete-time system with time-varying delay and nonlinearperturbationsrdquo ISA Transactions vol 60 pp 67ndash73 2016

[18] T I Fossen Handbook of Marine Craft Hydrodynamics andMotion Control John Wiley amp Sons New York USA 2011

[19] D Zhao F Ding L Zhou W Zhang and H Xu ldquoRobustH-infinity Control of Neutral System with Time-Delay forDynamic Positioning Shipsrdquo Mathematical Problems in Engi-neering vol 2015 Article ID 976925 11 pages 2015

[20] F Amato M Ariola and C Cosentino ldquoFinite-time stabilityof linear time-varying systems analysis and controller designrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 55 no 4 pp 1003ndash1008 2010

[21] L Sun andZ Zheng ldquoFinite-time slidingmode trajectory track-ing control of uncertain mechanical systemsrdquo Asian Journal ofControl vol 19 no 1 pp 399ndash404 2017

[22] P Shi Y Zhang and R Agarwal ldquoStochastic finite-time stateestimation for discrete time-delay neural networks withMarko-vian jumpsrdquo Neurocomputing vol 151 part 1 pp 168ndash174 2014

[23] L Zhang S Wang H R Karimi and A Jasra ldquoRobust Finite-Time Control of Switched Linear Systems and Application toa Class of Servomechanism Systemsrdquo IEEEASME Transactionson Mechatronics vol 20 no 5 pp 2476ndash2485 2015

[24] L Wu P Shi and H Gao ldquoState estimation and sliding-mode control of Markovian jump singular systemsrdquo Institute ofElectrical and Electronics Engineers Transactions on AutomaticControl vol 55 no 5 pp 1213ndash1219 2010

[25] Y Q Zhang P Shi S K Nguang and Y D Song ldquoRobust finite-time Hinfin control for uncertain discrete-time singular systemswithMarkovian jumpsrdquo IET ControlTheory ampApplications vol8 no 12 pp 1105ndash1111 2014

[26] S Wang T Shi L Zhang A Jasra and M Zeng ldquoExtendedfinite-time Hinfin control for uncertain switched linear neutralsystemswith time-varying delaysrdquoNeurocomputing vol 152 pp377ndash387 2015

[27] J Hu G Sui S Du and X Li ldquoFinite-time stability of uncer-tain nonlinear systems with time-varying delayrdquo MathematicalProblems in Engineering Article ID 2538904 9 pages 2017

[28] A Seuret and F Gouaisbaut ldquoWirtinger-based integral inequal-ity application to time-delay systemsrdquo Automatica vol 49 no9 pp 2860ndash2866 2013

[29] Z Li Y Bai C Huang and Y Cai ldquoNovel delay-partitioningstabilization approach for networked control system viaWirtinger-based inequalitiesrdquo ISA Transactions vol 61 pp75ndash86 2016

[30] P Park JWKo andC Jeong ldquoReciprocally convex approach tostability of systems with time-varying delaysrdquo Automatica vol47 no 1 pp 235ndash238 2011

[31] K Gu V L Kharitonov and J Chen Stability of Time DelaySystems Birkhauser Boston Mass USA 2003

[32] L El Ghaoui F Oustry and M AitRami ldquoA cone comple-mentarity linearization algorithm for static output-feedbackand related problemsrdquo Institute of Electrical and ElectronicsEngineers Transactions on Automatic Control vol 42 no 8 pp1171ndash1176 1997

[33] R Skjetne T I Fossen and P Kokotovic ldquoAdaptive maneu-vering with experiments for a model ship in a marine controllaboratoryrdquo Automatica vol 41 no 2 pp 289ndash298 2005

[34] N Wang C Qian J-C Sun and Y-C Liu ldquoAdaptive RobustFinite-Time Trajectory Tracking Control of Fully ActuatedMarine Surface Vehiclesrdquo IEEE Transactions on Control SystemsTechnology vol 24 no 4 pp 1454ndash1462 2016

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Page 5: Robust Finite-Time H-Infinity Control with Transients for ...downloads.hindawi.com/journals/mpe/2018/2838749.pdf · ResearchArticle Robust Finite-Time H-Infinity Control with Transients

Mathematical Problems in Engineering 5

Ω13 = 119875111198611119870 + 119860119879 12ℎ2 (2119882 + 1198801 + 1198802) 1198611119870 minus 2120588119882minus 1205881198851 minus 1205881198852Ω14 = 119875111198612 + 119860119879 12ℎ2 (2119882 + 1198801 + 1198802) 1198612Ω18 = minus181205881198802 + 61205881198801Ω23 = minus1205881198851119879 minus 1205881198852119879 minus 2120588119882Ω25 = minusℎ11987522 minus 61205881198802Ω26 = 21205881198852119879Ω27 = 6120588119882Ω28 = 121205881198802Ω34 = 1198701198791198611119879 12ℎ2 (2119882 + 1198801 + 1198802) 1198612Ω36 = 6120588119882 + 21205881198852119879Ω37 = 6120588119882 + 21205881198852Ω58 = 121205881198801 + 241205881198802Ω67 = minus41205881198852Ω24 = Ω35 = Ω38 = Ω45 = Ω46 = Ω47 = Ω48 = Ω56= Ω57 = Ω68 = Ω78 = 01205791 = 120582min (11)1205792 = 120582max (11) 1205793 = 120582max (22) 1205794 = 120582max (1198761) 1205795 = 120582max (1198762) 1205796 = 120582max () 1205797 = 120582max (1) 1205798 = 120582max (2) 1205799 = 120582max (119872) 120588 = 119890120572ℎ

11 = 119877minus1211987511119877minus1222 = 119877minus1211987522119877minus12119876119894 = 119877minus12119876119894119877minus12 (119894 = 1 2 3) = 119877minus12119882119877minus12119895 = 119877minus12119880119895119877minus12 (119895 = 1 2) 0 lt 1205791119868 lt 11987511 lt 12057921198680 lt 11987522 lt 12057931198680 lt 1198761 lt 12057941198680 lt 1198762 lt 12057951198680 lt 119882 lt 12057961198680 lt 1198801 lt 12057971198680 lt 1198802 lt 12057981198680 lt 119872 lt 1205799119868(21)

Proof Consider the candidate augment LKLF as follows119881 (119905) = 1198811 (119905) + 1198812 (119905) + 1198813 (119905) + 1198814 (119905) (22)1198811 (119905) = 120576 (119905)119879 119875120576 (119905) (23)1198812 (119905) = int119905119905minusℎ119890120572(119905minus119904)119909 (119904)1198791198761119909 (119904) 119889119904+ int119905119905minusℎ(119905)119890120572(119905minus119904)119909 (119904)1198791198762119909 (119904) 119889119904 (24)

1198813 (119905) = ℎint0minusℎint119905119905+120579119890120572(119905minus119904) (119904)119879119882 (119904) 119889119904119889120579 (25)

1198814 (119905) = int0minusℎint0120579int119905119905+120573119890120572(119905minus119904) (119904)1198791198801 (119904) 119889119904119889120573119889120579

+ int0minusℎint120579minusℎint119905119905+120573119890120572(119905minus119904) (119904)1198791198802 (119904) 119889119904119889120573119889120579 (26)

where 120576(119905) = (119909(119905) int119905119905minusℎ119909(119904)119889119904) 119875 = ( 11987511 0

lowast 11987522)

To deal with the formulas conveniently let us providesuch definitions

120585 (119905)= (119909 (119905) 119909 (119905 minus ℎ) 119909 (119905 minus ℎ (119905)) 120596 (119905) 1ℎ int119905119905minusℎ 119909 (119904) 119889119904 1ℎ (119905) int119905119905minusℎ(119905) 119909 (119904) 119889119904 1ℎ minus ℎ (119905) int119905minusℎ(119905)119905minusℎ119909 (119904) 119889119904 1ℎ2 int0minusℎ int119905119905+120579 119909 (119904) 119889119904119889120579)119897119896119879 = (0 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896minus1

1 0 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟8minus119896) (27)

6 Mathematical Problems in Engineering

The time-derivative of 119881(119905) along system (4) can bebounded as1 (119905) = 1205721198811 (119905) + 120576 (119905)119879 119875120576 (119905) + 120576 (119905)119879 119875 120576 (119905) minus 1205721198811 (119905) (28)2 (119905)= 1205721198812 (119905) + 119909 (119905)1198791198761119909 (119905)minus 119890120572ℎ119909 (119905 minus ℎ)1198791198761119909 (119905 minus ℎ) + 119909 (119905)1198791198762119909 (119905)minus (1 minus ℎ (119905)) 119890120572ℎ(119905)119909 (119905 minus ℎ (119905))1198791198762119909 (119905 minus ℎ (119905))le 1205721198812 (119905) + 119909 (119905)1198791198761119909 (119905)minus 119890120572ℎ119909 (119905 minus ℎ)1198791198761119909 (119905 minus ℎ) + 119909 (119905)1198791198762119909 (119905)minus (1 minus 120583) 119909 (119905 minus ℎ (119905))1198791198762119909 (119905 minus ℎ (119905))

(29)

3 (119905)= 1205721198813 (119905) + ℎ2 (119905)1198791198822 (119905)minus 119890120572ℎℎint119905119905minusℎ (119904)1198791198822 (119904) 119889119904 (30)

4 (119905)= 1205721198814 (119905) + 12ℎ2 (119905)1198791198801 (119905)minus 119890120572ℎ int0minusℎint119905119905+120573 (119904)1198791198801 (119904) 119889119904119889120573+ 12ℎ2 (119905)1198791198802 (119905)minus 119890120572ℎ int0

minusℎint119905+120573119905minusℎ (119904)1198791198802 (119904) 119889119904119889120573

(31)

Invoking Lemma 9 we can obtain3 (119905) = 1205721198813 (119905) + ℎ2 (119905)119879119882 (119905)minus 119890120572ℎℎint119905119905minusℎ(119905) (119904)119879119882 (119904) 119889119904

minus 119890120572ℎℎint119905minusℎ(119905)119905minusℎ

(119904)119879119882 (119904) 119889119904le 1205721198813 (119905) + ℎ2 (119905)119879119882 (119905)minus 119890120572ℎ ℎℎ (119905)1206031119879Λ 11206031 minus 119890120572ℎ ℎℎ minus ℎ (119905)1206032119879Λ 21206032(32)

where 1206031119879 = 120585119879 (119905)( (1198971119879 minus 1198973119879)119879(1198971119879 + 1198973119879 minus 21198976119879)119879)119879

1206032119879 = 120585119879 (119905)( (1198973119879 minus 1198972119879)119879(1198973119879 + 1198972119879 minus 21198977119879))119879

Λ 1 = Λ 2 = (119882 0lowast 3119882) (33)

In order to obtain a tighter bound of integral termLemma 11 is applied to (32) as follows3 (119905) le 1205721198813 (119905) + ℎ2 (119905)119879119882 (119905) minus 119890120572ℎ1206033119879Λ 31206033 (34)

where 1206033119879 = ( 12060311198791206032119879 ) Λ 3 = (119882 1198851 0 0

lowast 119882 0 0lowast lowast 3119882 1198852lowast lowast lowast 3119882

)By applying Lemma 10 to (31) we can obtain the inequal-

ity 4 (119905) le 1205721198814 (119905) + 12ℎ2 (119905)1198791198801 (119905)+ 12ℎ2 (119905)1198791198802 (119905) minus 119890120572ℎ1206034119879Λ 41206034 (35)

where

1206034119879 = 120585119879 (119905)(((((

(1198971119879 minus 1198972119879)119879(12 1198971119879 minus 1198978119879)119879(1198972119879 minus 1198975119879)119879(12 1198972119879 minus 1198975119879 + 1198978119879)119879)))))

119879

Λ 4 =(61198801 minus121198801 0 0lowast 361198801 0 0lowast lowast 61198802 minus121198802lowast lowast lowast 361198802)(36)

Combining (28) (29) (34) and (35) with the definitionof 120588 = 119890120572ℎ we can obtain minus 120572119881 minus 120573120596119879 (119905)119872120596 (119905) lt 120585119879 (119905) Ω120585 (119905) (37)

Assuming that Ω lt 0 and integrating the left part ofinequality (37)119881 (119905) lt 119890120572119905119881 (0) + 120573int119905

0119890120572(119905minus119904)120596119879 (119904)119872120596 (119904) 119889119904lt 119890120572119879 (119881 (0) + 120573120582max (119872) 119889) (38)

Invoking the Jensen inequality shown in Lemma 12 onehas 119881 (0) le 119909 (0)119879 11987511119909 (0) + ℎint0

minusℎ119909 (119904)119879 11987522119909 (119904) 119889119904+ 120588int0

minusℎ119909 (119904)1198791198761119909 (119904) 119889119904+ 120588int0

minusℎ(119905)119909 (119904)1198791198762119909 (119904) 119889119904

Mathematical Problems in Engineering 7

+ ℎ120588int0minusℎint0120579 (119904)119879119882 (119904) 119889119904119889120579

+ 120588int0minusℎint0120579int0120573 (119904)1198791198801 (119904) 119889119904119889120573119889120579

+ 120588int0minusℎint120579minusℎint0120573 (119904)1198791198802 (119904) 119889119904119889120573119889120579le 120582max (11) 1198881 + ℎ120582max (22) ℎ1198881+ 120588120582max (1198761) ℎ1198881 + 120588120582max (1198762) ℎ1198881+ 120588120582max () 12ℎ31198881 + 120588120582max (1) 16ℎ31198881+ 120588120582max (2) 13ℎ31198881

(39)

Based on inequalities (38) and (39) we have119881 (119905) gt 119909119879 (119905) 11987511119909 (119905) = 119909119879 (119905) 119877121111987712119909 (119905)gt 120582min (11) 119909119879 (119905) 119877119909 (119905) (40)

Therefore conditions (17) to (20) can guarantee the FTBof system (4) This completes the proof

Remark 14 Among the existing approaches there are twothreads one is to construct a novel LKLF that involves moreinformation of delay the other is to find a tighter estimationof upper bound for cross terms coming from the derivativeof the LKLF In this paper these two techniques are appliedto obtain the result In addition the less conservatism inpractical engineering will be shown in the simulations whichare always ignored in most literature sources

32 Controller Design In this subsection we focus on theproblem of finite-time 119867infin state feedback designing basedon Theorem 13 that is designing a state feedback controllerin the form of (7) such that the resulting DP system satisfiesthe control objective proposed in Section 1

Theorem 15 For given positive 120572 1198881 1198882 119879 120574 and matrices 119877 gt0 119878 ge 0 if there exist positive scalars 120579119897 (119897 = 1 2 sdot sdot sdot 8) andmatrices 11987511 gt 0 11987522 gt 0119882 gt 0 119876119894 gt 0 (119894 = 1 2) 119880119895 gt0 (119895 = 1 2) 119884 1198851 1198852 with appropriate dimensions satisfyingthe following conditions Π = (Π119894119895)12times12 lt 0 (41)Σ = (Σ119894119895)9times9 lt 0 (42)

(119882 1198851lowast 119882) gt 0 (43)

(3119882 1198852lowast 3119882) gt 0 (44)

119868 minus 120574211987811987511 lt 0 (45)0 lt 1198671 lt 11987511 lt 11986720 lt 11987522 lt 11986730 lt 1198761 lt 11986740 lt 1198762 lt 11986750 lt 119882 lt 11986760 lt 1198801 lt 11986770 lt 1198802 lt 1198678(46)

where Π11 = 11987511119860119879 + 11986011987511 minus 12057211987511 + 1198761 + 1198762 minus 4120588119882minus 31205881198801 minus 91205881198802Π44 = minus1205742119868Π22 = minus1205881198762 minus 4120588119882 minus 61205881198802Π33 = minus8120588119882 + 1205881198851119879 + 1205881198851 minus 1205881198852119879 minus 1205881198852minus (1 minus 120583)1198762Π13 = 1198611119884 minus 2120588119882 minus 1205881198851 minus 1205881198852Π14 = 1198612Π66 = minus12120588119882Π77 = minus12120588119882Π55 = 120572ℎ211987522 minus 181205881198802 minus 61205881198801Π88 = minus361205881198801 minus 361205881198802Π12 = 1205881198851 minus 1205881198852 + 61205881198802Π15 = ℎ11987522 + 121205881198802Π16 = 6120588119882Π17 = 21205881198852Π25 = minusℎ11987522 minus 61205881198802Π18 = minus181205881198802 + 61205881198801Π26 = 21205881198852119879Π23 = minus1205881198851119879 minus 1205881198852119879 minus 2120588119882Π27 = 6120588119882Π28 = 121205881198802

8 Mathematical Problems in Engineering

Π36 = 6120588119882 + 21205881198852119879Π37 = 6120588119882 + 21205881198852Π58 = 121205881198801 + 241205881198802Π67 = minus41205881198852Π19 = 12ℎ11987511119860119879Π39 = 12ℎ1198841198791198611119879Π49 = 12ℎ1198612119879Π110 = 12ℎ11987511119860119879Π310 = 12ℎ1198841198791198611119879Π410 = 12ℎ1198612119879Π111 = ℎ11987511119860119879Π311 = ℎ1198841198791198611119879Π411 = ℎ1198612119879Π112 = 11987511119862119879Π312 = 119884119879119863119879Π99 = 121198801 minus 11987511Π1010 = 121198802 minus 11987511Π1111 = 119882 minus 211987511Π1212 = minus119890minus120572119879119868(47)

others Π119894119895 = 0Σ11 = minus119890minus12057211987911988821198671Σ22 = minus1198672Σ33 = minus1198673Σ44 = minus1198674Σ55 = minus1198675Σ66 = minus1198676Σ77 = minus1198677Σ88 = minus1198678

Σ99 = minus1205742119889Σ12 = radic11988811198672Σ13 = radicℎ211988811198673Σ14 = radic120588ℎ11988811198674Σ15 = radic120588ℎ11988811198675Σ16 = radic12120588ℎ311988811198676Σ17 = radic16120588ℎ311988811198677Σ18 = radic13120588ℎ311988811198678Σ19 = 11987511(48)

others Σ119894119895 = 0 (49)

then a state feedback119867infin controller in form of (7) existssuch that

(1) the closed-loop system (8) with 119906(119905) = 0 is FTB(2) under given nonzero initial condition the closed

system (8) and (9) guarantees that 119869 = int1198790119911119879(119904)119911(119904) minus1205742120596119879(119904)120596(119904) minus 1205742(1119879)119909119879(0)119878119909(0)119889119904 lt 0 for all nonzero 120596(119905)

and 119905 isin [0 119879]The controller gain can be calculated by 119870 = 11988411987511minus1

Proof In preparation for designing we set119872 = 1205742119868 120573 = 1for (37) then (37) and (38) can be rewritten as follows minus 120572119881 minus 1205742120596119879 (119905) 120596 (119905) lt 120585119879 (119905) Ω120585 (119905) (50)119881 (119905) lt 119890120572119905119881 (0) + 1205742 int119905

0119890120572(119905minus119904)120596119879 (119904) 120596 (119904) 119889119904lt 119890120572119879 (119881 (0) + 1205742119889) (51)

where Ω44 = 1198612119879 12ℎ2 (2119882 + 1198801 + 1198802) 1198612 minus 1205742119868Ω119894119895 = Ω119894119895

(52)

Then two steps are provided to design the desiredcontroller

Step 1 Here we will give some conditions that can guaranteethe 119867infin performance over the finite-time interval undernonzero initial conditions firstly

Mathematical Problems in Engineering 9

Consider the following inequality minus 120572119881 + 119890120572119905119911119879 (119905) 119911 (119905) minus 1205742119890120572119905120596119879 (119905) 120596 (119905)lt minus 120572119881 + 119890120572119879119911119879 (119905) 119911 (119905) minus 1205742120596119879 (119905) 120596 (119905)lt 120585119879 (119905) (Ω minus Ψ119879 (minus119890minus120572119879119868)minus1Ψ) 120585 (119905) (53)

Noting that119882 1198801 and 1198802 are coupled with 119860 1198611119870 and1198612 in Ω to decouple these terms the Ω can be rewritten asΩ = Ω minus Ψ119879 (minus121198801)minus1 Ψ minus Ψ119879 (minus121198802)minus1Ψminus Ψ119879 (minus119882)minus1 Ψ (54)

whereΩ11 = 11986011987911987511 + 11987511119860 minus 12057211987511 + 1198761 + 1198762 minus 4120588119882minus 31205881198801 minus 91205881198802Ω33 = minus8120588119882 + 1205881198851119879 + 1205881198851 minus 1205881198852119879 minus 1205881198852minus (1 minus 120583)1198762Ω44 = minus1205742119868Ω13 = 119875111198611119870 minus 2120588119882 minus 1205881198851 minus 1205881198852Ω14 = 119875111198612Ω34 = 0(55)

othersΩ119894119895 = Ω119894119895Ψ = (119862 0 119863119870 0 0 0 0 0)Ψ= (ℎ121198801119879119860 0 ℎ1211988011198791198611119870 ℎ1211988011198791198612 0 0 0 0) Ψ= (ℎ121198802119879119860 0 ℎ1211988021198791198611119870 ℎ1211988021198791198612 0 0 0 0)Ψ = (ℎ119882119879119860 0 ℎ1198821198791198611119870 ℎ1198821198791198612 0 0 0 0) (56)

By Schur complementΩ lt 0 and Ω minusΨ119879(minus119890minus120572119879119868)minus1Ψ areequivalent to

Π =(((((

Ω Ψ119879 Ψ119879 Ψ119879 Ψ119879lowast minus121198801 0 0 0lowast lowast minus121198802 0 0lowast lowast lowast minus119882 0lowast lowast lowast lowast minus119890minus120572119879119868)))))lt 0 (57)

To ensure the proposed conditions satisfy the 119867infin per-formance we set the inequality like

11987511 minus 1205742119878 lt 0 (58)

from which we have

minus 120572119881 + 119890120572119905119911119879 (119905) 119911 (119905) minus 1205742119890120572119905120596119879 (119905) 120596 (119905)minus 1205742119890120572119905 1119879119909119879 (0) 119878119909 (0) lt minus 120572119881 + 119890120572119905119911119879 (119905) 119911 (119905)minus 1205742119890120572119905120596119879 (119905) 120596 (119905) minus 1205742119890120572119905 1119879119909119879 (0) 119878119909 (0)+ 119890120572119905 1119879119909119879 (0) 11987511119909 (0) lt minus 120572119881 + 119890120572119905119911119879 (119905) 119911 (119905)minus 1205742119890120572119905120596119879 (119905) 120596 (119905) + 119890120572119905 1119879119909119879 (0) (11987511 minus 1205742119878) 119909 (0)lt 0(59)

Integrating (58) from 0 to 119879 we can obtain

int1198790119911119879 (119904) 119911 (119904) 119889119904 minus int119879

01205742120596119879 (119904) 120596 (119904) 119889119904

minus 1205742119909119879 (0) 119878119909 (0) lt minusint1198790(119890minus120572119905 minus 120572119890minus120572119905119881)119889119904 lt 0 (60)

so conditions (43) (44) (57) and (58) with the inequali-ties as follows can ensure that systems (17) and (18) satisfy the119867infin performance

11988811205792 + ℎ211988811205793 + 120588ℎ11988811205794 + 120588ℎ11988811205795 + 12120588ℎ311988811205796+ 16120588ℎ311988811205797 + 13120588ℎ311988811205798 + 1205742119889 lt 119890minus120572119879119888212057910 lt 1205791119868 lt 11987511 lt 12057921198680 lt 11987522 lt 12057931198680 lt 1198761 lt 12057941198680 lt 1198762 lt 12057951198680 lt 119882 lt 12057961198680 lt 1198801 lt 12057971198680 lt 1198802 lt 1205798119868

(61)

10 Mathematical Problems in Engineering

Step 2 In this step the approach to design the gain matrix 119870of the controller is provided in forms of LMIs which can becalculated expediently

To linearize the nonlinear terms in (57) we make itmultiply by the followingmatrix fromboth left and right sidesfirstly

diag 11987511minus1 11987511minus1 11987511minus1 119868 11987511minus1 11987511minus1 11987511minus1 11987511minus1 1198801minus1 1198802minus1 119882minus1 119868 (62)

Let us do the definitions as follows11987511 = 11987511minus11198761 = 11987511minus1119876111987511minus11198762 = 11987511minus1119876211987511minus1119882 = 11987511minus111988211987511minus11198801 = 11987511minus1119880111987511minus11198802 = 11987511minus1119880211987511minus111987522 = 11987511minus11198752211987511minus11198851 = 11987511minus1119885111987511minus11198852 = 11987511minus1119885211987511minus1119884 = 11987011987511minus1

(63)

so the inequality Π lt 0 can guarantee Π lt 0 whereΠ99 = minus12119875111198801

minus111987511Π1010 = minus12119875111198802

minus111987511Π1111 = minus11987511119882minus111987511(64)

others Π119894119895 = Π119894119895 (65)

Then based on the inequalities minus(12)119875111198801

minus111987511 le(12)1198801 minus 11987511 minus(12)119875111198802

minus111987511 le (12)1198802 minus 11987511minus11987511119882minus111987511 le 119882 minus 211987511 condition (57) can be convertedinto (41) in form of LMIs

Accordingly similar to the above-mentioned procedures(61) can be transformed into (42) with the Schur complementand the definition of119867119894 = 11987511minus112057911989411987511minus1 (119894 = 1 2 sdot sdot sdot 8)

Therefore conditions (41) to (45) can obtain the con-troller in the form of (7) guaranteeing the desired perfor-mance This completes the proof

Remark 16 The119867infin performance with nonzero initial con-dition is studied in this literature In a practical view it is

hard to suffer the worst-case over a finite time for surfacevessel with nonzero initial condition and it will reducethe conservatism caused by the assumption of zero initialcondition Therefore it is reasonable to introduce such 119867infinperformance index to handle the problem of finite-time119867infincontrol for ship subject to exogenous signals

Remark 17 It is challenging to achieve the result satisfyingthe control objective while it is very different from thestandard 119867infin control because a weighting diagonal matrixis introduced especially to establish the form of LMIs Inthe proof of Theorem 15 inequality (58) is introduced toensure the 119867infin performance with nonzero initial conditionand make it convenient to construct the conditions in formof LMIs

Remark 18 By introducing some new additional matrixvariables Cone Complementarity Linearization algorithm(CCL) [32] can be utilized to linearize the nonlinear termsas an alternative method It will reduce the conservatism atthe expense of increasing the computational burden and thenumber of decision variables

4 Simulations

41 Simulation Settings To demonstrate the effectiveness andsuperiority of the robust finite-time 119867infin control schemesimulation studies are conducted on the parameters of theship CyberShip II [33] which is a 170 scale-replica of a supplyshipwhosemain parameters are shown inTable 1 In additionwe choose the time-varying function like ℎ(119905) = 3sin (1199056 minus1205876) to describe the time delay occurred in the input signal

Unknown external disturbances 120596(119905) with significantlylarge magnitude and high frequency which can be roughlyconsidered as complex environment including ocean windswaves and currents are governed by [34]

120596 (119905) = (9 sin (01120587119905 minus 1205875 )6 sin (03120587119905 + 1205876 )3 sin (02120587119905 + 1205873 )) (66)

with initial conditions 120578(0) = (1 1 01) and 120592(0) =(08 06 01)42 Numerical Simulations We choose 120574 = 13 119879 = 10 119878 =03lowast119868 1198881 = 302 1198882 = 10 120572 = 0001 By usingTheorem 15 thecontrol gain 119870 is calculated by LMIs (41) to (45)

Mathematical Problems in Engineering 11

Table 1 The main parameters of the CyberShip II119898 238000 119883 -20 119883119906 -07225119868119911 17600 119884V -100 119884V -08612119909119892 00460 119884 119903 -00 119884119903 01079119873 119903 -10 119873119903 05

0 20 40 60 80 100Time(s) Time(s)

minus2

0

2

4

6

8

10

12

0 2 4 6 8 100

2

4

6

8

10

12

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

xT(t)Rx(t)xT(t)Rx(t)

c=c=

c1=302

c1=302

Figure 1 State response 119909119879(119905)119877119909(119905) of DP system

119870 = (minus32743 0 0 minus313907 0 00 minus42784 minus01572 0 minus412136 minus129820 minus01389 minus03497 0 minus13362 minus33091) (67)

To demonstrate the superiority and validity three sce-narios as comparative cases are provided here It should bepointed out that the results within 100s are also provided toillustrate the steady-state performance

Case 1 (with different 119878) For comparison purposes we set119878 with different value to demonstrate the effect of 119878 on stateresponse Without loss of generality we set 119878 = 05 lowast 119868 and119878 = 01 lowast 119868 respectivelyCase 2 (119867infin with zero initial condition]) This scenario willshow that the control law proposed in this paper can achievea better transient performance and disturbance rejectioncapability over traditional119867infin control approach for dynamicpositioning system conducted in [11]

Case 3 (a control design based on existing methods) Basedon the approach in [17] applied for discrete-time system acontrol law for dynamic positioning ship with input delay isproposed to illustrate the less conservatism of our designingapproach in practical engineering subsequently

At first the effectiveness of the control law we designedis confirmed from Figures 1ndash6 Figure 1 shows the state

response of 119909119879(119905)119877119909(119905) Obviously from that curve we cansee that the value of 119909119879(119905)119877119909(119905) is far below the value of1198882 = 10 in the presence of disturbance as in Figure 2which means the finite-time full-state feedback controllerdesigned can guarantee the DP system (8) sim (9) FTB with(302 10 119889 10 3 119868) and have less conservatism where weset 119889 = int119879

0120596(119905)119879120596(119905)119889119905 Meanwhile the control law can

stabilize the dynamic positioning shipwithminimum steady-state error in 100s and the control input will be small enoughas shown in Figure 6 From Figures 3ndash5 we can find that allstates in DP system are within a small range As shown inFigures 2ndash4 the angle and the angular velocity in yaw havechanged within 01∘ which means that the heading of thevessel stays in a fixed direction Furthermore the velocitiesin surge and sway are low enough which illustrates thatassumption 2 is reasonable and it meets the requirement ofstabilization control for DP system

To demonstrate the effect of 119878 state responses of systemswith different 119878 are exhibited in Figures 7ndash9 As the valueof 119878 decreases the restriction for 11987511 is more exclusive andthe robustness is stronger during preset time interval as inFigures 7 and 8 while the required control input will increasedramatically as in Figure 9 When the time oversteps the

12 Mathematical Problems in Engineering

0 10 20 30 40 50 60 70 80 90 100Time(s)

minus10

minus5

0

5

10

Dist

urba

nces

(N)

XY

Figure 2 Disturbance in surge sway and yaw

0 50 100Time(s)

minus050

051

152

X(m

)

0 5 10Time(s)

005

115

2

X(m

)

0 50 100Time(s)

minus050

051

152

Y(m

)

0 5 10Time(s)

005

115

2Y(

m)

0 50 100Time(s)

minus01

0

01

02

0 5 10Time(s)

0

01

02

(∘

)

(∘

)

Figure 3 State responses of position and orientation for DP System

0 50 100Time(s)

minus1

0

1

2

0 5 10Time(s)

minus1

0

1

2

0 50 100Time(s)

minus1

0

1

2

v(m

s)

0 5 10Time(s)

minus1

0

1

2

v(m

s)

0 50 100Time(s)

minus01

0

01

02

0 5 10Time(s)

minus01

0

01

02

r(∘ s

)

r(∘ s

)

(ms

)

(m

s)

Figure 4 State responses of velocities for DP system

Mathematical Problems in Engineering 13

minus05 0 05 1 15 2X(m)

minus05

0

05

1

15

2

Y(m

)

Figure 5 Trajectory of the shiprsquos horizontal motion

0 50 100Time(s)

minus30minus20minus10

010

u1(N

)

0 5 10Time(s)

minus30minus20minus10

010

u1(N

)

0 50 100Time(s)

minus30minus20minus10

010

u2(N

)

0 5 10Time(s)

minus30minus20minus10

010

u2(N

)

0 50 100Time(s)

minus15minus1

minus050

05

u3(N

m)

0 5 10Time(s)

minus15minus1

minus050

05

u3(N

m)

Figure 6 Control inputs

0 20 40 60 80 100Time(s)

minus1

0

1

2

3

4

5

6

S=03S=05S=01

0 2 4 6 8 10Time(s)

0

2

4

6

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 7 State response 119909119879(119905)119877119909(119905) with different 119878

14 Mathematical Problems in Engineering

Figure 8 State responses of position and orientation with different 119878

0 50 100Time(s)

minus60minus30

03060

u1(N

)

0 5 10Time(s)

minus40minus20

02040

u1(N

)

0 50 100Time(s)

minus100

0

100

u2(N

)

0 5 10Time(s)

minus40minus20

02040

u2(N

)

0 50 100Time(s)

minus4minus2

024

u3(N

m)

0 5 10Time(s)

minus2

0

2

u3(N

m)

S=03S=05S=01

Figure 9 Control input with different 119878preset time interval all the states can be guaranteed withinthe given upper bound as before

As shown in Figures 10ndash12 the overconservatism oftraditional119867infin control approach applied inDP ship is solvedby the 119867infin control strategy with nonzero initial conditionproposed in this article As is well known the classical 119867infincontrol law is designed under the so-called worst-case normwhile the performance of antidisturbance and transient issacrificed to some degree From Figures 10 and 11 we cansee that the state response can be controlled in a smallerrange and stabilized in a shorter time with a lower overshootshown in Figure 12 Hence a conclusion can be drawn thatthe control law we present can achieve a better disturbance

rejection capability and transient performance while thematter of overconservatism can be resolved at the sametime

