Research Article Nonsingular Terminal Sliding Mode Control...

Research ArticleNonsingular Terminal Sliding Mode Control ofUncertain Second-Order Nonlinear Systems

Minh-Duc Tran1 and Hee-Jun Kang2

1University of Ulsan Ulsan 680-749 Republic of Korea2School of Electrical Engineering University of Ulsan Ulsan 680-749 Republic of Korea

Correspondence should be addressed to Hee-Jun Kang hjkangulsanackr

Received 27 April 2015 Revised 10 July 2015 Accepted 13 July 2015

Academic Editor Rongwei Guo

Copyright copy 2015 M-D Tran and H-J Kang This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper presents a high-performance nonsingular terminal sliding mode control method for uncertain second-order nonlinearsystems First a nonsingular terminal sliding mode surface is introduced to eliminate the singularity problem that exists inconventional terminal sliding mode control By using this method the system not only can guarantee that the tracking errorsreach the reference value in a finite time with high-precision tracking performance but also can overcome the complex-value andthe restrictions of the exponent (the exponent should be fractional number with an odd numerator and an odd denominator)in traditional terminal sliding mode Then in order to eliminate the chattering phenomenon a super-twisting higher-ordernonsingular terminal sliding mode control method is proposed The stability of the closed-loop system is established using theLyapunov theory Finally simulation results are presented to illustrate the effectiveness of the proposed method

1 Introduction

As the development of control schemes has progressed avariety of control systems have been developed for roboticmanipulators including proportional-integral-derivative(PID) control [1] adaptive control [2] computed torquecontrol [3 4] fuzzy control [5] and neural network control[6] Sliding mode control (SMC) is an efficient controlmethod that has been widely applied to control for bothlinear and nonlinear systems In order to design slidingmodecontrol systems establishment of suitable sliding surfacesto ensure the desired dynamics is considered first and thena sliding mode controller is designed to drive the systemstates to the sliding surface The main characteristic of SMCis to use discontinuous control effort to keep the systemstates on the sliding surfaces whereby SMC has strongrobustness with respect to system uncertainties and externaldisturbances fast response and good transient performanceHowever the conventional SMC method cannot guaranteethe invariance properties during the reaching phase and evenagainst disturbances can degrade the performance of system[7ndash9] Moreover this method adopts a linear sliding surface

which can only provide asymptotic stability of the system inthe sliding phase

Terminal sliding mode control (TSMC) methods whichuse nonlinear sliding surfaces instead of a linear surfacewere first introduced by Venkataraman and Gulati [10] andfurther developed by Man et al [11 12] and Wu et al [13]Compared with linear SMC TSMC schemes not only ensurethat the system states arrive at the equilibriumpoint in a finitetime but also offer some attractive properties such as theirfast response and higher precision However the traditionalTSMC methods may have slower convergence performancewhen the system states are not near the equilibrium pointand they also suffer from the singularity problem and haverestrictions on the range of the power function In orderto avoid these drawbacks some new TSMC methods havebeen proposed [14ndash16] Yu and Zhihong [14] have developedfast terminal sliding mode (FTSM) which can improve theconvergence speed when the system states are far fromthe equilibrium point This method however still has thesingularity problem To overcome this Feng et al [16] intro-duced nonsingular terminal sliding mode (NTSM) control

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 181737 8 pageshttpdxdoiorg1011552015181737

2 Mathematical Problems in Engineering

However this surface has a limitation on the power functionthat is 119901 and 119902must be positive odd integers

Discontinuous terminal sliding mode control (TSMC)has been widely applied to nonlinear systems Neverthelessthe main drawback of discontinuous TSMC is the chatteringphenomenon which comes from high frequency switchingof the control signal It shows undesirable oscillation on thesystem leads to low control accuracy causes high wear ofthe movingmechanical parts andmay damage the actuatorsTo deal with this problem the most common methodsreplace the sign function in the switching control with asaturating approximation [17] or boundary layer technique[18] The boundary layer method was proposed to eliminatethe chattering by defining a boundary layer around the slidingsurface and then approximate the discontinuous controlby continuous function within this boundary layer As aresult the chattering elimination is achieved however thereis a trade-off between chattering elimination and trackingperformance a thicker boundary layer can eliminate the chat-tering phenomenon but the tracking error will be increasedRecently intelligent control schemes (neural network andfuzzy logic) have been applied to attenuate the chatteringphenomenon [19ndash21] However some controller designsbased on intelligence techniques were quite complicated andfell into difficulties in stability analysis Therefore in thisstudy high-order sliding mode (HOSM) techniques havebeen studied and applied The main characteristic of HOSMis that they are working with the discontinuous control in thehigher-order time derivative [22ndash27] so the chattering canbe reduced because the control signal is continuous Further-more HOSM can bring better accuracy than conventionalSMC while the robustness of the control system is similar toSMC It has been presented in [23ndash25] for the control of rigidrobot manipulators

In this paper the above-mentioned problems are ad-dressed based on a proposed NTSM surface for second-ordernonlinear systems A control law is designed to drive thesystem states to reach the sliding surface and converge to zeroin a finite time It does not suffer from the singularity problemor the restriction on the power function Furthermore asuper-twisting second-order sliding mode is also used toreduce the chattering of the controller The global finite timestability of the closed-loop system is provenThe convergencetimes of the reaching phase and sliding phase are alsogiven The simulation results are presented to illustrate theeffectiveness of the proposed method on the two-link robotmanipulator

The remainder of this paper is arranged as follows Pre-liminaries and problem formulation are given in Section 2In Section 3 the structure of super-twisting nonsingularterminal sliding mode controller is presented and a stabilityanalysis is performed In Section 4 simulation results fora two-link robot manipulator are provided to demonstratethe performance of the proposed controller Finally someconcluding remarks are presented in Section 5

2 Preliminaries and Problem Formulation

Consider the following nonlinear second-order mechanicalsystems

1 = 1199092

2 = 119891 (119909 119905) + 119889 (119909 119905) + 119887 (119909 119905) 119906 (119905) (1)

where 119909 = [1199091 1199092]119879 denotes the system state vector 119891(119909 119905)

and 119887(119909 119905) are smooth nonlinear functions of 119909 119906(119905) isthe control input and 119889(119909 119905) presents the uncertainties anddisturbances

Assumption 1 Thematrices 119887(119909 119905) are invertible forall119909

Assumption 2 The uncertain term is bounded by

|119889 (119909 119905)| le 119863 (2)

where119863 is a known positive constant

Assumption 3 The desired state vector 119909119889(119905) isin 119877 is a twice

continuously differentiable function in terms of 119905The control objective of this paper is to design a controller

for system (1) to ensure that the error between the real statevector 119909 and the desired state vector 119909

119889(119905) converges to zero

in finite time

3 Main Results

In this section the design of super-twisting nonsingularterminal sliding mode controller is presented First a newnonsingular terminal sliding mode surface is proposed toeliminate the singularity problem Then the conventionalSMC and super-twisting nonsingular terminal sliding modecontroller are designed to ensure that the tracking errorconverges to zero in a finite amount time

31 New Form of NTSM Surface Wedefine the tracking erroras 120576(119905) = 1199091(119905)minus1199091119889(119905)Thus a newNTSMsurface is proposedas follows

119904 = 120576 + 1205731120576 + 1205732119890minus120582119905

(120576119879120576)minus120572

120576 (3)

where 119904 = [1199041 1199042 119904119899]119879 120576 = [1205761 1205762 120576119899]

119879 120576 = [ 1205761

1205762 120576119899]119879 1205731 = diag(12057311 12057312 1205731119899) 1205732 = diag(12057321

12057322 1205732119899) with 1205731119894 1205732119894 gt 0 for every 119894 = 1 2 119899 0 lt

120572 lt 1 and 120582 gt 0When the system operates in sliding mode the following

is true

119904 = 120576 + 1205731120576 + 1205732119890minus120582119905

(120576119879120576)minus120572

120576 = 0 (4)

120576 = minus 1205731120576 minus 1205732119890minus120582119905

(120576119879120576)minus120572

120576 (5)

Theorem 4 Considering the sliding mode dynamic equation(5) the system is finite time stable at the equilibrium point 120576 =

0 and the tracking error 120576 will converge to zero in finite time if2120572120582min(1205731) minus 120582 gt 0

Mathematical Problems in Engineering 3

The finite convergence time is

119879119904leln (1 + (119890

2120572120582min(1205731)119905 sdot 119881120572(0)) 1198862)

2120572120582min (1205731) minus 120582 (6)

where 1198862 is expressed by (15)

Proof Consider the Lyapunov function

119881 =12120576119879120576 (7)

Taking the derivative of 119881 in (7) and substituting (5) intoit yield

= 120576119879

120576 = 120576119879[minus1205731120576 minus 1205732119890

minus120582119905(120576119879120576)minus120572

120576] (8)

