Review ArticleA Review of Fuzzy Logic and Neural Network BasedIntelligent Control Design for Discrete-Time Systems

Yiming Jiang1 Chenguang Yang12 and Hongbin Ma3

1Key Lab of Autonomous Systems and Networked Control (MOE) School of Automation Science and EngineeringSouth China University of Technology Guangzhou 510640 China2Zienkiewicz Centre for Computational Engineering Swansea University Swansea SA1 8EN UK3State Key Lab of Intelligent Control and Decision of Complex Systems Beijing Institute of Technology Beijing 100081 China

Correspondence should be addressed to Chenguang Yang cyangieeeorg

Received 5 November 2015 Accepted 29 December 2015

Academic Editor Juan R Torregrosa

Copyright copy 2016 Yiming Jiang 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

Over the last few decades the intelligent control methods such as fuzzy logic control (FLC) and neural network (NN) control havebeen successfully used in various applications The rapid development of digital computer based control systems requires controlsignals to be calculated in a digital or discrete-time form In this background the intelligent controlmethods developed for discrete-time systems have drawn great attentions This survey aims to present a summary of the state of the art of the design of FLC andNN-based intelligent control for discrete-time systems For discrete-time FLC systems numerous remarkable design approachesare introduced and a series of efficient methods to deal with the robustness stability and time delay of FLC discrete-time systemsare recommended Techniques for NN-based intelligent control for discrete-time systems such as adaptive methods and adaptivedynamic programming approaches are also reviewed Overall this paper is devoted to make a brief summary for recent progressesin FLC and NN-based intelligent control design for discrete-time systems as well as to present our thoughts and considerations ofrecent trends and potential research directions in this area

1 Introduction

It is well known that control design is critical to theperformance of the closed-loop system response while anaccurate system model is usually necessary for a high qualitycontrol design But there are inevitable uncertainties duringmodeling of any practical systems These modeling uncer-tainties may result in poor performance and may even leadto instability of the closed-loop systems To improve controlperformance many control strategies have been developedto consider these uncertainties in the control design stageAs one of the major control approaches adaptive control hasbeen developed for more than half a century with intenseresearch activities involving rigorous problem formulationstability analysis robustness design performance analysisand applications [1ndash9]

Early progress of adaptive control focused on identifi-cation and closed-loop system analysis of linear systems

Self-tuning regulator and model reference adaptive controlare two typical adaptive controllers based on different designphilosophies and they have attracted renown scholars such asAstrom and Wittenmark [10] Ljung [11] and Goodwin et al[12] due to the challenges in closed-loop stability analysis Infact even for the discrete-time linear systems the closed-loopsystem will become highly nonlinear due to the injection ofadaptive controller such as Astrom-Wittenmark self-tuningregulator whose closed-loop stability was eventually resolvedby Guo based on his cooperation with Chen et al [6 13 14]

Besides linear systems discrete-time adaptive control hasalso been developed for various different classes of nonlinearsystems such as nonlinear systems with linear growth rate[15] nonlinear systems with convexconcave parameteri-zation [16 17] nonlinear systems with polynomial-formnonlinear growth rate [18ndash21] nonlinear systems with mul-tiplefinite model structure uncertainty [22ndash25] nonlinearsystems with nonlinear parameterization [26 27] nonlinear

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2016 Article ID 7217364 11 pageshttpdxdoiorg10115520167217364

2 Discrete Dynamics in Nature and Society

systems with semiparametric uncertainties [28] and so onMany challenges have been seen in these existing studiesto cope with the interaction of nonlinearity and adaptationNow it is clear that discrete-time adaptive control can beextended to nonlinear systems with linear growth rate byusing the modification technique such as projection [15]however we also know that there exist intrinsic difficultiesor even impossibility for discrete-time adaptive control whenthe nonlinear growth rate is high enough [29 30] whichis quantitatively characterized by fundamental limitations offeedback mechanism in terms of certain introduced uncer-tainty measure

In addition to the progress from linear systems to thenonlinear systems rigorous stability analysis of the closed-loop adaptive system has also been well established [31]Compared with robust control the advantage of adaptivecontrol lies in its ability in estimating and compensating forparametric uncertainties in a large range However towardsthe increasingly complexity of systems with complicatednonlinear functional uncertainties it is necessary to developmore powerful control design tools To this end researchersnoticed some limitations of traditional adaptive control inmainly copingwith parametric uncertainties And fuzzy logic(FL) and NN-based intelligent control have been introducedin the early 1990s [6]

The concept of fuzzy set was initially proposed byZadeh [32] and since then fuzzy technique gained a rapiddevelopment FL offers human reasoning capabilities tocapture nonlinearities and uncertainties which cannot bedescribed by precise mathematical model In [33] Wang andMendel found that linear combinations of a series of fuzzybasis functions are able to approximate smooth nonlinearfunctions and proved such feature of universal approximatorby using the Stone-Weierstrass theorem Based on this workan adaptive fuzzy control method was further proposed bycombining the universal approximation of the FL controllerand learning ability of the adaptive method [34] And later astable adaptive fuzzy controller was successfully designed fora class of nonlinear systems In this approach the adaptive lawwas constructed to update the parameters of the FL controllerduring the adaptation procedure while it was not necessary tohave the accurate mathematical model of the system Fromthen on fuzzy control has attracted ever increasing researchinterest since it is able to provide an effective solution tocontrol complex and uncertain plants by employing theknowledge of specific experts for the controller design

NN is inspired by biological neuronal systems witch con-sist of a number of simple processing neurons interconnectedto each other McCulloch and Pitts introduced an idea tostudy the computational abilities of networks composed ofsimple models of neurons in the 1940s [35] With highlyparallel structure NN is of great computational power andlearning ability to emulate various systems dynamics It is wellestablished that NN is capable of universally approximatingunknown smooth functions with errors that can be madearbitrarily small [36ndash38] In addition to system modelingand control NN has also been successfully applied in manyother fields such as learning pattern recognition and sig-nal processing In NN control methodology NN has been

extensively studied as function approximators to compensatefor the system uncertain nonlinearities in control design Inthe last two decades it has been verified that NN controlis very useful for controlling highly uncertain nonlinearsystems and NN control has demonstrated superiority overtraditional control methods Particularly the marriage ofadaptive control theories and NN techniques gives birth toadaptive NN control which guarantees stability robustnessand convergence of the closed-loop NN control systemswithout beforehand offline NN training [6]

Nowadays most of the control algorithms are realized bydigital computers and thus the desired control signals arecalculated in a digital manner Digital control systems haveadvantages such as easy to build less sensitive to environmentvariation flexible to change and less expensive A modelbased digital controller is designed in discrete-time form andoperated in continuous-time with analogy signals Since thedata are generally processed at discrete-time instants it isnecessary for us to build the system model in discrete-timefor ease of control design [6]

Yet in the past decades many significant progressesof FLC and NN control for nonlinear systems were incontinuous-time form and there is considerable lag in thedevelopment for nonlinear discrete-time systems [39 40]Although for linear systems there are numerous results ofNN and fuzzy controls in continuous-time systems theNN and fuzzy control of nonlinear discrete-time systemshave been considerately less studied than their counterpartsin continuous-time As a matter of fact many techniquesdeveloped for continuous-time systems cannot be directlyapplied in discrete-time especially when the systems arenonlinear Discrete-time systems can be expressed by variousequations which in great contrast to the differential equationsof continuous-time systems involve states at different timesteps Due to the different nature of difference equationand differential equation some concepts in discrete-timehave very different meanings from those in continuous-time and the stability analysis techniques become muchmore intractable For example the linearity property of thederivative of a continuous-time Lyapunov function is nolonger existed in a discrete-time Lyapunov function [41]Therefore in order to design discrete-time fuzzy or NNcontrollers and implement them into digital control systemsthey deserve in-depth investigation for fuzzy control and NNcontrol of nonlinear discrete-time systems

The remainder of this paper is organized as followsIn Section 2 a brief review of fuzzy control for discrete-time systems is presented Some research problems of fuzzycontrol and discrete-time systems such as adaptive controlrobustness issue stability analysis and time-delay systemsare concerned and introduced for they are both theoreti-cally challenging and practically meaningful In Section 3NN control methods for discrete-time systems are brieflyreviewed Research of adaptive NN control for discrete-time systems is introduced and NN-based adaptive dynamicprogramming methods for discrete-time systems are alsodiscussed Finally in Section 4 a brief conclusion is given

Discrete Dynamics in Nature and Society 3

2 Fuzzy Logic Control forDiscrete-Time Systems

Nowadays techniques for FLC have developed rapidly espe-cially in modeling complex nonlinear systems Since Wangfirst proved that the linear combinations of a series offuzzy basis functions are universal approximators of anynonlinear systems the universal approximation property ofthe fuzzy logic systems has been extensively studied [42ndash45]The combination of adaptive control and fuzzy logic systemallows adaptive laws to update parameters of the FL controllerduring the adaptation procedure while adaptive fuzzy controlprovides an efficient method to model complex nonlinearsystems In practice the difficulty of design a fuzzy controllerlies in the fact that there are usually various requirementimposed on the systems to ensure stability and performancewhile most of complex control systems today are using digitalcomputers to calculate the control signals in a digital formHence the continuous-time methods could not be directlyapplied to most practical systems The modelingcontrol fordiscrete-time systems is crucial to take the controllers intoreal plants Thus it is significant to study the discrete-timefuzzy systems as well as their molding and control To thisend researchers have paid efforts to develop the discrete-time FLCA typical procedure of themultiple-inputmultiple-output (MIMO) fuuzy logic system (FLS) approximatingan unknown function 119865(119909) is comprised of three primarycomponents as follows [46]

(1) Fuzzification Take 119883 as the input of the fuzzy systemand (119883) is the estimation of the output Both the input andthe output are fuzzified into fuzzy linguistic terms with fuzzymembership function

(2) Fuzzy Rules The collection of the fuzzy MIMO IF-THEN rules are designed to comprise the knowledge basefor constructing the FLS Using prior expert knowledge thefuzzy rules can be obtained as

Rule 119897 If 1199091is 119860119897

1 1199092is 119860119897

2 119909

119899is 119860119897

119899

then 1199101is 119861119897

1 1199102is 119861119897

2 119910

119898is 119861119897

119898

(1)

where 1199091

sim 119909119899and 119910

1sim 119910119898

are the premise variablesconsisting of 119883 and (119883) respectively 119860

119897 and 119861119897 are the

linguistic variables of the fuzzy sets 119897 = 1 2 119871

(3) Fuzzy Inference Engine and Defuzzification Using thestrategy of the sum-product inference engine and the center-average defuzzifier the output of the FLS can be described as

119910119895

=

sum119871

119897=1120593(119897)

119895(prod119899

119894=1120583(119897)

119860119894(119909119894))

sum119871

119897=1(prod119899

119894=1120583(119897)

119860119894(119909119894))

(2)

where 120593(119897)

119895is the point where fuzzy membership function

120583(119897)

119861119895(120593119895) achieves its maximum value Define

119908119895(119909) =

prod119899

119894=1120583(119897)

119860119894(119909119894)

sum119871

119897=1(prod119899

119894=1120583(119897)

119860119894(119909119894))

119882 (119909) = [1199081

(119909) 1199082

(119909) 119908119871

(119909)]119879

(3)

and (Φ)119895

= [120593(1)

119895 120593(2)

119895 120593

(119871)

119895]119879 Then the output of MIMO

FLS can be rewritten as

119910119895

= (Φ)119879

119895119882 (119883) (4)

where the subscript 119895 represents the 119895th column vector of thematrix Therefore the estimation of (119883) is

(119883)

= [(Φ)119879

1119882 (119883) (Φ)

119879

2119882 (119883) (Φ)

119879

119898119882 (119883)]

119879

(5)

Then 119865(119883) can be written as

(119883) = Φ119879

119882 (119883) (6)

21 Adaptive FLC of Discrete-Time Systems FLC has beenextensively investigated in both academic and industrialcommunities However problems with high dimensionalityalso existed in higher-dimensional fuzzy systems due to themathematical complications and intuitive limitations There-fore it required more systematic methods for advanced FLCdesign and synthesis such as the adaptive FLC Originallyadaptive control was proposed for aircraft autopilots to dealwith parameter variations during changing flight conditionsIn the 1960s the advances in stability theory and the progressof control theory improved the understanding of adaptivecontrol By the early 1980s several adaptive approaches havebeen proven to provide stable operation and asymptotictracking Since then the adaptive control problems have beenrigorously formulated

Generally adaptive FLC design for nonlinear discrete-time systems ismuchmore difficult than those in continuous-time The stability analysis techniques become much moreintractable for difference equations than those for differentialequations Many nice Lyapunov adaptive control designmethodologies developed in continuous-time are not appli-cable to discrete-time systems Thus it is significant toconsider the adaptive FLC in discrete-time form

In recent years the adaptive FLC for discrete-time non-linear systems has been extensively studied Jagannathanet al [45 47] developed an adaptive fuzzy logic controllerfor a general class of discrete-time nonlinear systems usingbasis vectors and provided rigorous proofs to stability of thediscrete-time adaptive FLC In this work a multiloop struc-ture controller was constructed with a fuzzy approximatelinearization loop an outer tracking loop and a robustifyingloop By selecting FLCrsquos structure and the adaptation laws andgiving a number of assumptions on a class of discrete-timenonlinear systems closed-loop signals can achieve uniform

4 Discrete Dynamics in Nature and Society

ultimate boundedness This method is significant in sensethat no persistence of excitation is used and no certaintyequivalence assumption is required The controller has alsobeen extended to a class of unknown feedback linearizablenonlinear dynamical systems under persistence of excitationin [48] in which rigorous stability proofs of discrete-timeadaptive FLC for feedback linearizable unknown nonlinearsystems were presented Indirect fuzzy control of uncertaindiscrete-time nonlinear system was shown by Qi and Brdys[49] This work presented an indirect adaptive FLC for theuncertainties in nonlinear plants and employed a Takagi-Sugeno (T-S) model to deal with the unknown dynamicsin input-output form A feedback linearization control lawwas designed by using structure states and parametersof the model Gradient descent algorithm and recursiveleast square estimation method were used to online updatethe modeling parameters The extension of sliding modecontrol (SMC) adaptive fuzzy SMC for a class of uncertaindiscrete-time nonlinear systems was investigated by Linet al [50] This work presented an adaptive interval type-2 fuzzy sliding mode controller for a class of unknownnonlinear discrete-time systems with training data disturbedby external disturbances It employed an adaptive intervaltype-2 fuzzy control scheme and SMC approach to controlthe plant tracking a reference trajectory and prevented bigchattering of the control effort While adaptive fuzzy controlhas also been applied in discrete-time chaotic systems In[51] Feng and Chen presented a novel adaptive controlalgorithm for discrete-time chaotic systems The basic ideais representing the chaotic system as a T-S fuzzy model anddesigning a local linear adaptive controller in each localregion Meanwhile a global adaptive controller on the entiredomainwas reconstructed and the stability of the closed-loopadaptive control system was proved

22 Robustness Issue in Discrete-Time Fuzzy Control Inpractice nonlinearities and uncertainties exist in almost allthe industrial plants while uncertainties in the modeling andcontrol of nonlinear systems are still one of themost challeng-ing problems in the control field The presence of nonlinear-ities and uncertainties brings difficulties to controller designIn order to conquer these problems a number of schemeshave been developed and among these robust control is oneof the most effective methods Robustness is regarded as oneof the most important requirements for a control system Inthe past two decades considerable attention has been paid tothe robustness in fuzzy model-based control of discrete-timesystems andmanymodification techniques were proposed toenhance the robustness of FLC for discrete-time systems [52ndash57]

The work of Lee et al [52] illustrated an approach ofrobust fuzzy control for nonlinear discrete-time systems Inthis work a systematic control structure was presented fornonlinear discrete-time T-S fuzzy systems with parametricuncertainties using the 119867

infincontrol design approach This

work also gave a novel solution to the robust stabilizationproblem of nonlinear systems by using basis-dependentLyapunov function

119867infindisturbance attenuation is one of the most important

requirements for a fuzzy control system Cao and Frank [53]have applied the 119867

infincontrol to address the robust stability

for a class of uncertain discrete-time fuzzy systems via linearmatrix inequality approach They studied both the robuststabilization and the 119867

infindisturbance attenuation while

they presented several sufficient conditions to ensure robuststability of the fuzzy models based on coupled linear matrixinequalities They also analyzed the robust 119867

infinperformance

of the fuzzy model-based discrete-time nonlinear systems byaddressing the robust 119867

infindisturbance attenuation problems

The robust stability of the models is achieved independentlyof the uncertainties

How to develop a robust 119867infin

controller for discrete-time systems using the basis-dependent Lyapunov functionwas also shown by Zhou et al [54] In their work anovel linear matrix inequalities characterization with 119867

infin

norm bound was presented for discrete-time fuzzy systemsWhile additional matrix variables were used to decouple theLyapunov function and system matrices as well as facilitatethe design approach of the controller this approach providessome sufficient results in the shape of strict linear matrixinequalities

Studied by Choi and Park the state feedback 119867infin

controlfor discrete-time systems was investigated by constructingLyapunov functions with fuzzy weights The fuzzy weightingLyapunov functions were designed with both current-timeand one-step-past information while designing controllerparameterized with linear matrix inequalities (PLMIs) Byselecting the structures of variables appropriately in thePLMIs a special case LMI formulation was obtained The119867infin

control with output feedback for discrete-time fuzzysystems has also been rigorously studied by Xu and Lam[55] while Wu et al [58] investigated the reliable 119867

infincontrol

for discrete-time systems with delays and stochastic actuatorfaults by representing the stochastic behavior with a discrete-time homogeneous Markov chain

Tseng andChen [56] used a fuzzy observer to estimate thepremise variables which depend on the state variables whilethe fuzzy observer was also used to address the nonlinear119897infin-gain control problem This work extended the 119897

infincontrol

from linear discrete-time systems to nonlinear discrete-time systems By using the T-S fuzzy model to representthe nonlinear discrete-time system an observer-based fuzzycontroller which minimized the upper bound of 119897

infin-gain and

attenuated the peak of perturbation was designed Xu et al[57] gave stability analysis for discrete-time singular fuzzysystems in the presence of time-varying uncertainties whilethey ensured the system to be regular causal and stable It hasbeen shown that for these systems robust stability conditioncan be obtained by giving a sufficient condition in terms of aset of linear matrix inequalities

23 Stability Issue in Discrete-Time Fuzzy Control Stabilityis one of the most important properties for a system whileFLC are used to address the stability problems for discrete-time systems as well The quadratic stability for uncertaindiscrete-time fuzzy dynamic systems was shown by Fengand Ma [59] This work gave some sufficient conditions of

Discrete Dynamics in Nature and Society 5

the quadratic stabilization for an uncertain fuzzy dynamicsystem Stability was achieved when the suitable Riccatiequation or a set of Riccati equations were solved Stabilizingfeedback control laws were also obtained by the developingalgorithms The nonquadratic stabilization conditions fornonlinear discrete-time fuzzy systems were also analyzedby Kruszewski et al [60] They considered discrete-timeuncertain nonlinear models in a T-S form and studied thestability through a nonquadratic Lyapunov function Thestabilization conditions were developed by considering theLyapunov function with a 119896-sample variation and extendedto uncertain T-S models Based on a switching fuzzy modeland piecewise Lyapunov function Wang et al [61] proposedtwo stabilization criteria for discrete-time T-S fuzzy systems

Zhou et al [54] used a basis-dependent Lyapunov-Krasovskii function to give stabilization analysis for discrete-time fuzzy systems A robust control design approach wasalso developed by using the Lyapunov-Krasovskii functionand facilitated by introducing additional instrumental matrixvariables Robust control problem for systemswith time delaywas concerned as well Feng [62] used a piecewise smoothLyapunov function to analyze the stability of discrete-time T-S fuzzy dynamic systems In this study stability of the systemguaranteed by constructing a piecewise Lyapunov functionMeanwhile the Lyapunov function was obtained using thelinear matrix inequalities methodThis work showed that thepiecewise quadratic Lyapunov functions based stability is lessconservative than the common quadratic Lyapunov functionbased stability

The stabilization for discrete-time T-S fuzzy systems withstate time-varying delay was shown by Gao et al [63] Intheir work a fuzzy Lyapunov function was constructed toimprove the delay-dependent stability condition By avoidingthe utilization of the bounding inequalities for the crossproducts between two vectors reduction of the conservatismof stability condition was achieved A delay-dependent stabi-lization algorithm was also developed for both state feedbackand observer-based output feedback cases using a paralleldistributed compensation scheme

24 Fuzzy Control for Discrete-Time Systemswith TimeDelaysIn practice time delays are the intrinsic nature of variousphysical systems such as communication hydraulic chem-ical processes and electronics The existence of time delayscould generally lower the system performance and even causeinstability and oscillation Hence remarkable attention hasbeen devoted to the analysis and synthesis of time delaynonlinear systems as well as the discrete-time fuzzy systems[55 64ndash69]

The stability and stabilization problems of systems time-varying delay were investigated by Wu et al [64] In theirwork the existing stability analysis approaches were dividedinto two types delay-dependent (including the delay size)and delay-independent (irrelevant to the delay size) Thestability of discrete-time T-S fuzzy systems with time-varyingstate delay was also investigated They developed a delay par-titioningmethod and employed a fuzzy Lyapunov-Krasovskiifunction to analyze the stability of delay-dependent time-varying state delay systems Through delay partitioning the

less conservative stability condition is obtained and solved byLMI optimization techniques

The work of Su et al [65] also gave a solution to thedynamic output feedback control problem for fuzzy sys-tems with time-varying delays They proposed a comparisonmodel by approximating the time-varying delay state whilethey gave a sufficient condition to ensure the asymptoti-cal stability by adopting the scaled small-gain theorem aswell as the delay partitioning method This method allowsdesigning desired dynamic output feedback controller byusing optimization techniques A filter design technique fordiscrete-time systems with time-varying delay has also beenpresented by Su et al [66] Both full-order and reduced-orderfilters were designed while approximators were employed toderive an input-output based sufficient condition Tseng [67]studied the time delay problem for a nonlinear discrete-timesystem based on model reference fuzzy tracking control Inthis work the T-S fuzzy model was applied to approximatea time delay discrete-time system while the fuzzy controllerwas designed to reduce the tracking error based on thisfuzzy model This method has advantage in that no feedbacklinearization or adaptive approaches are used in the controllerdesign

The robust 119867infin

control problem for uncertain discrete-time delay systems has been ivestigated by Xu and Lam [55]They studied the state-space T-S fuzzymodel with time delaysunder the assumption that the parameter uncertainties werenorm-boundedThe output feedback controller was designedwith full-order fuzzy dynamic to guarantee the closed-loopsystem is robust asymptotic stable as well as the admissibleuncertainties are norm bound constraint While solvabilityof this problem was addressed using a sufficient conditiongiven by the linear matrix inequalities The induced 119897

2filter

design of T-S fuzzy discrete-time stochastic systems withtime-varying delays has been investigated by Su et al [68]Wuet al [69] studied the 119867

infinmodel approximation for discrete-

time state delay fuzzy T-S systems

3 NN Control for Discrete-Time Systems

NN is of powerful computing ability and learning abilityto emulate various systems dynamics and is capable ofapproximating an unknown functionwith arbitrary accuracyNN is very successful in system modeling and controlby its capacity of universal approximate highly nonlinearand uncertain nonlinear and complex dynamic of systemsNNrsquos approximation ability has been shown by the Stone-Weierstrass theorem which states that a universal approxi-mator can approximate to an arbitrary degree of accuracyany real continuous function on a compact set Besides theuniversal approximation abilities NN also shows its excel-lence in parallel distributed processing abilities learningadaptation abilities natural fault tolerance and feasibilityfor hardware implementation These advantages make NNparticularly attractive and prospective for nonlinear controland modeling NN has been successfully applied to robotmanipulators control [70] distillation column control [71]chemical processes identification [72] flight control [73ndash75]

6 Discrete Dynamics in Nature and Society

and so forth To extend the NN technique from continuous-time to discrete-time many efforts have been made to studythe NN controlled discrete-time systems [76ndash78]

31 Adaptive NN Control for Discrete-Time Systems In theearly stage backpropagation (BP) algorithm [79] greatlyboosted the development of NN control It is noted that inthe early NN control design the control performances weredemonstrated through simulation or by particular exper-imental examples and consequently there were shortageof analytical analysis In addition an offline identificationprocedure was essential for achieving a stable NN controlsystem Thereafter the emergence of Lyapunov-based NNdesign makes it possible to use the available adaptive controltheories to rigorously guarantee stability robustness andconvergence of the closed-loop NN control systems Wecall the control design combining adaptive control theoriesand NN techniques as adaptive NN control It updates NNweights online and guarantees the stability of the closed-loop system Adaptive NN control design has been elegantlydeveloped for nonlinear systems with parametric uncertain-ties while many efforts have been dedicated in adaptive NNcontrol for nonlinear discrete-time systems [6]

For high-order affine nonlinear system in normal formadaptive NN controls using LPNN and MNN have beendeveloped in [80 81] using a filtered tracking error In [82]the controller is designed incorporating reinforcement learn-ing technique to improve control performance In this worka critic NN has been introduced to approximate the strategicutility function which is considered as the long-term systemperformance measure For discrete-time systems in strict-feedback form adaptive NN control has been developed viabackstepping design after system transformation [83] In [84]adaptive NN control has been investigated for discrete-timesystem in affine NARMAX form

In the above mentioned results the adaptive NN controldesigns were carried out through either feedback lineariza-tion or backstepping But these approaches are not applicableto nonaffine systems especially feedback linearization basedmethods which greatly depend on the affine appearance ofcontrol variables As a matter of fact adaptive NN controlmethods for nonaffine systems have been less studied incomparison with large amount of research work on affinenonlinear systems because of the difficulty of control designcaused by the nonaffine form of control input To overcomethe difficulty linearization based NN controls have beenput forward In [17] the nonaffine discrete-time system hasbeen divided into two parts one is linear and another isnonlinear and consequently a linear adaptive controller anda nonlinear adaptive NN controller have been designedwith a switching rule In [85] it directly utilized NN asemulator of the ldquoinverserdquo of the nonlinear discrete-timesystems Furthermore the study of discrete-time systems foradaptive NN control using implicit function to assert theexistence of an ideal inverse control was investigated in [86]Thereafter the implicit function based adaptive NN controlhas been widely studied in discrete-time form [87 88] Blocktriangular discrete-time systems with normal form subsys-tems have been studied in [80 81] For the block triangular

systems with strict-feedback subsystems state feedback andoutput feedback adaptive NN control have been developedin [89 90] by extending the systems transformation basedbackstepping technique proposed for SISO case in [83] In[91] adaptive NN control has been used for sampled-datanonlinear MIMO systems in general affine form based onlinearization The control scheme is an integration of an NNapproach and a variable structure method

An effort has been made in [92] to explore the adaptiveNN control of a class of nonaffine systems in discrete-timeThis work aimed to solve the nonaffine appearance andnoncausal problems of the following pure-feedback discrete-time system

120585119894(119896 + 1) = 119891

119894(120585119894 120585119894+ 1 (119896))

119894 = 1 2 119899 minus 1 119899 ge 2

120585119899

(119896 + 1) = 119891119899

(120585119899

(119896) 119906 (119896) 119889 (119896))

119910 (119896) = 1205851

(119896)

(7)

where 120585119894(119896) = [120585

1(119896) 1205852(119896) 120585

119899(119896)]119879 are system states

119891119894are unknown nonlinear functions and 119906(119896) and 119910(119896) are

system input and output respectively while 119889(119896) denotes theexternal disturbance

Using a states prediction technique the pure-feedbackdiscrete-time systems in (7) are shown to be transformableto an 119899-step-ahead predictor as below

119910 (119896 + 119899) = 120601 (120585119899

(119896) 119906 (119896) 119889 (119896))

= 120601119904(120585119899

(119896) 119906 (119896)) + 119889119904(119896)

(8)

And by future output predictions the above system (8)can be further transformed into an input-output model forthe output feedback control

119910 (119896 + 119899)

= 1198651198991

119910 (119896 + 1) 119906 (119896) 119889 (119896) 119889 (119896 minus 119899 + 2)

(9)

This shows that the system presentation (9) in NARMAXform is a transformation of the pure-feedback system (8)After transformation both state feedback and output feed-back controls only need to employ a single NN in thecontroller design rather than a number of NNs in previousresearches

Based on the SISO pure-feedback system (8) and itstransformation NARMAX model in (9) the control designhas been further investigated by using high-order neuralnetwork to approximate unknown functions [93] In theabove mentioned control design availability is assumed forknowledge of control directions which are defined as ldquothesigns of control variable gains in affine systems or the signsof partial derivatives over control variables in nonaffinesystemsrdquo [93]

The paper overcame the problems of output-feedbackcontrol when the prior knowledge of the control directions isunknownThemain idea is to introduce a discrete Nussbaum

Discrete Dynamics in Nature and Society 7

gain to counter the lack of knowledge on control gain inadaptive NN control

A rigorous definition of discrete Nussbaum gain in [93] isgiven as follows

Remark 1 Consider a discrete nonlinear function 119873(119909(119896))

defined on a sequence 119909(119896) with 119909119904(119896) = sup(119909

1015840

119896) 119873(119909(119896))

is a discrete Nussbaum gain if and only if it satisfies thefollowing two properties

(i) if 119909119904(119896) increases without bound then

sup 1

119909119904(119896)

119878119873

(119909 (119896)) = +infin

inf 1

119909119904(119896)

119878119873

(119909 (119896)) = minusinfin

(10)

(ii) if 119909119904(119896) ge (119896)120575

1 then |119878

119873(119909(119896))| ge 120575

2with some

positive constants 1205751and 1205752 where 119878

119873(119909(119896)) is defined as

119878119873

(119909 (119896)) =

119896

sum

1198961015840=0

119873 (119909 (1198961015840)) Δ119909 (119896

1015840) (11)

with Δ119909(119896) = 119909(119896 + 1) minus 119909(119896) In addition neither the upperbounds nor the lower bounds of the control gains are requiredto be known

Unknown control directions problem for MIMOdiscrete-time nonlinear systems was solved by using adaptiveoutput feedback NN control in [94] The studies in [92 93]all studied the discrete NN control for SISO systems and[94] extended the study to a class of MIMO discrete-timesystems with each subsystem in the nonaffine pure-feedbackas follows

120585119895119894119895

(119896 + 1) = 119891119895119894119895

(1205851119894119895minus1198981198951

(119896) 1205852119894119895minus1198981198952

(119896)

120585119899119894119895minus119898119895119899

(119896) 120585119895 119894119895+1

(119896)) 119894119895

= 1 2 119899119895

minus 1

120585119895119899119895

(119896 + 1) = 119891119895119899119895

(Ξ (119896) 119906119895(119896) 119889

119895(119896))

119910119895(119896) = 120585

1198951(119896)

(12)

where 120585119895119894119895

(119896) = [1205851198951

(119896) 1205851198952

(119896) 120585119895119894119895

(119896)]119879 are vectors

of states variable of subsystem Ξ(119896) is vector of all statevariables 119891

119895119894119895are unknown nonlinear functions 119906

119895(119896) and

119910119895(119896) are system inputs and outputs respectively while 119889

119895(119896)

denote the external disturbanceFor nonlinearMIMOdiscrete-time higher order systems

a major drawback is that there are too many online-tunedadaptive parameters and large online computation burden In[95] an adaptive neural output feedback adaptive controllerwas designed for MIMO nonlinear discrete-time systemswith fewer adaptive parameters The output feedback adap-tive control for a class of nonlinear discrete-time systemswithunknown control directions was investigated in [96]

32 NN-Based Dynamic Programming Algorithm for Discrete-Time Systems In the past several decades optimal control for

nonlinear systems has been a key focus in the control fieldsWhile adaptive dynamics programming (ADP) is a veryuseful approach to solve the optimal control problems ADPis a reinforcement learning method to give solution to thedynamic programming utilizing function approximating thevalue function based on adaptive approach [97] P Werbos[98] classified the discrete-time ADP into a number ofschemes action-dependentHDP dual HDP (DHP) heuristicdynamic programming (HDP) and action-dependent dualHDP while NN has been widely used to solve the adaptivedynamic programming problems of discrete-time systems[97 99ndash104]

He and Jagannathan [99] showed the reinforcementlearning with NN-based controller for nonlinear discrete-time systems This work developed an adaptive-critic-basediscrete-time NN controller to deliver a desired tracking per-formance for nonlinear systems in the presence of actuatorconstraints The critic NN and action NN were designed toapproximate the strategic utility function while uniformlyultimate boundedness of the closed-loop tracking error wasproved by Lyapunov approach The reinforcement learningNN control for SISO discrete-time pure-feedback systemswas studied in [100] The controller design is based on thetransformed predictor and two NNs of control architectureOne critic NN to approximate the strategic utility function isshown as follows

By introducing a utility function 119901(119896) based on thetracking error 119890(119896) = 119910(119896) minus 119910

119889(119896)

119901 (119896) = 1198860 |119890 (119896)| (13)

where 119901(119896) isin 119877 1198860

isin 119877 is the positive design parameter Andthe strategic utility function 119876 isin 119877 is defined as

119876 (119896) = 119886119873

(119901 + 1) + 119886119899minus1

119901 (119896 + 2) + sdot sdot sdot + 119886119896+1

119901 (119873)

+ sdot sdot sdot

(14)

where 119886 isin 119877 0 lt 119886 lt 1 119873 is the horizon Then the criticNN is used to approximate the strategic utility function119876(119896)

= 119879

119888(119896) 119878119888(119911 (119896)) 119878

119888(119911 (119896)) isin 119877

119897119888 (15)

where 119882119879

119888(119896) isin 119877

119897119888 is the estimation of optimal NN 119882lowast

119888

weights The uniformly ultimate boundedness of closed-loopsignals is established via Lyapunov stability analysis

Al-Tamimi et al [97] proved the convergence of a valuefunction solution for nonlinear dynamical systems usingHDP algorithm and NNs were employed to approximatethe value and the control action at each iteration Liu et al[101] used theHDP algorithm to develop aNN-based optimalcontroller for unknown discrete-time nonlinear systemsThemain idea of this work is to introduce an iterative ADPalgorithm and a globalizedHDP technique into the controllerdesign And by using the cost function and control law theconvergence of the optimal control is guaranteed In [102]an iterative ADP algorithm for discrete-time systems wasdeveloped as well An optimal controller was designed toaddress the infinite-horizon discrete-time nonlinear systems

8 Discrete Dynamics in Nature and Society

with finite approximation errors It was shown thatwhen con-vergence conditions are satisfied iterative performance indexfunction can converge to the performance index functionsrsquolower bound while the performance index functions wereapproximated by NNs to compute optimal control policy

The ADP control in the unknown discrete-time Markovjump systems was investigated by Zhong et al [103] AndWang et al [105] also studied the ADP for discrete-timesystems to obtain the 120576-optimal control by using neuralnetworks In [106] iterative ADP algorithmwas used to studythe near-optimal control with control constraints in discrete-time systems An iterative ADP for discrete-time systemswas studied in [104] In this work the optimal controllerwas designed with control constraints while NN was used toidentify the unknown dynamical systems with stability proof

4 Conclusion

In this short survey despite the impossibility in identifyingor listing all the related contributions best efforts have beenmade to summarize the major achievements in the area ofdiscrete-time adaptive control withmodern techniques basedon FL and NNs which are also conventionally termed asimportant part of ldquosoft-computingrdquo approaches or ldquointelli-gent controlrdquo in the control communities In particular themarriage of ldquoadaptive controlrdquo and ldquointelligent controlrdquo hasbeen reviewed for the purpose of clarifying main contribu-tions and outlining some possible trends for the developmentof this promising area

Generally speaking adaptive control for discrete-timelinear systems was extensively explored in the past decadesdespite the fact that the complete closed-loop stability analy-sis for the so-called self-tuning regulator was ever regarded asa long-term challenging problem Later it became clear thatdiscrete-time adaptive control can be extended to nonlinearsystems with linear growth rate However general nonlinearsystems with parametric andor nonparametric uncertaintiesare usually very difficult to cope with which motivatedvarious solutions for different certain classes of nonlinearsystems

Among the various solutions intelligent control emergedas one important way to resolve the challenges caused by thenonlinearityThe key for usability of intelligent control basedon FL or NNs lies in their universal approximator property atthe cost of tuning rules of FL or updating parameters of NNsFL and NNs are motivated by different background hencetheir design philosophies look different however essentiallyspeaking both of them can be expressed as weighted sumof some kernel functions where the weights can be tunedto approximate arbitrary smooth or continuous nonlinearfunction The ideas borrowed from adaptive estimation andadaptive control provide such a way to adaptively tune theweight parameters in FL orNNs thus the new area of adaptiveFLC or NN control emerged with extensive studies

In a summary a brief review on FLC for discrete-timesystems is provided by highlighting the adaptive FLC robust-ness issue and stability issue while NN control for discrete-time systems is also reviewed with focuses on adaptive NNcontrol and NN-based dynamic programming We believe

this topic would promote increasing investigations in boththeories and applications And some emerging techniquessuch as deep learning and big data could also bemerged withthe intelligent control for discrete-time systems and give birthto brand new design approaches of intelligent control in thefuture

Conflict of Interests

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

Acknowledgment

This work is partially supported by the National Natu-ral Science Foundation of China (NSFC) under Grants61473120 and 61473038 Guangdong Provincial Natural Sci-ence Foundation 2014A030313266 and International Scienceand Technology Collaboration Grant 2015A050502017 andFundamental Research Funds for the Central Universitiesunder Grant 2015ZM065 The authors would like to thankDr Bin Xu of Northwestern Polytechnical University for hisconstructive comments during the preparation of this paper

References

[1] G Goodwin and K Sin Adaptive Filtering Prediction andControl Prentice-Hall Englewood Cliffs NJ USA 1984

[2] K Astrom and B Wittenmark Adaptive Control Addison-Wesley 1989

[3] M Krstic I Kanellakopoulos and P V Kokotovic NonlinearandAdaptive Control Design JohnWileyamp Sons NewYork NYUSA 1995

[4] G Tao and P Kokotovic Aaptive Control of Systems with Actu-ator and Sensor Non-Linearities John Wiley amp Sons HobokenNJ USA 1996

[5] S S Ge C C Hang T H Lee and T Zhang Stable AdaptiveNeural Network Control Kluwer Academic Publishers NorwellMass USA 2001

[6] C Yang Adaptive control and neural network control of nonlin-ear discrete-time systems [PhD thesis] National University ofSingapore 2009

[7] C Yang H Ma and M Fu ldquoAdaptive predictive control ofperiodic non-linear auto-regressive moving average systemsusing nearest-neighbour compensationrdquo IET Control Theory ampApplications vol 7 no 7 pp 936ndash951 2013

[8] C Yang L Zhai S S Ge T Chai andTH Lee ldquoAdaptivemodelreference control of a class ofMIMOdiscrete-time systems withcompensation of nonparametric uncertaintyrdquo in Proceedings ofthe American Control Conference pp 4111ndash4116 IEEE SeattleWash USA June 2008

[9] S-L Dai C Yang S S Ge and T H Lee ldquoRobust adaptiveoutput feedback control of a class of discrete-time nonlinearsystems with nonlinear uncertainties and unknown controldirectionsrdquo International Journal of Robust and Nonlinear Con-trol vol 23 no 13 pp 1472ndash1495 2013

[10] K J Astrom and B Wittenmark ldquoOn self tuning regulatorsrdquoAutomatica vol 9 no 2 pp 185ndash199 1973

Discrete Dynamics in Nature and Society 9

[11] L Ljung ldquoAnalysis of recursive stochastic algorithmsrdquo IEEETransactions on Automatic Control vol 22 no 4 pp 551ndash5751977

[12] G C Goodwin P J Ramadge and P E Caines ldquoDiscretetime multivariable adaptive controlrdquo IEEE Transactions onAutomatic Control vol 25 no 3 pp 449ndash456 1980

[13] L Guo and H F Chen ldquoThe Astrom-Wittenmark self-tuningregulator revisited and ELS-based adaptive trackersrdquo IEEETransactions on Automatic Control vol 36 no 7 pp 802ndash8121991

[14] L Guo Time-Varing Stochastic Systems Jilin Science and Tech-nology Press Changchun China 1993 (Chinese)

[15] H F Chen and L Guo Identification and Stochastic AdaptiveControl Birkhauser Boston Mass USA 1991

[16] F P Skantze A Kojic A-P Loh and A M AnnaswamyldquoAdaptive estimation of discrete-time systems with nonlinearparameterizationrdquo Automatica vol 36 no 12 pp 1879ndash18872000

[17] L Chen and K S Narendra ldquoNonlinear adaptive control usingneural networks and multiple modelsrdquo Automatica vol 37 no8 pp 1245ndash1255 2001

[18] L Guo and C Wei ldquoLS-based discrete-time adaptive nonlinearcontrol feasibility and limitationsrdquo Science in China Series ETechnological Sciences vol 39 no 3 pp 255ndash269 1996

[19] L L Xie and L Guo ldquoAdaptive control of discrete-timenonlinear systems with structural uncertaintiesrdquo in Lectures onSystems Control and Information vol 17 of AMSIP Studiesin Advanced Mathematics American Mathematical SocietyInternational Press Providence RI USA 2000

[20] J D Boskovic ldquoStable adaptive control of a class of first-order nonlinearly parameterized plantsrdquo IEEE Transactions onAutomatic Control vol 40 no 2 pp 347ndash350 1995

[21] A L Fradkov I V Miroshnik and V O Nikiforov Nonlinearand Adaptive Control of Complex Systems Mathematics andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 2004

[22] D Angeli and E Mosca ldquoAdaptive switching supervisorycontrol of nonlinear systems with no prior knowledge of noiseboundsrdquo Automatica vol 40 no 3 pp 449ndash457 2004

[23] H B Ma ldquoFinite-model adaptive control using an LS-likealgorithmrdquo International Journal of Adaptive Control and SignalProcessing vol 21 no 5 pp 391ndash414 2007

[24] H B Ma ldquoFinite-model adaptive control using WLS-likealgorithmrdquo Automatica vol 43 no 4 pp 677ndash684 2007

[25] H B Ma ldquoSeveral algorithms for finite-model adaptive controlpartial answers to finite-model adaptive control problemrdquoMathematics of Control Signals and Systems vol 20 no 3 pp271ndash303 2008

[26] S S Ge C C Hang and T Zhang ldquoA direct adaptive controllerfor dynamic systems with a class of nonlinear parameteriza-tionsrdquo Automatica vol 35 no 4 pp 741ndash747 1999

[27] C Y Li and L Guo ldquoOn feedback capability in a class of nonlin-early parameterized uncertain systemsrdquo IEEE Transactions onAutomatic Control vol 56 no 12 pp 2946ndash2951 2011

[28] H Ma K-Y Lum and S S Ge ldquoAdaptive control for a discrete-time first-order nonlinear system with both parametric andnon-parametric uncertaintiesrdquo in Proceedings of the 46th IEEEConference on Decision and Control (CDC rsquo07) pp 4839ndash4844IEEE New Orleans La USA December 2007

[29] L Guo ldquoExploring the capability and limits of the feedbackmechanismrdquo in Proceedings of the International Congress ofMathematicians (ICM rsquo02) Beijing China August 2002

[30] H-B Ma ldquoAn lsquoimpossibilityrsquo theorem on a class of high-orderdiscrete-time nonlinear control systemsrdquo Systems and ControlLetters vol 57 no 6 pp 497ndash504 2008

[31] I Kanellakopoulos P V Kokotovic and A S Morse ldquoSys-tematic design of adaptive controllers for feedback linearizablesystemsrdquo IEEE Transactions on Automatic Control vol 36 no11 pp 1241ndash1253 1991

[32] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[33] L-X Wang and J M Mendel ldquoFuzzy basis functions universalapproximation and orthogonal least-squares learningrdquo IEEETransactions on Neural Networks vol 3 no 5 pp 807ndash814 1992

[34] L-X Wang ldquoStable adaptive fuzzy control of nonlinear sys-temsrdquo IEEE Transactions on Fuzzy Systems vol 1 no 2 pp 146ndash155 1993

[35] W S McCulloch and W Pitts ldquoA logical calculus of the ideasimmanent in nervous activityrdquo The Bulletin of MathematicalBiophysics vol 5 pp 115ndash133 1943

[36] K Hornik M Stinchcombe and HWhite ldquoMultilayer feedfor-ward networks are universal approximatorsrdquo Neural Networksvol 2 no 5 pp 359ndash366 1989

[37] T Khanna Foundations of Neural Networks Addison-WesleyReading Mass USA 1990

[38] R M Sanner and J-J E Slotine ldquoGaussian networks for directadaptive controlrdquo IEEE Transactions on Neural Networks vol 3no 6 pp 837ndash863 1992

[39] Y J Liu Y J Fang andM A Bao-Ping ldquoSliding-data-window-driven Bayesian-Gaussian neural network and its application tomodeling of nonlinear systemrdquo Control Theory amp Applicationsvol 26 no 12 pp 1435ndash1438 2009

[40] D Wang and J Huang ldquoAdaptive neural network control fora class of uncertain nonlinear systems in pure-feedback formrdquoAutomatica vol 38 no 8 pp 1365ndash1372 2002

[41] Y Song and J W Grizzle ldquoAdaptive output-feedback control ofa class of discrete-time nonlinear systemsrdquo in Proceedings of theAmerican Control Conference pp 1359ndash1363 June 1993

[42] B-S Chen C-S Tseng and H-J Uang ldquoRobustness designof nonlinear dynamic systems via fuzzy linear controlrdquo IEEETransactions on Fuzzy Systems vol 7 no 5 pp 571ndash585 1999

[43] T Chai and S Tong ldquoFuzzy direct adaptive control for a class ofnonlinear systemsrdquo Fuzzy Sets and Systems vol 103 no 3 pp379ndash387 1999

[44] K Tanaka T Ikeda and H O Wang ldquoRobust stabilizationof a class of uncertain nonlinear systems via fuzzy controlquadratic stabilizability 119867

infin control theory and linear matrixinequalitiesrdquo IEEE Transactions on Fuzzy Systems vol 4 no 1pp 1ndash13 1996

[45] S Jagannathan M W Vandegrift and F L Lewis ldquoAdaptivefuzzy logic control of discrete-time dynamical systemsrdquo Auto-matica vol 36 no 2 pp 229ndash241 2000

[46] Y Jiang Z Liu C Chen and Y Zhang ldquoAdaptive robust fuzzycontrol for dual arm robot with unknown input deadzonenonlinearityrdquo Nonlinear Dynamics vol 81 no 3 pp 1301ndash13142015

[47] MWVandegrift F L Lewis S Jagannathan andK Liu ldquoAdap-tive fuzzy logic control of discrete-time dynamical systemsrdquo inProceedings of the IEEE International Symposium on IntelligentControl pp 395ndash401 IEEE Monterey Calif USA August 1995

[48] S Jagannathan ldquoAdaptive fuzzy logic control of feedbacklinearizable discrete-time dynamical systems under persistenceof excitationrdquo Automatica vol 34 no 11 pp 1295ndash1310 1998

10 Discrete Dynamics in Nature and Society

[49] R Qi and M A Brdys ldquoStable indirect adaptive control basedon discrete-time T-S fuzzy modelrdquo Fuzzy Sets and Systems vol159 no 8 pp 900ndash925 2008

[50] T-C Lin S-W Chang and C-H Hsu ldquoRobust adaptivefuzzy sliding mode control for a class of uncertain discrete-time nonlinear systemsrdquo International Journal of InnovativeComputing Information and Control vol 8 no 1 pp 347ndash3592012

[51] G Feng andG Chen ldquoAdaptive control of discrete-time chaoticsystems a fuzzy control approachrdquoChaos Solitons and Fractalsvol 23 no 2 pp 459ndash467 2005

[52] H J Lee J B Park and G Chen ldquoRobust fuzzy controlof nonlinear systems with parametric uncertaintiesrdquo IEEETransactions on Fuzzy Systems vol 9 no 2 pp 369ndash379 2001

[53] Y-Y Cao and PM Frank ldquoRobust119867infindisturbance attenuation

for a class of uncertain discrete-time fuzzy systemsrdquo IEEETransactions on Fuzzy Systems vol 8 no 4 pp 406ndash415 2000

[54] S Zhou G Feng J Lam and S Xu ldquoRobust 119867infin

controlfor discrete-time fuzzy systems via basis-dependent Lyapunovfunctionsrdquo Information Sciences vol 174 no 3-4 pp 197ndash2172005

[55] S Xu and J Lam ldquoRobust 119867infin

control for uncertain discrete-time-delay fuzzy systems via output feedback controllersrdquo IEEETransactions on Fuzzy Systems vol 13 no 1 pp 82ndash93 2005

[56] C-S Tseng and B-S Chen ldquoRobust fuzzy observer-basedfuzzy control design for nonlinear discrete-time systems withpersistent bounded disturbancesrdquo IEEE Transactions on FuzzySystems vol 17 no 3 pp 711ndash723 2009

[57] S Xu B Song J Lu and J Lam ldquoRobust stability of uncertaindiscrete-time singular fuzzy systemsrdquo Fuzzy Sets and Systemsvol 158 no 20 pp 2306ndash2316 2007

[58] Z-G Wu P Shi H Su and J Chu ldquoReliable 119867infin

controlfor discrete-time fuzzy systems with infinite-distributed delayrdquoIEEE Transactions on Fuzzy Systems vol 20 no 1 pp 22ndash312012