Simulation results of Theorem 15 and Case 3 are shownin Figures 13 and 14 to show the better performance ofthe method we proposed We can find that the controllerbased on [17] and 119867infin control approach present in thisliterature can make all states of system within a smallenough region However compared with the controller weconstructed the state response has higher overshoot whichwill affect the control precision and transient performance inpractical engineering Based on the above simulation resultsthe developed controller can achieve a higher accuracy and

Mathematical Problems in Engineering 15

0 20 40 60 80 100Time(s)

minus2

0

2

4

6

8

10

12

Theorem 15Case 2

0 2 4 6 8 10Time(s)

minus2

0

2

4

6

8

10

12

Theorem 15Case 2

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 10 State response 119909119879(119905)119877119909(119905) of Theorem 15 and Case 2

0 50 100Time(s)

minus10123

X(m

)

0 5 10Time(s)

0123

X(m

)

0 50 100Time(s)

minus10123

Y(m

)

0 5 10Time(s)

0123

Y(m

)

0 50 100Time(s)

minus020

0204

0 5 10Time(s)

minus020

0204

Theorem 15Case 2

(∘

)

(∘

)

Figure 11 Position and orientation of Theorem 15 and Case 2

minus05 0 05 1 15 2 25X(m)

minus050

051

152

25

Y(m

)

Theorem 15Case 2

Figure 12 Trajectory of the shiprsquos horizontal motion of Theorem 15 and Case 2

16 Mathematical Problems in Engineering

0 20 40 60 80 100Time(s)

0

1

2

3

4

5

Theorem 15Case 3

0 2 4 6 8 10Time(s)

0

1

2

3

4

5

Theorem 15Case 3

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 13 State response 119909119879(119905)119877119909(119905) of Theorem 15 and Case 3

0 5 10Time(s)

05

1

15

X(m

)

Theorem 15Case 3

0 5 10Time(s)

05

1

15

Y(m

)

0 5 10Time(s)

0

005

01

015

02

(∘

)

Figure 14 Position and orientation of Theorem 15 and Case 3

lower overshoot which means that it is less conservative incomparison with existing result

5 Conclusion

In this paper the robust finite-time 119867infin control with tran-sients for DP system with input delay maintaining all statesof ocean surface vessel into a given threshold over a fixed timeinterval in presence of time-varying disturbance is proposedBased on a novel augmented LKLF with triple integral termsand a method combining theWirtinger inequality and recip-rocally convex approach a less conservative result is derivedIn particular a119867infin performance index with nonzero initialcondition attenuating the overconservatism caused by theassumption of zero initial condition and enhancing thetransient performance of ship subject to external disturbanceis adoptedThe obtained result is formulated in terms of LMIswhich can be easily solved by the standard numerical soft-ware Finally a numerical simulation for a ship is proposed toverify the effectiveness and less conservatism of the controller

we designed In the future the output feedback control can bedeveloped for the ocean surface vessel to avoid the use of thevessel-frame surge sway and yaw velocities

Data Availability

All the data supporting the conclusions of the study have beenprovided in Simulations and readers can access these data in[33 34]

Conflicts of Interest

The authors declare that there are no conflicts of interestrelated to this paper

Acknowledgments

This research was partially supported by the National Sci-ence Technology Support Program of China (Project no51609046)

Mathematical Problems in Engineering 17

References

[1] M Chen B Jiang and R Cui ldquoActuator fault-tolerant controlof ocean surface vessels with input saturationrdquo InternationalJournal of Robust amp Nonlinear Control vol 26 no 3 pp 542ndash564 2016

[2] J Du X Hu and Y Sun ldquoRobust dynamic positioning of shipswith disturbances under input saturationrdquo Automatica vol 73pp 207ndash214 2016

[3] H Ashrafiuon K R Muske L C McNinch and R ASoltan ldquoSliding-mode tracking control of surface vesselsrdquo IEEETransactions on Industrial Electronics vol 55 no 11 pp 4004ndash4012 2008

[4] W Si and X Dong ldquoNeural prescribed performance controlfor uncertain marine surface vessels without accurate initialerrorsrdquoMathematical Problems in Engineering Art ID 261832311 pages 2017

[5] T D Nguyen A J Soslashrensen and S T Quek ldquoDesign of hybridcontroller for dynamic positioning from calm to extreme seaconditionsrdquo Automatica vol 43 no 5 pp 768ndash785 2007

[6] J Du X Hu H Liu and C L Chen ldquoAdaptive robust outputfeedback control for a marine dynamic positioning systembased on a high-gain observerrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 11 pp 2775ndash27862015

[7] H Katayama ldquoNonlinear sampled-data stabilization of dynam-ically positioned shipsrdquo IEEE Transactions on Control SystemsTechnology vol 18 no 2 pp 463ndash468 2010

[8] K J Astrom andCG Kallstrom ldquoIdentification of ship steeringdynamicsrdquo Automatica vol 12 no 1 pp 9ndash22 1976

[9] Z Lei and C Guo ldquoDisturbance rejection control solutionfor ship steering system with uncertain time delayrdquo OceanEngineering vol 95 pp 78ndash83 2015

[10] J C Doyle K Glover P P Khargonekar and B A FrancisldquoState-space solutions to standard H-2 and Hinfin control prob-lemsrdquo IEEE Transactions on Automatic Control vol 34 no 8pp 831ndash847 1989

[11] M Katebi M Grimble and Y Zhang ldquoHinfin robust controldesign for dynamic ship positioningrdquo IEE Proceedings - ControlTheory and Applications vol 144 no 2 pp 110ndash120 1997

[12] D V Balandin and M M Kogan ldquoLMI-based Hinfin-optimalcontrol with transientsrdquo International Journal of Control vol 83no 8 pp 1664ndash1673 2010

[13] Z Feng J Lam S Xu and S Zhou ldquoHinfin Control withTransients for Singular Systemsrdquo Asian Journal of Control vol18 no 3 pp 817ndash827 2016

[14] P P Khargonekar K M Nagpal and K R Poolla ldquoHinfin controlwith transientsrdquo SIAM Journal onControl andOptimization vol29 no 6 pp 1373ndash1393 1991

[15] Z Zhang Z Zhang and H Zhang ldquoFinite-time stabilityanalysis and stabilization for uncertain continuous-time systemwith time-varying delayrdquo Journal of The Franklin Institute vol352 no 3 pp 1296ndash1317 2015

[16] J Cheng S Chen Z Liu H Wang and J Li ldquoRobust finite-time sampled-data control of linear systems subject to randomoccurring delays and its application to Four-Tank systemrdquoAppliedMathematics andComputation vol 281 pp 55ndash76 2016

[17] W Kang S Zhong K Shi and J Cheng ldquoFinite-time stabilityfor discrete-time system with time-varying delay and nonlinearperturbationsrdquo ISA Transactions vol 60 pp 67ndash73 2016

[18] T I Fossen Handbook of Marine Craft Hydrodynamics andMotion Control John Wiley amp Sons New York USA 2011

[19] D Zhao F Ding L Zhou W Zhang and H Xu ldquoRobustH-infinity Control of Neutral System with Time-Delay forDynamic Positioning Shipsrdquo Mathematical Problems in Engi-neering vol 2015 Article ID 976925 11 pages 2015

[20] F Amato M Ariola and C Cosentino ldquoFinite-time stabilityof linear time-varying systems analysis and controller designrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 55 no 4 pp 1003ndash1008 2010

[21] L Sun andZ Zheng ldquoFinite-time slidingmode trajectory track-ing control of uncertain mechanical systemsrdquo Asian Journal ofControl vol 19 no 1 pp 399ndash404 2017

[22] P Shi Y Zhang and R Agarwal ldquoStochastic finite-time stateestimation for discrete time-delay neural networks withMarko-vian jumpsrdquo Neurocomputing vol 151 part 1 pp 168ndash174 2014

[23] L Zhang S Wang H R Karimi and A Jasra ldquoRobust Finite-Time Control of Switched Linear Systems and Application toa Class of Servomechanism Systemsrdquo IEEEASME Transactionson Mechatronics vol 20 no 5 pp 2476ndash2485 2015

[24] L Wu P Shi and H Gao ldquoState estimation and sliding-mode control of Markovian jump singular systemsrdquo Institute ofElectrical and Electronics Engineers Transactions on AutomaticControl vol 55 no 5 pp 1213ndash1219 2010

[25] Y Q Zhang P Shi S K Nguang and Y D Song ldquoRobust finite-time Hinfin control for uncertain discrete-time singular systemswithMarkovian jumpsrdquo IET ControlTheory ampApplications vol8 no 12 pp 1105ndash1111 2014

[26] S Wang T Shi L Zhang A Jasra and M Zeng ldquoExtendedfinite-time Hinfin control for uncertain switched linear neutralsystemswith time-varying delaysrdquoNeurocomputing vol 152 pp377ndash387 2015

[27] J Hu G Sui S Du and X Li ldquoFinite-time stability of uncer-tain nonlinear systems with time-varying delayrdquo MathematicalProblems in Engineering Article ID 2538904 9 pages 2017

[28] A Seuret and F Gouaisbaut ldquoWirtinger-based integral inequal-ity application to time-delay systemsrdquo Automatica vol 49 no9 pp 2860ndash2866 2013

[29] Z Li Y Bai C Huang and Y Cai ldquoNovel delay-partitioningstabilization approach for networked control system viaWirtinger-based inequalitiesrdquo ISA Transactions vol 61 pp75ndash86 2016

[30] P Park JWKo andC Jeong ldquoReciprocally convex approach tostability of systems with time-varying delaysrdquo Automatica vol47 no 1 pp 235ndash238 2011

[31] K Gu V L Kharitonov and J Chen Stability of Time DelaySystems Birkhauser Boston Mass USA 2003

[32] L El Ghaoui F Oustry and M AitRami ldquoA cone comple-mentarity linearization algorithm for static output-feedbackand related problemsrdquo Institute of Electrical and ElectronicsEngineers Transactions on Automatic Control vol 42 no 8 pp1171ndash1176 1997

[33] R Skjetne T I Fossen and P Kokotovic ldquoAdaptive maneu-vering with experiments for a model ship in a marine controllaboratoryrdquo Automatica vol 41 no 2 pp 289ndash298 2005

[34] N Wang C Qian J-C Sun and Y-C Liu ldquoAdaptive RobustFinite-Time Trajectory Tracking Control of Fully ActuatedMarine Surface Vehiclesrdquo IEEE Transactions on Control SystemsTechnology vol 24 no 4 pp 1454ndash1462 2016

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Page 6: Robust Finite-Time H-Infinity Control with Transients for ...downloads.hindawi.com/journals/mpe/2018/2838749.pdf · ResearchArticle Robust Finite-Time H-Infinity Control with Transients

6 Mathematical Problems in Engineering

The time-derivative of 119881(119905) along system (4) can bebounded as1 (119905) = 1205721198811 (119905) + 120576 (119905)119879 119875120576 (119905) + 120576 (119905)119879 119875 120576 (119905) minus 1205721198811 (119905) (28)2 (119905)= 1205721198812 (119905) + 119909 (119905)1198791198761119909 (119905)minus 119890120572ℎ119909 (119905 minus ℎ)1198791198761119909 (119905 minus ℎ) + 119909 (119905)1198791198762119909 (119905)minus (1 minus ℎ (119905)) 119890120572ℎ(119905)119909 (119905 minus ℎ (119905))1198791198762119909 (119905 minus ℎ (119905))le 1205721198812 (119905) + 119909 (119905)1198791198761119909 (119905)minus 119890120572ℎ119909 (119905 minus ℎ)1198791198761119909 (119905 minus ℎ) + 119909 (119905)1198791198762119909 (119905)minus (1 minus 120583) 119909 (119905 minus ℎ (119905))1198791198762119909 (119905 minus ℎ (119905))

(29)

3 (119905)= 1205721198813 (119905) + ℎ2 (119905)1198791198822 (119905)minus 119890120572ℎℎint119905119905minusℎ (119904)1198791198822 (119904) 119889119904 (30)

4 (119905)= 1205721198814 (119905) + 12ℎ2 (119905)1198791198801 (119905)minus 119890120572ℎ int0minusℎint119905119905+120573 (119904)1198791198801 (119904) 119889119904119889120573+ 12ℎ2 (119905)1198791198802 (119905)minus 119890120572ℎ int0

minusℎint119905+120573119905minusℎ (119904)1198791198802 (119904) 119889119904119889120573

(31)

Invoking Lemma 9 we can obtain3 (119905) = 1205721198813 (119905) + ℎ2 (119905)119879119882 (119905)minus 119890120572ℎℎint119905119905minusℎ(119905) (119904)119879119882 (119904) 119889119904

minus 119890120572ℎℎint119905minusℎ(119905)119905minusℎ

(119904)119879119882 (119904) 119889119904le 1205721198813 (119905) + ℎ2 (119905)119879119882 (119905)minus 119890120572ℎ ℎℎ (119905)1206031119879Λ 11206031 minus 119890120572ℎ ℎℎ minus ℎ (119905)1206032119879Λ 21206032(32)

where 1206031119879 = 120585119879 (119905)( (1198971119879 minus 1198973119879)119879(1198971119879 + 1198973119879 minus 21198976119879)119879)119879

1206032119879 = 120585119879 (119905)( (1198973119879 minus 1198972119879)119879(1198973119879 + 1198972119879 minus 21198977119879))119879

Λ 1 = Λ 2 = (119882 0lowast 3119882) (33)

In order to obtain a tighter bound of integral termLemma 11 is applied to (32) as follows3 (119905) le 1205721198813 (119905) + ℎ2 (119905)119879119882 (119905) minus 119890120572ℎ1206033119879Λ 31206033 (34)

where 1206033119879 = ( 12060311198791206032119879 ) Λ 3 = (119882 1198851 0 0

lowast 119882 0 0lowast lowast 3119882 1198852lowast lowast lowast 3119882

)By applying Lemma 10 to (31) we can obtain the inequal-

ity 4 (119905) le 1205721198814 (119905) + 12ℎ2 (119905)1198791198801 (119905)+ 12ℎ2 (119905)1198791198802 (119905) minus 119890120572ℎ1206034119879Λ 41206034 (35)

where

1206034119879 = 120585119879 (119905)(((((

(1198971119879 minus 1198972119879)119879(12 1198971119879 minus 1198978119879)119879(1198972119879 minus 1198975119879)119879(12 1198972119879 minus 1198975119879 + 1198978119879)119879)))))

119879

Λ 4 =(61198801 minus121198801 0 0lowast 361198801 0 0lowast lowast 61198802 minus121198802lowast lowast lowast 361198802)(36)

Combining (28) (29) (34) and (35) with the definitionof 120588 = 119890120572ℎ we can obtain minus 120572119881 minus 120573120596119879 (119905)119872120596 (119905) lt 120585119879 (119905) Ω120585 (119905) (37)

Assuming that Ω lt 0 and integrating the left part ofinequality (37)119881 (119905) lt 119890120572119905119881 (0) + 120573int119905

0119890120572(119905minus119904)120596119879 (119904)119872120596 (119904) 119889119904lt 119890120572119879 (119881 (0) + 120573120582max (119872) 119889) (38)

Invoking the Jensen inequality shown in Lemma 12 onehas 119881 (0) le 119909 (0)119879 11987511119909 (0) + ℎint0

minusℎ119909 (119904)119879 11987522119909 (119904) 119889119904+ 120588int0

minusℎ119909 (119904)1198791198761119909 (119904) 119889119904+ 120588int0

minusℎ(119905)119909 (119904)1198791198762119909 (119904) 119889119904

Mathematical Problems in Engineering 7

+ ℎ120588int0minusℎint0120579 (119904)119879119882 (119904) 119889119904119889120579

+ 120588int0minusℎint0120579int0120573 (119904)1198791198801 (119904) 119889119904119889120573119889120579

+ 120588int0minusℎint120579minusℎint0120573 (119904)1198791198802 (119904) 119889119904119889120573119889120579le 120582max (11) 1198881 + ℎ120582max (22) ℎ1198881+ 120588120582max (1198761) ℎ1198881 + 120588120582max (1198762) ℎ1198881+ 120588120582max () 12ℎ31198881 + 120588120582max (1) 16ℎ31198881+ 120588120582max (2) 13ℎ31198881

(39)

Based on inequalities (38) and (39) we have119881 (119905) gt 119909119879 (119905) 11987511119909 (119905) = 119909119879 (119905) 119877121111987712119909 (119905)gt 120582min (11) 119909119879 (119905) 119877119909 (119905) (40)

Therefore conditions (17) to (20) can guarantee the FTBof system (4) This completes the proof

Remark 14 Among the existing approaches there are twothreads one is to construct a novel LKLF that involves moreinformation of delay the other is to find a tighter estimationof upper bound for cross terms coming from the derivativeof the LKLF In this paper these two techniques are appliedto obtain the result In addition the less conservatism inpractical engineering will be shown in the simulations whichare always ignored in most literature sources

32 Controller Design In this subsection we focus on theproblem of finite-time 119867infin state feedback designing basedon Theorem 13 that is designing a state feedback controllerin the form of (7) such that the resulting DP system satisfiesthe control objective proposed in Section 1

Theorem 15 For given positive 120572 1198881 1198882 119879 120574 and matrices 119877 gt0 119878 ge 0 if there exist positive scalars 120579119897 (119897 = 1 2 sdot sdot sdot 8) andmatrices 11987511 gt 0 11987522 gt 0119882 gt 0 119876119894 gt 0 (119894 = 1 2) 119880119895 gt0 (119895 = 1 2) 119884 1198851 1198852 with appropriate dimensions satisfyingthe following conditions Π = (Π119894119895)12times12 lt 0 (41)Σ = (Σ119894119895)9times9 lt 0 (42)

(119882 1198851lowast 119882) gt 0 (43)

(3119882 1198852lowast 3119882) gt 0 (44)

119868 minus 120574211987811987511 lt 0 (45)0 lt 1198671 lt 11987511 lt 11986720 lt 11987522 lt 11986730 lt 1198761 lt 11986740 lt 1198762 lt 11986750 lt 119882 lt 11986760 lt 1198801 lt 11986770 lt 1198802 lt 1198678(46)

where Π11 = 11987511119860119879 + 11986011987511 minus 12057211987511 + 1198761 + 1198762 minus 4120588119882minus 31205881198801 minus 91205881198802Π44 = minus1205742119868Π22 = minus1205881198762 minus 4120588119882 minus 61205881198802Π33 = minus8120588119882 + 1205881198851119879 + 1205881198851 minus 1205881198852119879 minus 1205881198852minus (1 minus 120583)1198762Π13 = 1198611119884 minus 2120588119882 minus 1205881198851 minus 1205881198852Π14 = 1198612Π66 = minus12120588119882Π77 = minus12120588119882Π55 = 120572ℎ211987522 minus 181205881198802 minus 61205881198801Π88 = minus361205881198801 minus 361205881198802Π12 = 1205881198851 minus 1205881198852 + 61205881198802Π15 = ℎ11987522 + 121205881198802Π16 = 6120588119882Π17 = 21205881198852Π25 = minusℎ11987522 minus 61205881198802Π18 = minus181205881198802 + 61205881198801Π26 = 21205881198852119879Π23 = minus1205881198851119879 minus 1205881198852119879 minus 2120588119882Π27 = 6120588119882Π28 = 121205881198802

8 Mathematical Problems in Engineering

Π36 = 6120588119882 + 21205881198852119879Π37 = 6120588119882 + 21205881198852Π58 = 121205881198801 + 241205881198802Π67 = minus41205881198852Π19 = 12ℎ11987511119860119879Π39 = 12ℎ1198841198791198611119879Π49 = 12ℎ1198612119879Π110 = 12ℎ11987511119860119879Π310 = 12ℎ1198841198791198611119879Π410 = 12ℎ1198612119879Π111 = ℎ11987511119860119879Π311 = ℎ1198841198791198611119879Π411 = ℎ1198612119879Π112 = 11987511119862119879Π312 = 119884119879119863119879Π99 = 121198801 minus 11987511Π1010 = 121198802 minus 11987511Π1111 = 119882 minus 211987511Π1212 = minus119890minus120572119879119868(47)

others Π119894119895 = 0Σ11 = minus119890minus12057211987911988821198671Σ22 = minus1198672Σ33 = minus1198673Σ44 = minus1198674Σ55 = minus1198675Σ66 = minus1198676Σ77 = minus1198677Σ88 = minus1198678

Σ99 = minus1205742119889Σ12 = radic11988811198672Σ13 = radicℎ211988811198673Σ14 = radic120588ℎ11988811198674Σ15 = radic120588ℎ11988811198675Σ16 = radic12120588ℎ311988811198676Σ17 = radic16120588ℎ311988811198677Σ18 = radic13120588ℎ311988811198678Σ19 = 11987511(48)

others Σ119894119895 = 0 (49)

then a state feedback119867infin controller in form of (7) existssuch that

(1) the closed-loop system (8) with 119906(119905) = 0 is FTB(2) under given nonzero initial condition the closed

system (8) and (9) guarantees that 119869 = int1198790119911119879(119904)119911(119904) minus1205742120596119879(119904)120596(119904) minus 1205742(1119879)119909119879(0)119878119909(0)119889119904 lt 0 for all nonzero 120596(119905)

and 119905 isin [0 119879]The controller gain can be calculated by 119870 = 11988411987511minus1

Proof In preparation for designing we set119872 = 1205742119868 120573 = 1for (37) then (37) and (38) can be rewritten as follows minus 120572119881 minus 1205742120596119879 (119905) 120596 (119905) lt 120585119879 (119905) Ω120585 (119905) (50)119881 (119905) lt 119890120572119905119881 (0) + 1205742 int119905

0119890120572(119905minus119904)120596119879 (119904) 120596 (119904) 119889119904lt 119890120572119879 (119881 (0) + 1205742119889) (51)

where Ω44 = 1198612119879 12ℎ2 (2119882 + 1198801 + 1198802) 1198612 minus 1205742119868Ω119894119895 = Ω119894119895

(52)

Then two steps are provided to design the desiredcontroller

Step 1 Here we will give some conditions that can guaranteethe 119867infin performance over the finite-time interval undernonzero initial conditions firstly

Mathematical Problems in Engineering 9

Consider the following inequality minus 120572119881 + 119890120572119905119911119879 (119905) 119911 (119905) minus 1205742119890120572119905120596119879 (119905) 120596 (119905)lt minus 120572119881 + 119890120572119879119911119879 (119905) 119911 (119905) minus 1205742120596119879 (119905) 120596 (119905)lt 120585119879 (119905) (Ω minus Ψ119879 (minus119890minus120572119879119868)minus1Ψ) 120585 (119905) (53)

Noting that119882 1198801 and 1198802 are coupled with 119860 1198611119870 and1198612 in Ω to decouple these terms the Ω can be rewritten asΩ = Ω minus Ψ119879 (minus121198801)minus1 Ψ minus Ψ119879 (minus121198802)minus1Ψminus Ψ119879 (minus119882)minus1 Ψ (54)

whereΩ11 = 11986011987911987511 + 11987511119860 minus 12057211987511 + 1198761 + 1198762 minus 4120588119882minus 31205881198801 minus 91205881198802Ω33 = minus8120588119882 + 1205881198851119879 + 1205881198851 minus 1205881198852119879 minus 1205881198852minus (1 minus 120583)1198762Ω44 = minus1205742119868Ω13 = 119875111198611119870 minus 2120588119882 minus 1205881198851 minus 1205881198852Ω14 = 119875111198612Ω34 = 0(55)

othersΩ119894119895 = Ω119894119895Ψ = (119862 0 119863119870 0 0 0 0 0)Ψ= (ℎ121198801119879119860 0 ℎ1211988011198791198611119870 ℎ1211988011198791198612 0 0 0 0) Ψ= (ℎ121198802119879119860 0 ℎ1211988021198791198611119870 ℎ1211988021198791198612 0 0 0 0)Ψ = (ℎ119882119879119860 0 ℎ1198821198791198611119870 ℎ1198821198791198612 0 0 0 0) (56)

By Schur complementΩ lt 0 and Ω minusΨ119879(minus119890minus120572119879119868)minus1Ψ areequivalent to

Π =(((((

Ω Ψ119879 Ψ119879 Ψ119879 Ψ119879lowast minus121198801 0 0 0lowast lowast minus121198802 0 0lowast lowast lowast minus119882 0lowast lowast lowast lowast minus119890minus120572119879119868)))))lt 0 (57)

To ensure the proposed conditions satisfy the 119867infin per-formance we set the inequality like

11987511 minus 1205742119878 lt 0 (58)

from which we have

minus 120572119881 + 119890120572119905119911119879 (119905) 119911 (119905) minus 1205742119890120572119905120596119879 (119905) 120596 (119905)minus 1205742119890120572119905 1119879119909119879 (0) 119878119909 (0) lt minus 120572119881 + 119890120572119905119911119879 (119905) 119911 (119905)minus 1205742119890120572119905120596119879 (119905) 120596 (119905) minus 1205742119890120572119905 1119879119909119879 (0) 119878119909 (0)+ 119890120572119905 1119879119909119879 (0) 11987511119909 (0) lt minus 120572119881 + 119890120572119905119911119879 (119905) 119911 (119905)minus 1205742119890120572119905120596119879 (119905) 120596 (119905) + 119890120572119905 1119879119909119879 (0) (11987511 minus 1205742119878) 119909 (0)lt 0(59)

Integrating (58) from 0 to 119879 we can obtain

int1198790119911119879 (119904) 119911 (119904) 119889119904 minus int119879

01205742120596119879 (119904) 120596 (119904) 119889119904

minus 1205742119909119879 (0) 119878119909 (0) lt minusint1198790(119890minus120572119905 minus 120572119890minus120572119905119881)119889119904 lt 0 (60)

so conditions (43) (44) (57) and (58) with the inequali-ties as follows can ensure that systems (17) and (18) satisfy the119867infin performance

11988811205792 + ℎ211988811205793 + 120588ℎ11988811205794 + 120588ℎ11988811205795 + 12120588ℎ311988811205796+ 16120588ℎ311988811205797 + 13120588ℎ311988811205798 + 1205742119889 lt 119890minus120572119879119888212057910 lt 1205791119868 lt 11987511 lt 12057921198680 lt 11987522 lt 12057931198680 lt 1198761 lt 12057941198680 lt 1198762 lt 12057951198680 lt 119882 lt 12057961198680 lt 1198801 lt 12057971198680 lt 1198802 lt 1205798119868

(61)

10 Mathematical Problems in Engineering

Step 2 In this step the approach to design the gain matrix 119870of the controller is provided in forms of LMIs which can becalculated expediently

To linearize the nonlinear terms in (57) we make itmultiply by the followingmatrix fromboth left and right sidesfirstly

diag 11987511minus1 11987511minus1 11987511minus1 119868 11987511minus1 11987511minus1 11987511minus1 11987511minus1 1198801minus1 1198802minus1 119882minus1 119868 (62)

Let us do the definitions as follows11987511 = 11987511minus11198761 = 11987511minus1119876111987511minus11198762 = 11987511minus1119876211987511minus1119882 = 11987511minus111988211987511minus11198801 = 11987511minus1119880111987511minus11198802 = 11987511minus1119880211987511minus111987522 = 11987511minus11198752211987511minus11198851 = 11987511minus1119885111987511minus11198852 = 11987511minus1119885211987511minus1119884 = 11987011987511minus1

(63)

so the inequality Π lt 0 can guarantee Π lt 0 whereΠ99 = minus12119875111198801

minus111987511Π1010 = minus12119875111198802

minus111987511Π1111 = minus11987511119882minus111987511(64)

others Π119894119895 = Π119894119895 (65)

Then based on the inequalities minus(12)119875111198801

minus111987511 le(12)1198801 minus 11987511 minus(12)119875111198802

minus111987511 le (12)1198802 minus 11987511minus11987511119882minus111987511 le 119882 minus 211987511 condition (57) can be convertedinto (41) in form of LMIs

Accordingly similar to the above-mentioned procedures(61) can be transformed into (42) with the Schur complementand the definition of119867119894 = 11987511minus112057911989411987511minus1 (119894 = 1 2 sdot sdot sdot 8)

Therefore conditions (41) to (45) can obtain the con-troller in the form of (7) guaranteeing the desired perfor-mance This completes the proof

Remark 16 The119867infin performance with nonzero initial con-dition is studied in this literature In a practical view it is

hard to suffer the worst-case over a finite time for surfacevessel with nonzero initial condition and it will reducethe conservatism caused by the assumption of zero initialcondition Therefore it is reasonable to introduce such 119867infinperformance index to handle the problem of finite-time119867infincontrol for ship subject to exogenous signals

Remark 17 It is challenging to achieve the result satisfyingthe control objective while it is very different from thestandard 119867infin control because a weighting diagonal matrixis introduced especially to establish the form of LMIs Inthe proof of Theorem 15 inequality (58) is introduced toensure the 119867infin performance with nonzero initial conditionand make it convenient to construct the conditions in formof LMIs

Remark 18 By introducing some new additional matrixvariables Cone Complementarity Linearization algorithm(CCL) [32] can be utilized to linearize the nonlinear termsas an alternative method It will reduce the conservatism atthe expense of increasing the computational burden and thenumber of decision variables

4 Simulations

41 Simulation Settings To demonstrate the effectiveness andsuperiority of the robust finite-time 119867infin control schemesimulation studies are conducted on the parameters of theship CyberShip II [33] which is a 170 scale-replica of a supplyshipwhosemain parameters are shown inTable 1 In additionwe choose the time-varying function like ℎ(119905) = 3sin (1199056 minus1205876) to describe the time delay occurred in the input signal

Unknown external disturbances 120596(119905) with significantlylarge magnitude and high frequency which can be roughlyconsidered as complex environment including ocean windswaves and currents are governed by [34]

120596 (119905) = (9 sin (01120587119905 minus 1205875 )6 sin (03120587119905 + 1205876 )3 sin (02120587119905 + 1205873 )) (66)

with initial conditions 120578(0) = (1 1 01) and 120592(0) =(08 06 01)42 Numerical Simulations We choose 120574 = 13 119879 = 10 119878 =03lowast119868 1198881 = 302 1198882 = 10 120572 = 0001 By usingTheorem 15 thecontrol gain 119870 is calculated by LMIs (41) to (45)

Mathematical Problems in Engineering 11

Table 1 The main parameters of the CyberShip II119898 238000 119883 -20 119883119906 -07225119868119911 17600 119884V -100 119884V -08612119909119892 00460 119884 119903 -00 119884119903 01079119873 119903 -10 119873119903 05

0 20 40 60 80 100Time(s) Time(s)

minus2

0

2

4

6

8

10

12

0 2 4 6 8 100

2

4

6

8

10

12

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

xT(t)Rx(t)xT(t)Rx(t)

c=c=

c1=302

c1=302

Figure 1 State response 119909119879(119905)119877119909(119905) of DP system

119870 = (minus32743 0 0 minus313907 0 00 minus42784 minus01572 0 minus412136 minus129820 minus01389 minus03497 0 minus13362 minus33091) (67)

To demonstrate the superiority and validity three sce-narios as comparative cases are provided here It should bepointed out that the results within 100s are also provided toillustrate the steady-state performance

Case 1 (with different 119878) For comparison purposes we set119878 with different value to demonstrate the effect of 119878 on stateresponse Without loss of generality we set 119878 = 05 lowast 119868 and119878 = 01 lowast 119868 respectivelyCase 2 (119867infin with zero initial condition]) This scenario willshow that the control law proposed in this paper can achievea better transient performance and disturbance rejectioncapability over traditional119867infin control approach for dynamicpositioning system conducted in [11]

Case 3 (a control design based on existing methods) Basedon the approach in [17] applied for discrete-time system acontrol law for dynamic positioning ship with input delay isproposed to illustrate the less conservatism of our designingapproach in practical engineering subsequently

At first the effectiveness of the control law we designedis confirmed from Figures 1ndash6 Figure 1 shows the state

response of 119909119879(119905)119877119909(119905) Obviously from that curve we cansee that the value of 119909119879(119905)119877119909(119905) is far below the value of1198882 = 10 in the presence of disturbance as in Figure 2which means the finite-time full-state feedback controllerdesigned can guarantee the DP system (8) sim (9) FTB with(302 10 119889 10 3 119868) and have less conservatism where weset 119889 = int119879

0120596(119905)119879120596(119905)119889119905 Meanwhile the control law can

stabilize the dynamic positioning shipwithminimum steady-state error in 100s and the control input will be small enoughas shown in Figure 6 From Figures 3ndash5 we can find that allstates in DP system are within a small range As shown inFigures 2ndash4 the angle and the angular velocity in yaw havechanged within 01∘ which means that the heading of thevessel stays in a fixed direction Furthermore the velocitiesin surge and sway are low enough which illustrates thatassumption 2 is reasonable and it meets the requirement ofstabilization control for DP system

To demonstrate the effect of 119878 state responses of systemswith different 119878 are exhibited in Figures 7ndash9 As the valueof 119878 decreases the restriction for 11987511 is more exclusive andthe robustness is stronger during preset time interval as inFigures 7 and 8 while the required control input will increasedramatically as in Figure 9 When the time oversteps the

12 Mathematical Problems in Engineering

0 10 20 30 40 50 60 70 80 90 100Time(s)

minus10

minus5

0

5

10

Dist

urba

nces

(N)

XY

Figure 2 Disturbance in surge sway and yaw

0 50 100Time(s)

minus050

051

152

X(m

)

0 5 10Time(s)

005

115

2

X(m

)

0 50 100Time(s)

minus050

051

152

Y(m

)

0 5 10Time(s)

005

115

2Y(

m)

0 50 100Time(s)

minus01

0

01

02

0 5 10Time(s)

0

01

02

(∘

)

(∘

)