= minus 1205761198791205731120576 minus 1205732119890

minus120582119905(120576119879120576)minus120572

120576119879120576

le minus 120582min (1205731) 120576119879120576 minus 120582min (1205732) 119890

minus120582119905(120576119879120576)

1minus120572

le minus 2120582min (1205731) 119881minus 21minus120572120582min (1205732) 119890minus120582119905

1198811minus120572

le 0

(9)

Therefore according to the Lyapunov stability it is obvi-ous that the origin is at globally stable equilibrium Next wewill show that the system states converge to zero in finite time

Multiplying both sides of (9) by 120572119881120572minus1 we have

120572119881120572minus1 119889119881

119889119905le minus 2120572120582min (1205731) 119881

120572minus 21minus120572120572120582min (1205732) 119890

minus120582119905

119889119881120572

119889119905+ 2120572120582min (1205731) 119881

120572le minus 21minus120572120572120582min (1205732) 119890

minus120582119905

(10)

Multiplying both sides of (10) by 1198902120572120582min(1205731)119905 yields

1198902120572120582min(1205731)119905 (

119889119881120572

119889119905+ 2120572120582min (1205731) 119881

120572)

le minus 21minus120572120572120582min (1205732) 119890[2120572120582min(1205731)minus120582]119905

(11)

119889 (1198902120572120582min(1205731)119905 sdot 119881

120572)


[2120572120582min(1205731)minus120582]119905 (12)

Taking the integral on both sides of (12) from 0 to 119879119904and

knowing 119881(119879119904) = 0 yield

minus 1198902120572120582min(1205731)119905 sdot 119881

120572(0) le minus 1198862 [119890

[2120572120582min(1205731)minus120582]119879119904 minus 1] (13)

119890[2120572120582min(1205731)minus120582]119879119904 le 1+ 119890

2120572120582min(1205731)119905 sdot 119881120572(0)

1198862 (14)

where

1198862 =21minus120572120572120582min (1205732)

2120572120582min (1205731) minus 120582gt 0 (15)

Taking the natural logarithm of both sides of (14) yields

[2120572120582min (1205731) minus 120582] 119879119904le ln(1+ 119890

2120572120582min(1205731)119905 sdot 119881120572(0)

1198862)

119879119904leln (1 + (119890

2120572120582min(1205731)119905 sdot 119881120572(0)) 1198862)

2120572120582min (1205731) minus 120582

(16)

This completes the proof

Remark 5 The expression in (3) is different from the pre-viously reported TSM and fast TSM in [14] which areexpressed respectively as

119904 = + 120573119909119902119901

119904 = + 120572119909+120573119909119902119901

(17)

where 120572 and 120573 are positive constants and 119901 and 119902 are positiveodd integers that satisfy the following condition 1 lt 119901119902 lt 2We can easily see that for119909 lt 0 the fractional power 119902119901maylead to the term 119909

119902119901notin 119877 which means notin 119877 In addition

the TSM control signals in [14] contain 1199091119902119901minus1

1199092 which maycause a singularity to occur if 1199092 = 0 when 1199091 = 0

To solve the complex-value problem in (17) Yu et al [28]proposed the TSM surface as

119904 = + 120573 |119909|120574 sign (119909)

119904 = + 120572119909+120573 |119909|120574 sign (119909)

(18)

The sliding surface in (18) could solve the complex-valuenumber but the control input can suffer from the singularityproblem if 1199092 = 0 when 1199091 = 0

Recently a nonsingular terminal sliding surface wasproposed to overcome the singularity problem [16]

119904 = 119909 +1120573119901119902

(19)

However this surface still has the limitation for theexponent of the power function that is 119901 and 119902 shouldbe positive odd integers Thus our proposed TSM surfacedoes not contain any of the mentioned singularities and theexponent can be any real number in the interval 0 lt 120572 lt 1

Remark 6 Comparing with linear sliding mode NTSMhas higher convergence rate when the system state is faraway from the equilibrium point while NTSM has lowerconvergence speed when the system state is close to theequilibrium point [29 30]

It is obvious that the term 119890minus120582119905 in the proposed surface

will go backward to zero after a certain time Thus thenonsingular terminal slidingmode surfacewill become linearsliding mode after a period of time By choosing a suitable 120582the proposed surface will have the advantage of both NTMSand linear sliding surface

32 NTSM Control (NTSMC) Design One suitable slidingmanifold is established The next step is to design the controlto drive the nonlinear system (1) to the expected slidingsurface (3) in a finite amount time The proposed controlmethod is summarized as follows

Theorem 7 For the system (1) if the control signal is designedas (20) and the gain 120578 of the controller is larger than the upper


bounds of the uncertainties the tracking error 120576(119905)will convergeto zero in finite time

119906 (119905) = minus 119887 (119909 119905)minus1

sdot [119891 (119909 119905) minus 119889+1205731 120576 + 1205732119860+120578 sign (119904)]

(20)

where 120578 = diag(1205781 1205782 120578119899) 120578119894 gt 0 Therefore

119860 = [(minus120582) 119890minus120582119905

(120576119879120576)minus120572

120576

+ 119890minus120582119905

(minus2120572) 119909 (120576119879120576)minus120572minus1

(120576119879

120576) 120576 + 119890minus120582119905

(120576119879120576)minus120572

120576]

(21)

Proof Consider the following Lyapunov candidate function

119881 =12119904119879119904 (22)

The time derivative of the sliding surface (3) with respectto time can be expressed as

119904 = 120576 + 1205731 120576 + 1205732119860

= 119891 (119909 119905) + 119889 (119909 119905) + 119887 (119909 119905) 119906 (119905) minus 119889 (119905) + 1205731 120576

+ 1205732119860

(23)

Differentiating 119881 with respect to time and substituting(20) and (23) into it yield

= 119904119879(minus120578 sign (119904) + 119889 (119905)) le minus (120578 minus119863) |119904| le 0 (24)

Therefore the condition for Lyapunov stability is satisfiedin the following we will show that the error converges to zeroin finite time

From (24) we have

le minusradic2 (120578 minus119863)11988112

119889119905 le minus119889119881

radic2 (120578 minus 119863)11988112= minus

radic211988911988112

(120578 minus 119863)

(25)

Taking the integral of both sides of (25) from 119879119903to 119879119904 we

have

119879119904minus119879119903le minusint

119881(119879119904)

119881(119879119903)

radic211988911988112

(120578 minus 119863)=

radic2(120578 minus 119863)

11988112

(119879119903) (26)

Note that 119881(119879119904) = 0 therefore the TSM will reach zero

in the finite time

119879119904le

radic2(120578 minus 119863)

11988112

(119879119903) +119879119903 (27)


Remark 8 In order to eliminate the chattering a saturationfunction sat or 119904(119904 + 120576) (120576 is a small positive constant) canbe used to replace the sign function

33 Super-Twisting NTSM Control (ST-NTSMC) Design Themain drawback of the conventional sliding mode is thechattering phenomenon which is caused by discontinu-ous control action when the system state operates nearthe sliding surface Even though the chattering reductioncan be achieved by using Remark 8 there is a trade-offbetween chattering elimination and tracking performanceincreasing the thickness of the boundary layer can eliminatethe chattering phenomenon but will increase the trackingerror Therefore in this subsection super-twisting control isapplied to attenuate chattering and to increase the trackingperformance

The ST-NSTSMC is designed as

119906 = 119906eq +119906STW (28)

where

119906eq = minus 119887 (119909 119905)minus1

[119891 (119909 119905) minus 119889+1205731 120576 + 1205732119860] (29)

Based on [27] the super-twisting controller is designed as

119906STW = minus 119887 (119909 119905)minus1

(1198961 |119904|12 sign (119904) + 119911)

= minus 1198962 sign (119904)

(30)

The differentiation of the sliding surface is now obtainedas

119904 = 119891 (119909 119905) + 119889 (119909 119905) + 119887 (119909 119905) (119906eq +119906SMW) minus 119889 (119905)

+ 1205731 120576 + 1205732119860

(31)

Substituting (29) and (30) into (31) yields

119904 = minus 1198961 |119904|12 sign (119904) + 119911 + 119889 (119909 119905)

= minus 1198962 sign (119904)

(32)

The stability and convergence of the closed-loop systemin (32) are given inTheorem 9

Theorem 9 Suppose that Assumption 1 is guaranteed and theuncertain terms are bounded by

119889 (119909 119905) le 120575 |119904|12

120575 = diag (1205751 1205752 120575119899) 120575119894gt 0

(33)

For system (1) with the terminal sliding mode surfacechosen as in (3) and the proposed control signal designedas in (28) if the sliding gains of 119906STW given in (30) satisfycondition (34) then the sliding surface 119904will converge to zeroin a finite time

1198961 gt 2120575

1198962 gt 119896151198961 + 41205752 (1198961 minus 2120575)

120575(34)

Proof Now referring to Morenorsquos work [27] let us considerthe Lyapunov candidate function

119881 = 120585119879119875120585 (35)


where

120585 = [|119904|12 sign (119904) 119911]