[59] G Feng and J Ma ldquoQuadratic stabilization of uncertaindiscrete-time fuzzy dynamic systemsrdquo IEEE Transactions onCircuits and Systems I Fundamental Theory and Applicationsvol 48 no 11 pp 1337ndash1344 2001

[60] A Kruszewski R Wang and T M Guerra ldquoNonquadratic sta-bilization conditions for a class of uncertain nonlinear discretetime TS fuzzy models a new approachrdquo IEEE Transactions onAutomatic Control vol 53 no 2 pp 606ndash611 2008

[61] W-J Wang Y-J Chen and C-H Sun ldquoRelaxed stabilizationcriteria for discrete-time T-S fuzzy control systems based ona switching fuzzy model and piecewise Lyapunov functionrdquoIEEE Transactions on Systems Man amp Cybernetics Part BCybernetics vol 37 no 3 pp 551ndash559 2007

[62] G Feng ldquoStability analysis of discrete-time fuzzy dynamicsystems based on piecewise Lyapunov functionsrdquo IEEE Trans-actions on Fuzzy Systems vol 12 no 1 pp 22ndash28 2004

[63] H Gao X Liu and J Lam ldquoStability analysis and stabilizationfor discrete-time fuzzy systems with time-varying delayrdquo IEEETransactions on Systems Man and Cybernetics Part B Cyber-netics vol 39 no 2 pp 306ndash317 2009

[64] L Wu X Su P Shi and J Qiu ldquoA new approach to stabilityanalysis and stabilization of discrete-time T-S fuzzy time-varying delay systemsrdquo IEEE Transactions on SystemsMan andCybernetics Part B Cybernetics vol 41 no 1 pp 273ndash286 2011

[65] X Su P Shi L Wu and Y-D Song ldquoA novel control design ondiscrete-time takagi-sugeno fuzzy systems with time-varying

delaysrdquo IEEE Transactions on Fuzzy Systems vol 21 no 4 pp655ndash671 2013

[66] X Su P Shi L Wu and Y-D Song ldquoA novel approach to filterdesign for T-S fuzzy discrete-time systems with time-varyingdelayrdquo IEEE Transactions on Fuzzy Systems vol 20 no 6 pp1114ndash1129 2012

[67] C-S Tseng ldquoModel reference output feedback fuzzy trackingcontrol design for nonlinear discrete-time systems with time-delayrdquo IEEE Transactions on Fuzzy Systems vol 14 no 1 pp58ndash70 2006

[68] X Su P Shi L Wu and S K Nguang ldquoInduced l2 filteringof fuzzy stochastic systems with time-varying delaysrdquo IEEETransactions on Cybernetics vol 43 no 4 pp 1257ndash1264 2013

[69] L Wu X Su P Shi and J Qiu ldquoModel approximation fordiscrete-time state-delay systems in the TS fuzzy frameworkrdquoIEEE Transactions on Fuzzy Systems vol 19 no 2 pp 366ndash3782011

[70] F L Lewis S Jagannathan and A Yesildirek Neural NetworkControl of Robot Manipulators and Nonlinear Systems Taylor ampFrancis London UK 1999

[71] A M Shaw and F J Doyle III ldquoMultivariable nonlinear controlapplications for a high purity distillation column using arecurrent dynamic neuron modelrdquo Journal of Process Controlvol 7 no 4 pp 255ndash268 1997

[72] K Najim Process Modeling and Control in Chemical Engineer-ing Marcel Dekker New York NY USA 1989

[73] B Xu D Wang F Sun and Z Shi ldquoDirect neural discretecontrol of hypersonic flight vehiclerdquo Nonlinear Dynamics vol70 no 1 pp 269ndash278 2012

[74] B Xu and Y Zhang ldquoNeural discrete back-stepping controlof hypersonic flight vehicle with equivalent prediction modelrdquoNeurocomputing vol 154 pp 337ndash346 2015

[75] B Xu F Sun H Liu and J Ren ldquoAdaptive Kriging controllerdesign for hypersonic flight vehicle via back-steppingrdquo IETControl Theory amp Applications vol 6 no 4 pp 487ndash497 2012

[76] B Xu ldquoRobust adaptive neural control of flexible hypersonicflight vehicle with dead-zone input nonlinearityrdquo NonlinearDynamics vol 80 no 3 pp 1509ndash1520 2015

[77] B Xu XHuangDWang and F Sun ldquoDynamic surface controlof constrained hypersonic flightmodels with parameter estima-tion and actuator compensationrdquo Asian Journal of Control vol16 no 1 pp 162ndash174 2014

[78] B Xu and Z Shi ldquoAn overview on flight dynamics and controlapproaches for hypersonic vehiclesrdquo Science China InformationSciences vol 58 no 7 pp 1ndash19 2015

[79] D E Rumelhart G E Hinton and R J Williams ldquoLearninginternal representations by error propagationrdquo in Parallel Dis-tributed Processing vol 1 pp 318ndash362 MIT Press 1986

[80] S Jagannathan and F L Lewis ldquoDiscrete-time neural netcontroller for a class of nonlinear dynamical systemsrdquo IEEETransactions on Automatic Control vol 41 no 11 pp 1693ndash16991996

[81] S Jagannathan and F L Lewis ldquoMultilayer discrete-timeneural-net controller with guaranteed performancerdquo IEEETransactions on Neural Network vol 7 no 1 pp 107ndash130 1996

[82] P He and S Jagannathan ldquoNeuro-controller for reducingcyclic variation in lean combustion spark ignition enginesrdquoAutomatica vol 41 no 7 pp 1133ndash1142 2005

[83] S S Ge G Y Li and T H Lee ldquoAdaptive NN controlfor a class of strict-feedback discrete-time nonlinear systemsrdquoAutomatica vol 39 no 5 pp 807ndash819 2003

Discrete Dynamics in Nature and Society 11

[84] S S Ge T H Lee G Y Li and J Zhang ldquoAdaptive NN controlfor a class of discrete-time non-linear systemsrdquo InternationalJournal of Control vol 76 no 4 pp 334ndash354 2003

[85] C J Goh ldquoModel reference control of non-linear systems viaimplicit function emulationrdquo International Journal of Controlvol 60 no 1 pp 91ndash115 1994

[86] C J Goh and T H Lee ldquoDirect adaptive control of nonlinearsystems via implicit function emulationrdquo Control Theory andAdvanced Technology vol 10 no 3 pp 539ndash552 1994

[87] A U Levin and K S Narendra ldquoControl of nonlinear dynami-cal systems using neural networksmdashpart II observability iden-tification and controlrdquo IEEE Transactions on Neural Networksvol 7 no 1 pp 30ndash42 1996

[88] S S Ge J Zhang and T H Lee ldquoAdaptive MNN control fora class of non-affine NARMAX systems with disturbancesrdquoSystems amp Control Letters vol 53 no 1 pp 1ndash12 2004

[89] S S Ge J Zhang and T H Lee ldquoAdaptive neural networkcontrol for a class of MIMO nonlinear systems with distur-bances in discrete-timerdquo IEEE Transactions on Systems Manand Cybernetics Part B Cybernetics vol 34 no 4 pp 1630ndash1645 2004

[90] J Zhang S S Ge and T H Lee ldquoOutput feedback control of aclass of discrete MIMO nonlinear systems with triangular forminputsrdquo IEEE Transactions onNeural Networks vol 16 no 6 pp1491ndash1503 2005

[91] F C Sun Z Sun and P-Y Woo ldquoStable neural-network-basedadaptive control for sampled-data nonlinear systemsrdquo IEEETransactions on Neural Networks vol 9 no 5 pp 956ndash9681998

[92] C Yang S S Ge C Xiang T Chai and T H Lee ldquoOutputfeedback NN control for two classes of discrete-time systemswith unknown control directions in a unified approachrdquo IEEETransactions on Neural Networks vol 19 no 11 pp 1873ndash18862008

[93] S S Ge C Yang and T H Lee ldquoAdaptive predictive controlusing neural network for a class of pure-feedback systems indiscrete timerdquo IEEE Transactions on Neural Networks vol 19no 9 pp 1599ndash1614 2008

[94] Y Li C Yang S S Ge and T H Lee ldquoAdaptive output feed-back NN control of a class of discrete-time MIMO nonlinearsystems with unknown control directionsrdquo IEEE Transactionson Systems Man and Cybernetics Part B Cybernetics vol 41no 2 pp 507ndash517 2011

[95] Y-J Liu C L P Chen G-X Wen and S Tong ldquoAdaptiveneural output feedback tracking control for a class of uncertaindiscrete-time nonlinear systemsrdquo IEEE Transactions on NeuralNetworks vol 22 no 7 pp 1162ndash1167 2011

[96] C Yang S S Ge and T H Lee ldquoOutput feedback adaptivecontrol of a class of nonlinear discrete-time systems withunknown control directionsrdquoAutomatica vol 45 no 1 pp 270ndash276 2009

[97] A Al-Tamimi F L Lewis and M Abu-Khalaf ldquoDiscrete-timenonlinear HJB solution using approximate dynamic program-ming convergence proofrdquo IEEE Transactions on Systems Manand Cybernetics Part B Cybernetics vol 38 no 4 pp 943ndash9492008

[98] P Werbos ldquoApproximate dynamic programming for real-timecontrol and neuralmodelingrdquo inHandbook of Intelligent ControlNeural Fuzzy amp Adaptive Approaches Van Nostrand Reinhold1992

[99] P He and S Jagannathan ldquoReinforcement learning neural-network-based controller for nonlinear discrete-time systems

with input constraintsrdquo IEEETransactions on SystemsMan andCybernetics Part B Cybernetics vol 37 no 2 pp 425ndash436 2007

[100] B Xu C Yang and Z Shi ldquoReinforcement learning outputfeedback NN control using deterministic learning techniquerdquoIEEE Transactions on Neural Networks and Learning Systemsvol 25 no 3 pp 635ndash641 2014

[101] D Liu DWang D Zhao QWei and N Jin ldquoNeural-network-based optimal control for a class of unknowndiscrete-time non-linear systems using globalized dual heuristic programmingrdquoIEEE Transactions on Automation Science and Engineering vol9 no 3 pp 628ndash634 2012

[102] D Liu and Q Wei ldquoFinite-approximation-error-based optimalcontrol approach for discrete-time nonlinear systemsrdquo IEEETransactions on Cybernetics vol 43 no 2 pp 779ndash789 2013

[103] X Zhong H He H Zhang and Z Wang ldquoOptimal control forunknown discrete-time nonlinear markov jump systems usingadaptive dynamic programmingrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 25 no 12 pp 2141ndash21552014

[104] D Liu D Wang and X Yang ldquoAn iterative adaptive dynamicprogramming algorithm for optimal control of unknowndiscrete-time nonlinear systemswith constrained inputsrdquo Infor-mation Sciences vol 220 pp 331ndash342 2013

[105] F-Y Wang N Jin D Liu and Q Wei ldquoAdaptive dynamicprogramming for finite-horizon optimal control of discrete-time nonlinear systems with 120576-error boundrdquo IEEE Transactionson Neural Networks vol 22 no 1 pp 24ndash36 2011

[106] H Zhang Y Luo and D Liu ldquoNeural-network-based near-optimal control for a class of discrete-time affine nonlinearsystems with control constraintsrdquo IEEE Transactions on NeuralNetworks vol 20 no 9 pp 1490ndash1503 2009

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2 Discrete Dynamics in Nature and Society

systems with semiparametric uncertainties [28] and so onMany challenges have been seen in these existing studiesto cope with the interaction of nonlinearity and adaptationNow it is clear that discrete-time adaptive control can beextended to nonlinear systems with linear growth rate byusing the modification technique such as projection [15]however we also know that there exist intrinsic difficultiesor even impossibility for discrete-time adaptive control whenthe nonlinear growth rate is high enough [29 30] whichis quantitatively characterized by fundamental limitations offeedback mechanism in terms of certain introduced uncer-tainty measure

In addition to the progress from linear systems to thenonlinear systems rigorous stability analysis of the closed-loop adaptive system has also been well established [31]Compared with robust control the advantage of adaptivecontrol lies in its ability in estimating and compensating forparametric uncertainties in a large range However towardsthe increasingly complexity of systems with complicatednonlinear functional uncertainties it is necessary to developmore powerful control design tools To this end researchersnoticed some limitations of traditional adaptive control inmainly copingwith parametric uncertainties And fuzzy logic(FL) and NN-based intelligent control have been introducedin the early 1990s [6]

The concept of fuzzy set was initially proposed byZadeh [32] and since then fuzzy technique gained a rapiddevelopment FL offers human reasoning capabilities tocapture nonlinearities and uncertainties which cannot bedescribed by precise mathematical model In [33] Wang andMendel found that linear combinations of a series of fuzzybasis functions are able to approximate smooth nonlinearfunctions and proved such feature of universal approximatorby using the Stone-Weierstrass theorem Based on this workan adaptive fuzzy control method was further proposed bycombining the universal approximation of the FL controllerand learning ability of the adaptive method [34] And later astable adaptive fuzzy controller was successfully designed fora class of nonlinear systems In this approach the adaptive lawwas constructed to update the parameters of the FL controllerduring the adaptation procedure while it was not necessary tohave the accurate mathematical model of the system Fromthen on fuzzy control has attracted ever increasing researchinterest since it is able to provide an effective solution tocontrol complex and uncertain plants by employing theknowledge of specific experts for the controller design

NN is inspired by biological neuronal systems witch con-sist of a number of simple processing neurons interconnectedto each other McCulloch and Pitts introduced an idea tostudy the computational abilities of networks composed ofsimple models of neurons in the 1940s [35] With highlyparallel structure NN is of great computational power andlearning ability to emulate various systems dynamics It is wellestablished that NN is capable of universally approximatingunknown smooth functions with errors that can be madearbitrarily small [36ndash38] In addition to system modelingand control NN has also been successfully applied in manyother fields such as learning pattern recognition and sig-nal processing In NN control methodology NN has been

extensively studied as function approximators to compensatefor the system uncertain nonlinearities in control design Inthe last two decades it has been verified that NN controlis very useful for controlling highly uncertain nonlinearsystems and NN control has demonstrated superiority overtraditional control methods Particularly the marriage ofadaptive control theories and NN techniques gives birth toadaptive NN control which guarantees stability robustnessand convergence of the closed-loop NN control systemswithout beforehand offline NN training [6]

Nowadays most of the control algorithms are realized bydigital computers and thus the desired control signals arecalculated in a digital manner Digital control systems haveadvantages such as easy to build less sensitive to environmentvariation flexible to change and less expensive A modelbased digital controller is designed in discrete-time form andoperated in continuous-time with analogy signals Since thedata are generally processed at discrete-time instants it isnecessary for us to build the system model in discrete-timefor ease of control design [6]

Yet in the past decades many significant progressesof FLC and NN control for nonlinear systems were incontinuous-time form and there is considerable lag in thedevelopment for nonlinear discrete-time systems [39 40]Although for linear systems there are numerous results ofNN and fuzzy controls in continuous-time systems theNN and fuzzy control of nonlinear discrete-time systemshave been considerately less studied than their counterpartsin continuous-time As a matter of fact many techniquesdeveloped for continuous-time systems cannot be directlyapplied in discrete-time especially when the systems arenonlinear Discrete-time systems can be expressed by variousequations which in great contrast to the differential equationsof continuous-time systems involve states at different timesteps Due to the different nature of difference equationand differential equation some concepts in discrete-timehave very different meanings from those in continuous-time and the stability analysis techniques become muchmore intractable For example the linearity property of thederivative of a continuous-time Lyapunov function is nolonger existed in a discrete-time Lyapunov function [41]Therefore in order to design discrete-time fuzzy or NNcontrollers and implement them into digital control systemsthey deserve in-depth investigation for fuzzy control and NNcontrol of nonlinear discrete-time systems

The remainder of this paper is organized as followsIn Section 2 a brief review of fuzzy control for discrete-time systems is presented Some research problems of fuzzycontrol and discrete-time systems such as adaptive controlrobustness issue stability analysis and time-delay systemsare concerned and introduced for they are both theoreti-cally challenging and practically meaningful In Section 3NN control methods for discrete-time systems are brieflyreviewed Research of adaptive NN control for discrete-time systems is introduced and NN-based adaptive dynamicprogramming methods for discrete-time systems are alsodiscussed Finally in Section 4 a brief conclusion is given

Discrete Dynamics in Nature and Society 3

2 Fuzzy Logic Control forDiscrete-Time Systems

Nowadays techniques for FLC have developed rapidly espe-cially in modeling complex nonlinear systems Since Wangfirst proved that the linear combinations of a series offuzzy basis functions are universal approximators of anynonlinear systems the universal approximation property ofthe fuzzy logic systems has been extensively studied [42ndash45]The combination of adaptive control and fuzzy logic systemallows adaptive laws to update parameters of the FL controllerduring the adaptation procedure while adaptive fuzzy controlprovides an efficient method to model complex nonlinearsystems In practice the difficulty of design a fuzzy controllerlies in the fact that there are usually various requirementimposed on the systems to ensure stability and performancewhile most of complex control systems today are using digitalcomputers to calculate the control signals in a digital formHence the continuous-time methods could not be directlyapplied to most practical systems The modelingcontrol fordiscrete-time systems is crucial to take the controllers intoreal plants Thus it is significant to study the discrete-timefuzzy systems as well as their molding and control To thisend researchers have paid efforts to develop the discrete-time FLCA typical procedure of themultiple-inputmultiple-output (MIMO) fuuzy logic system (FLS) approximatingan unknown function 119865(119909) is comprised of three primarycomponents as follows [46]

(1) Fuzzification Take 119883 as the input of the fuzzy systemand (119883) is the estimation of the output Both the input andthe output are fuzzified into fuzzy linguistic terms with fuzzymembership function

(2) Fuzzy Rules The collection of the fuzzy MIMO IF-THEN rules are designed to comprise the knowledge basefor constructing the FLS Using prior expert knowledge thefuzzy rules can be obtained as

Rule 119897 If 1199091is 119860119897

1 1199092is 119860119897

2 119909

119899is 119860119897

119899

then 1199101is 119861119897

1 1199102is 119861119897

2 119910

119898is 119861119897

119898

(1)

where 1199091

sim 119909119899and 119910

1sim 119910119898

are the premise variablesconsisting of 119883 and (119883) respectively 119860

119897 and 119861119897 are the

linguistic variables of the fuzzy sets 119897 = 1 2 119871

(3) Fuzzy Inference Engine and Defuzzification Using thestrategy of the sum-product inference engine and the center-average defuzzifier the output of the FLS can be described as

119910119895

=

sum119871

119897=1120593(119897)

119895(prod119899

119894=1120583(119897)

119860119894(119909119894))

sum119871

119897=1(prod119899

119894=1120583(119897)

119860119894(119909119894))

(2)

where 120593(119897)

119895is the point where fuzzy membership function

120583(119897)

119861119895(120593119895) achieves its maximum value Define

119908119895(119909) =

prod119899

119894=1120583(119897)

119860119894(119909119894)

sum119871

119897=1(prod119899

119894=1120583(119897)

119860119894(119909119894))

119882 (119909) = [1199081

(119909) 1199082

(119909) 119908119871

(119909)]119879

(3)

and (Φ)119895

= [120593(1)

119895 120593(2)

119895 120593

(119871)

119895]119879 Then the output of MIMO

FLS can be rewritten as

119910119895

= (Φ)119879

119895119882 (119883) (4)

where the subscript 119895 represents the 119895th column vector of thematrix Therefore the estimation of (119883) is

(119883)

= [(Φ)119879

1119882 (119883) (Φ)

119879

2119882 (119883) (Φ)

119879

119898119882 (119883)]

119879

(5)

Then 119865(119883) can be written as

(119883) = Φ119879

119882 (119883) (6)

21 Adaptive FLC of Discrete-Time Systems FLC has beenextensively investigated in both academic and industrialcommunities However problems with high dimensionalityalso existed in higher-dimensional fuzzy systems due to themathematical complications and intuitive limitations There-fore it required more systematic methods for advanced FLCdesign and synthesis such as the adaptive FLC Originallyadaptive control was proposed for aircraft autopilots to dealwith parameter variations during changing flight conditionsIn the 1960s the advances in stability theory and the progressof control theory improved the understanding of adaptivecontrol By the early 1980s several adaptive approaches havebeen proven to provide stable operation and asymptotictracking Since then the adaptive control problems have beenrigorously formulated

Generally adaptive FLC design for nonlinear discrete-time systems ismuchmore difficult than those in continuous-time The stability analysis techniques become much moreintractable for difference equations than those for differentialequations Many nice Lyapunov adaptive control designmethodologies developed in continuous-time are not appli-cable to discrete-time systems Thus it is significant toconsider the adaptive FLC in discrete-time form

In recent years the adaptive FLC for discrete-time non-linear systems has been extensively studied Jagannathanet al [45 47] developed an adaptive fuzzy logic controllerfor a general class of discrete-time nonlinear systems usingbasis vectors and provided rigorous proofs to stability of thediscrete-time adaptive FLC In this work a multiloop struc-ture controller was constructed with a fuzzy approximatelinearization loop an outer tracking loop and a robustifyingloop By selecting FLCrsquos structure and the adaptation laws andgiving a number of assumptions on a class of discrete-timenonlinear systems closed-loop signals can achieve uniform

4 Discrete Dynamics in Nature and Society

ultimate boundedness This method is significant in sensethat no persistence of excitation is used and no certaintyequivalence assumption is required The controller has alsobeen extended to a class of unknown feedback linearizablenonlinear dynamical systems under persistence of excitationin [48] in which rigorous stability proofs of discrete-timeadaptive FLC for feedback linearizable unknown nonlinearsystems were presented Indirect fuzzy control of uncertaindiscrete-time nonlinear system was shown by Qi and Brdys[49] This work presented an indirect adaptive FLC for theuncertainties in nonlinear plants and employed a Takagi-Sugeno (T-S) model to deal with the unknown dynamicsin input-output form A feedback linearization control lawwas designed by using structure states and parametersof the model Gradient descent algorithm and recursiveleast square estimation method were used to online updatethe modeling parameters The extension of sliding modecontrol (SMC) adaptive fuzzy SMC for a class of uncertaindiscrete-time nonlinear systems was investigated by Linet al [50] This work presented an adaptive interval type-2 fuzzy sliding mode controller for a class of unknownnonlinear discrete-time systems with training data disturbedby external disturbances It employed an adaptive intervaltype-2 fuzzy control scheme and SMC approach to controlthe plant tracking a reference trajectory and prevented bigchattering of the control effort While adaptive fuzzy controlhas also been applied in discrete-time chaotic systems In[51] Feng and Chen presented a novel adaptive controlalgorithm for discrete-time chaotic systems The basic ideais representing the chaotic system as a T-S fuzzy model anddesigning a local linear adaptive controller in each localregion Meanwhile a global adaptive controller on the entiredomainwas reconstructed and the stability of the closed-loopadaptive control system was proved

22 Robustness Issue in Discrete-Time Fuzzy Control Inpractice nonlinearities and uncertainties exist in almost allthe industrial plants while uncertainties in the modeling andcontrol of nonlinear systems are still one of themost challeng-ing problems in the control field The presence of nonlinear-ities and uncertainties brings difficulties to controller designIn order to conquer these problems a number of schemeshave been developed and among these robust control is oneof the most effective methods Robustness is regarded as oneof the most important requirements for a control system Inthe past two decades considerable attention has been paid tothe robustness in fuzzy model-based control of discrete-timesystems andmanymodification techniques were proposed toenhance the robustness of FLC for discrete-time systems [52ndash57]

The work of Lee et al [52] illustrated an approach ofrobust fuzzy control for nonlinear discrete-time systems Inthis work a systematic control structure was presented fornonlinear discrete-time T-S fuzzy systems with parametricuncertainties using the 119867

infincontrol design approach This

work also gave a novel solution to the robust stabilizationproblem of nonlinear systems by using basis-dependentLyapunov function

119867infindisturbance attenuation is one of the most important

requirements for a fuzzy control system Cao and Frank [53]have applied the 119867

infincontrol to address the robust stability

for a class of uncertain discrete-time fuzzy systems via linearmatrix inequality approach They studied both the robuststabilization and the 119867

infindisturbance attenuation while

they presented several sufficient conditions to ensure robuststability of the fuzzy models based on coupled linear matrixinequalities They also analyzed the robust 119867

infinperformance

of the fuzzy model-based discrete-time nonlinear systems byaddressing the robust 119867

infindisturbance attenuation problems

The robust stability of the models is achieved independentlyof the uncertainties

How to develop a robust 119867infin

controller for discrete-time systems using the basis-dependent Lyapunov functionwas also shown by Zhou et al [54] In their work anovel linear matrix inequalities characterization with 119867

infin

norm bound was presented for discrete-time fuzzy systemsWhile additional matrix variables were used to decouple theLyapunov function and system matrices as well as facilitatethe design approach of the controller this approach providessome sufficient results in the shape of strict linear matrixinequalities

Studied by Choi and Park the state feedback 119867infin

controlfor discrete-time systems was investigated by constructingLyapunov functions with fuzzy weights The fuzzy weightingLyapunov functions were designed with both current-timeand one-step-past information while designing controllerparameterized with linear matrix inequalities (PLMIs) Byselecting the structures of variables appropriately in thePLMIs a special case LMI formulation was obtained The119867infin

control with output feedback for discrete-time fuzzysystems has also been rigorously studied by Xu and Lam[55] while Wu et al [58] investigated the reliable 119867

infincontrol

for discrete-time systems with delays and stochastic actuatorfaults by representing the stochastic behavior with a discrete-time homogeneous Markov chain

Tseng andChen [56] used a fuzzy observer to estimate thepremise variables which depend on the state variables whilethe fuzzy observer was also used to address the nonlinear119897infin-gain control problem This work extended the 119897

infincontrol

from linear discrete-time systems to nonlinear discrete-time systems By using the T-S fuzzy model to representthe nonlinear discrete-time system an observer-based fuzzycontroller which minimized the upper bound of 119897

infin-gain and

attenuated the peak of perturbation was designed Xu et al[57] gave stability analysis for discrete-time singular fuzzysystems in the presence of time-varying uncertainties whilethey ensured the system to be regular causal and stable It hasbeen shown that for these systems robust stability conditioncan be obtained by giving a sufficient condition in terms of aset of linear matrix inequalities

23 Stability Issue in Discrete-Time Fuzzy Control Stabilityis one of the most important properties for a system whileFLC are used to address the stability problems for discrete-time systems as well The quadratic stability for uncertaindiscrete-time fuzzy dynamic systems was shown by Fengand Ma [59] This work gave some sufficient conditions of

Discrete Dynamics in Nature and Society 5

the quadratic stabilization for an uncertain fuzzy dynamicsystem Stability was achieved when the suitable Riccatiequation or a set of Riccati equations were solved Stabilizingfeedback control laws were also obtained by the developingalgorithms The nonquadratic stabilization conditions fornonlinear discrete-time fuzzy systems were also analyzedby Kruszewski et al [60] They considered discrete-timeuncertain nonlinear models in a T-S form and studied thestability through a nonquadratic Lyapunov function Thestabilization conditions were developed by considering theLyapunov function with a 119896-sample variation and extendedto uncertain T-S models Based on a switching fuzzy modeland piecewise Lyapunov function Wang et al [61] proposedtwo stabilization criteria for discrete-time T-S fuzzy systems

Zhou et al [54] used a basis-dependent Lyapunov-Krasovskii function to give stabilization analysis for discrete-time fuzzy systems A robust control design approach wasalso developed by using the Lyapunov-Krasovskii functionand facilitated by introducing additional instrumental matrixvariables Robust control problem for systemswith time delaywas concerned as well Feng [62] used a piecewise smoothLyapunov function to analyze the stability of discrete-time T-S fuzzy dynamic systems In this study stability of the systemguaranteed by constructing a piecewise Lyapunov functionMeanwhile the Lyapunov function was obtained using thelinear matrix inequalities methodThis work showed that thepiecewise quadratic Lyapunov functions based stability is lessconservative than the common quadratic Lyapunov functionbased stability

The stabilization for discrete-time T-S fuzzy systems withstate time-varying delay was shown by Gao et al [63] Intheir work a fuzzy Lyapunov function was constructed toimprove the delay-dependent stability condition By avoidingthe utilization of the bounding inequalities for the crossproducts between two vectors reduction of the conservatismof stability condition was achieved A delay-dependent stabi-lization algorithm was also developed for both state feedbackand observer-based output feedback cases using a paralleldistributed compensation scheme

24 Fuzzy Control for Discrete-Time Systemswith TimeDelaysIn practice time delays are the intrinsic nature of variousphysical systems such as communication hydraulic chem-ical processes and electronics The existence of time delayscould generally lower the system performance and even causeinstability and oscillation Hence remarkable attention hasbeen devoted to the analysis and synthesis of time delaynonlinear systems as well as the discrete-time fuzzy systems[55 64ndash69]

The stability and stabilization problems of systems time-varying delay were investigated by Wu et al [64] In theirwork the existing stability analysis approaches were dividedinto two types delay-dependent (including the delay size)and delay-independent (irrelevant to the delay size) Thestability of discrete-time T-S fuzzy systems with time-varyingstate delay was also investigated They developed a delay par-titioningmethod and employed a fuzzy Lyapunov-Krasovskiifunction to analyze the stability of delay-dependent time-varying state delay systems Through delay partitioning the

less conservative stability condition is obtained and solved byLMI optimization techniques

The work of Su et al [65] also gave a solution to thedynamic output feedback control problem for fuzzy sys-tems with time-varying delays They proposed a comparisonmodel by approximating the time-varying delay state whilethey gave a sufficient condition to ensure the asymptoti-cal stability by adopting the scaled small-gain theorem aswell as the delay partitioning method This method allowsdesigning desired dynamic output feedback controller byusing optimization techniques A filter design technique fordiscrete-time systems with time-varying delay has also beenpresented by Su et al [66] Both full-order and reduced-orderfilters were designed while approximators were employed toderive an input-output based sufficient condition Tseng [67]studied the time delay problem for a nonlinear discrete-timesystem based on model reference fuzzy tracking control Inthis work the T-S fuzzy model was applied to approximatea time delay discrete-time system while the fuzzy controllerwas designed to reduce the tracking error based on thisfuzzy model This method has advantage in that no feedbacklinearization or adaptive approaches are used in the controllerdesign

The robust 119867infin

control problem for uncertain discrete-time delay systems has been ivestigated by Xu and Lam [55]They studied the state-space T-S fuzzymodel with time delaysunder the assumption that the parameter uncertainties werenorm-boundedThe output feedback controller was designedwith full-order fuzzy dynamic to guarantee the closed-loopsystem is robust asymptotic stable as well as the admissibleuncertainties are norm bound constraint While solvabilityof this problem was addressed using a sufficient conditiongiven by the linear matrix inequalities The induced 119897

2filter

design of T-S fuzzy discrete-time stochastic systems withtime-varying delays has been investigated by Su et al [68]Wuet al [69] studied the 119867

infinmodel approximation for discrete-

time state delay fuzzy T-S systems

3 NN Control for Discrete-Time Systems

NN is of powerful computing ability and learning abilityto emulate various systems dynamics and is capable ofapproximating an unknown functionwith arbitrary accuracyNN is very successful in system modeling and controlby its capacity of universal approximate highly nonlinearand uncertain nonlinear and complex dynamic of systemsNNrsquos approximation ability has been shown by the Stone-Weierstrass theorem which states that a universal approxi-mator can approximate to an arbitrary degree of accuracyany real continuous function on a compact set Besides theuniversal approximation abilities NN also shows its excel-lence in parallel distributed processing abilities learningadaptation abilities natural fault tolerance and feasibilityfor hardware implementation These advantages make NNparticularly attractive and prospective for nonlinear controland modeling NN has been successfully applied to robotmanipulators control [70] distillation column control [71]chemical processes identification [72] flight control [73ndash75]

6 Discrete Dynamics in Nature and Society

and so forth To extend the NN technique from continuous-time to discrete-time many efforts have been made to studythe NN controlled discrete-time systems [76ndash78]

31 Adaptive NN Control for Discrete-Time Systems In theearly stage backpropagation (BP) algorithm [79] greatlyboosted the development of NN control It is noted that inthe early NN control design the control performances weredemonstrated through simulation or by particular exper-imental examples and consequently there were shortageof analytical analysis In addition an offline identificationprocedure was essential for achieving a stable NN controlsystem Thereafter the emergence of Lyapunov-based NNdesign makes it possible to use the available adaptive controltheories to rigorously guarantee stability robustness andconvergence of the closed-loop NN control systems Wecall the control design combining adaptive control theoriesand NN techniques as adaptive NN control It updates NNweights online and guarantees the stability of the closed-loop system Adaptive NN control design has been elegantlydeveloped for nonlinear systems with parametric uncertain-ties while many efforts have been dedicated in adaptive NNcontrol for nonlinear discrete-time systems [6]

For high-order affine nonlinear system in normal formadaptive NN controls using LPNN and MNN have beendeveloped in [80 81] using a filtered tracking error In [82]the controller is designed incorporating reinforcement learn-ing technique to improve control performance In this worka critic NN has been introduced to approximate the strategicutility function which is considered as the long-term systemperformance measure For discrete-time systems in strict-feedback form adaptive NN control has been developed viabackstepping design after system transformation [83] In [84]adaptive NN control has been investigated for discrete-timesystem in affine NARMAX form

In the above mentioned results the adaptive NN controldesigns were carried out through either feedback lineariza-tion or backstepping But these approaches are not applicableto nonaffine systems especially feedback linearization basedmethods which greatly depend on the affine appearance ofcontrol variables As a matter of fact adaptive NN controlmethods for nonaffine systems have been less studied incomparison with large amount of research work on affinenonlinear systems because of the difficulty of control designcaused by the nonaffine form of control input To overcomethe difficulty linearization based NN controls have beenput forward In [17] the nonaffine discrete-time system hasbeen divided into two parts one is linear and another isnonlinear and consequently a linear adaptive controller anda nonlinear adaptive NN controller have been designedwith a switching rule In [85] it directly utilized NN asemulator of the ldquoinverserdquo of the nonlinear discrete-timesystems Furthermore the study of discrete-time systems foradaptive NN control using implicit function to assert theexistence of an ideal inverse control was investigated in [86]Thereafter the implicit function based adaptive NN controlhas been widely studied in discrete-time form [87 88] Blocktriangular discrete-time systems with normal form subsys-tems have been studied in [80 81] For the block triangular

systems with strict-feedback subsystems state feedback andoutput feedback adaptive NN control have been developedin [89 90] by extending the systems transformation basedbackstepping technique proposed for SISO case in [83] In[91] adaptive NN control has been used for sampled-datanonlinear MIMO systems in general affine form based onlinearization The control scheme is an integration of an NNapproach and a variable structure method

An effort has been made in [92] to explore the adaptiveNN control of a class of nonaffine systems in discrete-timeThis work aimed to solve the nonaffine appearance andnoncausal problems of the following pure-feedback discrete-time system

120585119894(119896 + 1) = 119891

119894(120585119894 120585119894+ 1 (119896))

119894 = 1 2 119899 minus 1 119899 ge 2

120585119899

(119896 + 1) = 119891119899

(120585119899

(119896) 119906 (119896) 119889 (119896))

119910 (119896) = 1205851

(119896)

(7)

where 120585119894(119896) = [120585

1(119896) 1205852(119896) 120585

119899(119896)]119879 are system states

119891119894are unknown nonlinear functions and 119906(119896) and 119910(119896) are

system input and output respectively while 119889(119896) denotes theexternal disturbance

Using a states prediction technique the pure-feedbackdiscrete-time systems in (7) are shown to be transformableto an 119899-step-ahead predictor as below

119910 (119896 + 119899) = 120601 (120585119899

(119896) 119906 (119896) 119889 (119896))

= 120601119904(120585119899

(119896) 119906 (119896)) + 119889119904(119896)

(8)

And by future output predictions the above system (8)can be further transformed into an input-output model forthe output feedback control

119910 (119896 + 119899)

= 1198651198991

119910 (119896 + 1) 119906 (119896) 119889 (119896) 119889 (119896 minus 119899 + 2)

(9)

This shows that the system presentation (9) in NARMAXform is a transformation of the pure-feedback system (8)After transformation both state feedback and output feed-back controls only need to employ a single NN in thecontroller design rather than a number of NNs in previousresearches

Based on the SISO pure-feedback system (8) and itstransformation NARMAX model in (9) the control designhas been further investigated by using high-order neuralnetwork to approximate unknown functions [93] In theabove mentioned control design availability is assumed forknowledge of control directions which are defined as ldquothesigns of control variable gains in affine systems or the signsof partial derivatives over control variables in nonaffinesystemsrdquo [93]

The paper overcame the problems of output-feedbackcontrol when the prior knowledge of the control directions isunknownThemain idea is to introduce a discrete Nussbaum

Discrete Dynamics in Nature and Society 7

gain to counter the lack of knowledge on control gain inadaptive NN control

A rigorous definition of discrete Nussbaum gain in [93] isgiven as follows

Remark 1 Consider a discrete nonlinear function 119873(119909(119896))

defined on a sequence 119909(119896) with 119909119904(119896) = sup(119909

1015840

119896) 119873(119909(119896))

is a discrete Nussbaum gain if and only if it satisfies thefollowing two properties

(i) if 119909119904(119896) increases without bound then

sup 1

119909119904(119896)

119878119873

(119909 (119896)) = +infin

inf 1

119909119904(119896)

119878119873

(119909 (119896)) = minusinfin

(10)

(ii) if 119909119904(119896) ge (119896)120575

1 then |119878

119873(119909(119896))| ge 120575

2with some

positive constants 1205751and 1205752 where 119878

119873(119909(119896)) is defined as

119878119873

(119909 (119896)) =

119896

sum

1198961015840=0

119873 (119909 (1198961015840)) Δ119909 (119896

1015840) (11)

with Δ119909(119896) = 119909(119896 + 1) minus 119909(119896) In addition neither the upperbounds nor the lower bounds of the control gains are requiredto be known

Unknown control directions problem for MIMOdiscrete-time nonlinear systems was solved by using adaptiveoutput feedback NN control in [94] The studies in [92 93]all studied the discrete NN control for SISO systems and[94] extended the study to a class of MIMO discrete-timesystems with each subsystem in the nonaffine pure-feedbackas follows

120585119895119894119895

(119896 + 1) = 119891119895119894119895

(1205851119894119895minus1198981198951

(119896) 1205852119894119895minus1198981198952

(119896)

120585119899119894119895minus119898119895119899

(119896) 120585119895 119894119895+1

(119896)) 119894119895

= 1 2 119899119895

minus 1

120585119895119899119895

(119896 + 1) = 119891119895119899119895

(Ξ (119896) 119906119895(119896) 119889

119895(119896))

119910119895(119896) = 120585

1198951(119896)

(12)

where 120585119895119894119895

(119896) = [1205851198951

(119896) 1205851198952

(119896) 120585119895119894119895

(119896)]119879 are vectors

of states variable of subsystem Ξ(119896) is vector of all statevariables 119891

119895119894119895are unknown nonlinear functions 119906

119895(119896) and

119910119895(119896) are system inputs and outputs respectively while 119889

119895(119896)

denote the external disturbanceFor nonlinearMIMOdiscrete-time higher order systems

a major drawback is that there are too many online-tunedadaptive parameters and large online computation burden In[95] an adaptive neural output feedback adaptive controllerwas designed for MIMO nonlinear discrete-time systemswith fewer adaptive parameters The output feedback adap-tive control for a class of nonlinear discrete-time systemswithunknown control directions was investigated in [96]

32 NN-Based Dynamic Programming Algorithm for Discrete-Time Systems In the past several decades optimal control for

nonlinear systems has been a key focus in the control fieldsWhile adaptive dynamics programming (ADP) is a veryuseful approach to solve the optimal control problems ADPis a reinforcement learning method to give solution to thedynamic programming utilizing function approximating thevalue function based on adaptive approach [97] P Werbos[98] classified the discrete-time ADP into a number ofschemes action-dependentHDP dual HDP (DHP) heuristicdynamic programming (HDP) and action-dependent dualHDP while NN has been widely used to solve the adaptivedynamic programming problems of discrete-time systems[97 99ndash104]

He and Jagannathan [99] showed the reinforcementlearning with NN-based controller for nonlinear discrete-time systems This work developed an adaptive-critic-basediscrete-time NN controller to deliver a desired tracking per-formance for nonlinear systems in the presence of actuatorconstraints The critic NN and action NN were designed toapproximate the strategic utility function while uniformlyultimate boundedness of the closed-loop tracking error wasproved by Lyapunov approach The reinforcement learningNN control for SISO discrete-time pure-feedback systemswas studied in [100] The controller design is based on thetransformed predictor and two NNs of control architectureOne critic NN to approximate the strategic utility function isshown as follows

By introducing a utility function 119901(119896) based on thetracking error 119890(119896) = 119910(119896) minus 119910

119889(119896)

119901 (119896) = 1198860 |119890 (119896)| (13)

where 119901(119896) isin 119877 1198860

isin 119877 is the positive design parameter Andthe strategic utility function 119876 isin 119877 is defined as

119876 (119896) = 119886119873

(119901 + 1) + 119886119899minus1

119901 (119896 + 2) + sdot sdot sdot + 119886119896+1

119901 (119873)

+ sdot sdot sdot

(14)

where 119886 isin 119877 0 lt 119886 lt 1 119873 is the horizon Then the criticNN is used to approximate the strategic utility function119876(119896)

= 119879

119888(119896) 119878119888(119911 (119896)) 119878

119888(119911 (119896)) isin 119877

119897119888 (15)

where 119882119879

119888(119896) isin 119877

119897119888 is the estimation of optimal NN 119882lowast

119888

weights The uniformly ultimate boundedness of closed-loopsignals is established via Lyapunov stability analysis

Al-Tamimi et al [97] proved the convergence of a valuefunction solution for nonlinear dynamical systems usingHDP algorithm and NNs were employed to approximatethe value and the control action at each iteration Liu et al[101] used theHDP algorithm to develop aNN-based optimalcontroller for unknown discrete-time nonlinear systemsThemain idea of this work is to introduce an iterative ADPalgorithm and a globalizedHDP technique into the controllerdesign And by using the cost function and control law theconvergence of the optimal control is guaranteed In [102]an iterative ADP algorithm for discrete-time systems wasdeveloped as well An optimal controller was designed toaddress the infinite-horizon discrete-time nonlinear systems

8 Discrete Dynamics in Nature and Society

with finite approximation errors It was shown thatwhen con-vergence conditions are satisfied iterative performance indexfunction can converge to the performance index functionsrsquolower bound while the performance index functions wereapproximated by NNs to compute optimal control policy

The ADP control in the unknown discrete-time Markovjump systems was investigated by Zhong et al [103] AndWang et al [105] also studied the ADP for discrete-timesystems to obtain the 120576-optimal control by using neuralnetworks In [106] iterative ADP algorithmwas used to studythe near-optimal control with control constraints in discrete-time systems An iterative ADP for discrete-time systemswas studied in [104] In this work the optimal controllerwas designed with control constraints while NN was used toidentify the unknown dynamical systems with stability proof

4 Conclusion

In this short survey despite the impossibility in identifyingor listing all the related contributions best efforts have beenmade to summarize the major achievements in the area ofdiscrete-time adaptive control withmodern techniques basedon FL and NNs which are also conventionally termed asimportant part of ldquosoft-computingrdquo approaches or ldquointelli-gent controlrdquo in the control communities In particular themarriage of ldquoadaptive controlrdquo and ldquointelligent controlrdquo hasbeen reviewed for the purpose of clarifying main contribu-tions and outlining some possible trends for the developmentof this promising area

Generally speaking adaptive control for discrete-timelinear systems was extensively explored in the past decadesdespite the fact that the complete closed-loop stability analy-sis for the so-called self-tuning regulator was ever regarded asa long-term challenging problem Later it became clear thatdiscrete-time adaptive control can be extended to nonlinearsystems with linear growth rate However general nonlinearsystems with parametric andor nonparametric uncertaintiesare usually very difficult to cope with which motivatedvarious solutions for different certain classes of nonlinearsystems

Among the various solutions intelligent control emergedas one important way to resolve the challenges caused by thenonlinearityThe key for usability of intelligent control basedon FL or NNs lies in their universal approximator property atthe cost of tuning rules of FL or updating parameters of NNsFL and NNs are motivated by different background hencetheir design philosophies look different however essentiallyspeaking both of them can be expressed as weighted sumof some kernel functions where the weights can be tunedto approximate arbitrary smooth or continuous nonlinearfunction The ideas borrowed from adaptive estimation andadaptive control provide such a way to adaptively tune theweight parameters in FL orNNs thus the new area of adaptiveFLC or NN control emerged with extensive studies

In a summary a brief review on FLC for discrete-timesystems is provided by highlighting the adaptive FLC robust-ness issue and stability issue while NN control for discrete-time systems is also reviewed with focuses on adaptive NNcontrol and NN-based dynamic programming We believe

this topic would promote increasing investigations in boththeories and applications And some emerging techniquessuch as deep learning and big data could also bemerged withthe intelligent control for discrete-time systems and give birthto brand new design approaches of intelligent control in thefuture

Conflict of Interests

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

Acknowledgment

This work is partially supported by the National Natu-ral Science Foundation of China (NSFC) under Grants61473120 and 61473038 Guangdong Provincial Natural Sci-ence Foundation 2014A030313266 and International Scienceand Technology Collaboration Grant 2015A050502017 andFundamental Research Funds for the Central Universitiesunder Grant 2015ZM065 The authors would like to thankDr Bin Xu of Northwestern Polytechnical University for hisconstructive comments during the preparation of this paper

References

[1] G Goodwin and K Sin Adaptive Filtering Prediction andControl Prentice-Hall Englewood Cliffs NJ USA 1984

[2] K Astrom and B Wittenmark Adaptive Control Addison-Wesley 1989

[3] M Krstic I Kanellakopoulos and P V Kokotovic NonlinearandAdaptive Control Design JohnWileyamp Sons NewYork NYUSA 1995

[4] G Tao and P Kokotovic Aaptive Control of Systems with Actu-ator and Sensor Non-Linearities John Wiley amp Sons HobokenNJ USA 1996

[5] S S Ge C C Hang T H Lee and T Zhang Stable AdaptiveNeural Network Control Kluwer Academic Publishers NorwellMass USA 2001

[6] C Yang Adaptive control and neural network control of nonlin-ear discrete-time systems [PhD thesis] National University ofSingapore 2009

[7] C Yang H Ma and M Fu ldquoAdaptive predictive control ofperiodic non-linear auto-regressive moving average systemsusing nearest-neighbour compensationrdquo IET Control Theory ampApplications vol 7 no 7 pp 936ndash951 2013

[8] C Yang L Zhai S S Ge T Chai andTH Lee ldquoAdaptivemodelreference control of a class ofMIMOdiscrete-time systems withcompensation of nonparametric uncertaintyrdquo in Proceedings ofthe American Control Conference pp 4111ndash4116 IEEE SeattleWash USA June 2008

[9] S-L Dai C Yang S S Ge and T H Lee ldquoRobust adaptiveoutput feedback control of a class of discrete-time nonlinearsystems with nonlinear uncertainties and unknown controldirectionsrdquo International Journal of Robust and Nonlinear Con-trol vol 23 no 13 pp 1472ndash1495 2013

[10] K J Astrom and B Wittenmark ldquoOn self tuning regulatorsrdquoAutomatica vol 9 no 2 pp 185ndash199 1973

Discrete Dynamics in Nature and Society 9

[11] L Ljung ldquoAnalysis of recursive stochastic algorithmsrdquo IEEETransactions on Automatic Control vol 22 no 4 pp 551ndash5751977

[12] G C Goodwin P J Ramadge and P E Caines ldquoDiscretetime multivariable adaptive controlrdquo IEEE Transactions onAutomatic Control vol 25 no 3 pp 449ndash456 1980

[13] L Guo and H F Chen ldquoThe Astrom-Wittenmark self-tuningregulator revisited and ELS-based adaptive trackersrdquo IEEETransactions on Automatic Control vol 36 no 7 pp 802ndash8121991

[14] L Guo Time-Varing Stochastic Systems Jilin Science and Tech-nology Press Changchun China 1993 (Chinese)

[15] H F Chen and L Guo Identification and Stochastic AdaptiveControl Birkhauser Boston Mass USA 1991

[16] F P Skantze A Kojic A-P Loh and A M AnnaswamyldquoAdaptive estimation of discrete-time systems with nonlinearparameterizationrdquo Automatica vol 36 no 12 pp 1879ndash18872000

[17] L Chen and K S Narendra ldquoNonlinear adaptive control usingneural networks and multiple modelsrdquo Automatica vol 37 no8 pp 1245ndash1255 2001

[18] L Guo and C Wei ldquoLS-based discrete-time adaptive nonlinearcontrol feasibility and limitationsrdquo Science in China Series ETechnological Sciences vol 39 no 3 pp 255ndash269 1996

[19] L L Xie and L Guo ldquoAdaptive control of discrete-timenonlinear systems with structural uncertaintiesrdquo in Lectures onSystems Control and Information vol 17 of AMSIP Studiesin Advanced Mathematics American Mathematical SocietyInternational Press Providence RI USA 2000

[20] J D Boskovic ldquoStable adaptive control of a class of first-order nonlinearly parameterized plantsrdquo IEEE Transactions onAutomatic Control vol 40 no 2 pp 347ndash350 1995