Figure 3 State responses of position and orientation for DP System

0 50 100Time(s)

minus1

0

1

2

0 5 10Time(s)

minus1

0

1

2

0 50 100Time(s)

minus1

0

1

2

v(m

s)

0 5 10Time(s)

minus1

0

1

2

v(m

s)

0 50 100Time(s)

minus01

0

01

02

0 5 10Time(s)

minus01

0

01

02

r(∘ s

)

r(∘ s

)

(ms

)

(m

s)

Figure 4 State responses of velocities for DP system

Mathematical Problems in Engineering 13

minus05 0 05 1 15 2X(m)

minus05

0

05

1

15

2

Y(m

)

Figure 5 Trajectory of the shiprsquos horizontal motion

0 50 100Time(s)

minus30minus20minus10

010

u1(N

)

0 5 10Time(s)

minus30minus20minus10

010

u1(N

)

0 50 100Time(s)

minus30minus20minus10

010

u2(N

)

0 5 10Time(s)

minus30minus20minus10

010

u2(N

)

0 50 100Time(s)

minus15minus1

minus050

05

u3(N

m)

0 5 10Time(s)

minus15minus1

minus050

05

u3(N

m)

Figure 6 Control inputs

0 20 40 60 80 100Time(s)

minus1

0

1

2

3

4

5

6

S=03S=05S=01

0 2 4 6 8 10Time(s)

0

2

4

6

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 7 State response 119909119879(119905)119877119909(119905) with different 119878

14 Mathematical Problems in Engineering

Figure 8 State responses of position and orientation with different 119878

0 50 100Time(s)

minus60minus30

03060

u1(N

)

0 5 10Time(s)

minus40minus20

02040

u1(N

)

0 50 100Time(s)

minus100

0

100

u2(N

)

0 5 10Time(s)

minus40minus20

02040

u2(N

)

0 50 100Time(s)

minus4minus2

024

u3(N

m)

0 5 10Time(s)

minus2

0

2

u3(N

m)

S=03S=05S=01

Figure 9 Control input with different 119878preset time interval all the states can be guaranteed withinthe given upper bound as before

As shown in Figures 10ndash12 the overconservatism oftraditional119867infin control approach applied inDP ship is solvedby the 119867infin control strategy with nonzero initial conditionproposed in this article As is well known the classical 119867infincontrol law is designed under the so-called worst-case normwhile the performance of antidisturbance and transient issacrificed to some degree From Figures 10 and 11 we cansee that the state response can be controlled in a smallerrange and stabilized in a shorter time with a lower overshootshown in Figure 12 Hence a conclusion can be drawn thatthe control law we present can achieve a better disturbance

rejection capability and transient performance while thematter of overconservatism can be resolved at the sametime

Simulation results of Theorem 15 and Case 3 are shownin Figures 13 and 14 to show the better performance ofthe method we proposed We can find that the controllerbased on [17] and 119867infin control approach present in thisliterature can make all states of system within a smallenough region However compared with the controller weconstructed the state response has higher overshoot whichwill affect the control precision and transient performance inpractical engineering Based on the above simulation resultsthe developed controller can achieve a higher accuracy and

Mathematical Problems in Engineering 15

0 20 40 60 80 100Time(s)

minus2

0

2

4

6

8

10

12

Theorem 15Case 2

0 2 4 6 8 10Time(s)

minus2

0

2

4

6

8

10

12

Theorem 15Case 2

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 10 State response 119909119879(119905)119877119909(119905) of Theorem 15 and Case 2

0 50 100Time(s)

minus10123

X(m

)

0 5 10Time(s)

0123

X(m

)

0 50 100Time(s)

minus10123

Y(m

)

0 5 10Time(s)

0123

Y(m

)

0 50 100Time(s)

minus020

0204

0 5 10Time(s)

minus020

0204

Theorem 15Case 2

(∘

)

(∘

)

Figure 11 Position and orientation of Theorem 15 and Case 2

minus05 0 05 1 15 2 25X(m)

minus050

051

152

25

Y(m

)

Theorem 15Case 2

Figure 12 Trajectory of the shiprsquos horizontal motion of Theorem 15 and Case 2

16 Mathematical Problems in Engineering

0 20 40 60 80 100Time(s)

0

1

2

3

4

5

Theorem 15Case 3

0 2 4 6 8 10Time(s)

0

1

2

3

4

5

Theorem 15Case 3

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 13 State response 119909119879(119905)119877119909(119905) of Theorem 15 and Case 3

0 5 10Time(s)

05

1

15

X(m

)

Theorem 15Case 3

0 5 10Time(s)

05

1

15

Y(m

)

0 5 10Time(s)

0

005

01

015

02

(∘

)

Figure 14 Position and orientation of Theorem 15 and Case 3

lower overshoot which means that it is less conservative incomparison with existing result

5 Conclusion

In this paper the robust finite-time 119867infin control with tran-sients for DP system with input delay maintaining all statesof ocean surface vessel into a given threshold over a fixed timeinterval in presence of time-varying disturbance is proposedBased on a novel augmented LKLF with triple integral termsand a method combining theWirtinger inequality and recip-rocally convex approach a less conservative result is derivedIn particular a119867infin performance index with nonzero initialcondition attenuating the overconservatism caused by theassumption of zero initial condition and enhancing thetransient performance of ship subject to external disturbanceis adoptedThe obtained result is formulated in terms of LMIswhich can be easily solved by the standard numerical soft-ware Finally a numerical simulation for a ship is proposed toverify the effectiveness and less conservatism of the controller

we designed In the future the output feedback control can bedeveloped for the ocean surface vessel to avoid the use of thevessel-frame surge sway and yaw velocities

Data Availability

All the data supporting the conclusions of the study have beenprovided in Simulations and readers can access these data in[33 34]

Conflicts of Interest

The authors declare that there are no conflicts of interestrelated to this paper

Acknowledgments

This research was partially supported by the National Sci-ence Technology Support Program of China (Project no51609046)

Mathematical Problems in Engineering 17

References

[1] M Chen B Jiang and R Cui ldquoActuator fault-tolerant controlof ocean surface vessels with input saturationrdquo InternationalJournal of Robust amp Nonlinear Control vol 26 no 3 pp 542ndash564 2016

[2] J Du X Hu and Y Sun ldquoRobust dynamic positioning of shipswith disturbances under input saturationrdquo Automatica vol 73pp 207ndash214 2016

[3] H Ashrafiuon K R Muske L C McNinch and R ASoltan ldquoSliding-mode tracking control of surface vesselsrdquo IEEETransactions on Industrial Electronics vol 55 no 11 pp 4004ndash4012 2008

[4] W Si and X Dong ldquoNeural prescribed performance controlfor uncertain marine surface vessels without accurate initialerrorsrdquoMathematical Problems in Engineering Art ID 261832311 pages 2017

[5] T D Nguyen A J Soslashrensen and S T Quek ldquoDesign of hybridcontroller for dynamic positioning from calm to extreme seaconditionsrdquo Automatica vol 43 no 5 pp 768ndash785 2007

[6] J Du X Hu H Liu and C L Chen ldquoAdaptive robust outputfeedback control for a marine dynamic positioning systembased on a high-gain observerrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 11 pp 2775ndash27862015

[7] H Katayama ldquoNonlinear sampled-data stabilization of dynam-ically positioned shipsrdquo IEEE Transactions on Control SystemsTechnology vol 18 no 2 pp 463ndash468 2010

[8] K J Astrom andCG Kallstrom ldquoIdentification of ship steeringdynamicsrdquo Automatica vol 12 no 1 pp 9ndash22 1976

[9] Z Lei and C Guo ldquoDisturbance rejection control solutionfor ship steering system with uncertain time delayrdquo OceanEngineering vol 95 pp 78ndash83 2015

[10] J C Doyle K Glover P P Khargonekar and B A FrancisldquoState-space solutions to standard H-2 and Hinfin control prob-lemsrdquo IEEE Transactions on Automatic Control vol 34 no 8pp 831ndash847 1989

[11] M Katebi M Grimble and Y Zhang ldquoHinfin robust controldesign for dynamic ship positioningrdquo IEE Proceedings - ControlTheory and Applications vol 144 no 2 pp 110ndash120 1997

[12] D V Balandin and M M Kogan ldquoLMI-based Hinfin-optimalcontrol with transientsrdquo International Journal of Control vol 83no 8 pp 1664ndash1673 2010

[13] Z Feng J Lam S Xu and S Zhou ldquoHinfin Control withTransients for Singular Systemsrdquo Asian Journal of Control vol18 no 3 pp 817ndash827 2016

[14] P P Khargonekar K M Nagpal and K R Poolla ldquoHinfin controlwith transientsrdquo SIAM Journal onControl andOptimization vol29 no 6 pp 1373ndash1393 1991

[15] Z Zhang Z Zhang and H Zhang ldquoFinite-time stabilityanalysis and stabilization for uncertain continuous-time systemwith time-varying delayrdquo Journal of The Franklin Institute vol352 no 3 pp 1296ndash1317 2015

[16] J Cheng S Chen Z Liu H Wang and J Li ldquoRobust finite-time sampled-data control of linear systems subject to randomoccurring delays and its application to Four-Tank systemrdquoAppliedMathematics andComputation vol 281 pp 55ndash76 2016

[17] W Kang S Zhong K Shi and J Cheng ldquoFinite-time stabilityfor discrete-time system with time-varying delay and nonlinearperturbationsrdquo ISA Transactions vol 60 pp 67ndash73 2016

[18] T I Fossen Handbook of Marine Craft Hydrodynamics andMotion Control John Wiley amp Sons New York USA 2011

[19] D Zhao F Ding L Zhou W Zhang and H Xu ldquoRobustH-infinity Control of Neutral System with Time-Delay forDynamic Positioning Shipsrdquo Mathematical Problems in Engi-neering vol 2015 Article ID 976925 11 pages 2015

[20] F Amato M Ariola and C Cosentino ldquoFinite-time stabilityof linear time-varying systems analysis and controller designrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 55 no 4 pp 1003ndash1008 2010

[21] L Sun andZ Zheng ldquoFinite-time slidingmode trajectory track-ing control of uncertain mechanical systemsrdquo Asian Journal ofControl vol 19 no 1 pp 399ndash404 2017

[22] P Shi Y Zhang and R Agarwal ldquoStochastic finite-time stateestimation for discrete time-delay neural networks withMarko-vian jumpsrdquo Neurocomputing vol 151 part 1 pp 168ndash174 2014

[23] L Zhang S Wang H R Karimi and A Jasra ldquoRobust Finite-Time Control of Switched Linear Systems and Application toa Class of Servomechanism Systemsrdquo IEEEASME Transactionson Mechatronics vol 20 no 5 pp 2476ndash2485 2015

[24] L Wu P Shi and H Gao ldquoState estimation and sliding-mode control of Markovian jump singular systemsrdquo Institute ofElectrical and Electronics Engineers Transactions on AutomaticControl vol 55 no 5 pp 1213ndash1219 2010

[25] Y Q Zhang P Shi S K Nguang and Y D Song ldquoRobust finite-time Hinfin control for uncertain discrete-time singular systemswithMarkovian jumpsrdquo IET ControlTheory ampApplications vol8 no 12 pp 1105ndash1111 2014

[26] S Wang T Shi L Zhang A Jasra and M Zeng ldquoExtendedfinite-time Hinfin control for uncertain switched linear neutralsystemswith time-varying delaysrdquoNeurocomputing vol 152 pp377ndash387 2015

[27] J Hu G Sui S Du and X Li ldquoFinite-time stability of uncer-tain nonlinear systems with time-varying delayrdquo MathematicalProblems in Engineering Article ID 2538904 9 pages 2017

[28] A Seuret and F Gouaisbaut ldquoWirtinger-based integral inequal-ity application to time-delay systemsrdquo Automatica vol 49 no9 pp 2860ndash2866 2013

[29] Z Li Y Bai C Huang and Y Cai ldquoNovel delay-partitioningstabilization approach for networked control system viaWirtinger-based inequalitiesrdquo ISA Transactions vol 61 pp75ndash86 2016

[30] P Park JWKo andC Jeong ldquoReciprocally convex approach tostability of systems with time-varying delaysrdquo Automatica vol47 no 1 pp 235ndash238 2011

[31] K Gu V L Kharitonov and J Chen Stability of Time DelaySystems Birkhauser Boston Mass USA 2003

[32] L El Ghaoui F Oustry and M AitRami ldquoA cone comple-mentarity linearization algorithm for static output-feedbackand related problemsrdquo Institute of Electrical and ElectronicsEngineers Transactions on Automatic Control vol 42 no 8 pp1171ndash1176 1997

[33] R Skjetne T I Fossen and P Kokotovic ldquoAdaptive maneu-vering with experiments for a model ship in a marine controllaboratoryrdquo Automatica vol 41 no 2 pp 289ndash298 2005

[34] N Wang C Qian J-C Sun and Y-C Liu ldquoAdaptive RobustFinite-Time Trajectory Tracking Control of Fully ActuatedMarine Surface Vehiclesrdquo IEEE Transactions on Control SystemsTechnology vol 24 no 4 pp 1454ndash1462 2016

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Page 7: Robust Finite-Time H-Infinity Control with Transients for ...downloads.hindawi.com/journals/mpe/2018/2838749.pdf · ResearchArticle Robust Finite-Time H-Infinity Control with Transients

Mathematical Problems in Engineering 7

+ ℎ120588int0minusℎint0120579 (119904)119879119882 (119904) 119889119904119889120579

+ 120588int0minusℎint0120579int0120573 (119904)1198791198801 (119904) 119889119904119889120573119889120579

+ 120588int0minusℎint120579minusℎint0120573 (119904)1198791198802 (119904) 119889119904119889120573119889120579le 120582max (11) 1198881 + ℎ120582max (22) ℎ1198881+ 120588120582max (1198761) ℎ1198881 + 120588120582max (1198762) ℎ1198881+ 120588120582max () 12ℎ31198881 + 120588120582max (1) 16ℎ31198881+ 120588120582max (2) 13ℎ31198881

(39)

Based on inequalities (38) and (39) we have119881 (119905) gt 119909119879 (119905) 11987511119909 (119905) = 119909119879 (119905) 119877121111987712119909 (119905)gt 120582min (11) 119909119879 (119905) 119877119909 (119905) (40)

Therefore conditions (17) to (20) can guarantee the FTBof system (4) This completes the proof

Remark 14 Among the existing approaches there are twothreads one is to construct a novel LKLF that involves moreinformation of delay the other is to find a tighter estimationof upper bound for cross terms coming from the derivativeof the LKLF In this paper these two techniques are appliedto obtain the result In addition the less conservatism inpractical engineering will be shown in the simulations whichare always ignored in most literature sources

32 Controller Design In this subsection we focus on theproblem of finite-time 119867infin state feedback designing basedon Theorem 13 that is designing a state feedback controllerin the form of (7) such that the resulting DP system satisfiesthe control objective proposed in Section 1

Theorem 15 For given positive 120572 1198881 1198882 119879 120574 and matrices 119877 gt0 119878 ge 0 if there exist positive scalars 120579119897 (119897 = 1 2 sdot sdot sdot 8) andmatrices 11987511 gt 0 11987522 gt 0119882 gt 0 119876119894 gt 0 (119894 = 1 2) 119880119895 gt0 (119895 = 1 2) 119884 1198851 1198852 with appropriate dimensions satisfyingthe following conditions Π = (Π119894119895)12times12 lt 0 (41)Σ = (Σ119894119895)9times9 lt 0 (42)

(119882 1198851lowast 119882) gt 0 (43)

(3119882 1198852lowast 3119882) gt 0 (44)

119868 minus 120574211987811987511 lt 0 (45)0 lt 1198671 lt 11987511 lt 11986720 lt 11987522 lt 11986730 lt 1198761 lt 11986740 lt 1198762 lt 11986750 lt 119882 lt 11986760 lt 1198801 lt 11986770 lt 1198802 lt 1198678(46)

where Π11 = 11987511119860119879 + 11986011987511 minus 12057211987511 + 1198761 + 1198762 minus 4120588119882minus 31205881198801 minus 91205881198802Π44 = minus1205742119868Π22 = minus1205881198762 minus 4120588119882 minus 61205881198802Π33 = minus8120588119882 + 1205881198851119879 + 1205881198851 minus 1205881198852119879 minus 1205881198852minus (1 minus 120583)1198762Π13 = 1198611119884 minus 2120588119882 minus 1205881198851 minus 1205881198852Π14 = 1198612Π66 = minus12120588119882Π77 = minus12120588119882Π55 = 120572ℎ211987522 minus 181205881198802 minus 61205881198801Π88 = minus361205881198801 minus 361205881198802Π12 = 1205881198851 minus 1205881198852 + 61205881198802Π15 = ℎ11987522 + 121205881198802Π16 = 6120588119882Π17 = 21205881198852Π25 = minusℎ11987522 minus 61205881198802Π18 = minus181205881198802 + 61205881198801Π26 = 21205881198852119879Π23 = minus1205881198851119879 minus 1205881198852119879 minus 2120588119882Π27 = 6120588119882Π28 = 121205881198802

8 Mathematical Problems in Engineering

Π36 = 6120588119882 + 21205881198852119879Π37 = 6120588119882 + 21205881198852Π58 = 121205881198801 + 241205881198802Π67 = minus41205881198852Π19 = 12ℎ11987511119860119879Π39 = 12ℎ1198841198791198611119879Π49 = 12ℎ1198612119879Π110 = 12ℎ11987511119860119879Π310 = 12ℎ1198841198791198611119879Π410 = 12ℎ1198612119879Π111 = ℎ11987511119860119879Π311 = ℎ1198841198791198611119879Π411 = ℎ1198612119879Π112 = 11987511119862119879Π312 = 119884119879119863119879Π99 = 121198801 minus 11987511Π1010 = 121198802 minus 11987511Π1111 = 119882 minus 211987511Π1212 = minus119890minus120572119879119868(47)

others Π119894119895 = 0Σ11 = minus119890minus12057211987911988821198671Σ22 = minus1198672Σ33 = minus1198673Σ44 = minus1198674Σ55 = minus1198675Σ66 = minus1198676Σ77 = minus1198677Σ88 = minus1198678

Σ99 = minus1205742119889Σ12 = radic11988811198672Σ13 = radicℎ211988811198673Σ14 = radic120588ℎ11988811198674Σ15 = radic120588ℎ11988811198675Σ16 = radic12120588ℎ311988811198676Σ17 = radic16120588ℎ311988811198677Σ18 = radic13120588ℎ311988811198678Σ19 = 11987511(48)

others Σ119894119895 = 0 (49)

then a state feedback119867infin controller in form of (7) existssuch that

(1) the closed-loop system (8) with 119906(119905) = 0 is FTB(2) under given nonzero initial condition the closed

system (8) and (9) guarantees that 119869 = int1198790119911119879(119904)119911(119904) minus1205742120596119879(119904)120596(119904) minus 1205742(1119879)119909119879(0)119878119909(0)119889119904 lt 0 for all nonzero 120596(119905)

and 119905 isin [0 119879]The controller gain can be calculated by 119870 = 11988411987511minus1

Proof In preparation for designing we set119872 = 1205742119868 120573 = 1for (37) then (37) and (38) can be rewritten as follows minus 120572119881 minus 1205742120596119879 (119905) 120596 (119905) lt 120585119879 (119905) Ω120585 (119905) (50)119881 (119905) lt 119890120572119905119881 (0) + 1205742 int119905

0119890120572(119905minus119904)120596119879 (119904) 120596 (119904) 119889119904lt 119890120572119879 (119881 (0) + 1205742119889) (51)

where Ω44 = 1198612119879 12ℎ2 (2119882 + 1198801 + 1198802) 1198612 minus 1205742119868Ω119894119895 = Ω119894119895

(52)

Then two steps are provided to design the desiredcontroller

Step 1 Here we will give some conditions that can guaranteethe 119867infin performance over the finite-time interval undernonzero initial conditions firstly

Mathematical Problems in Engineering 9

Consider the following inequality minus 120572119881 + 119890120572119905119911119879 (119905) 119911 (119905) minus 1205742119890120572119905120596119879 (119905) 120596 (119905)lt minus 120572119881 + 119890120572119879119911119879 (119905) 119911 (119905) minus 1205742120596119879 (119905) 120596 (119905)lt 120585119879 (119905) (Ω minus Ψ119879 (minus119890minus120572119879119868)minus1Ψ) 120585 (119905) (53)

Noting that119882 1198801 and 1198802 are coupled with 119860 1198611119870 and1198612 in Ω to decouple these terms the Ω can be rewritten asΩ = Ω minus Ψ119879 (minus121198801)minus1 Ψ minus Ψ119879 (minus121198802)minus1Ψminus Ψ119879 (minus119882)minus1 Ψ (54)

whereΩ11 = 11986011987911987511 + 11987511119860 minus 12057211987511 + 1198761 + 1198762 minus 4120588119882minus 31205881198801 minus 91205881198802Ω33 = minus8120588119882 + 1205881198851119879 + 1205881198851 minus 1205881198852119879 minus 1205881198852minus (1 minus 120583)1198762Ω44 = minus1205742119868Ω13 = 119875111198611119870 minus 2120588119882 minus 1205881198851 minus 1205881198852Ω14 = 119875111198612Ω34 = 0(55)

othersΩ119894119895 = Ω119894119895Ψ = (119862 0 119863119870 0 0 0 0 0)Ψ= (ℎ121198801119879119860 0 ℎ1211988011198791198611119870 ℎ1211988011198791198612 0 0 0 0) Ψ= (ℎ121198802119879119860 0 ℎ1211988021198791198611119870 ℎ1211988021198791198612 0 0 0 0)Ψ = (ℎ119882119879119860 0 ℎ1198821198791198611119870 ℎ1198821198791198612 0 0 0 0) (56)

By Schur complementΩ lt 0 and Ω minusΨ119879(minus119890minus120572119879119868)minus1Ψ areequivalent to

Π =(((((

Ω Ψ119879 Ψ119879 Ψ119879 Ψ119879lowast minus121198801 0 0 0lowast lowast minus121198802 0 0lowast lowast lowast minus119882 0lowast lowast lowast lowast minus119890minus120572119879119868)))))lt 0 (57)

To ensure the proposed conditions satisfy the 119867infin per-formance we set the inequality like

11987511 minus 1205742119878 lt 0 (58)

from which we have

minus 120572119881 + 119890120572119905119911119879 (119905) 119911 (119905) minus 1205742119890120572119905120596119879 (119905) 120596 (119905)minus 1205742119890120572119905 1119879119909119879 (0) 119878119909 (0) lt minus 120572119881 + 119890120572119905119911119879 (119905) 119911 (119905)minus 1205742119890120572119905120596119879 (119905) 120596 (119905) minus 1205742119890120572119905 1119879119909119879 (0) 119878119909 (0)+ 119890120572119905 1119879119909119879 (0) 11987511119909 (0) lt minus 120572119881 + 119890120572119905119911119879 (119905) 119911 (119905)minus 1205742119890120572119905120596119879 (119905) 120596 (119905) + 119890120572119905 1119879119909119879 (0) (11987511 minus 1205742119878) 119909 (0)lt 0(59)

Integrating (58) from 0 to 119879 we can obtain

int1198790119911119879 (119904) 119911 (119904) 119889119904 minus int119879

01205742120596119879 (119904) 120596 (119904) 119889119904

minus 1205742119909119879 (0) 119878119909 (0) lt minusint1198790(119890minus120572119905 minus 120572119890minus120572119905119881)119889119904 lt 0 (60)

so conditions (43) (44) (57) and (58) with the inequali-ties as follows can ensure that systems (17) and (18) satisfy the119867infin performance

11988811205792 + ℎ211988811205793 + 120588ℎ11988811205794 + 120588ℎ11988811205795 + 12120588ℎ311988811205796+ 16120588ℎ311988811205797 + 13120588ℎ311988811205798 + 1205742119889 lt 119890minus120572119879119888212057910 lt 1205791119868 lt 11987511 lt 12057921198680 lt 11987522 lt 12057931198680 lt 1198761 lt 12057941198680 lt 1198762 lt 12057951198680 lt 119882 lt 12057961198680 lt 1198801 lt 12057971198680 lt 1198802 lt 1205798119868

(61)

10 Mathematical Problems in Engineering

Step 2 In this step the approach to design the gain matrix 119870of the controller is provided in forms of LMIs which can becalculated expediently

To linearize the nonlinear terms in (57) we make itmultiply by the followingmatrix fromboth left and right sidesfirstly

diag 11987511minus1 11987511minus1 11987511minus1 119868 11987511minus1 11987511minus1 11987511minus1 11987511minus1 1198801minus1 1198802minus1 119882minus1 119868 (62)

Let us do the definitions as follows11987511 = 11987511minus11198761 = 11987511minus1119876111987511minus11198762 = 11987511minus1119876211987511minus1119882 = 11987511minus111988211987511minus11198801 = 11987511minus1119880111987511minus11198802 = 11987511minus1119880211987511minus111987522 = 11987511minus11198752211987511minus11198851 = 11987511minus1119885111987511minus11198852 = 11987511minus1119885211987511minus1119884 = 11987011987511minus1

(63)

so the inequality Π lt 0 can guarantee Π lt 0 whereΠ99 = minus12119875111198801

minus111987511Π1010 = minus12119875111198802

minus111987511Π1111 = minus11987511119882minus111987511(64)

others Π119894119895 = Π119894119895 (65)

Then based on the inequalities minus(12)119875111198801

minus111987511 le(12)1198801 minus 11987511 minus(12)119875111198802

minus111987511 le (12)1198802 minus 11987511minus11987511119882minus111987511 le 119882 minus 211987511 condition (57) can be convertedinto (41) in form of LMIs

Accordingly similar to the above-mentioned procedures(61) can be transformed into (42) with the Schur complementand the definition of119867119894 = 11987511minus112057911989411987511minus1 (119894 = 1 2 sdot sdot sdot 8)

Therefore conditions (41) to (45) can obtain the con-troller in the form of (7) guaranteeing the desired perfor-mance This completes the proof

Remark 16 The119867infin performance with nonzero initial con-dition is studied in this literature In a practical view it is

hard to suffer the worst-case over a finite time for surfacevessel with nonzero initial condition and it will reducethe conservatism caused by the assumption of zero initialcondition Therefore it is reasonable to introduce such 119867infinperformance index to handle the problem of finite-time119867infincontrol for ship subject to exogenous signals

Remark 17 It is challenging to achieve the result satisfyingthe control objective while it is very different from thestandard 119867infin control because a weighting diagonal matrixis introduced especially to establish the form of LMIs Inthe proof of Theorem 15 inequality (58) is introduced toensure the 119867infin performance with nonzero initial conditionand make it convenient to construct the conditions in formof LMIs

Remark 18 By introducing some new additional matrixvariables Cone Complementarity Linearization algorithm(CCL) [32] can be utilized to linearize the nonlinear termsas an alternative method It will reduce the conservatism atthe expense of increasing the computational burden and thenumber of decision variables

4 Simulations

41 Simulation Settings To demonstrate the effectiveness andsuperiority of the robust finite-time 119867infin control schemesimulation studies are conducted on the parameters of theship CyberShip II [33] which is a 170 scale-replica of a supplyshipwhosemain parameters are shown inTable 1 In additionwe choose the time-varying function like ℎ(119905) = 3sin (1199056 minus1205876) to describe the time delay occurred in the input signal

Unknown external disturbances 120596(119905) with significantlylarge magnitude and high frequency which can be roughlyconsidered as complex environment including ocean windswaves and currents are governed by [34]

120596 (119905) = (9 sin (01120587119905 minus 1205875 )6 sin (03120587119905 + 1205876 )3 sin (02120587119905 + 1205873 )) (66)

with initial conditions 120578(0) = (1 1 01) and 120592(0) =(08 06 01)42 Numerical Simulations We choose 120574 = 13 119879 = 10 119878 =03lowast119868 1198881 = 302 1198882 = 10 120572 = 0001 By usingTheorem 15 thecontrol gain 119870 is calculated by LMIs (41) to (45)

Mathematical Problems in Engineering 11

Table 1 The main parameters of the CyberShip II119898 238000 119883 -20 119883119906 -07225119868119911 17600 119884V -100 119884V -08612119909119892 00460 119884 119903 -00 119884119903 01079119873 119903 -10 119873119903 05

0 20 40 60 80 100Time(s) Time(s)

minus2

0

2

4

6

8

10

12

0 2 4 6 8 100

2

4

6

8

10

12

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

xT(t)Rx(t)xT(t)Rx(t)

c=c=

c1=302

c1=302

Figure 1 State response 119909119879(119905)119877119909(119905) of DP system

119870 = (minus32743 0 0 minus313907 0 00 minus42784 minus01572 0 minus412136 minus129820 minus01389 minus03497 0 minus13362 minus33091) (67)

To demonstrate the superiority and validity three sce-narios as comparative cases are provided here It should bepointed out that the results within 100s are also provided toillustrate the steady-state performance

Case 1 (with different 119878) For comparison purposes we set119878 with different value to demonstrate the effect of 119878 on stateresponse Without loss of generality we set 119878 = 05 lowast 119868 and119878 = 01 lowast 119868 respectivelyCase 2 (119867infin with zero initial condition]) This scenario willshow that the control law proposed in this paper can achievea better transient performance and disturbance rejectioncapability over traditional119867infin control approach for dynamicpositioning system conducted in [11]

Case 3 (a control design based on existing methods) Basedon the approach in [17] applied for discrete-time system acontrol law for dynamic positioning ship with input delay isproposed to illustrate the less conservatism of our designingapproach in practical engineering subsequently

At first the effectiveness of the control law we designedis confirmed from Figures 1ndash6 Figure 1 shows the state

response of 119909119879(119905)119877119909(119905) Obviously from that curve we cansee that the value of 119909119879(119905)119877119909(119905) is far below the value of1198882 = 10 in the presence of disturbance as in Figure 2which means the finite-time full-state feedback controllerdesigned can guarantee the DP system (8) sim (9) FTB with(302 10 119889 10 3 119868) and have less conservatism where weset 119889 = int119879

0120596(119905)119879120596(119905)119889119905 Meanwhile the control law can

stabilize the dynamic positioning shipwithminimum steady-state error in 100s and the control input will be small enoughas shown in Figure 6 From Figures 3ndash5 we can find that allstates in DP system are within a small range As shown inFigures 2ndash4 the angle and the angular velocity in yaw havechanged within 01∘ which means that the heading of thevessel stays in a fixed direction Furthermore the velocitiesin surge and sway are low enough which illustrates thatassumption 2 is reasonable and it meets the requirement ofstabilization control for DP system

To demonstrate the effect of 119878 state responses of systemswith different 119878 are exhibited in Figures 7ndash9 As the valueof 119878 decreases the restriction for 11987511 is more exclusive andthe robustness is stronger during preset time interval as inFigures 7 and 8 while the required control input will increasedramatically as in Figure 9 When the time oversteps the

12 Mathematical Problems in Engineering

0 10 20 30 40 50 60 70 80 90 100Time(s)

minus10

minus5

0

5

10

Dist

urba

nces

(N)

XY

Figure 2 Disturbance in surge sway and yaw

0 50 100Time(s)

minus050

051

152

X(m

)

0 5 10Time(s)

005

115

2

X(m

)

0 50 100Time(s)

minus050

051

152

Y(m

)

0 5 10Time(s)

005

115

2Y(

m)

0 50 100Time(s)

minus01

0

01

02

0 5 10Time(s)

0

01

02

(∘

)

(∘

)

Figure 3 State responses of position and orientation for DP System

0 50 100Time(s)

minus1

0

1

2

0 5 10Time(s)

minus1

0

1

2

0 50 100Time(s)

minus1

0

1

2

v(m

s)

0 5 10Time(s)

minus1

0

1

2

v(m

s)

0 50 100Time(s)

minus01

0

01

02

0 5 10Time(s)

minus01

0

01

02

r(∘ s

)

r(∘ s

)

(ms

)

(m

s)

Figure 4 State responses of velocities for DP system

Mathematical Problems in Engineering 13

minus05 0 05 1 15 2X(m)

minus05

0

05

1

15

2

Y(m

)

Figure 5 Trajectory of the shiprsquos horizontal motion

0 50 100Time(s)

minus30minus20minus10

010

u1(N

)

0 5 10Time(s)

minus30minus20minus10

010

u1(N

)

0 50 100Time(s)

minus30minus20minus10

010

u2(N

)

0 5 10Time(s)

minus30minus20minus10

010

u2(N

)

0 50 100Time(s)

minus15minus1

minus050

05

u3(N

m)

0 5 10Time(s)

minus15minus1

minus050

05

u3(N

m)

Figure 6 Control inputs

0 20 40 60 80 100Time(s)

minus1

0

1

2

3

4

5

6

S=03S=05S=01

0 2 4 6 8 10Time(s)

0

2

4

6

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 7 State response 119909119879(119905)119877119909(119905) with different 119878

14 Mathematical Problems in Engineering

Figure 8 State responses of position and orientation with different 119878

0 50 100Time(s)

minus60minus30

03060

u1(N

)

0 5 10Time(s)

minus40minus20

02040

u1(N

)

0 50 100Time(s)

minus100

0

100

u2(N

)

0 5 10Time(s)

minus40minus20

02040

u2(N

)

0 50 100Time(s)

minus4minus2

024

u3(N

m)

0 5 10Time(s)

minus2

0

2

u3(N

m)

S=03S=05S=01

Figure 9 Control input with different 119878preset time interval all the states can be guaranteed withinthe given upper bound as before