119879

119875 =12[1198961

2+ 41198962 minus1198961

minus1198961 2]

(36)

As we know 119881 is positive definite and radially un-bounded

120582min (119875)100381710038171003817100381712057710038171003817100381710038172le 119881 le 120582max (119875)

100381710038171003817100381712057710038171003817100381710038172 (37)

where 1205772 = |119904| + 1199112 The time derivative of 119881 becomes

= minus1

|119904|12 (1205851198791198761120585 minus 119889 (119909 119905) 1198762

119879120585) (38)

where

1198761 =11989612

[1198961

2+ 21198962 minus1198961

minus1198961 1]

1198762119879= [

11989612

2+ 21198962 minus

11989612]

(39)

Using condition (33) it can be shown that

le minus1

|119904|12 120585119879119876120585 le minus

1|119904|

12 120582min (119876)100381710038171003817100381712058510038171003817100381710038172 (40)

where

119876 =11989612

[[

[

11989612+ 21198962 minus (

411989621198961

+ 1198961)120575 minus (1198961 + 2120575)

minus (1198961 + 2120575) 1

]]

]

(41)

In the case in which the condition in (34) is satisfied119876 gt

0 so is negative definiteWe can use (37) and the fact that

|119904|12

le10038171003817100381710038171205771003817100381710038171003817 le

11988112

120582min12

(119875)

10038171003817100381710038171205771003817100381710038171003817 ge

11988112

120582max12

(119875)

(42)

Then substituting (42) into (40) yields

le minus 12058111988112

(43)

where

120581 =120582min

12(119875) 120582min (119876)

120582max (119875) (44)

Since the solution of the differential equation

V le minus 120581V12

V (0) = V0 gt 0(45)

l1

l2

m2

m1

q2

q1

Figure 1 Configuration of the two-link robotic system [3]

is given as

V (119905) = (V012

minus120581

2119905)

2 (46)

here V(119905) converges to zero in a finite time and reacheszero after 119879 = 211988112

(1199090)120581 It follows from the comparisonprinciple [18] that 119881(119905) le V(119905) when 119881(1199090) le V0 From(46) we can determine that 119881(119905) and therefore 119904 convergeto zero in a finite time and reach that value at most after119879 = 211988112

(1199090)120581

4 Simulation Results

In this section to verify the validity and effectiveness of theproposed method the two-link planar robot manipulatorshown in Figure 1 is considered

The dynamic equation of the two-link robot is describedas follows [3]

119872(119902) 119902 +119862 (119902 119902) +119866 (119902) = 120591 (119905) + 120591119889+119865 ( 119902) (47)

where

119872(119902)

= [119897221198982 + 21198971119897211989821198882 + 119897

21 (1198981 + 1198982) 119897

221198982 + 1198971119897211989821198882

119897221198982 + 1198971119897211989821198882 119897

221198982

]

119862 (119902 119902) = [minus1198982119897111989721199042 119902

22 minus 21198982119897111989721199042 1199021 1199022

1198982119897111989721199042 11990221

]

119866 (119902) = [1198982119897211989211988812 + (1198981 + 1198982) 11989711198921198881

1198982119897211989211988812]

(48)

and 119902 = (1199021 1199022)119879 is the joint variable vector 119872(119902) is the

inertial matrix119862(119902 119902) represents the centripetal and Coriolistorque matrix 119866(119902) represents the gravity torque vector 120591

119889is

the vector of the bounded external disturbance 119865( 119902) is thefriction and 120591 is the control torque 1198981 and 1198982 are the linkmasses 1198971 and 1198972 are the link lengths gravity 119892 = 981(119898119904

2)

and the symbols 1199041 1199042 11990412 and 1198881 1198882 11988812 are respectivelydefined as 1199041 = sin(1199021) 1199042 = sin(1199022) 11990412 = sin(11990212) 1198881 =

cos(1199021) 1198882 = cos(1199022) and 11988812 = cos(11990212)


0 1 2 3 4 5 6 7 8 9 10

08

1

12

14

Time (s)

ReferenceC-TSMCST-NTSMC

q1

(rad

)

(a)

Time (s)


0 1 2 3 4 5 6 7 8 9 100

05

1

15

q2

(rad

)

(b)

Figure 2 Tracking performance of two-link robot manipulator (a) at joint 1 (b) at joint 2

Time (s)0 1 2 3 4 5 6 7 8 9 10

0

01

02

03

C-TSMCST-NTSMC

2 4 6 8 10

0

5

e1

(rad

)

minus01

minus5

times10minus6

(a)

Time (s)

C-TSMCST-NTSMC

0 1 2 3 4 5 6 7 8 9 10

002

2 4 6 8 10

0

5

e2

(rad

)minus06

minus04

minus02

minus5

times10minus5

(b)

Figure 3 Tracking errors of two-link robot manipulator (a) at joint 1 (b) at joint 2

The friction and external disturbance are chosen as

119865 ( 119902) = [1199021 + 2 sin (1199021)

05 sin (1199022)]

120591119889= [

02 sin (119905)

02 cos (2119905)]

(49)

The parameter values employed to simulate the robot aregiven as1198981 = 1198982 = 1 (m) and 1198971 = 1198972 = 1 (kg) and the designreference signals are given by

1199021119889 = 1+ 02 sin (05120587119905)

1199022119889 = 1minus 02 cos (05120587119905) (50)

The initial states of the system are chosen as

1199021 (0) = 13

1199022 (0) = 03

1199021 (0) = 0

1199022 (0) = 0

(51)

To this end MatlabSimulink is used to perform all ofthe simulations and with the sampling time set to 10minus4 sthe simulation compares the proposed ST-NTSMC control

Table 1 Control parameters

Control schemes ParametersC-TSMC [28] 120573 = 101198682 1205781 = 251198682 1205782 = 451198682 120574 = 15 120593 = 03

ST-NTSMC 1205731 = 121198682 1205732 = 101198682 1198961 = 9 1198962 = 5 120582 = 3120572 = 03

scheme with the previously proposed control method in [28]Yu et al [28] suggested the continuous terminal sliding modecontrol (C-TSMC) which was designed for a two-link robotmanipulator as follows

120591 = 1198620 (119902 119902) +1198660 (119902) +1198720 (119902) 119902119889

minus1198720120573minus1120574minus1sig ( 119890)

(2minus120574)minus1198720 (1205781119904 + 1205782sig (119904)

120593)

(52)

where 119904 = 119890 + 120573 sig ( 119890)120574 sig (119909)120574 = [|1199091|

120574 sign (1199091)|1199092|120574 sign (1199092) |119909119899|

120574 sign (119909119899)] 120573 = diag(1205731 1205732 120573119899)

1205781 = diag(12057811 12057812 1205781119899) 1205782 = diag(12057821 12057822 1205782119899) 1205731198941205781119894 1205782119894 gt 0 0 lt 120593 lt 1 and 1 lt 120574 lt 2

The control parameters are selected as shown in Table 1The simulation results are shown in Figures 2ndash5 In

Figure 2 the tracking results of the robot manipulator usingthe two control laws above are compared It shows that thestate trajectories can reach the design reference signals inthe presence of model parameter uncertainties and external


0 1 2 3 4 5 6 7 8 9 10

minus50

0

50

U1

Time (s)

C-TSMCST-NTSMC

(a)

0 1 2 3 4 5 6 7 8 9 10minus20

minus10

0

10

U2

Time (s)

C-TSMCST-NTSMC

(b)

Figure 4 Control inputs (a) at joint 1 (b) at joint 2

0 1 2 3 4 5 6 7 8 9 10

2 4 6 8 10

0

2

minus1

0

1

2

3

4

S1

minus2

S2

Time (s)

C-TSMCST-NTSMC

times10minus4

(a)

0 1 2 3 4 5 6 7 8 9 10

2 4 6 8 10

0

2

S1

minus10

minus8

minus6

minus4

minus2

minus2

0

2

S2

Time (s)

C-TSMCST-NTSMC

times10minus3

(b)

Figure 5 Time responses of the terminal sliding mode surface (a) at joint 1 (b) at joint 2

disturbances The tracking errors via two controllers arecompared in Figure 3 One can easily see that the ST-NTSMCproduces tracking performance with faster convergence andhigher precision Figure 4 shows the time histories of theapplied control inputs and shows that the proposed ST-NTSMC method achieves superior control input perfor-mance with smaller control efforts higher precision trackingand smoother than the C-TSMCmethodThe time responsesof the sliding manifolds are shown in Figure 5 Clearlythe sliding surface of the proposed method was also muchsmaller than C-TSMC

5 Conclusions

In this paper we presented the ST-NTSMC method forsecond-order nonlinear systems This method has beensuccessfully applied in a two-link robot manipulator Thedesigned nonsingular terminal sliding surface not only avoidsthe singularity problem but also can overcome the complex-value and the restriction on the exponent of a power functionin conventional TSMC The performance of the proposedmethod was evaluated in comparison with recently pro-posed approaches [28] The simulation results show that theproposed method achieves highly precise tracking fast andfinite time convergence and robustness against parameteruncertainties and external disturbances Furthermore ST-NTSMC is used to smooth the discontinuous control termin order to attenuate the chattering phenomenon