[21] A L Fradkov I V Miroshnik and V O Nikiforov Nonlinearand Adaptive Control of Complex Systems Mathematics andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 2004

[22] D Angeli and E Mosca ldquoAdaptive switching supervisorycontrol of nonlinear systems with no prior knowledge of noiseboundsrdquo Automatica vol 40 no 3 pp 449ndash457 2004

[23] H B Ma ldquoFinite-model adaptive control using an LS-likealgorithmrdquo International Journal of Adaptive Control and SignalProcessing vol 21 no 5 pp 391ndash414 2007

[24] H B Ma ldquoFinite-model adaptive control using WLS-likealgorithmrdquo Automatica vol 43 no 4 pp 677ndash684 2007

[25] H B Ma ldquoSeveral algorithms for finite-model adaptive controlpartial answers to finite-model adaptive control problemrdquoMathematics of Control Signals and Systems vol 20 no 3 pp271ndash303 2008

[26] S S Ge C C Hang and T Zhang ldquoA direct adaptive controllerfor dynamic systems with a class of nonlinear parameteriza-tionsrdquo Automatica vol 35 no 4 pp 741ndash747 1999

[27] C Y Li and L Guo ldquoOn feedback capability in a class of nonlin-early parameterized uncertain systemsrdquo IEEE Transactions onAutomatic Control vol 56 no 12 pp 2946ndash2951 2011

[28] H Ma K-Y Lum and S S Ge ldquoAdaptive control for a discrete-time first-order nonlinear system with both parametric andnon-parametric uncertaintiesrdquo in Proceedings of the 46th IEEEConference on Decision and Control (CDC rsquo07) pp 4839ndash4844IEEE New Orleans La USA December 2007

[29] L Guo ldquoExploring the capability and limits of the feedbackmechanismrdquo in Proceedings of the International Congress ofMathematicians (ICM rsquo02) Beijing China August 2002

[30] H-B Ma ldquoAn lsquoimpossibilityrsquo theorem on a class of high-orderdiscrete-time nonlinear control systemsrdquo Systems and ControlLetters vol 57 no 6 pp 497ndash504 2008

[31] I Kanellakopoulos P V Kokotovic and A S Morse ldquoSys-tematic design of adaptive controllers for feedback linearizablesystemsrdquo IEEE Transactions on Automatic Control vol 36 no11 pp 1241ndash1253 1991

[32] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[33] L-X Wang and J M Mendel ldquoFuzzy basis functions universalapproximation and orthogonal least-squares learningrdquo IEEETransactions on Neural Networks vol 3 no 5 pp 807ndash814 1992

[34] L-X Wang ldquoStable adaptive fuzzy control of nonlinear sys-temsrdquo IEEE Transactions on Fuzzy Systems vol 1 no 2 pp 146ndash155 1993

[35] W S McCulloch and W Pitts ldquoA logical calculus of the ideasimmanent in nervous activityrdquo The Bulletin of MathematicalBiophysics vol 5 pp 115ndash133 1943

[36] K Hornik M Stinchcombe and HWhite ldquoMultilayer feedfor-ward networks are universal approximatorsrdquo Neural Networksvol 2 no 5 pp 359ndash366 1989

[37] T Khanna Foundations of Neural Networks Addison-WesleyReading Mass USA 1990

[38] R M Sanner and J-J E Slotine ldquoGaussian networks for directadaptive controlrdquo IEEE Transactions on Neural Networks vol 3no 6 pp 837ndash863 1992

[39] Y J Liu Y J Fang andM A Bao-Ping ldquoSliding-data-window-driven Bayesian-Gaussian neural network and its application tomodeling of nonlinear systemrdquo Control Theory amp Applicationsvol 26 no 12 pp 1435ndash1438 2009

[40] D Wang and J Huang ldquoAdaptive neural network control fora class of uncertain nonlinear systems in pure-feedback formrdquoAutomatica vol 38 no 8 pp 1365ndash1372 2002

[41] Y Song and J W Grizzle ldquoAdaptive output-feedback control ofa class of discrete-time nonlinear systemsrdquo in Proceedings of theAmerican Control Conference pp 1359ndash1363 June 1993

[42] B-S Chen C-S Tseng and H-J Uang ldquoRobustness designof nonlinear dynamic systems via fuzzy linear controlrdquo IEEETransactions on Fuzzy Systems vol 7 no 5 pp 571ndash585 1999

[43] T Chai and S Tong ldquoFuzzy direct adaptive control for a class ofnonlinear systemsrdquo Fuzzy Sets and Systems vol 103 no 3 pp379ndash387 1999

[44] K Tanaka T Ikeda and H O Wang ldquoRobust stabilizationof a class of uncertain nonlinear systems via fuzzy controlquadratic stabilizability 119867

infin control theory and linear matrixinequalitiesrdquo IEEE Transactions on Fuzzy Systems vol 4 no 1pp 1ndash13 1996

[45] S Jagannathan M W Vandegrift and F L Lewis ldquoAdaptivefuzzy logic control of discrete-time dynamical systemsrdquo Auto-matica vol 36 no 2 pp 229ndash241 2000

[46] Y Jiang Z Liu C Chen and Y Zhang ldquoAdaptive robust fuzzycontrol for dual arm robot with unknown input deadzonenonlinearityrdquo Nonlinear Dynamics vol 81 no 3 pp 1301ndash13142015

[47] MWVandegrift F L Lewis S Jagannathan andK Liu ldquoAdap-tive fuzzy logic control of discrete-time dynamical systemsrdquo inProceedings of the IEEE International Symposium on IntelligentControl pp 395ndash401 IEEE Monterey Calif USA August 1995

[48] S Jagannathan ldquoAdaptive fuzzy logic control of feedbacklinearizable discrete-time dynamical systems under persistenceof excitationrdquo Automatica vol 34 no 11 pp 1295ndash1310 1998

10 Discrete Dynamics in Nature and Society

[49] R Qi and M A Brdys ldquoStable indirect adaptive control basedon discrete-time T-S fuzzy modelrdquo Fuzzy Sets and Systems vol159 no 8 pp 900ndash925 2008

[50] T-C Lin S-W Chang and C-H Hsu ldquoRobust adaptivefuzzy sliding mode control for a class of uncertain discrete-time nonlinear systemsrdquo International Journal of InnovativeComputing Information and Control vol 8 no 1 pp 347ndash3592012

[51] G Feng andG Chen ldquoAdaptive control of discrete-time chaoticsystems a fuzzy control approachrdquoChaos Solitons and Fractalsvol 23 no 2 pp 459ndash467 2005

[52] H J Lee J B Park and G Chen ldquoRobust fuzzy controlof nonlinear systems with parametric uncertaintiesrdquo IEEETransactions on Fuzzy Systems vol 9 no 2 pp 369ndash379 2001

[53] Y-Y Cao and PM Frank ldquoRobust119867infindisturbance attenuation

for a class of uncertain discrete-time fuzzy systemsrdquo IEEETransactions on Fuzzy Systems vol 8 no 4 pp 406ndash415 2000

[54] S Zhou G Feng J Lam and S Xu ldquoRobust 119867infin

controlfor discrete-time fuzzy systems via basis-dependent Lyapunovfunctionsrdquo Information Sciences vol 174 no 3-4 pp 197ndash2172005

[55] S Xu and J Lam ldquoRobust 119867infin

control for uncertain discrete-time-delay fuzzy systems via output feedback controllersrdquo IEEETransactions on Fuzzy Systems vol 13 no 1 pp 82ndash93 2005

[56] C-S Tseng and B-S Chen ldquoRobust fuzzy observer-basedfuzzy control design for nonlinear discrete-time systems withpersistent bounded disturbancesrdquo IEEE Transactions on FuzzySystems vol 17 no 3 pp 711ndash723 2009

[57] S Xu B Song J Lu and J Lam ldquoRobust stability of uncertaindiscrete-time singular fuzzy systemsrdquo Fuzzy Sets and Systemsvol 158 no 20 pp 2306ndash2316 2007

[58] Z-G Wu P Shi H Su and J Chu ldquoReliable 119867infin

controlfor discrete-time fuzzy systems with infinite-distributed delayrdquoIEEE Transactions on Fuzzy Systems vol 20 no 1 pp 22ndash312012

[59] G Feng and J Ma ldquoQuadratic stabilization of uncertaindiscrete-time fuzzy dynamic systemsrdquo IEEE Transactions onCircuits and Systems I Fundamental Theory and Applicationsvol 48 no 11 pp 1337ndash1344 2001

[60] A Kruszewski R Wang and T M Guerra ldquoNonquadratic sta-bilization conditions for a class of uncertain nonlinear discretetime TS fuzzy models a new approachrdquo IEEE Transactions onAutomatic Control vol 53 no 2 pp 606ndash611 2008

[61] W-J Wang Y-J Chen and C-H Sun ldquoRelaxed stabilizationcriteria for discrete-time T-S fuzzy control systems based ona switching fuzzy model and piecewise Lyapunov functionrdquoIEEE Transactions on Systems Man amp Cybernetics Part BCybernetics vol 37 no 3 pp 551ndash559 2007

[62] G Feng ldquoStability analysis of discrete-time fuzzy dynamicsystems based on piecewise Lyapunov functionsrdquo IEEE Trans-actions on Fuzzy Systems vol 12 no 1 pp 22ndash28 2004

[63] H Gao X Liu and J Lam ldquoStability analysis and stabilizationfor discrete-time fuzzy systems with time-varying delayrdquo IEEETransactions on Systems Man and Cybernetics Part B Cyber-netics vol 39 no 2 pp 306ndash317 2009

[64] L Wu X Su P Shi and J Qiu ldquoA new approach to stabilityanalysis and stabilization of discrete-time T-S fuzzy time-varying delay systemsrdquo IEEE Transactions on SystemsMan andCybernetics Part B Cybernetics vol 41 no 1 pp 273ndash286 2011

[65] X Su P Shi L Wu and Y-D Song ldquoA novel control design ondiscrete-time takagi-sugeno fuzzy systems with time-varying

delaysrdquo IEEE Transactions on Fuzzy Systems vol 21 no 4 pp655ndash671 2013

[66] X Su P Shi L Wu and Y-D Song ldquoA novel approach to filterdesign for T-S fuzzy discrete-time systems with time-varyingdelayrdquo IEEE Transactions on Fuzzy Systems vol 20 no 6 pp1114ndash1129 2012

[67] C-S Tseng ldquoModel reference output feedback fuzzy trackingcontrol design for nonlinear discrete-time systems with time-delayrdquo IEEE Transactions on Fuzzy Systems vol 14 no 1 pp58ndash70 2006

[68] X Su P Shi L Wu and S K Nguang ldquoInduced l2 filteringof fuzzy stochastic systems with time-varying delaysrdquo IEEETransactions on Cybernetics vol 43 no 4 pp 1257ndash1264 2013

[69] L Wu X Su P Shi and J Qiu ldquoModel approximation fordiscrete-time state-delay systems in the TS fuzzy frameworkrdquoIEEE Transactions on Fuzzy Systems vol 19 no 2 pp 366ndash3782011

[70] F L Lewis S Jagannathan and A Yesildirek Neural NetworkControl of Robot Manipulators and Nonlinear Systems Taylor ampFrancis London UK 1999

[71] A M Shaw and F J Doyle III ldquoMultivariable nonlinear controlapplications for a high purity distillation column using arecurrent dynamic neuron modelrdquo Journal of Process Controlvol 7 no 4 pp 255ndash268 1997

[72] K Najim Process Modeling and Control in Chemical Engineer-ing Marcel Dekker New York NY USA 1989

[73] B Xu D Wang F Sun and Z Shi ldquoDirect neural discretecontrol of hypersonic flight vehiclerdquo Nonlinear Dynamics vol70 no 1 pp 269ndash278 2012

[74] B Xu and Y Zhang ldquoNeural discrete back-stepping controlof hypersonic flight vehicle with equivalent prediction modelrdquoNeurocomputing vol 154 pp 337ndash346 2015

[75] B Xu F Sun H Liu and J Ren ldquoAdaptive Kriging controllerdesign for hypersonic flight vehicle via back-steppingrdquo IETControl Theory amp Applications vol 6 no 4 pp 487ndash497 2012

[76] B Xu ldquoRobust adaptive neural control of flexible hypersonicflight vehicle with dead-zone input nonlinearityrdquo NonlinearDynamics vol 80 no 3 pp 1509ndash1520 2015

[77] B Xu XHuangDWang and F Sun ldquoDynamic surface controlof constrained hypersonic flightmodels with parameter estima-tion and actuator compensationrdquo Asian Journal of Control vol16 no 1 pp 162ndash174 2014

[78] B Xu and Z Shi ldquoAn overview on flight dynamics and controlapproaches for hypersonic vehiclesrdquo Science China InformationSciences vol 58 no 7 pp 1ndash19 2015

[79] D E Rumelhart G E Hinton and R J Williams ldquoLearninginternal representations by error propagationrdquo in Parallel Dis-tributed Processing vol 1 pp 318ndash362 MIT Press 1986

[80] S Jagannathan and F L Lewis ldquoDiscrete-time neural netcontroller for a class of nonlinear dynamical systemsrdquo IEEETransactions on Automatic Control vol 41 no 11 pp 1693ndash16991996

[81] S Jagannathan and F L Lewis ldquoMultilayer discrete-timeneural-net controller with guaranteed performancerdquo IEEETransactions on Neural Network vol 7 no 1 pp 107ndash130 1996

[82] P He and S Jagannathan ldquoNeuro-controller for reducingcyclic variation in lean combustion spark ignition enginesrdquoAutomatica vol 41 no 7 pp 1133ndash1142 2005

[83] S S Ge G Y Li and T H Lee ldquoAdaptive NN controlfor a class of strict-feedback discrete-time nonlinear systemsrdquoAutomatica vol 39 no 5 pp 807ndash819 2003

Discrete Dynamics in Nature and Society 11

[84] S S Ge T H Lee G Y Li and J Zhang ldquoAdaptive NN controlfor a class of discrete-time non-linear systemsrdquo InternationalJournal of Control vol 76 no 4 pp 334ndash354 2003

[85] C J Goh ldquoModel reference control of non-linear systems viaimplicit function emulationrdquo International Journal of Controlvol 60 no 1 pp 91ndash115 1994

[86] C J Goh and T H Lee ldquoDirect adaptive control of nonlinearsystems via implicit function emulationrdquo Control Theory andAdvanced Technology vol 10 no 3 pp 539ndash552 1994

[87] A U Levin and K S Narendra ldquoControl of nonlinear dynami-cal systems using neural networksmdashpart II observability iden-tification and controlrdquo IEEE Transactions on Neural Networksvol 7 no 1 pp 30ndash42 1996

[88] S S Ge J Zhang and T H Lee ldquoAdaptive MNN control fora class of non-affine NARMAX systems with disturbancesrdquoSystems amp Control Letters vol 53 no 1 pp 1ndash12 2004

[89] S S Ge J Zhang and T H Lee ldquoAdaptive neural networkcontrol for a class of MIMO nonlinear systems with distur-bances in discrete-timerdquo IEEE Transactions on Systems Manand Cybernetics Part B Cybernetics vol 34 no 4 pp 1630ndash1645 2004

[90] J Zhang S S Ge and T H Lee ldquoOutput feedback control of aclass of discrete MIMO nonlinear systems with triangular forminputsrdquo IEEE Transactions onNeural Networks vol 16 no 6 pp1491ndash1503 2005

[91] F C Sun Z Sun and P-Y Woo ldquoStable neural-network-basedadaptive control for sampled-data nonlinear systemsrdquo IEEETransactions on Neural Networks vol 9 no 5 pp 956ndash9681998

[92] C Yang S S Ge C Xiang T Chai and T H Lee ldquoOutputfeedback NN control for two classes of discrete-time systemswith unknown control directions in a unified approachrdquo IEEETransactions on Neural Networks vol 19 no 11 pp 1873ndash18862008

[93] S S Ge C Yang and T H Lee ldquoAdaptive predictive controlusing neural network for a class of pure-feedback systems indiscrete timerdquo IEEE Transactions on Neural Networks vol 19no 9 pp 1599ndash1614 2008

[94] Y Li C Yang S S Ge and T H Lee ldquoAdaptive output feed-back NN control of a class of discrete-time MIMO nonlinearsystems with unknown control directionsrdquo IEEE Transactionson Systems Man and Cybernetics Part B Cybernetics vol 41no 2 pp 507ndash517 2011

[95] Y-J Liu C L P Chen G-X Wen and S Tong ldquoAdaptiveneural output feedback tracking control for a class of uncertaindiscrete-time nonlinear systemsrdquo IEEE Transactions on NeuralNetworks vol 22 no 7 pp 1162ndash1167 2011

[96] C Yang S S Ge and T H Lee ldquoOutput feedback adaptivecontrol of a class of nonlinear discrete-time systems withunknown control directionsrdquoAutomatica vol 45 no 1 pp 270ndash276 2009

[97] A Al-Tamimi F L Lewis and M Abu-Khalaf ldquoDiscrete-timenonlinear HJB solution using approximate dynamic program-ming convergence proofrdquo IEEE Transactions on Systems Manand Cybernetics Part B Cybernetics vol 38 no 4 pp 943ndash9492008

[98] P Werbos ldquoApproximate dynamic programming for real-timecontrol and neuralmodelingrdquo inHandbook of Intelligent ControlNeural Fuzzy amp Adaptive Approaches Van Nostrand Reinhold1992

[99] P He and S Jagannathan ldquoReinforcement learning neural-network-based controller for nonlinear discrete-time systems

with input constraintsrdquo IEEETransactions on SystemsMan andCybernetics Part B Cybernetics vol 37 no 2 pp 425ndash436 2007

[100] B Xu C Yang and Z Shi ldquoReinforcement learning outputfeedback NN control using deterministic learning techniquerdquoIEEE Transactions on Neural Networks and Learning Systemsvol 25 no 3 pp 635ndash641 2014

[101] D Liu DWang D Zhao QWei and N Jin ldquoNeural-network-based optimal control for a class of unknowndiscrete-time non-linear systems using globalized dual heuristic programmingrdquoIEEE Transactions on Automation Science and Engineering vol9 no 3 pp 628ndash634 2012

[102] D Liu and Q Wei ldquoFinite-approximation-error-based optimalcontrol approach for discrete-time nonlinear systemsrdquo IEEETransactions on Cybernetics vol 43 no 2 pp 779ndash789 2013

[103] X Zhong H He H Zhang and Z Wang ldquoOptimal control forunknown discrete-time nonlinear markov jump systems usingadaptive dynamic programmingrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 25 no 12 pp 2141ndash21552014

[104] D Liu D Wang and X Yang ldquoAn iterative adaptive dynamicprogramming algorithm for optimal control of unknowndiscrete-time nonlinear systemswith constrained inputsrdquo Infor-mation Sciences vol 220 pp 331ndash342 2013

[105] F-Y Wang N Jin D Liu and Q Wei ldquoAdaptive dynamicprogramming for finite-horizon optimal control of discrete-time nonlinear systems with 120576-error boundrdquo IEEE Transactionson Neural Networks vol 22 no 1 pp 24ndash36 2011

[106] H Zhang Y Luo and D Liu ldquoNeural-network-based near-optimal control for a class of discrete-time affine nonlinearsystems with control constraintsrdquo IEEE Transactions on NeuralNetworks vol 20 no 9 pp 1490ndash1503 2009

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

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Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 3

2 Fuzzy Logic Control forDiscrete-Time Systems

Nowadays techniques for FLC have developed rapidly espe-cially in modeling complex nonlinear systems Since Wangfirst proved that the linear combinations of a series offuzzy basis functions are universal approximators of anynonlinear systems the universal approximation property ofthe fuzzy logic systems has been extensively studied [42ndash45]The combination of adaptive control and fuzzy logic systemallows adaptive laws to update parameters of the FL controllerduring the adaptation procedure while adaptive fuzzy controlprovides an efficient method to model complex nonlinearsystems In practice the difficulty of design a fuzzy controllerlies in the fact that there are usually various requirementimposed on the systems to ensure stability and performancewhile most of complex control systems today are using digitalcomputers to calculate the control signals in a digital formHence the continuous-time methods could not be directlyapplied to most practical systems The modelingcontrol fordiscrete-time systems is crucial to take the controllers intoreal plants Thus it is significant to study the discrete-timefuzzy systems as well as their molding and control To thisend researchers have paid efforts to develop the discrete-time FLCA typical procedure of themultiple-inputmultiple-output (MIMO) fuuzy logic system (FLS) approximatingan unknown function 119865(119909) is comprised of three primarycomponents as follows [46]

(1) Fuzzification Take 119883 as the input of the fuzzy systemand (119883) is the estimation of the output Both the input andthe output are fuzzified into fuzzy linguistic terms with fuzzymembership function

(2) Fuzzy Rules The collection of the fuzzy MIMO IF-THEN rules are designed to comprise the knowledge basefor constructing the FLS Using prior expert knowledge thefuzzy rules can be obtained as

Rule 119897 If 1199091is 119860119897

1 1199092is 119860119897

2 119909

119899is 119860119897

119899

then 1199101is 119861119897

1 1199102is 119861119897

2 119910

119898is 119861119897

119898

(1)

where 1199091

sim 119909119899and 119910

1sim 119910119898

are the premise variablesconsisting of 119883 and (119883) respectively 119860

119897 and 119861119897 are the

linguistic variables of the fuzzy sets 119897 = 1 2 119871

(3) Fuzzy Inference Engine and Defuzzification Using thestrategy of the sum-product inference engine and the center-average defuzzifier the output of the FLS can be described as

119910119895

=

sum119871

119897=1120593(119897)

119895(prod119899

119894=1120583(119897)

119860119894(119909119894))

sum119871

119897=1(prod119899

119894=1120583(119897)

119860119894(119909119894))

(2)

where 120593(119897)

119895is the point where fuzzy membership function

120583(119897)

119861119895(120593119895) achieves its maximum value Define

119908119895(119909) =

prod119899

119894=1120583(119897)

119860119894(119909119894)

sum119871

119897=1(prod119899

119894=1120583(119897)

119860119894(119909119894))

119882 (119909) = [1199081

(119909) 1199082

(119909) 119908119871

(119909)]119879

(3)

and (Φ)119895

= [120593(1)

119895 120593(2)

119895 120593

(119871)

119895]119879 Then the output of MIMO

FLS can be rewritten as

119910119895

= (Φ)119879

119895119882 (119883) (4)

where the subscript 119895 represents the 119895th column vector of thematrix Therefore the estimation of (119883) is

(119883)

= [(Φ)119879

1119882 (119883) (Φ)

119879

2119882 (119883) (Φ)

119879

119898119882 (119883)]

119879

(5)

Then 119865(119883) can be written as

(119883) = Φ119879

119882 (119883) (6)

21 Adaptive FLC of Discrete-Time Systems FLC has beenextensively investigated in both academic and industrialcommunities However problems with high dimensionalityalso existed in higher-dimensional fuzzy systems due to themathematical complications and intuitive limitations There-fore it required more systematic methods for advanced FLCdesign and synthesis such as the adaptive FLC Originallyadaptive control was proposed for aircraft autopilots to dealwith parameter variations during changing flight conditionsIn the 1960s the advances in stability theory and the progressof control theory improved the understanding of adaptivecontrol By the early 1980s several adaptive approaches havebeen proven to provide stable operation and asymptotictracking Since then the adaptive control problems have beenrigorously formulated

Generally adaptive FLC design for nonlinear discrete-time systems ismuchmore difficult than those in continuous-time The stability analysis techniques become much moreintractable for difference equations than those for differentialequations Many nice Lyapunov adaptive control designmethodologies developed in continuous-time are not appli-cable to discrete-time systems Thus it is significant toconsider the adaptive FLC in discrete-time form

In recent years the adaptive FLC for discrete-time non-linear systems has been extensively studied Jagannathanet al [45 47] developed an adaptive fuzzy logic controllerfor a general class of discrete-time nonlinear systems usingbasis vectors and provided rigorous proofs to stability of thediscrete-time adaptive FLC In this work a multiloop struc-ture controller was constructed with a fuzzy approximatelinearization loop an outer tracking loop and a robustifyingloop By selecting FLCrsquos structure and the adaptation laws andgiving a number of assumptions on a class of discrete-timenonlinear systems closed-loop signals can achieve uniform

4 Discrete Dynamics in Nature and Society

ultimate boundedness This method is significant in sensethat no persistence of excitation is used and no certaintyequivalence assumption is required The controller has alsobeen extended to a class of unknown feedback linearizablenonlinear dynamical systems under persistence of excitationin [48] in which rigorous stability proofs of discrete-timeadaptive FLC for feedback linearizable unknown nonlinearsystems were presented Indirect fuzzy control of uncertaindiscrete-time nonlinear system was shown by Qi and Brdys[49] This work presented an indirect adaptive FLC for theuncertainties in nonlinear plants and employed a Takagi-Sugeno (T-S) model to deal with the unknown dynamicsin input-output form A feedback linearization control lawwas designed by using structure states and parametersof the model Gradient descent algorithm and recursiveleast square estimation method were used to online updatethe modeling parameters The extension of sliding modecontrol (SMC) adaptive fuzzy SMC for a class of uncertaindiscrete-time nonlinear systems was investigated by Linet al [50] This work presented an adaptive interval type-2 fuzzy sliding mode controller for a class of unknownnonlinear discrete-time systems with training data disturbedby external disturbances It employed an adaptive intervaltype-2 fuzzy control scheme and SMC approach to controlthe plant tracking a reference trajectory and prevented bigchattering of the control effort While adaptive fuzzy controlhas also been applied in discrete-time chaotic systems In[51] Feng and Chen presented a novel adaptive controlalgorithm for discrete-time chaotic systems The basic ideais representing the chaotic system as a T-S fuzzy model anddesigning a local linear adaptive controller in each localregion Meanwhile a global adaptive controller on the entiredomainwas reconstructed and the stability of the closed-loopadaptive control system was proved

22 Robustness Issue in Discrete-Time Fuzzy Control Inpractice nonlinearities and uncertainties exist in almost allthe industrial plants while uncertainties in the modeling andcontrol of nonlinear systems are still one of themost challeng-ing problems in the control field The presence of nonlinear-ities and uncertainties brings difficulties to controller designIn order to conquer these problems a number of schemeshave been developed and among these robust control is oneof the most effective methods Robustness is regarded as oneof the most important requirements for a control system Inthe past two decades considerable attention has been paid tothe robustness in fuzzy model-based control of discrete-timesystems andmanymodification techniques were proposed toenhance the robustness of FLC for discrete-time systems [52ndash57]

The work of Lee et al [52] illustrated an approach ofrobust fuzzy control for nonlinear discrete-time systems Inthis work a systematic control structure was presented fornonlinear discrete-time T-S fuzzy systems with parametricuncertainties using the 119867

infincontrol design approach This

work also gave a novel solution to the robust stabilizationproblem of nonlinear systems by using basis-dependentLyapunov function

119867infindisturbance attenuation is one of the most important

requirements for a fuzzy control system Cao and Frank [53]have applied the 119867

infincontrol to address the robust stability

for a class of uncertain discrete-time fuzzy systems via linearmatrix inequality approach They studied both the robuststabilization and the 119867

infindisturbance attenuation while

they presented several sufficient conditions to ensure robuststability of the fuzzy models based on coupled linear matrixinequalities They also analyzed the robust 119867

infinperformance

of the fuzzy model-based discrete-time nonlinear systems byaddressing the robust 119867

infindisturbance attenuation problems

The robust stability of the models is achieved independentlyof the uncertainties

How to develop a robust 119867infin

controller for discrete-time systems using the basis-dependent Lyapunov functionwas also shown by Zhou et al [54] In their work anovel linear matrix inequalities characterization with 119867

infin

norm bound was presented for discrete-time fuzzy systemsWhile additional matrix variables were used to decouple theLyapunov function and system matrices as well as facilitatethe design approach of the controller this approach providessome sufficient results in the shape of strict linear matrixinequalities

Studied by Choi and Park the state feedback 119867infin

controlfor discrete-time systems was investigated by constructingLyapunov functions with fuzzy weights The fuzzy weightingLyapunov functions were designed with both current-timeand one-step-past information while designing controllerparameterized with linear matrix inequalities (PLMIs) Byselecting the structures of variables appropriately in thePLMIs a special case LMI formulation was obtained The119867infin

control with output feedback for discrete-time fuzzysystems has also been rigorously studied by Xu and Lam[55] while Wu et al [58] investigated the reliable 119867

infincontrol

for discrete-time systems with delays and stochastic actuatorfaults by representing the stochastic behavior with a discrete-time homogeneous Markov chain

Tseng andChen [56] used a fuzzy observer to estimate thepremise variables which depend on the state variables whilethe fuzzy observer was also used to address the nonlinear119897infin-gain control problem This work extended the 119897

infincontrol

from linear discrete-time systems to nonlinear discrete-time systems By using the T-S fuzzy model to representthe nonlinear discrete-time system an observer-based fuzzycontroller which minimized the upper bound of 119897

infin-gain and

attenuated the peak of perturbation was designed Xu et al[57] gave stability analysis for discrete-time singular fuzzysystems in the presence of time-varying uncertainties whilethey ensured the system to be regular causal and stable It hasbeen shown that for these systems robust stability conditioncan be obtained by giving a sufficient condition in terms of aset of linear matrix inequalities

23 Stability Issue in Discrete-Time Fuzzy Control Stabilityis one of the most important properties for a system whileFLC are used to address the stability problems for discrete-time systems as well The quadratic stability for uncertaindiscrete-time fuzzy dynamic systems was shown by Fengand Ma [59] This work gave some sufficient conditions of

Discrete Dynamics in Nature and Society 5

the quadratic stabilization for an uncertain fuzzy dynamicsystem Stability was achieved when the suitable Riccatiequation or a set of Riccati equations were solved Stabilizingfeedback control laws were also obtained by the developingalgorithms The nonquadratic stabilization conditions fornonlinear discrete-time fuzzy systems were also analyzedby Kruszewski et al [60] They considered discrete-timeuncertain nonlinear models in a T-S form and studied thestability through a nonquadratic Lyapunov function Thestabilization conditions were developed by considering theLyapunov function with a 119896-sample variation and extendedto uncertain T-S models Based on a switching fuzzy modeland piecewise Lyapunov function Wang et al [61] proposedtwo stabilization criteria for discrete-time T-S fuzzy systems

Zhou et al [54] used a basis-dependent Lyapunov-Krasovskii function to give stabilization analysis for discrete-time fuzzy systems A robust control design approach wasalso developed by using the Lyapunov-Krasovskii functionand facilitated by introducing additional instrumental matrixvariables Robust control problem for systemswith time delaywas concerned as well Feng [62] used a piecewise smoothLyapunov function to analyze the stability of discrete-time T-S fuzzy dynamic systems In this study stability of the systemguaranteed by constructing a piecewise Lyapunov functionMeanwhile the Lyapunov function was obtained using thelinear matrix inequalities methodThis work showed that thepiecewise quadratic Lyapunov functions based stability is lessconservative than the common quadratic Lyapunov functionbased stability

The stabilization for discrete-time T-S fuzzy systems withstate time-varying delay was shown by Gao et al [63] Intheir work a fuzzy Lyapunov function was constructed toimprove the delay-dependent stability condition By avoidingthe utilization of the bounding inequalities for the crossproducts between two vectors reduction of the conservatismof stability condition was achieved A delay-dependent stabi-lization algorithm was also developed for both state feedbackand observer-based output feedback cases using a paralleldistributed compensation scheme

24 Fuzzy Control for Discrete-Time Systemswith TimeDelaysIn practice time delays are the intrinsic nature of variousphysical systems such as communication hydraulic chem-ical processes and electronics The existence of time delayscould generally lower the system performance and even causeinstability and oscillation Hence remarkable attention hasbeen devoted to the analysis and synthesis of time delaynonlinear systems as well as the discrete-time fuzzy systems[55 64ndash69]

The stability and stabilization problems of systems time-varying delay were investigated by Wu et al [64] In theirwork the existing stability analysis approaches were dividedinto two types delay-dependent (including the delay size)and delay-independent (irrelevant to the delay size) Thestability of discrete-time T-S fuzzy systems with time-varyingstate delay was also investigated They developed a delay par-titioningmethod and employed a fuzzy Lyapunov-Krasovskiifunction to analyze the stability of delay-dependent time-varying state delay systems Through delay partitioning the

less conservative stability condition is obtained and solved byLMI optimization techniques

The work of Su et al [65] also gave a solution to thedynamic output feedback control problem for fuzzy sys-tems with time-varying delays They proposed a comparisonmodel by approximating the time-varying delay state whilethey gave a sufficient condition to ensure the asymptoti-cal stability by adopting the scaled small-gain theorem aswell as the delay partitioning method This method allowsdesigning desired dynamic output feedback controller byusing optimization techniques A filter design technique fordiscrete-time systems with time-varying delay has also beenpresented by Su et al [66] Both full-order and reduced-orderfilters were designed while approximators were employed toderive an input-output based sufficient condition Tseng [67]studied the time delay problem for a nonlinear discrete-timesystem based on model reference fuzzy tracking control Inthis work the T-S fuzzy model was applied to approximatea time delay discrete-time system while the fuzzy controllerwas designed to reduce the tracking error based on thisfuzzy model This method has advantage in that no feedbacklinearization or adaptive approaches are used in the controllerdesign

The robust 119867infin

control problem for uncertain discrete-time delay systems has been ivestigated by Xu and Lam [55]They studied the state-space T-S fuzzymodel with time delaysunder the assumption that the parameter uncertainties werenorm-boundedThe output feedback controller was designedwith full-order fuzzy dynamic to guarantee the closed-loopsystem is robust asymptotic stable as well as the admissibleuncertainties are norm bound constraint While solvabilityof this problem was addressed using a sufficient conditiongiven by the linear matrix inequalities The induced 119897

2filter

design of T-S fuzzy discrete-time stochastic systems withtime-varying delays has been investigated by Su et al [68]Wuet al [69] studied the 119867

infinmodel approximation for discrete-

time state delay fuzzy T-S systems

3 NN Control for Discrete-Time Systems

NN is of powerful computing ability and learning abilityto emulate various systems dynamics and is capable ofapproximating an unknown functionwith arbitrary accuracyNN is very successful in system modeling and controlby its capacity of universal approximate highly nonlinearand uncertain nonlinear and complex dynamic of systemsNNrsquos approximation ability has been shown by the Stone-Weierstrass theorem which states that a universal approxi-mator can approximate to an arbitrary degree of accuracyany real continuous function on a compact set Besides theuniversal approximation abilities NN also shows its excel-lence in parallel distributed processing abilities learningadaptation abilities natural fault tolerance and feasibilityfor hardware implementation These advantages make NNparticularly attractive and prospective for nonlinear controland modeling NN has been successfully applied to robotmanipulators control [70] distillation column control [71]chemical processes identification [72] flight control [73ndash75]

6 Discrete Dynamics in Nature and Society

and so forth To extend the NN technique from continuous-time to discrete-time many efforts have been made to studythe NN controlled discrete-time systems [76ndash78]

31 Adaptive NN Control for Discrete-Time Systems In theearly stage backpropagation (BP) algorithm [79] greatlyboosted the development of NN control It is noted that inthe early NN control design the control performances weredemonstrated through simulation or by particular exper-imental examples and consequently there were shortageof analytical analysis In addition an offline identificationprocedure was essential for achieving a stable NN controlsystem Thereafter the emergence of Lyapunov-based NNdesign makes it possible to use the available adaptive controltheories to rigorously guarantee stability robustness andconvergence of the closed-loop NN control systems Wecall the control design combining adaptive control theoriesand NN techniques as adaptive NN control It updates NNweights online and guarantees the stability of the closed-loop system Adaptive NN control design has been elegantlydeveloped for nonlinear systems with parametric uncertain-ties while many efforts have been dedicated in adaptive NNcontrol for nonlinear discrete-time systems [6]

For high-order affine nonlinear system in normal formadaptive NN controls using LPNN and MNN have beendeveloped in [80 81] using a filtered tracking error In [82]the controller is designed incorporating reinforcement learn-ing technique to improve control performance In this worka critic NN has been introduced to approximate the strategicutility function which is considered as the long-term systemperformance measure For discrete-time systems in strict-feedback form adaptive NN control has been developed viabackstepping design after system transformation [83] In [84]adaptive NN control has been investigated for discrete-timesystem in affine NARMAX form

In the above mentioned results the adaptive NN controldesigns were carried out through either feedback lineariza-tion or backstepping But these approaches are not applicableto nonaffine systems especially feedback linearization basedmethods which greatly depend on the affine appearance ofcontrol variables As a matter of fact adaptive NN controlmethods for nonaffine systems have been less studied incomparison with large amount of research work on affinenonlinear systems because of the difficulty of control designcaused by the nonaffine form of control input To overcomethe difficulty linearization based NN controls have beenput forward In [17] the nonaffine discrete-time system hasbeen divided into two parts one is linear and another isnonlinear and consequently a linear adaptive controller anda nonlinear adaptive NN controller have been designedwith a switching rule In [85] it directly utilized NN asemulator of the ldquoinverserdquo of the nonlinear discrete-timesystems Furthermore the study of discrete-time systems foradaptive NN control using implicit function to assert theexistence of an ideal inverse control was investigated in [86]Thereafter the implicit function based adaptive NN controlhas been widely studied in discrete-time form [87 88] Blocktriangular discrete-time systems with normal form subsys-tems have been studied in [80 81] For the block triangular

systems with strict-feedback subsystems state feedback andoutput feedback adaptive NN control have been developedin [89 90] by extending the systems transformation basedbackstepping technique proposed for SISO case in [83] In[91] adaptive NN control has been used for sampled-datanonlinear MIMO systems in general affine form based onlinearization The control scheme is an integration of an NNapproach and a variable structure method

An effort has been made in [92] to explore the adaptiveNN control of a class of nonaffine systems in discrete-timeThis work aimed to solve the nonaffine appearance andnoncausal problems of the following pure-feedback discrete-time system

120585119894(119896 + 1) = 119891

119894(120585119894 120585119894+ 1 (119896))

119894 = 1 2 119899 minus 1 119899 ge 2

120585119899

(119896 + 1) = 119891119899

(120585119899

(119896) 119906 (119896) 119889 (119896))

119910 (119896) = 1205851

(119896)

(7)

where 120585119894(119896) = [120585

1(119896) 1205852(119896) 120585

119899(119896)]119879 are system states

119891119894are unknown nonlinear functions and 119906(119896) and 119910(119896) are

system input and output respectively while 119889(119896) denotes theexternal disturbance

Using a states prediction technique the pure-feedbackdiscrete-time systems in (7) are shown to be transformableto an 119899-step-ahead predictor as below

119910 (119896 + 119899) = 120601 (120585119899

(119896) 119906 (119896) 119889 (119896))

= 120601119904(120585119899

(119896) 119906 (119896)) + 119889119904(119896)

(8)

And by future output predictions the above system (8)can be further transformed into an input-output model forthe output feedback control

119910 (119896 + 119899)

= 1198651198991

119910 (119896 + 1) 119906 (119896) 119889 (119896) 119889 (119896 minus 119899 + 2)

(9)

This shows that the system presentation (9) in NARMAXform is a transformation of the pure-feedback system (8)After transformation both state feedback and output feed-back controls only need to employ a single NN in thecontroller design rather than a number of NNs in previousresearches

Based on the SISO pure-feedback system (8) and itstransformation NARMAX model in (9) the control designhas been further investigated by using high-order neuralnetwork to approximate unknown functions [93] In theabove mentioned control design availability is assumed forknowledge of control directions which are defined as ldquothesigns of control variable gains in affine systems or the signsof partial derivatives over control variables in nonaffinesystemsrdquo [93]

The paper overcame the problems of output-feedbackcontrol when the prior knowledge of the control directions isunknownThemain idea is to introduce a discrete Nussbaum

Discrete Dynamics in Nature and Society 7

gain to counter the lack of knowledge on control gain inadaptive NN control

A rigorous definition of discrete Nussbaum gain in [93] isgiven as follows

Remark 1 Consider a discrete nonlinear function 119873(119909(119896))

defined on a sequence 119909(119896) with 119909119904(119896) = sup(119909

1015840

119896) 119873(119909(119896))

is a discrete Nussbaum gain if and only if it satisfies thefollowing two properties

(i) if 119909119904(119896) increases without bound then

sup 1

119909119904(119896)

119878119873

(119909 (119896)) = +infin

inf 1

119909119904(119896)

119878119873

(119909 (119896)) = minusinfin

(10)

(ii) if 119909119904(119896) ge (119896)120575

1 then |119878

119873(119909(119896))| ge 120575

2with some

positive constants 1205751and 1205752 where 119878

119873(119909(119896)) is defined as

119878119873

(119909 (119896)) =

119896

sum

1198961015840=0

119873 (119909 (1198961015840)) Δ119909 (119896

1015840) (11)

with Δ119909(119896) = 119909(119896 + 1) minus 119909(119896) In addition neither the upperbounds nor the lower bounds of the control gains are requiredto be known

Unknown control directions problem for MIMOdiscrete-time nonlinear systems was solved by using adaptiveoutput feedback NN control in [94] The studies in [92 93]all studied the discrete NN control for SISO systems and[94] extended the study to a class of MIMO discrete-timesystems with each subsystem in the nonaffine pure-feedbackas follows

120585119895119894119895

(119896 + 1) = 119891119895119894119895

(1205851119894119895minus1198981198951

(119896) 1205852119894119895minus1198981198952

(119896)

120585119899119894119895minus119898119895119899

(119896) 120585119895 119894119895+1

(119896)) 119894119895

= 1 2 119899119895

minus 1

120585119895119899119895

(119896 + 1) = 119891119895119899119895

(Ξ (119896) 119906119895(119896) 119889

119895(119896))

119910119895(119896) = 120585

1198951(119896)

(12)

where 120585119895119894119895

(119896) = [1205851198951

(119896) 1205851198952

(119896) 120585119895119894119895

(119896)]119879 are vectors

of states variable of subsystem Ξ(119896) is vector of all statevariables 119891

119895119894119895are unknown nonlinear functions 119906

119895(119896) and

119910119895(119896) are system inputs and outputs respectively while 119889

119895(119896)

denote the external disturbanceFor nonlinearMIMOdiscrete-time higher order systems

a major drawback is that there are too many online-tunedadaptive parameters and large online computation burden In[95] an adaptive neural output feedback adaptive controllerwas designed for MIMO nonlinear discrete-time systemswith fewer adaptive parameters The output feedback adap-tive control for a class of nonlinear discrete-time systemswithunknown control directions was investigated in [96]

32 NN-Based Dynamic Programming Algorithm for Discrete-Time Systems In the past several decades optimal control for

nonlinear systems has been a key focus in the control fieldsWhile adaptive dynamics programming (ADP) is a veryuseful approach to solve the optimal control problems ADPis a reinforcement learning method to give solution to thedynamic programming utilizing function approximating thevalue function based on adaptive approach [97] P Werbos[98] classified the discrete-time ADP into a number ofschemes action-dependentHDP dual HDP (DHP) heuristicdynamic programming (HDP) and action-dependent dualHDP while NN has been widely used to solve the adaptivedynamic programming problems of discrete-time systems[97 99ndash104]

He and Jagannathan [99] showed the reinforcementlearning with NN-based controller for nonlinear discrete-time systems This work developed an adaptive-critic-basediscrete-time NN controller to deliver a desired tracking per-formance for nonlinear systems in the presence of actuatorconstraints The critic NN and action NN were designed toapproximate the strategic utility function while uniformlyultimate boundedness of the closed-loop tracking error wasproved by Lyapunov approach The reinforcement learningNN control for SISO discrete-time pure-feedback systemswas studied in [100] The controller design is based on thetransformed predictor and two NNs of control architectureOne critic NN to approximate the strategic utility function isshown as follows

By introducing a utility function 119901(119896) based on thetracking error 119890(119896) = 119910(119896) minus 119910

119889(119896)

119901 (119896) = 1198860 |119890 (119896)| (13)

where 119901(119896) isin 119877 1198860

isin 119877 is the positive design parameter Andthe strategic utility function 119876 isin 119877 is defined as

119876 (119896) = 119886119873

(119901 + 1) + 119886119899minus1

119901 (119896 + 2) + sdot sdot sdot + 119886119896+1

119901 (119873)

+ sdot sdot sdot

(14)

where 119886 isin 119877 0 lt 119886 lt 1 119873 is the horizon Then the criticNN is used to approximate the strategic utility function119876(119896)

= 119879

119888(119896) 119878119888(119911 (119896)) 119878

119888(119911 (119896)) isin 119877

119897119888 (15)

where 119882119879

119888(119896) isin 119877

119897119888 is the estimation of optimal NN 119882lowast

119888

weights The uniformly ultimate boundedness of closed-loopsignals is established via Lyapunov stability analysis

Al-Tamimi et al [97] proved the convergence of a valuefunction solution for nonlinear dynamical systems usingHDP algorithm and NNs were employed to approximatethe value and the control action at each iteration Liu et al[101] used theHDP algorithm to develop aNN-based optimalcontroller for unknown discrete-time nonlinear systemsThemain idea of this work is to introduce an iterative ADPalgorithm and a globalizedHDP technique into the controllerdesign And by using the cost function and control law theconvergence of the optimal control is guaranteed In [102]an iterative ADP algorithm for discrete-time systems wasdeveloped as well An optimal controller was designed toaddress the infinite-horizon discrete-time nonlinear systems

8 Discrete Dynamics in Nature and Society

with finite approximation errors It was shown thatwhen con-vergence conditions are satisfied iterative performance indexfunction can converge to the performance index functionsrsquolower bound while the performance index functions wereapproximated by NNs to compute optimal control policy

The ADP control in the unknown discrete-time Markovjump systems was investigated by Zhong et al [103] AndWang et al [105] also studied the ADP for discrete-timesystems to obtain the 120576-optimal control by using neuralnetworks In [106] iterative ADP algorithmwas used to studythe near-optimal control with control constraints in discrete-time systems An iterative ADP for discrete-time systemswas studied in [104] In this work the optimal controllerwas designed with control constraints while NN was used toidentify the unknown dynamical systems with stability proof

4 Conclusion

In this short survey despite the impossibility in identifyingor listing all the related contributions best efforts have beenmade to summarize the major achievements in the area ofdiscrete-time adaptive control withmodern techniques basedon FL and NNs which are also conventionally termed asimportant part of ldquosoft-computingrdquo approaches or ldquointelli-gent controlrdquo in the control communities In particular themarriage of ldquoadaptive controlrdquo and ldquointelligent controlrdquo hasbeen reviewed for the purpose of clarifying main contribu-tions and outlining some possible trends for the developmentof this promising area

Generally speaking adaptive control for discrete-timelinear systems was extensively explored in the past decadesdespite the fact that the complete closed-loop stability analy-sis for the so-called self-tuning regulator was ever regarded asa long-term challenging problem Later it became clear thatdiscrete-time adaptive control can be extended to nonlinearsystems with linear growth rate However general nonlinearsystems with parametric andor nonparametric uncertaintiesare usually very difficult to cope with which motivatedvarious solutions for different certain classes of nonlinearsystems

Among the various solutions intelligent control emergedas one important way to resolve the challenges caused by thenonlinearityThe key for usability of intelligent control basedon FL or NNs lies in their universal approximator property atthe cost of tuning rules of FL or updating parameters of NNsFL and NNs are motivated by different background hencetheir design philosophies look different however essentiallyspeaking both of them can be expressed as weighted sumof some kernel functions where the weights can be tunedto approximate arbitrary smooth or continuous nonlinearfunction The ideas borrowed from adaptive estimation andadaptive control provide such a way to adaptively tune theweight parameters in FL orNNs thus the new area of adaptiveFLC or NN control emerged with extensive studies

In a summary a brief review on FLC for discrete-timesystems is provided by highlighting the adaptive FLC robust-ness issue and stability issue while NN control for discrete-time systems is also reviewed with focuses on adaptive NNcontrol and NN-based dynamic programming We believe

this topic would promote increasing investigations in boththeories and applications And some emerging techniquessuch as deep learning and big data could also bemerged withthe intelligent control for discrete-time systems and give birthto brand new design approaches of intelligent control in thefuture

Conflict of Interests

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

Acknowledgment

This work is partially supported by the National Natu-ral Science Foundation of China (NSFC) under Grants61473120 and 61473038 Guangdong Provincial Natural Sci-ence Foundation 2014A030313266 and International Scienceand Technology Collaboration Grant 2015A050502017 andFundamental Research Funds for the Central Universitiesunder Grant 2015ZM065 The authors would like to thankDr Bin Xu of Northwestern Polytechnical University for hisconstructive comments during the preparation of this paper

References

[1] G Goodwin and K Sin Adaptive Filtering Prediction andControl Prentice-Hall Englewood Cliffs NJ USA 1984

[2] K Astrom and B Wittenmark Adaptive Control Addison-Wesley 1989

[3] M Krstic I Kanellakopoulos and P V Kokotovic NonlinearandAdaptive Control Design JohnWileyamp Sons NewYork NYUSA 1995

[4] G Tao and P Kokotovic Aaptive Control of Systems with Actu-ator and Sensor Non-Linearities John Wiley amp Sons HobokenNJ USA 1996

[5] S S Ge C C Hang T H Lee and T Zhang Stable AdaptiveNeural Network Control Kluwer Academic Publishers NorwellMass USA 2001