As shown in Figures 10ndash12 the overconservatism oftraditional119867infin control approach applied inDP ship is solvedby the 119867infin control strategy with nonzero initial conditionproposed in this article As is well known the classical 119867infincontrol law is designed under the so-called worst-case normwhile the performance of antidisturbance and transient issacrificed to some degree From Figures 10 and 11 we cansee that the state response can be controlled in a smallerrange and stabilized in a shorter time with a lower overshootshown in Figure 12 Hence a conclusion can be drawn thatthe control law we present can achieve a better disturbance

rejection capability and transient performance while thematter of overconservatism can be resolved at the sametime

Simulation results of Theorem 15 and Case 3 are shownin Figures 13 and 14 to show the better performance ofthe method we proposed We can find that the controllerbased on [17] and 119867infin control approach present in thisliterature can make all states of system within a smallenough region However compared with the controller weconstructed the state response has higher overshoot whichwill affect the control precision and transient performance inpractical engineering Based on the above simulation resultsthe developed controller can achieve a higher accuracy and

Mathematical Problems in Engineering 15

0 20 40 60 80 100Time(s)

minus2

0

2

4

6

8

10

12

Theorem 15Case 2

0 2 4 6 8 10Time(s)

minus2

0

2

4

6

8

10

12

Theorem 15Case 2

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 10 State response 119909119879(119905)119877119909(119905) of Theorem 15 and Case 2

0 50 100Time(s)

minus10123

X(m

)

0 5 10Time(s)

0123

X(m

)

0 50 100Time(s)

minus10123

Y(m

)

0 5 10Time(s)

0123

Y(m

)

0 50 100Time(s)

minus020

0204

0 5 10Time(s)

minus020

0204

Theorem 15Case 2

(∘

)

(∘

)

Figure 11 Position and orientation of Theorem 15 and Case 2

minus05 0 05 1 15 2 25X(m)

minus050

051

152

25

Y(m

)

Theorem 15Case 2

Figure 12 Trajectory of the shiprsquos horizontal motion of Theorem 15 and Case 2

16 Mathematical Problems in Engineering

0 20 40 60 80 100Time(s)

0

1

2

3

4

5

Theorem 15Case 3

0 2 4 6 8 10Time(s)

0

1

2

3

4

5

Theorem 15Case 3

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 13 State response 119909119879(119905)119877119909(119905) of Theorem 15 and Case 3

0 5 10Time(s)

05

1

15

X(m

)

Theorem 15Case 3

0 5 10Time(s)

05

1

15

Y(m

)

0 5 10Time(s)

0

005

01

015

02

(∘

)

Figure 14 Position and orientation of Theorem 15 and Case 3

lower overshoot which means that it is less conservative incomparison with existing result

5 Conclusion

In this paper the robust finite-time 119867infin control with tran-sients for DP system with input delay maintaining all statesof ocean surface vessel into a given threshold over a fixed timeinterval in presence of time-varying disturbance is proposedBased on a novel augmented LKLF with triple integral termsand a method combining theWirtinger inequality and recip-rocally convex approach a less conservative result is derivedIn particular a119867infin performance index with nonzero initialcondition attenuating the overconservatism caused by theassumption of zero initial condition and enhancing thetransient performance of ship subject to external disturbanceis adoptedThe obtained result is formulated in terms of LMIswhich can be easily solved by the standard numerical soft-ware Finally a numerical simulation for a ship is proposed toverify the effectiveness and less conservatism of the controller

we designed In the future the output feedback control can bedeveloped for the ocean surface vessel to avoid the use of thevessel-frame surge sway and yaw velocities

Data Availability

All the data supporting the conclusions of the study have beenprovided in Simulations and readers can access these data in[33 34]

Conflicts of Interest

The authors declare that there are no conflicts of interestrelated to this paper

Acknowledgments

This research was partially supported by the National Sci-ence Technology Support Program of China (Project no51609046)

Mathematical Problems in Engineering 17

References

[1] M Chen B Jiang and R Cui ldquoActuator fault-tolerant controlof ocean surface vessels with input saturationrdquo InternationalJournal of Robust amp Nonlinear Control vol 26 no 3 pp 542ndash564 2016

[2] J Du X Hu and Y Sun ldquoRobust dynamic positioning of shipswith disturbances under input saturationrdquo Automatica vol 73pp 207ndash214 2016

[3] H Ashrafiuon K R Muske L C McNinch and R ASoltan ldquoSliding-mode tracking control of surface vesselsrdquo IEEETransactions on Industrial Electronics vol 55 no 11 pp 4004ndash4012 2008

[4] W Si and X Dong ldquoNeural prescribed performance controlfor uncertain marine surface vessels without accurate initialerrorsrdquoMathematical Problems in Engineering Art ID 261832311 pages 2017

[5] T D Nguyen A J Soslashrensen and S T Quek ldquoDesign of hybridcontroller for dynamic positioning from calm to extreme seaconditionsrdquo Automatica vol 43 no 5 pp 768ndash785 2007

[6] J Du X Hu H Liu and C L Chen ldquoAdaptive robust outputfeedback control for a marine dynamic positioning systembased on a high-gain observerrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 11 pp 2775ndash27862015

[7] H Katayama ldquoNonlinear sampled-data stabilization of dynam-ically positioned shipsrdquo IEEE Transactions on Control SystemsTechnology vol 18 no 2 pp 463ndash468 2010

[8] K J Astrom andCG Kallstrom ldquoIdentification of ship steeringdynamicsrdquo Automatica vol 12 no 1 pp 9ndash22 1976

[9] Z Lei and C Guo ldquoDisturbance rejection control solutionfor ship steering system with uncertain time delayrdquo OceanEngineering vol 95 pp 78ndash83 2015

[10] J C Doyle K Glover P P Khargonekar and B A FrancisldquoState-space solutions to standard H-2 and Hinfin control prob-lemsrdquo IEEE Transactions on Automatic Control vol 34 no 8pp 831ndash847 1989

[11] M Katebi M Grimble and Y Zhang ldquoHinfin robust controldesign for dynamic ship positioningrdquo IEE Proceedings - ControlTheory and Applications vol 144 no 2 pp 110ndash120 1997

[12] D V Balandin and M M Kogan ldquoLMI-based Hinfin-optimalcontrol with transientsrdquo International Journal of Control vol 83no 8 pp 1664ndash1673 2010

[13] Z Feng J Lam S Xu and S Zhou ldquoHinfin Control withTransients for Singular Systemsrdquo Asian Journal of Control vol18 no 3 pp 817ndash827 2016

[14] P P Khargonekar K M Nagpal and K R Poolla ldquoHinfin controlwith transientsrdquo SIAM Journal onControl andOptimization vol29 no 6 pp 1373ndash1393 1991

[15] Z Zhang Z Zhang and H Zhang ldquoFinite-time stabilityanalysis and stabilization for uncertain continuous-time systemwith time-varying delayrdquo Journal of The Franklin Institute vol352 no 3 pp 1296ndash1317 2015

[16] J Cheng S Chen Z Liu H Wang and J Li ldquoRobust finite-time sampled-data control of linear systems subject to randomoccurring delays and its application to Four-Tank systemrdquoAppliedMathematics andComputation vol 281 pp 55ndash76 2016

[17] W Kang S Zhong K Shi and J Cheng ldquoFinite-time stabilityfor discrete-time system with time-varying delay and nonlinearperturbationsrdquo ISA Transactions vol 60 pp 67ndash73 2016

[18] T I Fossen Handbook of Marine Craft Hydrodynamics andMotion Control John Wiley amp Sons New York USA 2011

[19] D Zhao F Ding L Zhou W Zhang and H Xu ldquoRobustH-infinity Control of Neutral System with Time-Delay forDynamic Positioning Shipsrdquo Mathematical Problems in Engi-neering vol 2015 Article ID 976925 11 pages 2015

[20] F Amato M Ariola and C Cosentino ldquoFinite-time stabilityof linear time-varying systems analysis and controller designrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 55 no 4 pp 1003ndash1008 2010

[21] L Sun andZ Zheng ldquoFinite-time slidingmode trajectory track-ing control of uncertain mechanical systemsrdquo Asian Journal ofControl vol 19 no 1 pp 399ndash404 2017

[22] P Shi Y Zhang and R Agarwal ldquoStochastic finite-time stateestimation for discrete time-delay neural networks withMarko-vian jumpsrdquo Neurocomputing vol 151 part 1 pp 168ndash174 2014

[23] L Zhang S Wang H R Karimi and A Jasra ldquoRobust Finite-Time Control of Switched Linear Systems and Application toa Class of Servomechanism Systemsrdquo IEEEASME Transactionson Mechatronics vol 20 no 5 pp 2476ndash2485 2015

[24] L Wu P Shi and H Gao ldquoState estimation and sliding-mode control of Markovian jump singular systemsrdquo Institute ofElectrical and Electronics Engineers Transactions on AutomaticControl vol 55 no 5 pp 1213ndash1219 2010

[25] Y Q Zhang P Shi S K Nguang and Y D Song ldquoRobust finite-time Hinfin control for uncertain discrete-time singular systemswithMarkovian jumpsrdquo IET ControlTheory ampApplications vol8 no 12 pp 1105ndash1111 2014

[26] S Wang T Shi L Zhang A Jasra and M Zeng ldquoExtendedfinite-time Hinfin control for uncertain switched linear neutralsystemswith time-varying delaysrdquoNeurocomputing vol 152 pp377ndash387 2015

[27] J Hu G Sui S Du and X Li ldquoFinite-time stability of uncer-tain nonlinear systems with time-varying delayrdquo MathematicalProblems in Engineering Article ID 2538904 9 pages 2017

[28] A Seuret and F Gouaisbaut ldquoWirtinger-based integral inequal-ity application to time-delay systemsrdquo Automatica vol 49 no9 pp 2860ndash2866 2013

[29] Z Li Y Bai C Huang and Y Cai ldquoNovel delay-partitioningstabilization approach for networked control system viaWirtinger-based inequalitiesrdquo ISA Transactions vol 61 pp75ndash86 2016

[30] P Park JWKo andC Jeong ldquoReciprocally convex approach tostability of systems with time-varying delaysrdquo Automatica vol47 no 1 pp 235ndash238 2011

[31] K Gu V L Kharitonov and J Chen Stability of Time DelaySystems Birkhauser Boston Mass USA 2003

[32] L El Ghaoui F Oustry and M AitRami ldquoA cone comple-mentarity linearization algorithm for static output-feedbackand related problemsrdquo Institute of Electrical and ElectronicsEngineers Transactions on Automatic Control vol 42 no 8 pp1171ndash1176 1997

[33] R Skjetne T I Fossen and P Kokotovic ldquoAdaptive maneu-vering with experiments for a model ship in a marine controllaboratoryrdquo Automatica vol 41 no 2 pp 289ndash298 2005

[34] N Wang C Qian J-C Sun and Y-C Liu ldquoAdaptive RobustFinite-Time Trajectory Tracking Control of Fully ActuatedMarine Surface Vehiclesrdquo IEEE Transactions on Control SystemsTechnology vol 24 no 4 pp 1454ndash1462 2016

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Page 8: Robust Finite-Time H-Infinity Control with Transients for ...downloads.hindawi.com/journals/mpe/2018/2838749.pdf · ResearchArticle Robust Finite-Time H-Infinity Control with Transients

8 Mathematical Problems in Engineering

Π36 = 6120588119882 + 21205881198852119879Π37 = 6120588119882 + 21205881198852Π58 = 121205881198801 + 241205881198802Π67 = minus41205881198852Π19 = 12ℎ11987511119860119879Π39 = 12ℎ1198841198791198611119879Π49 = 12ℎ1198612119879Π110 = 12ℎ11987511119860119879Π310 = 12ℎ1198841198791198611119879Π410 = 12ℎ1198612119879Π111 = ℎ11987511119860119879Π311 = ℎ1198841198791198611119879Π411 = ℎ1198612119879Π112 = 11987511119862119879Π312 = 119884119879119863119879Π99 = 121198801 minus 11987511Π1010 = 121198802 minus 11987511Π1111 = 119882 minus 211987511Π1212 = minus119890minus120572119879119868(47)

others Π119894119895 = 0Σ11 = minus119890minus12057211987911988821198671Σ22 = minus1198672Σ33 = minus1198673Σ44 = minus1198674Σ55 = minus1198675Σ66 = minus1198676Σ77 = minus1198677Σ88 = minus1198678

Σ99 = minus1205742119889Σ12 = radic11988811198672Σ13 = radicℎ211988811198673Σ14 = radic120588ℎ11988811198674Σ15 = radic120588ℎ11988811198675Σ16 = radic12120588ℎ311988811198676Σ17 = radic16120588ℎ311988811198677Σ18 = radic13120588ℎ311988811198678Σ19 = 11987511(48)

others Σ119894119895 = 0 (49)

then a state feedback119867infin controller in form of (7) existssuch that

(1) the closed-loop system (8) with 119906(119905) = 0 is FTB(2) under given nonzero initial condition the closed

system (8) and (9) guarantees that 119869 = int1198790119911119879(119904)119911(119904) minus1205742120596119879(119904)120596(119904) minus 1205742(1119879)119909119879(0)119878119909(0)119889119904 lt 0 for all nonzero 120596(119905)

and 119905 isin [0 119879]The controller gain can be calculated by 119870 = 11988411987511minus1

Proof In preparation for designing we set119872 = 1205742119868 120573 = 1for (37) then (37) and (38) can be rewritten as follows minus 120572119881 minus 1205742120596119879 (119905) 120596 (119905) lt 120585119879 (119905) Ω120585 (119905) (50)119881 (119905) lt 119890120572119905119881 (0) + 1205742 int119905

0119890120572(119905minus119904)120596119879 (119904) 120596 (119904) 119889119904lt 119890120572119879 (119881 (0) + 1205742119889) (51)

where Ω44 = 1198612119879 12ℎ2 (2119882 + 1198801 + 1198802) 1198612 minus 1205742119868Ω119894119895 = Ω119894119895

(52)

Then two steps are provided to design the desiredcontroller

Step 1 Here we will give some conditions that can guaranteethe 119867infin performance over the finite-time interval undernonzero initial conditions firstly

Mathematical Problems in Engineering 9

Consider the following inequality minus 120572119881 + 119890120572119905119911119879 (119905) 119911 (119905) minus 1205742119890120572119905120596119879 (119905) 120596 (119905)lt minus 120572119881 + 119890120572119879119911119879 (119905) 119911 (119905) minus 1205742120596119879 (119905) 120596 (119905)lt 120585119879 (119905) (Ω minus Ψ119879 (minus119890minus120572119879119868)minus1Ψ) 120585 (119905) (53)

Noting that119882 1198801 and 1198802 are coupled with 119860 1198611119870 and1198612 in Ω to decouple these terms the Ω can be rewritten asΩ = Ω minus Ψ119879 (minus121198801)minus1 Ψ minus Ψ119879 (minus121198802)minus1Ψminus Ψ119879 (minus119882)minus1 Ψ (54)

whereΩ11 = 11986011987911987511 + 11987511119860 minus 12057211987511 + 1198761 + 1198762 minus 4120588119882minus 31205881198801 minus 91205881198802Ω33 = minus8120588119882 + 1205881198851119879 + 1205881198851 minus 1205881198852119879 minus 1205881198852minus (1 minus 120583)1198762Ω44 = minus1205742119868Ω13 = 119875111198611119870 minus 2120588119882 minus 1205881198851 minus 1205881198852Ω14 = 119875111198612Ω34 = 0(55)

othersΩ119894119895 = Ω119894119895Ψ = (119862 0 119863119870 0 0 0 0 0)Ψ= (ℎ121198801119879119860 0 ℎ1211988011198791198611119870 ℎ1211988011198791198612 0 0 0 0) Ψ= (ℎ121198802119879119860 0 ℎ1211988021198791198611119870 ℎ1211988021198791198612 0 0 0 0)Ψ = (ℎ119882119879119860 0 ℎ1198821198791198611119870 ℎ1198821198791198612 0 0 0 0) (56)

By Schur complementΩ lt 0 and Ω minusΨ119879(minus119890minus120572119879119868)minus1Ψ areequivalent to

Π =(((((

Ω Ψ119879 Ψ119879 Ψ119879 Ψ119879lowast minus121198801 0 0 0lowast lowast minus121198802 0 0lowast lowast lowast minus119882 0lowast lowast lowast lowast minus119890minus120572119879119868)))))lt 0 (57)

To ensure the proposed conditions satisfy the 119867infin per-formance we set the inequality like

11987511 minus 1205742119878 lt 0 (58)

from which we have

minus 120572119881 + 119890120572119905119911119879 (119905) 119911 (119905) minus 1205742119890120572119905120596119879 (119905) 120596 (119905)minus 1205742119890120572119905 1119879119909119879 (0) 119878119909 (0) lt minus 120572119881 + 119890120572119905119911119879 (119905) 119911 (119905)minus 1205742119890120572119905120596119879 (119905) 120596 (119905) minus 1205742119890120572119905 1119879119909119879 (0) 119878119909 (0)+ 119890120572119905 1119879119909119879 (0) 11987511119909 (0) lt minus 120572119881 + 119890120572119905119911119879 (119905) 119911 (119905)minus 1205742119890120572119905120596119879 (119905) 120596 (119905) + 119890120572119905 1119879119909119879 (0) (11987511 minus 1205742119878) 119909 (0)lt 0(59)

Integrating (58) from 0 to 119879 we can obtain

int1198790119911119879 (119904) 119911 (119904) 119889119904 minus int119879

01205742120596119879 (119904) 120596 (119904) 119889119904

minus 1205742119909119879 (0) 119878119909 (0) lt minusint1198790(119890minus120572119905 minus 120572119890minus120572119905119881)119889119904 lt 0 (60)

so conditions (43) (44) (57) and (58) with the inequali-ties as follows can ensure that systems (17) and (18) satisfy the119867infin performance

11988811205792 + ℎ211988811205793 + 120588ℎ11988811205794 + 120588ℎ11988811205795 + 12120588ℎ311988811205796+ 16120588ℎ311988811205797 + 13120588ℎ311988811205798 + 1205742119889 lt 119890minus120572119879119888212057910 lt 1205791119868 lt 11987511 lt 12057921198680 lt 11987522 lt 12057931198680 lt 1198761 lt 12057941198680 lt 1198762 lt 12057951198680 lt 119882 lt 12057961198680 lt 1198801 lt 12057971198680 lt 1198802 lt 1205798119868

(61)

10 Mathematical Problems in Engineering

Step 2 In this step the approach to design the gain matrix 119870of the controller is provided in forms of LMIs which can becalculated expediently

To linearize the nonlinear terms in (57) we make itmultiply by the followingmatrix fromboth left and right sidesfirstly

diag 11987511minus1 11987511minus1 11987511minus1 119868 11987511minus1 11987511minus1 11987511minus1 11987511minus1 1198801minus1 1198802minus1 119882minus1 119868 (62)

Let us do the definitions as follows11987511 = 11987511minus11198761 = 11987511minus1119876111987511minus11198762 = 11987511minus1119876211987511minus1119882 = 11987511minus111988211987511minus11198801 = 11987511minus1119880111987511minus11198802 = 11987511minus1119880211987511minus111987522 = 11987511minus11198752211987511minus11198851 = 11987511minus1119885111987511minus11198852 = 11987511minus1119885211987511minus1119884 = 11987011987511minus1

(63)

so the inequality Π lt 0 can guarantee Π lt 0 whereΠ99 = minus12119875111198801

minus111987511Π1010 = minus12119875111198802

minus111987511Π1111 = minus11987511119882minus111987511(64)

others Π119894119895 = Π119894119895 (65)

Then based on the inequalities minus(12)119875111198801

minus111987511 le(12)1198801 minus 11987511 minus(12)119875111198802

minus111987511 le (12)1198802 minus 11987511minus11987511119882minus111987511 le 119882 minus 211987511 condition (57) can be convertedinto (41) in form of LMIs

Accordingly similar to the above-mentioned procedures(61) can be transformed into (42) with the Schur complementand the definition of119867119894 = 11987511minus112057911989411987511minus1 (119894 = 1 2 sdot sdot sdot 8)

Therefore conditions (41) to (45) can obtain the con-troller in the form of (7) guaranteeing the desired perfor-mance This completes the proof

Remark 16 The119867infin performance with nonzero initial con-dition is studied in this literature In a practical view it is

hard to suffer the worst-case over a finite time for surfacevessel with nonzero initial condition and it will reducethe conservatism caused by the assumption of zero initialcondition Therefore it is reasonable to introduce such 119867infinperformance index to handle the problem of finite-time119867infincontrol for ship subject to exogenous signals

Remark 17 It is challenging to achieve the result satisfyingthe control objective while it is very different from thestandard 119867infin control because a weighting diagonal matrixis introduced especially to establish the form of LMIs Inthe proof of Theorem 15 inequality (58) is introduced toensure the 119867infin performance with nonzero initial conditionand make it convenient to construct the conditions in formof LMIs

Remark 18 By introducing some new additional matrixvariables Cone Complementarity Linearization algorithm(CCL) [32] can be utilized to linearize the nonlinear termsas an alternative method It will reduce the conservatism atthe expense of increasing the computational burden and thenumber of decision variables

4 Simulations

41 Simulation Settings To demonstrate the effectiveness andsuperiority of the robust finite-time 119867infin control schemesimulation studies are conducted on the parameters of theship CyberShip II [33] which is a 170 scale-replica of a supplyshipwhosemain parameters are shown inTable 1 In additionwe choose the time-varying function like ℎ(119905) = 3sin (1199056 minus1205876) to describe the time delay occurred in the input signal

Unknown external disturbances 120596(119905) with significantlylarge magnitude and high frequency which can be roughlyconsidered as complex environment including ocean windswaves and currents are governed by [34]

120596 (119905) = (9 sin (01120587119905 minus 1205875 )6 sin (03120587119905 + 1205876 )3 sin (02120587119905 + 1205873 )) (66)

with initial conditions 120578(0) = (1 1 01) and 120592(0) =(08 06 01)42 Numerical Simulations We choose 120574 = 13 119879 = 10 119878 =03lowast119868 1198881 = 302 1198882 = 10 120572 = 0001 By usingTheorem 15 thecontrol gain 119870 is calculated by LMIs (41) to (45)

Mathematical Problems in Engineering 11

Table 1 The main parameters of the CyberShip II119898 238000 119883 -20 119883119906 -07225119868119911 17600 119884V -100 119884V -08612119909119892 00460 119884 119903 -00 119884119903 01079119873 119903 -10 119873119903 05

0 20 40 60 80 100Time(s) Time(s)

minus2

0

2

4

6

8

10

12

0 2 4 6 8 100

2

4

6

8

10

12

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

xT(t)Rx(t)xT(t)Rx(t)

c=c=

c1=302

c1=302

Figure 1 State response 119909119879(119905)119877119909(119905) of DP system

119870 = (minus32743 0 0 minus313907 0 00 minus42784 minus01572 0 minus412136 minus129820 minus01389 minus03497 0 minus13362 minus33091) (67)

To demonstrate the superiority and validity three sce-narios as comparative cases are provided here It should bepointed out that the results within 100s are also provided toillustrate the steady-state performance

Case 1 (with different 119878) For comparison purposes we set119878 with different value to demonstrate the effect of 119878 on stateresponse Without loss of generality we set 119878 = 05 lowast 119868 and119878 = 01 lowast 119868 respectivelyCase 2 (119867infin with zero initial condition]) This scenario willshow that the control law proposed in this paper can achievea better transient performance and disturbance rejectioncapability over traditional119867infin control approach for dynamicpositioning system conducted in [11]

Case 3 (a control design based on existing methods) Basedon the approach in [17] applied for discrete-time system acontrol law for dynamic positioning ship with input delay isproposed to illustrate the less conservatism of our designingapproach in practical engineering subsequently

At first the effectiveness of the control law we designedis confirmed from Figures 1ndash6 Figure 1 shows the state

response of 119909119879(119905)119877119909(119905) Obviously from that curve we cansee that the value of 119909119879(119905)119877119909(119905) is far below the value of1198882 = 10 in the presence of disturbance as in Figure 2which means the finite-time full-state feedback controllerdesigned can guarantee the DP system (8) sim (9) FTB with(302 10 119889 10 3 119868) and have less conservatism where weset 119889 = int119879

0120596(119905)119879120596(119905)119889119905 Meanwhile the control law can

stabilize the dynamic positioning shipwithminimum steady-state error in 100s and the control input will be small enoughas shown in Figure 6 From Figures 3ndash5 we can find that allstates in DP system are within a small range As shown inFigures 2ndash4 the angle and the angular velocity in yaw havechanged within 01∘ which means that the heading of thevessel stays in a fixed direction Furthermore the velocitiesin surge and sway are low enough which illustrates thatassumption 2 is reasonable and it meets the requirement ofstabilization control for DP system

To demonstrate the effect of 119878 state responses of systemswith different 119878 are exhibited in Figures 7ndash9 As the valueof 119878 decreases the restriction for 11987511 is more exclusive andthe robustness is stronger during preset time interval as inFigures 7 and 8 while the required control input will increasedramatically as in Figure 9 When the time oversteps the

12 Mathematical Problems in Engineering

0 10 20 30 40 50 60 70 80 90 100Time(s)

minus10

minus5

0

5

10

Dist

urba

nces

(N)

XY

Figure 2 Disturbance in surge sway and yaw

0 50 100Time(s)

minus050

051

152

X(m

)

0 5 10Time(s)

005

115

2

X(m

)

0 50 100Time(s)

minus050

051

152

Y(m

)

0 5 10Time(s)

005

115

2Y(

m)

0 50 100Time(s)

minus01

0

01

02

0 5 10Time(s)

0

01

02

(∘

)

(∘

)

Figure 3 State responses of position and orientation for DP System

0 50 100Time(s)

minus1

0

1

2

0 5 10Time(s)

minus1

0

1

2

0 50 100Time(s)

minus1

0

1

2

v(m

s)

0 5 10Time(s)

minus1

0

1

2

v(m

s)

0 50 100Time(s)

minus01

0

01

02

0 5 10Time(s)

minus01

0

01

02

r(∘ s

)

r(∘ s

)

(ms

)

(m

s)

Figure 4 State responses of velocities for DP system

Mathematical Problems in Engineering 13

minus05 0 05 1 15 2X(m)

minus05

0

05

1

15

2

Y(m

)

Figure 5 Trajectory of the shiprsquos horizontal motion

0 50 100Time(s)

minus30minus20minus10

010

u1(N

)

0 5 10Time(s)

minus30minus20minus10

010

u1(N

)

0 50 100Time(s)

minus30minus20minus10

010

u2(N

)

0 5 10Time(s)

minus30minus20minus10

010

u2(N

)

0 50 100Time(s)

minus15minus1

minus050

05

u3(N

m)

0 5 10Time(s)

minus15minus1

minus050

05

u3(N

m)

Figure 6 Control inputs

0 20 40 60 80 100Time(s)

minus1

0

1

2

3

4

5

6

S=03S=05S=01

0 2 4 6 8 10Time(s)

0

2

4

6

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 7 State response 119909119879(119905)119877119909(119905) with different 119878

14 Mathematical Problems in Engineering

Figure 8 State responses of position and orientation with different 119878

0 50 100Time(s)

minus60minus30

03060

u1(N

)

0 5 10Time(s)

minus40minus20

02040

u1(N

)

0 50 100Time(s)

minus100

0

100

u2(N

)

0 5 10Time(s)

minus40minus20

02040

u2(N

)

0 50 100Time(s)

minus4minus2

024

u3(N

m)

0 5 10Time(s)

minus2

0

2

u3(N

m)

S=03S=05S=01

Figure 9 Control input with different 119878preset time interval all the states can be guaranteed withinthe given upper bound as before

As shown in Figures 10ndash12 the overconservatism oftraditional119867infin control approach applied inDP ship is solvedby the 119867infin control strategy with nonzero initial conditionproposed in this article As is well known the classical 119867infincontrol law is designed under the so-called worst-case normwhile the performance of antidisturbance and transient issacrificed to some degree From Figures 10 and 11 we cansee that the state response can be controlled in a smallerrange and stabilized in a shorter time with a lower overshootshown in Figure 12 Hence a conclusion can be drawn thatthe control law we present can achieve a better disturbance

rejection capability and transient performance while thematter of overconservatism can be resolved at the sametime

Simulation results of Theorem 15 and Case 3 are shownin Figures 13 and 14 to show the better performance ofthe method we proposed We can find that the controllerbased on [17] and 119867infin control approach present in thisliterature can make all states of system within a smallenough region However compared with the controller weconstructed the state response has higher overshoot whichwill affect the control precision and transient performance inpractical engineering Based on the above simulation resultsthe developed controller can achieve a higher accuracy and

Mathematical Problems in Engineering 15

0 20 40 60 80 100Time(s)

minus2

0

2

4

6

8

10

12

Theorem 15Case 2

0 2 4 6 8 10Time(s)

minus2

0

2

4

6

8

10

12

Theorem 15Case 2

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 10 State response 119909119879(119905)119877119909(119905) of Theorem 15 and Case 2

0 50 100Time(s)

minus10123

X(m

)

0 5 10Time(s)

0123

X(m

)

0 50 100Time(s)

minus10123

Y(m

)

0 5 10Time(s)

0123

Y(m

)

0 50 100Time(s)

minus020

0204

0 5 10Time(s)

minus020

0204

Theorem 15Case 2

(∘

)

(∘

)

Figure 11 Position and orientation of Theorem 15 and Case 2

minus05 0 05 1 15 2 25X(m)

minus050

051

152

25

Y(m

)

Theorem 15Case 2

Figure 12 Trajectory of the shiprsquos horizontal motion of Theorem 15 and Case 2

16 Mathematical Problems in Engineering

0 20 40 60 80 100Time(s)

0

1

2

3

4

5

Theorem 15Case 3

0 2 4 6 8 10Time(s)

0

1

2

3

4

5

Theorem 15Case 3

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 13 State response 119909119879(119905)119877119909(119905) of Theorem 15 and Case 3

0 5 10Time(s)

05

1

15

X(m

)

Theorem 15Case 3

0 5 10Time(s)

05

1

15

Y(m

)

0 5 10Time(s)

0

005

01

015

02

(∘

)

Figure 14 Position and orientation of Theorem 15 and Case 3

lower overshoot which means that it is less conservative incomparison with existing result

5 Conclusion

In this paper the robust finite-time 119867infin control with tran-sients for DP system with input delay maintaining all statesof ocean surface vessel into a given threshold over a fixed timeinterval in presence of time-varying disturbance is proposedBased on a novel augmented LKLF with triple integral termsand a method combining theWirtinger inequality and recip-rocally convex approach a less conservative result is derivedIn particular a119867infin performance index with nonzero initialcondition attenuating the overconservatism caused by theassumption of zero initial condition and enhancing thetransient performance of ship subject to external disturbanceis adoptedThe obtained result is formulated in terms of LMIswhich can be easily solved by the standard numerical soft-ware Finally a numerical simulation for a ship is proposed toverify the effectiveness and less conservatism of the controller

we designed In the future the output feedback control can bedeveloped for the ocean surface vessel to avoid the use of thevessel-frame surge sway and yaw velocities

Data Availability

All the data supporting the conclusions of the study have beenprovided in Simulations and readers can access these data in[33 34]

Conflicts of Interest

The authors declare that there are no conflicts of interestrelated to this paper

Acknowledgments

This research was partially supported by the National Sci-ence Technology Support Program of China (Project no51609046)

Mathematical Problems in Engineering 17

References

[1] M Chen B Jiang and R Cui ldquoActuator fault-tolerant controlof ocean surface vessels with input saturationrdquo InternationalJournal of Robust amp Nonlinear Control vol 26 no 3 pp 542ndash564 2016

[2] J Du X Hu and Y Sun ldquoRobust dynamic positioning of shipswith disturbances under input saturationrdquo Automatica vol 73pp 207ndash214 2016

[3] H Ashrafiuon K R Muske L C McNinch and R ASoltan ldquoSliding-mode tracking control of surface vesselsrdquo IEEETransactions on Industrial Electronics vol 55 no 11 pp 4004ndash4012 2008

[4] W Si and X Dong ldquoNeural prescribed performance controlfor uncertain marine surface vessels without accurate initialerrorsrdquoMathematical Problems in Engineering Art ID 261832311 pages 2017

[5] T D Nguyen A J Soslashrensen and S T Quek ldquoDesign of hybridcontroller for dynamic positioning from calm to extreme seaconditionsrdquo Automatica vol 43 no 5 pp 768ndash785 2007

[6] J Du X Hu H Liu and C L Chen ldquoAdaptive robust outputfeedback control for a marine dynamic positioning systembased on a high-gain observerrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 11 pp 2775ndash27862015

[7] H Katayama ldquoNonlinear sampled-data stabilization of dynam-ically positioned shipsrdquo IEEE Transactions on Control SystemsTechnology vol 18 no 2 pp 463ndash468 2010

[8] K J Astrom andCG Kallstrom ldquoIdentification of ship steeringdynamicsrdquo Automatica vol 12 no 1 pp 9ndash22 1976

[9] Z Lei and C Guo ldquoDisturbance rejection control solutionfor ship steering system with uncertain time delayrdquo OceanEngineering vol 95 pp 78ndash83 2015

[10] J C Doyle K Glover P P Khargonekar and B A FrancisldquoState-space solutions to standard H-2 and Hinfin control prob-lemsrdquo IEEE Transactions on Automatic Control vol 34 no 8pp 831ndash847 1989

[11] M Katebi M Grimble and Y Zhang ldquoHinfin robust controldesign for dynamic ship positioningrdquo IEE Proceedings - ControlTheory and Applications vol 144 no 2 pp 110ndash120 1997

[12] D V Balandin and M M Kogan ldquoLMI-based Hinfin-optimalcontrol with transientsrdquo International Journal of Control vol 83no 8 pp 1664ndash1673 2010