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This paper is a result of a study on the ldquoLeaders Industry-University Cooperationrdquo Project supported by the Ministryof Education (MOE)

References

[1] S Arimoto and F Miyazaki ldquoStability and robustness of PIDfeedback control for robot manipulators of sensory capabilityrdquoin Robotic Research M Brady and R P Paul Eds MIT PressCambridge Mass USA 1984

[2] J-J E Slotine and W Li ldquoOn the adaptive control of robotmanipulatorsrdquo International Journal of Robotics Research vol 6no 3 pp 49ndash59 1987

[3] J J Craig Introduction to Robotics Addion-Wesley ReadingMass USA 1989

[4] J J Spong and M Vidyasagar Robot Dynamics and ControlWiley New York NY USA 1989

[5] J H Lilly Fuzzy Control and Identification Wiley 2010[6] L Jinkun Radial Basis Function Neural Network Control for

Mechanical Systems Tsinghua University Press Beijing China2013


[7] V I Utkin ldquoVariable structure systems with sliding modesrdquoIEEE Transactions on Automatic Control vol 22 no 2 pp 212ndash222 1977

[8] J Y HungWGao and J CHung ldquoVariable structure control asurveyrdquo IEEE Transactions on Industrial Electronics vol 40 no1 pp 2ndash22 1993

[9] A Sabanovic ldquoVariable structure systems with sliding modesin motion controlmdasha surveyrdquo IEEE Transactions on IndustrialInformatics vol 7 no 2 pp 212ndash223 2011

[10] S T Venkataraman and S Gulati ldquoControl of nonlinear systemsusing terminal sliding modesrdquo Transactions of the ASMEmdashJournal of Dynamic Systems Measurement and Control vol 115no 3 pp 554ndash560 1993

[11] ZManA P Paplinski andHRWu ldquoA robustMIMO terminalsliding mode control scheme for rigid robotic manipulatorsrdquoIEEE Transactions on Automatic Control vol 39 no 12 pp2464ndash2469 1994

[12] Z Man and X Yu ldquoTerminal sliding mode control of MIMOlinear systemsrdquo IEEE Transactions on Circuits and Systems IFundamental Theory and Applications vol 44 no 11 pp 1065ndash1070 1997

[13] Y Wu X Yu and Z Man ldquoTerminal sliding mode controldesign for uncertain dynamic systemsrdquo Systems amp ControlLetters vol 34 no 5 pp 281ndash287 1998

[14] X Yu and M Zhihong ldquoFast terminal sliding-mode controldesign for nonlinear dynamical systemsrdquo IEEE Transactions onCircuits and Systems I Fundamental Theory and Applicationsvol 49 no 2 pp 261ndash264 2002

[15] L Yang and J Yang ldquoNonsingular fast terminal sliding-modecontrol for nonlinear dynamical systemsrdquo International Journalof Robust and Nonlinear Control vol 21 no 16 pp 1865ndash18792011

[16] Y Feng X Yu and Z Man ldquoNon-singular terminal slidingmode control of rigid manipulatorsrdquo Automatica vol 38 no12 pp 2159ndash2167 2002

[17] T H S Li and Y-C Huang ldquoMIMO adaptive fuzzy terminalsliding-mode controller for robotic manipulatorsrdquo InformationSciences vol 180 no 23 pp 4641ndash4660 2010

[18] J J E Slotine and W Li Applied Nonlinear Control Prentice-Hall Englewood Cliffs NJ USA 1991

[19] Y Jiang QWang and C Dong ldquoA reaching law neural networkterminal sliding mode guidance law designrdquo in Proceedings ofthe IEEE Region 10 Conference (TENCON rsquo13) Xirsquoan ChinaOctober 2013

[20] B Yoo and W Ham ldquoAdaptive fuzzy sliding mode control ofnonlinear systemrdquo IEEE Transactions on Fuzzy Systems vol 6no 2 pp 315ndash321 1998

[21] M Roopaei M Zolghadri Jahromi and S Jafari ldquoAdaptive gainfuzzy slidingmode control for the synchronization of nonlinearchaotic gyrosrdquo Chaos vol 19 no 1 Article ID 013125 2009

[22] J Rivera C Mora J J Raygoza and S Ortega ldquoSupper-twisting sliding mode in motion control systemsrdquo in SlidingMode Control A Bartoszewicz Ed pp 978ndash953 InTech 2011

[23] D Hernandez W Yu and M A Moreno-Amendariz ldquoNeuralPD control with second-order sliding mode compensation forrobot manipulatorsrdquo in Proceedings of the International JointConference on Neural Networks (IJCNN rsquo11) pp 2392ndash2402 SanJose Calif USA August 2011

[24] L M Capisani A Ferrara and L Magnani ldquoSecond ordersliding mode motion control of rigid robot manipulatorsrdquo inProceedings of the 46th IEEEConference onDecision and Control(CDC rsquo07) pp 3691ndash3696 December 2007

[25] M Van H-J Kang and Y-S Suh ldquoSecond order slidingmode-based output feedback tracking control for uncertainrobot manipulatorsrdquo International Journal of Advanced RoboticSystems vol 10 article 16 2013

[26] J Davila L Fridman and A Levant ldquoSecond-order sliding-mode observer for mechanical systemsrdquo IEEE Transactions onAutomatic Control vol 50 no 11 pp 1785ndash1789 2005

[27] J A Moreno and M Osorio ldquoA Lyapunov approach to second-order sliding mode controllers and observersrdquo in Proceedings ofthe 47th IEEE Conference on Decision and Control pp 2856ndash2861 December 2008

[28] S Yu X Yu B Shirinzadeh and Z Man ldquoContinuous finite-time control for robotic manipulators with terminal slidingmoderdquo Automatica vol 41 no 11 pp 1957ndash1964 2005

[29] X Zhao Y X Jiang Y J Wu and Y Q Zhou ldquoFast nonsingularterminal sliding mode control based on multi-slide-moderdquoJournal of BeijingUniversity of Aeronautics andAstronautics vol37 no 1 pp 110ndash113 2011

[30] H J Shi L F Qian Y D Xu and L M Chen ldquoFuzzy movingfast terminal sliding mode control for robotic manipulatorsrdquoin Proceedings of the IEEE International Conference on Roboticsand Biomimetics (ROBIO rsquo12) pp 1943ndash1949 IEEE GuangzhouChina December 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


However this surface has a limitation on the power functionthat is 119901 and 119902must be positive odd integers

Discontinuous terminal sliding mode control (TSMC)has been widely applied to nonlinear systems Neverthelessthe main drawback of discontinuous TSMC is the chatteringphenomenon which comes from high frequency switchingof the control signal It shows undesirable oscillation on thesystem leads to low control accuracy causes high wear ofthe movingmechanical parts andmay damage the actuatorsTo deal with this problem the most common methodsreplace the sign function in the switching control with asaturating approximation [17] or boundary layer technique[18] The boundary layer method was proposed to eliminatethe chattering by defining a boundary layer around the slidingsurface and then approximate the discontinuous controlby continuous function within this boundary layer As aresult the chattering elimination is achieved however thereis a trade-off between chattering elimination and trackingperformance a thicker boundary layer can eliminate the chat-tering phenomenon but the tracking error will be increasedRecently intelligent control schemes (neural network andfuzzy logic) have been applied to attenuate the chatteringphenomenon [19ndash21] However some controller designsbased on intelligence techniques were quite complicated andfell into difficulties in stability analysis Therefore in thisstudy high-order sliding mode (HOSM) techniques havebeen studied and applied The main characteristic of HOSMis that they are working with the discontinuous control in thehigher-order time derivative [22ndash27] so the chattering canbe reduced because the control signal is continuous Further-more HOSM can bring better accuracy than conventionalSMC while the robustness of the control system is similar toSMC It has been presented in [23ndash25] for the control of rigidrobot manipulators

In this paper the above-mentioned problems are ad-dressed based on a proposed NTSM surface for second-ordernonlinear systems A control law is designed to drive thesystem states to reach the sliding surface and converge to zeroin a finite time It does not suffer from the singularity problemor the restriction on the power function Furthermore asuper-twisting second-order sliding mode is also used toreduce the chattering of the controller The global finite timestability of the closed-loop system is provenThe convergencetimes of the reaching phase and sliding phase are alsogiven The simulation results are presented to illustrate theeffectiveness of the proposed method on the two-link robotmanipulator

The remainder of this paper is arranged as follows Pre-liminaries and problem formulation are given in Section 2In Section 3 the structure of super-twisting nonsingularterminal sliding mode controller is presented and a stabilityanalysis is performed In Section 4 simulation results fora two-link robot manipulator are provided to demonstratethe performance of the proposed controller Finally someconcluding remarks are presented in Section 5