[6] C Yang Adaptive control and neural network control of nonlin-ear discrete-time systems [PhD thesis] National University ofSingapore 2009

[7] C Yang H Ma and M Fu ldquoAdaptive predictive control ofperiodic non-linear auto-regressive moving average systemsusing nearest-neighbour compensationrdquo IET Control Theory ampApplications vol 7 no 7 pp 936ndash951 2013

[8] C Yang L Zhai S S Ge T Chai andTH Lee ldquoAdaptivemodelreference control of a class ofMIMOdiscrete-time systems withcompensation of nonparametric uncertaintyrdquo in Proceedings ofthe American Control Conference pp 4111ndash4116 IEEE SeattleWash USA June 2008

[9] S-L Dai C Yang S S Ge and T H Lee ldquoRobust adaptiveoutput feedback control of a class of discrete-time nonlinearsystems with nonlinear uncertainties and unknown controldirectionsrdquo International Journal of Robust and Nonlinear Con-trol vol 23 no 13 pp 1472ndash1495 2013

[10] K J Astrom and B Wittenmark ldquoOn self tuning regulatorsrdquoAutomatica vol 9 no 2 pp 185ndash199 1973

Discrete Dynamics in Nature and Society 9

[11] L Ljung ldquoAnalysis of recursive stochastic algorithmsrdquo IEEETransactions on Automatic Control vol 22 no 4 pp 551ndash5751977

[12] G C Goodwin P J Ramadge and P E Caines ldquoDiscretetime multivariable adaptive controlrdquo IEEE Transactions onAutomatic Control vol 25 no 3 pp 449ndash456 1980

[13] L Guo and H F Chen ldquoThe Astrom-Wittenmark self-tuningregulator revisited and ELS-based adaptive trackersrdquo IEEETransactions on Automatic Control vol 36 no 7 pp 802ndash8121991

[14] L Guo Time-Varing Stochastic Systems Jilin Science and Tech-nology Press Changchun China 1993 (Chinese)

[15] H F Chen and L Guo Identification and Stochastic AdaptiveControl Birkhauser Boston Mass USA 1991

[16] F P Skantze A Kojic A-P Loh and A M AnnaswamyldquoAdaptive estimation of discrete-time systems with nonlinearparameterizationrdquo Automatica vol 36 no 12 pp 1879ndash18872000

[17] L Chen and K S Narendra ldquoNonlinear adaptive control usingneural networks and multiple modelsrdquo Automatica vol 37 no8 pp 1245ndash1255 2001

[18] L Guo and C Wei ldquoLS-based discrete-time adaptive nonlinearcontrol feasibility and limitationsrdquo Science in China Series ETechnological Sciences vol 39 no 3 pp 255ndash269 1996

[19] L L Xie and L Guo ldquoAdaptive control of discrete-timenonlinear systems with structural uncertaintiesrdquo in Lectures onSystems Control and Information vol 17 of AMSIP Studiesin Advanced Mathematics American Mathematical SocietyInternational Press Providence RI USA 2000

[20] J D Boskovic ldquoStable adaptive control of a class of first-order nonlinearly parameterized plantsrdquo IEEE Transactions onAutomatic Control vol 40 no 2 pp 347ndash350 1995

[21] A L Fradkov I V Miroshnik and V O Nikiforov Nonlinearand Adaptive Control of Complex Systems Mathematics andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 2004

[22] D Angeli and E Mosca ldquoAdaptive switching supervisorycontrol of nonlinear systems with no prior knowledge of noiseboundsrdquo Automatica vol 40 no 3 pp 449ndash457 2004

[23] H B Ma ldquoFinite-model adaptive control using an LS-likealgorithmrdquo International Journal of Adaptive Control and SignalProcessing vol 21 no 5 pp 391ndash414 2007

[24] H B Ma ldquoFinite-model adaptive control using WLS-likealgorithmrdquo Automatica vol 43 no 4 pp 677ndash684 2007

[25] H B Ma ldquoSeveral algorithms for finite-model adaptive controlpartial answers to finite-model adaptive control problemrdquoMathematics of Control Signals and Systems vol 20 no 3 pp271ndash303 2008

[26] S S Ge C C Hang and T Zhang ldquoA direct adaptive controllerfor dynamic systems with a class of nonlinear parameteriza-tionsrdquo Automatica vol 35 no 4 pp 741ndash747 1999

[27] C Y Li and L Guo ldquoOn feedback capability in a class of nonlin-early parameterized uncertain systemsrdquo IEEE Transactions onAutomatic Control vol 56 no 12 pp 2946ndash2951 2011

[28] H Ma K-Y Lum and S S Ge ldquoAdaptive control for a discrete-time first-order nonlinear system with both parametric andnon-parametric uncertaintiesrdquo in Proceedings of the 46th IEEEConference on Decision and Control (CDC rsquo07) pp 4839ndash4844IEEE New Orleans La USA December 2007

[29] L Guo ldquoExploring the capability and limits of the feedbackmechanismrdquo in Proceedings of the International Congress ofMathematicians (ICM rsquo02) Beijing China August 2002

[30] H-B Ma ldquoAn lsquoimpossibilityrsquo theorem on a class of high-orderdiscrete-time nonlinear control systemsrdquo Systems and ControlLetters vol 57 no 6 pp 497ndash504 2008

[31] I Kanellakopoulos P V Kokotovic and A S Morse ldquoSys-tematic design of adaptive controllers for feedback linearizablesystemsrdquo IEEE Transactions on Automatic Control vol 36 no11 pp 1241ndash1253 1991

[32] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[33] L-X Wang and J M Mendel ldquoFuzzy basis functions universalapproximation and orthogonal least-squares learningrdquo IEEETransactions on Neural Networks vol 3 no 5 pp 807ndash814 1992

[34] L-X Wang ldquoStable adaptive fuzzy control of nonlinear sys-temsrdquo IEEE Transactions on Fuzzy Systems vol 1 no 2 pp 146ndash155 1993

[35] W S McCulloch and W Pitts ldquoA logical calculus of the ideasimmanent in nervous activityrdquo The Bulletin of MathematicalBiophysics vol 5 pp 115ndash133 1943

[36] K Hornik M Stinchcombe and HWhite ldquoMultilayer feedfor-ward networks are universal approximatorsrdquo Neural Networksvol 2 no 5 pp 359ndash366 1989

[37] T Khanna Foundations of Neural Networks Addison-WesleyReading Mass USA 1990

[38] R M Sanner and J-J E Slotine ldquoGaussian networks for directadaptive controlrdquo IEEE Transactions on Neural Networks vol 3no 6 pp 837ndash863 1992

[39] Y J Liu Y J Fang andM A Bao-Ping ldquoSliding-data-window-driven Bayesian-Gaussian neural network and its application tomodeling of nonlinear systemrdquo Control Theory amp Applicationsvol 26 no 12 pp 1435ndash1438 2009

[40] D Wang and J Huang ldquoAdaptive neural network control fora class of uncertain nonlinear systems in pure-feedback formrdquoAutomatica vol 38 no 8 pp 1365ndash1372 2002

[41] Y Song and J W Grizzle ldquoAdaptive output-feedback control ofa class of discrete-time nonlinear systemsrdquo in Proceedings of theAmerican Control Conference pp 1359ndash1363 June 1993

[42] B-S Chen C-S Tseng and H-J Uang ldquoRobustness designof nonlinear dynamic systems via fuzzy linear controlrdquo IEEETransactions on Fuzzy Systems vol 7 no 5 pp 571ndash585 1999

[43] T Chai and S Tong ldquoFuzzy direct adaptive control for a class ofnonlinear systemsrdquo Fuzzy Sets and Systems vol 103 no 3 pp379ndash387 1999

[44] K Tanaka T Ikeda and H O Wang ldquoRobust stabilizationof a class of uncertain nonlinear systems via fuzzy controlquadratic stabilizability 119867

infin control theory and linear matrixinequalitiesrdquo IEEE Transactions on Fuzzy Systems vol 4 no 1pp 1ndash13 1996

[45] S Jagannathan M W Vandegrift and F L Lewis ldquoAdaptivefuzzy logic control of discrete-time dynamical systemsrdquo Auto-matica vol 36 no 2 pp 229ndash241 2000

[46] Y Jiang Z Liu C Chen and Y Zhang ldquoAdaptive robust fuzzycontrol for dual arm robot with unknown input deadzonenonlinearityrdquo Nonlinear Dynamics vol 81 no 3 pp 1301ndash13142015

[47] MWVandegrift F L Lewis S Jagannathan andK Liu ldquoAdap-tive fuzzy logic control of discrete-time dynamical systemsrdquo inProceedings of the IEEE International Symposium on IntelligentControl pp 395ndash401 IEEE Monterey Calif USA August 1995

[48] S Jagannathan ldquoAdaptive fuzzy logic control of feedbacklinearizable discrete-time dynamical systems under persistenceof excitationrdquo Automatica vol 34 no 11 pp 1295ndash1310 1998

10 Discrete Dynamics in Nature and Society

[49] R Qi and M A Brdys ldquoStable indirect adaptive control basedon discrete-time T-S fuzzy modelrdquo Fuzzy Sets and Systems vol159 no 8 pp 900ndash925 2008

[50] T-C Lin S-W Chang and C-H Hsu ldquoRobust adaptivefuzzy sliding mode control for a class of uncertain discrete-time nonlinear systemsrdquo International Journal of InnovativeComputing Information and Control vol 8 no 1 pp 347ndash3592012

[51] G Feng andG Chen ldquoAdaptive control of discrete-time chaoticsystems a fuzzy control approachrdquoChaos Solitons and Fractalsvol 23 no 2 pp 459ndash467 2005

[52] H J Lee J B Park and G Chen ldquoRobust fuzzy controlof nonlinear systems with parametric uncertaintiesrdquo IEEETransactions on Fuzzy Systems vol 9 no 2 pp 369ndash379 2001

[53] Y-Y Cao and PM Frank ldquoRobust119867infindisturbance attenuation

for a class of uncertain discrete-time fuzzy systemsrdquo IEEETransactions on Fuzzy Systems vol 8 no 4 pp 406ndash415 2000

[54] S Zhou G Feng J Lam and S Xu ldquoRobust 119867infin

controlfor discrete-time fuzzy systems via basis-dependent Lyapunovfunctionsrdquo Information Sciences vol 174 no 3-4 pp 197ndash2172005

[55] S Xu and J Lam ldquoRobust 119867infin

control for uncertain discrete-time-delay fuzzy systems via output feedback controllersrdquo IEEETransactions on Fuzzy Systems vol 13 no 1 pp 82ndash93 2005

[56] C-S Tseng and B-S Chen ldquoRobust fuzzy observer-basedfuzzy control design for nonlinear discrete-time systems withpersistent bounded disturbancesrdquo IEEE Transactions on FuzzySystems vol 17 no 3 pp 711ndash723 2009

[57] S Xu B Song J Lu and J Lam ldquoRobust stability of uncertaindiscrete-time singular fuzzy systemsrdquo Fuzzy Sets and Systemsvol 158 no 20 pp 2306ndash2316 2007

[58] Z-G Wu P Shi H Su and J Chu ldquoReliable 119867infin

controlfor discrete-time fuzzy systems with infinite-distributed delayrdquoIEEE Transactions on Fuzzy Systems vol 20 no 1 pp 22ndash312012

[59] G Feng and J Ma ldquoQuadratic stabilization of uncertaindiscrete-time fuzzy dynamic systemsrdquo IEEE Transactions onCircuits and Systems I Fundamental Theory and Applicationsvol 48 no 11 pp 1337ndash1344 2001

[60] A Kruszewski R Wang and T M Guerra ldquoNonquadratic sta-bilization conditions for a class of uncertain nonlinear discretetime TS fuzzy models a new approachrdquo IEEE Transactions onAutomatic Control vol 53 no 2 pp 606ndash611 2008

[61] W-J Wang Y-J Chen and C-H Sun ldquoRelaxed stabilizationcriteria for discrete-time T-S fuzzy control systems based ona switching fuzzy model and piecewise Lyapunov functionrdquoIEEE Transactions on Systems Man amp Cybernetics Part BCybernetics vol 37 no 3 pp 551ndash559 2007

[62] G Feng ldquoStability analysis of discrete-time fuzzy dynamicsystems based on piecewise Lyapunov functionsrdquo IEEE Trans-actions on Fuzzy Systems vol 12 no 1 pp 22ndash28 2004

[63] H Gao X Liu and J Lam ldquoStability analysis and stabilizationfor discrete-time fuzzy systems with time-varying delayrdquo IEEETransactions on Systems Man and Cybernetics Part B Cyber-netics vol 39 no 2 pp 306ndash317 2009

[64] L Wu X Su P Shi and J Qiu ldquoA new approach to stabilityanalysis and stabilization of discrete-time T-S fuzzy time-varying delay systemsrdquo IEEE Transactions on SystemsMan andCybernetics Part B Cybernetics vol 41 no 1 pp 273ndash286 2011

[65] X Su P Shi L Wu and Y-D Song ldquoA novel control design ondiscrete-time takagi-sugeno fuzzy systems with time-varying

delaysrdquo IEEE Transactions on Fuzzy Systems vol 21 no 4 pp655ndash671 2013

[66] X Su P Shi L Wu and Y-D Song ldquoA novel approach to filterdesign for T-S fuzzy discrete-time systems with time-varyingdelayrdquo IEEE Transactions on Fuzzy Systems vol 20 no 6 pp1114ndash1129 2012

[67] C-S Tseng ldquoModel reference output feedback fuzzy trackingcontrol design for nonlinear discrete-time systems with time-delayrdquo IEEE Transactions on Fuzzy Systems vol 14 no 1 pp58ndash70 2006

[68] X Su P Shi L Wu and S K Nguang ldquoInduced l2 filteringof fuzzy stochastic systems with time-varying delaysrdquo IEEETransactions on Cybernetics vol 43 no 4 pp 1257ndash1264 2013

[69] L Wu X Su P Shi and J Qiu ldquoModel approximation fordiscrete-time state-delay systems in the TS fuzzy frameworkrdquoIEEE Transactions on Fuzzy Systems vol 19 no 2 pp 366ndash3782011

[70] F L Lewis S Jagannathan and A Yesildirek Neural NetworkControl of Robot Manipulators and Nonlinear Systems Taylor ampFrancis London UK 1999

[71] A M Shaw and F J Doyle III ldquoMultivariable nonlinear controlapplications for a high purity distillation column using arecurrent dynamic neuron modelrdquo Journal of Process Controlvol 7 no 4 pp 255ndash268 1997

[72] K Najim Process Modeling and Control in Chemical Engineer-ing Marcel Dekker New York NY USA 1989

[73] B Xu D Wang F Sun and Z Shi ldquoDirect neural discretecontrol of hypersonic flight vehiclerdquo Nonlinear Dynamics vol70 no 1 pp 269ndash278 2012

[74] B Xu and Y Zhang ldquoNeural discrete back-stepping controlof hypersonic flight vehicle with equivalent prediction modelrdquoNeurocomputing vol 154 pp 337ndash346 2015

[75] B Xu F Sun H Liu and J Ren ldquoAdaptive Kriging controllerdesign for hypersonic flight vehicle via back-steppingrdquo IETControl Theory amp Applications vol 6 no 4 pp 487ndash497 2012

[76] B Xu ldquoRobust adaptive neural control of flexible hypersonicflight vehicle with dead-zone input nonlinearityrdquo NonlinearDynamics vol 80 no 3 pp 1509ndash1520 2015

[77] B Xu XHuangDWang and F Sun ldquoDynamic surface controlof constrained hypersonic flightmodels with parameter estima-tion and actuator compensationrdquo Asian Journal of Control vol16 no 1 pp 162ndash174 2014

[78] B Xu and Z Shi ldquoAn overview on flight dynamics and controlapproaches for hypersonic vehiclesrdquo Science China InformationSciences vol 58 no 7 pp 1ndash19 2015

[79] D E Rumelhart G E Hinton and R J Williams ldquoLearninginternal representations by error propagationrdquo in Parallel Dis-tributed Processing vol 1 pp 318ndash362 MIT Press 1986

[80] S Jagannathan and F L Lewis ldquoDiscrete-time neural netcontroller for a class of nonlinear dynamical systemsrdquo IEEETransactions on Automatic Control vol 41 no 11 pp 1693ndash16991996

[81] S Jagannathan and F L Lewis ldquoMultilayer discrete-timeneural-net controller with guaranteed performancerdquo IEEETransactions on Neural Network vol 7 no 1 pp 107ndash130 1996

[82] P He and S Jagannathan ldquoNeuro-controller for reducingcyclic variation in lean combustion spark ignition enginesrdquoAutomatica vol 41 no 7 pp 1133ndash1142 2005

[83] S S Ge G Y Li and T H Lee ldquoAdaptive NN controlfor a class of strict-feedback discrete-time nonlinear systemsrdquoAutomatica vol 39 no 5 pp 807ndash819 2003

Discrete Dynamics in Nature and Society 11

[84] S S Ge T H Lee G Y Li and J Zhang ldquoAdaptive NN controlfor a class of discrete-time non-linear systemsrdquo InternationalJournal of Control vol 76 no 4 pp 334ndash354 2003

[85] C J Goh ldquoModel reference control of non-linear systems viaimplicit function emulationrdquo International Journal of Controlvol 60 no 1 pp 91ndash115 1994

[86] C J Goh and T H Lee ldquoDirect adaptive control of nonlinearsystems via implicit function emulationrdquo Control Theory andAdvanced Technology vol 10 no 3 pp 539ndash552 1994

[87] A U Levin and K S Narendra ldquoControl of nonlinear dynami-cal systems using neural networksmdashpart II observability iden-tification and controlrdquo IEEE Transactions on Neural Networksvol 7 no 1 pp 30ndash42 1996

[88] S S Ge J Zhang and T H Lee ldquoAdaptive MNN control fora class of non-affine NARMAX systems with disturbancesrdquoSystems amp Control Letters vol 53 no 1 pp 1ndash12 2004

[89] S S Ge J Zhang and T H Lee ldquoAdaptive neural networkcontrol for a class of MIMO nonlinear systems with distur-bances in discrete-timerdquo IEEE Transactions on Systems Manand Cybernetics Part B Cybernetics vol 34 no 4 pp 1630ndash1645 2004

[90] J Zhang S S Ge and T H Lee ldquoOutput feedback control of aclass of discrete MIMO nonlinear systems with triangular forminputsrdquo IEEE Transactions onNeural Networks vol 16 no 6 pp1491ndash1503 2005

[91] F C Sun Z Sun and P-Y Woo ldquoStable neural-network-basedadaptive control for sampled-data nonlinear systemsrdquo IEEETransactions on Neural Networks vol 9 no 5 pp 956ndash9681998

[92] C Yang S S Ge C Xiang T Chai and T H Lee ldquoOutputfeedback NN control for two classes of discrete-time systemswith unknown control directions in a unified approachrdquo IEEETransactions on Neural Networks vol 19 no 11 pp 1873ndash18862008

[93] S S Ge C Yang and T H Lee ldquoAdaptive predictive controlusing neural network for a class of pure-feedback systems indiscrete timerdquo IEEE Transactions on Neural Networks vol 19no 9 pp 1599ndash1614 2008

[94] Y Li C Yang S S Ge and T H Lee ldquoAdaptive output feed-back NN control of a class of discrete-time MIMO nonlinearsystems with unknown control directionsrdquo IEEE Transactionson Systems Man and Cybernetics Part B Cybernetics vol 41no 2 pp 507ndash517 2011

[95] Y-J Liu C L P Chen G-X Wen and S Tong ldquoAdaptiveneural output feedback tracking control for a class of uncertaindiscrete-time nonlinear systemsrdquo IEEE Transactions on NeuralNetworks vol 22 no 7 pp 1162ndash1167 2011

[96] C Yang S S Ge and T H Lee ldquoOutput feedback adaptivecontrol of a class of nonlinear discrete-time systems withunknown control directionsrdquoAutomatica vol 45 no 1 pp 270ndash276 2009

[97] A Al-Tamimi F L Lewis and M Abu-Khalaf ldquoDiscrete-timenonlinear HJB solution using approximate dynamic program-ming convergence proofrdquo IEEE Transactions on Systems Manand Cybernetics Part B Cybernetics vol 38 no 4 pp 943ndash9492008

[98] P Werbos ldquoApproximate dynamic programming for real-timecontrol and neuralmodelingrdquo inHandbook of Intelligent ControlNeural Fuzzy amp Adaptive Approaches Van Nostrand Reinhold1992

[99] P He and S Jagannathan ldquoReinforcement learning neural-network-based controller for nonlinear discrete-time systems

with input constraintsrdquo IEEETransactions on SystemsMan andCybernetics Part B Cybernetics vol 37 no 2 pp 425ndash436 2007

[100] B Xu C Yang and Z Shi ldquoReinforcement learning outputfeedback NN control using deterministic learning techniquerdquoIEEE Transactions on Neural Networks and Learning Systemsvol 25 no 3 pp 635ndash641 2014

[101] D Liu DWang D Zhao QWei and N Jin ldquoNeural-network-based optimal control for a class of unknowndiscrete-time non-linear systems using globalized dual heuristic programmingrdquoIEEE Transactions on Automation Science and Engineering vol9 no 3 pp 628ndash634 2012

[102] D Liu and Q Wei ldquoFinite-approximation-error-based optimalcontrol approach for discrete-time nonlinear systemsrdquo IEEETransactions on Cybernetics vol 43 no 2 pp 779ndash789 2013

[103] X Zhong H He H Zhang and Z Wang ldquoOptimal control forunknown discrete-time nonlinear markov jump systems usingadaptive dynamic programmingrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 25 no 12 pp 2141ndash21552014

[104] D Liu D Wang and X Yang ldquoAn iterative adaptive dynamicprogramming algorithm for optimal control of unknowndiscrete-time nonlinear systemswith constrained inputsrdquo Infor-mation Sciences vol 220 pp 331ndash342 2013

[105] F-Y Wang N Jin D Liu and Q Wei ldquoAdaptive dynamicprogramming for finite-horizon optimal control of discrete-time nonlinear systems with 120576-error boundrdquo IEEE Transactionson Neural Networks vol 22 no 1 pp 24ndash36 2011

[106] H Zhang Y Luo and D Liu ldquoNeural-network-based near-optimal control for a class of discrete-time affine nonlinearsystems with control constraintsrdquo IEEE Transactions on NeuralNetworks vol 20 no 9 pp 1490ndash1503 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Mathematical PhysicsAdvances in

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Discrete Dynamics in Nature and Society

ultimate boundedness This method is significant in sensethat no persistence of excitation is used and no certaintyequivalence assumption is required The controller has alsobeen extended to a class of unknown feedback linearizablenonlinear dynamical systems under persistence of excitationin [48] in which rigorous stability proofs of discrete-timeadaptive FLC for feedback linearizable unknown nonlinearsystems were presented Indirect fuzzy control of uncertaindiscrete-time nonlinear system was shown by Qi and Brdys[49] This work presented an indirect adaptive FLC for theuncertainties in nonlinear plants and employed a Takagi-Sugeno (T-S) model to deal with the unknown dynamicsin input-output form A feedback linearization control lawwas designed by using structure states and parametersof the model Gradient descent algorithm and recursiveleast square estimation method were used to online updatethe modeling parameters The extension of sliding modecontrol (SMC) adaptive fuzzy SMC for a class of uncertaindiscrete-time nonlinear systems was investigated by Linet al [50] This work presented an adaptive interval type-2 fuzzy sliding mode controller for a class of unknownnonlinear discrete-time systems with training data disturbedby external disturbances It employed an adaptive intervaltype-2 fuzzy control scheme and SMC approach to controlthe plant tracking a reference trajectory and prevented bigchattering of the control effort While adaptive fuzzy controlhas also been applied in discrete-time chaotic systems In[51] Feng and Chen presented a novel adaptive controlalgorithm for discrete-time chaotic systems The basic ideais representing the chaotic system as a T-S fuzzy model anddesigning a local linear adaptive controller in each localregion Meanwhile a global adaptive controller on the entiredomainwas reconstructed and the stability of the closed-loopadaptive control system was proved

22 Robustness Issue in Discrete-Time Fuzzy Control Inpractice nonlinearities and uncertainties exist in almost allthe industrial plants while uncertainties in the modeling andcontrol of nonlinear systems are still one of themost challeng-ing problems in the control field The presence of nonlinear-ities and uncertainties brings difficulties to controller designIn order to conquer these problems a number of schemeshave been developed and among these robust control is oneof the most effective methods Robustness is regarded as oneof the most important requirements for a control system Inthe past two decades considerable attention has been paid tothe robustness in fuzzy model-based control of discrete-timesystems andmanymodification techniques were proposed toenhance the robustness of FLC for discrete-time systems [52ndash57]

The work of Lee et al [52] illustrated an approach ofrobust fuzzy control for nonlinear discrete-time systems Inthis work a systematic control structure was presented fornonlinear discrete-time T-S fuzzy systems with parametricuncertainties using the 119867

infincontrol design approach This

work also gave a novel solution to the robust stabilizationproblem of nonlinear systems by using basis-dependentLyapunov function

119867infindisturbance attenuation is one of the most important

requirements for a fuzzy control system Cao and Frank [53]have applied the 119867

infincontrol to address the robust stability

for a class of uncertain discrete-time fuzzy systems via linearmatrix inequality approach They studied both the robuststabilization and the 119867

infindisturbance attenuation while

they presented several sufficient conditions to ensure robuststability of the fuzzy models based on coupled linear matrixinequalities They also analyzed the robust 119867

infinperformance

of the fuzzy model-based discrete-time nonlinear systems byaddressing the robust 119867

infindisturbance attenuation problems

The robust stability of the models is achieved independentlyof the uncertainties

How to develop a robust 119867infin

controller for discrete-time systems using the basis-dependent Lyapunov functionwas also shown by Zhou et al [54] In their work anovel linear matrix inequalities characterization with 119867

infin

norm bound was presented for discrete-time fuzzy systemsWhile additional matrix variables were used to decouple theLyapunov function and system matrices as well as facilitatethe design approach of the controller this approach providessome sufficient results in the shape of strict linear matrixinequalities

Studied by Choi and Park the state feedback 119867infin

controlfor discrete-time systems was investigated by constructingLyapunov functions with fuzzy weights The fuzzy weightingLyapunov functions were designed with both current-timeand one-step-past information while designing controllerparameterized with linear matrix inequalities (PLMIs) Byselecting the structures of variables appropriately in thePLMIs a special case LMI formulation was obtained The119867infin

control with output feedback for discrete-time fuzzysystems has also been rigorously studied by Xu and Lam[55] while Wu et al [58] investigated the reliable 119867

infincontrol

for discrete-time systems with delays and stochastic actuatorfaults by representing the stochastic behavior with a discrete-time homogeneous Markov chain

Tseng andChen [56] used a fuzzy observer to estimate thepremise variables which depend on the state variables whilethe fuzzy observer was also used to address the nonlinear119897infin-gain control problem This work extended the 119897

infincontrol

from linear discrete-time systems to nonlinear discrete-time systems By using the T-S fuzzy model to representthe nonlinear discrete-time system an observer-based fuzzycontroller which minimized the upper bound of 119897

infin-gain and

attenuated the peak of perturbation was designed Xu et al[57] gave stability analysis for discrete-time singular fuzzysystems in the presence of time-varying uncertainties whilethey ensured the system to be regular causal and stable It hasbeen shown that for these systems robust stability conditioncan be obtained by giving a sufficient condition in terms of aset of linear matrix inequalities

23 Stability Issue in Discrete-Time Fuzzy Control Stabilityis one of the most important properties for a system whileFLC are used to address the stability problems for discrete-time systems as well The quadratic stability for uncertaindiscrete-time fuzzy dynamic systems was shown by Fengand Ma [59] This work gave some sufficient conditions of

Discrete Dynamics in Nature and Society 5

the quadratic stabilization for an uncertain fuzzy dynamicsystem Stability was achieved when the suitable Riccatiequation or a set of Riccati equations were solved Stabilizingfeedback control laws were also obtained by the developingalgorithms The nonquadratic stabilization conditions fornonlinear discrete-time fuzzy systems were also analyzedby Kruszewski et al [60] They considered discrete-timeuncertain nonlinear models in a T-S form and studied thestability through a nonquadratic Lyapunov function Thestabilization conditions were developed by considering theLyapunov function with a 119896-sample variation and extendedto uncertain T-S models Based on a switching fuzzy modeland piecewise Lyapunov function Wang et al [61] proposedtwo stabilization criteria for discrete-time T-S fuzzy systems

Zhou et al [54] used a basis-dependent Lyapunov-Krasovskii function to give stabilization analysis for discrete-time fuzzy systems A robust control design approach wasalso developed by using the Lyapunov-Krasovskii functionand facilitated by introducing additional instrumental matrixvariables Robust control problem for systemswith time delaywas concerned as well Feng [62] used a piecewise smoothLyapunov function to analyze the stability of discrete-time T-S fuzzy dynamic systems In this study stability of the systemguaranteed by constructing a piecewise Lyapunov functionMeanwhile the Lyapunov function was obtained using thelinear matrix inequalities methodThis work showed that thepiecewise quadratic Lyapunov functions based stability is lessconservative than the common quadratic Lyapunov functionbased stability

The stabilization for discrete-time T-S fuzzy systems withstate time-varying delay was shown by Gao et al [63] Intheir work a fuzzy Lyapunov function was constructed toimprove the delay-dependent stability condition By avoidingthe utilization of the bounding inequalities for the crossproducts between two vectors reduction of the conservatismof stability condition was achieved A delay-dependent stabi-lization algorithm was also developed for both state feedbackand observer-based output feedback cases using a paralleldistributed compensation scheme

24 Fuzzy Control for Discrete-Time Systemswith TimeDelaysIn practice time delays are the intrinsic nature of variousphysical systems such as communication hydraulic chem-ical processes and electronics The existence of time delayscould generally lower the system performance and even causeinstability and oscillation Hence remarkable attention hasbeen devoted to the analysis and synthesis of time delaynonlinear systems as well as the discrete-time fuzzy systems[55 64ndash69]

The stability and stabilization problems of systems time-varying delay were investigated by Wu et al [64] In theirwork the existing stability analysis approaches were dividedinto two types delay-dependent (including the delay size)and delay-independent (irrelevant to the delay size) Thestability of discrete-time T-S fuzzy systems with time-varyingstate delay was also investigated They developed a delay par-titioningmethod and employed a fuzzy Lyapunov-Krasovskiifunction to analyze the stability of delay-dependent time-varying state delay systems Through delay partitioning the

less conservative stability condition is obtained and solved byLMI optimization techniques

The work of Su et al [65] also gave a solution to thedynamic output feedback control problem for fuzzy sys-tems with time-varying delays They proposed a comparisonmodel by approximating the time-varying delay state whilethey gave a sufficient condition to ensure the asymptoti-cal stability by adopting the scaled small-gain theorem aswell as the delay partitioning method This method allowsdesigning desired dynamic output feedback controller byusing optimization techniques A filter design technique fordiscrete-time systems with time-varying delay has also beenpresented by Su et al [66] Both full-order and reduced-orderfilters were designed while approximators were employed toderive an input-output based sufficient condition Tseng [67]studied the time delay problem for a nonlinear discrete-timesystem based on model reference fuzzy tracking control Inthis work the T-S fuzzy model was applied to approximatea time delay discrete-time system while the fuzzy controllerwas designed to reduce the tracking error based on thisfuzzy model This method has advantage in that no feedbacklinearization or adaptive approaches are used in the controllerdesign

The robust 119867infin

control problem for uncertain discrete-time delay systems has been ivestigated by Xu and Lam [55]They studied the state-space T-S fuzzymodel with time delaysunder the assumption that the parameter uncertainties werenorm-boundedThe output feedback controller was designedwith full-order fuzzy dynamic to guarantee the closed-loopsystem is robust asymptotic stable as well as the admissibleuncertainties are norm bound constraint While solvabilityof this problem was addressed using a sufficient conditiongiven by the linear matrix inequalities The induced 119897

2filter

design of T-S fuzzy discrete-time stochastic systems withtime-varying delays has been investigated by Su et al [68]Wuet al [69] studied the 119867

infinmodel approximation for discrete-

time state delay fuzzy T-S systems

3 NN Control for Discrete-Time Systems

NN is of powerful computing ability and learning abilityto emulate various systems dynamics and is capable ofapproximating an unknown functionwith arbitrary accuracyNN is very successful in system modeling and controlby its capacity of universal approximate highly nonlinearand uncertain nonlinear and complex dynamic of systemsNNrsquos approximation ability has been shown by the Stone-Weierstrass theorem which states that a universal approxi-mator can approximate to an arbitrary degree of accuracyany real continuous function on a compact set Besides theuniversal approximation abilities NN also shows its excel-lence in parallel distributed processing abilities learningadaptation abilities natural fault tolerance and feasibilityfor hardware implementation These advantages make NNparticularly attractive and prospective for nonlinear controland modeling NN has been successfully applied to robotmanipulators control [70] distillation column control [71]chemical processes identification [72] flight control [73ndash75]

6 Discrete Dynamics in Nature and Society

and so forth To extend the NN technique from continuous-time to discrete-time many efforts have been made to studythe NN controlled discrete-time systems [76ndash78]

31 Adaptive NN Control for Discrete-Time Systems In theearly stage backpropagation (BP) algorithm [79] greatlyboosted the development of NN control It is noted that inthe early NN control design the control performances weredemonstrated through simulation or by particular exper-imental examples and consequently there were shortageof analytical analysis In addition an offline identificationprocedure was essential for achieving a stable NN controlsystem Thereafter the emergence of Lyapunov-based NNdesign makes it possible to use the available adaptive controltheories to rigorously guarantee stability robustness andconvergence of the closed-loop NN control systems Wecall the control design combining adaptive control theoriesand NN techniques as adaptive NN control It updates NNweights online and guarantees the stability of the closed-loop system Adaptive NN control design has been elegantlydeveloped for nonlinear systems with parametric uncertain-ties while many efforts have been dedicated in adaptive NNcontrol for nonlinear discrete-time systems [6]

For high-order affine nonlinear system in normal formadaptive NN controls using LPNN and MNN have beendeveloped in [80 81] using a filtered tracking error In [82]the controller is designed incorporating reinforcement learn-ing technique to improve control performance In this worka critic NN has been introduced to approximate the strategicutility function which is considered as the long-term systemperformance measure For discrete-time systems in strict-feedback form adaptive NN control has been developed viabackstepping design after system transformation [83] In [84]adaptive NN control has been investigated for discrete-timesystem in affine NARMAX form

In the above mentioned results the adaptive NN controldesigns were carried out through either feedback lineariza-tion or backstepping But these approaches are not applicableto nonaffine systems especially feedback linearization basedmethods which greatly depend on the affine appearance ofcontrol variables As a matter of fact adaptive NN controlmethods for nonaffine systems have been less studied incomparison with large amount of research work on affinenonlinear systems because of the difficulty of control designcaused by the nonaffine form of control input To overcomethe difficulty linearization based NN controls have beenput forward In [17] the nonaffine discrete-time system hasbeen divided into two parts one is linear and another isnonlinear and consequently a linear adaptive controller anda nonlinear adaptive NN controller have been designedwith a switching rule In [85] it directly utilized NN asemulator of the ldquoinverserdquo of the nonlinear discrete-timesystems Furthermore the study of discrete-time systems foradaptive NN control using implicit function to assert theexistence of an ideal inverse control was investigated in [86]Thereafter the implicit function based adaptive NN controlhas been widely studied in discrete-time form [87 88] Blocktriangular discrete-time systems with normal form subsys-tems have been studied in [80 81] For the block triangular

systems with strict-feedback subsystems state feedback andoutput feedback adaptive NN control have been developedin [89 90] by extending the systems transformation basedbackstepping technique proposed for SISO case in [83] In[91] adaptive NN control has been used for sampled-datanonlinear MIMO systems in general affine form based onlinearization The control scheme is an integration of an NNapproach and a variable structure method

An effort has been made in [92] to explore the adaptiveNN control of a class of nonaffine systems in discrete-timeThis work aimed to solve the nonaffine appearance andnoncausal problems of the following pure-feedback discrete-time system

120585119894(119896 + 1) = 119891

119894(120585119894 120585119894+ 1 (119896))

119894 = 1 2 119899 minus 1 119899 ge 2

120585119899

(119896 + 1) = 119891119899

(120585119899

(119896) 119906 (119896) 119889 (119896))

119910 (119896) = 1205851

(119896)

(7)

where 120585119894(119896) = [120585

1(119896) 1205852(119896) 120585

119899(119896)]119879 are system states

119891119894are unknown nonlinear functions and 119906(119896) and 119910(119896) are

system input and output respectively while 119889(119896) denotes theexternal disturbance

Using a states prediction technique the pure-feedbackdiscrete-time systems in (7) are shown to be transformableto an 119899-step-ahead predictor as below

119910 (119896 + 119899) = 120601 (120585119899

(119896) 119906 (119896) 119889 (119896))

= 120601119904(120585119899

(119896) 119906 (119896)) + 119889119904(119896)

(8)

And by future output predictions the above system (8)can be further transformed into an input-output model forthe output feedback control

119910 (119896 + 119899)

= 1198651198991

119910 (119896 + 1) 119906 (119896) 119889 (119896) 119889 (119896 minus 119899 + 2)

(9)

This shows that the system presentation (9) in NARMAXform is a transformation of the pure-feedback system (8)After transformation both state feedback and output feed-back controls only need to employ a single NN in thecontroller design rather than a number of NNs in previousresearches

Based on the SISO pure-feedback system (8) and itstransformation NARMAX model in (9) the control designhas been further investigated by using high-order neuralnetwork to approximate unknown functions [93] In theabove mentioned control design availability is assumed forknowledge of control directions which are defined as ldquothesigns of control variable gains in affine systems or the signsof partial derivatives over control variables in nonaffinesystemsrdquo [93]

The paper overcame the problems of output-feedbackcontrol when the prior knowledge of the control directions isunknownThemain idea is to introduce a discrete Nussbaum

Discrete Dynamics in Nature and Society 7

gain to counter the lack of knowledge on control gain inadaptive NN control

A rigorous definition of discrete Nussbaum gain in [93] isgiven as follows

Remark 1 Consider a discrete nonlinear function 119873(119909(119896))

defined on a sequence 119909(119896) with 119909119904(119896) = sup(119909

1015840

119896) 119873(119909(119896))

is a discrete Nussbaum gain if and only if it satisfies thefollowing two properties

(i) if 119909119904(119896) increases without bound then

sup 1

119909119904(119896)

119878119873

(119909 (119896)) = +infin

inf 1

119909119904(119896)

119878119873

(119909 (119896)) = minusinfin

(10)

(ii) if 119909119904(119896) ge (119896)120575

1 then |119878

119873(119909(119896))| ge 120575

2with some

positive constants 1205751and 1205752 where 119878

119873(119909(119896)) is defined as

119878119873

(119909 (119896)) =

119896

sum

1198961015840=0

119873 (119909 (1198961015840)) Δ119909 (119896

1015840) (11)

with Δ119909(119896) = 119909(119896 + 1) minus 119909(119896) In addition neither the upperbounds nor the lower bounds of the control gains are requiredto be known

Unknown control directions problem for MIMOdiscrete-time nonlinear systems was solved by using adaptiveoutput feedback NN control in [94] The studies in [92 93]all studied the discrete NN control for SISO systems and[94] extended the study to a class of MIMO discrete-timesystems with each subsystem in the nonaffine pure-feedbackas follows

120585119895119894119895

(119896 + 1) = 119891119895119894119895

(1205851119894119895minus1198981198951

(119896) 1205852119894119895minus1198981198952

(119896)

120585119899119894119895minus119898119895119899

(119896) 120585119895 119894119895+1

(119896)) 119894119895

= 1 2 119899119895

minus 1

120585119895119899119895

(119896 + 1) = 119891119895119899119895

(Ξ (119896) 119906119895(119896) 119889

119895(119896))

119910119895(119896) = 120585

1198951(119896)

(12)

where 120585119895119894119895

(119896) = [1205851198951

(119896) 1205851198952

(119896) 120585119895119894119895

(119896)]119879 are vectors

of states variable of subsystem Ξ(119896) is vector of all statevariables 119891

119895119894119895are unknown nonlinear functions 119906

119895(119896) and

119910119895(119896) are system inputs and outputs respectively while 119889

119895(119896)

denote the external disturbanceFor nonlinearMIMOdiscrete-time higher order systems

a major drawback is that there are too many online-tunedadaptive parameters and large online computation burden In[95] an adaptive neural output feedback adaptive controllerwas designed for MIMO nonlinear discrete-time systemswith fewer adaptive parameters The output feedback adap-tive control for a class of nonlinear discrete-time systemswithunknown control directions was investigated in [96]

32 NN-Based Dynamic Programming Algorithm for Discrete-Time Systems In the past several decades optimal control for

nonlinear systems has been a key focus in the control fieldsWhile adaptive dynamics programming (ADP) is a veryuseful approach to solve the optimal control problems ADPis a reinforcement learning method to give solution to thedynamic programming utilizing function approximating thevalue function based on adaptive approach [97] P Werbos[98] classified the discrete-time ADP into a number ofschemes action-dependentHDP dual HDP (DHP) heuristicdynamic programming (HDP) and action-dependent dualHDP while NN has been widely used to solve the adaptivedynamic programming problems of discrete-time systems[97 99ndash104]

He and Jagannathan [99] showed the reinforcementlearning with NN-based controller for nonlinear discrete-time systems This work developed an adaptive-critic-basediscrete-time NN controller to deliver a desired tracking per-formance for nonlinear systems in the presence of actuatorconstraints The critic NN and action NN were designed toapproximate the strategic utility function while uniformlyultimate boundedness of the closed-loop tracking error wasproved by Lyapunov approach The reinforcement learningNN control for SISO discrete-time pure-feedback systemswas studied in [100] The controller design is based on thetransformed predictor and two NNs of control architectureOne critic NN to approximate the strategic utility function isshown as follows

By introducing a utility function 119901(119896) based on thetracking error 119890(119896) = 119910(119896) minus 119910

119889(119896)

119901 (119896) = 1198860 |119890 (119896)| (13)

where 119901(119896) isin 119877 1198860

isin 119877 is the positive design parameter Andthe strategic utility function 119876 isin 119877 is defined as

119876 (119896) = 119886119873

(119901 + 1) + 119886119899minus1

119901 (119896 + 2) + sdot sdot sdot + 119886119896+1

119901 (119873)

+ sdot sdot sdot

(14)

where 119886 isin 119877 0 lt 119886 lt 1 119873 is the horizon Then the criticNN is used to approximate the strategic utility function119876(119896)

= 119879

119888(119896) 119878119888(119911 (119896)) 119878

119888(119911 (119896)) isin 119877

119897119888 (15)

where 119882119879

119888(119896) isin 119877

119897119888 is the estimation of optimal NN 119882lowast

119888

weights The uniformly ultimate boundedness of closed-loopsignals is established via Lyapunov stability analysis

Al-Tamimi et al [97] proved the convergence of a valuefunction solution for nonlinear dynamical systems usingHDP algorithm and NNs were employed to approximatethe value and the control action at each iteration Liu et al[101] used theHDP algorithm to develop aNN-based optimalcontroller for unknown discrete-time nonlinear systemsThemain idea of this work is to introduce an iterative ADPalgorithm and a globalizedHDP technique into the controllerdesign And by using the cost function and control law theconvergence of the optimal control is guaranteed In [102]an iterative ADP algorithm for discrete-time systems wasdeveloped as well An optimal controller was designed toaddress the infinite-horizon discrete-time nonlinear systems

8 Discrete Dynamics in Nature and Society

with finite approximation errors It was shown thatwhen con-vergence conditions are satisfied iterative performance indexfunction can converge to the performance index functionsrsquolower bound while the performance index functions wereapproximated by NNs to compute optimal control policy

The ADP control in the unknown discrete-time Markovjump systems was investigated by Zhong et al [103] AndWang et al [105] also studied the ADP for discrete-timesystems to obtain the 120576-optimal control by using neuralnetworks In [106] iterative ADP algorithmwas used to studythe near-optimal control with control constraints in discrete-time systems An iterative ADP for discrete-time systemswas studied in [104] In this work the optimal controllerwas designed with control constraints while NN was used toidentify the unknown dynamical systems with stability proof

4 Conclusion

In this short survey despite the impossibility in identifyingor listing all the related contributions best efforts have beenmade to summarize the major achievements in the area ofdiscrete-time adaptive control withmodern techniques basedon FL and NNs which are also conventionally termed asimportant part of ldquosoft-computingrdquo approaches or ldquointelli-gent controlrdquo in the control communities In particular themarriage of ldquoadaptive controlrdquo and ldquointelligent controlrdquo hasbeen reviewed for the purpose of clarifying main contribu-tions and outlining some possible trends for the developmentof this promising area

Generally speaking adaptive control for discrete-timelinear systems was extensively explored in the past decadesdespite the fact that the complete closed-loop stability analy-sis for the so-called self-tuning regulator was ever regarded asa long-term challenging problem Later it became clear thatdiscrete-time adaptive control can be extended to nonlinearsystems with linear growth rate However general nonlinearsystems with parametric andor nonparametric uncertaintiesare usually very difficult to cope with which motivatedvarious solutions for different certain classes of nonlinearsystems

Among the various solutions intelligent control emergedas one important way to resolve the challenges caused by thenonlinearityThe key for usability of intelligent control basedon FL or NNs lies in their universal approximator property atthe cost of tuning rules of FL or updating parameters of NNsFL and NNs are motivated by different background hencetheir design philosophies look different however essentiallyspeaking both of them can be expressed as weighted sumof some kernel functions where the weights can be tunedto approximate arbitrary smooth or continuous nonlinearfunction The ideas borrowed from adaptive estimation andadaptive control provide such a way to adaptively tune theweight parameters in FL orNNs thus the new area of adaptiveFLC or NN control emerged with extensive studies

In a summary a brief review on FLC for discrete-timesystems is provided by highlighting the adaptive FLC robust-ness issue and stability issue while NN control for discrete-time systems is also reviewed with focuses on adaptive NNcontrol and NN-based dynamic programming We believe

this topic would promote increasing investigations in boththeories and applications And some emerging techniquessuch as deep learning and big data could also bemerged withthe intelligent control for discrete-time systems and give birthto brand new design approaches of intelligent control in thefuture

Conflict of Interests

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

Acknowledgment

This work is partially supported by the National Natu-ral Science Foundation of China (NSFC) under Grants61473120 and 61473038 Guangdong Provincial Natural Sci-ence Foundation 2014A030313266 and International Scienceand Technology Collaboration Grant 2015A050502017 andFundamental Research Funds for the Central Universitiesunder Grant 2015ZM065 The authors would like to thankDr Bin Xu of Northwestern Polytechnical University for hisconstructive comments during the preparation of this paper

References

[1] G Goodwin and K Sin Adaptive Filtering Prediction andControl Prentice-Hall Englewood Cliffs NJ USA 1984

[2] K Astrom and B Wittenmark Adaptive Control Addison-Wesley 1989

[3] M Krstic I Kanellakopoulos and P V Kokotovic NonlinearandAdaptive Control Design JohnWileyamp Sons NewYork NYUSA 1995

[4] G Tao and P Kokotovic Aaptive Control of Systems with Actu-ator and Sensor Non-Linearities John Wiley amp Sons HobokenNJ USA 1996

[5] S S Ge C C Hang T H Lee and T Zhang Stable AdaptiveNeural Network Control Kluwer Academic Publishers NorwellMass USA 2001

[6] C Yang Adaptive control and neural network control of nonlin-ear discrete-time systems [PhD thesis] National University ofSingapore 2009

[7] C Yang H Ma and M Fu ldquoAdaptive predictive control ofperiodic non-linear auto-regressive moving average systemsusing nearest-neighbour compensationrdquo IET Control Theory ampApplications vol 7 no 7 pp 936ndash951 2013

[8] C Yang L Zhai S S Ge T Chai andTH Lee ldquoAdaptivemodelreference control of a class ofMIMOdiscrete-time systems withcompensation of nonparametric uncertaintyrdquo in Proceedings ofthe American Control Conference pp 4111ndash4116 IEEE SeattleWash USA June 2008

[9] S-L Dai C Yang S S Ge and T H Lee ldquoRobust adaptiveoutput feedback control of a class of discrete-time nonlinearsystems with nonlinear uncertainties and unknown controldirectionsrdquo International Journal of Robust and Nonlinear Con-trol vol 23 no 13 pp 1472ndash1495 2013

[10] K J Astrom and B Wittenmark ldquoOn self tuning regulatorsrdquoAutomatica vol 9 no 2 pp 185ndash199 1973

Discrete Dynamics in Nature and Society 9

[11] L Ljung ldquoAnalysis of recursive stochastic algorithmsrdquo IEEETransactions on Automatic Control vol 22 no 4 pp 551ndash5751977

[12] G C Goodwin P J Ramadge and P E Caines ldquoDiscretetime multivariable adaptive controlrdquo IEEE Transactions onAutomatic Control vol 25 no 3 pp 449ndash456 1980

[13] L Guo and H F Chen ldquoThe Astrom-Wittenmark self-tuningregulator revisited and ELS-based adaptive trackersrdquo IEEETransactions on Automatic Control vol 36 no 7 pp 802ndash8121991

[14] L Guo Time-Varing Stochastic Systems Jilin Science and Tech-nology Press Changchun China 1993 (Chinese)