[13] Z Feng J Lam S Xu and S Zhou ldquoHinfin Control withTransients for Singular Systemsrdquo Asian Journal of Control vol18 no 3 pp 817ndash827 2016

[14] P P Khargonekar K M Nagpal and K R Poolla ldquoHinfin controlwith transientsrdquo SIAM Journal onControl andOptimization vol29 no 6 pp 1373ndash1393 1991

[15] Z Zhang Z Zhang and H Zhang ldquoFinite-time stabilityanalysis and stabilization for uncertain continuous-time systemwith time-varying delayrdquo Journal of The Franklin Institute vol352 no 3 pp 1296ndash1317 2015

[16] J Cheng S Chen Z Liu H Wang and J Li ldquoRobust finite-time sampled-data control of linear systems subject to randomoccurring delays and its application to Four-Tank systemrdquoAppliedMathematics andComputation vol 281 pp 55ndash76 2016

[17] W Kang S Zhong K Shi and J Cheng ldquoFinite-time stabilityfor discrete-time system with time-varying delay and nonlinearperturbationsrdquo ISA Transactions vol 60 pp 67ndash73 2016

[18] T I Fossen Handbook of Marine Craft Hydrodynamics andMotion Control John Wiley amp Sons New York USA 2011

[19] D Zhao F Ding L Zhou W Zhang and H Xu ldquoRobustH-infinity Control of Neutral System with Time-Delay forDynamic Positioning Shipsrdquo Mathematical Problems in Engi-neering vol 2015 Article ID 976925 11 pages 2015

[20] F Amato M Ariola and C Cosentino ldquoFinite-time stabilityof linear time-varying systems analysis and controller designrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 55 no 4 pp 1003ndash1008 2010

[21] L Sun andZ Zheng ldquoFinite-time slidingmode trajectory track-ing control of uncertain mechanical systemsrdquo Asian Journal ofControl vol 19 no 1 pp 399ndash404 2017

[22] P Shi Y Zhang and R Agarwal ldquoStochastic finite-time stateestimation for discrete time-delay neural networks withMarko-vian jumpsrdquo Neurocomputing vol 151 part 1 pp 168ndash174 2014

[23] L Zhang S Wang H R Karimi and A Jasra ldquoRobust Finite-Time Control of Switched Linear Systems and Application toa Class of Servomechanism Systemsrdquo IEEEASME Transactionson Mechatronics vol 20 no 5 pp 2476ndash2485 2015

[24] L Wu P Shi and H Gao ldquoState estimation and sliding-mode control of Markovian jump singular systemsrdquo Institute ofElectrical and Electronics Engineers Transactions on AutomaticControl vol 55 no 5 pp 1213ndash1219 2010

[25] Y Q Zhang P Shi S K Nguang and Y D Song ldquoRobust finite-time Hinfin control for uncertain discrete-time singular systemswithMarkovian jumpsrdquo IET ControlTheory ampApplications vol8 no 12 pp 1105ndash1111 2014

[26] S Wang T Shi L Zhang A Jasra and M Zeng ldquoExtendedfinite-time Hinfin control for uncertain switched linear neutralsystemswith time-varying delaysrdquoNeurocomputing vol 152 pp377ndash387 2015

[27] J Hu G Sui S Du and X Li ldquoFinite-time stability of uncer-tain nonlinear systems with time-varying delayrdquo MathematicalProblems in Engineering Article ID 2538904 9 pages 2017

[28] A Seuret and F Gouaisbaut ldquoWirtinger-based integral inequal-ity application to time-delay systemsrdquo Automatica vol 49 no9 pp 2860ndash2866 2013

[29] Z Li Y Bai C Huang and Y Cai ldquoNovel delay-partitioningstabilization approach for networked control system viaWirtinger-based inequalitiesrdquo ISA Transactions vol 61 pp75ndash86 2016

[30] P Park JWKo andC Jeong ldquoReciprocally convex approach tostability of systems with time-varying delaysrdquo Automatica vol47 no 1 pp 235ndash238 2011

[31] K Gu V L Kharitonov and J Chen Stability of Time DelaySystems Birkhauser Boston Mass USA 2003

[32] L El Ghaoui F Oustry and M AitRami ldquoA cone comple-mentarity linearization algorithm for static output-feedbackand related problemsrdquo Institute of Electrical and ElectronicsEngineers Transactions on Automatic Control vol 42 no 8 pp1171ndash1176 1997

[33] R Skjetne T I Fossen and P Kokotovic ldquoAdaptive maneu-vering with experiments for a model ship in a marine controllaboratoryrdquo Automatica vol 41 no 2 pp 289ndash298 2005

[34] N Wang C Qian J-C Sun and Y-C Liu ldquoAdaptive RobustFinite-Time Trajectory Tracking Control of Fully ActuatedMarine Surface Vehiclesrdquo IEEE Transactions on Control SystemsTechnology vol 24 no 4 pp 1454ndash1462 2016

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Page 9: Robust Finite-Time H-Infinity Control with Transients for ...downloads.hindawi.com/journals/mpe/2018/2838749.pdf · ResearchArticle Robust Finite-Time H-Infinity Control with Transients

Mathematical Problems in Engineering 9

Consider the following inequality minus 120572119881 + 119890120572119905119911119879 (119905) 119911 (119905) minus 1205742119890120572119905120596119879 (119905) 120596 (119905)lt minus 120572119881 + 119890120572119879119911119879 (119905) 119911 (119905) minus 1205742120596119879 (119905) 120596 (119905)lt 120585119879 (119905) (Ω minus Ψ119879 (minus119890minus120572119879119868)minus1Ψ) 120585 (119905) (53)

Noting that119882 1198801 and 1198802 are coupled with 119860 1198611119870 and1198612 in Ω to decouple these terms the Ω can be rewritten asΩ = Ω minus Ψ119879 (minus121198801)minus1 Ψ minus Ψ119879 (minus121198802)minus1Ψminus Ψ119879 (minus119882)minus1 Ψ (54)

whereΩ11 = 11986011987911987511 + 11987511119860 minus 12057211987511 + 1198761 + 1198762 minus 4120588119882minus 31205881198801 minus 91205881198802Ω33 = minus8120588119882 + 1205881198851119879 + 1205881198851 minus 1205881198852119879 minus 1205881198852minus (1 minus 120583)1198762Ω44 = minus1205742119868Ω13 = 119875111198611119870 minus 2120588119882 minus 1205881198851 minus 1205881198852Ω14 = 119875111198612Ω34 = 0(55)

othersΩ119894119895 = Ω119894119895Ψ = (119862 0 119863119870 0 0 0 0 0)Ψ= (ℎ121198801119879119860 0 ℎ1211988011198791198611119870 ℎ1211988011198791198612 0 0 0 0) Ψ= (ℎ121198802119879119860 0 ℎ1211988021198791198611119870 ℎ1211988021198791198612 0 0 0 0)Ψ = (ℎ119882119879119860 0 ℎ1198821198791198611119870 ℎ1198821198791198612 0 0 0 0) (56)

By Schur complementΩ lt 0 and Ω minusΨ119879(minus119890minus120572119879119868)minus1Ψ areequivalent to

Π =(((((

Ω Ψ119879 Ψ119879 Ψ119879 Ψ119879lowast minus121198801 0 0 0lowast lowast minus121198802 0 0lowast lowast lowast minus119882 0lowast lowast lowast lowast minus119890minus120572119879119868)))))lt 0 (57)

To ensure the proposed conditions satisfy the 119867infin per-formance we set the inequality like

11987511 minus 1205742119878 lt 0 (58)

from which we have

minus 120572119881 + 119890120572119905119911119879 (119905) 119911 (119905) minus 1205742119890120572119905120596119879 (119905) 120596 (119905)minus 1205742119890120572119905 1119879119909119879 (0) 119878119909 (0) lt minus 120572119881 + 119890120572119905119911119879 (119905) 119911 (119905)minus 1205742119890120572119905120596119879 (119905) 120596 (119905) minus 1205742119890120572119905 1119879119909119879 (0) 119878119909 (0)+ 119890120572119905 1119879119909119879 (0) 11987511119909 (0) lt minus 120572119881 + 119890120572119905119911119879 (119905) 119911 (119905)minus 1205742119890120572119905120596119879 (119905) 120596 (119905) + 119890120572119905 1119879119909119879 (0) (11987511 minus 1205742119878) 119909 (0)lt 0(59)

Integrating (58) from 0 to 119879 we can obtain

int1198790119911119879 (119904) 119911 (119904) 119889119904 minus int119879

01205742120596119879 (119904) 120596 (119904) 119889119904

minus 1205742119909119879 (0) 119878119909 (0) lt minusint1198790(119890minus120572119905 minus 120572119890minus120572119905119881)119889119904 lt 0 (60)

so conditions (43) (44) (57) and (58) with the inequali-ties as follows can ensure that systems (17) and (18) satisfy the119867infin performance

11988811205792 + ℎ211988811205793 + 120588ℎ11988811205794 + 120588ℎ11988811205795 + 12120588ℎ311988811205796+ 16120588ℎ311988811205797 + 13120588ℎ311988811205798 + 1205742119889 lt 119890minus120572119879119888212057910 lt 1205791119868 lt 11987511 lt 12057921198680 lt 11987522 lt 12057931198680 lt 1198761 lt 12057941198680 lt 1198762 lt 12057951198680 lt 119882 lt 12057961198680 lt 1198801 lt 12057971198680 lt 1198802 lt 1205798119868

(61)

10 Mathematical Problems in Engineering

Step 2 In this step the approach to design the gain matrix 119870of the controller is provided in forms of LMIs which can becalculated expediently

To linearize the nonlinear terms in (57) we make itmultiply by the followingmatrix fromboth left and right sidesfirstly

diag 11987511minus1 11987511minus1 11987511minus1 119868 11987511minus1 11987511minus1 11987511minus1 11987511minus1 1198801minus1 1198802minus1 119882minus1 119868 (62)

Let us do the definitions as follows11987511 = 11987511minus11198761 = 11987511minus1119876111987511minus11198762 = 11987511minus1119876211987511minus1119882 = 11987511minus111988211987511minus11198801 = 11987511minus1119880111987511minus11198802 = 11987511minus1119880211987511minus111987522 = 11987511minus11198752211987511minus11198851 = 11987511minus1119885111987511minus11198852 = 11987511minus1119885211987511minus1119884 = 11987011987511minus1

(63)

so the inequality Π lt 0 can guarantee Π lt 0 whereΠ99 = minus12119875111198801

minus111987511Π1010 = minus12119875111198802

minus111987511Π1111 = minus11987511119882minus111987511(64)

others Π119894119895 = Π119894119895 (65)

Then based on the inequalities minus(12)119875111198801

minus111987511 le(12)1198801 minus 11987511 minus(12)119875111198802

minus111987511 le (12)1198802 minus 11987511minus11987511119882minus111987511 le 119882 minus 211987511 condition (57) can be convertedinto (41) in form of LMIs

Accordingly similar to the above-mentioned procedures(61) can be transformed into (42) with the Schur complementand the definition of119867119894 = 11987511minus112057911989411987511minus1 (119894 = 1 2 sdot sdot sdot 8)

Therefore conditions (41) to (45) can obtain the con-troller in the form of (7) guaranteeing the desired perfor-mance This completes the proof

Remark 16 The119867infin performance with nonzero initial con-dition is studied in this literature In a practical view it is

hard to suffer the worst-case over a finite time for surfacevessel with nonzero initial condition and it will reducethe conservatism caused by the assumption of zero initialcondition Therefore it is reasonable to introduce such 119867infinperformance index to handle the problem of finite-time119867infincontrol for ship subject to exogenous signals

Remark 17 It is challenging to achieve the result satisfyingthe control objective while it is very different from thestandard 119867infin control because a weighting diagonal matrixis introduced especially to establish the form of LMIs Inthe proof of Theorem 15 inequality (58) is introduced toensure the 119867infin performance with nonzero initial conditionand make it convenient to construct the conditions in formof LMIs

Remark 18 By introducing some new additional matrixvariables Cone Complementarity Linearization algorithm(CCL) [32] can be utilized to linearize the nonlinear termsas an alternative method It will reduce the conservatism atthe expense of increasing the computational burden and thenumber of decision variables

4 Simulations

41 Simulation Settings To demonstrate the effectiveness andsuperiority of the robust finite-time 119867infin control schemesimulation studies are conducted on the parameters of theship CyberShip II [33] which is a 170 scale-replica of a supplyshipwhosemain parameters are shown inTable 1 In additionwe choose the time-varying function like ℎ(119905) = 3sin (1199056 minus1205876) to describe the time delay occurred in the input signal

Unknown external disturbances 120596(119905) with significantlylarge magnitude and high frequency which can be roughlyconsidered as complex environment including ocean windswaves and currents are governed by [34]

120596 (119905) = (9 sin (01120587119905 minus 1205875 )6 sin (03120587119905 + 1205876 )3 sin (02120587119905 + 1205873 )) (66)

with initial conditions 120578(0) = (1 1 01) and 120592(0) =(08 06 01)42 Numerical Simulations We choose 120574 = 13 119879 = 10 119878 =03lowast119868 1198881 = 302 1198882 = 10 120572 = 0001 By usingTheorem 15 thecontrol gain 119870 is calculated by LMIs (41) to (45)

Mathematical Problems in Engineering 11

Table 1 The main parameters of the CyberShip II119898 238000 119883 -20 119883119906 -07225119868119911 17600 119884V -100 119884V -08612119909119892 00460 119884 119903 -00 119884119903 01079119873 119903 -10 119873119903 05

0 20 40 60 80 100Time(s) Time(s)

minus2

0

2

4

6

8

10

12

0 2 4 6 8 100

2

4

6

8

10

12

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

xT(t)Rx(t)xT(t)Rx(t)

c=c=

c1=302

c1=302

Figure 1 State response 119909119879(119905)119877119909(119905) of DP system

119870 = (minus32743 0 0 minus313907 0 00 minus42784 minus01572 0 minus412136 minus129820 minus01389 minus03497 0 minus13362 minus33091) (67)

To demonstrate the superiority and validity three sce-narios as comparative cases are provided here It should bepointed out that the results within 100s are also provided toillustrate the steady-state performance

Case 1 (with different 119878) For comparison purposes we set119878 with different value to demonstrate the effect of 119878 on stateresponse Without loss of generality we set 119878 = 05 lowast 119868 and119878 = 01 lowast 119868 respectivelyCase 2 (119867infin with zero initial condition]) This scenario willshow that the control law proposed in this paper can achievea better transient performance and disturbance rejectioncapability over traditional119867infin control approach for dynamicpositioning system conducted in [11]

Case 3 (a control design based on existing methods) Basedon the approach in [17] applied for discrete-time system acontrol law for dynamic positioning ship with input delay isproposed to illustrate the less conservatism of our designingapproach in practical engineering subsequently

At first the effectiveness of the control law we designedis confirmed from Figures 1ndash6 Figure 1 shows the state

response of 119909119879(119905)119877119909(119905) Obviously from that curve we cansee that the value of 119909119879(119905)119877119909(119905) is far below the value of1198882 = 10 in the presence of disturbance as in Figure 2which means the finite-time full-state feedback controllerdesigned can guarantee the DP system (8) sim (9) FTB with(302 10 119889 10 3 119868) and have less conservatism where weset 119889 = int119879

0120596(119905)119879120596(119905)119889119905 Meanwhile the control law can

stabilize the dynamic positioning shipwithminimum steady-state error in 100s and the control input will be small enoughas shown in Figure 6 From Figures 3ndash5 we can find that allstates in DP system are within a small range As shown inFigures 2ndash4 the angle and the angular velocity in yaw havechanged within 01∘ which means that the heading of thevessel stays in a fixed direction Furthermore the velocitiesin surge and sway are low enough which illustrates thatassumption 2 is reasonable and it meets the requirement ofstabilization control for DP system

To demonstrate the effect of 119878 state responses of systemswith different 119878 are exhibited in Figures 7ndash9 As the valueof 119878 decreases the restriction for 11987511 is more exclusive andthe robustness is stronger during preset time interval as inFigures 7 and 8 while the required control input will increasedramatically as in Figure 9 When the time oversteps the

12 Mathematical Problems in Engineering

0 10 20 30 40 50 60 70 80 90 100Time(s)

minus10

minus5

0

5

10

Dist

urba

nces

(N)

XY

Figure 2 Disturbance in surge sway and yaw

0 50 100Time(s)

minus050

051

152

X(m

)

0 5 10Time(s)

005

115

2

X(m

)

0 50 100Time(s)

minus050

051

152

Y(m

)

0 5 10Time(s)

005

115

2Y(

m)

0 50 100Time(s)

minus01

0

01

02

0 5 10Time(s)

0

01

02

(∘

)

(∘

)

Figure 3 State responses of position and orientation for DP System

0 50 100Time(s)

minus1

0

1

2

0 5 10Time(s)

minus1

0

1

2

0 50 100Time(s)

minus1

0

1

2

v(m

s)

0 5 10Time(s)

minus1

0

1

2

v(m

s)

0 50 100Time(s)

minus01

0

01

02

0 5 10Time(s)

minus01

0

01

02

r(∘ s

)

r(∘ s

)

(ms

)

(m

s)

Figure 4 State responses of velocities for DP system

Mathematical Problems in Engineering 13

minus05 0 05 1 15 2X(m)

minus05

0

05

1

15

2

Y(m

)

Figure 5 Trajectory of the shiprsquos horizontal motion

0 50 100Time(s)

minus30minus20minus10

010

u1(N

)

0 5 10Time(s)

minus30minus20minus10

010

u1(N

)

0 50 100Time(s)

minus30minus20minus10

010

u2(N

)

0 5 10Time(s)

minus30minus20minus10

010

u2(N

)

0 50 100Time(s)

minus15minus1

minus050

05

u3(N

m)

0 5 10Time(s)

minus15minus1

minus050

05

u3(N

m)

Figure 6 Control inputs

0 20 40 60 80 100Time(s)

minus1

0

1

2

3

4

5

6

S=03S=05S=01

0 2 4 6 8 10Time(s)

0

2

4

6

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 7 State response 119909119879(119905)119877119909(119905) with different 119878

14 Mathematical Problems in Engineering

Figure 8 State responses of position and orientation with different 119878

0 50 100Time(s)

minus60minus30

03060

u1(N

)

0 5 10Time(s)

minus40minus20

02040

u1(N

)

0 50 100Time(s)

minus100

0

100

u2(N

)

0 5 10Time(s)

minus40minus20

02040

u2(N

)

0 50 100Time(s)

minus4minus2

024

u3(N

m)

0 5 10Time(s)

minus2

0

2

u3(N

m)

S=03S=05S=01

Figure 9 Control input with different 119878preset time interval all the states can be guaranteed withinthe given upper bound as before

As shown in Figures 10ndash12 the overconservatism oftraditional119867infin control approach applied inDP ship is solvedby the 119867infin control strategy with nonzero initial conditionproposed in this article As is well known the classical 119867infincontrol law is designed under the so-called worst-case normwhile the performance of antidisturbance and transient issacrificed to some degree From Figures 10 and 11 we cansee that the state response can be controlled in a smallerrange and stabilized in a shorter time with a lower overshootshown in Figure 12 Hence a conclusion can be drawn thatthe control law we present can achieve a better disturbance

rejection capability and transient performance while thematter of overconservatism can be resolved at the sametime

Simulation results of Theorem 15 and Case 3 are shownin Figures 13 and 14 to show the better performance ofthe method we proposed We can find that the controllerbased on [17] and 119867infin control approach present in thisliterature can make all states of system within a smallenough region However compared with the controller weconstructed the state response has higher overshoot whichwill affect the control precision and transient performance inpractical engineering Based on the above simulation resultsthe developed controller can achieve a higher accuracy and

Mathematical Problems in Engineering 15

0 20 40 60 80 100Time(s)

minus2

0

2

4

6

8

10

12

Theorem 15Case 2

0 2 4 6 8 10Time(s)

minus2

0

2

4

6

8

10

12

Theorem 15Case 2

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 10 State response 119909119879(119905)119877119909(119905) of Theorem 15 and Case 2

0 50 100Time(s)

minus10123

X(m

)

0 5 10Time(s)

0123

X(m

)

0 50 100Time(s)

minus10123

Y(m

)

0 5 10Time(s)

0123

Y(m

)

0 50 100Time(s)

minus020

0204

0 5 10Time(s)

minus020

0204

Theorem 15Case 2

(∘

)

(∘

)

Figure 11 Position and orientation of Theorem 15 and Case 2

minus05 0 05 1 15 2 25X(m)

minus050

051

152

25

Y(m

)

Theorem 15Case 2

Figure 12 Trajectory of the shiprsquos horizontal motion of Theorem 15 and Case 2

16 Mathematical Problems in Engineering

0 20 40 60 80 100Time(s)

0

1

2

3

4

5

Theorem 15Case 3

0 2 4 6 8 10Time(s)

0

1

2

3

4

5

Theorem 15Case 3

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 13 State response 119909119879(119905)119877119909(119905) of Theorem 15 and Case 3

0 5 10Time(s)

05

1

15

X(m

)

Theorem 15Case 3

0 5 10Time(s)

05

1

15

Y(m

)

0 5 10Time(s)

0

005

01

015

02

(∘

)

Figure 14 Position and orientation of Theorem 15 and Case 3

lower overshoot which means that it is less conservative incomparison with existing result

5 Conclusion

In this paper the robust finite-time 119867infin control with tran-sients for DP system with input delay maintaining all statesof ocean surface vessel into a given threshold over a fixed timeinterval in presence of time-varying disturbance is proposedBased on a novel augmented LKLF with triple integral termsand a method combining theWirtinger inequality and recip-rocally convex approach a less conservative result is derivedIn particular a119867infin performance index with nonzero initialcondition attenuating the overconservatism caused by theassumption of zero initial condition and enhancing thetransient performance of ship subject to external disturbanceis adoptedThe obtained result is formulated in terms of LMIswhich can be easily solved by the standard numerical soft-ware Finally a numerical simulation for a ship is proposed toverify the effectiveness and less conservatism of the controller

we designed In the future the output feedback control can bedeveloped for the ocean surface vessel to avoid the use of thevessel-frame surge sway and yaw velocities

Data Availability

All the data supporting the conclusions of the study have beenprovided in Simulations and readers can access these data in[33 34]

Conflicts of Interest

The authors declare that there are no conflicts of interestrelated to this paper

Acknowledgments

This research was partially supported by the National Sci-ence Technology Support Program of China (Project no51609046)

Mathematical Problems in Engineering 17

References

[1] M Chen B Jiang and R Cui ldquoActuator fault-tolerant controlof ocean surface vessels with input saturationrdquo InternationalJournal of Robust amp Nonlinear Control vol 26 no 3 pp 542ndash564 2016

[2] J Du X Hu and Y Sun ldquoRobust dynamic positioning of shipswith disturbances under input saturationrdquo Automatica vol 73pp 207ndash214 2016

[3] H Ashrafiuon K R Muske L C McNinch and R ASoltan ldquoSliding-mode tracking control of surface vesselsrdquo IEEETransactions on Industrial Electronics vol 55 no 11 pp 4004ndash4012 2008

[4] W Si and X Dong ldquoNeural prescribed performance controlfor uncertain marine surface vessels without accurate initialerrorsrdquoMathematical Problems in Engineering Art ID 261832311 pages 2017

[5] T D Nguyen A J Soslashrensen and S T Quek ldquoDesign of hybridcontroller for dynamic positioning from calm to extreme seaconditionsrdquo Automatica vol 43 no 5 pp 768ndash785 2007

[6] J Du X Hu H Liu and C L Chen ldquoAdaptive robust outputfeedback control for a marine dynamic positioning systembased on a high-gain observerrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 11 pp 2775ndash27862015

[7] H Katayama ldquoNonlinear sampled-data stabilization of dynam-ically positioned shipsrdquo IEEE Transactions on Control SystemsTechnology vol 18 no 2 pp 463ndash468 2010

[8] K J Astrom andCG Kallstrom ldquoIdentification of ship steeringdynamicsrdquo Automatica vol 12 no 1 pp 9ndash22 1976

[9] Z Lei and C Guo ldquoDisturbance rejection control solutionfor ship steering system with uncertain time delayrdquo OceanEngineering vol 95 pp 78ndash83 2015

[10] J C Doyle K Glover P P Khargonekar and B A FrancisldquoState-space solutions to standard H-2 and Hinfin control prob-lemsrdquo IEEE Transactions on Automatic Control vol 34 no 8pp 831ndash847 1989

[11] M Katebi M Grimble and Y Zhang ldquoHinfin robust controldesign for dynamic ship positioningrdquo IEE Proceedings - ControlTheory and Applications vol 144 no 2 pp 110ndash120 1997

[12] D V Balandin and M M Kogan ldquoLMI-based Hinfin-optimalcontrol with transientsrdquo International Journal of Control vol 83no 8 pp 1664ndash1673 2010

[13] Z Feng J Lam S Xu and S Zhou ldquoHinfin Control withTransients for Singular Systemsrdquo Asian Journal of Control vol18 no 3 pp 817ndash827 2016

[14] P P Khargonekar K M Nagpal and K R Poolla ldquoHinfin controlwith transientsrdquo SIAM Journal onControl andOptimization vol29 no 6 pp 1373ndash1393 1991

[15] Z Zhang Z Zhang and H Zhang ldquoFinite-time stabilityanalysis and stabilization for uncertain continuous-time systemwith time-varying delayrdquo Journal of The Franklin Institute vol352 no 3 pp 1296ndash1317 2015

[16] J Cheng S Chen Z Liu H Wang and J Li ldquoRobust finite-time sampled-data control of linear systems subject to randomoccurring delays and its application to Four-Tank systemrdquoAppliedMathematics andComputation vol 281 pp 55ndash76 2016

[17] W Kang S Zhong K Shi and J Cheng ldquoFinite-time stabilityfor discrete-time system with time-varying delay and nonlinearperturbationsrdquo ISA Transactions vol 60 pp 67ndash73 2016

[18] T I Fossen Handbook of Marine Craft Hydrodynamics andMotion Control John Wiley amp Sons New York USA 2011

[19] D Zhao F Ding L Zhou W Zhang and H Xu ldquoRobustH-infinity Control of Neutral System with Time-Delay forDynamic Positioning Shipsrdquo Mathematical Problems in Engi-neering vol 2015 Article ID 976925 11 pages 2015

[20] F Amato M Ariola and C Cosentino ldquoFinite-time stabilityof linear time-varying systems analysis and controller designrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 55 no 4 pp 1003ndash1008 2010

[21] L Sun andZ Zheng ldquoFinite-time slidingmode trajectory track-ing control of uncertain mechanical systemsrdquo Asian Journal ofControl vol 19 no 1 pp 399ndash404 2017

[22] P Shi Y Zhang and R Agarwal ldquoStochastic finite-time stateestimation for discrete time-delay neural networks withMarko-vian jumpsrdquo Neurocomputing vol 151 part 1 pp 168ndash174 2014

[23] L Zhang S Wang H R Karimi and A Jasra ldquoRobust Finite-Time Control of Switched Linear Systems and Application toa Class of Servomechanism Systemsrdquo IEEEASME Transactionson Mechatronics vol 20 no 5 pp 2476ndash2485 2015

[24] L Wu P Shi and H Gao ldquoState estimation and sliding-mode control of Markovian jump singular systemsrdquo Institute ofElectrical and Electronics Engineers Transactions on AutomaticControl vol 55 no 5 pp 1213ndash1219 2010

[25] Y Q Zhang P Shi S K Nguang and Y D Song ldquoRobust finite-time Hinfin control for uncertain discrete-time singular systemswithMarkovian jumpsrdquo IET ControlTheory ampApplications vol8 no 12 pp 1105ndash1111 2014

[26] S Wang T Shi L Zhang A Jasra and M Zeng ldquoExtendedfinite-time Hinfin control for uncertain switched linear neutralsystemswith time-varying delaysrdquoNeurocomputing vol 152 pp377ndash387 2015

[27] J Hu G Sui S Du and X Li ldquoFinite-time stability of uncer-tain nonlinear systems with time-varying delayrdquo MathematicalProblems in Engineering Article ID 2538904 9 pages 2017

[28] A Seuret and F Gouaisbaut ldquoWirtinger-based integral inequal-ity application to time-delay systemsrdquo Automatica vol 49 no9 pp 2860ndash2866 2013

[29] Z Li Y Bai C Huang and Y Cai ldquoNovel delay-partitioningstabilization approach for networked control system viaWirtinger-based inequalitiesrdquo ISA Transactions vol 61 pp75ndash86 2016

[30] P Park JWKo andC Jeong ldquoReciprocally convex approach tostability of systems with time-varying delaysrdquo Automatica vol47 no 1 pp 235ndash238 2011

[31] K Gu V L Kharitonov and J Chen Stability of Time DelaySystems Birkhauser Boston Mass USA 2003

[32] L El Ghaoui F Oustry and M AitRami ldquoA cone comple-mentarity linearization algorithm for static output-feedbackand related problemsrdquo Institute of Electrical and ElectronicsEngineers Transactions on Automatic Control vol 42 no 8 pp1171ndash1176 1997

[33] R Skjetne T I Fossen and P Kokotovic ldquoAdaptive maneu-vering with experiments for a model ship in a marine controllaboratoryrdquo Automatica vol 41 no 2 pp 289ndash298 2005

[34] N Wang C Qian J-C Sun and Y-C Liu ldquoAdaptive RobustFinite-Time Trajectory Tracking Control of Fully ActuatedMarine Surface Vehiclesrdquo IEEE Transactions on Control SystemsTechnology vol 24 no 4 pp 1454ndash1462 2016

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Page 10: Robust Finite-Time H-Infinity Control with Transients for ...downloads.hindawi.com/journals/mpe/2018/2838749.pdf · ResearchArticle Robust Finite-Time H-Infinity Control with Transients

10 Mathematical Problems in Engineering

Step 2 In this step the approach to design the gain matrix 119870of the controller is provided in forms of LMIs which can becalculated expediently

To linearize the nonlinear terms in (57) we make itmultiply by the followingmatrix fromboth left and right sidesfirstly

diag 11987511minus1 11987511minus1 11987511minus1 119868 11987511minus1 11987511minus1 11987511minus1 11987511minus1 1198801minus1 1198802minus1 119882minus1 119868 (62)

Let us do the definitions as follows11987511 = 11987511minus11198761 = 11987511minus1119876111987511minus11198762 = 11987511minus1119876211987511minus1119882 = 11987511minus111988211987511minus11198801 = 11987511minus1119880111987511minus11198802 = 11987511minus1119880211987511minus111987522 = 11987511minus11198752211987511minus11198851 = 11987511minus1119885111987511minus11198852 = 11987511minus1119885211987511minus1119884 = 11987011987511minus1

(63)

so the inequality Π lt 0 can guarantee Π lt 0 whereΠ99 = minus12119875111198801

minus111987511Π1010 = minus12119875111198802

minus111987511Π1111 = minus11987511119882minus111987511(64)

others Π119894119895 = Π119894119895 (65)

Then based on the inequalities minus(12)119875111198801

minus111987511 le(12)1198801 minus 11987511 minus(12)119875111198802

minus111987511 le (12)1198802 minus 11987511minus11987511119882minus111987511 le 119882 minus 211987511 condition (57) can be convertedinto (41) in form of LMIs

Accordingly similar to the above-mentioned procedures(61) can be transformed into (42) with the Schur complementand the definition of119867119894 = 11987511minus112057911989411987511minus1 (119894 = 1 2 sdot sdot sdot 8)

Therefore conditions (41) to (45) can obtain the con-troller in the form of (7) guaranteeing the desired perfor-mance This completes the proof

Remark 16 The119867infin performance with nonzero initial con-dition is studied in this literature In a practical view it is

hard to suffer the worst-case over a finite time for surfacevessel with nonzero initial condition and it will reducethe conservatism caused by the assumption of zero initialcondition Therefore it is reasonable to introduce such 119867infinperformance index to handle the problem of finite-time119867infincontrol for ship subject to exogenous signals

Remark 17 It is challenging to achieve the result satisfyingthe control objective while it is very different from thestandard 119867infin control because a weighting diagonal matrixis introduced especially to establish the form of LMIs Inthe proof of Theorem 15 inequality (58) is introduced toensure the 119867infin performance with nonzero initial conditionand make it convenient to construct the conditions in formof LMIs

Remark 18 By introducing some new additional matrixvariables Cone Complementarity Linearization algorithm(CCL) [32] can be utilized to linearize the nonlinear termsas an alternative method It will reduce the conservatism atthe expense of increasing the computational burden and thenumber of decision variables

4 Simulations

41 Simulation Settings To demonstrate the effectiveness andsuperiority of the robust finite-time 119867infin control schemesimulation studies are conducted on the parameters of theship CyberShip II [33] which is a 170 scale-replica of a supplyshipwhosemain parameters are shown inTable 1 In additionwe choose the time-varying function like ℎ(119905) = 3sin (1199056 minus1205876) to describe the time delay occurred in the input signal

Unknown external disturbances 120596(119905) with significantlylarge magnitude and high frequency which can be roughlyconsidered as complex environment including ocean windswaves and currents are governed by [34]

120596 (119905) = (9 sin (01120587119905 minus 1205875 )6 sin (03120587119905 + 1205876 )3 sin (02120587119905 + 1205873 )) (66)

with initial conditions 120578(0) = (1 1 01) and 120592(0) =(08 06 01)42 Numerical Simulations We choose 120574 = 13 119879 = 10 119878 =03lowast119868 1198881 = 302 1198882 = 10 120572 = 0001 By usingTheorem 15 thecontrol gain 119870 is calculated by LMIs (41) to (45)