2 Preliminaries and Problem Formulation

Consider the following nonlinear second-order mechanicalsystems

1 = 1199092

2 = 119891 (119909 119905) + 119889 (119909 119905) + 119887 (119909 119905) 119906 (119905) (1)

where 119909 = [1199091 1199092]119879 denotes the system state vector 119891(119909 119905)

and 119887(119909 119905) are smooth nonlinear functions of 119909 119906(119905) isthe control input and 119889(119909 119905) presents the uncertainties anddisturbances

Assumption 1 Thematrices 119887(119909 119905) are invertible forall119909

Assumption 2 The uncertain term is bounded by

|119889 (119909 119905)| le 119863 (2)

where119863 is a known positive constant

Assumption 3 The desired state vector 119909119889(119905) isin 119877 is a twice

continuously differentiable function in terms of 119905The control objective of this paper is to design a controller

for system (1) to ensure that the error between the real statevector 119909 and the desired state vector 119909

119889(119905) converges to zero

in finite time

3 Main Results

In this section the design of super-twisting nonsingularterminal sliding mode controller is presented First a newnonsingular terminal sliding mode surface is proposed toeliminate the singularity problem Then the conventionalSMC and super-twisting nonsingular terminal sliding modecontroller are designed to ensure that the tracking errorconverges to zero in a finite amount time

31 New Form of NTSM Surface Wedefine the tracking erroras 120576(119905) = 1199091(119905)minus1199091119889(119905)Thus a newNTSMsurface is proposedas follows

119904 = 120576 + 1205731120576 + 1205732119890minus120582119905

(120576119879120576)minus120572

120576 (3)

where 119904 = [1199041 1199042 119904119899]119879 120576 = [1205761 1205762 120576119899]

119879 120576 = [ 1205761

1205762 120576119899]119879 1205731 = diag(12057311 12057312 1205731119899) 1205732 = diag(12057321

12057322 1205732119899) with 1205731119894 1205732119894 gt 0 for every 119894 = 1 2 119899 0 lt

120572 lt 1 and 120582 gt 0When the system operates in sliding mode the following

is true

119904 = 120576 + 1205731120576 + 1205732119890minus120582119905

(120576119879120576)minus120572

120576 = 0 (4)

120576 = minus 1205731120576 minus 1205732119890minus120582119905

(120576119879120576)minus120572

120576 (5)

Theorem 4 Considering the sliding mode dynamic equation(5) the system is finite time stable at the equilibrium point 120576 =

0 and the tracking error 120576 will converge to zero in finite time if2120572120582min(1205731) minus 120582 gt 0



119879119904leln (1 + (119890

2120572120582min(1205731)119905 sdot 119881120572(0)) 1198862)

2120572120582min (1205731) minus 120582 (6)



119881 =12120576119879120576 (7)


= 120576119879

120576 = 120576119879[minus1205731120576 minus 1205732119890

minus120582119905(120576119879120576)minus120572

120576] (8)

= minus 1205761198791205731120576 minus 1205732119890

minus120582119905(120576119879120576)minus120572

120576119879120576


minus120582119905(120576119879120576)

1minus120572


1198811minus120572

le 0

(9)



120572119881120572minus1 119889119881

119889119905le minus 2120572120582min (1205731) 119881

120572minus 21minus120572120572120582min (1205732) 119890

minus120582119905

119889119881120572

119889119905+ 2120572120582min (1205731) 119881


minus120582119905

(10)


1198902120572120582min(1205731)119905 (

119889119881120572

119889119905+ 2120572120582min (1205731) 119881

120572)


(11)

119889 (1198902120572120582min(1205731)119905 sdot 119881

120572)


[2120572120582min(1205731)minus120582]119905 (12)


knowing 119881(119879119904) = 0 yield

minus 1198902120572120582min(1205731)119905 sdot 119881

120572(0) le minus 1198862 [119890

[2120572120582min(1205731)minus120582]119879119904 minus 1] (13)

119890[2120572120582min(1205731)minus120582]119879119904 le 1+ 119890

2120572120582min(1205731)119905 sdot 119881120572(0)

1198862 (14)

where

1198862 =21minus120572120572120582min (1205732)

2120572120582min (1205731) minus 120582gt 0 (15)


[2120572120582min (1205731) minus 120582] 119879119904le ln(1+ 119890

2120572120582min(1205731)119905 sdot 119881120572(0)

1198862)

119879119904leln (1 + (119890

2120572120582min(1205731)119905 sdot 119881120572(0)) 1198862)

2120572120582min (1205731) minus 120582

(16)



119904 = + 120573119909119902119901

119904 = + 120572119909+120573119909119902119901

(17)






119904 = + 120573 |119909|120574 sign (119909)

119904 = + 120572119909+120573 |119909|120574 sign (119909)

(18)



119904 = 119909 +1120573119901119902

(19)









119906 (119905) = minus 119887 (119909 119905)minus1

sdot [119891 (119909 119905) minus 119889+1205731 120576 + 1205732119860+120578 sign (119904)]

(20)


119860 = [(minus120582) 119890minus120582119905

(120576119879120576)minus120572

120576

+ 119890minus120582119905


(120576119879

120576) 120576 + 119890minus120582119905

(120576119879120576)minus120572

120576]

(21)


119881 =12119904119879119904 (22)


119904 = 120576 + 1205731 120576 + 1205732119860

= 119891 (119909 119905) + 119889 (119909 119905) + 119887 (119909 119905) 119906 (119905) minus 119889 (119905) + 1205731 120576

+ 1205732119860

(23)




From (24) we have


119889119905 le minus119889119881


radic211988911988112

(120578 minus 119863)

(25)


have


119881(119879119904)

119881(119879119903)

radic211988911988112

(120578 minus 119863)=

radic2(120578 minus 119863)

11988112

(119879119903) (26)


in the finite time

119879119904le

radic2(120578 minus 119863)

11988112

(119879119903) +119879119903 (27)





119906 = 119906eq +119906STW (28)

where

119906eq = minus 119887 (119909 119905)minus1

[119891 (119909 119905) minus 119889+1205731 120576 + 1205732119860] (29)


119906STW = minus 119887 (119909 119905)minus1

(1198961 |119904|12 sign (119904) + 119911)

= minus 1198962 sign (119904)

(30)


119904 = 119891 (119909 119905) + 119889 (119909 119905) + 119887 (119909 119905) (119906eq +119906SMW) minus 119889 (119905)

+ 1205731 120576 + 1205732119860

(31)


119904 = minus 1198961 |119904|12 sign (119904) + 119911 + 119889 (119909 119905)

= minus 1198962 sign (119904)

(32)



119889 (119909 119905) le 120575 |119904|12

120575 = diag (1205751 1205752 120575119899) 120575119894gt 0

(33)


1198961 gt 2120575

1198962 gt 119896151198961 + 41205752 (1198961 minus 2120575)

120575(34)


119881 = 120585119879119875120585 (35)


where

120585 = [|119904|12 sign (119904) 119911]

119879

119875 =12[1198961

2+ 41198962 minus1198961

minus1198961 2]

(36)


120582min (119875)100381710038171003817100381712057710038171003817100381710038172le 119881 le 120582max (119875)

100381710038171003817100381712057710038171003817100381710038172 (37)


= minus1

|119904|12 (1205851198791198761120585 minus 119889 (119909 119905) 1198762

119879120585) (38)

where

1198761 =11989612

[1198961

2+ 21198962 minus1198961

minus1198961 1]

1198762119879= [

11989612

2+ 21198962 minus

11989612]

(39)


le minus1

|119904|12 120585119879119876120585 le minus

1|119904|

12 120582min (119876)100381710038171003817100381712058510038171003817100381710038172 (40)

where

119876 =11989612

[[

[

11989612+ 21198962 minus (

411989621198961

+ 1198961)120575 minus (1198961 + 2120575)

minus (1198961 + 2120575) 1

]]

]

(41)



|119904|12

le10038171003817100381710038171205771003817100381710038171003817 le

11988112

120582min12

(119875)

10038171003817100381710038171205771003817100381710038171003817 ge

11988112

120582max12

(119875)

(42)


le minus 12058111988112

(43)

where

120581 =120582min

12(119875) 120582min (119876)

120582max (119875) (44)



V (0) = V0 gt 0(45)

l1

l2

m2

m1

q2

q1


is given as

V (119905) = (V012

minus120581

2119905)

2 (46)



(1199090)120581




119872(119902) 119902 +119862 (119902 119902) +119866 (119902) = 120591 (119905) + 120591119889+119865 ( 119902) (47)

where

119872(119902)

= [119897221198982 + 21198971119897211989821198882 + 119897

21 (1198981 + 1198982) 119897

221198982 + 1198971119897211989821198882

119897221198982 + 1198971119897211989821198882 119897

221198982

]

119862 (119902 119902) = [minus1198982119897111989721199042 119902

22 minus 21198982119897111989721199042 1199021 1199022

1198982119897111989721199042 11990221

]

119866 (119902) = [1198982119897211989211988812 + (1198981 + 1198982) 11989711198921198881