[15] H F Chen and L Guo Identification and Stochastic AdaptiveControl Birkhauser Boston Mass USA 1991

[16] F P Skantze A Kojic A-P Loh and A M AnnaswamyldquoAdaptive estimation of discrete-time systems with nonlinearparameterizationrdquo Automatica vol 36 no 12 pp 1879ndash18872000

[17] L Chen and K S Narendra ldquoNonlinear adaptive control usingneural networks and multiple modelsrdquo Automatica vol 37 no8 pp 1245ndash1255 2001

[18] L Guo and C Wei ldquoLS-based discrete-time adaptive nonlinearcontrol feasibility and limitationsrdquo Science in China Series ETechnological Sciences vol 39 no 3 pp 255ndash269 1996

[19] L L Xie and L Guo ldquoAdaptive control of discrete-timenonlinear systems with structural uncertaintiesrdquo in Lectures onSystems Control and Information vol 17 of AMSIP Studiesin Advanced Mathematics American Mathematical SocietyInternational Press Providence RI USA 2000

[20] J D Boskovic ldquoStable adaptive control of a class of first-order nonlinearly parameterized plantsrdquo IEEE Transactions onAutomatic Control vol 40 no 2 pp 347ndash350 1995

[21] A L Fradkov I V Miroshnik and V O Nikiforov Nonlinearand Adaptive Control of Complex Systems Mathematics andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 2004

[22] D Angeli and E Mosca ldquoAdaptive switching supervisorycontrol of nonlinear systems with no prior knowledge of noiseboundsrdquo Automatica vol 40 no 3 pp 449ndash457 2004

[23] H B Ma ldquoFinite-model adaptive control using an LS-likealgorithmrdquo International Journal of Adaptive Control and SignalProcessing vol 21 no 5 pp 391ndash414 2007

[24] H B Ma ldquoFinite-model adaptive control using WLS-likealgorithmrdquo Automatica vol 43 no 4 pp 677ndash684 2007

[25] H B Ma ldquoSeveral algorithms for finite-model adaptive controlpartial answers to finite-model adaptive control problemrdquoMathematics of Control Signals and Systems vol 20 no 3 pp271ndash303 2008

[26] S S Ge C C Hang and T Zhang ldquoA direct adaptive controllerfor dynamic systems with a class of nonlinear parameteriza-tionsrdquo Automatica vol 35 no 4 pp 741ndash747 1999

[27] C Y Li and L Guo ldquoOn feedback capability in a class of nonlin-early parameterized uncertain systemsrdquo IEEE Transactions onAutomatic Control vol 56 no 12 pp 2946ndash2951 2011

[28] H Ma K-Y Lum and S S Ge ldquoAdaptive control for a discrete-time first-order nonlinear system with both parametric andnon-parametric uncertaintiesrdquo in Proceedings of the 46th IEEEConference on Decision and Control (CDC rsquo07) pp 4839ndash4844IEEE New Orleans La USA December 2007

[29] L Guo ldquoExploring the capability and limits of the feedbackmechanismrdquo in Proceedings of the International Congress ofMathematicians (ICM rsquo02) Beijing China August 2002

[30] H-B Ma ldquoAn lsquoimpossibilityrsquo theorem on a class of high-orderdiscrete-time nonlinear control systemsrdquo Systems and ControlLetters vol 57 no 6 pp 497ndash504 2008

[31] I Kanellakopoulos P V Kokotovic and A S Morse ldquoSys-tematic design of adaptive controllers for feedback linearizablesystemsrdquo IEEE Transactions on Automatic Control vol 36 no11 pp 1241ndash1253 1991

[32] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[33] L-X Wang and J M Mendel ldquoFuzzy basis functions universalapproximation and orthogonal least-squares learningrdquo IEEETransactions on Neural Networks vol 3 no 5 pp 807ndash814 1992

[34] L-X Wang ldquoStable adaptive fuzzy control of nonlinear sys-temsrdquo IEEE Transactions on Fuzzy Systems vol 1 no 2 pp 146ndash155 1993

[35] W S McCulloch and W Pitts ldquoA logical calculus of the ideasimmanent in nervous activityrdquo The Bulletin of MathematicalBiophysics vol 5 pp 115ndash133 1943

[36] K Hornik M Stinchcombe and HWhite ldquoMultilayer feedfor-ward networks are universal approximatorsrdquo Neural Networksvol 2 no 5 pp 359ndash366 1989

[37] T Khanna Foundations of Neural Networks Addison-WesleyReading Mass USA 1990

[38] R M Sanner and J-J E Slotine ldquoGaussian networks for directadaptive controlrdquo IEEE Transactions on Neural Networks vol 3no 6 pp 837ndash863 1992

[39] Y J Liu Y J Fang andM A Bao-Ping ldquoSliding-data-window-driven Bayesian-Gaussian neural network and its application tomodeling of nonlinear systemrdquo Control Theory amp Applicationsvol 26 no 12 pp 1435ndash1438 2009

[40] D Wang and J Huang ldquoAdaptive neural network control fora class of uncertain nonlinear systems in pure-feedback formrdquoAutomatica vol 38 no 8 pp 1365ndash1372 2002

[41] Y Song and J W Grizzle ldquoAdaptive output-feedback control ofa class of discrete-time nonlinear systemsrdquo in Proceedings of theAmerican Control Conference pp 1359ndash1363 June 1993

[42] B-S Chen C-S Tseng and H-J Uang ldquoRobustness designof nonlinear dynamic systems via fuzzy linear controlrdquo IEEETransactions on Fuzzy Systems vol 7 no 5 pp 571ndash585 1999

[43] T Chai and S Tong ldquoFuzzy direct adaptive control for a class ofnonlinear systemsrdquo Fuzzy Sets and Systems vol 103 no 3 pp379ndash387 1999

[44] K Tanaka T Ikeda and H O Wang ldquoRobust stabilizationof a class of uncertain nonlinear systems via fuzzy controlquadratic stabilizability 119867

infin control theory and linear matrixinequalitiesrdquo IEEE Transactions on Fuzzy Systems vol 4 no 1pp 1ndash13 1996

[45] S Jagannathan M W Vandegrift and F L Lewis ldquoAdaptivefuzzy logic control of discrete-time dynamical systemsrdquo Auto-matica vol 36 no 2 pp 229ndash241 2000

[46] Y Jiang Z Liu C Chen and Y Zhang ldquoAdaptive robust fuzzycontrol for dual arm robot with unknown input deadzonenonlinearityrdquo Nonlinear Dynamics vol 81 no 3 pp 1301ndash13142015

[47] MWVandegrift F L Lewis S Jagannathan andK Liu ldquoAdap-tive fuzzy logic control of discrete-time dynamical systemsrdquo inProceedings of the IEEE International Symposium on IntelligentControl pp 395ndash401 IEEE Monterey Calif USA August 1995

[48] S Jagannathan ldquoAdaptive fuzzy logic control of feedbacklinearizable discrete-time dynamical systems under persistenceof excitationrdquo Automatica vol 34 no 11 pp 1295ndash1310 1998

10 Discrete Dynamics in Nature and Society

[49] R Qi and M A Brdys ldquoStable indirect adaptive control basedon discrete-time T-S fuzzy modelrdquo Fuzzy Sets and Systems vol159 no 8 pp 900ndash925 2008

[50] T-C Lin S-W Chang and C-H Hsu ldquoRobust adaptivefuzzy sliding mode control for a class of uncertain discrete-time nonlinear systemsrdquo International Journal of InnovativeComputing Information and Control vol 8 no 1 pp 347ndash3592012

[51] G Feng andG Chen ldquoAdaptive control of discrete-time chaoticsystems a fuzzy control approachrdquoChaos Solitons and Fractalsvol 23 no 2 pp 459ndash467 2005

[52] H J Lee J B Park and G Chen ldquoRobust fuzzy controlof nonlinear systems with parametric uncertaintiesrdquo IEEETransactions on Fuzzy Systems vol 9 no 2 pp 369ndash379 2001

[53] Y-Y Cao and PM Frank ldquoRobust119867infindisturbance attenuation

for a class of uncertain discrete-time fuzzy systemsrdquo IEEETransactions on Fuzzy Systems vol 8 no 4 pp 406ndash415 2000

[54] S Zhou G Feng J Lam and S Xu ldquoRobust 119867infin

controlfor discrete-time fuzzy systems via basis-dependent Lyapunovfunctionsrdquo Information Sciences vol 174 no 3-4 pp 197ndash2172005

[55] S Xu and J Lam ldquoRobust 119867infin

control for uncertain discrete-time-delay fuzzy systems via output feedback controllersrdquo IEEETransactions on Fuzzy Systems vol 13 no 1 pp 82ndash93 2005

[56] C-S Tseng and B-S Chen ldquoRobust fuzzy observer-basedfuzzy control design for nonlinear discrete-time systems withpersistent bounded disturbancesrdquo IEEE Transactions on FuzzySystems vol 17 no 3 pp 711ndash723 2009

[57] S Xu B Song J Lu and J Lam ldquoRobust stability of uncertaindiscrete-time singular fuzzy systemsrdquo Fuzzy Sets and Systemsvol 158 no 20 pp 2306ndash2316 2007

[58] Z-G Wu P Shi H Su and J Chu ldquoReliable 119867infin

controlfor discrete-time fuzzy systems with infinite-distributed delayrdquoIEEE Transactions on Fuzzy Systems vol 20 no 1 pp 22ndash312012

[59] G Feng and J Ma ldquoQuadratic stabilization of uncertaindiscrete-time fuzzy dynamic systemsrdquo IEEE Transactions onCircuits and Systems I Fundamental Theory and Applicationsvol 48 no 11 pp 1337ndash1344 2001

[60] A Kruszewski R Wang and T M Guerra ldquoNonquadratic sta-bilization conditions for a class of uncertain nonlinear discretetime TS fuzzy models a new approachrdquo IEEE Transactions onAutomatic Control vol 53 no 2 pp 606ndash611 2008

[61] W-J Wang Y-J Chen and C-H Sun ldquoRelaxed stabilizationcriteria for discrete-time T-S fuzzy control systems based ona switching fuzzy model and piecewise Lyapunov functionrdquoIEEE Transactions on Systems Man amp Cybernetics Part BCybernetics vol 37 no 3 pp 551ndash559 2007

[62] G Feng ldquoStability analysis of discrete-time fuzzy dynamicsystems based on piecewise Lyapunov functionsrdquo IEEE Trans-actions on Fuzzy Systems vol 12 no 1 pp 22ndash28 2004

[63] H Gao X Liu and J Lam ldquoStability analysis and stabilizationfor discrete-time fuzzy systems with time-varying delayrdquo IEEETransactions on Systems Man and Cybernetics Part B Cyber-netics vol 39 no 2 pp 306ndash317 2009

[64] L Wu X Su P Shi and J Qiu ldquoA new approach to stabilityanalysis and stabilization of discrete-time T-S fuzzy time-varying delay systemsrdquo IEEE Transactions on SystemsMan andCybernetics Part B Cybernetics vol 41 no 1 pp 273ndash286 2011

[65] X Su P Shi L Wu and Y-D Song ldquoA novel control design ondiscrete-time takagi-sugeno fuzzy systems with time-varying

delaysrdquo IEEE Transactions on Fuzzy Systems vol 21 no 4 pp655ndash671 2013

[66] X Su P Shi L Wu and Y-D Song ldquoA novel approach to filterdesign for T-S fuzzy discrete-time systems with time-varyingdelayrdquo IEEE Transactions on Fuzzy Systems vol 20 no 6 pp1114ndash1129 2012

[67] C-S Tseng ldquoModel reference output feedback fuzzy trackingcontrol design for nonlinear discrete-time systems with time-delayrdquo IEEE Transactions on Fuzzy Systems vol 14 no 1 pp58ndash70 2006

[68] X Su P Shi L Wu and S K Nguang ldquoInduced l2 filteringof fuzzy stochastic systems with time-varying delaysrdquo IEEETransactions on Cybernetics vol 43 no 4 pp 1257ndash1264 2013

[69] L Wu X Su P Shi and J Qiu ldquoModel approximation fordiscrete-time state-delay systems in the TS fuzzy frameworkrdquoIEEE Transactions on Fuzzy Systems vol 19 no 2 pp 366ndash3782011

[70] F L Lewis S Jagannathan and A Yesildirek Neural NetworkControl of Robot Manipulators and Nonlinear Systems Taylor ampFrancis London UK 1999

[71] A M Shaw and F J Doyle III ldquoMultivariable nonlinear controlapplications for a high purity distillation column using arecurrent dynamic neuron modelrdquo Journal of Process Controlvol 7 no 4 pp 255ndash268 1997

[72] K Najim Process Modeling and Control in Chemical Engineer-ing Marcel Dekker New York NY USA 1989

[73] B Xu D Wang F Sun and Z Shi ldquoDirect neural discretecontrol of hypersonic flight vehiclerdquo Nonlinear Dynamics vol70 no 1 pp 269ndash278 2012

[74] B Xu and Y Zhang ldquoNeural discrete back-stepping controlof hypersonic flight vehicle with equivalent prediction modelrdquoNeurocomputing vol 154 pp 337ndash346 2015

[75] B Xu F Sun H Liu and J Ren ldquoAdaptive Kriging controllerdesign for hypersonic flight vehicle via back-steppingrdquo IETControl Theory amp Applications vol 6 no 4 pp 487ndash497 2012

[76] B Xu ldquoRobust adaptive neural control of flexible hypersonicflight vehicle with dead-zone input nonlinearityrdquo NonlinearDynamics vol 80 no 3 pp 1509ndash1520 2015

[77] B Xu XHuangDWang and F Sun ldquoDynamic surface controlof constrained hypersonic flightmodels with parameter estima-tion and actuator compensationrdquo Asian Journal of Control vol16 no 1 pp 162ndash174 2014

[78] B Xu and Z Shi ldquoAn overview on flight dynamics and controlapproaches for hypersonic vehiclesrdquo Science China InformationSciences vol 58 no 7 pp 1ndash19 2015

[79] D E Rumelhart G E Hinton and R J Williams ldquoLearninginternal representations by error propagationrdquo in Parallel Dis-tributed Processing vol 1 pp 318ndash362 MIT Press 1986

[80] S Jagannathan and F L Lewis ldquoDiscrete-time neural netcontroller for a class of nonlinear dynamical systemsrdquo IEEETransactions on Automatic Control vol 41 no 11 pp 1693ndash16991996

[81] S Jagannathan and F L Lewis ldquoMultilayer discrete-timeneural-net controller with guaranteed performancerdquo IEEETransactions on Neural Network vol 7 no 1 pp 107ndash130 1996

[82] P He and S Jagannathan ldquoNeuro-controller for reducingcyclic variation in lean combustion spark ignition enginesrdquoAutomatica vol 41 no 7 pp 1133ndash1142 2005

[83] S S Ge G Y Li and T H Lee ldquoAdaptive NN controlfor a class of strict-feedback discrete-time nonlinear systemsrdquoAutomatica vol 39 no 5 pp 807ndash819 2003

Discrete Dynamics in Nature and Society 11

[84] S S Ge T H Lee G Y Li and J Zhang ldquoAdaptive NN controlfor a class of discrete-time non-linear systemsrdquo InternationalJournal of Control vol 76 no 4 pp 334ndash354 2003

[85] C J Goh ldquoModel reference control of non-linear systems viaimplicit function emulationrdquo International Journal of Controlvol 60 no 1 pp 91ndash115 1994

[86] C J Goh and T H Lee ldquoDirect adaptive control of nonlinearsystems via implicit function emulationrdquo Control Theory andAdvanced Technology vol 10 no 3 pp 539ndash552 1994

[87] A U Levin and K S Narendra ldquoControl of nonlinear dynami-cal systems using neural networksmdashpart II observability iden-tification and controlrdquo IEEE Transactions on Neural Networksvol 7 no 1 pp 30ndash42 1996

[88] S S Ge J Zhang and T H Lee ldquoAdaptive MNN control fora class of non-affine NARMAX systems with disturbancesrdquoSystems amp Control Letters vol 53 no 1 pp 1ndash12 2004

[89] S S Ge J Zhang and T H Lee ldquoAdaptive neural networkcontrol for a class of MIMO nonlinear systems with distur-bances in discrete-timerdquo IEEE Transactions on Systems Manand Cybernetics Part B Cybernetics vol 34 no 4 pp 1630ndash1645 2004

[90] J Zhang S S Ge and T H Lee ldquoOutput feedback control of aclass of discrete MIMO nonlinear systems with triangular forminputsrdquo IEEE Transactions onNeural Networks vol 16 no 6 pp1491ndash1503 2005

[91] F C Sun Z Sun and P-Y Woo ldquoStable neural-network-basedadaptive control for sampled-data nonlinear systemsrdquo IEEETransactions on Neural Networks vol 9 no 5 pp 956ndash9681998

[92] C Yang S S Ge C Xiang T Chai and T H Lee ldquoOutputfeedback NN control for two classes of discrete-time systemswith unknown control directions in a unified approachrdquo IEEETransactions on Neural Networks vol 19 no 11 pp 1873ndash18862008

[93] S S Ge C Yang and T H Lee ldquoAdaptive predictive controlusing neural network for a class of pure-feedback systems indiscrete timerdquo IEEE Transactions on Neural Networks vol 19no 9 pp 1599ndash1614 2008

[94] Y Li C Yang S S Ge and T H Lee ldquoAdaptive output feed-back NN control of a class of discrete-time MIMO nonlinearsystems with unknown control directionsrdquo IEEE Transactionson Systems Man and Cybernetics Part B Cybernetics vol 41no 2 pp 507ndash517 2011

[95] Y-J Liu C L P Chen G-X Wen and S Tong ldquoAdaptiveneural output feedback tracking control for a class of uncertaindiscrete-time nonlinear systemsrdquo IEEE Transactions on NeuralNetworks vol 22 no 7 pp 1162ndash1167 2011

[96] C Yang S S Ge and T H Lee ldquoOutput feedback adaptivecontrol of a class of nonlinear discrete-time systems withunknown control directionsrdquoAutomatica vol 45 no 1 pp 270ndash276 2009

[97] A Al-Tamimi F L Lewis and M Abu-Khalaf ldquoDiscrete-timenonlinear HJB solution using approximate dynamic program-ming convergence proofrdquo IEEE Transactions on Systems Manand Cybernetics Part B Cybernetics vol 38 no 4 pp 943ndash9492008

[98] P Werbos ldquoApproximate dynamic programming for real-timecontrol and neuralmodelingrdquo inHandbook of Intelligent ControlNeural Fuzzy amp Adaptive Approaches Van Nostrand Reinhold1992

[99] P He and S Jagannathan ldquoReinforcement learning neural-network-based controller for nonlinear discrete-time systems

with input constraintsrdquo IEEETransactions on SystemsMan andCybernetics Part B Cybernetics vol 37 no 2 pp 425ndash436 2007

[100] B Xu C Yang and Z Shi ldquoReinforcement learning outputfeedback NN control using deterministic learning techniquerdquoIEEE Transactions on Neural Networks and Learning Systemsvol 25 no 3 pp 635ndash641 2014

[101] D Liu DWang D Zhao QWei and N Jin ldquoNeural-network-based optimal control for a class of unknowndiscrete-time non-linear systems using globalized dual heuristic programmingrdquoIEEE Transactions on Automation Science and Engineering vol9 no 3 pp 628ndash634 2012

[102] D Liu and Q Wei ldquoFinite-approximation-error-based optimalcontrol approach for discrete-time nonlinear systemsrdquo IEEETransactions on Cybernetics vol 43 no 2 pp 779ndash789 2013

[103] X Zhong H He H Zhang and Z Wang ldquoOptimal control forunknown discrete-time nonlinear markov jump systems usingadaptive dynamic programmingrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 25 no 12 pp 2141ndash21552014

[104] D Liu D Wang and X Yang ldquoAn iterative adaptive dynamicprogramming algorithm for optimal control of unknowndiscrete-time nonlinear systemswith constrained inputsrdquo Infor-mation Sciences vol 220 pp 331ndash342 2013

[105] F-Y Wang N Jin D Liu and Q Wei ldquoAdaptive dynamicprogramming for finite-horizon optimal control of discrete-time nonlinear systems with 120576-error boundrdquo IEEE Transactionson Neural Networks vol 22 no 1 pp 24ndash36 2011

[106] H Zhang Y Luo and D Liu ldquoNeural-network-based near-optimal control for a class of discrete-time affine nonlinearsystems with control constraintsrdquo IEEE Transactions on NeuralNetworks vol 20 no 9 pp 1490ndash1503 2009

Submit your manuscripts athttpwwwhindawicom

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Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 5

the quadratic stabilization for an uncertain fuzzy dynamicsystem Stability was achieved when the suitable Riccatiequation or a set of Riccati equations were solved Stabilizingfeedback control laws were also obtained by the developingalgorithms The nonquadratic stabilization conditions fornonlinear discrete-time fuzzy systems were also analyzedby Kruszewski et al [60] They considered discrete-timeuncertain nonlinear models in a T-S form and studied thestability through a nonquadratic Lyapunov function Thestabilization conditions were developed by considering theLyapunov function with a 119896-sample variation and extendedto uncertain T-S models Based on a switching fuzzy modeland piecewise Lyapunov function Wang et al [61] proposedtwo stabilization criteria for discrete-time T-S fuzzy systems

Zhou et al [54] used a basis-dependent Lyapunov-Krasovskii function to give stabilization analysis for discrete-time fuzzy systems A robust control design approach wasalso developed by using the Lyapunov-Krasovskii functionand facilitated by introducing additional instrumental matrixvariables Robust control problem for systemswith time delaywas concerned as well Feng [62] used a piecewise smoothLyapunov function to analyze the stability of discrete-time T-S fuzzy dynamic systems In this study stability of the systemguaranteed by constructing a piecewise Lyapunov functionMeanwhile the Lyapunov function was obtained using thelinear matrix inequalities methodThis work showed that thepiecewise quadratic Lyapunov functions based stability is lessconservative than the common quadratic Lyapunov functionbased stability

The stabilization for discrete-time T-S fuzzy systems withstate time-varying delay was shown by Gao et al [63] Intheir work a fuzzy Lyapunov function was constructed toimprove the delay-dependent stability condition By avoidingthe utilization of the bounding inequalities for the crossproducts between two vectors reduction of the conservatismof stability condition was achieved A delay-dependent stabi-lization algorithm was also developed for both state feedbackand observer-based output feedback cases using a paralleldistributed compensation scheme

24 Fuzzy Control for Discrete-Time Systemswith TimeDelaysIn practice time delays are the intrinsic nature of variousphysical systems such as communication hydraulic chem-ical processes and electronics The existence of time delayscould generally lower the system performance and even causeinstability and oscillation Hence remarkable attention hasbeen devoted to the analysis and synthesis of time delaynonlinear systems as well as the discrete-time fuzzy systems[55 64ndash69]

The stability and stabilization problems of systems time-varying delay were investigated by Wu et al [64] In theirwork the existing stability analysis approaches were dividedinto two types delay-dependent (including the delay size)and delay-independent (irrelevant to the delay size) Thestability of discrete-time T-S fuzzy systems with time-varyingstate delay was also investigated They developed a delay par-titioningmethod and employed a fuzzy Lyapunov-Krasovskiifunction to analyze the stability of delay-dependent time-varying state delay systems Through delay partitioning the

less conservative stability condition is obtained and solved byLMI optimization techniques

The work of Su et al [65] also gave a solution to thedynamic output feedback control problem for fuzzy sys-tems with time-varying delays They proposed a comparisonmodel by approximating the time-varying delay state whilethey gave a sufficient condition to ensure the asymptoti-cal stability by adopting the scaled small-gain theorem aswell as the delay partitioning method This method allowsdesigning desired dynamic output feedback controller byusing optimization techniques A filter design technique fordiscrete-time systems with time-varying delay has also beenpresented by Su et al [66] Both full-order and reduced-orderfilters were designed while approximators were employed toderive an input-output based sufficient condition Tseng [67]studied the time delay problem for a nonlinear discrete-timesystem based on model reference fuzzy tracking control Inthis work the T-S fuzzy model was applied to approximatea time delay discrete-time system while the fuzzy controllerwas designed to reduce the tracking error based on thisfuzzy model This method has advantage in that no feedbacklinearization or adaptive approaches are used in the controllerdesign

The robust 119867infin

control problem for uncertain discrete-time delay systems has been ivestigated by Xu and Lam [55]They studied the state-space T-S fuzzymodel with time delaysunder the assumption that the parameter uncertainties werenorm-boundedThe output feedback controller was designedwith full-order fuzzy dynamic to guarantee the closed-loopsystem is robust asymptotic stable as well as the admissibleuncertainties are norm bound constraint While solvabilityof this problem was addressed using a sufficient conditiongiven by the linear matrix inequalities The induced 119897

2filter

design of T-S fuzzy discrete-time stochastic systems withtime-varying delays has been investigated by Su et al [68]Wuet al [69] studied the 119867

infinmodel approximation for discrete-

time state delay fuzzy T-S systems

3 NN Control for Discrete-Time Systems

NN is of powerful computing ability and learning abilityto emulate various systems dynamics and is capable ofapproximating an unknown functionwith arbitrary accuracyNN is very successful in system modeling and controlby its capacity of universal approximate highly nonlinearand uncertain nonlinear and complex dynamic of systemsNNrsquos approximation ability has been shown by the Stone-Weierstrass theorem which states that a universal approxi-mator can approximate to an arbitrary degree of accuracyany real continuous function on a compact set Besides theuniversal approximation abilities NN also shows its excel-lence in parallel distributed processing abilities learningadaptation abilities natural fault tolerance and feasibilityfor hardware implementation These advantages make NNparticularly attractive and prospective for nonlinear controland modeling NN has been successfully applied to robotmanipulators control [70] distillation column control [71]chemical processes identification [72] flight control [73ndash75]

6 Discrete Dynamics in Nature and Society

and so forth To extend the NN technique from continuous-time to discrete-time many efforts have been made to studythe NN controlled discrete-time systems [76ndash78]

31 Adaptive NN Control for Discrete-Time Systems In theearly stage backpropagation (BP) algorithm [79] greatlyboosted the development of NN control It is noted that inthe early NN control design the control performances weredemonstrated through simulation or by particular exper-imental examples and consequently there were shortageof analytical analysis In addition an offline identificationprocedure was essential for achieving a stable NN controlsystem Thereafter the emergence of Lyapunov-based NNdesign makes it possible to use the available adaptive controltheories to rigorously guarantee stability robustness andconvergence of the closed-loop NN control systems Wecall the control design combining adaptive control theoriesand NN techniques as adaptive NN control It updates NNweights online and guarantees the stability of the closed-loop system Adaptive NN control design has been elegantlydeveloped for nonlinear systems with parametric uncertain-ties while many efforts have been dedicated in adaptive NNcontrol for nonlinear discrete-time systems [6]

For high-order affine nonlinear system in normal formadaptive NN controls using LPNN and MNN have beendeveloped in [80 81] using a filtered tracking error In [82]the controller is designed incorporating reinforcement learn-ing technique to improve control performance In this worka critic NN has been introduced to approximate the strategicutility function which is considered as the long-term systemperformance measure For discrete-time systems in strict-feedback form adaptive NN control has been developed viabackstepping design after system transformation [83] In [84]adaptive NN control has been investigated for discrete-timesystem in affine NARMAX form

In the above mentioned results the adaptive NN controldesigns were carried out through either feedback lineariza-tion or backstepping But these approaches are not applicableto nonaffine systems especially feedback linearization basedmethods which greatly depend on the affine appearance ofcontrol variables As a matter of fact adaptive NN controlmethods for nonaffine systems have been less studied incomparison with large amount of research work on affinenonlinear systems because of the difficulty of control designcaused by the nonaffine form of control input To overcomethe difficulty linearization based NN controls have beenput forward In [17] the nonaffine discrete-time system hasbeen divided into two parts one is linear and another isnonlinear and consequently a linear adaptive controller anda nonlinear adaptive NN controller have been designedwith a switching rule In [85] it directly utilized NN asemulator of the ldquoinverserdquo of the nonlinear discrete-timesystems Furthermore the study of discrete-time systems foradaptive NN control using implicit function to assert theexistence of an ideal inverse control was investigated in [86]Thereafter the implicit function based adaptive NN controlhas been widely studied in discrete-time form [87 88] Blocktriangular discrete-time systems with normal form subsys-tems have been studied in [80 81] For the block triangular

systems with strict-feedback subsystems state feedback andoutput feedback adaptive NN control have been developedin [89 90] by extending the systems transformation basedbackstepping technique proposed for SISO case in [83] In[91] adaptive NN control has been used for sampled-datanonlinear MIMO systems in general affine form based onlinearization The control scheme is an integration of an NNapproach and a variable structure method

An effort has been made in [92] to explore the adaptiveNN control of a class of nonaffine systems in discrete-timeThis work aimed to solve the nonaffine appearance andnoncausal problems of the following pure-feedback discrete-time system

120585119894(119896 + 1) = 119891

119894(120585119894 120585119894+ 1 (119896))

119894 = 1 2 119899 minus 1 119899 ge 2

120585119899

(119896 + 1) = 119891119899

(120585119899

(119896) 119906 (119896) 119889 (119896))

119910 (119896) = 1205851

(119896)

(7)

where 120585119894(119896) = [120585

1(119896) 1205852(119896) 120585

119899(119896)]119879 are system states

119891119894are unknown nonlinear functions and 119906(119896) and 119910(119896) are

system input and output respectively while 119889(119896) denotes theexternal disturbance

Using a states prediction technique the pure-feedbackdiscrete-time systems in (7) are shown to be transformableto an 119899-step-ahead predictor as below

119910 (119896 + 119899) = 120601 (120585119899

(119896) 119906 (119896) 119889 (119896))

= 120601119904(120585119899

(119896) 119906 (119896)) + 119889119904(119896)

(8)

And by future output predictions the above system (8)can be further transformed into an input-output model forthe output feedback control

119910 (119896 + 119899)

= 1198651198991

119910 (119896 + 1) 119906 (119896) 119889 (119896) 119889 (119896 minus 119899 + 2)

(9)

This shows that the system presentation (9) in NARMAXform is a transformation of the pure-feedback system (8)After transformation both state feedback and output feed-back controls only need to employ a single NN in thecontroller design rather than a number of NNs in previousresearches

Based on the SISO pure-feedback system (8) and itstransformation NARMAX model in (9) the control designhas been further investigated by using high-order neuralnetwork to approximate unknown functions [93] In theabove mentioned control design availability is assumed forknowledge of control directions which are defined as ldquothesigns of control variable gains in affine systems or the signsof partial derivatives over control variables in nonaffinesystemsrdquo [93]

The paper overcame the problems of output-feedbackcontrol when the prior knowledge of the control directions isunknownThemain idea is to introduce a discrete Nussbaum

Discrete Dynamics in Nature and Society 7

gain to counter the lack of knowledge on control gain inadaptive NN control

A rigorous definition of discrete Nussbaum gain in [93] isgiven as follows

Remark 1 Consider a discrete nonlinear function 119873(119909(119896))

defined on a sequence 119909(119896) with 119909119904(119896) = sup(119909

1015840

119896) 119873(119909(119896))

is a discrete Nussbaum gain if and only if it satisfies thefollowing two properties

(i) if 119909119904(119896) increases without bound then

sup 1

119909119904(119896)

119878119873

(119909 (119896)) = +infin

inf 1

119909119904(119896)

119878119873

(119909 (119896)) = minusinfin

(10)

(ii) if 119909119904(119896) ge (119896)120575

1 then |119878

119873(119909(119896))| ge 120575

2with some

positive constants 1205751and 1205752 where 119878

119873(119909(119896)) is defined as

119878119873

(119909 (119896)) =

119896

sum

1198961015840=0

119873 (119909 (1198961015840)) Δ119909 (119896

1015840) (11)

with Δ119909(119896) = 119909(119896 + 1) minus 119909(119896) In addition neither the upperbounds nor the lower bounds of the control gains are requiredto be known

Unknown control directions problem for MIMOdiscrete-time nonlinear systems was solved by using adaptiveoutput feedback NN control in [94] The studies in [92 93]all studied the discrete NN control for SISO systems and[94] extended the study to a class of MIMO discrete-timesystems with each subsystem in the nonaffine pure-feedbackas follows

120585119895119894119895

(119896 + 1) = 119891119895119894119895

(1205851119894119895minus1198981198951

(119896) 1205852119894119895minus1198981198952

(119896)

120585119899119894119895minus119898119895119899

(119896) 120585119895 119894119895+1

(119896)) 119894119895

= 1 2 119899119895

minus 1

120585119895119899119895

(119896 + 1) = 119891119895119899119895

(Ξ (119896) 119906119895(119896) 119889

119895(119896))

119910119895(119896) = 120585

1198951(119896)

(12)

where 120585119895119894119895

(119896) = [1205851198951

(119896) 1205851198952

(119896) 120585119895119894119895

(119896)]119879 are vectors

of states variable of subsystem Ξ(119896) is vector of all statevariables 119891

119895119894119895are unknown nonlinear functions 119906

119895(119896) and

119910119895(119896) are system inputs and outputs respectively while 119889

119895(119896)

denote the external disturbanceFor nonlinearMIMOdiscrete-time higher order systems

a major drawback is that there are too many online-tunedadaptive parameters and large online computation burden In[95] an adaptive neural output feedback adaptive controllerwas designed for MIMO nonlinear discrete-time systemswith fewer adaptive parameters The output feedback adap-tive control for a class of nonlinear discrete-time systemswithunknown control directions was investigated in [96]

32 NN-Based Dynamic Programming Algorithm for Discrete-Time Systems In the past several decades optimal control for

nonlinear systems has been a key focus in the control fieldsWhile adaptive dynamics programming (ADP) is a veryuseful approach to solve the optimal control problems ADPis a reinforcement learning method to give solution to thedynamic programming utilizing function approximating thevalue function based on adaptive approach [97] P Werbos[98] classified the discrete-time ADP into a number ofschemes action-dependentHDP dual HDP (DHP) heuristicdynamic programming (HDP) and action-dependent dualHDP while NN has been widely used to solve the adaptivedynamic programming problems of discrete-time systems[97 99ndash104]

He and Jagannathan [99] showed the reinforcementlearning with NN-based controller for nonlinear discrete-time systems This work developed an adaptive-critic-basediscrete-time NN controller to deliver a desired tracking per-formance for nonlinear systems in the presence of actuatorconstraints The critic NN and action NN were designed toapproximate the strategic utility function while uniformlyultimate boundedness of the closed-loop tracking error wasproved by Lyapunov approach The reinforcement learningNN control for SISO discrete-time pure-feedback systemswas studied in [100] The controller design is based on thetransformed predictor and two NNs of control architectureOne critic NN to approximate the strategic utility function isshown as follows

By introducing a utility function 119901(119896) based on thetracking error 119890(119896) = 119910(119896) minus 119910

119889(119896)

119901 (119896) = 1198860 |119890 (119896)| (13)

where 119901(119896) isin 119877 1198860

isin 119877 is the positive design parameter Andthe strategic utility function 119876 isin 119877 is defined as

119876 (119896) = 119886119873

(119901 + 1) + 119886119899minus1

119901 (119896 + 2) + sdot sdot sdot + 119886119896+1

119901 (119873)

+ sdot sdot sdot

(14)

where 119886 isin 119877 0 lt 119886 lt 1 119873 is the horizon Then the criticNN is used to approximate the strategic utility function119876(119896)

= 119879

119888(119896) 119878119888(119911 (119896)) 119878

119888(119911 (119896)) isin 119877

119897119888 (15)

where 119882119879

119888(119896) isin 119877

119897119888 is the estimation of optimal NN 119882lowast

119888

weights The uniformly ultimate boundedness of closed-loopsignals is established via Lyapunov stability analysis

Al-Tamimi et al [97] proved the convergence of a valuefunction solution for nonlinear dynamical systems usingHDP algorithm and NNs were employed to approximatethe value and the control action at each iteration Liu et al[101] used theHDP algorithm to develop aNN-based optimalcontroller for unknown discrete-time nonlinear systemsThemain idea of this work is to introduce an iterative ADPalgorithm and a globalizedHDP technique into the controllerdesign And by using the cost function and control law theconvergence of the optimal control is guaranteed In [102]an iterative ADP algorithm for discrete-time systems wasdeveloped as well An optimal controller was designed toaddress the infinite-horizon discrete-time nonlinear systems

8 Discrete Dynamics in Nature and Society

with finite approximation errors It was shown thatwhen con-vergence conditions are satisfied iterative performance indexfunction can converge to the performance index functionsrsquolower bound while the performance index functions wereapproximated by NNs to compute optimal control policy

The ADP control in the unknown discrete-time Markovjump systems was investigated by Zhong et al [103] AndWang et al [105] also studied the ADP for discrete-timesystems to obtain the 120576-optimal control by using neuralnetworks In [106] iterative ADP algorithmwas used to studythe near-optimal control with control constraints in discrete-time systems An iterative ADP for discrete-time systemswas studied in [104] In this work the optimal controllerwas designed with control constraints while NN was used toidentify the unknown dynamical systems with stability proof

4 Conclusion

In this short survey despite the impossibility in identifyingor listing all the related contributions best efforts have beenmade to summarize the major achievements in the area ofdiscrete-time adaptive control withmodern techniques basedon FL and NNs which are also conventionally termed asimportant part of ldquosoft-computingrdquo approaches or ldquointelli-gent controlrdquo in the control communities In particular themarriage of ldquoadaptive controlrdquo and ldquointelligent controlrdquo hasbeen reviewed for the purpose of clarifying main contribu-tions and outlining some possible trends for the developmentof this promising area

Generally speaking adaptive control for discrete-timelinear systems was extensively explored in the past decadesdespite the fact that the complete closed-loop stability analy-sis for the so-called self-tuning regulator was ever regarded asa long-term challenging problem Later it became clear thatdiscrete-time adaptive control can be extended to nonlinearsystems with linear growth rate However general nonlinearsystems with parametric andor nonparametric uncertaintiesare usually very difficult to cope with which motivatedvarious solutions for different certain classes of nonlinearsystems

Among the various solutions intelligent control emergedas one important way to resolve the challenges caused by thenonlinearityThe key for usability of intelligent control basedon FL or NNs lies in their universal approximator property atthe cost of tuning rules of FL or updating parameters of NNsFL and NNs are motivated by different background hencetheir design philosophies look different however essentiallyspeaking both of them can be expressed as weighted sumof some kernel functions where the weights can be tunedto approximate arbitrary smooth or continuous nonlinearfunction The ideas borrowed from adaptive estimation andadaptive control provide such a way to adaptively tune theweight parameters in FL orNNs thus the new area of adaptiveFLC or NN control emerged with extensive studies

In a summary a brief review on FLC for discrete-timesystems is provided by highlighting the adaptive FLC robust-ness issue and stability issue while NN control for discrete-time systems is also reviewed with focuses on adaptive NNcontrol and NN-based dynamic programming We believe

this topic would promote increasing investigations in boththeories and applications And some emerging techniquessuch as deep learning and big data could also bemerged withthe intelligent control for discrete-time systems and give birthto brand new design approaches of intelligent control in thefuture

Conflict of Interests

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

Acknowledgment

This work is partially supported by the National Natu-ral Science Foundation of China (NSFC) under Grants61473120 and 61473038 Guangdong Provincial Natural Sci-ence Foundation 2014A030313266 and International Scienceand Technology Collaboration Grant 2015A050502017 andFundamental Research Funds for the Central Universitiesunder Grant 2015ZM065 The authors would like to thankDr Bin Xu of Northwestern Polytechnical University for hisconstructive comments during the preparation of this paper

References

[1] G Goodwin and K Sin Adaptive Filtering Prediction andControl Prentice-Hall Englewood Cliffs NJ USA 1984

[2] K Astrom and B Wittenmark Adaptive Control Addison-Wesley 1989

[3] M Krstic I Kanellakopoulos and P V Kokotovic NonlinearandAdaptive Control Design JohnWileyamp Sons NewYork NYUSA 1995

[4] G Tao and P Kokotovic Aaptive Control of Systems with Actu-ator and Sensor Non-Linearities John Wiley amp Sons HobokenNJ USA 1996

[5] S S Ge C C Hang T H Lee and T Zhang Stable AdaptiveNeural Network Control Kluwer Academic Publishers NorwellMass USA 2001

[6] C Yang Adaptive control and neural network control of nonlin-ear discrete-time systems [PhD thesis] National University ofSingapore 2009

[7] C Yang H Ma and M Fu ldquoAdaptive predictive control ofperiodic non-linear auto-regressive moving average systemsusing nearest-neighbour compensationrdquo IET Control Theory ampApplications vol 7 no 7 pp 936ndash951 2013

[8] C Yang L Zhai S S Ge T Chai andTH Lee ldquoAdaptivemodelreference control of a class ofMIMOdiscrete-time systems withcompensation of nonparametric uncertaintyrdquo in Proceedings ofthe American Control Conference pp 4111ndash4116 IEEE SeattleWash USA June 2008

[9] S-L Dai C Yang S S Ge and T H Lee ldquoRobust adaptiveoutput feedback control of a class of discrete-time nonlinearsystems with nonlinear uncertainties and unknown controldirectionsrdquo International Journal of Robust and Nonlinear Con-trol vol 23 no 13 pp 1472ndash1495 2013

[10] K J Astrom and B Wittenmark ldquoOn self tuning regulatorsrdquoAutomatica vol 9 no 2 pp 185ndash199 1973

Discrete Dynamics in Nature and Society 9

[11] L Ljung ldquoAnalysis of recursive stochastic algorithmsrdquo IEEETransactions on Automatic Control vol 22 no 4 pp 551ndash5751977

[12] G C Goodwin P J Ramadge and P E Caines ldquoDiscretetime multivariable adaptive controlrdquo IEEE Transactions onAutomatic Control vol 25 no 3 pp 449ndash456 1980

[13] L Guo and H F Chen ldquoThe Astrom-Wittenmark self-tuningregulator revisited and ELS-based adaptive trackersrdquo IEEETransactions on Automatic Control vol 36 no 7 pp 802ndash8121991

[14] L Guo Time-Varing Stochastic Systems Jilin Science and Tech-nology Press Changchun China 1993 (Chinese)

[15] H F Chen and L Guo Identification and Stochastic AdaptiveControl Birkhauser Boston Mass USA 1991

[16] F P Skantze A Kojic A-P Loh and A M AnnaswamyldquoAdaptive estimation of discrete-time systems with nonlinearparameterizationrdquo Automatica vol 36 no 12 pp 1879ndash18872000

[17] L Chen and K S Narendra ldquoNonlinear adaptive control usingneural networks and multiple modelsrdquo Automatica vol 37 no8 pp 1245ndash1255 2001

[18] L Guo and C Wei ldquoLS-based discrete-time adaptive nonlinearcontrol feasibility and limitationsrdquo Science in China Series ETechnological Sciences vol 39 no 3 pp 255ndash269 1996

[19] L L Xie and L Guo ldquoAdaptive control of discrete-timenonlinear systems with structural uncertaintiesrdquo in Lectures onSystems Control and Information vol 17 of AMSIP Studiesin Advanced Mathematics American Mathematical SocietyInternational Press Providence RI USA 2000

[20] J D Boskovic ldquoStable adaptive control of a class of first-order nonlinearly parameterized plantsrdquo IEEE Transactions onAutomatic Control vol 40 no 2 pp 347ndash350 1995

[21] A L Fradkov I V Miroshnik and V O Nikiforov Nonlinearand Adaptive Control of Complex Systems Mathematics andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 2004

[22] D Angeli and E Mosca ldquoAdaptive switching supervisorycontrol of nonlinear systems with no prior knowledge of noiseboundsrdquo Automatica vol 40 no 3 pp 449ndash457 2004

[23] H B Ma ldquoFinite-model adaptive control using an LS-likealgorithmrdquo International Journal of Adaptive Control and SignalProcessing vol 21 no 5 pp 391ndash414 2007

[24] H B Ma ldquoFinite-model adaptive control using WLS-likealgorithmrdquo Automatica vol 43 no 4 pp 677ndash684 2007

[25] H B Ma ldquoSeveral algorithms for finite-model adaptive controlpartial answers to finite-model adaptive control problemrdquoMathematics of Control Signals and Systems vol 20 no 3 pp271ndash303 2008

[26] S S Ge C C Hang and T Zhang ldquoA direct adaptive controllerfor dynamic systems with a class of nonlinear parameteriza-tionsrdquo Automatica vol 35 no 4 pp 741ndash747 1999

[27] C Y Li and L Guo ldquoOn feedback capability in a class of nonlin-early parameterized uncertain systemsrdquo IEEE Transactions onAutomatic Control vol 56 no 12 pp 2946ndash2951 2011

[28] H Ma K-Y Lum and S S Ge ldquoAdaptive control for a discrete-time first-order nonlinear system with both parametric andnon-parametric uncertaintiesrdquo in Proceedings of the 46th IEEEConference on Decision and Control (CDC rsquo07) pp 4839ndash4844IEEE New Orleans La USA December 2007

[29] L Guo ldquoExploring the capability and limits of the feedbackmechanismrdquo in Proceedings of the International Congress ofMathematicians (ICM rsquo02) Beijing China August 2002

[30] H-B Ma ldquoAn lsquoimpossibilityrsquo theorem on a class of high-orderdiscrete-time nonlinear control systemsrdquo Systems and ControlLetters vol 57 no 6 pp 497ndash504 2008

[31] I Kanellakopoulos P V Kokotovic and A S Morse ldquoSys-tematic design of adaptive controllers for feedback linearizablesystemsrdquo IEEE Transactions on Automatic Control vol 36 no11 pp 1241ndash1253 1991

[32] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[33] L-X Wang and J M Mendel ldquoFuzzy basis functions universalapproximation and orthogonal least-squares learningrdquo IEEETransactions on Neural Networks vol 3 no 5 pp 807ndash814 1992

[34] L-X Wang ldquoStable adaptive fuzzy control of nonlinear sys-temsrdquo IEEE Transactions on Fuzzy Systems vol 1 no 2 pp 146ndash155 1993

[35] W S McCulloch and W Pitts ldquoA logical calculus of the ideasimmanent in nervous activityrdquo The Bulletin of MathematicalBiophysics vol 5 pp 115ndash133 1943

[36] K Hornik M Stinchcombe and HWhite ldquoMultilayer feedfor-ward networks are universal approximatorsrdquo Neural Networksvol 2 no 5 pp 359ndash366 1989

[37] T Khanna Foundations of Neural Networks Addison-WesleyReading Mass USA 1990

[38] R M Sanner and J-J E Slotine ldquoGaussian networks for directadaptive controlrdquo IEEE Transactions on Neural Networks vol 3no 6 pp 837ndash863 1992

[39] Y J Liu Y J Fang andM A Bao-Ping ldquoSliding-data-window-driven Bayesian-Gaussian neural network and its application tomodeling of nonlinear systemrdquo Control Theory amp Applicationsvol 26 no 12 pp 1435ndash1438 2009

[40] D Wang and J Huang ldquoAdaptive neural network control fora class of uncertain nonlinear systems in pure-feedback formrdquoAutomatica vol 38 no 8 pp 1365ndash1372 2002

[41] Y Song and J W Grizzle ldquoAdaptive output-feedback control ofa class of discrete-time nonlinear systemsrdquo in Proceedings of theAmerican Control Conference pp 1359ndash1363 June 1993

[42] B-S Chen C-S Tseng and H-J Uang ldquoRobustness designof nonlinear dynamic systems via fuzzy linear controlrdquo IEEETransactions on Fuzzy Systems vol 7 no 5 pp 571ndash585 1999

[43] T Chai and S Tong ldquoFuzzy direct adaptive control for a class ofnonlinear systemsrdquo Fuzzy Sets and Systems vol 103 no 3 pp379ndash387 1999

[44] K Tanaka T Ikeda and H O Wang ldquoRobust stabilizationof a class of uncertain nonlinear systems via fuzzy controlquadratic stabilizability 119867

infin control theory and linear matrixinequalitiesrdquo IEEE Transactions on Fuzzy Systems vol 4 no 1pp 1ndash13 1996

[45] S Jagannathan M W Vandegrift and F L Lewis ldquoAdaptivefuzzy logic control of discrete-time dynamical systemsrdquo Auto-matica vol 36 no 2 pp 229ndash241 2000

[46] Y Jiang Z Liu C Chen and Y Zhang ldquoAdaptive robust fuzzycontrol for dual arm robot with unknown input deadzonenonlinearityrdquo Nonlinear Dynamics vol 81 no 3 pp 1301ndash13142015

[47] MWVandegrift F L Lewis S Jagannathan andK Liu ldquoAdap-tive fuzzy logic control of discrete-time dynamical systemsrdquo inProceedings of the IEEE International Symposium on IntelligentControl pp 395ndash401 IEEE Monterey Calif USA August 1995

[48] S Jagannathan ldquoAdaptive fuzzy logic control of feedbacklinearizable discrete-time dynamical systems under persistenceof excitationrdquo Automatica vol 34 no 11 pp 1295ndash1310 1998

10 Discrete Dynamics in Nature and Society

[49] R Qi and M A Brdys ldquoStable indirect adaptive control basedon discrete-time T-S fuzzy modelrdquo Fuzzy Sets and Systems vol159 no 8 pp 900ndash925 2008

[50] T-C Lin S-W Chang and C-H Hsu ldquoRobust adaptivefuzzy sliding mode control for a class of uncertain discrete-time nonlinear systemsrdquo International Journal of InnovativeComputing Information and Control vol 8 no 1 pp 347ndash3592012