Mathematical Problems in Engineering 11

Table 1 The main parameters of the CyberShip II119898 238000 119883 -20 119883119906 -07225119868119911 17600 119884V -100 119884V -08612119909119892 00460 119884 119903 -00 119884119903 01079119873 119903 -10 119873119903 05

0 20 40 60 80 100Time(s) Time(s)

minus2

0

2

4

6

8

10

12

0 2 4 6 8 100

2

4

6

8

10

12

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

xT(t)Rx(t)xT(t)Rx(t)

c=c=

c1=302

c1=302

Figure 1 State response 119909119879(119905)119877119909(119905) of DP system

119870 = (minus32743 0 0 minus313907 0 00 minus42784 minus01572 0 minus412136 minus129820 minus01389 minus03497 0 minus13362 minus33091) (67)

To demonstrate the superiority and validity three sce-narios as comparative cases are provided here It should bepointed out that the results within 100s are also provided toillustrate the steady-state performance

Case 1 (with different 119878) For comparison purposes we set119878 with different value to demonstrate the effect of 119878 on stateresponse Without loss of generality we set 119878 = 05 lowast 119868 and119878 = 01 lowast 119868 respectivelyCase 2 (119867infin with zero initial condition]) This scenario willshow that the control law proposed in this paper can achievea better transient performance and disturbance rejectioncapability over traditional119867infin control approach for dynamicpositioning system conducted in [11]

Case 3 (a control design based on existing methods) Basedon the approach in [17] applied for discrete-time system acontrol law for dynamic positioning ship with input delay isproposed to illustrate the less conservatism of our designingapproach in practical engineering subsequently

At first the effectiveness of the control law we designedis confirmed from Figures 1ndash6 Figure 1 shows the state

response of 119909119879(119905)119877119909(119905) Obviously from that curve we cansee that the value of 119909119879(119905)119877119909(119905) is far below the value of1198882 = 10 in the presence of disturbance as in Figure 2which means the finite-time full-state feedback controllerdesigned can guarantee the DP system (8) sim (9) FTB with(302 10 119889 10 3 119868) and have less conservatism where weset 119889 = int119879

0120596(119905)119879120596(119905)119889119905 Meanwhile the control law can

stabilize the dynamic positioning shipwithminimum steady-state error in 100s and the control input will be small enoughas shown in Figure 6 From Figures 3ndash5 we can find that allstates in DP system are within a small range As shown inFigures 2ndash4 the angle and the angular velocity in yaw havechanged within 01∘ which means that the heading of thevessel stays in a fixed direction Furthermore the velocitiesin surge and sway are low enough which illustrates thatassumption 2 is reasonable and it meets the requirement ofstabilization control for DP system

To demonstrate the effect of 119878 state responses of systemswith different 119878 are exhibited in Figures 7ndash9 As the valueof 119878 decreases the restriction for 11987511 is more exclusive andthe robustness is stronger during preset time interval as inFigures 7 and 8 while the required control input will increasedramatically as in Figure 9 When the time oversteps the

12 Mathematical Problems in Engineering

0 10 20 30 40 50 60 70 80 90 100Time(s)

minus10

minus5

0

5

10

Dist

urba

nces

(N)

XY

Figure 2 Disturbance in surge sway and yaw

0 50 100Time(s)

minus050

051

152

X(m

)

0 5 10Time(s)

005

115

2

X(m

)

0 50 100Time(s)

minus050

051

152

Y(m

)

0 5 10Time(s)

005

115

2Y(

m)

0 50 100Time(s)

minus01

0

01

02

0 5 10Time(s)

0

01

02

(∘

)

(∘

)

Figure 3 State responses of position and orientation for DP System

0 50 100Time(s)

minus1

0

1

2

0 5 10Time(s)

minus1

0

1

2

0 50 100Time(s)

minus1

0

1

2

v(m

s)

0 5 10Time(s)

minus1

0

1

2

v(m

s)

0 50 100Time(s)

minus01

0

01

02

0 5 10Time(s)

minus01

0

01

02

r(∘ s

)

r(∘ s

)

(ms

)

(m

s)

Figure 4 State responses of velocities for DP system

Mathematical Problems in Engineering 13

minus05 0 05 1 15 2X(m)

minus05

0

05

1

15

2

Y(m

)

Figure 5 Trajectory of the shiprsquos horizontal motion

0 50 100Time(s)

minus30minus20minus10

010

u1(N

)

0 5 10Time(s)

minus30minus20minus10

010

u1(N

)

0 50 100Time(s)

minus30minus20minus10

010

u2(N

)

0 5 10Time(s)

minus30minus20minus10

010

u2(N

)

0 50 100Time(s)

minus15minus1

minus050

05

u3(N

m)

0 5 10Time(s)

minus15minus1

minus050

05

u3(N

m)

Figure 6 Control inputs

0 20 40 60 80 100Time(s)

minus1

0

1

2

3

4

5

6

S=03S=05S=01

0 2 4 6 8 10Time(s)

0

2

4

6

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 7 State response 119909119879(119905)119877119909(119905) with different 119878

14 Mathematical Problems in Engineering

Figure 8 State responses of position and orientation with different 119878

0 50 100Time(s)

minus60minus30

03060

u1(N

)

0 5 10Time(s)

minus40minus20

02040

u1(N

)

0 50 100Time(s)

minus100

0

100

u2(N

)

0 5 10Time(s)

minus40minus20

02040

u2(N

)

0 50 100Time(s)

minus4minus2

024

u3(N

m)

0 5 10Time(s)

minus2

0

2

u3(N

m)

S=03S=05S=01

Figure 9 Control input with different 119878preset time interval all the states can be guaranteed withinthe given upper bound as before

As shown in Figures 10ndash12 the overconservatism oftraditional119867infin control approach applied inDP ship is solvedby the 119867infin control strategy with nonzero initial conditionproposed in this article As is well known the classical 119867infincontrol law is designed under the so-called worst-case normwhile the performance of antidisturbance and transient issacrificed to some degree From Figures 10 and 11 we cansee that the state response can be controlled in a smallerrange and stabilized in a shorter time with a lower overshootshown in Figure 12 Hence a conclusion can be drawn thatthe control law we present can achieve a better disturbance

rejection capability and transient performance while thematter of overconservatism can be resolved at the sametime

Simulation results of Theorem 15 and Case 3 are shownin Figures 13 and 14 to show the better performance ofthe method we proposed We can find that the controllerbased on [17] and 119867infin control approach present in thisliterature can make all states of system within a smallenough region However compared with the controller weconstructed the state response has higher overshoot whichwill affect the control precision and transient performance inpractical engineering Based on the above simulation resultsthe developed controller can achieve a higher accuracy and

Mathematical Problems in Engineering 15

0 20 40 60 80 100Time(s)

minus2

0

2

4

6

8

10

12

Theorem 15Case 2

0 2 4 6 8 10Time(s)

minus2

0

2

4

6

8

10

12

Theorem 15Case 2

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 10 State response 119909119879(119905)119877119909(119905) of Theorem 15 and Case 2

0 50 100Time(s)

minus10123

X(m

)

0 5 10Time(s)

0123

X(m

)

0 50 100Time(s)

minus10123

Y(m

)

0 5 10Time(s)

0123

Y(m

)

0 50 100Time(s)

minus020

0204

0 5 10Time(s)

minus020

0204

Theorem 15Case 2

(∘

)

(∘

)

Figure 11 Position and orientation of Theorem 15 and Case 2

minus05 0 05 1 15 2 25X(m)

minus050

051

152

25

Y(m

)

Theorem 15Case 2

Figure 12 Trajectory of the shiprsquos horizontal motion of Theorem 15 and Case 2

16 Mathematical Problems in Engineering

0 20 40 60 80 100Time(s)

0

1

2

3

4

5

Theorem 15Case 3

0 2 4 6 8 10Time(s)

0

1

2

3

4

5

Theorem 15Case 3

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 13 State response 119909119879(119905)119877119909(119905) of Theorem 15 and Case 3

0 5 10Time(s)

05

1

15

X(m

)

Theorem 15Case 3

0 5 10Time(s)

05

1

15

Y(m

)

0 5 10Time(s)

0

005

01

015

02

(∘

)

Figure 14 Position and orientation of Theorem 15 and Case 3

lower overshoot which means that it is less conservative incomparison with existing result

5 Conclusion

In this paper the robust finite-time 119867infin control with tran-sients for DP system with input delay maintaining all statesof ocean surface vessel into a given threshold over a fixed timeinterval in presence of time-varying disturbance is proposedBased on a novel augmented LKLF with triple integral termsand a method combining theWirtinger inequality and recip-rocally convex approach a less conservative result is derivedIn particular a119867infin performance index with nonzero initialcondition attenuating the overconservatism caused by theassumption of zero initial condition and enhancing thetransient performance of ship subject to external disturbanceis adoptedThe obtained result is formulated in terms of LMIswhich can be easily solved by the standard numerical soft-ware Finally a numerical simulation for a ship is proposed toverify the effectiveness and less conservatism of the controller

we designed In the future the output feedback control can bedeveloped for the ocean surface vessel to avoid the use of thevessel-frame surge sway and yaw velocities

Data Availability

All the data supporting the conclusions of the study have beenprovided in Simulations and readers can access these data in[33 34]

Conflicts of Interest

The authors declare that there are no conflicts of interestrelated to this paper

Acknowledgments

This research was partially supported by the National Sci-ence Technology Support Program of China (Project no51609046)

Mathematical Problems in Engineering 17

References

[1] M Chen B Jiang and R Cui ldquoActuator fault-tolerant controlof ocean surface vessels with input saturationrdquo InternationalJournal of Robust amp Nonlinear Control vol 26 no 3 pp 542ndash564 2016

[2] J Du X Hu and Y Sun ldquoRobust dynamic positioning of shipswith disturbances under input saturationrdquo Automatica vol 73pp 207ndash214 2016

[3] H Ashrafiuon K R Muske L C McNinch and R ASoltan ldquoSliding-mode tracking control of surface vesselsrdquo IEEETransactions on Industrial Electronics vol 55 no 11 pp 4004ndash4012 2008

[4] W Si and X Dong ldquoNeural prescribed performance controlfor uncertain marine surface vessels without accurate initialerrorsrdquoMathematical Problems in Engineering Art ID 261832311 pages 2017

[5] T D Nguyen A J Soslashrensen and S T Quek ldquoDesign of hybridcontroller for dynamic positioning from calm to extreme seaconditionsrdquo Automatica vol 43 no 5 pp 768ndash785 2007

[6] J Du X Hu H Liu and C L Chen ldquoAdaptive robust outputfeedback control for a marine dynamic positioning systembased on a high-gain observerrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 11 pp 2775ndash27862015

[7] H Katayama ldquoNonlinear sampled-data stabilization of dynam-ically positioned shipsrdquo IEEE Transactions on Control SystemsTechnology vol 18 no 2 pp 463ndash468 2010

[8] K J Astrom andCG Kallstrom ldquoIdentification of ship steeringdynamicsrdquo Automatica vol 12 no 1 pp 9ndash22 1976

[9] Z Lei and C Guo ldquoDisturbance rejection control solutionfor ship steering system with uncertain time delayrdquo OceanEngineering vol 95 pp 78ndash83 2015

[10] J C Doyle K Glover P P Khargonekar and B A FrancisldquoState-space solutions to standard H-2 and Hinfin control prob-lemsrdquo IEEE Transactions on Automatic Control vol 34 no 8pp 831ndash847 1989

[11] M Katebi M Grimble and Y Zhang ldquoHinfin robust controldesign for dynamic ship positioningrdquo IEE Proceedings - ControlTheory and Applications vol 144 no 2 pp 110ndash120 1997

[12] D V Balandin and M M Kogan ldquoLMI-based Hinfin-optimalcontrol with transientsrdquo International Journal of Control vol 83no 8 pp 1664ndash1673 2010

[13] Z Feng J Lam S Xu and S Zhou ldquoHinfin Control withTransients for Singular Systemsrdquo Asian Journal of Control vol18 no 3 pp 817ndash827 2016

[14] P P Khargonekar K M Nagpal and K R Poolla ldquoHinfin controlwith transientsrdquo SIAM Journal onControl andOptimization vol29 no 6 pp 1373ndash1393 1991

[15] Z Zhang Z Zhang and H Zhang ldquoFinite-time stabilityanalysis and stabilization for uncertain continuous-time systemwith time-varying delayrdquo Journal of The Franklin Institute vol352 no 3 pp 1296ndash1317 2015

[16] J Cheng S Chen Z Liu H Wang and J Li ldquoRobust finite-time sampled-data control of linear systems subject to randomoccurring delays and its application to Four-Tank systemrdquoAppliedMathematics andComputation vol 281 pp 55ndash76 2016

[17] W Kang S Zhong K Shi and J Cheng ldquoFinite-time stabilityfor discrete-time system with time-varying delay and nonlinearperturbationsrdquo ISA Transactions vol 60 pp 67ndash73 2016

[18] T I Fossen Handbook of Marine Craft Hydrodynamics andMotion Control John Wiley amp Sons New York USA 2011

[19] D Zhao F Ding L Zhou W Zhang and H Xu ldquoRobustH-infinity Control of Neutral System with Time-Delay forDynamic Positioning Shipsrdquo Mathematical Problems in Engi-neering vol 2015 Article ID 976925 11 pages 2015

[20] F Amato M Ariola and C Cosentino ldquoFinite-time stabilityof linear time-varying systems analysis and controller designrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 55 no 4 pp 1003ndash1008 2010

[21] L Sun andZ Zheng ldquoFinite-time slidingmode trajectory track-ing control of uncertain mechanical systemsrdquo Asian Journal ofControl vol 19 no 1 pp 399ndash404 2017

[22] P Shi Y Zhang and R Agarwal ldquoStochastic finite-time stateestimation for discrete time-delay neural networks withMarko-vian jumpsrdquo Neurocomputing vol 151 part 1 pp 168ndash174 2014

[23] L Zhang S Wang H R Karimi and A Jasra ldquoRobust Finite-Time Control of Switched Linear Systems and Application toa Class of Servomechanism Systemsrdquo IEEEASME Transactionson Mechatronics vol 20 no 5 pp 2476ndash2485 2015

[24] L Wu P Shi and H Gao ldquoState estimation and sliding-mode control of Markovian jump singular systemsrdquo Institute ofElectrical and Electronics Engineers Transactions on AutomaticControl vol 55 no 5 pp 1213ndash1219 2010

[25] Y Q Zhang P Shi S K Nguang and Y D Song ldquoRobust finite-time Hinfin control for uncertain discrete-time singular systemswithMarkovian jumpsrdquo IET ControlTheory ampApplications vol8 no 12 pp 1105ndash1111 2014

[26] S Wang T Shi L Zhang A Jasra and M Zeng ldquoExtendedfinite-time Hinfin control for uncertain switched linear neutralsystemswith time-varying delaysrdquoNeurocomputing vol 152 pp377ndash387 2015

[27] J Hu G Sui S Du and X Li ldquoFinite-time stability of uncer-tain nonlinear systems with time-varying delayrdquo MathematicalProblems in Engineering Article ID 2538904 9 pages 2017

[28] A Seuret and F Gouaisbaut ldquoWirtinger-based integral inequal-ity application to time-delay systemsrdquo Automatica vol 49 no9 pp 2860ndash2866 2013

[29] Z Li Y Bai C Huang and Y Cai ldquoNovel delay-partitioningstabilization approach for networked control system viaWirtinger-based inequalitiesrdquo ISA Transactions vol 61 pp75ndash86 2016

[30] P Park JWKo andC Jeong ldquoReciprocally convex approach tostability of systems with time-varying delaysrdquo Automatica vol47 no 1 pp 235ndash238 2011

[31] K Gu V L Kharitonov and J Chen Stability of Time DelaySystems Birkhauser Boston Mass USA 2003

[32] L El Ghaoui F Oustry and M AitRami ldquoA cone comple-mentarity linearization algorithm for static output-feedbackand related problemsrdquo Institute of Electrical and ElectronicsEngineers Transactions on Automatic Control vol 42 no 8 pp1171ndash1176 1997

[33] R Skjetne T I Fossen and P Kokotovic ldquoAdaptive maneu-vering with experiments for a model ship in a marine controllaboratoryrdquo Automatica vol 41 no 2 pp 289ndash298 2005

[34] N Wang C Qian J-C Sun and Y-C Liu ldquoAdaptive RobustFinite-Time Trajectory Tracking Control of Fully ActuatedMarine Surface Vehiclesrdquo IEEE Transactions on Control SystemsTechnology vol 24 no 4 pp 1454ndash1462 2016

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 11: Robust Finite-Time H-Infinity Control with Transients for ...downloads.hindawi.com/journals/mpe/2018/2838749.pdf · ResearchArticle Robust Finite-Time H-Infinity Control with Transients

Mathematical Problems in Engineering 11

Table 1 The main parameters of the CyberShip II119898 238000 119883 -20 119883119906 -07225119868119911 17600 119884V -100 119884V -08612119909119892 00460 119884 119903 -00 119884119903 01079119873 119903 -10 119873119903 05

0 20 40 60 80 100Time(s) Time(s)

minus2

0

2

4

6

8

10

12

0 2 4 6 8 100

2

4

6

8

10

12

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

xT(t)Rx(t)xT(t)Rx(t)

c=c=

c1=302

c1=302

Figure 1 State response 119909119879(119905)119877119909(119905) of DP system

119870 = (minus32743 0 0 minus313907 0 00 minus42784 minus01572 0 minus412136 minus129820 minus01389 minus03497 0 minus13362 minus33091) (67)

To demonstrate the superiority and validity three sce-narios as comparative cases are provided here It should bepointed out that the results within 100s are also provided toillustrate the steady-state performance

Case 1 (with different 119878) For comparison purposes we set119878 with different value to demonstrate the effect of 119878 on stateresponse Without loss of generality we set 119878 = 05 lowast 119868 and119878 = 01 lowast 119868 respectivelyCase 2 (119867infin with zero initial condition]) This scenario willshow that the control law proposed in this paper can achievea better transient performance and disturbance rejectioncapability over traditional119867infin control approach for dynamicpositioning system conducted in [11]

Case 3 (a control design based on existing methods) Basedon the approach in [17] applied for discrete-time system acontrol law for dynamic positioning ship with input delay isproposed to illustrate the less conservatism of our designingapproach in practical engineering subsequently

At first the effectiveness of the control law we designedis confirmed from Figures 1ndash6 Figure 1 shows the state

response of 119909119879(119905)119877119909(119905) Obviously from that curve we cansee that the value of 119909119879(119905)119877119909(119905) is far below the value of1198882 = 10 in the presence of disturbance as in Figure 2which means the finite-time full-state feedback controllerdesigned can guarantee the DP system (8) sim (9) FTB with(302 10 119889 10 3 119868) and have less conservatism where weset 119889 = int119879

0120596(119905)119879120596(119905)119889119905 Meanwhile the control law can

stabilize the dynamic positioning shipwithminimum steady-state error in 100s and the control input will be small enoughas shown in Figure 6 From Figures 3ndash5 we can find that allstates in DP system are within a small range As shown inFigures 2ndash4 the angle and the angular velocity in yaw havechanged within 01∘ which means that the heading of thevessel stays in a fixed direction Furthermore the velocitiesin surge and sway are low enough which illustrates thatassumption 2 is reasonable and it meets the requirement ofstabilization control for DP system

To demonstrate the effect of 119878 state responses of systemswith different 119878 are exhibited in Figures 7ndash9 As the valueof 119878 decreases the restriction for 11987511 is more exclusive andthe robustness is stronger during preset time interval as inFigures 7 and 8 while the required control input will increasedramatically as in Figure 9 When the time oversteps the

12 Mathematical Problems in Engineering

0 10 20 30 40 50 60 70 80 90 100Time(s)

minus10

minus5

0

5

10

Dist

urba

nces

(N)

XY

Figure 2 Disturbance in surge sway and yaw

0 50 100Time(s)

minus050

051

152

X(m

)

0 5 10Time(s)

005

115

2

X(m

)

0 50 100Time(s)

minus050

051

152

Y(m

)

0 5 10Time(s)

005

115

2Y(

m)

0 50 100Time(s)

minus01

0

01

02

0 5 10Time(s)

0

01

02

(∘

)

(∘

)

Figure 3 State responses of position and orientation for DP System

0 50 100Time(s)

minus1

0

1

2

0 5 10Time(s)

minus1

0

1

2

0 50 100Time(s)

minus1

0

1

2

v(m

s)

0 5 10Time(s)

minus1

0

1

2

v(m

s)

0 50 100Time(s)

minus01

0

01

02

0 5 10Time(s)

minus01

0

01

02

r(∘ s

)

r(∘ s

)

(ms

)

(m

s)

Figure 4 State responses of velocities for DP system

Mathematical Problems in Engineering 13

minus05 0 05 1 15 2X(m)

minus05

0

05

1

15

2

Y(m

)

Figure 5 Trajectory of the shiprsquos horizontal motion

0 50 100Time(s)

minus30minus20minus10

010

u1(N

)

0 5 10Time(s)

minus30minus20minus10

010

u1(N

)

0 50 100Time(s)

minus30minus20minus10

010

u2(N

)

0 5 10Time(s)

minus30minus20minus10

010

u2(N

)

0 50 100Time(s)

minus15minus1

minus050

05

u3(N

m)

0 5 10Time(s)

minus15minus1

minus050

05

u3(N

m)

Figure 6 Control inputs

0 20 40 60 80 100Time(s)

minus1

0

1

2

3

4

5

6

S=03S=05S=01

0 2 4 6 8 10Time(s)

0

2

4

6

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 7 State response 119909119879(119905)119877119909(119905) with different 119878

14 Mathematical Problems in Engineering

Figure 8 State responses of position and orientation with different 119878

0 50 100Time(s)

minus60minus30

03060

u1(N

)

0 5 10Time(s)

minus40minus20

02040

u1(N

)

0 50 100Time(s)

minus100

0

100

u2(N

)

0 5 10Time(s)

minus40minus20

02040

u2(N

)

0 50 100Time(s)

minus4minus2

024

u3(N

m)

0 5 10Time(s)

minus2

0

2

u3(N

m)

S=03S=05S=01

Figure 9 Control input with different 119878preset time interval all the states can be guaranteed withinthe given upper bound as before

As shown in Figures 10ndash12 the overconservatism oftraditional119867infin control approach applied inDP ship is solvedby the 119867infin control strategy with nonzero initial conditionproposed in this article As is well known the classical 119867infincontrol law is designed under the so-called worst-case normwhile the performance of antidisturbance and transient issacrificed to some degree From Figures 10 and 11 we cansee that the state response can be controlled in a smallerrange and stabilized in a shorter time with a lower overshootshown in Figure 12 Hence a conclusion can be drawn thatthe control law we present can achieve a better disturbance

rejection capability and transient performance while thematter of overconservatism can be resolved at the sametime

Simulation results of Theorem 15 and Case 3 are shownin Figures 13 and 14 to show the better performance ofthe method we proposed We can find that the controllerbased on [17] and 119867infin control approach present in thisliterature can make all states of system within a smallenough region However compared with the controller weconstructed the state response has higher overshoot whichwill affect the control precision and transient performance inpractical engineering Based on the above simulation resultsthe developed controller can achieve a higher accuracy and

Mathematical Problems in Engineering 15

0 20 40 60 80 100Time(s)

minus2

0

2

4

6

8

10

12

Theorem 15Case 2

0 2 4 6 8 10Time(s)

minus2

0

2

4

6

8

10

12

Theorem 15Case 2

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 10 State response 119909119879(119905)119877119909(119905) of Theorem 15 and Case 2

0 50 100Time(s)

minus10123

X(m

)

0 5 10Time(s)

0123

X(m

)

0 50 100Time(s)

minus10123

Y(m

)

0 5 10Time(s)

0123

Y(m

)

0 50 100Time(s)

minus020

0204

0 5 10Time(s)

minus020

0204

Theorem 15Case 2

(∘

)

(∘

)

Figure 11 Position and orientation of Theorem 15 and Case 2

minus05 0 05 1 15 2 25X(m)

minus050

051

152

25

Y(m

)

Theorem 15Case 2

Figure 12 Trajectory of the shiprsquos horizontal motion of Theorem 15 and Case 2

16 Mathematical Problems in Engineering

0 20 40 60 80 100Time(s)

0

1

2

3

4

5

Theorem 15Case 3

0 2 4 6 8 10Time(s)

0

1

2

3

4

5

Theorem 15Case 3

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 13 State response 119909119879(119905)119877119909(119905) of Theorem 15 and Case 3

0 5 10Time(s)

05

1

15

X(m

)

Theorem 15Case 3

0 5 10Time(s)

05

1

15

Y(m

)

0 5 10Time(s)

0

005

01

015

02

(∘

)

Figure 14 Position and orientation of Theorem 15 and Case 3

lower overshoot which means that it is less conservative incomparison with existing result

5 Conclusion

In this paper the robust finite-time 119867infin control with tran-sients for DP system with input delay maintaining all statesof ocean surface vessel into a given threshold over a fixed timeinterval in presence of time-varying disturbance is proposedBased on a novel augmented LKLF with triple integral termsand a method combining theWirtinger inequality and recip-rocally convex approach a less conservative result is derivedIn particular a119867infin performance index with nonzero initialcondition attenuating the overconservatism caused by theassumption of zero initial condition and enhancing thetransient performance of ship subject to external disturbanceis adoptedThe obtained result is formulated in terms of LMIswhich can be easily solved by the standard numerical soft-ware Finally a numerical simulation for a ship is proposed toverify the effectiveness and less conservatism of the controller

we designed In the future the output feedback control can bedeveloped for the ocean surface vessel to avoid the use of thevessel-frame surge sway and yaw velocities

Data Availability

All the data supporting the conclusions of the study have beenprovided in Simulations and readers can access these data in[33 34]

Conflicts of Interest

The authors declare that there are no conflicts of interestrelated to this paper

Acknowledgments

This research was partially supported by the National Sci-ence Technology Support Program of China (Project no51609046)

Mathematical Problems in Engineering 17

References

[1] M Chen B Jiang and R Cui ldquoActuator fault-tolerant controlof ocean surface vessels with input saturationrdquo InternationalJournal of Robust amp Nonlinear Control vol 26 no 3 pp 542ndash564 2016

[2] J Du X Hu and Y Sun ldquoRobust dynamic positioning of shipswith disturbances under input saturationrdquo Automatica vol 73pp 207ndash214 2016

[3] H Ashrafiuon K R Muske L C McNinch and R ASoltan ldquoSliding-mode tracking control of surface vesselsrdquo IEEETransactions on Industrial Electronics vol 55 no 11 pp 4004ndash4012 2008

[4] W Si and X Dong ldquoNeural prescribed performance controlfor uncertain marine surface vessels without accurate initialerrorsrdquoMathematical Problems in Engineering Art ID 261832311 pages 2017

[5] T D Nguyen A J Soslashrensen and S T Quek ldquoDesign of hybridcontroller for dynamic positioning from calm to extreme seaconditionsrdquo Automatica vol 43 no 5 pp 768ndash785 2007

[6] J Du X Hu H Liu and C L Chen ldquoAdaptive robust outputfeedback control for a marine dynamic positioning systembased on a high-gain observerrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 11 pp 2775ndash27862015

[7] H Katayama ldquoNonlinear sampled-data stabilization of dynam-ically positioned shipsrdquo IEEE Transactions on Control SystemsTechnology vol 18 no 2 pp 463ndash468 2010

[8] K J Astrom andCG Kallstrom ldquoIdentification of ship steeringdynamicsrdquo Automatica vol 12 no 1 pp 9ndash22 1976

[9] Z Lei and C Guo ldquoDisturbance rejection control solutionfor ship steering system with uncertain time delayrdquo OceanEngineering vol 95 pp 78ndash83 2015

[10] J C Doyle K Glover P P Khargonekar and B A FrancisldquoState-space solutions to standard H-2 and Hinfin control prob-lemsrdquo IEEE Transactions on Automatic Control vol 34 no 8pp 831ndash847 1989

[11] M Katebi M Grimble and Y Zhang ldquoHinfin robust controldesign for dynamic ship positioningrdquo IEE Proceedings - ControlTheory and Applications vol 144 no 2 pp 110ndash120 1997

[12] D V Balandin and M M Kogan ldquoLMI-based Hinfin-optimalcontrol with transientsrdquo International Journal of Control vol 83no 8 pp 1664ndash1673 2010

[13] Z Feng J Lam S Xu and S Zhou ldquoHinfin Control withTransients for Singular Systemsrdquo Asian Journal of Control vol18 no 3 pp 817ndash827 2016

[14] P P Khargonekar K M Nagpal and K R Poolla ldquoHinfin controlwith transientsrdquo SIAM Journal onControl andOptimization vol29 no 6 pp 1373ndash1393 1991

[15] Z Zhang Z Zhang and H Zhang ldquoFinite-time stabilityanalysis and stabilization for uncertain continuous-time systemwith time-varying delayrdquo Journal of The Franklin Institute vol352 no 3 pp 1296ndash1317 2015

[16] J Cheng S Chen Z Liu H Wang and J Li ldquoRobust finite-time sampled-data control of linear systems subject to randomoccurring delays and its application to Four-Tank systemrdquoAppliedMathematics andComputation vol 281 pp 55ndash76 2016

[17] W Kang S Zhong K Shi and J Cheng ldquoFinite-time stabilityfor discrete-time system with time-varying delay and nonlinearperturbationsrdquo ISA Transactions vol 60 pp 67ndash73 2016

[18] T I Fossen Handbook of Marine Craft Hydrodynamics andMotion Control John Wiley amp Sons New York USA 2011

[19] D Zhao F Ding L Zhou W Zhang and H Xu ldquoRobustH-infinity Control of Neutral System with Time-Delay forDynamic Positioning Shipsrdquo Mathematical Problems in Engi-neering vol 2015 Article ID 976925 11 pages 2015

[20] F Amato M Ariola and C Cosentino ldquoFinite-time stabilityof linear time-varying systems analysis and controller designrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 55 no 4 pp 1003ndash1008 2010

[21] L Sun andZ Zheng ldquoFinite-time slidingmode trajectory track-ing control of uncertain mechanical systemsrdquo Asian Journal ofControl vol 19 no 1 pp 399ndash404 2017

[22] P Shi Y Zhang and R Agarwal ldquoStochastic finite-time stateestimation for discrete time-delay neural networks withMarko-vian jumpsrdquo Neurocomputing vol 151 part 1 pp 168ndash174 2014

[23] L Zhang S Wang H R Karimi and A Jasra ldquoRobust Finite-Time Control of Switched Linear Systems and Application toa Class of Servomechanism Systemsrdquo IEEEASME Transactionson Mechatronics vol 20 no 5 pp 2476ndash2485 2015

[24] L Wu P Shi and H Gao ldquoState estimation and sliding-mode control of Markovian jump singular systemsrdquo Institute ofElectrical and Electronics Engineers Transactions on AutomaticControl vol 55 no 5 pp 1213ndash1219 2010

[25] Y Q Zhang P Shi S K Nguang and Y D Song ldquoRobust finite-time Hinfin control for uncertain discrete-time singular systemswithMarkovian jumpsrdquo IET ControlTheory ampApplications vol8 no 12 pp 1105ndash1111 2014

[26] S Wang T Shi L Zhang A Jasra and M Zeng ldquoExtendedfinite-time Hinfin control for uncertain switched linear neutralsystemswith time-varying delaysrdquoNeurocomputing vol 152 pp377ndash387 2015

[27] J Hu G Sui S Du and X Li ldquoFinite-time stability of uncer-tain nonlinear systems with time-varying delayrdquo MathematicalProblems in Engineering Article ID 2538904 9 pages 2017

[28] A Seuret and F Gouaisbaut ldquoWirtinger-based integral inequal-ity application to time-delay systemsrdquo Automatica vol 49 no9 pp 2860ndash2866 2013

[29] Z Li Y Bai C Huang and Y Cai ldquoNovel delay-partitioningstabilization approach for networked control system viaWirtinger-based inequalitiesrdquo ISA Transactions vol 61 pp75ndash86 2016

[30] P Park JWKo andC Jeong ldquoReciprocally convex approach tostability of systems with time-varying delaysrdquo Automatica vol47 no 1 pp 235ndash238 2011

[31] K Gu V L Kharitonov and J Chen Stability of Time DelaySystems Birkhauser Boston Mass USA 2003

[32] L El Ghaoui F Oustry and M AitRami ldquoA cone comple-mentarity linearization algorithm for static output-feedbackand related problemsrdquo Institute of Electrical and ElectronicsEngineers Transactions on Automatic Control vol 42 no 8 pp1171ndash1176 1997

[33] R Skjetne T I Fossen and P Kokotovic ldquoAdaptive maneu-vering with experiments for a model ship in a marine controllaboratoryrdquo Automatica vol 41 no 2 pp 289ndash298 2005

[34] N Wang C Qian J-C Sun and Y-C Liu ldquoAdaptive RobustFinite-Time Trajectory Tracking Control of Fully ActuatedMarine Surface Vehiclesrdquo IEEE Transactions on Control SystemsTechnology vol 24 no 4 pp 1454ndash1462 2016

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 12: Robust Finite-Time H-Infinity Control with Transients for ...downloads.hindawi.com/journals/mpe/2018/2838749.pdf · ResearchArticle Robust Finite-Time H-Infinity Control with Transients

12 Mathematical Problems in Engineering

0 10 20 30 40 50 60 70 80 90 100Time(s)

minus10

minus5

0

5

10

Dist

urba

nces

(N)

XY

Figure 2 Disturbance in surge sway and yaw

0 50 100Time(s)

minus050

051

152

X(m

)

0 5 10Time(s)

005

115

2

X(m

)