1198982119897211989211988812]

(48)



119889is


2)


cos(1199021) 1198882 = cos(1199022) and 11988812 = cos(11990212)


0 1 2 3 4 5 6 7 8 9 10

08

1

12

14

Time (s)


q1

(rad

)

(a)

Time (s)


0 1 2 3 4 5 6 7 8 9 100

05

1

15

q2

(rad

)

(b)


Time (s)0 1 2 3 4 5 6 7 8 9 10

0

01

02

03

C-TSMCST-NTSMC

2 4 6 8 10

0

5

e1

(rad

)

minus01

minus5

times10minus6

(a)

Time (s)

C-TSMCST-NTSMC

0 1 2 3 4 5 6 7 8 9 10

002

2 4 6 8 10

0

5

e2

(rad

)minus06

minus04

minus02

minus5

times10minus5

(b)



119865 ( 119902) = [1199021 + 2 sin (1199021)

05 sin (1199022)]

120591119889= [

02 sin (119905)

02 cos (2119905)]

(49)


1199021119889 = 1+ 02 sin (05120587119905)

1199022119889 = 1minus 02 cos (05120587119905) (50)


1199021 (0) = 13

1199022 (0) = 03

1199021 (0) = 0

1199022 (0) = 0

(51)




ST-NTSMC 1205731 = 121198682 1205732 = 101198682 1198961 = 9 1198962 = 5 120582 = 3120572 = 03


120591 = 1198620 (119902 119902) +1198660 (119902) +1198720 (119902) 119902119889


(2minus120574)minus1198720 (1205781119904 + 1205782sig (119904)

120593)

(52)

where 119904 = 119890 + 120573 sig ( 119890)120574 sig (119909)120574 = [|1199091|

120574 sign (1199091)|1199092|120574 sign (1199092) |119909119899|

120574 sign (119909119899)] 120573 = diag(1205731 1205732 120573119899)





0 1 2 3 4 5 6 7 8 9 10

minus50

0

50

U1

Time (s)

C-TSMCST-NTSMC

(a)

0 1 2 3 4 5 6 7 8 9 10minus20

minus10

0

10

U2

Time (s)

C-TSMCST-NTSMC

(b)


0 1 2 3 4 5 6 7 8 9 10

2 4 6 8 10

0

2

minus1

0

1

2

3

4

S1

minus2

S2

Time (s)

C-TSMCST-NTSMC

times10minus4

(a)

0 1 2 3 4 5 6 7 8 9 10

2 4 6 8 10

0

2

S1

minus10

minus8

minus6

minus4

minus2

minus2

0

2

S2

Time (s)

C-TSMCST-NTSMC

times10minus3

(b)



5 Conclusions




Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119879119904leln (1 + (119890

2120572120582min(1205731)119905 sdot 119881120572(0)) 1198862)

2120572120582min (1205731) minus 120582 (6)



119881 =12120576119879120576 (7)


= 120576119879

120576 = 120576119879[minus1205731120576 minus 1205732119890

minus120582119905(120576119879120576)minus120572

120576] (8)

= minus 1205761198791205731120576 minus 1205732119890

minus120582119905(120576119879120576)minus120572

120576119879120576


minus120582119905(120576119879120576)

1minus120572


1198811minus120572

le 0

(9)



120572119881120572minus1 119889119881

119889119905le minus 2120572120582min (1205731) 119881

120572minus 21minus120572120572120582min (1205732) 119890

minus120582119905

119889119881120572

119889119905+ 2120572120582min (1205731) 119881


minus120582119905

(10)


1198902120572120582min(1205731)119905 (

119889119881120572

119889119905+ 2120572120582min (1205731) 119881

120572)


(11)

119889 (1198902120572120582min(1205731)119905 sdot 119881

120572)


[2120572120582min(1205731)minus120582]119905 (12)


knowing 119881(119879119904) = 0 yield

minus 1198902120572120582min(1205731)119905 sdot 119881

120572(0) le minus 1198862 [119890

[2120572120582min(1205731)minus120582]119879119904 minus 1] (13)

119890[2120572120582min(1205731)minus120582]119879119904 le 1+ 119890

2120572120582min(1205731)119905 sdot 119881120572(0)

1198862 (14)

where

1198862 =21minus120572120572120582min (1205732)

2120572120582min (1205731) minus 120582gt 0 (15)


[2120572120582min (1205731) minus 120582] 119879119904le ln(1+ 119890

2120572120582min(1205731)119905 sdot 119881120572(0)

1198862)

119879119904leln (1 + (119890

2120572120582min(1205731)119905 sdot 119881120572(0)) 1198862)

2120572120582min (1205731) minus 120582

(16)



119904 = + 120573119909119902119901

119904 = + 120572119909+120573119909119902119901

(17)






119904 = + 120573 |119909|120574 sign (119909)

119904 = + 120572119909+120573 |119909|120574 sign (119909)

(18)



119904 = 119909 +1120573119901119902

(19)









119906 (119905) = minus 119887 (119909 119905)minus1

sdot [119891 (119909 119905) minus 119889+1205731 120576 + 1205732119860+120578 sign (119904)]

(20)


119860 = [(minus120582) 119890minus120582119905

(120576119879120576)minus120572

120576

+ 119890minus120582119905


(120576119879

120576) 120576 + 119890minus120582119905

(120576119879120576)minus120572

120576]

(21)


119881 =12119904119879119904 (22)


119904 = 120576 + 1205731 120576 + 1205732119860

= 119891 (119909 119905) + 119889 (119909 119905) + 119887 (119909 119905) 119906 (119905) minus 119889 (119905) + 1205731 120576

+ 1205732119860

(23)




From (24) we have


119889119905 le minus119889119881


radic211988911988112

(120578 minus 119863)

(25)


have


119881(119879119904)

119881(119879119903)

radic211988911988112

(120578 minus 119863)=

radic2(120578 minus 119863)

11988112

(119879119903) (26)


in the finite time

119879119904le

radic2(120578 minus 119863)

11988112

(119879119903) +119879119903 (27)





119906 = 119906eq +119906STW (28)

where

119906eq = minus 119887 (119909 119905)minus1

[119891 (119909 119905) minus 119889+1205731 120576 + 1205732119860] (29)


119906STW = minus 119887 (119909 119905)minus1

(1198961 |119904|12 sign (119904) + 119911)

= minus 1198962 sign (119904)

(30)


119904 = 119891 (119909 119905) + 119889 (119909 119905) + 119887 (119909 119905) (119906eq +119906SMW) minus 119889 (119905)

+ 1205731 120576 + 1205732119860

(31)


119904 = minus 1198961 |119904|12 sign (119904) + 119911 + 119889 (119909 119905)

= minus 1198962 sign (119904)

(32)



119889 (119909 119905) le 120575 |119904|12

120575 = diag (1205751 1205752 120575119899) 120575119894gt 0

(33)


1198961 gt 2120575

1198962 gt 119896151198961 + 41205752 (1198961 minus 2120575)

120575(34)


119881 = 120585119879119875120585 (35)


where

120585 = [|119904|12 sign (119904) 119911]

119879

119875 =12[1198961

2+ 41198962 minus1198961

minus1198961 2]

(36)


120582min (119875)100381710038171003817100381712057710038171003817100381710038172le 119881 le 120582max (119875)

100381710038171003817100381712057710038171003817100381710038172 (37)


= minus1

|119904|12 (1205851198791198761120585 minus 119889 (119909 119905) 1198762

119879120585) (38)

where

1198761 =11989612

[1198961

2+ 21198962 minus1198961

minus1198961 1]

1198762119879= [

11989612

2+ 21198962 minus

11989612]

(39)


le minus1

|119904|12 120585119879119876120585 le minus

1|119904|

12 120582min (119876)100381710038171003817100381712058510038171003817100381710038172 (40)

where

119876 =11989612

[[

[

11989612+ 21198962 minus (

411989621198961

+ 1198961)120575 minus (1198961 + 2120575)

minus (1198961 + 2120575) 1

]]

]

(41)



|119904|12

le10038171003817100381710038171205771003817100381710038171003817 le

11988112

120582min12

(119875)

10038171003817100381710038171205771003817100381710038171003817 ge

11988112

120582max12

(119875)

(42)


le minus 12058111988112

(43)

where

120581 =120582min

12(119875) 120582min (119876)

120582max (119875) (44)



V (0) = V0 gt 0(45)

l1

l2

m2

m1

q2

q1


is given as

V (119905) = (V012

minus120581

2119905)

2 (46)



(1199090)120581




119872(119902) 119902 +119862 (119902 119902) +119866 (119902) = 120591 (119905) + 120591119889+119865 ( 119902) (47)

where

119872(119902)

= [119897221198982 + 21198971119897211989821198882 + 119897

21 (1198981 + 1198982) 119897

221198982 + 1198971119897211989821198882

119897221198982 + 1198971119897211989821198882 119897

221198982

]