[51] G Feng andG Chen ldquoAdaptive control of discrete-time chaoticsystems a fuzzy control approachrdquoChaos Solitons and Fractalsvol 23 no 2 pp 459ndash467 2005

[52] H J Lee J B Park and G Chen ldquoRobust fuzzy controlof nonlinear systems with parametric uncertaintiesrdquo IEEETransactions on Fuzzy Systems vol 9 no 2 pp 369ndash379 2001

[53] Y-Y Cao and PM Frank ldquoRobust119867infindisturbance attenuation

for a class of uncertain discrete-time fuzzy systemsrdquo IEEETransactions on Fuzzy Systems vol 8 no 4 pp 406ndash415 2000

[54] S Zhou G Feng J Lam and S Xu ldquoRobust 119867infin

controlfor discrete-time fuzzy systems via basis-dependent Lyapunovfunctionsrdquo Information Sciences vol 174 no 3-4 pp 197ndash2172005

[55] S Xu and J Lam ldquoRobust 119867infin

control for uncertain discrete-time-delay fuzzy systems via output feedback controllersrdquo IEEETransactions on Fuzzy Systems vol 13 no 1 pp 82ndash93 2005

[56] C-S Tseng and B-S Chen ldquoRobust fuzzy observer-basedfuzzy control design for nonlinear discrete-time systems withpersistent bounded disturbancesrdquo IEEE Transactions on FuzzySystems vol 17 no 3 pp 711ndash723 2009

[57] S Xu B Song J Lu and J Lam ldquoRobust stability of uncertaindiscrete-time singular fuzzy systemsrdquo Fuzzy Sets and Systemsvol 158 no 20 pp 2306ndash2316 2007

[58] Z-G Wu P Shi H Su and J Chu ldquoReliable 119867infin

controlfor discrete-time fuzzy systems with infinite-distributed delayrdquoIEEE Transactions on Fuzzy Systems vol 20 no 1 pp 22ndash312012

[59] G Feng and J Ma ldquoQuadratic stabilization of uncertaindiscrete-time fuzzy dynamic systemsrdquo IEEE Transactions onCircuits and Systems I Fundamental Theory and Applicationsvol 48 no 11 pp 1337ndash1344 2001

[60] A Kruszewski R Wang and T M Guerra ldquoNonquadratic sta-bilization conditions for a class of uncertain nonlinear discretetime TS fuzzy models a new approachrdquo IEEE Transactions onAutomatic Control vol 53 no 2 pp 606ndash611 2008

[61] W-J Wang Y-J Chen and C-H Sun ldquoRelaxed stabilizationcriteria for discrete-time T-S fuzzy control systems based ona switching fuzzy model and piecewise Lyapunov functionrdquoIEEE Transactions on Systems Man amp Cybernetics Part BCybernetics vol 37 no 3 pp 551ndash559 2007

[62] G Feng ldquoStability analysis of discrete-time fuzzy dynamicsystems based on piecewise Lyapunov functionsrdquo IEEE Trans-actions on Fuzzy Systems vol 12 no 1 pp 22ndash28 2004

[63] H Gao X Liu and J Lam ldquoStability analysis and stabilizationfor discrete-time fuzzy systems with time-varying delayrdquo IEEETransactions on Systems Man and Cybernetics Part B Cyber-netics vol 39 no 2 pp 306ndash317 2009

[64] L Wu X Su P Shi and J Qiu ldquoA new approach to stabilityanalysis and stabilization of discrete-time T-S fuzzy time-varying delay systemsrdquo IEEE Transactions on SystemsMan andCybernetics Part B Cybernetics vol 41 no 1 pp 273ndash286 2011

[65] X Su P Shi L Wu and Y-D Song ldquoA novel control design ondiscrete-time takagi-sugeno fuzzy systems with time-varying

delaysrdquo IEEE Transactions on Fuzzy Systems vol 21 no 4 pp655ndash671 2013

[66] X Su P Shi L Wu and Y-D Song ldquoA novel approach to filterdesign for T-S fuzzy discrete-time systems with time-varyingdelayrdquo IEEE Transactions on Fuzzy Systems vol 20 no 6 pp1114ndash1129 2012

[67] C-S Tseng ldquoModel reference output feedback fuzzy trackingcontrol design for nonlinear discrete-time systems with time-delayrdquo IEEE Transactions on Fuzzy Systems vol 14 no 1 pp58ndash70 2006

[68] X Su P Shi L Wu and S K Nguang ldquoInduced l2 filteringof fuzzy stochastic systems with time-varying delaysrdquo IEEETransactions on Cybernetics vol 43 no 4 pp 1257ndash1264 2013

[69] L Wu X Su P Shi and J Qiu ldquoModel approximation fordiscrete-time state-delay systems in the TS fuzzy frameworkrdquoIEEE Transactions on Fuzzy Systems vol 19 no 2 pp 366ndash3782011

[70] F L Lewis S Jagannathan and A Yesildirek Neural NetworkControl of Robot Manipulators and Nonlinear Systems Taylor ampFrancis London UK 1999

[71] A M Shaw and F J Doyle III ldquoMultivariable nonlinear controlapplications for a high purity distillation column using arecurrent dynamic neuron modelrdquo Journal of Process Controlvol 7 no 4 pp 255ndash268 1997

[72] K Najim Process Modeling and Control in Chemical Engineer-ing Marcel Dekker New York NY USA 1989

[73] B Xu D Wang F Sun and Z Shi ldquoDirect neural discretecontrol of hypersonic flight vehiclerdquo Nonlinear Dynamics vol70 no 1 pp 269ndash278 2012

[74] B Xu and Y Zhang ldquoNeural discrete back-stepping controlof hypersonic flight vehicle with equivalent prediction modelrdquoNeurocomputing vol 154 pp 337ndash346 2015

[75] B Xu F Sun H Liu and J Ren ldquoAdaptive Kriging controllerdesign for hypersonic flight vehicle via back-steppingrdquo IETControl Theory amp Applications vol 6 no 4 pp 487ndash497 2012

[76] B Xu ldquoRobust adaptive neural control of flexible hypersonicflight vehicle with dead-zone input nonlinearityrdquo NonlinearDynamics vol 80 no 3 pp 1509ndash1520 2015

[77] B Xu XHuangDWang and F Sun ldquoDynamic surface controlof constrained hypersonic flightmodels with parameter estima-tion and actuator compensationrdquo Asian Journal of Control vol16 no 1 pp 162ndash174 2014

[78] B Xu and Z Shi ldquoAn overview on flight dynamics and controlapproaches for hypersonic vehiclesrdquo Science China InformationSciences vol 58 no 7 pp 1ndash19 2015

[79] D E Rumelhart G E Hinton and R J Williams ldquoLearninginternal representations by error propagationrdquo in Parallel Dis-tributed Processing vol 1 pp 318ndash362 MIT Press 1986

[80] S Jagannathan and F L Lewis ldquoDiscrete-time neural netcontroller for a class of nonlinear dynamical systemsrdquo IEEETransactions on Automatic Control vol 41 no 11 pp 1693ndash16991996

[81] S Jagannathan and F L Lewis ldquoMultilayer discrete-timeneural-net controller with guaranteed performancerdquo IEEETransactions on Neural Network vol 7 no 1 pp 107ndash130 1996

[82] P He and S Jagannathan ldquoNeuro-controller for reducingcyclic variation in lean combustion spark ignition enginesrdquoAutomatica vol 41 no 7 pp 1133ndash1142 2005

[83] S S Ge G Y Li and T H Lee ldquoAdaptive NN controlfor a class of strict-feedback discrete-time nonlinear systemsrdquoAutomatica vol 39 no 5 pp 807ndash819 2003

Discrete Dynamics in Nature and Society 11

[84] S S Ge T H Lee G Y Li and J Zhang ldquoAdaptive NN controlfor a class of discrete-time non-linear systemsrdquo InternationalJournal of Control vol 76 no 4 pp 334ndash354 2003

[85] C J Goh ldquoModel reference control of non-linear systems viaimplicit function emulationrdquo International Journal of Controlvol 60 no 1 pp 91ndash115 1994

[86] C J Goh and T H Lee ldquoDirect adaptive control of nonlinearsystems via implicit function emulationrdquo Control Theory andAdvanced Technology vol 10 no 3 pp 539ndash552 1994

[87] A U Levin and K S Narendra ldquoControl of nonlinear dynami-cal systems using neural networksmdashpart II observability iden-tification and controlrdquo IEEE Transactions on Neural Networksvol 7 no 1 pp 30ndash42 1996

[88] S S Ge J Zhang and T H Lee ldquoAdaptive MNN control fora class of non-affine NARMAX systems with disturbancesrdquoSystems amp Control Letters vol 53 no 1 pp 1ndash12 2004

[89] S S Ge J Zhang and T H Lee ldquoAdaptive neural networkcontrol for a class of MIMO nonlinear systems with distur-bances in discrete-timerdquo IEEE Transactions on Systems Manand Cybernetics Part B Cybernetics vol 34 no 4 pp 1630ndash1645 2004

[90] J Zhang S S Ge and T H Lee ldquoOutput feedback control of aclass of discrete MIMO nonlinear systems with triangular forminputsrdquo IEEE Transactions onNeural Networks vol 16 no 6 pp1491ndash1503 2005

[91] F C Sun Z Sun and P-Y Woo ldquoStable neural-network-basedadaptive control for sampled-data nonlinear systemsrdquo IEEETransactions on Neural Networks vol 9 no 5 pp 956ndash9681998

[92] C Yang S S Ge C Xiang T Chai and T H Lee ldquoOutputfeedback NN control for two classes of discrete-time systemswith unknown control directions in a unified approachrdquo IEEETransactions on Neural Networks vol 19 no 11 pp 1873ndash18862008

[93] S S Ge C Yang and T H Lee ldquoAdaptive predictive controlusing neural network for a class of pure-feedback systems indiscrete timerdquo IEEE Transactions on Neural Networks vol 19no 9 pp 1599ndash1614 2008

[94] Y Li C Yang S S Ge and T H Lee ldquoAdaptive output feed-back NN control of a class of discrete-time MIMO nonlinearsystems with unknown control directionsrdquo IEEE Transactionson Systems Man and Cybernetics Part B Cybernetics vol 41no 2 pp 507ndash517 2011

[95] Y-J Liu C L P Chen G-X Wen and S Tong ldquoAdaptiveneural output feedback tracking control for a class of uncertaindiscrete-time nonlinear systemsrdquo IEEE Transactions on NeuralNetworks vol 22 no 7 pp 1162ndash1167 2011

[96] C Yang S S Ge and T H Lee ldquoOutput feedback adaptivecontrol of a class of nonlinear discrete-time systems withunknown control directionsrdquoAutomatica vol 45 no 1 pp 270ndash276 2009

[97] A Al-Tamimi F L Lewis and M Abu-Khalaf ldquoDiscrete-timenonlinear HJB solution using approximate dynamic program-ming convergence proofrdquo IEEE Transactions on Systems Manand Cybernetics Part B Cybernetics vol 38 no 4 pp 943ndash9492008

[98] P Werbos ldquoApproximate dynamic programming for real-timecontrol and neuralmodelingrdquo inHandbook of Intelligent ControlNeural Fuzzy amp Adaptive Approaches Van Nostrand Reinhold1992

[99] P He and S Jagannathan ldquoReinforcement learning neural-network-based controller for nonlinear discrete-time systems

with input constraintsrdquo IEEETransactions on SystemsMan andCybernetics Part B Cybernetics vol 37 no 2 pp 425ndash436 2007

[100] B Xu C Yang and Z Shi ldquoReinforcement learning outputfeedback NN control using deterministic learning techniquerdquoIEEE Transactions on Neural Networks and Learning Systemsvol 25 no 3 pp 635ndash641 2014

[101] D Liu DWang D Zhao QWei and N Jin ldquoNeural-network-based optimal control for a class of unknowndiscrete-time non-linear systems using globalized dual heuristic programmingrdquoIEEE Transactions on Automation Science and Engineering vol9 no 3 pp 628ndash634 2012

[102] D Liu and Q Wei ldquoFinite-approximation-error-based optimalcontrol approach for discrete-time nonlinear systemsrdquo IEEETransactions on Cybernetics vol 43 no 2 pp 779ndash789 2013

[103] X Zhong H He H Zhang and Z Wang ldquoOptimal control forunknown discrete-time nonlinear markov jump systems usingadaptive dynamic programmingrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 25 no 12 pp 2141ndash21552014

[104] D Liu D Wang and X Yang ldquoAn iterative adaptive dynamicprogramming algorithm for optimal control of unknowndiscrete-time nonlinear systemswith constrained inputsrdquo Infor-mation Sciences vol 220 pp 331ndash342 2013

[105] F-Y Wang N Jin D Liu and Q Wei ldquoAdaptive dynamicprogramming for finite-horizon optimal control of discrete-time nonlinear systems with 120576-error boundrdquo IEEE Transactionson Neural Networks vol 22 no 1 pp 24ndash36 2011

[106] H Zhang Y Luo and D Liu ldquoNeural-network-based near-optimal control for a class of discrete-time affine nonlinearsystems with control constraintsrdquo IEEE Transactions on NeuralNetworks vol 20 no 9 pp 1490ndash1503 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Discrete Dynamics in Nature and Society

and so forth To extend the NN technique from continuous-time to discrete-time many efforts have been made to studythe NN controlled discrete-time systems [76ndash78]

31 Adaptive NN Control for Discrete-Time Systems In theearly stage backpropagation (BP) algorithm [79] greatlyboosted the development of NN control It is noted that inthe early NN control design the control performances weredemonstrated through simulation or by particular exper-imental examples and consequently there were shortageof analytical analysis In addition an offline identificationprocedure was essential for achieving a stable NN controlsystem Thereafter the emergence of Lyapunov-based NNdesign makes it possible to use the available adaptive controltheories to rigorously guarantee stability robustness andconvergence of the closed-loop NN control systems Wecall the control design combining adaptive control theoriesand NN techniques as adaptive NN control It updates NNweights online and guarantees the stability of the closed-loop system Adaptive NN control design has been elegantlydeveloped for nonlinear systems with parametric uncertain-ties while many efforts have been dedicated in adaptive NNcontrol for nonlinear discrete-time systems [6]

For high-order affine nonlinear system in normal formadaptive NN controls using LPNN and MNN have beendeveloped in [80 81] using a filtered tracking error In [82]the controller is designed incorporating reinforcement learn-ing technique to improve control performance In this worka critic NN has been introduced to approximate the strategicutility function which is considered as the long-term systemperformance measure For discrete-time systems in strict-feedback form adaptive NN control has been developed viabackstepping design after system transformation [83] In [84]adaptive NN control has been investigated for discrete-timesystem in affine NARMAX form

In the above mentioned results the adaptive NN controldesigns were carried out through either feedback lineariza-tion or backstepping But these approaches are not applicableto nonaffine systems especially feedback linearization basedmethods which greatly depend on the affine appearance ofcontrol variables As a matter of fact adaptive NN controlmethods for nonaffine systems have been less studied incomparison with large amount of research work on affinenonlinear systems because of the difficulty of control designcaused by the nonaffine form of control input To overcomethe difficulty linearization based NN controls have beenput forward In [17] the nonaffine discrete-time system hasbeen divided into two parts one is linear and another isnonlinear and consequently a linear adaptive controller anda nonlinear adaptive NN controller have been designedwith a switching rule In [85] it directly utilized NN asemulator of the ldquoinverserdquo of the nonlinear discrete-timesystems Furthermore the study of discrete-time systems foradaptive NN control using implicit function to assert theexistence of an ideal inverse control was investigated in [86]Thereafter the implicit function based adaptive NN controlhas been widely studied in discrete-time form [87 88] Blocktriangular discrete-time systems with normal form subsys-tems have been studied in [80 81] For the block triangular

systems with strict-feedback subsystems state feedback andoutput feedback adaptive NN control have been developedin [89 90] by extending the systems transformation basedbackstepping technique proposed for SISO case in [83] In[91] adaptive NN control has been used for sampled-datanonlinear MIMO systems in general affine form based onlinearization The control scheme is an integration of an NNapproach and a variable structure method

An effort has been made in [92] to explore the adaptiveNN control of a class of nonaffine systems in discrete-timeThis work aimed to solve the nonaffine appearance andnoncausal problems of the following pure-feedback discrete-time system

120585119894(119896 + 1) = 119891

119894(120585119894 120585119894+ 1 (119896))

119894 = 1 2 119899 minus 1 119899 ge 2

120585119899

(119896 + 1) = 119891119899

(120585119899

(119896) 119906 (119896) 119889 (119896))

119910 (119896) = 1205851

(119896)

(7)

where 120585119894(119896) = [120585

1(119896) 1205852(119896) 120585

119899(119896)]119879 are system states

119891119894are unknown nonlinear functions and 119906(119896) and 119910(119896) are

system input and output respectively while 119889(119896) denotes theexternal disturbance

Using a states prediction technique the pure-feedbackdiscrete-time systems in (7) are shown to be transformableto an 119899-step-ahead predictor as below

119910 (119896 + 119899) = 120601 (120585119899

(119896) 119906 (119896) 119889 (119896))

= 120601119904(120585119899

(119896) 119906 (119896)) + 119889119904(119896)

(8)

And by future output predictions the above system (8)can be further transformed into an input-output model forthe output feedback control

119910 (119896 + 119899)

= 1198651198991

119910 (119896 + 1) 119906 (119896) 119889 (119896) 119889 (119896 minus 119899 + 2)

(9)

This shows that the system presentation (9) in NARMAXform is a transformation of the pure-feedback system (8)After transformation both state feedback and output feed-back controls only need to employ a single NN in thecontroller design rather than a number of NNs in previousresearches

Based on the SISO pure-feedback system (8) and itstransformation NARMAX model in (9) the control designhas been further investigated by using high-order neuralnetwork to approximate unknown functions [93] In theabove mentioned control design availability is assumed forknowledge of control directions which are defined as ldquothesigns of control variable gains in affine systems or the signsof partial derivatives over control variables in nonaffinesystemsrdquo [93]

The paper overcame the problems of output-feedbackcontrol when the prior knowledge of the control directions isunknownThemain idea is to introduce a discrete Nussbaum

Discrete Dynamics in Nature and Society 7

gain to counter the lack of knowledge on control gain inadaptive NN control

A rigorous definition of discrete Nussbaum gain in [93] isgiven as follows

Remark 1 Consider a discrete nonlinear function 119873(119909(119896))

defined on a sequence 119909(119896) with 119909119904(119896) = sup(119909

1015840

119896) 119873(119909(119896))

is a discrete Nussbaum gain if and only if it satisfies thefollowing two properties

(i) if 119909119904(119896) increases without bound then

sup 1

119909119904(119896)

119878119873

(119909 (119896)) = +infin

inf 1

119909119904(119896)

119878119873

(119909 (119896)) = minusinfin

(10)

(ii) if 119909119904(119896) ge (119896)120575

1 then |119878

119873(119909(119896))| ge 120575

2with some

positive constants 1205751and 1205752 where 119878

119873(119909(119896)) is defined as

119878119873

(119909 (119896)) =

119896

sum

1198961015840=0

119873 (119909 (1198961015840)) Δ119909 (119896

1015840) (11)

with Δ119909(119896) = 119909(119896 + 1) minus 119909(119896) In addition neither the upperbounds nor the lower bounds of the control gains are requiredto be known

Unknown control directions problem for MIMOdiscrete-time nonlinear systems was solved by using adaptiveoutput feedback NN control in [94] The studies in [92 93]all studied the discrete NN control for SISO systems and[94] extended the study to a class of MIMO discrete-timesystems with each subsystem in the nonaffine pure-feedbackas follows

120585119895119894119895

(119896 + 1) = 119891119895119894119895

(1205851119894119895minus1198981198951

(119896) 1205852119894119895minus1198981198952

(119896)

120585119899119894119895minus119898119895119899

(119896) 120585119895 119894119895+1

(119896)) 119894119895

= 1 2 119899119895

minus 1

120585119895119899119895

(119896 + 1) = 119891119895119899119895

(Ξ (119896) 119906119895(119896) 119889

119895(119896))

119910119895(119896) = 120585

1198951(119896)

(12)

where 120585119895119894119895

(119896) = [1205851198951

(119896) 1205851198952

(119896) 120585119895119894119895

(119896)]119879 are vectors

of states variable of subsystem Ξ(119896) is vector of all statevariables 119891

119895119894119895are unknown nonlinear functions 119906

119895(119896) and

119910119895(119896) are system inputs and outputs respectively while 119889

119895(119896)

denote the external disturbanceFor nonlinearMIMOdiscrete-time higher order systems

a major drawback is that there are too many online-tunedadaptive parameters and large online computation burden In[95] an adaptive neural output feedback adaptive controllerwas designed for MIMO nonlinear discrete-time systemswith fewer adaptive parameters The output feedback adap-tive control for a class of nonlinear discrete-time systemswithunknown control directions was investigated in [96]

32 NN-Based Dynamic Programming Algorithm for Discrete-Time Systems In the past several decades optimal control for

nonlinear systems has been a key focus in the control fieldsWhile adaptive dynamics programming (ADP) is a veryuseful approach to solve the optimal control problems ADPis a reinforcement learning method to give solution to thedynamic programming utilizing function approximating thevalue function based on adaptive approach [97] P Werbos[98] classified the discrete-time ADP into a number ofschemes action-dependentHDP dual HDP (DHP) heuristicdynamic programming (HDP) and action-dependent dualHDP while NN has been widely used to solve the adaptivedynamic programming problems of discrete-time systems[97 99ndash104]

He and Jagannathan [99] showed the reinforcementlearning with NN-based controller for nonlinear discrete-time systems This work developed an adaptive-critic-basediscrete-time NN controller to deliver a desired tracking per-formance for nonlinear systems in the presence of actuatorconstraints The critic NN and action NN were designed toapproximate the strategic utility function while uniformlyultimate boundedness of the closed-loop tracking error wasproved by Lyapunov approach The reinforcement learningNN control for SISO discrete-time pure-feedback systemswas studied in [100] The controller design is based on thetransformed predictor and two NNs of control architectureOne critic NN to approximate the strategic utility function isshown as follows

By introducing a utility function 119901(119896) based on thetracking error 119890(119896) = 119910(119896) minus 119910

119889(119896)

119901 (119896) = 1198860 |119890 (119896)| (13)

where 119901(119896) isin 119877 1198860

isin 119877 is the positive design parameter Andthe strategic utility function 119876 isin 119877 is defined as

119876 (119896) = 119886119873

(119901 + 1) + 119886119899minus1

119901 (119896 + 2) + sdot sdot sdot + 119886119896+1

119901 (119873)

+ sdot sdot sdot

(14)

where 119886 isin 119877 0 lt 119886 lt 1 119873 is the horizon Then the criticNN is used to approximate the strategic utility function119876(119896)

= 119879

119888(119896) 119878119888(119911 (119896)) 119878

119888(119911 (119896)) isin 119877

119897119888 (15)

where 119882119879

119888(119896) isin 119877

119897119888 is the estimation of optimal NN 119882lowast

119888

weights The uniformly ultimate boundedness of closed-loopsignals is established via Lyapunov stability analysis

Al-Tamimi et al [97] proved the convergence of a valuefunction solution for nonlinear dynamical systems usingHDP algorithm and NNs were employed to approximatethe value and the control action at each iteration Liu et al[101] used theHDP algorithm to develop aNN-based optimalcontroller for unknown discrete-time nonlinear systemsThemain idea of this work is to introduce an iterative ADPalgorithm and a globalizedHDP technique into the controllerdesign And by using the cost function and control law theconvergence of the optimal control is guaranteed In [102]an iterative ADP algorithm for discrete-time systems wasdeveloped as well An optimal controller was designed toaddress the infinite-horizon discrete-time nonlinear systems

8 Discrete Dynamics in Nature and Society

with finite approximation errors It was shown thatwhen con-vergence conditions are satisfied iterative performance indexfunction can converge to the performance index functionsrsquolower bound while the performance index functions wereapproximated by NNs to compute optimal control policy

The ADP control in the unknown discrete-time Markovjump systems was investigated by Zhong et al [103] AndWang et al [105] also studied the ADP for discrete-timesystems to obtain the 120576-optimal control by using neuralnetworks In [106] iterative ADP algorithmwas used to studythe near-optimal control with control constraints in discrete-time systems An iterative ADP for discrete-time systemswas studied in [104] In this work the optimal controllerwas designed with control constraints while NN was used toidentify the unknown dynamical systems with stability proof

4 Conclusion

In this short survey despite the impossibility in identifyingor listing all the related contributions best efforts have beenmade to summarize the major achievements in the area ofdiscrete-time adaptive control withmodern techniques basedon FL and NNs which are also conventionally termed asimportant part of ldquosoft-computingrdquo approaches or ldquointelli-gent controlrdquo in the control communities In particular themarriage of ldquoadaptive controlrdquo and ldquointelligent controlrdquo hasbeen reviewed for the purpose of clarifying main contribu-tions and outlining some possible trends for the developmentof this promising area

Generally speaking adaptive control for discrete-timelinear systems was extensively explored in the past decadesdespite the fact that the complete closed-loop stability analy-sis for the so-called self-tuning regulator was ever regarded asa long-term challenging problem Later it became clear thatdiscrete-time adaptive control can be extended to nonlinearsystems with linear growth rate However general nonlinearsystems with parametric andor nonparametric uncertaintiesare usually very difficult to cope with which motivatedvarious solutions for different certain classes of nonlinearsystems

Among the various solutions intelligent control emergedas one important way to resolve the challenges caused by thenonlinearityThe key for usability of intelligent control basedon FL or NNs lies in their universal approximator property atthe cost of tuning rules of FL or updating parameters of NNsFL and NNs are motivated by different background hencetheir design philosophies look different however essentiallyspeaking both of them can be expressed as weighted sumof some kernel functions where the weights can be tunedto approximate arbitrary smooth or continuous nonlinearfunction The ideas borrowed from adaptive estimation andadaptive control provide such a way to adaptively tune theweight parameters in FL orNNs thus the new area of adaptiveFLC or NN control emerged with extensive studies

In a summary a brief review on FLC for discrete-timesystems is provided by highlighting the adaptive FLC robust-ness issue and stability issue while NN control for discrete-time systems is also reviewed with focuses on adaptive NNcontrol and NN-based dynamic programming We believe

this topic would promote increasing investigations in boththeories and applications And some emerging techniquessuch as deep learning and big data could also bemerged withthe intelligent control for discrete-time systems and give birthto brand new design approaches of intelligent control in thefuture

Conflict of Interests

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

Acknowledgment

This work is partially supported by the National Natu-ral Science Foundation of China (NSFC) under Grants61473120 and 61473038 Guangdong Provincial Natural Sci-ence Foundation 2014A030313266 and International Scienceand Technology Collaboration Grant 2015A050502017 andFundamental Research Funds for the Central Universitiesunder Grant 2015ZM065 The authors would like to thankDr Bin Xu of Northwestern Polytechnical University for hisconstructive comments during the preparation of this paper

References

[1] G Goodwin and K Sin Adaptive Filtering Prediction andControl Prentice-Hall Englewood Cliffs NJ USA 1984

[2] K Astrom and B Wittenmark Adaptive Control Addison-Wesley 1989

[3] M Krstic I Kanellakopoulos and P V Kokotovic NonlinearandAdaptive Control Design JohnWileyamp Sons NewYork NYUSA 1995

[4] G Tao and P Kokotovic Aaptive Control of Systems with Actu-ator and Sensor Non-Linearities John Wiley amp Sons HobokenNJ USA 1996

[5] S S Ge C C Hang T H Lee and T Zhang Stable AdaptiveNeural Network Control Kluwer Academic Publishers NorwellMass USA 2001

[6] C Yang Adaptive control and neural network control of nonlin-ear discrete-time systems [PhD thesis] National University ofSingapore 2009

[7] C Yang H Ma and M Fu ldquoAdaptive predictive control ofperiodic non-linear auto-regressive moving average systemsusing nearest-neighbour compensationrdquo IET Control Theory ampApplications vol 7 no 7 pp 936ndash951 2013

[8] C Yang L Zhai S S Ge T Chai andTH Lee ldquoAdaptivemodelreference control of a class ofMIMOdiscrete-time systems withcompensation of nonparametric uncertaintyrdquo in Proceedings ofthe American Control Conference pp 4111ndash4116 IEEE SeattleWash USA June 2008

[9] S-L Dai C Yang S S Ge and T H Lee ldquoRobust adaptiveoutput feedback control of a class of discrete-time nonlinearsystems with nonlinear uncertainties and unknown controldirectionsrdquo International Journal of Robust and Nonlinear Con-trol vol 23 no 13 pp 1472ndash1495 2013

[10] K J Astrom and B Wittenmark ldquoOn self tuning regulatorsrdquoAutomatica vol 9 no 2 pp 185ndash199 1973

Discrete Dynamics in Nature and Society 9

[11] L Ljung ldquoAnalysis of recursive stochastic algorithmsrdquo IEEETransactions on Automatic Control vol 22 no 4 pp 551ndash5751977

[12] G C Goodwin P J Ramadge and P E Caines ldquoDiscretetime multivariable adaptive controlrdquo IEEE Transactions onAutomatic Control vol 25 no 3 pp 449ndash456 1980

[13] L Guo and H F Chen ldquoThe Astrom-Wittenmark self-tuningregulator revisited and ELS-based adaptive trackersrdquo IEEETransactions on Automatic Control vol 36 no 7 pp 802ndash8121991

[14] L Guo Time-Varing Stochastic Systems Jilin Science and Tech-nology Press Changchun China 1993 (Chinese)

[15] H F Chen and L Guo Identification and Stochastic AdaptiveControl Birkhauser Boston Mass USA 1991

[16] F P Skantze A Kojic A-P Loh and A M AnnaswamyldquoAdaptive estimation of discrete-time systems with nonlinearparameterizationrdquo Automatica vol 36 no 12 pp 1879ndash18872000

[17] L Chen and K S Narendra ldquoNonlinear adaptive control usingneural networks and multiple modelsrdquo Automatica vol 37 no8 pp 1245ndash1255 2001

[18] L Guo and C Wei ldquoLS-based discrete-time adaptive nonlinearcontrol feasibility and limitationsrdquo Science in China Series ETechnological Sciences vol 39 no 3 pp 255ndash269 1996

[19] L L Xie and L Guo ldquoAdaptive control of discrete-timenonlinear systems with structural uncertaintiesrdquo in Lectures onSystems Control and Information vol 17 of AMSIP Studiesin Advanced Mathematics American Mathematical SocietyInternational Press Providence RI USA 2000

[20] J D Boskovic ldquoStable adaptive control of a class of first-order nonlinearly parameterized plantsrdquo IEEE Transactions onAutomatic Control vol 40 no 2 pp 347ndash350 1995

[21] A L Fradkov I V Miroshnik and V O Nikiforov Nonlinearand Adaptive Control of Complex Systems Mathematics andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 2004

[22] D Angeli and E Mosca ldquoAdaptive switching supervisorycontrol of nonlinear systems with no prior knowledge of noiseboundsrdquo Automatica vol 40 no 3 pp 449ndash457 2004

[23] H B Ma ldquoFinite-model adaptive control using an LS-likealgorithmrdquo International Journal of Adaptive Control and SignalProcessing vol 21 no 5 pp 391ndash414 2007

[24] H B Ma ldquoFinite-model adaptive control using WLS-likealgorithmrdquo Automatica vol 43 no 4 pp 677ndash684 2007

[25] H B Ma ldquoSeveral algorithms for finite-model adaptive controlpartial answers to finite-model adaptive control problemrdquoMathematics of Control Signals and Systems vol 20 no 3 pp271ndash303 2008

[26] S S Ge C C Hang and T Zhang ldquoA direct adaptive controllerfor dynamic systems with a class of nonlinear parameteriza-tionsrdquo Automatica vol 35 no 4 pp 741ndash747 1999

[27] C Y Li and L Guo ldquoOn feedback capability in a class of nonlin-early parameterized uncertain systemsrdquo IEEE Transactions onAutomatic Control vol 56 no 12 pp 2946ndash2951 2011

[28] H Ma K-Y Lum and S S Ge ldquoAdaptive control for a discrete-time first-order nonlinear system with both parametric andnon-parametric uncertaintiesrdquo in Proceedings of the 46th IEEEConference on Decision and Control (CDC rsquo07) pp 4839ndash4844IEEE New Orleans La USA December 2007

[29] L Guo ldquoExploring the capability and limits of the feedbackmechanismrdquo in Proceedings of the International Congress ofMathematicians (ICM rsquo02) Beijing China August 2002

[30] H-B Ma ldquoAn lsquoimpossibilityrsquo theorem on a class of high-orderdiscrete-time nonlinear control systemsrdquo Systems and ControlLetters vol 57 no 6 pp 497ndash504 2008

[31] I Kanellakopoulos P V Kokotovic and A S Morse ldquoSys-tematic design of adaptive controllers for feedback linearizablesystemsrdquo IEEE Transactions on Automatic Control vol 36 no11 pp 1241ndash1253 1991

[32] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[33] L-X Wang and J M Mendel ldquoFuzzy basis functions universalapproximation and orthogonal least-squares learningrdquo IEEETransactions on Neural Networks vol 3 no 5 pp 807ndash814 1992

[34] L-X Wang ldquoStable adaptive fuzzy control of nonlinear sys-temsrdquo IEEE Transactions on Fuzzy Systems vol 1 no 2 pp 146ndash155 1993

[35] W S McCulloch and W Pitts ldquoA logical calculus of the ideasimmanent in nervous activityrdquo The Bulletin of MathematicalBiophysics vol 5 pp 115ndash133 1943

[36] K Hornik M Stinchcombe and HWhite ldquoMultilayer feedfor-ward networks are universal approximatorsrdquo Neural Networksvol 2 no 5 pp 359ndash366 1989

[37] T Khanna Foundations of Neural Networks Addison-WesleyReading Mass USA 1990

[38] R M Sanner and J-J E Slotine ldquoGaussian networks for directadaptive controlrdquo IEEE Transactions on Neural Networks vol 3no 6 pp 837ndash863 1992

[39] Y J Liu Y J Fang andM A Bao-Ping ldquoSliding-data-window-driven Bayesian-Gaussian neural network and its application tomodeling of nonlinear systemrdquo Control Theory amp Applicationsvol 26 no 12 pp 1435ndash1438 2009

[40] D Wang and J Huang ldquoAdaptive neural network control fora class of uncertain nonlinear systems in pure-feedback formrdquoAutomatica vol 38 no 8 pp 1365ndash1372 2002

[41] Y Song and J W Grizzle ldquoAdaptive output-feedback control ofa class of discrete-time nonlinear systemsrdquo in Proceedings of theAmerican Control Conference pp 1359ndash1363 June 1993

[42] B-S Chen C-S Tseng and H-J Uang ldquoRobustness designof nonlinear dynamic systems via fuzzy linear controlrdquo IEEETransactions on Fuzzy Systems vol 7 no 5 pp 571ndash585 1999

[43] T Chai and S Tong ldquoFuzzy direct adaptive control for a class ofnonlinear systemsrdquo Fuzzy Sets and Systems vol 103 no 3 pp379ndash387 1999

[44] K Tanaka T Ikeda and H O Wang ldquoRobust stabilizationof a class of uncertain nonlinear systems via fuzzy controlquadratic stabilizability 119867

infin control theory and linear matrixinequalitiesrdquo IEEE Transactions on Fuzzy Systems vol 4 no 1pp 1ndash13 1996

[45] S Jagannathan M W Vandegrift and F L Lewis ldquoAdaptivefuzzy logic control of discrete-time dynamical systemsrdquo Auto-matica vol 36 no 2 pp 229ndash241 2000

[46] Y Jiang Z Liu C Chen and Y Zhang ldquoAdaptive robust fuzzycontrol for dual arm robot with unknown input deadzonenonlinearityrdquo Nonlinear Dynamics vol 81 no 3 pp 1301ndash13142015

[47] MWVandegrift F L Lewis S Jagannathan andK Liu ldquoAdap-tive fuzzy logic control of discrete-time dynamical systemsrdquo inProceedings of the IEEE International Symposium on IntelligentControl pp 395ndash401 IEEE Monterey Calif USA August 1995

[48] S Jagannathan ldquoAdaptive fuzzy logic control of feedbacklinearizable discrete-time dynamical systems under persistenceof excitationrdquo Automatica vol 34 no 11 pp 1295ndash1310 1998

10 Discrete Dynamics in Nature and Society

[49] R Qi and M A Brdys ldquoStable indirect adaptive control basedon discrete-time T-S fuzzy modelrdquo Fuzzy Sets and Systems vol159 no 8 pp 900ndash925 2008

[50] T-C Lin S-W Chang and C-H Hsu ldquoRobust adaptivefuzzy sliding mode control for a class of uncertain discrete-time nonlinear systemsrdquo International Journal of InnovativeComputing Information and Control vol 8 no 1 pp 347ndash3592012

[51] G Feng andG Chen ldquoAdaptive control of discrete-time chaoticsystems a fuzzy control approachrdquoChaos Solitons and Fractalsvol 23 no 2 pp 459ndash467 2005

[52] H J Lee J B Park and G Chen ldquoRobust fuzzy controlof nonlinear systems with parametric uncertaintiesrdquo IEEETransactions on Fuzzy Systems vol 9 no 2 pp 369ndash379 2001

[53] Y-Y Cao and PM Frank ldquoRobust119867infindisturbance attenuation

for a class of uncertain discrete-time fuzzy systemsrdquo IEEETransactions on Fuzzy Systems vol 8 no 4 pp 406ndash415 2000

[54] S Zhou G Feng J Lam and S Xu ldquoRobust 119867infin

controlfor discrete-time fuzzy systems via basis-dependent Lyapunovfunctionsrdquo Information Sciences vol 174 no 3-4 pp 197ndash2172005

[55] S Xu and J Lam ldquoRobust 119867infin

control for uncertain discrete-time-delay fuzzy systems via output feedback controllersrdquo IEEETransactions on Fuzzy Systems vol 13 no 1 pp 82ndash93 2005

[56] C-S Tseng and B-S Chen ldquoRobust fuzzy observer-basedfuzzy control design for nonlinear discrete-time systems withpersistent bounded disturbancesrdquo IEEE Transactions on FuzzySystems vol 17 no 3 pp 711ndash723 2009

[57] S Xu B Song J Lu and J Lam ldquoRobust stability of uncertaindiscrete-time singular fuzzy systemsrdquo Fuzzy Sets and Systemsvol 158 no 20 pp 2306ndash2316 2007

[58] Z-G Wu P Shi H Su and J Chu ldquoReliable 119867infin

controlfor discrete-time fuzzy systems with infinite-distributed delayrdquoIEEE Transactions on Fuzzy Systems vol 20 no 1 pp 22ndash312012

[59] G Feng and J Ma ldquoQuadratic stabilization of uncertaindiscrete-time fuzzy dynamic systemsrdquo IEEE Transactions onCircuits and Systems I Fundamental Theory and Applicationsvol 48 no 11 pp 1337ndash1344 2001

[60] A Kruszewski R Wang and T M Guerra ldquoNonquadratic sta-bilization conditions for a class of uncertain nonlinear discretetime TS fuzzy models a new approachrdquo IEEE Transactions onAutomatic Control vol 53 no 2 pp 606ndash611 2008

[61] W-J Wang Y-J Chen and C-H Sun ldquoRelaxed stabilizationcriteria for discrete-time T-S fuzzy control systems based ona switching fuzzy model and piecewise Lyapunov functionrdquoIEEE Transactions on Systems Man amp Cybernetics Part BCybernetics vol 37 no 3 pp 551ndash559 2007

[62] G Feng ldquoStability analysis of discrete-time fuzzy dynamicsystems based on piecewise Lyapunov functionsrdquo IEEE Trans-actions on Fuzzy Systems vol 12 no 1 pp 22ndash28 2004

[63] H Gao X Liu and J Lam ldquoStability analysis and stabilizationfor discrete-time fuzzy systems with time-varying delayrdquo IEEETransactions on Systems Man and Cybernetics Part B Cyber-netics vol 39 no 2 pp 306ndash317 2009

[64] L Wu X Su P Shi and J Qiu ldquoA new approach to stabilityanalysis and stabilization of discrete-time T-S fuzzy time-varying delay systemsrdquo IEEE Transactions on SystemsMan andCybernetics Part B Cybernetics vol 41 no 1 pp 273ndash286 2011

[65] X Su P Shi L Wu and Y-D Song ldquoA novel control design ondiscrete-time takagi-sugeno fuzzy systems with time-varying

delaysrdquo IEEE Transactions on Fuzzy Systems vol 21 no 4 pp655ndash671 2013

[66] X Su P Shi L Wu and Y-D Song ldquoA novel approach to filterdesign for T-S fuzzy discrete-time systems with time-varyingdelayrdquo IEEE Transactions on Fuzzy Systems vol 20 no 6 pp1114ndash1129 2012

[67] C-S Tseng ldquoModel reference output feedback fuzzy trackingcontrol design for nonlinear discrete-time systems with time-delayrdquo IEEE Transactions on Fuzzy Systems vol 14 no 1 pp58ndash70 2006

[68] X Su P Shi L Wu and S K Nguang ldquoInduced l2 filteringof fuzzy stochastic systems with time-varying delaysrdquo IEEETransactions on Cybernetics vol 43 no 4 pp 1257ndash1264 2013

[69] L Wu X Su P Shi and J Qiu ldquoModel approximation fordiscrete-time state-delay systems in the TS fuzzy frameworkrdquoIEEE Transactions on Fuzzy Systems vol 19 no 2 pp 366ndash3782011

[70] F L Lewis S Jagannathan and A Yesildirek Neural NetworkControl of Robot Manipulators and Nonlinear Systems Taylor ampFrancis London UK 1999

[71] A M Shaw and F J Doyle III ldquoMultivariable nonlinear controlapplications for a high purity distillation column using arecurrent dynamic neuron modelrdquo Journal of Process Controlvol 7 no 4 pp 255ndash268 1997

[72] K Najim Process Modeling and Control in Chemical Engineer-ing Marcel Dekker New York NY USA 1989

[73] B Xu D Wang F Sun and Z Shi ldquoDirect neural discretecontrol of hypersonic flight vehiclerdquo Nonlinear Dynamics vol70 no 1 pp 269ndash278 2012

[74] B Xu and Y Zhang ldquoNeural discrete back-stepping controlof hypersonic flight vehicle with equivalent prediction modelrdquoNeurocomputing vol 154 pp 337ndash346 2015

[75] B Xu F Sun H Liu and J Ren ldquoAdaptive Kriging controllerdesign for hypersonic flight vehicle via back-steppingrdquo IETControl Theory amp Applications vol 6 no 4 pp 487ndash497 2012

[76] B Xu ldquoRobust adaptive neural control of flexible hypersonicflight vehicle with dead-zone input nonlinearityrdquo NonlinearDynamics vol 80 no 3 pp 1509ndash1520 2015

[77] B Xu XHuangDWang and F Sun ldquoDynamic surface controlof constrained hypersonic flightmodels with parameter estima-tion and actuator compensationrdquo Asian Journal of Control vol16 no 1 pp 162ndash174 2014

[78] B Xu and Z Shi ldquoAn overview on flight dynamics and controlapproaches for hypersonic vehiclesrdquo Science China InformationSciences vol 58 no 7 pp 1ndash19 2015

[79] D E Rumelhart G E Hinton and R J Williams ldquoLearninginternal representations by error propagationrdquo in Parallel Dis-tributed Processing vol 1 pp 318ndash362 MIT Press 1986

[80] S Jagannathan and F L Lewis ldquoDiscrete-time neural netcontroller for a class of nonlinear dynamical systemsrdquo IEEETransactions on Automatic Control vol 41 no 11 pp 1693ndash16991996

[81] S Jagannathan and F L Lewis ldquoMultilayer discrete-timeneural-net controller with guaranteed performancerdquo IEEETransactions on Neural Network vol 7 no 1 pp 107ndash130 1996

[82] P He and S Jagannathan ldquoNeuro-controller for reducingcyclic variation in lean combustion spark ignition enginesrdquoAutomatica vol 41 no 7 pp 1133ndash1142 2005

[83] S S Ge G Y Li and T H Lee ldquoAdaptive NN controlfor a class of strict-feedback discrete-time nonlinear systemsrdquoAutomatica vol 39 no 5 pp 807ndash819 2003

Discrete Dynamics in Nature and Society 11

[84] S S Ge T H Lee G Y Li and J Zhang ldquoAdaptive NN controlfor a class of discrete-time non-linear systemsrdquo InternationalJournal of Control vol 76 no 4 pp 334ndash354 2003

[85] C J Goh ldquoModel reference control of non-linear systems viaimplicit function emulationrdquo International Journal of Controlvol 60 no 1 pp 91ndash115 1994

[86] C J Goh and T H Lee ldquoDirect adaptive control of nonlinearsystems via implicit function emulationrdquo Control Theory andAdvanced Technology vol 10 no 3 pp 539ndash552 1994

[87] A U Levin and K S Narendra ldquoControl of nonlinear dynami-cal systems using neural networksmdashpart II observability iden-tification and controlrdquo IEEE Transactions on Neural Networksvol 7 no 1 pp 30ndash42 1996

[88] S S Ge J Zhang and T H Lee ldquoAdaptive MNN control fora class of non-affine NARMAX systems with disturbancesrdquoSystems amp Control Letters vol 53 no 1 pp 1ndash12 2004

[89] S S Ge J Zhang and T H Lee ldquoAdaptive neural networkcontrol for a class of MIMO nonlinear systems with distur-bances in discrete-timerdquo IEEE Transactions on Systems Manand Cybernetics Part B Cybernetics vol 34 no 4 pp 1630ndash1645 2004

[90] J Zhang S S Ge and T H Lee ldquoOutput feedback control of aclass of discrete MIMO nonlinear systems with triangular forminputsrdquo IEEE Transactions onNeural Networks vol 16 no 6 pp1491ndash1503 2005

[91] F C Sun Z Sun and P-Y Woo ldquoStable neural-network-basedadaptive control for sampled-data nonlinear systemsrdquo IEEETransactions on Neural Networks vol 9 no 5 pp 956ndash9681998

[92] C Yang S S Ge C Xiang T Chai and T H Lee ldquoOutputfeedback NN control for two classes of discrete-time systemswith unknown control directions in a unified approachrdquo IEEETransactions on Neural Networks vol 19 no 11 pp 1873ndash18862008

[93] S S Ge C Yang and T H Lee ldquoAdaptive predictive controlusing neural network for a class of pure-feedback systems indiscrete timerdquo IEEE Transactions on Neural Networks vol 19no 9 pp 1599ndash1614 2008

[94] Y Li C Yang S S Ge and T H Lee ldquoAdaptive output feed-back NN control of a class of discrete-time MIMO nonlinearsystems with unknown control directionsrdquo IEEE Transactionson Systems Man and Cybernetics Part B Cybernetics vol 41no 2 pp 507ndash517 2011

[95] Y-J Liu C L P Chen G-X Wen and S Tong ldquoAdaptiveneural output feedback tracking control for a class of uncertaindiscrete-time nonlinear systemsrdquo IEEE Transactions on NeuralNetworks vol 22 no 7 pp 1162ndash1167 2011

[96] C Yang S S Ge and T H Lee ldquoOutput feedback adaptivecontrol of a class of nonlinear discrete-time systems withunknown control directionsrdquoAutomatica vol 45 no 1 pp 270ndash276 2009

[97] A Al-Tamimi F L Lewis and M Abu-Khalaf ldquoDiscrete-timenonlinear HJB solution using approximate dynamic program-ming convergence proofrdquo IEEE Transactions on Systems Manand Cybernetics Part B Cybernetics vol 38 no 4 pp 943ndash9492008

[98] P Werbos ldquoApproximate dynamic programming for real-timecontrol and neuralmodelingrdquo inHandbook of Intelligent ControlNeural Fuzzy amp Adaptive Approaches Van Nostrand Reinhold1992

[99] P He and S Jagannathan ldquoReinforcement learning neural-network-based controller for nonlinear discrete-time systems

with input constraintsrdquo IEEETransactions on SystemsMan andCybernetics Part B Cybernetics vol 37 no 2 pp 425ndash436 2007

[100] B Xu C Yang and Z Shi ldquoReinforcement learning outputfeedback NN control using deterministic learning techniquerdquoIEEE Transactions on Neural Networks and Learning Systemsvol 25 no 3 pp 635ndash641 2014

[101] D Liu DWang D Zhao QWei and N Jin ldquoNeural-network-based optimal control for a class of unknowndiscrete-time non-linear systems using globalized dual heuristic programmingrdquoIEEE Transactions on Automation Science and Engineering vol9 no 3 pp 628ndash634 2012

[102] D Liu and Q Wei ldquoFinite-approximation-error-based optimalcontrol approach for discrete-time nonlinear systemsrdquo IEEETransactions on Cybernetics vol 43 no 2 pp 779ndash789 2013

[103] X Zhong H He H Zhang and Z Wang ldquoOptimal control forunknown discrete-time nonlinear markov jump systems usingadaptive dynamic programmingrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 25 no 12 pp 2141ndash21552014

[104] D Liu D Wang and X Yang ldquoAn iterative adaptive dynamicprogramming algorithm for optimal control of unknowndiscrete-time nonlinear systemswith constrained inputsrdquo Infor-mation Sciences vol 220 pp 331ndash342 2013