0 50 100Time(s)

minus050

051

152

Y(m

)

0 5 10Time(s)

005

115

2Y(

m)

0 50 100Time(s)

minus01

0

01

02

0 5 10Time(s)

0

01

02

(∘

)

(∘

)

Figure 3 State responses of position and orientation for DP System

0 50 100Time(s)

minus1

0

1

2

0 5 10Time(s)

minus1

0

1

2

0 50 100Time(s)

minus1

0

1

2

v(m

s)

0 5 10Time(s)

minus1

0

1

2

v(m

s)

0 50 100Time(s)

minus01

0

01

02

0 5 10Time(s)

minus01

0

01

02

r(∘ s

)

r(∘ s

)

(ms

)

(m

s)

Figure 4 State responses of velocities for DP system

Mathematical Problems in Engineering 13

minus05 0 05 1 15 2X(m)

minus05

0

05

1

15

2

Y(m

)

Figure 5 Trajectory of the shiprsquos horizontal motion

0 50 100Time(s)

minus30minus20minus10

010

u1(N

)

0 5 10Time(s)

minus30minus20minus10

010

u1(N

)

0 50 100Time(s)

minus30minus20minus10

010

u2(N

)

0 5 10Time(s)

minus30minus20minus10

010

u2(N

)

0 50 100Time(s)

minus15minus1

minus050

05

u3(N

m)

0 5 10Time(s)

minus15minus1

minus050

05

u3(N

m)

Figure 6 Control inputs

0 20 40 60 80 100Time(s)

minus1

0

1

2

3

4

5

6

S=03S=05S=01

0 2 4 6 8 10Time(s)

0

2

4

6

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 7 State response 119909119879(119905)119877119909(119905) with different 119878

14 Mathematical Problems in Engineering

Figure 8 State responses of position and orientation with different 119878

0 50 100Time(s)

minus60minus30

03060

u1(N

)

0 5 10Time(s)

minus40minus20

02040

u1(N

)

0 50 100Time(s)

minus100

0

100

u2(N

)

0 5 10Time(s)

minus40minus20

02040

u2(N

)

0 50 100Time(s)

minus4minus2

024

u3(N

m)

0 5 10Time(s)

minus2

0

2

u3(N

m)

S=03S=05S=01

Figure 9 Control input with different 119878preset time interval all the states can be guaranteed withinthe given upper bound as before

As shown in Figures 10ndash12 the overconservatism oftraditional119867infin control approach applied inDP ship is solvedby the 119867infin control strategy with nonzero initial conditionproposed in this article As is well known the classical 119867infincontrol law is designed under the so-called worst-case normwhile the performance of antidisturbance and transient issacrificed to some degree From Figures 10 and 11 we cansee that the state response can be controlled in a smallerrange and stabilized in a shorter time with a lower overshootshown in Figure 12 Hence a conclusion can be drawn thatthe control law we present can achieve a better disturbance

rejection capability and transient performance while thematter of overconservatism can be resolved at the sametime

Simulation results of Theorem 15 and Case 3 are shownin Figures 13 and 14 to show the better performance ofthe method we proposed We can find that the controllerbased on [17] and 119867infin control approach present in thisliterature can make all states of system within a smallenough region However compared with the controller weconstructed the state response has higher overshoot whichwill affect the control precision and transient performance inpractical engineering Based on the above simulation resultsthe developed controller can achieve a higher accuracy and

Mathematical Problems in Engineering 15

0 20 40 60 80 100Time(s)

minus2

0

2

4

6

8

10

12

Theorem 15Case 2

0 2 4 6 8 10Time(s)

minus2

0

2

4

6

8

10

12

Theorem 15Case 2

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 10 State response 119909119879(119905)119877119909(119905) of Theorem 15 and Case 2

0 50 100Time(s)

minus10123

X(m

)

0 5 10Time(s)

0123

X(m

)

0 50 100Time(s)

minus10123

Y(m

)

0 5 10Time(s)

0123

Y(m

)

0 50 100Time(s)

minus020

0204

0 5 10Time(s)

minus020

0204

Theorem 15Case 2

(∘

)

(∘

)

Figure 11 Position and orientation of Theorem 15 and Case 2

minus05 0 05 1 15 2 25X(m)

minus050

051

152

25

Y(m

)

Theorem 15Case 2

Figure 12 Trajectory of the shiprsquos horizontal motion of Theorem 15 and Case 2

16 Mathematical Problems in Engineering

0 20 40 60 80 100Time(s)

0

1

2

3

4

5

Theorem 15Case 3

0 2 4 6 8 10Time(s)

0

1

2

3

4

5

Theorem 15Case 3

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 13 State response 119909119879(119905)119877119909(119905) of Theorem 15 and Case 3

0 5 10Time(s)

05

1

15

X(m

)

Theorem 15Case 3

0 5 10Time(s)

05

1

15

Y(m

)

0 5 10Time(s)

0

005

01

015

02

(∘

)

Figure 14 Position and orientation of Theorem 15 and Case 3

lower overshoot which means that it is less conservative incomparison with existing result

5 Conclusion

In this paper the robust finite-time 119867infin control with tran-sients for DP system with input delay maintaining all statesof ocean surface vessel into a given threshold over a fixed timeinterval in presence of time-varying disturbance is proposedBased on a novel augmented LKLF with triple integral termsand a method combining theWirtinger inequality and recip-rocally convex approach a less conservative result is derivedIn particular a119867infin performance index with nonzero initialcondition attenuating the overconservatism caused by theassumption of zero initial condition and enhancing thetransient performance of ship subject to external disturbanceis adoptedThe obtained result is formulated in terms of LMIswhich can be easily solved by the standard numerical soft-ware Finally a numerical simulation for a ship is proposed toverify the effectiveness and less conservatism of the controller

we designed In the future the output feedback control can bedeveloped for the ocean surface vessel to avoid the use of thevessel-frame surge sway and yaw velocities

Data Availability

All the data supporting the conclusions of the study have beenprovided in Simulations and readers can access these data in[33 34]

Conflicts of Interest

The authors declare that there are no conflicts of interestrelated to this paper

Acknowledgments

This research was partially supported by the National Sci-ence Technology Support Program of China (Project no51609046)

Mathematical Problems in Engineering 17

References

[1] M Chen B Jiang and R Cui ldquoActuator fault-tolerant controlof ocean surface vessels with input saturationrdquo InternationalJournal of Robust amp Nonlinear Control vol 26 no 3 pp 542ndash564 2016

[2] J Du X Hu and Y Sun ldquoRobust dynamic positioning of shipswith disturbances under input saturationrdquo Automatica vol 73pp 207ndash214 2016

[3] H Ashrafiuon K R Muske L C McNinch and R ASoltan ldquoSliding-mode tracking control of surface vesselsrdquo IEEETransactions on Industrial Electronics vol 55 no 11 pp 4004ndash4012 2008

[4] W Si and X Dong ldquoNeural prescribed performance controlfor uncertain marine surface vessels without accurate initialerrorsrdquoMathematical Problems in Engineering Art ID 261832311 pages 2017

[5] T D Nguyen A J Soslashrensen and S T Quek ldquoDesign of hybridcontroller for dynamic positioning from calm to extreme seaconditionsrdquo Automatica vol 43 no 5 pp 768ndash785 2007

[6] J Du X Hu H Liu and C L Chen ldquoAdaptive robust outputfeedback control for a marine dynamic positioning systembased on a high-gain observerrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 11 pp 2775ndash27862015

[7] H Katayama ldquoNonlinear sampled-data stabilization of dynam-ically positioned shipsrdquo IEEE Transactions on Control SystemsTechnology vol 18 no 2 pp 463ndash468 2010

[8] K J Astrom andCG Kallstrom ldquoIdentification of ship steeringdynamicsrdquo Automatica vol 12 no 1 pp 9ndash22 1976

[9] Z Lei and C Guo ldquoDisturbance rejection control solutionfor ship steering system with uncertain time delayrdquo OceanEngineering vol 95 pp 78ndash83 2015

[10] J C Doyle K Glover P P Khargonekar and B A FrancisldquoState-space solutions to standard H-2 and Hinfin control prob-lemsrdquo IEEE Transactions on Automatic Control vol 34 no 8pp 831ndash847 1989

[11] M Katebi M Grimble and Y Zhang ldquoHinfin robust controldesign for dynamic ship positioningrdquo IEE Proceedings - ControlTheory and Applications vol 144 no 2 pp 110ndash120 1997

[12] D V Balandin and M M Kogan ldquoLMI-based Hinfin-optimalcontrol with transientsrdquo International Journal of Control vol 83no 8 pp 1664ndash1673 2010

[13] Z Feng J Lam S Xu and S Zhou ldquoHinfin Control withTransients for Singular Systemsrdquo Asian Journal of Control vol18 no 3 pp 817ndash827 2016

[14] P P Khargonekar K M Nagpal and K R Poolla ldquoHinfin controlwith transientsrdquo SIAM Journal onControl andOptimization vol29 no 6 pp 1373ndash1393 1991

[15] Z Zhang Z Zhang and H Zhang ldquoFinite-time stabilityanalysis and stabilization for uncertain continuous-time systemwith time-varying delayrdquo Journal of The Franklin Institute vol352 no 3 pp 1296ndash1317 2015

[16] J Cheng S Chen Z Liu H Wang and J Li ldquoRobust finite-time sampled-data control of linear systems subject to randomoccurring delays and its application to Four-Tank systemrdquoAppliedMathematics andComputation vol 281 pp 55ndash76 2016

[17] W Kang S Zhong K Shi and J Cheng ldquoFinite-time stabilityfor discrete-time system with time-varying delay and nonlinearperturbationsrdquo ISA Transactions vol 60 pp 67ndash73 2016

[18] T I Fossen Handbook of Marine Craft Hydrodynamics andMotion Control John Wiley amp Sons New York USA 2011

[19] D Zhao F Ding L Zhou W Zhang and H Xu ldquoRobustH-infinity Control of Neutral System with Time-Delay forDynamic Positioning Shipsrdquo Mathematical Problems in Engi-neering vol 2015 Article ID 976925 11 pages 2015

[20] F Amato M Ariola and C Cosentino ldquoFinite-time stabilityof linear time-varying systems analysis and controller designrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 55 no 4 pp 1003ndash1008 2010

[21] L Sun andZ Zheng ldquoFinite-time slidingmode trajectory track-ing control of uncertain mechanical systemsrdquo Asian Journal ofControl vol 19 no 1 pp 399ndash404 2017

[22] P Shi Y Zhang and R Agarwal ldquoStochastic finite-time stateestimation for discrete time-delay neural networks withMarko-vian jumpsrdquo Neurocomputing vol 151 part 1 pp 168ndash174 2014

[23] L Zhang S Wang H R Karimi and A Jasra ldquoRobust Finite-Time Control of Switched Linear Systems and Application toa Class of Servomechanism Systemsrdquo IEEEASME Transactionson Mechatronics vol 20 no 5 pp 2476ndash2485 2015

[24] L Wu P Shi and H Gao ldquoState estimation and sliding-mode control of Markovian jump singular systemsrdquo Institute ofElectrical and Electronics Engineers Transactions on AutomaticControl vol 55 no 5 pp 1213ndash1219 2010

[25] Y Q Zhang P Shi S K Nguang and Y D Song ldquoRobust finite-time Hinfin control for uncertain discrete-time singular systemswithMarkovian jumpsrdquo IET ControlTheory ampApplications vol8 no 12 pp 1105ndash1111 2014

[26] S Wang T Shi L Zhang A Jasra and M Zeng ldquoExtendedfinite-time Hinfin control for uncertain switched linear neutralsystemswith time-varying delaysrdquoNeurocomputing vol 152 pp377ndash387 2015

[27] J Hu G Sui S Du and X Li ldquoFinite-time stability of uncer-tain nonlinear systems with time-varying delayrdquo MathematicalProblems in Engineering Article ID 2538904 9 pages 2017

[28] A Seuret and F Gouaisbaut ldquoWirtinger-based integral inequal-ity application to time-delay systemsrdquo Automatica vol 49 no9 pp 2860ndash2866 2013

[29] Z Li Y Bai C Huang and Y Cai ldquoNovel delay-partitioningstabilization approach for networked control system viaWirtinger-based inequalitiesrdquo ISA Transactions vol 61 pp75ndash86 2016

[30] P Park JWKo andC Jeong ldquoReciprocally convex approach tostability of systems with time-varying delaysrdquo Automatica vol47 no 1 pp 235ndash238 2011

[31] K Gu V L Kharitonov and J Chen Stability of Time DelaySystems Birkhauser Boston Mass USA 2003

[32] L El Ghaoui F Oustry and M AitRami ldquoA cone comple-mentarity linearization algorithm for static output-feedbackand related problemsrdquo Institute of Electrical and ElectronicsEngineers Transactions on Automatic Control vol 42 no 8 pp1171ndash1176 1997

[33] R Skjetne T I Fossen and P Kokotovic ldquoAdaptive maneu-vering with experiments for a model ship in a marine controllaboratoryrdquo Automatica vol 41 no 2 pp 289ndash298 2005

[34] N Wang C Qian J-C Sun and Y-C Liu ldquoAdaptive RobustFinite-Time Trajectory Tracking Control of Fully ActuatedMarine Surface Vehiclesrdquo IEEE Transactions on Control SystemsTechnology vol 24 no 4 pp 1454ndash1462 2016

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 13: Robust Finite-Time H-Infinity Control with Transients for ...downloads.hindawi.com/journals/mpe/2018/2838749.pdf · ResearchArticle Robust Finite-Time H-Infinity Control with Transients

Mathematical Problems in Engineering 13

minus05 0 05 1 15 2X(m)

minus05

0

05

1

15

2

Y(m

)

Figure 5 Trajectory of the shiprsquos horizontal motion

0 50 100Time(s)

minus30minus20minus10

010

u1(N

)

0 5 10Time(s)

minus30minus20minus10

010

u1(N

)

0 50 100Time(s)

minus30minus20minus10

010

u2(N

)

0 5 10Time(s)

minus30minus20minus10

010

u2(N

)

0 50 100Time(s)

minus15minus1

minus050

05

u3(N

m)

0 5 10Time(s)

minus15minus1

minus050

05

u3(N

m)

Figure 6 Control inputs

0 20 40 60 80 100Time(s)

minus1

0

1

2

3

4

5

6

S=03S=05S=01

0 2 4 6 8 10Time(s)

0

2

4

6

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 7 State response 119909119879(119905)119877119909(119905) with different 119878

14 Mathematical Problems in Engineering

Figure 8 State responses of position and orientation with different 119878

0 50 100Time(s)

minus60minus30

03060

u1(N

)

0 5 10Time(s)

minus40minus20

02040

u1(N

)

0 50 100Time(s)

minus100

0

100

u2(N

)

0 5 10Time(s)

minus40minus20

02040

u2(N

)

0 50 100Time(s)

minus4minus2

024

u3(N

m)

0 5 10Time(s)

minus2

0

2

u3(N

m)

S=03S=05S=01

Figure 9 Control input with different 119878preset time interval all the states can be guaranteed withinthe given upper bound as before

As shown in Figures 10ndash12 the overconservatism oftraditional119867infin control approach applied inDP ship is solvedby the 119867infin control strategy with nonzero initial conditionproposed in this article As is well known the classical 119867infincontrol law is designed under the so-called worst-case normwhile the performance of antidisturbance and transient issacrificed to some degree From Figures 10 and 11 we cansee that the state response can be controlled in a smallerrange and stabilized in a shorter time with a lower overshootshown in Figure 12 Hence a conclusion can be drawn thatthe control law we present can achieve a better disturbance

rejection capability and transient performance while thematter of overconservatism can be resolved at the sametime

Simulation results of Theorem 15 and Case 3 are shownin Figures 13 and 14 to show the better performance ofthe method we proposed We can find that the controllerbased on [17] and 119867infin control approach present in thisliterature can make all states of system within a smallenough region However compared with the controller weconstructed the state response has higher overshoot whichwill affect the control precision and transient performance inpractical engineering Based on the above simulation resultsthe developed controller can achieve a higher accuracy and

Mathematical Problems in Engineering 15

0 20 40 60 80 100Time(s)

minus2

0

2

4

6

8

10

12

Theorem 15Case 2

0 2 4 6 8 10Time(s)

minus2

0

2

4

6

8

10

12

Theorem 15Case 2

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 10 State response 119909119879(119905)119877119909(119905) of Theorem 15 and Case 2

0 50 100Time(s)

minus10123

X(m

)

0 5 10Time(s)

0123

X(m

)

0 50 100Time(s)

minus10123

Y(m

)

0 5 10Time(s)

0123

Y(m

)

0 50 100Time(s)

minus020

0204

0 5 10Time(s)

minus020

0204

Theorem 15Case 2

(∘

)

(∘

)

Figure 11 Position and orientation of Theorem 15 and Case 2

minus05 0 05 1 15 2 25X(m)

minus050

051

152

25

Y(m

)

Theorem 15Case 2

Figure 12 Trajectory of the shiprsquos horizontal motion of Theorem 15 and Case 2

16 Mathematical Problems in Engineering

0 20 40 60 80 100Time(s)

0

1

2

3

4

5

Theorem 15Case 3

0 2 4 6 8 10Time(s)

0

1

2

3

4

5

Theorem 15Case 3

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 13 State response 119909119879(119905)119877119909(119905) of Theorem 15 and Case 3

0 5 10Time(s)

05

1

15

X(m

)

Theorem 15Case 3

0 5 10Time(s)

05

1

15

Y(m

)

0 5 10Time(s)

0

005

01

015

02

(∘

)

Figure 14 Position and orientation of Theorem 15 and Case 3

lower overshoot which means that it is less conservative incomparison with existing result

5 Conclusion

In this paper the robust finite-time 119867infin control with tran-sients for DP system with input delay maintaining all statesof ocean surface vessel into a given threshold over a fixed timeinterval in presence of time-varying disturbance is proposedBased on a novel augmented LKLF with triple integral termsand a method combining theWirtinger inequality and recip-rocally convex approach a less conservative result is derivedIn particular a119867infin performance index with nonzero initialcondition attenuating the overconservatism caused by theassumption of zero initial condition and enhancing thetransient performance of ship subject to external disturbanceis adoptedThe obtained result is formulated in terms of LMIswhich can be easily solved by the standard numerical soft-ware Finally a numerical simulation for a ship is proposed toverify the effectiveness and less conservatism of the controller

we designed In the future the output feedback control can bedeveloped for the ocean surface vessel to avoid the use of thevessel-frame surge sway and yaw velocities

Data Availability

All the data supporting the conclusions of the study have beenprovided in Simulations and readers can access these data in[33 34]

Conflicts of Interest

The authors declare that there are no conflicts of interestrelated to this paper

Acknowledgments

This research was partially supported by the National Sci-ence Technology Support Program of China (Project no51609046)

Mathematical Problems in Engineering 17

References

[1] M Chen B Jiang and R Cui ldquoActuator fault-tolerant controlof ocean surface vessels with input saturationrdquo InternationalJournal of Robust amp Nonlinear Control vol 26 no 3 pp 542ndash564 2016

[2] J Du X Hu and Y Sun ldquoRobust dynamic positioning of shipswith disturbances under input saturationrdquo Automatica vol 73pp 207ndash214 2016

[3] H Ashrafiuon K R Muske L C McNinch and R ASoltan ldquoSliding-mode tracking control of surface vesselsrdquo IEEETransactions on Industrial Electronics vol 55 no 11 pp 4004ndash4012 2008

[4] W Si and X Dong ldquoNeural prescribed performance controlfor uncertain marine surface vessels without accurate initialerrorsrdquoMathematical Problems in Engineering Art ID 261832311 pages 2017

[5] T D Nguyen A J Soslashrensen and S T Quek ldquoDesign of hybridcontroller for dynamic positioning from calm to extreme seaconditionsrdquo Automatica vol 43 no 5 pp 768ndash785 2007

[6] J Du X Hu H Liu and C L Chen ldquoAdaptive robust outputfeedback control for a marine dynamic positioning systembased on a high-gain observerrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 11 pp 2775ndash27862015

[7] H Katayama ldquoNonlinear sampled-data stabilization of dynam-ically positioned shipsrdquo IEEE Transactions on Control SystemsTechnology vol 18 no 2 pp 463ndash468 2010

[8] K J Astrom andCG Kallstrom ldquoIdentification of ship steeringdynamicsrdquo Automatica vol 12 no 1 pp 9ndash22 1976

[9] Z Lei and C Guo ldquoDisturbance rejection control solutionfor ship steering system with uncertain time delayrdquo OceanEngineering vol 95 pp 78ndash83 2015

[10] J C Doyle K Glover P P Khargonekar and B A FrancisldquoState-space solutions to standard H-2 and Hinfin control prob-lemsrdquo IEEE Transactions on Automatic Control vol 34 no 8pp 831ndash847 1989

[11] M Katebi M Grimble and Y Zhang ldquoHinfin robust controldesign for dynamic ship positioningrdquo IEE Proceedings - ControlTheory and Applications vol 144 no 2 pp 110ndash120 1997

[12] D V Balandin and M M Kogan ldquoLMI-based Hinfin-optimalcontrol with transientsrdquo International Journal of Control vol 83no 8 pp 1664ndash1673 2010

[13] Z Feng J Lam S Xu and S Zhou ldquoHinfin Control withTransients for Singular Systemsrdquo Asian Journal of Control vol18 no 3 pp 817ndash827 2016

[14] P P Khargonekar K M Nagpal and K R Poolla ldquoHinfin controlwith transientsrdquo SIAM Journal onControl andOptimization vol29 no 6 pp 1373ndash1393 1991

[15] Z Zhang Z Zhang and H Zhang ldquoFinite-time stabilityanalysis and stabilization for uncertain continuous-time systemwith time-varying delayrdquo Journal of The Franklin Institute vol352 no 3 pp 1296ndash1317 2015

[16] J Cheng S Chen Z Liu H Wang and J Li ldquoRobust finite-time sampled-data control of linear systems subject to randomoccurring delays and its application to Four-Tank systemrdquoAppliedMathematics andComputation vol 281 pp 55ndash76 2016

[17] W Kang S Zhong K Shi and J Cheng ldquoFinite-time stabilityfor discrete-time system with time-varying delay and nonlinearperturbationsrdquo ISA Transactions vol 60 pp 67ndash73 2016

[18] T I Fossen Handbook of Marine Craft Hydrodynamics andMotion Control John Wiley amp Sons New York USA 2011

[19] D Zhao F Ding L Zhou W Zhang and H Xu ldquoRobustH-infinity Control of Neutral System with Time-Delay forDynamic Positioning Shipsrdquo Mathematical Problems in Engi-neering vol 2015 Article ID 976925 11 pages 2015

[20] F Amato M Ariola and C Cosentino ldquoFinite-time stabilityof linear time-varying systems analysis and controller designrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 55 no 4 pp 1003ndash1008 2010

[21] L Sun andZ Zheng ldquoFinite-time slidingmode trajectory track-ing control of uncertain mechanical systemsrdquo Asian Journal ofControl vol 19 no 1 pp 399ndash404 2017

[22] P Shi Y Zhang and R Agarwal ldquoStochastic finite-time stateestimation for discrete time-delay neural networks withMarko-vian jumpsrdquo Neurocomputing vol 151 part 1 pp 168ndash174 2014

[23] L Zhang S Wang H R Karimi and A Jasra ldquoRobust Finite-Time Control of Switched Linear Systems and Application toa Class of Servomechanism Systemsrdquo IEEEASME Transactionson Mechatronics vol 20 no 5 pp 2476ndash2485 2015

[24] L Wu P Shi and H Gao ldquoState estimation and sliding-mode control of Markovian jump singular systemsrdquo Institute ofElectrical and Electronics Engineers Transactions on AutomaticControl vol 55 no 5 pp 1213ndash1219 2010

[25] Y Q Zhang P Shi S K Nguang and Y D Song ldquoRobust finite-time Hinfin control for uncertain discrete-time singular systemswithMarkovian jumpsrdquo IET ControlTheory ampApplications vol8 no 12 pp 1105ndash1111 2014

[26] S Wang T Shi L Zhang A Jasra and M Zeng ldquoExtendedfinite-time Hinfin control for uncertain switched linear neutralsystemswith time-varying delaysrdquoNeurocomputing vol 152 pp377ndash387 2015

[27] J Hu G Sui S Du and X Li ldquoFinite-time stability of uncer-tain nonlinear systems with time-varying delayrdquo MathematicalProblems in Engineering Article ID 2538904 9 pages 2017

[28] A Seuret and F Gouaisbaut ldquoWirtinger-based integral inequal-ity application to time-delay systemsrdquo Automatica vol 49 no9 pp 2860ndash2866 2013

[29] Z Li Y Bai C Huang and Y Cai ldquoNovel delay-partitioningstabilization approach for networked control system viaWirtinger-based inequalitiesrdquo ISA Transactions vol 61 pp75ndash86 2016

[30] P Park JWKo andC Jeong ldquoReciprocally convex approach tostability of systems with time-varying delaysrdquo Automatica vol47 no 1 pp 235ndash238 2011

[31] K Gu V L Kharitonov and J Chen Stability of Time DelaySystems Birkhauser Boston Mass USA 2003

[32] L El Ghaoui F Oustry and M AitRami ldquoA cone comple-mentarity linearization algorithm for static output-feedbackand related problemsrdquo Institute of Electrical and ElectronicsEngineers Transactions on Automatic Control vol 42 no 8 pp1171ndash1176 1997

[33] R Skjetne T I Fossen and P Kokotovic ldquoAdaptive maneu-vering with experiments for a model ship in a marine controllaboratoryrdquo Automatica vol 41 no 2 pp 289ndash298 2005

[34] N Wang C Qian J-C Sun and Y-C Liu ldquoAdaptive RobustFinite-Time Trajectory Tracking Control of Fully ActuatedMarine Surface Vehiclesrdquo IEEE Transactions on Control SystemsTechnology vol 24 no 4 pp 1454ndash1462 2016

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 14: Robust Finite-Time H-Infinity Control with Transients for ...downloads.hindawi.com/journals/mpe/2018/2838749.pdf · ResearchArticle Robust Finite-Time H-Infinity Control with Transients

14 Mathematical Problems in Engineering

Figure 8 State responses of position and orientation with different 119878

0 50 100Time(s)

minus60minus30

03060

u1(N

)

0 5 10Time(s)

minus40minus20

02040

u1(N

)

0 50 100Time(s)

minus100

0

100

u2(N

)

0 5 10Time(s)

minus40minus20

02040

u2(N

)

0 50 100Time(s)

minus4minus2

024

u3(N

m)

0 5 10Time(s)

minus2

0

2

u3(N

m)

S=03S=05S=01

Figure 9 Control input with different 119878preset time interval all the states can be guaranteed withinthe given upper bound as before

As shown in Figures 10ndash12 the overconservatism oftraditional119867infin control approach applied inDP ship is solvedby the 119867infin control strategy with nonzero initial conditionproposed in this article As is well known the classical 119867infincontrol law is designed under the so-called worst-case normwhile the performance of antidisturbance and transient issacrificed to some degree From Figures 10 and 11 we cansee that the state response can be controlled in a smallerrange and stabilized in a shorter time with a lower overshootshown in Figure 12 Hence a conclusion can be drawn thatthe control law we present can achieve a better disturbance

rejection capability and transient performance while thematter of overconservatism can be resolved at the sametime

Simulation results of Theorem 15 and Case 3 are shownin Figures 13 and 14 to show the better performance ofthe method we proposed We can find that the controllerbased on [17] and 119867infin control approach present in thisliterature can make all states of system within a smallenough region However compared with the controller weconstructed the state response has higher overshoot whichwill affect the control precision and transient performance inpractical engineering Based on the above simulation resultsthe developed controller can achieve a higher accuracy and

Mathematical Problems in Engineering 15

0 20 40 60 80 100Time(s)

minus2

0

2

4

6

8

10

12

Theorem 15Case 2

0 2 4 6 8 10Time(s)

minus2

0

2

4

6

8

10

12

Theorem 15Case 2

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 10 State response 119909119879(119905)119877119909(119905) of Theorem 15 and Case 2

0 50 100Time(s)

minus10123

X(m

)

0 5 10Time(s)

0123

X(m

)

0 50 100Time(s)

minus10123

Y(m

)

0 5 10Time(s)

0123

Y(m

)

0 50 100Time(s)

minus020

0204

0 5 10Time(s)

minus020

0204

Theorem 15Case 2

(∘

)

(∘

)

Figure 11 Position and orientation of Theorem 15 and Case 2

minus05 0 05 1 15 2 25X(m)

minus050

051

152

25

Y(m

)

Theorem 15Case 2

Figure 12 Trajectory of the shiprsquos horizontal motion of Theorem 15 and Case 2

16 Mathematical Problems in Engineering

0 20 40 60 80 100Time(s)

0

1

2

3

4

5

Theorem 15Case 3

0 2 4 6 8 10Time(s)

0

1

2

3

4

5

Theorem 15Case 3

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 13 State response 119909119879(119905)119877119909(119905) of Theorem 15 and Case 3

0 5 10Time(s)

05

1

15

X(m

)

Theorem 15Case 3

0 5 10Time(s)

05

1

15

Y(m

)

0 5 10Time(s)

0

005

01

015

02

(∘

)

Figure 14 Position and orientation of Theorem 15 and Case 3

lower overshoot which means that it is less conservative incomparison with existing result

5 Conclusion

In this paper the robust finite-time 119867infin control with tran-sients for DP system with input delay maintaining all statesof ocean surface vessel into a given threshold over a fixed timeinterval in presence of time-varying disturbance is proposedBased on a novel augmented LKLF with triple integral termsand a method combining theWirtinger inequality and recip-rocally convex approach a less conservative result is derivedIn particular a119867infin performance index with nonzero initialcondition attenuating the overconservatism caused by theassumption of zero initial condition and enhancing thetransient performance of ship subject to external disturbanceis adoptedThe obtained result is formulated in terms of LMIswhich can be easily solved by the standard numerical soft-ware Finally a numerical simulation for a ship is proposed toverify the effectiveness and less conservatism of the controller

we designed In the future the output feedback control can bedeveloped for the ocean surface vessel to avoid the use of thevessel-frame surge sway and yaw velocities

Data Availability

All the data supporting the conclusions of the study have beenprovided in Simulations and readers can access these data in[33 34]

Conflicts of Interest

The authors declare that there are no conflicts of interestrelated to this paper

Acknowledgments

This research was partially supported by the National Sci-ence Technology Support Program of China (Project no51609046)

Mathematical Problems in Engineering 17

References

[1] M Chen B Jiang and R Cui ldquoActuator fault-tolerant controlof ocean surface vessels with input saturationrdquo InternationalJournal of Robust amp Nonlinear Control vol 26 no 3 pp 542ndash564 2016

[2] J Du X Hu and Y Sun ldquoRobust dynamic positioning of shipswith disturbances under input saturationrdquo Automatica vol 73pp 207ndash214 2016

[3] H Ashrafiuon K R Muske L C McNinch and R ASoltan ldquoSliding-mode tracking control of surface vesselsrdquo IEEETransactions on Industrial Electronics vol 55 no 11 pp 4004ndash4012 2008

[4] W Si and X Dong ldquoNeural prescribed performance controlfor uncertain marine surface vessels without accurate initialerrorsrdquoMathematical Problems in Engineering Art ID 261832311 pages 2017

[5] T D Nguyen A J Soslashrensen and S T Quek ldquoDesign of hybridcontroller for dynamic positioning from calm to extreme seaconditionsrdquo Automatica vol 43 no 5 pp 768ndash785 2007

[6] J Du X Hu H Liu and C L Chen ldquoAdaptive robust outputfeedback control for a marine dynamic positioning systembased on a high-gain observerrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 11 pp 2775ndash27862015

[7] H Katayama ldquoNonlinear sampled-data stabilization of dynam-ically positioned shipsrdquo IEEE Transactions on Control SystemsTechnology vol 18 no 2 pp 463ndash468 2010

[8] K J Astrom andCG Kallstrom ldquoIdentification of ship steeringdynamicsrdquo Automatica vol 12 no 1 pp 9ndash22 1976

[9] Z Lei and C Guo ldquoDisturbance rejection control solutionfor ship steering system with uncertain time delayrdquo OceanEngineering vol 95 pp 78ndash83 2015

[10] J C Doyle K Glover P P Khargonekar and B A FrancisldquoState-space solutions to standard H-2 and Hinfin control prob-lemsrdquo IEEE Transactions on Automatic Control vol 34 no 8pp 831ndash847 1989

[11] M Katebi M Grimble and Y Zhang ldquoHinfin robust controldesign for dynamic ship positioningrdquo IEE Proceedings - ControlTheory and Applications vol 144 no 2 pp 110ndash120 1997

[12] D V Balandin and M M Kogan ldquoLMI-based Hinfin-optimalcontrol with transientsrdquo International Journal of Control vol 83no 8 pp 1664ndash1673 2010

[13] Z Feng J Lam S Xu and S Zhou ldquoHinfin Control withTransients for Singular Systemsrdquo Asian Journal of Control vol18 no 3 pp 817ndash827 2016

[14] P P Khargonekar K M Nagpal and K R Poolla ldquoHinfin controlwith transientsrdquo SIAM Journal onControl andOptimization vol29 no 6 pp 1373ndash1393 1991

[15] Z Zhang Z Zhang and H Zhang ldquoFinite-time stabilityanalysis and stabilization for uncertain continuous-time systemwith time-varying delayrdquo Journal of The Franklin Institute vol352 no 3 pp 1296ndash1317 2015

[16] J Cheng S Chen Z Liu H Wang and J Li ldquoRobust finite-time sampled-data control of linear systems subject to randomoccurring delays and its application to Four-Tank systemrdquoAppliedMathematics andComputation vol 281 pp 55ndash76 2016