119862 (119902 119902) = [minus1198982119897111989721199042 119902

22 minus 21198982119897111989721199042 1199021 1199022

1198982119897111989721199042 11990221

]

119866 (119902) = [1198982119897211989211988812 + (1198981 + 1198982) 11989711198921198881

1198982119897211989211988812]

(48)



119889is


2)


cos(1199021) 1198882 = cos(1199022) and 11988812 = cos(11990212)


0 1 2 3 4 5 6 7 8 9 10

08

1

12

14

Time (s)


q1

(rad

)

(a)

Time (s)


0 1 2 3 4 5 6 7 8 9 100

05

1

15

q2

(rad

)

(b)


Time (s)0 1 2 3 4 5 6 7 8 9 10

0

01

02

03

C-TSMCST-NTSMC

2 4 6 8 10

0

5

e1

(rad

)

minus01

minus5

times10minus6

(a)

Time (s)

C-TSMCST-NTSMC

0 1 2 3 4 5 6 7 8 9 10

002

2 4 6 8 10

0

5

e2

(rad

)minus06

minus04

minus02

minus5

times10minus5

(b)



119865 ( 119902) = [1199021 + 2 sin (1199021)

05 sin (1199022)]

120591119889= [

02 sin (119905)

02 cos (2119905)]

(49)


1199021119889 = 1+ 02 sin (05120587119905)

1199022119889 = 1minus 02 cos (05120587119905) (50)


1199021 (0) = 13

1199022 (0) = 03

1199021 (0) = 0

1199022 (0) = 0

(51)




ST-NTSMC 1205731 = 121198682 1205732 = 101198682 1198961 = 9 1198962 = 5 120582 = 3120572 = 03


120591 = 1198620 (119902 119902) +1198660 (119902) +1198720 (119902) 119902119889


(2minus120574)minus1198720 (1205781119904 + 1205782sig (119904)

120593)

(52)

where 119904 = 119890 + 120573 sig ( 119890)120574 sig (119909)120574 = [|1199091|

120574 sign (1199091)|1199092|120574 sign (1199092) |119909119899|

120574 sign (119909119899)] 120573 = diag(1205731 1205732 120573119899)





0 1 2 3 4 5 6 7 8 9 10

minus50

0

50

U1

Time (s)

C-TSMCST-NTSMC

(a)

0 1 2 3 4 5 6 7 8 9 10minus20

minus10

0

10

U2

Time (s)

C-TSMCST-NTSMC

(b)


0 1 2 3 4 5 6 7 8 9 10

2 4 6 8 10

0

2

minus1

0

1

2

3

4

S1

minus2

S2

Time (s)

C-TSMCST-NTSMC

times10minus4

(a)

0 1 2 3 4 5 6 7 8 9 10

2 4 6 8 10

0

2

S1

minus10

minus8

minus6

minus4

minus2

minus2

0

2

S2

Time (s)

C-TSMCST-NTSMC

times10minus3

(b)



5 Conclusions




Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119906 (119905) = minus 119887 (119909 119905)minus1

sdot [119891 (119909 119905) minus 119889+1205731 120576 + 1205732119860+120578 sign (119904)]

(20)


119860 = [(minus120582) 119890minus120582119905

(120576119879120576)minus120572

120576

+ 119890minus120582119905


(120576119879

120576) 120576 + 119890minus120582119905

(120576119879120576)minus120572

120576]

(21)


119881 =12119904119879119904 (22)


119904 = 120576 + 1205731 120576 + 1205732119860

= 119891 (119909 119905) + 119889 (119909 119905) + 119887 (119909 119905) 119906 (119905) minus 119889 (119905) + 1205731 120576

+ 1205732119860

(23)




From (24) we have


119889119905 le minus119889119881


radic211988911988112

(120578 minus 119863)

(25)


have


119881(119879119904)

119881(119879119903)

radic211988911988112

(120578 minus 119863)=

radic2(120578 minus 119863)

11988112

(119879119903) (26)


in the finite time

119879119904le

radic2(120578 minus 119863)

11988112

(119879119903) +119879119903 (27)





119906 = 119906eq +119906STW (28)

where

119906eq = minus 119887 (119909 119905)minus1

[119891 (119909 119905) minus 119889+1205731 120576 + 1205732119860] (29)


119906STW = minus 119887 (119909 119905)minus1

(1198961 |119904|12 sign (119904) + 119911)

= minus 1198962 sign (119904)

(30)


119904 = 119891 (119909 119905) + 119889 (119909 119905) + 119887 (119909 119905) (119906eq +119906SMW) minus 119889 (119905)

+ 1205731 120576 + 1205732119860

(31)


119904 = minus 1198961 |119904|12 sign (119904) + 119911 + 119889 (119909 119905)

= minus 1198962 sign (119904)

(32)



119889 (119909 119905) le 120575 |119904|12

120575 = diag (1205751 1205752 120575119899) 120575119894gt 0

(33)


1198961 gt 2120575

1198962 gt 119896151198961 + 41205752 (1198961 minus 2120575)

120575(34)


119881 = 120585119879119875120585 (35)


where

120585 = [|119904|12 sign (119904) 119911]

119879

119875 =12[1198961

2+ 41198962 minus1198961

minus1198961 2]

(36)


120582min (119875)100381710038171003817100381712057710038171003817100381710038172le 119881 le 120582max (119875)

100381710038171003817100381712057710038171003817100381710038172 (37)


= minus1

|119904|12 (1205851198791198761120585 minus 119889 (119909 119905) 1198762

119879120585) (38)

where

1198761 =11989612

[1198961

2+ 21198962 minus1198961

minus1198961 1]

1198762119879= [

11989612

2+ 21198962 minus

11989612]

(39)


le minus1

|119904|12 120585119879119876120585 le minus

1|119904|

12 120582min (119876)100381710038171003817100381712058510038171003817100381710038172 (40)

where

119876 =11989612

[[

[

11989612+ 21198962 minus (

411989621198961

+ 1198961)120575 minus (1198961 + 2120575)

minus (1198961 + 2120575) 1

]]

]

(41)



|119904|12

le10038171003817100381710038171205771003817100381710038171003817 le

11988112

120582min12

(119875)

10038171003817100381710038171205771003817100381710038171003817 ge

11988112

120582max12

(119875)

(42)


le minus 12058111988112

(43)

where

120581 =120582min

12(119875) 120582min (119876)

120582max (119875) (44)



V (0) = V0 gt 0(45)

l1

l2

m2

m1

q2

q1


is given as

V (119905) = (V012

minus120581

2119905)

2 (46)



(1199090)120581




119872(119902) 119902 +119862 (119902 119902) +119866 (119902) = 120591 (119905) + 120591119889+119865 ( 119902) (47)

where

119872(119902)

= [119897221198982 + 21198971119897211989821198882 + 119897

21 (1198981 + 1198982) 119897

221198982 + 1198971119897211989821198882

119897221198982 + 1198971119897211989821198882 119897

221198982

]

119862 (119902 119902) = [minus1198982119897111989721199042 119902

22 minus 21198982119897111989721199042 1199021 1199022

1198982119897111989721199042 11990221

]

119866 (119902) = [1198982119897211989211988812 + (1198981 + 1198982) 11989711198921198881

1198982119897211989211988812]

(48)



119889is


2)


cos(1199021) 1198882 = cos(1199022) and 11988812 = cos(11990212)


0 1 2 3 4 5 6 7 8 9 10

08

1

12

14

Time (s)


q1

(rad

)

(a)

Time (s)


0 1 2 3 4 5 6 7 8 9 100

05

1

15

q2

(rad

)

(b)


Time (s)0 1 2 3 4 5 6 7 8 9 10

0

01

02

03

C-TSMCST-NTSMC

2 4 6 8 10

0

5

e1

(rad

)

minus01

minus5

times10minus6

(a)

Time (s)

C-TSMCST-NTSMC

0 1 2 3 4 5 6 7 8 9 10

002

2 4 6 8 10

0

5

e2

(rad

)minus06

minus04

minus02

minus5

times10minus5

(b)



119865 ( 119902) = [1199021 + 2 sin (1199021)

05 sin (1199022)]

120591119889= [

02 sin (119905)

02 cos (2119905)]

(49)


1199021119889 = 1+ 02 sin (05120587119905)

1199022119889 = 1minus 02 cos (05120587119905) (50)


1199021 (0) = 13

1199022 (0) = 03

1199021 (0) = 0

1199022 (0) = 0

(51)




ST-NTSMC 1205731 = 121198682 1205732 = 101198682 1198961 = 9 1198962 = 5 120582 = 3120572 = 03


120591 = 1198620 (119902 119902) +1198660 (119902) +1198720 (119902) 119902119889


(2minus120574)minus1198720 (1205781119904 + 1205782sig (119904)

120593)

(52)

where 119904 = 119890 + 120573 sig ( 119890)120574 sig (119909)120574 = [|1199091|

120574 sign (1199091)|1199092|120574 sign (1199092) |119909119899|

120574 sign (119909119899)] 120573 = diag(1205731 1205732 120573119899)