[105] F-Y Wang N Jin D Liu and Q Wei ldquoAdaptive dynamicprogramming for finite-horizon optimal control of discrete-time nonlinear systems with 120576-error boundrdquo IEEE Transactionson Neural Networks vol 22 no 1 pp 24ndash36 2011

[106] H Zhang Y Luo and D Liu ldquoNeural-network-based near-optimal control for a class of discrete-time affine nonlinearsystems with control constraintsrdquo IEEE Transactions on NeuralNetworks vol 20 no 9 pp 1490ndash1503 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 7

gain to counter the lack of knowledge on control gain inadaptive NN control

A rigorous definition of discrete Nussbaum gain in [93] isgiven as follows

Remark 1 Consider a discrete nonlinear function 119873(119909(119896))

defined on a sequence 119909(119896) with 119909119904(119896) = sup(119909

1015840

119896) 119873(119909(119896))

is a discrete Nussbaum gain if and only if it satisfies thefollowing two properties

(i) if 119909119904(119896) increases without bound then

sup 1

119909119904(119896)

119878119873

(119909 (119896)) = +infin

inf 1

119909119904(119896)

119878119873

(119909 (119896)) = minusinfin

(10)

(ii) if 119909119904(119896) ge (119896)120575

1 then |119878

119873(119909(119896))| ge 120575

2with some

positive constants 1205751and 1205752 where 119878

119873(119909(119896)) is defined as

119878119873

(119909 (119896)) =

119896

sum

1198961015840=0

119873 (119909 (1198961015840)) Δ119909 (119896

1015840) (11)

with Δ119909(119896) = 119909(119896 + 1) minus 119909(119896) In addition neither the upperbounds nor the lower bounds of the control gains are requiredto be known

Unknown control directions problem for MIMOdiscrete-time nonlinear systems was solved by using adaptiveoutput feedback NN control in [94] The studies in [92 93]all studied the discrete NN control for SISO systems and[94] extended the study to a class of MIMO discrete-timesystems with each subsystem in the nonaffine pure-feedbackas follows

120585119895119894119895

(119896 + 1) = 119891119895119894119895

(1205851119894119895minus1198981198951

(119896) 1205852119894119895minus1198981198952

(119896)

120585119899119894119895minus119898119895119899

(119896) 120585119895 119894119895+1

(119896)) 119894119895

= 1 2 119899119895

minus 1

120585119895119899119895

(119896 + 1) = 119891119895119899119895

(Ξ (119896) 119906119895(119896) 119889

119895(119896))

119910119895(119896) = 120585

1198951(119896)

(12)

where 120585119895119894119895

(119896) = [1205851198951

(119896) 1205851198952

(119896) 120585119895119894119895

(119896)]119879 are vectors

of states variable of subsystem Ξ(119896) is vector of all statevariables 119891

119895119894119895are unknown nonlinear functions 119906

119895(119896) and

119910119895(119896) are system inputs and outputs respectively while 119889

119895(119896)

denote the external disturbanceFor nonlinearMIMOdiscrete-time higher order systems

a major drawback is that there are too many online-tunedadaptive parameters and large online computation burden In[95] an adaptive neural output feedback adaptive controllerwas designed for MIMO nonlinear discrete-time systemswith fewer adaptive parameters The output feedback adap-tive control for a class of nonlinear discrete-time systemswithunknown control directions was investigated in [96]

32 NN-Based Dynamic Programming Algorithm for Discrete-Time Systems In the past several decades optimal control for

nonlinear systems has been a key focus in the control fieldsWhile adaptive dynamics programming (ADP) is a veryuseful approach to solve the optimal control problems ADPis a reinforcement learning method to give solution to thedynamic programming utilizing function approximating thevalue function based on adaptive approach [97] P Werbos[98] classified the discrete-time ADP into a number ofschemes action-dependentHDP dual HDP (DHP) heuristicdynamic programming (HDP) and action-dependent dualHDP while NN has been widely used to solve the adaptivedynamic programming problems of discrete-time systems[97 99ndash104]

He and Jagannathan [99] showed the reinforcementlearning with NN-based controller for nonlinear discrete-time systems This work developed an adaptive-critic-basediscrete-time NN controller to deliver a desired tracking per-formance for nonlinear systems in the presence of actuatorconstraints The critic NN and action NN were designed toapproximate the strategic utility function while uniformlyultimate boundedness of the closed-loop tracking error wasproved by Lyapunov approach The reinforcement learningNN control for SISO discrete-time pure-feedback systemswas studied in [100] The controller design is based on thetransformed predictor and two NNs of control architectureOne critic NN to approximate the strategic utility function isshown as follows

By introducing a utility function 119901(119896) based on thetracking error 119890(119896) = 119910(119896) minus 119910

119889(119896)

119901 (119896) = 1198860 |119890 (119896)| (13)

where 119901(119896) isin 119877 1198860

isin 119877 is the positive design parameter Andthe strategic utility function 119876 isin 119877 is defined as

119876 (119896) = 119886119873

(119901 + 1) + 119886119899minus1

119901 (119896 + 2) + sdot sdot sdot + 119886119896+1

119901 (119873)

+ sdot sdot sdot

(14)

where 119886 isin 119877 0 lt 119886 lt 1 119873 is the horizon Then the criticNN is used to approximate the strategic utility function119876(119896)

= 119879

119888(119896) 119878119888(119911 (119896)) 119878

119888(119911 (119896)) isin 119877

119897119888 (15)

where 119882119879

119888(119896) isin 119877

119897119888 is the estimation of optimal NN 119882lowast

119888

weights The uniformly ultimate boundedness of closed-loopsignals is established via Lyapunov stability analysis

Al-Tamimi et al [97] proved the convergence of a valuefunction solution for nonlinear dynamical systems usingHDP algorithm and NNs were employed to approximatethe value and the control action at each iteration Liu et al[101] used theHDP algorithm to develop aNN-based optimalcontroller for unknown discrete-time nonlinear systemsThemain idea of this work is to introduce an iterative ADPalgorithm and a globalizedHDP technique into the controllerdesign And by using the cost function and control law theconvergence of the optimal control is guaranteed In [102]an iterative ADP algorithm for discrete-time systems wasdeveloped as well An optimal controller was designed toaddress the infinite-horizon discrete-time nonlinear systems

8 Discrete Dynamics in Nature and Society

with finite approximation errors It was shown thatwhen con-vergence conditions are satisfied iterative performance indexfunction can converge to the performance index functionsrsquolower bound while the performance index functions wereapproximated by NNs to compute optimal control policy

The ADP control in the unknown discrete-time Markovjump systems was investigated by Zhong et al [103] AndWang et al [105] also studied the ADP for discrete-timesystems to obtain the 120576-optimal control by using neuralnetworks In [106] iterative ADP algorithmwas used to studythe near-optimal control with control constraints in discrete-time systems An iterative ADP for discrete-time systemswas studied in [104] In this work the optimal controllerwas designed with control constraints while NN was used toidentify the unknown dynamical systems with stability proof

4 Conclusion

In this short survey despite the impossibility in identifyingor listing all the related contributions best efforts have beenmade to summarize the major achievements in the area ofdiscrete-time adaptive control withmodern techniques basedon FL and NNs which are also conventionally termed asimportant part of ldquosoft-computingrdquo approaches or ldquointelli-gent controlrdquo in the control communities In particular themarriage of ldquoadaptive controlrdquo and ldquointelligent controlrdquo hasbeen reviewed for the purpose of clarifying main contribu-tions and outlining some possible trends for the developmentof this promising area

Generally speaking adaptive control for discrete-timelinear systems was extensively explored in the past decadesdespite the fact that the complete closed-loop stability analy-sis for the so-called self-tuning regulator was ever regarded asa long-term challenging problem Later it became clear thatdiscrete-time adaptive control can be extended to nonlinearsystems with linear growth rate However general nonlinearsystems with parametric andor nonparametric uncertaintiesare usually very difficult to cope with which motivatedvarious solutions for different certain classes of nonlinearsystems

Among the various solutions intelligent control emergedas one important way to resolve the challenges caused by thenonlinearityThe key for usability of intelligent control basedon FL or NNs lies in their universal approximator property atthe cost of tuning rules of FL or updating parameters of NNsFL and NNs are motivated by different background hencetheir design philosophies look different however essentiallyspeaking both of them can be expressed as weighted sumof some kernel functions where the weights can be tunedto approximate arbitrary smooth or continuous nonlinearfunction The ideas borrowed from adaptive estimation andadaptive control provide such a way to adaptively tune theweight parameters in FL orNNs thus the new area of adaptiveFLC or NN control emerged with extensive studies

In a summary a brief review on FLC for discrete-timesystems is provided by highlighting the adaptive FLC robust-ness issue and stability issue while NN control for discrete-time systems is also reviewed with focuses on adaptive NNcontrol and NN-based dynamic programming We believe

this topic would promote increasing investigations in boththeories and applications And some emerging techniquessuch as deep learning and big data could also bemerged withthe intelligent control for discrete-time systems and give birthto brand new design approaches of intelligent control in thefuture

Conflict of Interests

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

Acknowledgment

This work is partially supported by the National Natu-ral Science Foundation of China (NSFC) under Grants61473120 and 61473038 Guangdong Provincial Natural Sci-ence Foundation 2014A030313266 and International Scienceand Technology Collaboration Grant 2015A050502017 andFundamental Research Funds for the Central Universitiesunder Grant 2015ZM065 The authors would like to thankDr Bin Xu of Northwestern Polytechnical University for hisconstructive comments during the preparation of this paper

References

[1] G Goodwin and K Sin Adaptive Filtering Prediction andControl Prentice-Hall Englewood Cliffs NJ USA 1984

[2] K Astrom and B Wittenmark Adaptive Control Addison-Wesley 1989

[3] M Krstic I Kanellakopoulos and P V Kokotovic NonlinearandAdaptive Control Design JohnWileyamp Sons NewYork NYUSA 1995

[4] G Tao and P Kokotovic Aaptive Control of Systems with Actu-ator and Sensor Non-Linearities John Wiley amp Sons HobokenNJ USA 1996

[5] S S Ge C C Hang T H Lee and T Zhang Stable AdaptiveNeural Network Control Kluwer Academic Publishers NorwellMass USA 2001

[6] C Yang Adaptive control and neural network control of nonlin-ear discrete-time systems [PhD thesis] National University ofSingapore 2009

[7] C Yang H Ma and M Fu ldquoAdaptive predictive control ofperiodic non-linear auto-regressive moving average systemsusing nearest-neighbour compensationrdquo IET Control Theory ampApplications vol 7 no 7 pp 936ndash951 2013

[8] C Yang L Zhai S S Ge T Chai andTH Lee ldquoAdaptivemodelreference control of a class ofMIMOdiscrete-time systems withcompensation of nonparametric uncertaintyrdquo in Proceedings ofthe American Control Conference pp 4111ndash4116 IEEE SeattleWash USA June 2008

[9] S-L Dai C Yang S S Ge and T H Lee ldquoRobust adaptiveoutput feedback control of a class of discrete-time nonlinearsystems with nonlinear uncertainties and unknown controldirectionsrdquo International Journal of Robust and Nonlinear Con-trol vol 23 no 13 pp 1472ndash1495 2013

[10] K J Astrom and B Wittenmark ldquoOn self tuning regulatorsrdquoAutomatica vol 9 no 2 pp 185ndash199 1973

Discrete Dynamics in Nature and Society 9

[11] L Ljung ldquoAnalysis of recursive stochastic algorithmsrdquo IEEETransactions on Automatic Control vol 22 no 4 pp 551ndash5751977

[12] G C Goodwin P J Ramadge and P E Caines ldquoDiscretetime multivariable adaptive controlrdquo IEEE Transactions onAutomatic Control vol 25 no 3 pp 449ndash456 1980

[13] L Guo and H F Chen ldquoThe Astrom-Wittenmark self-tuningregulator revisited and ELS-based adaptive trackersrdquo IEEETransactions on Automatic Control vol 36 no 7 pp 802ndash8121991

[14] L Guo Time-Varing Stochastic Systems Jilin Science and Tech-nology Press Changchun China 1993 (Chinese)

[15] H F Chen and L Guo Identification and Stochastic AdaptiveControl Birkhauser Boston Mass USA 1991

[16] F P Skantze A Kojic A-P Loh and A M AnnaswamyldquoAdaptive estimation of discrete-time systems with nonlinearparameterizationrdquo Automatica vol 36 no 12 pp 1879ndash18872000

[17] L Chen and K S Narendra ldquoNonlinear adaptive control usingneural networks and multiple modelsrdquo Automatica vol 37 no8 pp 1245ndash1255 2001

[18] L Guo and C Wei ldquoLS-based discrete-time adaptive nonlinearcontrol feasibility and limitationsrdquo Science in China Series ETechnological Sciences vol 39 no 3 pp 255ndash269 1996

[19] L L Xie and L Guo ldquoAdaptive control of discrete-timenonlinear systems with structural uncertaintiesrdquo in Lectures onSystems Control and Information vol 17 of AMSIP Studiesin Advanced Mathematics American Mathematical SocietyInternational Press Providence RI USA 2000

[20] J D Boskovic ldquoStable adaptive control of a class of first-order nonlinearly parameterized plantsrdquo IEEE Transactions onAutomatic Control vol 40 no 2 pp 347ndash350 1995

[21] A L Fradkov I V Miroshnik and V O Nikiforov Nonlinearand Adaptive Control of Complex Systems Mathematics andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 2004

[22] D Angeli and E Mosca ldquoAdaptive switching supervisorycontrol of nonlinear systems with no prior knowledge of noiseboundsrdquo Automatica vol 40 no 3 pp 449ndash457 2004

[23] H B Ma ldquoFinite-model adaptive control using an LS-likealgorithmrdquo International Journal of Adaptive Control and SignalProcessing vol 21 no 5 pp 391ndash414 2007

[24] H B Ma ldquoFinite-model adaptive control using WLS-likealgorithmrdquo Automatica vol 43 no 4 pp 677ndash684 2007

[25] H B Ma ldquoSeveral algorithms for finite-model adaptive controlpartial answers to finite-model adaptive control problemrdquoMathematics of Control Signals and Systems vol 20 no 3 pp271ndash303 2008

[26] S S Ge C C Hang and T Zhang ldquoA direct adaptive controllerfor dynamic systems with a class of nonlinear parameteriza-tionsrdquo Automatica vol 35 no 4 pp 741ndash747 1999

[27] C Y Li and L Guo ldquoOn feedback capability in a class of nonlin-early parameterized uncertain systemsrdquo IEEE Transactions onAutomatic Control vol 56 no 12 pp 2946ndash2951 2011

[28] H Ma K-Y Lum and S S Ge ldquoAdaptive control for a discrete-time first-order nonlinear system with both parametric andnon-parametric uncertaintiesrdquo in Proceedings of the 46th IEEEConference on Decision and Control (CDC rsquo07) pp 4839ndash4844IEEE New Orleans La USA December 2007

[29] L Guo ldquoExploring the capability and limits of the feedbackmechanismrdquo in Proceedings of the International Congress ofMathematicians (ICM rsquo02) Beijing China August 2002

[30] H-B Ma ldquoAn lsquoimpossibilityrsquo theorem on a class of high-orderdiscrete-time nonlinear control systemsrdquo Systems and ControlLetters vol 57 no 6 pp 497ndash504 2008

[31] I Kanellakopoulos P V Kokotovic and A S Morse ldquoSys-tematic design of adaptive controllers for feedback linearizablesystemsrdquo IEEE Transactions on Automatic Control vol 36 no11 pp 1241ndash1253 1991

[32] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[33] L-X Wang and J M Mendel ldquoFuzzy basis functions universalapproximation and orthogonal least-squares learningrdquo IEEETransactions on Neural Networks vol 3 no 5 pp 807ndash814 1992

[34] L-X Wang ldquoStable adaptive fuzzy control of nonlinear sys-temsrdquo IEEE Transactions on Fuzzy Systems vol 1 no 2 pp 146ndash155 1993

[35] W S McCulloch and W Pitts ldquoA logical calculus of the ideasimmanent in nervous activityrdquo The Bulletin of MathematicalBiophysics vol 5 pp 115ndash133 1943

[36] K Hornik M Stinchcombe and HWhite ldquoMultilayer feedfor-ward networks are universal approximatorsrdquo Neural Networksvol 2 no 5 pp 359ndash366 1989

[37] T Khanna Foundations of Neural Networks Addison-WesleyReading Mass USA 1990

[38] R M Sanner and J-J E Slotine ldquoGaussian networks for directadaptive controlrdquo IEEE Transactions on Neural Networks vol 3no 6 pp 837ndash863 1992

[39] Y J Liu Y J Fang andM A Bao-Ping ldquoSliding-data-window-driven Bayesian-Gaussian neural network and its application tomodeling of nonlinear systemrdquo Control Theory amp Applicationsvol 26 no 12 pp 1435ndash1438 2009

[40] D Wang and J Huang ldquoAdaptive neural network control fora class of uncertain nonlinear systems in pure-feedback formrdquoAutomatica vol 38 no 8 pp 1365ndash1372 2002

[41] Y Song and J W Grizzle ldquoAdaptive output-feedback control ofa class of discrete-time nonlinear systemsrdquo in Proceedings of theAmerican Control Conference pp 1359ndash1363 June 1993

[42] B-S Chen C-S Tseng and H-J Uang ldquoRobustness designof nonlinear dynamic systems via fuzzy linear controlrdquo IEEETransactions on Fuzzy Systems vol 7 no 5 pp 571ndash585 1999

[43] T Chai and S Tong ldquoFuzzy direct adaptive control for a class ofnonlinear systemsrdquo Fuzzy Sets and Systems vol 103 no 3 pp379ndash387 1999

[44] K Tanaka T Ikeda and H O Wang ldquoRobust stabilizationof a class of uncertain nonlinear systems via fuzzy controlquadratic stabilizability 119867

infin control theory and linear matrixinequalitiesrdquo IEEE Transactions on Fuzzy Systems vol 4 no 1pp 1ndash13 1996

[45] S Jagannathan M W Vandegrift and F L Lewis ldquoAdaptivefuzzy logic control of discrete-time dynamical systemsrdquo Auto-matica vol 36 no 2 pp 229ndash241 2000

[46] Y Jiang Z Liu C Chen and Y Zhang ldquoAdaptive robust fuzzycontrol for dual arm robot with unknown input deadzonenonlinearityrdquo Nonlinear Dynamics vol 81 no 3 pp 1301ndash13142015

[47] MWVandegrift F L Lewis S Jagannathan andK Liu ldquoAdap-tive fuzzy logic control of discrete-time dynamical systemsrdquo inProceedings of the IEEE International Symposium on IntelligentControl pp 395ndash401 IEEE Monterey Calif USA August 1995

[48] S Jagannathan ldquoAdaptive fuzzy logic control of feedbacklinearizable discrete-time dynamical systems under persistenceof excitationrdquo Automatica vol 34 no 11 pp 1295ndash1310 1998

10 Discrete Dynamics in Nature and Society

[49] R Qi and M A Brdys ldquoStable indirect adaptive control basedon discrete-time T-S fuzzy modelrdquo Fuzzy Sets and Systems vol159 no 8 pp 900ndash925 2008

[50] T-C Lin S-W Chang and C-H Hsu ldquoRobust adaptivefuzzy sliding mode control for a class of uncertain discrete-time nonlinear systemsrdquo International Journal of InnovativeComputing Information and Control vol 8 no 1 pp 347ndash3592012

[51] G Feng andG Chen ldquoAdaptive control of discrete-time chaoticsystems a fuzzy control approachrdquoChaos Solitons and Fractalsvol 23 no 2 pp 459ndash467 2005

[52] H J Lee J B Park and G Chen ldquoRobust fuzzy controlof nonlinear systems with parametric uncertaintiesrdquo IEEETransactions on Fuzzy Systems vol 9 no 2 pp 369ndash379 2001

[53] Y-Y Cao and PM Frank ldquoRobust119867infindisturbance attenuation

for a class of uncertain discrete-time fuzzy systemsrdquo IEEETransactions on Fuzzy Systems vol 8 no 4 pp 406ndash415 2000

[54] S Zhou G Feng J Lam and S Xu ldquoRobust 119867infin

controlfor discrete-time fuzzy systems via basis-dependent Lyapunovfunctionsrdquo Information Sciences vol 174 no 3-4 pp 197ndash2172005

[55] S Xu and J Lam ldquoRobust 119867infin

control for uncertain discrete-time-delay fuzzy systems via output feedback controllersrdquo IEEETransactions on Fuzzy Systems vol 13 no 1 pp 82ndash93 2005

[56] C-S Tseng and B-S Chen ldquoRobust fuzzy observer-basedfuzzy control design for nonlinear discrete-time systems withpersistent bounded disturbancesrdquo IEEE Transactions on FuzzySystems vol 17 no 3 pp 711ndash723 2009

[57] S Xu B Song J Lu and J Lam ldquoRobust stability of uncertaindiscrete-time singular fuzzy systemsrdquo Fuzzy Sets and Systemsvol 158 no 20 pp 2306ndash2316 2007

[58] Z-G Wu P Shi H Su and J Chu ldquoReliable 119867infin

controlfor discrete-time fuzzy systems with infinite-distributed delayrdquoIEEE Transactions on Fuzzy Systems vol 20 no 1 pp 22ndash312012

[59] G Feng and J Ma ldquoQuadratic stabilization of uncertaindiscrete-time fuzzy dynamic systemsrdquo IEEE Transactions onCircuits and Systems I Fundamental Theory and Applicationsvol 48 no 11 pp 1337ndash1344 2001

[60] A Kruszewski R Wang and T M Guerra ldquoNonquadratic sta-bilization conditions for a class of uncertain nonlinear discretetime TS fuzzy models a new approachrdquo IEEE Transactions onAutomatic Control vol 53 no 2 pp 606ndash611 2008

[61] W-J Wang Y-J Chen and C-H Sun ldquoRelaxed stabilizationcriteria for discrete-time T-S fuzzy control systems based ona switching fuzzy model and piecewise Lyapunov functionrdquoIEEE Transactions on Systems Man amp Cybernetics Part BCybernetics vol 37 no 3 pp 551ndash559 2007

[62] G Feng ldquoStability analysis of discrete-time fuzzy dynamicsystems based on piecewise Lyapunov functionsrdquo IEEE Trans-actions on Fuzzy Systems vol 12 no 1 pp 22ndash28 2004

[63] H Gao X Liu and J Lam ldquoStability analysis and stabilizationfor discrete-time fuzzy systems with time-varying delayrdquo IEEETransactions on Systems Man and Cybernetics Part B Cyber-netics vol 39 no 2 pp 306ndash317 2009

[64] L Wu X Su P Shi and J Qiu ldquoA new approach to stabilityanalysis and stabilization of discrete-time T-S fuzzy time-varying delay systemsrdquo IEEE Transactions on SystemsMan andCybernetics Part B Cybernetics vol 41 no 1 pp 273ndash286 2011

[65] X Su P Shi L Wu and Y-D Song ldquoA novel control design ondiscrete-time takagi-sugeno fuzzy systems with time-varying

delaysrdquo IEEE Transactions on Fuzzy Systems vol 21 no 4 pp655ndash671 2013

[66] X Su P Shi L Wu and Y-D Song ldquoA novel approach to filterdesign for T-S fuzzy discrete-time systems with time-varyingdelayrdquo IEEE Transactions on Fuzzy Systems vol 20 no 6 pp1114ndash1129 2012

[67] C-S Tseng ldquoModel reference output feedback fuzzy trackingcontrol design for nonlinear discrete-time systems with time-delayrdquo IEEE Transactions on Fuzzy Systems vol 14 no 1 pp58ndash70 2006

[68] X Su P Shi L Wu and S K Nguang ldquoInduced l2 filteringof fuzzy stochastic systems with time-varying delaysrdquo IEEETransactions on Cybernetics vol 43 no 4 pp 1257ndash1264 2013

[69] L Wu X Su P Shi and J Qiu ldquoModel approximation fordiscrete-time state-delay systems in the TS fuzzy frameworkrdquoIEEE Transactions on Fuzzy Systems vol 19 no 2 pp 366ndash3782011

[70] F L Lewis S Jagannathan and A Yesildirek Neural NetworkControl of Robot Manipulators and Nonlinear Systems Taylor ampFrancis London UK 1999

[71] A M Shaw and F J Doyle III ldquoMultivariable nonlinear controlapplications for a high purity distillation column using arecurrent dynamic neuron modelrdquo Journal of Process Controlvol 7 no 4 pp 255ndash268 1997

[72] K Najim Process Modeling and Control in Chemical Engineer-ing Marcel Dekker New York NY USA 1989

[73] B Xu D Wang F Sun and Z Shi ldquoDirect neural discretecontrol of hypersonic flight vehiclerdquo Nonlinear Dynamics vol70 no 1 pp 269ndash278 2012

[74] B Xu and Y Zhang ldquoNeural discrete back-stepping controlof hypersonic flight vehicle with equivalent prediction modelrdquoNeurocomputing vol 154 pp 337ndash346 2015

[75] B Xu F Sun H Liu and J Ren ldquoAdaptive Kriging controllerdesign for hypersonic flight vehicle via back-steppingrdquo IETControl Theory amp Applications vol 6 no 4 pp 487ndash497 2012

[76] B Xu ldquoRobust adaptive neural control of flexible hypersonicflight vehicle with dead-zone input nonlinearityrdquo NonlinearDynamics vol 80 no 3 pp 1509ndash1520 2015

[77] B Xu XHuangDWang and F Sun ldquoDynamic surface controlof constrained hypersonic flightmodels with parameter estima-tion and actuator compensationrdquo Asian Journal of Control vol16 no 1 pp 162ndash174 2014

[78] B Xu and Z Shi ldquoAn overview on flight dynamics and controlapproaches for hypersonic vehiclesrdquo Science China InformationSciences vol 58 no 7 pp 1ndash19 2015

[79] D E Rumelhart G E Hinton and R J Williams ldquoLearninginternal representations by error propagationrdquo in Parallel Dis-tributed Processing vol 1 pp 318ndash362 MIT Press 1986

[80] S Jagannathan and F L Lewis ldquoDiscrete-time neural netcontroller for a class of nonlinear dynamical systemsrdquo IEEETransactions on Automatic Control vol 41 no 11 pp 1693ndash16991996

[81] S Jagannathan and F L Lewis ldquoMultilayer discrete-timeneural-net controller with guaranteed performancerdquo IEEETransactions on Neural Network vol 7 no 1 pp 107ndash130 1996

[82] P He and S Jagannathan ldquoNeuro-controller for reducingcyclic variation in lean combustion spark ignition enginesrdquoAutomatica vol 41 no 7 pp 1133ndash1142 2005

[83] S S Ge G Y Li and T H Lee ldquoAdaptive NN controlfor a class of strict-feedback discrete-time nonlinear systemsrdquoAutomatica vol 39 no 5 pp 807ndash819 2003

Discrete Dynamics in Nature and Society 11

[84] S S Ge T H Lee G Y Li and J Zhang ldquoAdaptive NN controlfor a class of discrete-time non-linear systemsrdquo InternationalJournal of Control vol 76 no 4 pp 334ndash354 2003

[85] C J Goh ldquoModel reference control of non-linear systems viaimplicit function emulationrdquo International Journal of Controlvol 60 no 1 pp 91ndash115 1994

[86] C J Goh and T H Lee ldquoDirect adaptive control of nonlinearsystems via implicit function emulationrdquo Control Theory andAdvanced Technology vol 10 no 3 pp 539ndash552 1994

[87] A U Levin and K S Narendra ldquoControl of nonlinear dynami-cal systems using neural networksmdashpart II observability iden-tification and controlrdquo IEEE Transactions on Neural Networksvol 7 no 1 pp 30ndash42 1996

[88] S S Ge J Zhang and T H Lee ldquoAdaptive MNN control fora class of non-affine NARMAX systems with disturbancesrdquoSystems amp Control Letters vol 53 no 1 pp 1ndash12 2004

[89] S S Ge J Zhang and T H Lee ldquoAdaptive neural networkcontrol for a class of MIMO nonlinear systems with distur-bances in discrete-timerdquo IEEE Transactions on Systems Manand Cybernetics Part B Cybernetics vol 34 no 4 pp 1630ndash1645 2004

[90] J Zhang S S Ge and T H Lee ldquoOutput feedback control of aclass of discrete MIMO nonlinear systems with triangular forminputsrdquo IEEE Transactions onNeural Networks vol 16 no 6 pp1491ndash1503 2005

[91] F C Sun Z Sun and P-Y Woo ldquoStable neural-network-basedadaptive control for sampled-data nonlinear systemsrdquo IEEETransactions on Neural Networks vol 9 no 5 pp 956ndash9681998

[92] C Yang S S Ge C Xiang T Chai and T H Lee ldquoOutputfeedback NN control for two classes of discrete-time systemswith unknown control directions in a unified approachrdquo IEEETransactions on Neural Networks vol 19 no 11 pp 1873ndash18862008

[93] S S Ge C Yang and T H Lee ldquoAdaptive predictive controlusing neural network for a class of pure-feedback systems indiscrete timerdquo IEEE Transactions on Neural Networks vol 19no 9 pp 1599ndash1614 2008

[94] Y Li C Yang S S Ge and T H Lee ldquoAdaptive output feed-back NN control of a class of discrete-time MIMO nonlinearsystems with unknown control directionsrdquo IEEE Transactionson Systems Man and Cybernetics Part B Cybernetics vol 41no 2 pp 507ndash517 2011

[95] Y-J Liu C L P Chen G-X Wen and S Tong ldquoAdaptiveneural output feedback tracking control for a class of uncertaindiscrete-time nonlinear systemsrdquo IEEE Transactions on NeuralNetworks vol 22 no 7 pp 1162ndash1167 2011

[96] C Yang S S Ge and T H Lee ldquoOutput feedback adaptivecontrol of a class of nonlinear discrete-time systems withunknown control directionsrdquoAutomatica vol 45 no 1 pp 270ndash276 2009

[97] A Al-Tamimi F L Lewis and M Abu-Khalaf ldquoDiscrete-timenonlinear HJB solution using approximate dynamic program-ming convergence proofrdquo IEEE Transactions on Systems Manand Cybernetics Part B Cybernetics vol 38 no 4 pp 943ndash9492008

[98] P Werbos ldquoApproximate dynamic programming for real-timecontrol and neuralmodelingrdquo inHandbook of Intelligent ControlNeural Fuzzy amp Adaptive Approaches Van Nostrand Reinhold1992

[99] P He and S Jagannathan ldquoReinforcement learning neural-network-based controller for nonlinear discrete-time systems

with input constraintsrdquo IEEETransactions on SystemsMan andCybernetics Part B Cybernetics vol 37 no 2 pp 425ndash436 2007

[100] B Xu C Yang and Z Shi ldquoReinforcement learning outputfeedback NN control using deterministic learning techniquerdquoIEEE Transactions on Neural Networks and Learning Systemsvol 25 no 3 pp 635ndash641 2014

[101] D Liu DWang D Zhao QWei and N Jin ldquoNeural-network-based optimal control for a class of unknowndiscrete-time non-linear systems using globalized dual heuristic programmingrdquoIEEE Transactions on Automation Science and Engineering vol9 no 3 pp 628ndash634 2012

[102] D Liu and Q Wei ldquoFinite-approximation-error-based optimalcontrol approach for discrete-time nonlinear systemsrdquo IEEETransactions on Cybernetics vol 43 no 2 pp 779ndash789 2013

[103] X Zhong H He H Zhang and Z Wang ldquoOptimal control forunknown discrete-time nonlinear markov jump systems usingadaptive dynamic programmingrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 25 no 12 pp 2141ndash21552014

[104] D Liu D Wang and X Yang ldquoAn iterative adaptive dynamicprogramming algorithm for optimal control of unknowndiscrete-time nonlinear systemswith constrained inputsrdquo Infor-mation Sciences vol 220 pp 331ndash342 2013

[105] F-Y Wang N Jin D Liu and Q Wei ldquoAdaptive dynamicprogramming for finite-horizon optimal control of discrete-time nonlinear systems with 120576-error boundrdquo IEEE Transactionson Neural Networks vol 22 no 1 pp 24ndash36 2011

[106] H Zhang Y Luo and D Liu ldquoNeural-network-based near-optimal control for a class of discrete-time affine nonlinearsystems with control constraintsrdquo IEEE Transactions on NeuralNetworks vol 20 no 9 pp 1490ndash1503 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Discrete Dynamics in Nature and Society

with finite approximation errors It was shown thatwhen con-vergence conditions are satisfied iterative performance indexfunction can converge to the performance index functionsrsquolower bound while the performance index functions wereapproximated by NNs to compute optimal control policy

The ADP control in the unknown discrete-time Markovjump systems was investigated by Zhong et al [103] AndWang et al [105] also studied the ADP for discrete-timesystems to obtain the 120576-optimal control by using neuralnetworks In [106] iterative ADP algorithmwas used to studythe near-optimal control with control constraints in discrete-time systems An iterative ADP for discrete-time systemswas studied in [104] In this work the optimal controllerwas designed with control constraints while NN was used toidentify the unknown dynamical systems with stability proof

4 Conclusion

In this short survey despite the impossibility in identifyingor listing all the related contributions best efforts have beenmade to summarize the major achievements in the area ofdiscrete-time adaptive control withmodern techniques basedon FL and NNs which are also conventionally termed asimportant part of ldquosoft-computingrdquo approaches or ldquointelli-gent controlrdquo in the control communities In particular themarriage of ldquoadaptive controlrdquo and ldquointelligent controlrdquo hasbeen reviewed for the purpose of clarifying main contribu-tions and outlining some possible trends for the developmentof this promising area

Generally speaking adaptive control for discrete-timelinear systems was extensively explored in the past decadesdespite the fact that the complete closed-loop stability analy-sis for the so-called self-tuning regulator was ever regarded asa long-term challenging problem Later it became clear thatdiscrete-time adaptive control can be extended to nonlinearsystems with linear growth rate However general nonlinearsystems with parametric andor nonparametric uncertaintiesare usually very difficult to cope with which motivatedvarious solutions for different certain classes of nonlinearsystems

Among the various solutions intelligent control emergedas one important way to resolve the challenges caused by thenonlinearityThe key for usability of intelligent control basedon FL or NNs lies in their universal approximator property atthe cost of tuning rules of FL or updating parameters of NNsFL and NNs are motivated by different background hencetheir design philosophies look different however essentiallyspeaking both of them can be expressed as weighted sumof some kernel functions where the weights can be tunedto approximate arbitrary smooth or continuous nonlinearfunction The ideas borrowed from adaptive estimation andadaptive control provide such a way to adaptively tune theweight parameters in FL orNNs thus the new area of adaptiveFLC or NN control emerged with extensive studies

In a summary a brief review on FLC for discrete-timesystems is provided by highlighting the adaptive FLC robust-ness issue and stability issue while NN control for discrete-time systems is also reviewed with focuses on adaptive NNcontrol and NN-based dynamic programming We believe

this topic would promote increasing investigations in boththeories and applications And some emerging techniquessuch as deep learning and big data could also bemerged withthe intelligent control for discrete-time systems and give birthto brand new design approaches of intelligent control in thefuture

Conflict of Interests

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

Acknowledgment

This work is partially supported by the National Natu-ral Science Foundation of China (NSFC) under Grants61473120 and 61473038 Guangdong Provincial Natural Sci-ence Foundation 2014A030313266 and International Scienceand Technology Collaboration Grant 2015A050502017 andFundamental Research Funds for the Central Universitiesunder Grant 2015ZM065 The authors would like to thankDr Bin Xu of Northwestern Polytechnical University for hisconstructive comments during the preparation of this paper

References

[1] G Goodwin and K Sin Adaptive Filtering Prediction andControl Prentice-Hall Englewood Cliffs NJ USA 1984

[2] K Astrom and B Wittenmark Adaptive Control Addison-Wesley 1989

[3] M Krstic I Kanellakopoulos and P V Kokotovic NonlinearandAdaptive Control Design JohnWileyamp Sons NewYork NYUSA 1995

[4] G Tao and P Kokotovic Aaptive Control of Systems with Actu-ator and Sensor Non-Linearities John Wiley amp Sons HobokenNJ USA 1996

[5] S S Ge C C Hang T H Lee and T Zhang Stable AdaptiveNeural Network Control Kluwer Academic Publishers NorwellMass USA 2001

[6] C Yang Adaptive control and neural network control of nonlin-ear discrete-time systems [PhD thesis] National University ofSingapore 2009

[7] C Yang H Ma and M Fu ldquoAdaptive predictive control ofperiodic non-linear auto-regressive moving average systemsusing nearest-neighbour compensationrdquo IET Control Theory ampApplications vol 7 no 7 pp 936ndash951 2013

[8] C Yang L Zhai S S Ge T Chai andTH Lee ldquoAdaptivemodelreference control of a class ofMIMOdiscrete-time systems withcompensation of nonparametric uncertaintyrdquo in Proceedings ofthe American Control Conference pp 4111ndash4116 IEEE SeattleWash USA June 2008

[9] S-L Dai C Yang S S Ge and T H Lee ldquoRobust adaptiveoutput feedback control of a class of discrete-time nonlinearsystems with nonlinear uncertainties and unknown controldirectionsrdquo International Journal of Robust and Nonlinear Con-trol vol 23 no 13 pp 1472ndash1495 2013

[10] K J Astrom and B Wittenmark ldquoOn self tuning regulatorsrdquoAutomatica vol 9 no 2 pp 185ndash199 1973

Discrete Dynamics in Nature and Society 9

[11] L Ljung ldquoAnalysis of recursive stochastic algorithmsrdquo IEEETransactions on Automatic Control vol 22 no 4 pp 551ndash5751977

[12] G C Goodwin P J Ramadge and P E Caines ldquoDiscretetime multivariable adaptive controlrdquo IEEE Transactions onAutomatic Control vol 25 no 3 pp 449ndash456 1980

[13] L Guo and H F Chen ldquoThe Astrom-Wittenmark self-tuningregulator revisited and ELS-based adaptive trackersrdquo IEEETransactions on Automatic Control vol 36 no 7 pp 802ndash8121991

[14] L Guo Time-Varing Stochastic Systems Jilin Science and Tech-nology Press Changchun China 1993 (Chinese)

[15] H F Chen and L Guo Identification and Stochastic AdaptiveControl Birkhauser Boston Mass USA 1991

[16] F P Skantze A Kojic A-P Loh and A M AnnaswamyldquoAdaptive estimation of discrete-time systems with nonlinearparameterizationrdquo Automatica vol 36 no 12 pp 1879ndash18872000

[17] L Chen and K S Narendra ldquoNonlinear adaptive control usingneural networks and multiple modelsrdquo Automatica vol 37 no8 pp 1245ndash1255 2001

[18] L Guo and C Wei ldquoLS-based discrete-time adaptive nonlinearcontrol feasibility and limitationsrdquo Science in China Series ETechnological Sciences vol 39 no 3 pp 255ndash269 1996

[19] L L Xie and L Guo ldquoAdaptive control of discrete-timenonlinear systems with structural uncertaintiesrdquo in Lectures onSystems Control and Information vol 17 of AMSIP Studiesin Advanced Mathematics American Mathematical SocietyInternational Press Providence RI USA 2000

[20] J D Boskovic ldquoStable adaptive control of a class of first-order nonlinearly parameterized plantsrdquo IEEE Transactions onAutomatic Control vol 40 no 2 pp 347ndash350 1995

[21] A L Fradkov I V Miroshnik and V O Nikiforov Nonlinearand Adaptive Control of Complex Systems Mathematics andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 2004

[22] D Angeli and E Mosca ldquoAdaptive switching supervisorycontrol of nonlinear systems with no prior knowledge of noiseboundsrdquo Automatica vol 40 no 3 pp 449ndash457 2004

[23] H B Ma ldquoFinite-model adaptive control using an LS-likealgorithmrdquo International Journal of Adaptive Control and SignalProcessing vol 21 no 5 pp 391ndash414 2007

[24] H B Ma ldquoFinite-model adaptive control using WLS-likealgorithmrdquo Automatica vol 43 no 4 pp 677ndash684 2007

[25] H B Ma ldquoSeveral algorithms for finite-model adaptive controlpartial answers to finite-model adaptive control problemrdquoMathematics of Control Signals and Systems vol 20 no 3 pp271ndash303 2008

[26] S S Ge C C Hang and T Zhang ldquoA direct adaptive controllerfor dynamic systems with a class of nonlinear parameteriza-tionsrdquo Automatica vol 35 no 4 pp 741ndash747 1999

[27] C Y Li and L Guo ldquoOn feedback capability in a class of nonlin-early parameterized uncertain systemsrdquo IEEE Transactions onAutomatic Control vol 56 no 12 pp 2946ndash2951 2011

[28] H Ma K-Y Lum and S S Ge ldquoAdaptive control for a discrete-time first-order nonlinear system with both parametric andnon-parametric uncertaintiesrdquo in Proceedings of the 46th IEEEConference on Decision and Control (CDC rsquo07) pp 4839ndash4844IEEE New Orleans La USA December 2007

[29] L Guo ldquoExploring the capability and limits of the feedbackmechanismrdquo in Proceedings of the International Congress ofMathematicians (ICM rsquo02) Beijing China August 2002

[30] H-B Ma ldquoAn lsquoimpossibilityrsquo theorem on a class of high-orderdiscrete-time nonlinear control systemsrdquo Systems and ControlLetters vol 57 no 6 pp 497ndash504 2008

[31] I Kanellakopoulos P V Kokotovic and A S Morse ldquoSys-tematic design of adaptive controllers for feedback linearizablesystemsrdquo IEEE Transactions on Automatic Control vol 36 no11 pp 1241ndash1253 1991

[32] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[33] L-X Wang and J M Mendel ldquoFuzzy basis functions universalapproximation and orthogonal least-squares learningrdquo IEEETransactions on Neural Networks vol 3 no 5 pp 807ndash814 1992

[34] L-X Wang ldquoStable adaptive fuzzy control of nonlinear sys-temsrdquo IEEE Transactions on Fuzzy Systems vol 1 no 2 pp 146ndash155 1993

[35] W S McCulloch and W Pitts ldquoA logical calculus of the ideasimmanent in nervous activityrdquo The Bulletin of MathematicalBiophysics vol 5 pp 115ndash133 1943

[36] K Hornik M Stinchcombe and HWhite ldquoMultilayer feedfor-ward networks are universal approximatorsrdquo Neural Networksvol 2 no 5 pp 359ndash366 1989

[37] T Khanna Foundations of Neural Networks Addison-WesleyReading Mass USA 1990

[38] R M Sanner and J-J E Slotine ldquoGaussian networks for directadaptive controlrdquo IEEE Transactions on Neural Networks vol 3no 6 pp 837ndash863 1992

[39] Y J Liu Y J Fang andM A Bao-Ping ldquoSliding-data-window-driven Bayesian-Gaussian neural network and its application tomodeling of nonlinear systemrdquo Control Theory amp Applicationsvol 26 no 12 pp 1435ndash1438 2009

[40] D Wang and J Huang ldquoAdaptive neural network control fora class of uncertain nonlinear systems in pure-feedback formrdquoAutomatica vol 38 no 8 pp 1365ndash1372 2002

[41] Y Song and J W Grizzle ldquoAdaptive output-feedback control ofa class of discrete-time nonlinear systemsrdquo in Proceedings of theAmerican Control Conference pp 1359ndash1363 June 1993

[42] B-S Chen C-S Tseng and H-J Uang ldquoRobustness designof nonlinear dynamic systems via fuzzy linear controlrdquo IEEETransactions on Fuzzy Systems vol 7 no 5 pp 571ndash585 1999

[43] T Chai and S Tong ldquoFuzzy direct adaptive control for a class ofnonlinear systemsrdquo Fuzzy Sets and Systems vol 103 no 3 pp379ndash387 1999

[44] K Tanaka T Ikeda and H O Wang ldquoRobust stabilizationof a class of uncertain nonlinear systems via fuzzy controlquadratic stabilizability 119867

infin control theory and linear matrixinequalitiesrdquo IEEE Transactions on Fuzzy Systems vol 4 no 1pp 1ndash13 1996

[45] S Jagannathan M W Vandegrift and F L Lewis ldquoAdaptivefuzzy logic control of discrete-time dynamical systemsrdquo Auto-matica vol 36 no 2 pp 229ndash241 2000

[46] Y Jiang Z Liu C Chen and Y Zhang ldquoAdaptive robust fuzzycontrol for dual arm robot with unknown input deadzonenonlinearityrdquo Nonlinear Dynamics vol 81 no 3 pp 1301ndash13142015

[47] MWVandegrift F L Lewis S Jagannathan andK Liu ldquoAdap-tive fuzzy logic control of discrete-time dynamical systemsrdquo inProceedings of the IEEE International Symposium on IntelligentControl pp 395ndash401 IEEE Monterey Calif USA August 1995

[48] S Jagannathan ldquoAdaptive fuzzy logic control of feedbacklinearizable discrete-time dynamical systems under persistenceof excitationrdquo Automatica vol 34 no 11 pp 1295ndash1310 1998

10 Discrete Dynamics in Nature and Society

[49] R Qi and M A Brdys ldquoStable indirect adaptive control basedon discrete-time T-S fuzzy modelrdquo Fuzzy Sets and Systems vol159 no 8 pp 900ndash925 2008

[50] T-C Lin S-W Chang and C-H Hsu ldquoRobust adaptivefuzzy sliding mode control for a class of uncertain discrete-time nonlinear systemsrdquo International Journal of InnovativeComputing Information and Control vol 8 no 1 pp 347ndash3592012

[51] G Feng andG Chen ldquoAdaptive control of discrete-time chaoticsystems a fuzzy control approachrdquoChaos Solitons and Fractalsvol 23 no 2 pp 459ndash467 2005

[52] H J Lee J B Park and G Chen ldquoRobust fuzzy controlof nonlinear systems with parametric uncertaintiesrdquo IEEETransactions on Fuzzy Systems vol 9 no 2 pp 369ndash379 2001

[53] Y-Y Cao and PM Frank ldquoRobust119867infindisturbance attenuation

for a class of uncertain discrete-time fuzzy systemsrdquo IEEETransactions on Fuzzy Systems vol 8 no 4 pp 406ndash415 2000

[54] S Zhou G Feng J Lam and S Xu ldquoRobust 119867infin

controlfor discrete-time fuzzy systems via basis-dependent Lyapunovfunctionsrdquo Information Sciences vol 174 no 3-4 pp 197ndash2172005

[55] S Xu and J Lam ldquoRobust 119867infin

control for uncertain discrete-time-delay fuzzy systems via output feedback controllersrdquo IEEETransactions on Fuzzy Systems vol 13 no 1 pp 82ndash93 2005

[56] C-S Tseng and B-S Chen ldquoRobust fuzzy observer-basedfuzzy control design for nonlinear discrete-time systems withpersistent bounded disturbancesrdquo IEEE Transactions on FuzzySystems vol 17 no 3 pp 711ndash723 2009

[57] S Xu B Song J Lu and J Lam ldquoRobust stability of uncertaindiscrete-time singular fuzzy systemsrdquo Fuzzy Sets and Systemsvol 158 no 20 pp 2306ndash2316 2007

[58] Z-G Wu P Shi H Su and J Chu ldquoReliable 119867infin

controlfor discrete-time fuzzy systems with infinite-distributed delayrdquoIEEE Transactions on Fuzzy Systems vol 20 no 1 pp 22ndash312012

[59] G Feng and J Ma ldquoQuadratic stabilization of uncertaindiscrete-time fuzzy dynamic systemsrdquo IEEE Transactions onCircuits and Systems I Fundamental Theory and Applicationsvol 48 no 11 pp 1337ndash1344 2001

[60] A Kruszewski R Wang and T M Guerra ldquoNonquadratic sta-bilization conditions for a class of uncertain nonlinear discretetime TS fuzzy models a new approachrdquo IEEE Transactions onAutomatic Control vol 53 no 2 pp 606ndash611 2008

[61] W-J Wang Y-J Chen and C-H Sun ldquoRelaxed stabilizationcriteria for discrete-time T-S fuzzy control systems based ona switching fuzzy model and piecewise Lyapunov functionrdquoIEEE Transactions on Systems Man amp Cybernetics Part BCybernetics vol 37 no 3 pp 551ndash559 2007

[62] G Feng ldquoStability analysis of discrete-time fuzzy dynamicsystems based on piecewise Lyapunov functionsrdquo IEEE Trans-actions on Fuzzy Systems vol 12 no 1 pp 22ndash28 2004

[63] H Gao X Liu and J Lam ldquoStability analysis and stabilizationfor discrete-time fuzzy systems with time-varying delayrdquo IEEETransactions on Systems Man and Cybernetics Part B Cyber-netics vol 39 no 2 pp 306ndash317 2009

[64] L Wu X Su P Shi and J Qiu ldquoA new approach to stabilityanalysis and stabilization of discrete-time T-S fuzzy time-varying delay systemsrdquo IEEE Transactions on SystemsMan andCybernetics Part B Cybernetics vol 41 no 1 pp 273ndash286 2011

[65] X Su P Shi L Wu and Y-D Song ldquoA novel control design ondiscrete-time takagi-sugeno fuzzy systems with time-varying

delaysrdquo IEEE Transactions on Fuzzy Systems vol 21 no 4 pp655ndash671 2013

[66] X Su P Shi L Wu and Y-D Song ldquoA novel approach to filterdesign for T-S fuzzy discrete-time systems with time-varyingdelayrdquo IEEE Transactions on Fuzzy Systems vol 20 no 6 pp1114ndash1129 2012