[17] W Kang S Zhong K Shi and J Cheng ldquoFinite-time stabilityfor discrete-time system with time-varying delay and nonlinearperturbationsrdquo ISA Transactions vol 60 pp 67ndash73 2016

[18] T I Fossen Handbook of Marine Craft Hydrodynamics andMotion Control John Wiley amp Sons New York USA 2011

[19] D Zhao F Ding L Zhou W Zhang and H Xu ldquoRobustH-infinity Control of Neutral System with Time-Delay forDynamic Positioning Shipsrdquo Mathematical Problems in Engi-neering vol 2015 Article ID 976925 11 pages 2015

[20] F Amato M Ariola and C Cosentino ldquoFinite-time stabilityof linear time-varying systems analysis and controller designrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 55 no 4 pp 1003ndash1008 2010

[21] L Sun andZ Zheng ldquoFinite-time slidingmode trajectory track-ing control of uncertain mechanical systemsrdquo Asian Journal ofControl vol 19 no 1 pp 399ndash404 2017

[22] P Shi Y Zhang and R Agarwal ldquoStochastic finite-time stateestimation for discrete time-delay neural networks withMarko-vian jumpsrdquo Neurocomputing vol 151 part 1 pp 168ndash174 2014

[23] L Zhang S Wang H R Karimi and A Jasra ldquoRobust Finite-Time Control of Switched Linear Systems and Application toa Class of Servomechanism Systemsrdquo IEEEASME Transactionson Mechatronics vol 20 no 5 pp 2476ndash2485 2015

[24] L Wu P Shi and H Gao ldquoState estimation and sliding-mode control of Markovian jump singular systemsrdquo Institute ofElectrical and Electronics Engineers Transactions on AutomaticControl vol 55 no 5 pp 1213ndash1219 2010

[25] Y Q Zhang P Shi S K Nguang and Y D Song ldquoRobust finite-time Hinfin control for uncertain discrete-time singular systemswithMarkovian jumpsrdquo IET ControlTheory ampApplications vol8 no 12 pp 1105ndash1111 2014

[26] S Wang T Shi L Zhang A Jasra and M Zeng ldquoExtendedfinite-time Hinfin control for uncertain switched linear neutralsystemswith time-varying delaysrdquoNeurocomputing vol 152 pp377ndash387 2015

[27] J Hu G Sui S Du and X Li ldquoFinite-time stability of uncer-tain nonlinear systems with time-varying delayrdquo MathematicalProblems in Engineering Article ID 2538904 9 pages 2017

[28] A Seuret and F Gouaisbaut ldquoWirtinger-based integral inequal-ity application to time-delay systemsrdquo Automatica vol 49 no9 pp 2860ndash2866 2013

[29] Z Li Y Bai C Huang and Y Cai ldquoNovel delay-partitioningstabilization approach for networked control system viaWirtinger-based inequalitiesrdquo ISA Transactions vol 61 pp75ndash86 2016

[30] P Park JWKo andC Jeong ldquoReciprocally convex approach tostability of systems with time-varying delaysrdquo Automatica vol47 no 1 pp 235ndash238 2011

[31] K Gu V L Kharitonov and J Chen Stability of Time DelaySystems Birkhauser Boston Mass USA 2003

[32] L El Ghaoui F Oustry and M AitRami ldquoA cone comple-mentarity linearization algorithm for static output-feedbackand related problemsrdquo Institute of Electrical and ElectronicsEngineers Transactions on Automatic Control vol 42 no 8 pp1171ndash1176 1997

[33] R Skjetne T I Fossen and P Kokotovic ldquoAdaptive maneu-vering with experiments for a model ship in a marine controllaboratoryrdquo Automatica vol 41 no 2 pp 289ndash298 2005

[34] N Wang C Qian J-C Sun and Y-C Liu ldquoAdaptive RobustFinite-Time Trajectory Tracking Control of Fully ActuatedMarine Surface Vehiclesrdquo IEEE Transactions on Control SystemsTechnology vol 24 no 4 pp 1454ndash1462 2016

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 15: Robust Finite-Time H-Infinity Control with Transients for ...downloads.hindawi.com/journals/mpe/2018/2838749.pdf · ResearchArticle Robust Finite-Time H-Infinity Control with Transients

Mathematical Problems in Engineering 15

0 20 40 60 80 100Time(s)

minus2

0

2

4

6

8

10

12

Theorem 15Case 2

0 2 4 6 8 10Time(s)

minus2

0

2

4

6

8

10

12

Theorem 15Case 2

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 10 State response 119909119879(119905)119877119909(119905) of Theorem 15 and Case 2

0 50 100Time(s)

minus10123

X(m

)

0 5 10Time(s)

0123

X(m

)

0 50 100Time(s)

minus10123

Y(m

)

0 5 10Time(s)

0123

Y(m

)

0 50 100Time(s)

minus020

0204

0 5 10Time(s)

minus020

0204

Theorem 15Case 2

(∘

)

(∘

)

Figure 11 Position and orientation of Theorem 15 and Case 2

minus05 0 05 1 15 2 25X(m)

minus050

051

152

25

Y(m

)

Theorem 15Case 2

Figure 12 Trajectory of the shiprsquos horizontal motion of Theorem 15 and Case 2

16 Mathematical Problems in Engineering

0 20 40 60 80 100Time(s)

0

1

2

3

4

5

Theorem 15Case 3

0 2 4 6 8 10Time(s)

0

1

2

3

4

5

Theorem 15Case 3

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 13 State response 119909119879(119905)119877119909(119905) of Theorem 15 and Case 3

0 5 10Time(s)

05

1

15

X(m

)

Theorem 15Case 3

0 5 10Time(s)

05

1

15

Y(m

)

0 5 10Time(s)

0

005

01

015

02

(∘

)

Figure 14 Position and orientation of Theorem 15 and Case 3

lower overshoot which means that it is less conservative incomparison with existing result

5 Conclusion

In this paper the robust finite-time 119867infin control with tran-sients for DP system with input delay maintaining all statesof ocean surface vessel into a given threshold over a fixed timeinterval in presence of time-varying disturbance is proposedBased on a novel augmented LKLF with triple integral termsand a method combining theWirtinger inequality and recip-rocally convex approach a less conservative result is derivedIn particular a119867infin performance index with nonzero initialcondition attenuating the overconservatism caused by theassumption of zero initial condition and enhancing thetransient performance of ship subject to external disturbanceis adoptedThe obtained result is formulated in terms of LMIswhich can be easily solved by the standard numerical soft-ware Finally a numerical simulation for a ship is proposed toverify the effectiveness and less conservatism of the controller

we designed In the future the output feedback control can bedeveloped for the ocean surface vessel to avoid the use of thevessel-frame surge sway and yaw velocities

Data Availability

All the data supporting the conclusions of the study have beenprovided in Simulations and readers can access these data in[33 34]

Conflicts of Interest

The authors declare that there are no conflicts of interestrelated to this paper

Acknowledgments

This research was partially supported by the National Sci-ence Technology Support Program of China (Project no51609046)

Mathematical Problems in Engineering 17

References

[1] M Chen B Jiang and R Cui ldquoActuator fault-tolerant controlof ocean surface vessels with input saturationrdquo InternationalJournal of Robust amp Nonlinear Control vol 26 no 3 pp 542ndash564 2016

[2] J Du X Hu and Y Sun ldquoRobust dynamic positioning of shipswith disturbances under input saturationrdquo Automatica vol 73pp 207ndash214 2016

[3] H Ashrafiuon K R Muske L C McNinch and R ASoltan ldquoSliding-mode tracking control of surface vesselsrdquo IEEETransactions on Industrial Electronics vol 55 no 11 pp 4004ndash4012 2008

[4] W Si and X Dong ldquoNeural prescribed performance controlfor uncertain marine surface vessels without accurate initialerrorsrdquoMathematical Problems in Engineering Art ID 261832311 pages 2017

[5] T D Nguyen A J Soslashrensen and S T Quek ldquoDesign of hybridcontroller for dynamic positioning from calm to extreme seaconditionsrdquo Automatica vol 43 no 5 pp 768ndash785 2007

[6] J Du X Hu H Liu and C L Chen ldquoAdaptive robust outputfeedback control for a marine dynamic positioning systembased on a high-gain observerrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 11 pp 2775ndash27862015

[7] H Katayama ldquoNonlinear sampled-data stabilization of dynam-ically positioned shipsrdquo IEEE Transactions on Control SystemsTechnology vol 18 no 2 pp 463ndash468 2010

[8] K J Astrom andCG Kallstrom ldquoIdentification of ship steeringdynamicsrdquo Automatica vol 12 no 1 pp 9ndash22 1976

[9] Z Lei and C Guo ldquoDisturbance rejection control solutionfor ship steering system with uncertain time delayrdquo OceanEngineering vol 95 pp 78ndash83 2015

[10] J C Doyle K Glover P P Khargonekar and B A FrancisldquoState-space solutions to standard H-2 and Hinfin control prob-lemsrdquo IEEE Transactions on Automatic Control vol 34 no 8pp 831ndash847 1989

[11] M Katebi M Grimble and Y Zhang ldquoHinfin robust controldesign for dynamic ship positioningrdquo IEE Proceedings - ControlTheory and Applications vol 144 no 2 pp 110ndash120 1997

[12] D V Balandin and M M Kogan ldquoLMI-based Hinfin-optimalcontrol with transientsrdquo International Journal of Control vol 83no 8 pp 1664ndash1673 2010

[13] Z Feng J Lam S Xu and S Zhou ldquoHinfin Control withTransients for Singular Systemsrdquo Asian Journal of Control vol18 no 3 pp 817ndash827 2016

[14] P P Khargonekar K M Nagpal and K R Poolla ldquoHinfin controlwith transientsrdquo SIAM Journal onControl andOptimization vol29 no 6 pp 1373ndash1393 1991

[15] Z Zhang Z Zhang and H Zhang ldquoFinite-time stabilityanalysis and stabilization for uncertain continuous-time systemwith time-varying delayrdquo Journal of The Franklin Institute vol352 no 3 pp 1296ndash1317 2015

[16] J Cheng S Chen Z Liu H Wang and J Li ldquoRobust finite-time sampled-data control of linear systems subject to randomoccurring delays and its application to Four-Tank systemrdquoAppliedMathematics andComputation vol 281 pp 55ndash76 2016

[17] W Kang S Zhong K Shi and J Cheng ldquoFinite-time stabilityfor discrete-time system with time-varying delay and nonlinearperturbationsrdquo ISA Transactions vol 60 pp 67ndash73 2016

[18] T I Fossen Handbook of Marine Craft Hydrodynamics andMotion Control John Wiley amp Sons New York USA 2011

[19] D Zhao F Ding L Zhou W Zhang and H Xu ldquoRobustH-infinity Control of Neutral System with Time-Delay forDynamic Positioning Shipsrdquo Mathematical Problems in Engi-neering vol 2015 Article ID 976925 11 pages 2015

[20] F Amato M Ariola and C Cosentino ldquoFinite-time stabilityof linear time-varying systems analysis and controller designrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 55 no 4 pp 1003ndash1008 2010

[21] L Sun andZ Zheng ldquoFinite-time slidingmode trajectory track-ing control of uncertain mechanical systemsrdquo Asian Journal ofControl vol 19 no 1 pp 399ndash404 2017

[22] P Shi Y Zhang and R Agarwal ldquoStochastic finite-time stateestimation for discrete time-delay neural networks withMarko-vian jumpsrdquo Neurocomputing vol 151 part 1 pp 168ndash174 2014

[23] L Zhang S Wang H R Karimi and A Jasra ldquoRobust Finite-Time Control of Switched Linear Systems and Application toa Class of Servomechanism Systemsrdquo IEEEASME Transactionson Mechatronics vol 20 no 5 pp 2476ndash2485 2015

[24] L Wu P Shi and H Gao ldquoState estimation and sliding-mode control of Markovian jump singular systemsrdquo Institute ofElectrical and Electronics Engineers Transactions on AutomaticControl vol 55 no 5 pp 1213ndash1219 2010

[25] Y Q Zhang P Shi S K Nguang and Y D Song ldquoRobust finite-time Hinfin control for uncertain discrete-time singular systemswithMarkovian jumpsrdquo IET ControlTheory ampApplications vol8 no 12 pp 1105ndash1111 2014

[26] S Wang T Shi L Zhang A Jasra and M Zeng ldquoExtendedfinite-time Hinfin control for uncertain switched linear neutralsystemswith time-varying delaysrdquoNeurocomputing vol 152 pp377ndash387 2015

[27] J Hu G Sui S Du and X Li ldquoFinite-time stability of uncer-tain nonlinear systems with time-varying delayrdquo MathematicalProblems in Engineering Article ID 2538904 9 pages 2017

[28] A Seuret and F Gouaisbaut ldquoWirtinger-based integral inequal-ity application to time-delay systemsrdquo Automatica vol 49 no9 pp 2860ndash2866 2013

[29] Z Li Y Bai C Huang and Y Cai ldquoNovel delay-partitioningstabilization approach for networked control system viaWirtinger-based inequalitiesrdquo ISA Transactions vol 61 pp75ndash86 2016

[30] P Park JWKo andC Jeong ldquoReciprocally convex approach tostability of systems with time-varying delaysrdquo Automatica vol47 no 1 pp 235ndash238 2011

[31] K Gu V L Kharitonov and J Chen Stability of Time DelaySystems Birkhauser Boston Mass USA 2003

[32] L El Ghaoui F Oustry and M AitRami ldquoA cone comple-mentarity linearization algorithm for static output-feedbackand related problemsrdquo Institute of Electrical and ElectronicsEngineers Transactions on Automatic Control vol 42 no 8 pp1171ndash1176 1997

[33] R Skjetne T I Fossen and P Kokotovic ldquoAdaptive maneu-vering with experiments for a model ship in a marine controllaboratoryrdquo Automatica vol 41 no 2 pp 289ndash298 2005

[34] N Wang C Qian J-C Sun and Y-C Liu ldquoAdaptive RobustFinite-Time Trajectory Tracking Control of Fully ActuatedMarine Surface Vehiclesrdquo IEEE Transactions on Control SystemsTechnology vol 24 no 4 pp 1454ndash1462 2016

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 16: Robust Finite-Time H-Infinity Control with Transients for ...downloads.hindawi.com/journals/mpe/2018/2838749.pdf · ResearchArticle Robust Finite-Time H-Infinity Control with Transients

16 Mathematical Problems in Engineering

0 20 40 60 80 100Time(s)

0

1

2

3

4

5

Theorem 15Case 3

0 2 4 6 8 10Time(s)

0

1

2

3

4

5

Theorem 15Case 3

Stat

e Res

pons

e of x

T(t)R

x(t)

Stat

e Res

pons

e of x

T(t)R

x(t)

Figure 13 State response 119909119879(119905)119877119909(119905) of Theorem 15 and Case 3

0 5 10Time(s)

05

1

15

X(m

)

Theorem 15Case 3

0 5 10Time(s)

05

1

15

Y(m

)

0 5 10Time(s)

0

005

01

015

02

(∘

)

Figure 14 Position and orientation of Theorem 15 and Case 3

lower overshoot which means that it is less conservative incomparison with existing result

5 Conclusion

In this paper the robust finite-time 119867infin control with tran-sients for DP system with input delay maintaining all statesof ocean surface vessel into a given threshold over a fixed timeinterval in presence of time-varying disturbance is proposedBased on a novel augmented LKLF with triple integral termsand a method combining theWirtinger inequality and recip-rocally convex approach a less conservative result is derivedIn particular a119867infin performance index with nonzero initialcondition attenuating the overconservatism caused by theassumption of zero initial condition and enhancing thetransient performance of ship subject to external disturbanceis adoptedThe obtained result is formulated in terms of LMIswhich can be easily solved by the standard numerical soft-ware Finally a numerical simulation for a ship is proposed toverify the effectiveness and less conservatism of the controller

we designed In the future the output feedback control can bedeveloped for the ocean surface vessel to avoid the use of thevessel-frame surge sway and yaw velocities

Data Availability

All the data supporting the conclusions of the study have beenprovided in Simulations and readers can access these data in[33 34]

Conflicts of Interest

The authors declare that there are no conflicts of interestrelated to this paper

Acknowledgments

This research was partially supported by the National Sci-ence Technology Support Program of China (Project no51609046)

Mathematical Problems in Engineering 17

References

[1] M Chen B Jiang and R Cui ldquoActuator fault-tolerant controlof ocean surface vessels with input saturationrdquo InternationalJournal of Robust amp Nonlinear Control vol 26 no 3 pp 542ndash564 2016

[2] J Du X Hu and Y Sun ldquoRobust dynamic positioning of shipswith disturbances under input saturationrdquo Automatica vol 73pp 207ndash214 2016

[3] H Ashrafiuon K R Muske L C McNinch and R ASoltan ldquoSliding-mode tracking control of surface vesselsrdquo IEEETransactions on Industrial Electronics vol 55 no 11 pp 4004ndash4012 2008

[4] W Si and X Dong ldquoNeural prescribed performance controlfor uncertain marine surface vessels without accurate initialerrorsrdquoMathematical Problems in Engineering Art ID 261832311 pages 2017

[5] T D Nguyen A J Soslashrensen and S T Quek ldquoDesign of hybridcontroller for dynamic positioning from calm to extreme seaconditionsrdquo Automatica vol 43 no 5 pp 768ndash785 2007

[6] J Du X Hu H Liu and C L Chen ldquoAdaptive robust outputfeedback control for a marine dynamic positioning systembased on a high-gain observerrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 11 pp 2775ndash27862015

[7] H Katayama ldquoNonlinear sampled-data stabilization of dynam-ically positioned shipsrdquo IEEE Transactions on Control SystemsTechnology vol 18 no 2 pp 463ndash468 2010

[8] K J Astrom andCG Kallstrom ldquoIdentification of ship steeringdynamicsrdquo Automatica vol 12 no 1 pp 9ndash22 1976

[9] Z Lei and C Guo ldquoDisturbance rejection control solutionfor ship steering system with uncertain time delayrdquo OceanEngineering vol 95 pp 78ndash83 2015

[10] J C Doyle K Glover P P Khargonekar and B A FrancisldquoState-space solutions to standard H-2 and Hinfin control prob-lemsrdquo IEEE Transactions on Automatic Control vol 34 no 8pp 831ndash847 1989

[11] M Katebi M Grimble and Y Zhang ldquoHinfin robust controldesign for dynamic ship positioningrdquo IEE Proceedings - ControlTheory and Applications vol 144 no 2 pp 110ndash120 1997

[12] D V Balandin and M M Kogan ldquoLMI-based Hinfin-optimalcontrol with transientsrdquo International Journal of Control vol 83no 8 pp 1664ndash1673 2010

[13] Z Feng J Lam S Xu and S Zhou ldquoHinfin Control withTransients for Singular Systemsrdquo Asian Journal of Control vol18 no 3 pp 817ndash827 2016

[14] P P Khargonekar K M Nagpal and K R Poolla ldquoHinfin controlwith transientsrdquo SIAM Journal onControl andOptimization vol29 no 6 pp 1373ndash1393 1991

[15] Z Zhang Z Zhang and H Zhang ldquoFinite-time stabilityanalysis and stabilization for uncertain continuous-time systemwith time-varying delayrdquo Journal of The Franklin Institute vol352 no 3 pp 1296ndash1317 2015

[16] J Cheng S Chen Z Liu H Wang and J Li ldquoRobust finite-time sampled-data control of linear systems subject to randomoccurring delays and its application to Four-Tank systemrdquoAppliedMathematics andComputation vol 281 pp 55ndash76 2016

[17] W Kang S Zhong K Shi and J Cheng ldquoFinite-time stabilityfor discrete-time system with time-varying delay and nonlinearperturbationsrdquo ISA Transactions vol 60 pp 67ndash73 2016

[18] T I Fossen Handbook of Marine Craft Hydrodynamics andMotion Control John Wiley amp Sons New York USA 2011

[19] D Zhao F Ding L Zhou W Zhang and H Xu ldquoRobustH-infinity Control of Neutral System with Time-Delay forDynamic Positioning Shipsrdquo Mathematical Problems in Engi-neering vol 2015 Article ID 976925 11 pages 2015

[20] F Amato M Ariola and C Cosentino ldquoFinite-time stabilityof linear time-varying systems analysis and controller designrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 55 no 4 pp 1003ndash1008 2010

[21] L Sun andZ Zheng ldquoFinite-time slidingmode trajectory track-ing control of uncertain mechanical systemsrdquo Asian Journal ofControl vol 19 no 1 pp 399ndash404 2017

[22] P Shi Y Zhang and R Agarwal ldquoStochastic finite-time stateestimation for discrete time-delay neural networks withMarko-vian jumpsrdquo Neurocomputing vol 151 part 1 pp 168ndash174 2014

[23] L Zhang S Wang H R Karimi and A Jasra ldquoRobust Finite-Time Control of Switched Linear Systems and Application toa Class of Servomechanism Systemsrdquo IEEEASME Transactionson Mechatronics vol 20 no 5 pp 2476ndash2485 2015

[24] L Wu P Shi and H Gao ldquoState estimation and sliding-mode control of Markovian jump singular systemsrdquo Institute ofElectrical and Electronics Engineers Transactions on AutomaticControl vol 55 no 5 pp 1213ndash1219 2010

[25] Y Q Zhang P Shi S K Nguang and Y D Song ldquoRobust finite-time Hinfin control for uncertain discrete-time singular systemswithMarkovian jumpsrdquo IET ControlTheory ampApplications vol8 no 12 pp 1105ndash1111 2014

[26] S Wang T Shi L Zhang A Jasra and M Zeng ldquoExtendedfinite-time Hinfin control for uncertain switched linear neutralsystemswith time-varying delaysrdquoNeurocomputing vol 152 pp377ndash387 2015

[27] J Hu G Sui S Du and X Li ldquoFinite-time stability of uncer-tain nonlinear systems with time-varying delayrdquo MathematicalProblems in Engineering Article ID 2538904 9 pages 2017

[28] A Seuret and F Gouaisbaut ldquoWirtinger-based integral inequal-ity application to time-delay systemsrdquo Automatica vol 49 no9 pp 2860ndash2866 2013

[29] Z Li Y Bai C Huang and Y Cai ldquoNovel delay-partitioningstabilization approach for networked control system viaWirtinger-based inequalitiesrdquo ISA Transactions vol 61 pp75ndash86 2016

[30] P Park JWKo andC Jeong ldquoReciprocally convex approach tostability of systems with time-varying delaysrdquo Automatica vol47 no 1 pp 235ndash238 2011

[31] K Gu V L Kharitonov and J Chen Stability of Time DelaySystems Birkhauser Boston Mass USA 2003

[32] L El Ghaoui F Oustry and M AitRami ldquoA cone comple-mentarity linearization algorithm for static output-feedbackand related problemsrdquo Institute of Electrical and ElectronicsEngineers Transactions on Automatic Control vol 42 no 8 pp1171ndash1176 1997

[33] R Skjetne T I Fossen and P Kokotovic ldquoAdaptive maneu-vering with experiments for a model ship in a marine controllaboratoryrdquo Automatica vol 41 no 2 pp 289ndash298 2005

[34] N Wang C Qian J-C Sun and Y-C Liu ldquoAdaptive RobustFinite-Time Trajectory Tracking Control of Fully ActuatedMarine Surface Vehiclesrdquo IEEE Transactions on Control SystemsTechnology vol 24 no 4 pp 1454ndash1462 2016

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 17: Robust Finite-Time H-Infinity Control with Transients for ...downloads.hindawi.com/journals/mpe/2018/2838749.pdf · ResearchArticle Robust Finite-Time H-Infinity Control with Transients

Mathematical Problems in Engineering 17

References

[1] M Chen B Jiang and R Cui ldquoActuator fault-tolerant controlof ocean surface vessels with input saturationrdquo InternationalJournal of Robust amp Nonlinear Control vol 26 no 3 pp 542ndash564 2016

[2] J Du X Hu and Y Sun ldquoRobust dynamic positioning of shipswith disturbances under input saturationrdquo Automatica vol 73pp 207ndash214 2016

[3] H Ashrafiuon K R Muske L C McNinch and R ASoltan ldquoSliding-mode tracking control of surface vesselsrdquo IEEETransactions on Industrial Electronics vol 55 no 11 pp 4004ndash4012 2008

[4] W Si and X Dong ldquoNeural prescribed performance controlfor uncertain marine surface vessels without accurate initialerrorsrdquoMathematical Problems in Engineering Art ID 261832311 pages 2017

[5] T D Nguyen A J Soslashrensen and S T Quek ldquoDesign of hybridcontroller for dynamic positioning from calm to extreme seaconditionsrdquo Automatica vol 43 no 5 pp 768ndash785 2007

[6] J Du X Hu H Liu and C L Chen ldquoAdaptive robust outputfeedback control for a marine dynamic positioning systembased on a high-gain observerrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 11 pp 2775ndash27862015

[7] H Katayama ldquoNonlinear sampled-data stabilization of dynam-ically positioned shipsrdquo IEEE Transactions on Control SystemsTechnology vol 18 no 2 pp 463ndash468 2010

[8] K J Astrom andCG Kallstrom ldquoIdentification of ship steeringdynamicsrdquo Automatica vol 12 no 1 pp 9ndash22 1976

[9] Z Lei and C Guo ldquoDisturbance rejection control solutionfor ship steering system with uncertain time delayrdquo OceanEngineering vol 95 pp 78ndash83 2015

[10] J C Doyle K Glover P P Khargonekar and B A FrancisldquoState-space solutions to standard H-2 and Hinfin control prob-lemsrdquo IEEE Transactions on Automatic Control vol 34 no 8pp 831ndash847 1989

[11] M Katebi M Grimble and Y Zhang ldquoHinfin robust controldesign for dynamic ship positioningrdquo IEE Proceedings - ControlTheory and Applications vol 144 no 2 pp 110ndash120 1997

[12] D V Balandin and M M Kogan ldquoLMI-based Hinfin-optimalcontrol with transientsrdquo International Journal of Control vol 83no 8 pp 1664ndash1673 2010

[13] Z Feng J Lam S Xu and S Zhou ldquoHinfin Control withTransients for Singular Systemsrdquo Asian Journal of Control vol18 no 3 pp 817ndash827 2016

[14] P P Khargonekar K M Nagpal and K R Poolla ldquoHinfin controlwith transientsrdquo SIAM Journal onControl andOptimization vol29 no 6 pp 1373ndash1393 1991

[15] Z Zhang Z Zhang and H Zhang ldquoFinite-time stabilityanalysis and stabilization for uncertain continuous-time systemwith time-varying delayrdquo Journal of The Franklin Institute vol352 no 3 pp 1296ndash1317 2015

[16] J Cheng S Chen Z Liu H Wang and J Li ldquoRobust finite-time sampled-data control of linear systems subject to randomoccurring delays and its application to Four-Tank systemrdquoAppliedMathematics andComputation vol 281 pp 55ndash76 2016

[17] W Kang S Zhong K Shi and J Cheng ldquoFinite-time stabilityfor discrete-time system with time-varying delay and nonlinearperturbationsrdquo ISA Transactions vol 60 pp 67ndash73 2016

[18] T I Fossen Handbook of Marine Craft Hydrodynamics andMotion Control John Wiley amp Sons New York USA 2011

[19] D Zhao F Ding L Zhou W Zhang and H Xu ldquoRobustH-infinity Control of Neutral System with Time-Delay forDynamic Positioning Shipsrdquo Mathematical Problems in Engi-neering vol 2015 Article ID 976925 11 pages 2015

[20] F Amato M Ariola and C Cosentino ldquoFinite-time stabilityof linear time-varying systems analysis and controller designrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 55 no 4 pp 1003ndash1008 2010

[21] L Sun andZ Zheng ldquoFinite-time slidingmode trajectory track-ing control of uncertain mechanical systemsrdquo Asian Journal ofControl vol 19 no 1 pp 399ndash404 2017

[22] P Shi Y Zhang and R Agarwal ldquoStochastic finite-time stateestimation for discrete time-delay neural networks withMarko-vian jumpsrdquo Neurocomputing vol 151 part 1 pp 168ndash174 2014

[23] L Zhang S Wang H R Karimi and A Jasra ldquoRobust Finite-Time Control of Switched Linear Systems and Application toa Class of Servomechanism Systemsrdquo IEEEASME Transactionson Mechatronics vol 20 no 5 pp 2476ndash2485 2015

[24] L Wu P Shi and H Gao ldquoState estimation and sliding-mode control of Markovian jump singular systemsrdquo Institute ofElectrical and Electronics Engineers Transactions on AutomaticControl vol 55 no 5 pp 1213ndash1219 2010

[25] Y Q Zhang P Shi S K Nguang and Y D Song ldquoRobust finite-time Hinfin control for uncertain discrete-time singular systemswithMarkovian jumpsrdquo IET ControlTheory ampApplications vol8 no 12 pp 1105ndash1111 2014

[26] S Wang T Shi L Zhang A Jasra and M Zeng ldquoExtendedfinite-time Hinfin control for uncertain switched linear neutralsystemswith time-varying delaysrdquoNeurocomputing vol 152 pp377ndash387 2015

[27] J Hu G Sui S Du and X Li ldquoFinite-time stability of uncer-tain nonlinear systems with time-varying delayrdquo MathematicalProblems in Engineering Article ID 2538904 9 pages 2017

[28] A Seuret and F Gouaisbaut ldquoWirtinger-based integral inequal-ity application to time-delay systemsrdquo Automatica vol 49 no9 pp 2860ndash2866 2013

[29] Z Li Y Bai C Huang and Y Cai ldquoNovel delay-partitioningstabilization approach for networked control system viaWirtinger-based inequalitiesrdquo ISA Transactions vol 61 pp75ndash86 2016

[30] P Park JWKo andC Jeong ldquoReciprocally convex approach tostability of systems with time-varying delaysrdquo Automatica vol47 no 1 pp 235ndash238 2011

[31] K Gu V L Kharitonov and J Chen Stability of Time DelaySystems Birkhauser Boston Mass USA 2003

[32] L El Ghaoui F Oustry and M AitRami ldquoA cone comple-mentarity linearization algorithm for static output-feedbackand related problemsrdquo Institute of Electrical and ElectronicsEngineers Transactions on Automatic Control vol 42 no 8 pp1171ndash1176 1997

[33] R Skjetne T I Fossen and P Kokotovic ldquoAdaptive maneu-vering with experiments for a model ship in a marine controllaboratoryrdquo Automatica vol 41 no 2 pp 289ndash298 2005

[34] N Wang C Qian J-C Sun and Y-C Liu ldquoAdaptive RobustFinite-Time Trajectory Tracking Control of Fully ActuatedMarine Surface Vehiclesrdquo IEEE Transactions on Control SystemsTechnology vol 24 no 4 pp 1454ndash1462 2016

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 18: Robust Finite-Time H-Infinity Control with Transients for ...downloads.hindawi.com/journals/mpe/2018/2838749.pdf · ResearchArticle Robust Finite-Time H-Infinity Control with Transients

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom


RL Transients

RL Transients

Quantum transients

Quantum transients

Variables and Transients

Variables and Transients

Solved Problems on Limits at Infinity, Asymptotes and ... Solved Problems... · In all limits at infinity or at a singular finite point, where the function is undefined, we try to

Solved Problems on Limits at Infinity, Asymptotes and ... Solved Problems... · In all limits at infinity or at a singular finite point, where the function is undefined, we try to

Transients in

Transients in

Electrical Fast Transients

Electrical Fast Transients

Inductive Transients

Inductive Transients

IMPERIA - mojnabytok.sk · Imperia 04 Imperia 10 . Infinity IS Infinity 20 Infinity 21 Infinity 01 Infinity 07 Infinity 13 Infinity 02 Infinity 08 Infinity 14 Infinity 03 Infinity

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+DHANALAKSHMI SRINIVASAN ENGINEERING COLLEGE- … POWER SYSTEM TRANSIENTS(KP).pdf · Ultra transients 2. Medium fast transients 3. Slow transients Power system transients based on

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Finite Infinity: Paradox in Emily Dickinsons Poetry · 2019-02-06 · 1 Finite Infinity: Paradox in Emily Dickinsons Poetry Johanna Schakenraad In a letter to T.W. Higginson Emily

Finite Infinity: Paradox in Emily Dickinsons Poetry · 2019-02-06 · 1 Finite Infinity: Paradox in Emily Dickinsons Poetry Johanna Schakenraad In a letter to T.W. Higginson Emily

An adaptive finite element method for the infinity Laplacian

An adaptive finite element method for the infinity Laplacian

Transients - STFC

Transients - STFC

Arc Transients

Arc Transients

90400 Transients

90400 Transients

Auto Transients

Auto Transients

Chapter 4 Transients

Chapter 4 Transients

Viscoelastic finite element modeling of deformation transients of … · 2017. 2. 17. · Viscoelastic finite element modeling of deformation transients of single cells 3 Acknowledgement

Viscoelastic finite element modeling of deformation transients of … · 2017. 2. 17. · Viscoelastic finite element modeling of deformation transients of single cells 3 Acknowledgement

Chapter02 Transients

Chapter02 Transients

DC Transients

DC Transients

Microscope & Telescope Fundamentals - NTUAusers.ntua.gr/eglytsis/OptEng/Microscope_Telescope_p.pdf · Compound Microscope Objective . Infinity Correction . Finite and infinity-corrected

Microscope & Telescope Fundamentals - NTUAusers.ntua.gr/eglytsis/OptEng/Microscope_Telescope_p.pdf · Compound Microscope Objective . Infinity Correction . Finite and infinity-corrected