0 1 2 3 4 5 6 7 8 9 10

minus50

0

50

U1

Time (s)

C-TSMCST-NTSMC

(a)

0 1 2 3 4 5 6 7 8 9 10minus20

minus10

0

10

U2

Time (s)

C-TSMCST-NTSMC

(b)


0 1 2 3 4 5 6 7 8 9 10

2 4 6 8 10

0

2

minus1

0

1

2

3

4

S1

minus2

S2

Time (s)

C-TSMCST-NTSMC

times10minus4

(a)

0 1 2 3 4 5 6 7 8 9 10

2 4 6 8 10

0

2

S1

minus10

minus8

minus6

minus4

minus2

minus2

0

2

S2

Time (s)

C-TSMCST-NTSMC

times10minus3

(b)



5 Conclusions




Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where

120585 = [|119904|12 sign (119904) 119911]

119879

119875 =12[1198961

2+ 41198962 minus1198961

minus1198961 2]

(36)


120582min (119875)100381710038171003817100381712057710038171003817100381710038172le 119881 le 120582max (119875)

100381710038171003817100381712057710038171003817100381710038172 (37)


= minus1

|119904|12 (1205851198791198761120585 minus 119889 (119909 119905) 1198762

119879120585) (38)

where

1198761 =11989612

[1198961

2+ 21198962 minus1198961

minus1198961 1]

1198762119879= [

11989612

2+ 21198962 minus

11989612]

(39)


le minus1

|119904|12 120585119879119876120585 le minus

1|119904|

12 120582min (119876)100381710038171003817100381712058510038171003817100381710038172 (40)

where

119876 =11989612

[[

[

11989612+ 21198962 minus (

411989621198961

+ 1198961)120575 minus (1198961 + 2120575)

minus (1198961 + 2120575) 1

]]

]

(41)



|119904|12

le10038171003817100381710038171205771003817100381710038171003817 le

11988112

120582min12

(119875)

10038171003817100381710038171205771003817100381710038171003817 ge

11988112

120582max12

(119875)

(42)


le minus 12058111988112

(43)

where

120581 =120582min

12(119875) 120582min (119876)

120582max (119875) (44)



V (0) = V0 gt 0(45)

l1

l2

m2

m1

q2

q1


is given as

V (119905) = (V012

minus120581

2119905)

2 (46)



(1199090)120581




119872(119902) 119902 +119862 (119902 119902) +119866 (119902) = 120591 (119905) + 120591119889+119865 ( 119902) (47)

where

119872(119902)

= [119897221198982 + 21198971119897211989821198882 + 119897

21 (1198981 + 1198982) 119897

221198982 + 1198971119897211989821198882

119897221198982 + 1198971119897211989821198882 119897

221198982

]

119862 (119902 119902) = [minus1198982119897111989721199042 119902

22 minus 21198982119897111989721199042 1199021 1199022

1198982119897111989721199042 11990221

]

119866 (119902) = [1198982119897211989211988812 + (1198981 + 1198982) 11989711198921198881

1198982119897211989211988812]

(48)



119889is


2)


cos(1199021) 1198882 = cos(1199022) and 11988812 = cos(11990212)


0 1 2 3 4 5 6 7 8 9 10

08

1

12

14

Time (s)


q1

(rad

)

(a)

Time (s)


0 1 2 3 4 5 6 7 8 9 100

05

1

15

q2

(rad

)

(b)


Time (s)0 1 2 3 4 5 6 7 8 9 10

0

01

02

03

C-TSMCST-NTSMC

2 4 6 8 10

0

5

e1

(rad

)

minus01

minus5

times10minus6

(a)

Time (s)

C-TSMCST-NTSMC

0 1 2 3 4 5 6 7 8 9 10

002

2 4 6 8 10

0

5

e2

(rad

)minus06

minus04

minus02

minus5

times10minus5

(b)



119865 ( 119902) = [1199021 + 2 sin (1199021)

05 sin (1199022)]

120591119889= [

02 sin (119905)

02 cos (2119905)]

(49)


1199021119889 = 1+ 02 sin (05120587119905)

1199022119889 = 1minus 02 cos (05120587119905) (50)


1199021 (0) = 13

1199022 (0) = 03

1199021 (0) = 0

1199022 (0) = 0

(51)




ST-NTSMC 1205731 = 121198682 1205732 = 101198682 1198961 = 9 1198962 = 5 120582 = 3120572 = 03


120591 = 1198620 (119902 119902) +1198660 (119902) +1198720 (119902) 119902119889


(2minus120574)minus1198720 (1205781119904 + 1205782sig (119904)

120593)

(52)

where 119904 = 119890 + 120573 sig ( 119890)120574 sig (119909)120574 = [|1199091|

120574 sign (1199091)|1199092|120574 sign (1199092) |119909119899|

120574 sign (119909119899)] 120573 = diag(1205731 1205732 120573119899)





0 1 2 3 4 5 6 7 8 9 10

minus50

0

50

U1

Time (s)

C-TSMCST-NTSMC

(a)

0 1 2 3 4 5 6 7 8 9 10minus20

minus10

0

10

U2

Time (s)

C-TSMCST-NTSMC

(b)


0 1 2 3 4 5 6 7 8 9 10

2 4 6 8 10

0

2

minus1

0

1

2

3

4

S1

minus2

S2

Time (s)

C-TSMCST-NTSMC

times10minus4

(a)

0 1 2 3 4 5 6 7 8 9 10

2 4 6 8 10

0

2

S1

minus10

minus8

minus6

minus4

minus2

minus2

0

2

S2

Time (s)

C-TSMCST-NTSMC

times10minus3

(b)



5 Conclusions




Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5 6 7 8 9 10

08

1

12

14

Time (s)


q1

(rad

)

(a)

Time (s)


0 1 2 3 4 5 6 7 8 9 100

05

1

15

q2

(rad

)

(b)


Time (s)0 1 2 3 4 5 6 7 8 9 10

0

01

02

03

C-TSMCST-NTSMC

2 4 6 8 10

0

5

e1

(rad

)

minus01

minus5

times10minus6

(a)

Time (s)

C-TSMCST-NTSMC

0 1 2 3 4 5 6 7 8 9 10

002

2 4 6 8 10

0

5

e2

(rad

)minus06

minus04

minus02

minus5

times10minus5

(b)



119865 ( 119902) = [1199021 + 2 sin (1199021)

05 sin (1199022)]

120591119889= [

02 sin (119905)

02 cos (2119905)]

(49)


1199021119889 = 1+ 02 sin (05120587119905)

1199022119889 = 1minus 02 cos (05120587119905) (50)


1199021 (0) = 13

1199022 (0) = 03

1199021 (0) = 0

1199022 (0) = 0

(51)




ST-NTSMC 1205731 = 121198682 1205732 = 101198682 1198961 = 9 1198962 = 5 120582 = 3120572 = 03


120591 = 1198620 (119902 119902) +1198660 (119902) +1198720 (119902) 119902119889


(2minus120574)minus1198720 (1205781119904 + 1205782sig (119904)

120593)

(52)

where 119904 = 119890 + 120573 sig ( 119890)120574 sig (119909)120574 = [|1199091|

120574 sign (1199091)|1199092|120574 sign (1199092) |119909119899|

120574 sign (119909119899)] 120573 = diag(1205731 1205732 120573119899)





0 1 2 3 4 5 6 7 8 9 10

minus50

0

50

U1

Time (s)

C-TSMCST-NTSMC

(a)

0 1 2 3 4 5 6 7 8 9 10minus20

minus10

0

10

U2

Time (s)

C-TSMCST-NTSMC

(b)


0 1 2 3 4 5 6 7 8 9 10

2 4 6 8 10

0

2

minus1

0

1

2

3

4

S1

minus2

S2

Time (s)

C-TSMCST-NTSMC

times10minus4

(a)

0 1 2 3 4 5 6 7 8 9 10

2 4 6 8 10

0

2

S1

minus10

minus8

minus6

minus4

minus2

minus2

0

2

S2

Time (s)

C-TSMCST-NTSMC

times10minus3

(b)



5 Conclusions




Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5 6 7 8 9 10

minus50

0

50

U1

Time (s)

C-TSMCST-NTSMC

(a)

0 1 2 3 4 5 6 7 8 9 10minus20

minus10

0

10

U2

Time (s)

C-TSMCST-NTSMC

(b)


0 1 2 3 4 5 6 7 8 9 10

2 4 6 8 10

0

2

minus1

0

1

2

3

4

S1

minus2

S2

Time (s)

C-TSMCST-NTSMC

times10minus4

(a)

0 1 2 3 4 5 6 7 8 9 10

2 4 6 8 10

0

2

S1

minus10

minus8

minus6

minus4

minus2

minus2

0

2

S2

Time (s)

C-TSMCST-NTSMC

times10minus3

(b)



5 Conclusions




Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Nonsingular Terminal Sliding Mode Control...

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