[67] C-S Tseng ldquoModel reference output feedback fuzzy trackingcontrol design for nonlinear discrete-time systems with time-delayrdquo IEEE Transactions on Fuzzy Systems vol 14 no 1 pp58ndash70 2006

[68] X Su P Shi L Wu and S K Nguang ldquoInduced l2 filteringof fuzzy stochastic systems with time-varying delaysrdquo IEEETransactions on Cybernetics vol 43 no 4 pp 1257ndash1264 2013

[69] L Wu X Su P Shi and J Qiu ldquoModel approximation fordiscrete-time state-delay systems in the TS fuzzy frameworkrdquoIEEE Transactions on Fuzzy Systems vol 19 no 2 pp 366ndash3782011

[70] F L Lewis S Jagannathan and A Yesildirek Neural NetworkControl of Robot Manipulators and Nonlinear Systems Taylor ampFrancis London UK 1999

[71] A M Shaw and F J Doyle III ldquoMultivariable nonlinear controlapplications for a high purity distillation column using arecurrent dynamic neuron modelrdquo Journal of Process Controlvol 7 no 4 pp 255ndash268 1997

[72] K Najim Process Modeling and Control in Chemical Engineer-ing Marcel Dekker New York NY USA 1989

[73] B Xu D Wang F Sun and Z Shi ldquoDirect neural discretecontrol of hypersonic flight vehiclerdquo Nonlinear Dynamics vol70 no 1 pp 269ndash278 2012

[74] B Xu and Y Zhang ldquoNeural discrete back-stepping controlof hypersonic flight vehicle with equivalent prediction modelrdquoNeurocomputing vol 154 pp 337ndash346 2015

[75] B Xu F Sun H Liu and J Ren ldquoAdaptive Kriging controllerdesign for hypersonic flight vehicle via back-steppingrdquo IETControl Theory amp Applications vol 6 no 4 pp 487ndash497 2012

[76] B Xu ldquoRobust adaptive neural control of flexible hypersonicflight vehicle with dead-zone input nonlinearityrdquo NonlinearDynamics vol 80 no 3 pp 1509ndash1520 2015

[77] B Xu XHuangDWang and F Sun ldquoDynamic surface controlof constrained hypersonic flightmodels with parameter estima-tion and actuator compensationrdquo Asian Journal of Control vol16 no 1 pp 162ndash174 2014

[78] B Xu and Z Shi ldquoAn overview on flight dynamics and controlapproaches for hypersonic vehiclesrdquo Science China InformationSciences vol 58 no 7 pp 1ndash19 2015

[79] D E Rumelhart G E Hinton and R J Williams ldquoLearninginternal representations by error propagationrdquo in Parallel Dis-tributed Processing vol 1 pp 318ndash362 MIT Press 1986

[80] S Jagannathan and F L Lewis ldquoDiscrete-time neural netcontroller for a class of nonlinear dynamical systemsrdquo IEEETransactions on Automatic Control vol 41 no 11 pp 1693ndash16991996

[81] S Jagannathan and F L Lewis ldquoMultilayer discrete-timeneural-net controller with guaranteed performancerdquo IEEETransactions on Neural Network vol 7 no 1 pp 107ndash130 1996

[82] P He and S Jagannathan ldquoNeuro-controller for reducingcyclic variation in lean combustion spark ignition enginesrdquoAutomatica vol 41 no 7 pp 1133ndash1142 2005

[83] S S Ge G Y Li and T H Lee ldquoAdaptive NN controlfor a class of strict-feedback discrete-time nonlinear systemsrdquoAutomatica vol 39 no 5 pp 807ndash819 2003

Discrete Dynamics in Nature and Society 11

[84] S S Ge T H Lee G Y Li and J Zhang ldquoAdaptive NN controlfor a class of discrete-time non-linear systemsrdquo InternationalJournal of Control vol 76 no 4 pp 334ndash354 2003

[85] C J Goh ldquoModel reference control of non-linear systems viaimplicit function emulationrdquo International Journal of Controlvol 60 no 1 pp 91ndash115 1994

[86] C J Goh and T H Lee ldquoDirect adaptive control of nonlinearsystems via implicit function emulationrdquo Control Theory andAdvanced Technology vol 10 no 3 pp 539ndash552 1994

[87] A U Levin and K S Narendra ldquoControl of nonlinear dynami-cal systems using neural networksmdashpart II observability iden-tification and controlrdquo IEEE Transactions on Neural Networksvol 7 no 1 pp 30ndash42 1996

[88] S S Ge J Zhang and T H Lee ldquoAdaptive MNN control fora class of non-affine NARMAX systems with disturbancesrdquoSystems amp Control Letters vol 53 no 1 pp 1ndash12 2004

[89] S S Ge J Zhang and T H Lee ldquoAdaptive neural networkcontrol for a class of MIMO nonlinear systems with distur-bances in discrete-timerdquo IEEE Transactions on Systems Manand Cybernetics Part B Cybernetics vol 34 no 4 pp 1630ndash1645 2004

[90] J Zhang S S Ge and T H Lee ldquoOutput feedback control of aclass of discrete MIMO nonlinear systems with triangular forminputsrdquo IEEE Transactions onNeural Networks vol 16 no 6 pp1491ndash1503 2005

[91] F C Sun Z Sun and P-Y Woo ldquoStable neural-network-basedadaptive control for sampled-data nonlinear systemsrdquo IEEETransactions on Neural Networks vol 9 no 5 pp 956ndash9681998

[92] C Yang S S Ge C Xiang T Chai and T H Lee ldquoOutputfeedback NN control for two classes of discrete-time systemswith unknown control directions in a unified approachrdquo IEEETransactions on Neural Networks vol 19 no 11 pp 1873ndash18862008

[93] S S Ge C Yang and T H Lee ldquoAdaptive predictive controlusing neural network for a class of pure-feedback systems indiscrete timerdquo IEEE Transactions on Neural Networks vol 19no 9 pp 1599ndash1614 2008

[94] Y Li C Yang S S Ge and T H Lee ldquoAdaptive output feed-back NN control of a class of discrete-time MIMO nonlinearsystems with unknown control directionsrdquo IEEE Transactionson Systems Man and Cybernetics Part B Cybernetics vol 41no 2 pp 507ndash517 2011

[95] Y-J Liu C L P Chen G-X Wen and S Tong ldquoAdaptiveneural output feedback tracking control for a class of uncertaindiscrete-time nonlinear systemsrdquo IEEE Transactions on NeuralNetworks vol 22 no 7 pp 1162ndash1167 2011

[96] C Yang S S Ge and T H Lee ldquoOutput feedback adaptivecontrol of a class of nonlinear discrete-time systems withunknown control directionsrdquoAutomatica vol 45 no 1 pp 270ndash276 2009

[97] A Al-Tamimi F L Lewis and M Abu-Khalaf ldquoDiscrete-timenonlinear HJB solution using approximate dynamic program-ming convergence proofrdquo IEEE Transactions on Systems Manand Cybernetics Part B Cybernetics vol 38 no 4 pp 943ndash9492008

[98] P Werbos ldquoApproximate dynamic programming for real-timecontrol and neuralmodelingrdquo inHandbook of Intelligent ControlNeural Fuzzy amp Adaptive Approaches Van Nostrand Reinhold1992

[99] P He and S Jagannathan ldquoReinforcement learning neural-network-based controller for nonlinear discrete-time systems

with input constraintsrdquo IEEETransactions on SystemsMan andCybernetics Part B Cybernetics vol 37 no 2 pp 425ndash436 2007

[100] B Xu C Yang and Z Shi ldquoReinforcement learning outputfeedback NN control using deterministic learning techniquerdquoIEEE Transactions on Neural Networks and Learning Systemsvol 25 no 3 pp 635ndash641 2014

[101] D Liu DWang D Zhao QWei and N Jin ldquoNeural-network-based optimal control for a class of unknowndiscrete-time non-linear systems using globalized dual heuristic programmingrdquoIEEE Transactions on Automation Science and Engineering vol9 no 3 pp 628ndash634 2012

[102] D Liu and Q Wei ldquoFinite-approximation-error-based optimalcontrol approach for discrete-time nonlinear systemsrdquo IEEETransactions on Cybernetics vol 43 no 2 pp 779ndash789 2013

[103] X Zhong H He H Zhang and Z Wang ldquoOptimal control forunknown discrete-time nonlinear markov jump systems usingadaptive dynamic programmingrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 25 no 12 pp 2141ndash21552014

[104] D Liu D Wang and X Yang ldquoAn iterative adaptive dynamicprogramming algorithm for optimal control of unknowndiscrete-time nonlinear systemswith constrained inputsrdquo Infor-mation Sciences vol 220 pp 331ndash342 2013

[105] F-Y Wang N Jin D Liu and Q Wei ldquoAdaptive dynamicprogramming for finite-horizon optimal control of discrete-time nonlinear systems with 120576-error boundrdquo IEEE Transactionson Neural Networks vol 22 no 1 pp 24ndash36 2011

[106] H Zhang Y Luo and D Liu ldquoNeural-network-based near-optimal control for a class of discrete-time affine nonlinearsystems with control constraintsrdquo IEEE Transactions on NeuralNetworks vol 20 no 9 pp 1490ndash1503 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 9

[11] L Ljung ldquoAnalysis of recursive stochastic algorithmsrdquo IEEETransactions on Automatic Control vol 22 no 4 pp 551ndash5751977

[12] G C Goodwin P J Ramadge and P E Caines ldquoDiscretetime multivariable adaptive controlrdquo IEEE Transactions onAutomatic Control vol 25 no 3 pp 449ndash456 1980

[13] L Guo and H F Chen ldquoThe Astrom-Wittenmark self-tuningregulator revisited and ELS-based adaptive trackersrdquo IEEETransactions on Automatic Control vol 36 no 7 pp 802ndash8121991

[14] L Guo Time-Varing Stochastic Systems Jilin Science and Tech-nology Press Changchun China 1993 (Chinese)

[15] H F Chen and L Guo Identification and Stochastic AdaptiveControl Birkhauser Boston Mass USA 1991

[16] F P Skantze A Kojic A-P Loh and A M AnnaswamyldquoAdaptive estimation of discrete-time systems with nonlinearparameterizationrdquo Automatica vol 36 no 12 pp 1879ndash18872000

[17] L Chen and K S Narendra ldquoNonlinear adaptive control usingneural networks and multiple modelsrdquo Automatica vol 37 no8 pp 1245ndash1255 2001

[18] L Guo and C Wei ldquoLS-based discrete-time adaptive nonlinearcontrol feasibility and limitationsrdquo Science in China Series ETechnological Sciences vol 39 no 3 pp 255ndash269 1996

[19] L L Xie and L Guo ldquoAdaptive control of discrete-timenonlinear systems with structural uncertaintiesrdquo in Lectures onSystems Control and Information vol 17 of AMSIP Studiesin Advanced Mathematics American Mathematical SocietyInternational Press Providence RI USA 2000

[20] J D Boskovic ldquoStable adaptive control of a class of first-order nonlinearly parameterized plantsrdquo IEEE Transactions onAutomatic Control vol 40 no 2 pp 347ndash350 1995

[21] A L Fradkov I V Miroshnik and V O Nikiforov Nonlinearand Adaptive Control of Complex Systems Mathematics andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 2004

[22] D Angeli and E Mosca ldquoAdaptive switching supervisorycontrol of nonlinear systems with no prior knowledge of noiseboundsrdquo Automatica vol 40 no 3 pp 449ndash457 2004

[23] H B Ma ldquoFinite-model adaptive control using an LS-likealgorithmrdquo International Journal of Adaptive Control and SignalProcessing vol 21 no 5 pp 391ndash414 2007

[24] H B Ma ldquoFinite-model adaptive control using WLS-likealgorithmrdquo Automatica vol 43 no 4 pp 677ndash684 2007

[25] H B Ma ldquoSeveral algorithms for finite-model adaptive controlpartial answers to finite-model adaptive control problemrdquoMathematics of Control Signals and Systems vol 20 no 3 pp271ndash303 2008

[26] S S Ge C C Hang and T Zhang ldquoA direct adaptive controllerfor dynamic systems with a class of nonlinear parameteriza-tionsrdquo Automatica vol 35 no 4 pp 741ndash747 1999

[27] C Y Li and L Guo ldquoOn feedback capability in a class of nonlin-early parameterized uncertain systemsrdquo IEEE Transactions onAutomatic Control vol 56 no 12 pp 2946ndash2951 2011

[28] H Ma K-Y Lum and S S Ge ldquoAdaptive control for a discrete-time first-order nonlinear system with both parametric andnon-parametric uncertaintiesrdquo in Proceedings of the 46th IEEEConference on Decision and Control (CDC rsquo07) pp 4839ndash4844IEEE New Orleans La USA December 2007

[29] L Guo ldquoExploring the capability and limits of the feedbackmechanismrdquo in Proceedings of the International Congress ofMathematicians (ICM rsquo02) Beijing China August 2002

[30] H-B Ma ldquoAn lsquoimpossibilityrsquo theorem on a class of high-orderdiscrete-time nonlinear control systemsrdquo Systems and ControlLetters vol 57 no 6 pp 497ndash504 2008

[31] I Kanellakopoulos P V Kokotovic and A S Morse ldquoSys-tematic design of adaptive controllers for feedback linearizablesystemsrdquo IEEE Transactions on Automatic Control vol 36 no11 pp 1241ndash1253 1991

[32] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[33] L-X Wang and J M Mendel ldquoFuzzy basis functions universalapproximation and orthogonal least-squares learningrdquo IEEETransactions on Neural Networks vol 3 no 5 pp 807ndash814 1992

[34] L-X Wang ldquoStable adaptive fuzzy control of nonlinear sys-temsrdquo IEEE Transactions on Fuzzy Systems vol 1 no 2 pp 146ndash155 1993

[35] W S McCulloch and W Pitts ldquoA logical calculus of the ideasimmanent in nervous activityrdquo The Bulletin of MathematicalBiophysics vol 5 pp 115ndash133 1943

[36] K Hornik M Stinchcombe and HWhite ldquoMultilayer feedfor-ward networks are universal approximatorsrdquo Neural Networksvol 2 no 5 pp 359ndash366 1989

[37] T Khanna Foundations of Neural Networks Addison-WesleyReading Mass USA 1990

[38] R M Sanner and J-J E Slotine ldquoGaussian networks for directadaptive controlrdquo IEEE Transactions on Neural Networks vol 3no 6 pp 837ndash863 1992

[39] Y J Liu Y J Fang andM A Bao-Ping ldquoSliding-data-window-driven Bayesian-Gaussian neural network and its application tomodeling of nonlinear systemrdquo Control Theory amp Applicationsvol 26 no 12 pp 1435ndash1438 2009

[40] D Wang and J Huang ldquoAdaptive neural network control fora class of uncertain nonlinear systems in pure-feedback formrdquoAutomatica vol 38 no 8 pp 1365ndash1372 2002

[41] Y Song and J W Grizzle ldquoAdaptive output-feedback control ofa class of discrete-time nonlinear systemsrdquo in Proceedings of theAmerican Control Conference pp 1359ndash1363 June 1993

[42] B-S Chen C-S Tseng and H-J Uang ldquoRobustness designof nonlinear dynamic systems via fuzzy linear controlrdquo IEEETransactions on Fuzzy Systems vol 7 no 5 pp 571ndash585 1999

[43] T Chai and S Tong ldquoFuzzy direct adaptive control for a class ofnonlinear systemsrdquo Fuzzy Sets and Systems vol 103 no 3 pp379ndash387 1999

[44] K Tanaka T Ikeda and H O Wang ldquoRobust stabilizationof a class of uncertain nonlinear systems via fuzzy controlquadratic stabilizability 119867

infin control theory and linear matrixinequalitiesrdquo IEEE Transactions on Fuzzy Systems vol 4 no 1pp 1ndash13 1996

[45] S Jagannathan M W Vandegrift and F L Lewis ldquoAdaptivefuzzy logic control of discrete-time dynamical systemsrdquo Auto-matica vol 36 no 2 pp 229ndash241 2000

[46] Y Jiang Z Liu C Chen and Y Zhang ldquoAdaptive robust fuzzycontrol for dual arm robot with unknown input deadzonenonlinearityrdquo Nonlinear Dynamics vol 81 no 3 pp 1301ndash13142015

[47] MWVandegrift F L Lewis S Jagannathan andK Liu ldquoAdap-tive fuzzy logic control of discrete-time dynamical systemsrdquo inProceedings of the IEEE International Symposium on IntelligentControl pp 395ndash401 IEEE Monterey Calif USA August 1995

[48] S Jagannathan ldquoAdaptive fuzzy logic control of feedbacklinearizable discrete-time dynamical systems under persistenceof excitationrdquo Automatica vol 34 no 11 pp 1295ndash1310 1998

10 Discrete Dynamics in Nature and Society

[49] R Qi and M A Brdys ldquoStable indirect adaptive control basedon discrete-time T-S fuzzy modelrdquo Fuzzy Sets and Systems vol159 no 8 pp 900ndash925 2008

[50] T-C Lin S-W Chang and C-H Hsu ldquoRobust adaptivefuzzy sliding mode control for a class of uncertain discrete-time nonlinear systemsrdquo International Journal of InnovativeComputing Information and Control vol 8 no 1 pp 347ndash3592012

[51] G Feng andG Chen ldquoAdaptive control of discrete-time chaoticsystems a fuzzy control approachrdquoChaos Solitons and Fractalsvol 23 no 2 pp 459ndash467 2005

[52] H J Lee J B Park and G Chen ldquoRobust fuzzy controlof nonlinear systems with parametric uncertaintiesrdquo IEEETransactions on Fuzzy Systems vol 9 no 2 pp 369ndash379 2001

[53] Y-Y Cao and PM Frank ldquoRobust119867infindisturbance attenuation

for a class of uncertain discrete-time fuzzy systemsrdquo IEEETransactions on Fuzzy Systems vol 8 no 4 pp 406ndash415 2000

[54] S Zhou G Feng J Lam and S Xu ldquoRobust 119867infin

controlfor discrete-time fuzzy systems via basis-dependent Lyapunovfunctionsrdquo Information Sciences vol 174 no 3-4 pp 197ndash2172005

[55] S Xu and J Lam ldquoRobust 119867infin

control for uncertain discrete-time-delay fuzzy systems via output feedback controllersrdquo IEEETransactions on Fuzzy Systems vol 13 no 1 pp 82ndash93 2005

[56] C-S Tseng and B-S Chen ldquoRobust fuzzy observer-basedfuzzy control design for nonlinear discrete-time systems withpersistent bounded disturbancesrdquo IEEE Transactions on FuzzySystems vol 17 no 3 pp 711ndash723 2009

[57] S Xu B Song J Lu and J Lam ldquoRobust stability of uncertaindiscrete-time singular fuzzy systemsrdquo Fuzzy Sets and Systemsvol 158 no 20 pp 2306ndash2316 2007

[58] Z-G Wu P Shi H Su and J Chu ldquoReliable 119867infin

controlfor discrete-time fuzzy systems with infinite-distributed delayrdquoIEEE Transactions on Fuzzy Systems vol 20 no 1 pp 22ndash312012

[59] G Feng and J Ma ldquoQuadratic stabilization of uncertaindiscrete-time fuzzy dynamic systemsrdquo IEEE Transactions onCircuits and Systems I Fundamental Theory and Applicationsvol 48 no 11 pp 1337ndash1344 2001

[60] A Kruszewski R Wang and T M Guerra ldquoNonquadratic sta-bilization conditions for a class of uncertain nonlinear discretetime TS fuzzy models a new approachrdquo IEEE Transactions onAutomatic Control vol 53 no 2 pp 606ndash611 2008

[61] W-J Wang Y-J Chen and C-H Sun ldquoRelaxed stabilizationcriteria for discrete-time T-S fuzzy control systems based ona switching fuzzy model and piecewise Lyapunov functionrdquoIEEE Transactions on Systems Man amp Cybernetics Part BCybernetics vol 37 no 3 pp 551ndash559 2007

[62] G Feng ldquoStability analysis of discrete-time fuzzy dynamicsystems based on piecewise Lyapunov functionsrdquo IEEE Trans-actions on Fuzzy Systems vol 12 no 1 pp 22ndash28 2004

[63] H Gao X Liu and J Lam ldquoStability analysis and stabilizationfor discrete-time fuzzy systems with time-varying delayrdquo IEEETransactions on Systems Man and Cybernetics Part B Cyber-netics vol 39 no 2 pp 306ndash317 2009

[64] L Wu X Su P Shi and J Qiu ldquoA new approach to stabilityanalysis and stabilization of discrete-time T-S fuzzy time-varying delay systemsrdquo IEEE Transactions on SystemsMan andCybernetics Part B Cybernetics vol 41 no 1 pp 273ndash286 2011

[65] X Su P Shi L Wu and Y-D Song ldquoA novel control design ondiscrete-time takagi-sugeno fuzzy systems with time-varying

delaysrdquo IEEE Transactions on Fuzzy Systems vol 21 no 4 pp655ndash671 2013

[66] X Su P Shi L Wu and Y-D Song ldquoA novel approach to filterdesign for T-S fuzzy discrete-time systems with time-varyingdelayrdquo IEEE Transactions on Fuzzy Systems vol 20 no 6 pp1114ndash1129 2012

[67] C-S Tseng ldquoModel reference output feedback fuzzy trackingcontrol design for nonlinear discrete-time systems with time-delayrdquo IEEE Transactions on Fuzzy Systems vol 14 no 1 pp58ndash70 2006

[68] X Su P Shi L Wu and S K Nguang ldquoInduced l2 filteringof fuzzy stochastic systems with time-varying delaysrdquo IEEETransactions on Cybernetics vol 43 no 4 pp 1257ndash1264 2013

[69] L Wu X Su P Shi and J Qiu ldquoModel approximation fordiscrete-time state-delay systems in the TS fuzzy frameworkrdquoIEEE Transactions on Fuzzy Systems vol 19 no 2 pp 366ndash3782011

[70] F L Lewis S Jagannathan and A Yesildirek Neural NetworkControl of Robot Manipulators and Nonlinear Systems Taylor ampFrancis London UK 1999

[71] A M Shaw and F J Doyle III ldquoMultivariable nonlinear controlapplications for a high purity distillation column using arecurrent dynamic neuron modelrdquo Journal of Process Controlvol 7 no 4 pp 255ndash268 1997

[72] K Najim Process Modeling and Control in Chemical Engineer-ing Marcel Dekker New York NY USA 1989

[73] B Xu D Wang F Sun and Z Shi ldquoDirect neural discretecontrol of hypersonic flight vehiclerdquo Nonlinear Dynamics vol70 no 1 pp 269ndash278 2012

[74] B Xu and Y Zhang ldquoNeural discrete back-stepping controlof hypersonic flight vehicle with equivalent prediction modelrdquoNeurocomputing vol 154 pp 337ndash346 2015

[75] B Xu F Sun H Liu and J Ren ldquoAdaptive Kriging controllerdesign for hypersonic flight vehicle via back-steppingrdquo IETControl Theory amp Applications vol 6 no 4 pp 487ndash497 2012

[76] B Xu ldquoRobust adaptive neural control of flexible hypersonicflight vehicle with dead-zone input nonlinearityrdquo NonlinearDynamics vol 80 no 3 pp 1509ndash1520 2015

[77] B Xu XHuangDWang and F Sun ldquoDynamic surface controlof constrained hypersonic flightmodels with parameter estima-tion and actuator compensationrdquo Asian Journal of Control vol16 no 1 pp 162ndash174 2014

[78] B Xu and Z Shi ldquoAn overview on flight dynamics and controlapproaches for hypersonic vehiclesrdquo Science China InformationSciences vol 58 no 7 pp 1ndash19 2015

[79] D E Rumelhart G E Hinton and R J Williams ldquoLearninginternal representations by error propagationrdquo in Parallel Dis-tributed Processing vol 1 pp 318ndash362 MIT Press 1986

[80] S Jagannathan and F L Lewis ldquoDiscrete-time neural netcontroller for a class of nonlinear dynamical systemsrdquo IEEETransactions on Automatic Control vol 41 no 11 pp 1693ndash16991996

[81] S Jagannathan and F L Lewis ldquoMultilayer discrete-timeneural-net controller with guaranteed performancerdquo IEEETransactions on Neural Network vol 7 no 1 pp 107ndash130 1996

[82] P He and S Jagannathan ldquoNeuro-controller for reducingcyclic variation in lean combustion spark ignition enginesrdquoAutomatica vol 41 no 7 pp 1133ndash1142 2005

[83] S S Ge G Y Li and T H Lee ldquoAdaptive NN controlfor a class of strict-feedback discrete-time nonlinear systemsrdquoAutomatica vol 39 no 5 pp 807ndash819 2003

Discrete Dynamics in Nature and Society 11

[84] S S Ge T H Lee G Y Li and J Zhang ldquoAdaptive NN controlfor a class of discrete-time non-linear systemsrdquo InternationalJournal of Control vol 76 no 4 pp 334ndash354 2003

[85] C J Goh ldquoModel reference control of non-linear systems viaimplicit function emulationrdquo International Journal of Controlvol 60 no 1 pp 91ndash115 1994

[86] C J Goh and T H Lee ldquoDirect adaptive control of nonlinearsystems via implicit function emulationrdquo Control Theory andAdvanced Technology vol 10 no 3 pp 539ndash552 1994

[87] A U Levin and K S Narendra ldquoControl of nonlinear dynami-cal systems using neural networksmdashpart II observability iden-tification and controlrdquo IEEE Transactions on Neural Networksvol 7 no 1 pp 30ndash42 1996

[88] S S Ge J Zhang and T H Lee ldquoAdaptive MNN control fora class of non-affine NARMAX systems with disturbancesrdquoSystems amp Control Letters vol 53 no 1 pp 1ndash12 2004

[89] S S Ge J Zhang and T H Lee ldquoAdaptive neural networkcontrol for a class of MIMO nonlinear systems with distur-bances in discrete-timerdquo IEEE Transactions on Systems Manand Cybernetics Part B Cybernetics vol 34 no 4 pp 1630ndash1645 2004

[90] J Zhang S S Ge and T H Lee ldquoOutput feedback control of aclass of discrete MIMO nonlinear systems with triangular forminputsrdquo IEEE Transactions onNeural Networks vol 16 no 6 pp1491ndash1503 2005

[91] F C Sun Z Sun and P-Y Woo ldquoStable neural-network-basedadaptive control for sampled-data nonlinear systemsrdquo IEEETransactions on Neural Networks vol 9 no 5 pp 956ndash9681998

[92] C Yang S S Ge C Xiang T Chai and T H Lee ldquoOutputfeedback NN control for two classes of discrete-time systemswith unknown control directions in a unified approachrdquo IEEETransactions on Neural Networks vol 19 no 11 pp 1873ndash18862008

[93] S S Ge C Yang and T H Lee ldquoAdaptive predictive controlusing neural network for a class of pure-feedback systems indiscrete timerdquo IEEE Transactions on Neural Networks vol 19no 9 pp 1599ndash1614 2008

[94] Y Li C Yang S S Ge and T H Lee ldquoAdaptive output feed-back NN control of a class of discrete-time MIMO nonlinearsystems with unknown control directionsrdquo IEEE Transactionson Systems Man and Cybernetics Part B Cybernetics vol 41no 2 pp 507ndash517 2011

[95] Y-J Liu C L P Chen G-X Wen and S Tong ldquoAdaptiveneural output feedback tracking control for a class of uncertaindiscrete-time nonlinear systemsrdquo IEEE Transactions on NeuralNetworks vol 22 no 7 pp 1162ndash1167 2011

[96] C Yang S S Ge and T H Lee ldquoOutput feedback adaptivecontrol of a class of nonlinear discrete-time systems withunknown control directionsrdquoAutomatica vol 45 no 1 pp 270ndash276 2009

[97] A Al-Tamimi F L Lewis and M Abu-Khalaf ldquoDiscrete-timenonlinear HJB solution using approximate dynamic program-ming convergence proofrdquo IEEE Transactions on Systems Manand Cybernetics Part B Cybernetics vol 38 no 4 pp 943ndash9492008

[98] P Werbos ldquoApproximate dynamic programming for real-timecontrol and neuralmodelingrdquo inHandbook of Intelligent ControlNeural Fuzzy amp Adaptive Approaches Van Nostrand Reinhold1992

[99] P He and S Jagannathan ldquoReinforcement learning neural-network-based controller for nonlinear discrete-time systems

with input constraintsrdquo IEEETransactions on SystemsMan andCybernetics Part B Cybernetics vol 37 no 2 pp 425ndash436 2007

[100] B Xu C Yang and Z Shi ldquoReinforcement learning outputfeedback NN control using deterministic learning techniquerdquoIEEE Transactions on Neural Networks and Learning Systemsvol 25 no 3 pp 635ndash641 2014

[101] D Liu DWang D Zhao QWei and N Jin ldquoNeural-network-based optimal control for a class of unknowndiscrete-time non-linear systems using globalized dual heuristic programmingrdquoIEEE Transactions on Automation Science and Engineering vol9 no 3 pp 628ndash634 2012

[102] D Liu and Q Wei ldquoFinite-approximation-error-based optimalcontrol approach for discrete-time nonlinear systemsrdquo IEEETransactions on Cybernetics vol 43 no 2 pp 779ndash789 2013

[103] X Zhong H He H Zhang and Z Wang ldquoOptimal control forunknown discrete-time nonlinear markov jump systems usingadaptive dynamic programmingrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 25 no 12 pp 2141ndash21552014

[104] D Liu D Wang and X Yang ldquoAn iterative adaptive dynamicprogramming algorithm for optimal control of unknowndiscrete-time nonlinear systemswith constrained inputsrdquo Infor-mation Sciences vol 220 pp 331ndash342 2013

[105] F-Y Wang N Jin D Liu and Q Wei ldquoAdaptive dynamicprogramming for finite-horizon optimal control of discrete-time nonlinear systems with 120576-error boundrdquo IEEE Transactionson Neural Networks vol 22 no 1 pp 24ndash36 2011

[106] H Zhang Y Luo and D Liu ldquoNeural-network-based near-optimal control for a class of discrete-time affine nonlinearsystems with control constraintsrdquo IEEE Transactions on NeuralNetworks vol 20 no 9 pp 1490ndash1503 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Discrete Dynamics in Nature and Society

[49] R Qi and M A Brdys ldquoStable indirect adaptive control basedon discrete-time T-S fuzzy modelrdquo Fuzzy Sets and Systems vol159 no 8 pp 900ndash925 2008

[50] T-C Lin S-W Chang and C-H Hsu ldquoRobust adaptivefuzzy sliding mode control for a class of uncertain discrete-time nonlinear systemsrdquo International Journal of InnovativeComputing Information and Control vol 8 no 1 pp 347ndash3592012

[51] G Feng andG Chen ldquoAdaptive control of discrete-time chaoticsystems a fuzzy control approachrdquoChaos Solitons and Fractalsvol 23 no 2 pp 459ndash467 2005

[52] H J Lee J B Park and G Chen ldquoRobust fuzzy controlof nonlinear systems with parametric uncertaintiesrdquo IEEETransactions on Fuzzy Systems vol 9 no 2 pp 369ndash379 2001

[53] Y-Y Cao and PM Frank ldquoRobust119867infindisturbance attenuation

for a class of uncertain discrete-time fuzzy systemsrdquo IEEETransactions on Fuzzy Systems vol 8 no 4 pp 406ndash415 2000

[54] S Zhou G Feng J Lam and S Xu ldquoRobust 119867infin

controlfor discrete-time fuzzy systems via basis-dependent Lyapunovfunctionsrdquo Information Sciences vol 174 no 3-4 pp 197ndash2172005

[55] S Xu and J Lam ldquoRobust 119867infin

control for uncertain discrete-time-delay fuzzy systems via output feedback controllersrdquo IEEETransactions on Fuzzy Systems vol 13 no 1 pp 82ndash93 2005

[56] C-S Tseng and B-S Chen ldquoRobust fuzzy observer-basedfuzzy control design for nonlinear discrete-time systems withpersistent bounded disturbancesrdquo IEEE Transactions on FuzzySystems vol 17 no 3 pp 711ndash723 2009

[57] S Xu B Song J Lu and J Lam ldquoRobust stability of uncertaindiscrete-time singular fuzzy systemsrdquo Fuzzy Sets and Systemsvol 158 no 20 pp 2306ndash2316 2007

[58] Z-G Wu P Shi H Su and J Chu ldquoReliable 119867infin

controlfor discrete-time fuzzy systems with infinite-distributed delayrdquoIEEE Transactions on Fuzzy Systems vol 20 no 1 pp 22ndash312012

[59] G Feng and J Ma ldquoQuadratic stabilization of uncertaindiscrete-time fuzzy dynamic systemsrdquo IEEE Transactions onCircuits and Systems I Fundamental Theory and Applicationsvol 48 no 11 pp 1337ndash1344 2001

[60] A Kruszewski R Wang and T M Guerra ldquoNonquadratic sta-bilization conditions for a class of uncertain nonlinear discretetime TS fuzzy models a new approachrdquo IEEE Transactions onAutomatic Control vol 53 no 2 pp 606ndash611 2008

[61] W-J Wang Y-J Chen and C-H Sun ldquoRelaxed stabilizationcriteria for discrete-time T-S fuzzy control systems based ona switching fuzzy model and piecewise Lyapunov functionrdquoIEEE Transactions on Systems Man amp Cybernetics Part BCybernetics vol 37 no 3 pp 551ndash559 2007

[62] G Feng ldquoStability analysis of discrete-time fuzzy dynamicsystems based on piecewise Lyapunov functionsrdquo IEEE Trans-actions on Fuzzy Systems vol 12 no 1 pp 22ndash28 2004

[63] H Gao X Liu and J Lam ldquoStability analysis and stabilizationfor discrete-time fuzzy systems with time-varying delayrdquo IEEETransactions on Systems Man and Cybernetics Part B Cyber-netics vol 39 no 2 pp 306ndash317 2009

[64] L Wu X Su P Shi and J Qiu ldquoA new approach to stabilityanalysis and stabilization of discrete-time T-S fuzzy time-varying delay systemsrdquo IEEE Transactions on SystemsMan andCybernetics Part B Cybernetics vol 41 no 1 pp 273ndash286 2011

[65] X Su P Shi L Wu and Y-D Song ldquoA novel control design ondiscrete-time takagi-sugeno fuzzy systems with time-varying

delaysrdquo IEEE Transactions on Fuzzy Systems vol 21 no 4 pp655ndash671 2013

[66] X Su P Shi L Wu and Y-D Song ldquoA novel approach to filterdesign for T-S fuzzy discrete-time systems with time-varyingdelayrdquo IEEE Transactions on Fuzzy Systems vol 20 no 6 pp1114ndash1129 2012

[67] C-S Tseng ldquoModel reference output feedback fuzzy trackingcontrol design for nonlinear discrete-time systems with time-delayrdquo IEEE Transactions on Fuzzy Systems vol 14 no 1 pp58ndash70 2006

[68] X Su P Shi L Wu and S K Nguang ldquoInduced l2 filteringof fuzzy stochastic systems with time-varying delaysrdquo IEEETransactions on Cybernetics vol 43 no 4 pp 1257ndash1264 2013

[69] L Wu X Su P Shi and J Qiu ldquoModel approximation fordiscrete-time state-delay systems in the TS fuzzy frameworkrdquoIEEE Transactions on Fuzzy Systems vol 19 no 2 pp 366ndash3782011

[70] F L Lewis S Jagannathan and A Yesildirek Neural NetworkControl of Robot Manipulators and Nonlinear Systems Taylor ampFrancis London UK 1999

[71] A M Shaw and F J Doyle III ldquoMultivariable nonlinear controlapplications for a high purity distillation column using arecurrent dynamic neuron modelrdquo Journal of Process Controlvol 7 no 4 pp 255ndash268 1997

[72] K Najim Process Modeling and Control in Chemical Engineer-ing Marcel Dekker New York NY USA 1989

[73] B Xu D Wang F Sun and Z Shi ldquoDirect neural discretecontrol of hypersonic flight vehiclerdquo Nonlinear Dynamics vol70 no 1 pp 269ndash278 2012

[74] B Xu and Y Zhang ldquoNeural discrete back-stepping controlof hypersonic flight vehicle with equivalent prediction modelrdquoNeurocomputing vol 154 pp 337ndash346 2015

[75] B Xu F Sun H Liu and J Ren ldquoAdaptive Kriging controllerdesign for hypersonic flight vehicle via back-steppingrdquo IETControl Theory amp Applications vol 6 no 4 pp 487ndash497 2012

[76] B Xu ldquoRobust adaptive neural control of flexible hypersonicflight vehicle with dead-zone input nonlinearityrdquo NonlinearDynamics vol 80 no 3 pp 1509ndash1520 2015

[77] B Xu XHuangDWang and F Sun ldquoDynamic surface controlof constrained hypersonic flightmodels with parameter estima-tion and actuator compensationrdquo Asian Journal of Control vol16 no 1 pp 162ndash174 2014

[78] B Xu and Z Shi ldquoAn overview on flight dynamics and controlapproaches for hypersonic vehiclesrdquo Science China InformationSciences vol 58 no 7 pp 1ndash19 2015

[79] D E Rumelhart G E Hinton and R J Williams ldquoLearninginternal representations by error propagationrdquo in Parallel Dis-tributed Processing vol 1 pp 318ndash362 MIT Press 1986

[80] S Jagannathan and F L Lewis ldquoDiscrete-time neural netcontroller for a class of nonlinear dynamical systemsrdquo IEEETransactions on Automatic Control vol 41 no 11 pp 1693ndash16991996

[81] S Jagannathan and F L Lewis ldquoMultilayer discrete-timeneural-net controller with guaranteed performancerdquo IEEETransactions on Neural Network vol 7 no 1 pp 107ndash130 1996

[82] P He and S Jagannathan ldquoNeuro-controller for reducingcyclic variation in lean combustion spark ignition enginesrdquoAutomatica vol 41 no 7 pp 1133ndash1142 2005

[83] S S Ge G Y Li and T H Lee ldquoAdaptive NN controlfor a class of strict-feedback discrete-time nonlinear systemsrdquoAutomatica vol 39 no 5 pp 807ndash819 2003

Discrete Dynamics in Nature and Society 11

[84] S S Ge T H Lee G Y Li and J Zhang ldquoAdaptive NN controlfor a class of discrete-time non-linear systemsrdquo InternationalJournal of Control vol 76 no 4 pp 334ndash354 2003

[85] C J Goh ldquoModel reference control of non-linear systems viaimplicit function emulationrdquo International Journal of Controlvol 60 no 1 pp 91ndash115 1994

[86] C J Goh and T H Lee ldquoDirect adaptive control of nonlinearsystems via implicit function emulationrdquo Control Theory andAdvanced Technology vol 10 no 3 pp 539ndash552 1994

[87] A U Levin and K S Narendra ldquoControl of nonlinear dynami-cal systems using neural networksmdashpart II observability iden-tification and controlrdquo IEEE Transactions on Neural Networksvol 7 no 1 pp 30ndash42 1996

[88] S S Ge J Zhang and T H Lee ldquoAdaptive MNN control fora class of non-affine NARMAX systems with disturbancesrdquoSystems amp Control Letters vol 53 no 1 pp 1ndash12 2004

[89] S S Ge J Zhang and T H Lee ldquoAdaptive neural networkcontrol for a class of MIMO nonlinear systems with distur-bances in discrete-timerdquo IEEE Transactions on Systems Manand Cybernetics Part B Cybernetics vol 34 no 4 pp 1630ndash1645 2004

[90] J Zhang S S Ge and T H Lee ldquoOutput feedback control of aclass of discrete MIMO nonlinear systems with triangular forminputsrdquo IEEE Transactions onNeural Networks vol 16 no 6 pp1491ndash1503 2005

[91] F C Sun Z Sun and P-Y Woo ldquoStable neural-network-basedadaptive control for sampled-data nonlinear systemsrdquo IEEETransactions on Neural Networks vol 9 no 5 pp 956ndash9681998

[92] C Yang S S Ge C Xiang T Chai and T H Lee ldquoOutputfeedback NN control for two classes of discrete-time systemswith unknown control directions in a unified approachrdquo IEEETransactions on Neural Networks vol 19 no 11 pp 1873ndash18862008

[93] S S Ge C Yang and T H Lee ldquoAdaptive predictive controlusing neural network for a class of pure-feedback systems indiscrete timerdquo IEEE Transactions on Neural Networks vol 19no 9 pp 1599ndash1614 2008

[94] Y Li C Yang S S Ge and T H Lee ldquoAdaptive output feed-back NN control of a class of discrete-time MIMO nonlinearsystems with unknown control directionsrdquo IEEE Transactionson Systems Man and Cybernetics Part B Cybernetics vol 41no 2 pp 507ndash517 2011

[95] Y-J Liu C L P Chen G-X Wen and S Tong ldquoAdaptiveneural output feedback tracking control for a class of uncertaindiscrete-time nonlinear systemsrdquo IEEE Transactions on NeuralNetworks vol 22 no 7 pp 1162ndash1167 2011

[96] C Yang S S Ge and T H Lee ldquoOutput feedback adaptivecontrol of a class of nonlinear discrete-time systems withunknown control directionsrdquoAutomatica vol 45 no 1 pp 270ndash276 2009

[97] A Al-Tamimi F L Lewis and M Abu-Khalaf ldquoDiscrete-timenonlinear HJB solution using approximate dynamic program-ming convergence proofrdquo IEEE Transactions on Systems Manand Cybernetics Part B Cybernetics vol 38 no 4 pp 943ndash9492008

[98] P Werbos ldquoApproximate dynamic programming for real-timecontrol and neuralmodelingrdquo inHandbook of Intelligent ControlNeural Fuzzy amp Adaptive Approaches Van Nostrand Reinhold1992

[99] P He and S Jagannathan ldquoReinforcement learning neural-network-based controller for nonlinear discrete-time systems

with input constraintsrdquo IEEETransactions on SystemsMan andCybernetics Part B Cybernetics vol 37 no 2 pp 425ndash436 2007

[100] B Xu C Yang and Z Shi ldquoReinforcement learning outputfeedback NN control using deterministic learning techniquerdquoIEEE Transactions on Neural Networks and Learning Systemsvol 25 no 3 pp 635ndash641 2014

[101] D Liu DWang D Zhao QWei and N Jin ldquoNeural-network-based optimal control for a class of unknowndiscrete-time non-linear systems using globalized dual heuristic programmingrdquoIEEE Transactions on Automation Science and Engineering vol9 no 3 pp 628ndash634 2012

[102] D Liu and Q Wei ldquoFinite-approximation-error-based optimalcontrol approach for discrete-time nonlinear systemsrdquo IEEETransactions on Cybernetics vol 43 no 2 pp 779ndash789 2013

[103] X Zhong H He H Zhang and Z Wang ldquoOptimal control forunknown discrete-time nonlinear markov jump systems usingadaptive dynamic programmingrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 25 no 12 pp 2141ndash21552014

[104] D Liu D Wang and X Yang ldquoAn iterative adaptive dynamicprogramming algorithm for optimal control of unknowndiscrete-time nonlinear systemswith constrained inputsrdquo Infor-mation Sciences vol 220 pp 331ndash342 2013

[105] F-Y Wang N Jin D Liu and Q Wei ldquoAdaptive dynamicprogramming for finite-horizon optimal control of discrete-time nonlinear systems with 120576-error boundrdquo IEEE Transactionson Neural Networks vol 22 no 1 pp 24ndash36 2011

[106] H Zhang Y Luo and D Liu ldquoNeural-network-based near-optimal control for a class of discrete-time affine nonlinearsystems with control constraintsrdquo IEEE Transactions on NeuralNetworks vol 20 no 9 pp 1490ndash1503 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 11

[84] S S Ge T H Lee G Y Li and J Zhang ldquoAdaptive NN controlfor a class of discrete-time non-linear systemsrdquo InternationalJournal of Control vol 76 no 4 pp 334ndash354 2003

[85] C J Goh ldquoModel reference control of non-linear systems viaimplicit function emulationrdquo International Journal of Controlvol 60 no 1 pp 91ndash115 1994

[86] C J Goh and T H Lee ldquoDirect adaptive control of nonlinearsystems via implicit function emulationrdquo Control Theory andAdvanced Technology vol 10 no 3 pp 539ndash552 1994

[87] A U Levin and K S Narendra ldquoControl of nonlinear dynami-cal systems using neural networksmdashpart II observability iden-tification and controlrdquo IEEE Transactions on Neural Networksvol 7 no 1 pp 30ndash42 1996

[88] S S Ge J Zhang and T H Lee ldquoAdaptive MNN control fora class of non-affine NARMAX systems with disturbancesrdquoSystems amp Control Letters vol 53 no 1 pp 1ndash12 2004

[89] S S Ge J Zhang and T H Lee ldquoAdaptive neural networkcontrol for a class of MIMO nonlinear systems with distur-bances in discrete-timerdquo IEEE Transactions on Systems Manand Cybernetics Part B Cybernetics vol 34 no 4 pp 1630ndash1645 2004

[90] J Zhang S S Ge and T H Lee ldquoOutput feedback control of aclass of discrete MIMO nonlinear systems with triangular forminputsrdquo IEEE Transactions onNeural Networks vol 16 no 6 pp1491ndash1503 2005

[91] F C Sun Z Sun and P-Y Woo ldquoStable neural-network-basedadaptive control for sampled-data nonlinear systemsrdquo IEEETransactions on Neural Networks vol 9 no 5 pp 956ndash9681998

[92] C Yang S S Ge C Xiang T Chai and T H Lee ldquoOutputfeedback NN control for two classes of discrete-time systemswith unknown control directions in a unified approachrdquo IEEETransactions on Neural Networks vol 19 no 11 pp 1873ndash18862008

[93] S S Ge C Yang and T H Lee ldquoAdaptive predictive controlusing neural network for a class of pure-feedback systems indiscrete timerdquo IEEE Transactions on Neural Networks vol 19no 9 pp 1599ndash1614 2008

[94] Y Li C Yang S S Ge and T H Lee ldquoAdaptive output feed-back NN control of a class of discrete-time MIMO nonlinearsystems with unknown control directionsrdquo IEEE Transactionson Systems Man and Cybernetics Part B Cybernetics vol 41no 2 pp 507ndash517 2011

[95] Y-J Liu C L P Chen G-X Wen and S Tong ldquoAdaptiveneural output feedback tracking control for a class of uncertaindiscrete-time nonlinear systemsrdquo IEEE Transactions on NeuralNetworks vol 22 no 7 pp 1162ndash1167 2011

[96] C Yang S S Ge and T H Lee ldquoOutput feedback adaptivecontrol of a class of nonlinear discrete-time systems withunknown control directionsrdquoAutomatica vol 45 no 1 pp 270ndash276 2009

[97] A Al-Tamimi F L Lewis and M Abu-Khalaf ldquoDiscrete-timenonlinear HJB solution using approximate dynamic program-ming convergence proofrdquo IEEE Transactions on Systems Manand Cybernetics Part B Cybernetics vol 38 no 4 pp 943ndash9492008

[98] P Werbos ldquoApproximate dynamic programming for real-timecontrol and neuralmodelingrdquo inHandbook of Intelligent ControlNeural Fuzzy amp Adaptive Approaches Van Nostrand Reinhold1992

[99] P He and S Jagannathan ldquoReinforcement learning neural-network-based controller for nonlinear discrete-time systems

with input constraintsrdquo IEEETransactions on SystemsMan andCybernetics Part B Cybernetics vol 37 no 2 pp 425ndash436 2007

[100] B Xu C Yang and Z Shi ldquoReinforcement learning outputfeedback NN control using deterministic learning techniquerdquoIEEE Transactions on Neural Networks and Learning Systemsvol 25 no 3 pp 635ndash641 2014

[101] D Liu DWang D Zhao QWei and N Jin ldquoNeural-network-based optimal control for a class of unknowndiscrete-time non-linear systems using globalized dual heuristic programmingrdquoIEEE Transactions on Automation Science and Engineering vol9 no 3 pp 628ndash634 2012

[102] D Liu and Q Wei ldquoFinite-approximation-error-based optimalcontrol approach for discrete-time nonlinear systemsrdquo IEEETransactions on Cybernetics vol 43 no 2 pp 779ndash789 2013

[103] X Zhong H He H Zhang and Z Wang ldquoOptimal control forunknown discrete-time nonlinear markov jump systems usingadaptive dynamic programmingrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 25 no 12 pp 2141ndash21552014

[104] D Liu D Wang and X Yang ldquoAn iterative adaptive dynamicprogramming algorithm for optimal control of unknowndiscrete-time nonlinear systemswith constrained inputsrdquo Infor-mation Sciences vol 220 pp 331ndash342 2013

[105] F-Y Wang N Jin D Liu and Q Wei ldquoAdaptive dynamicprogramming for finite-horizon optimal control of discrete-time nonlinear systems with 120576-error boundrdquo IEEE Transactionson Neural Networks vol 22 no 1 pp 24ndash36 2011

[106] H Zhang Y Luo and D Liu ldquoNeural-network-based near-optimal control for a class of discrete-time affine nonlinearsystems with control constraintsrdquo IEEE Transactions on NeuralNetworks vol 20 no 9 pp 1490ndash1503 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of