Robust Control for Nonlinear Time-Delay Systems

299 Pages·2017·4.762 MB·english

by  Changchun Hua, Liuliu Zhang, Xinping Guan

Changchun Hua Liuliu Zhang
(cid:129)
Xinping Guan
Robust Control for Nonlinear
Time-Delay Systems
123
Changchun Hua XinpingGuan
Institute of Electrical Engineering Department ofAutomation
Yanshan University ShanghaiJiao Tong University
Qinhuangdao Shanghai
China China
Liuliu Zhang
Institute of Electrical Engineering
Yanshan University
Qinhuangdao
China
ISBN978-981-10-5130-2 ISBN978-981-10-5131-9 (eBook)
DOI 10.1007/978-981-10-5131-9
LibraryofCongressControlNumber:2017943106
©SpringerNatureSingaporePteLtd.2018
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart
of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission
orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar
methodologynowknownorhereafterdeveloped.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom
therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.
The publisher, the authors and the editors are safe to assume that the advice and information in this
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor
for any errors or omissions that may have been made. The publisher remains neutral with regard to
jurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations.
Printedonacid-freepaper
ThisSpringerimprintispublishedbySpringerNature
TheregisteredcompanyisSpringerNatureSingaporePteLtd.
Theregisteredcompanyaddressis:152BeachRoad,#21-01/04GatewayEast,Singapore189721,Singapore
Preface
Timedelayappearsinmanyphysicalprocessesduetotheperiodoftimeittakesfor
the signals to transmit. Time-delay systems are largely encountered in modeling
propagation phenomena, population dynamics, interconnected systems, supply
chains,andsystemscontrolledovercommunicationnetworks.Itiswellknownthat
time delay in control systems may lead to deterioration of the closed-loop perfor-
mance or even destabilize the systems; therefore, specific analysis techniques and
design methods are needed to be developed for control systems in the presence of
time delay. The time-delay systems can be divided into linear time-delay systems
and nonlinear time-delay systems. Recently, the stability analysis and control
designoflineartime-delaysystemshavebeenextensivelystudiedwiththepopular
tools—Lyapunov–Krasovskii functional method and Lyapunov–Razumikhin
method.Thestabilityandstabilizationconditionscanbetransformed into solvable
linear matrix inequalities (LMIs) with the help of Schur complement lemma.
Compared with linear time-delay systems, the study of nonlinear time-delay
systems is more important for control theory and control applications, as most of
practical systems have nonlinear dynamics and nonlinear uncertainties generally
exist in practical engineering systems due to the modeling error and external dis-
turbances.However,theanalysisandsynthesisofnonlineartime-delaysystemsare
moredifficultandchallenging.Themainreasonsareasfollows:(i)Itisnoteasyto
select Lyapunov functional for nonlinear time-delay systems. The Lyapunov
functionalforlineartime-delaysystemsisgenerallychosentobequadratic.Butfor
nonlinear time-delay systems, we should construct Lyapunov functional based on
the specific system structure, which increases the difficulties of stability analysis
and control design. (ii) It is difficult to compensate for the time-delay effect while
designing nonlinear controllers. For the control design of time-delay systems, we
aim to design memoryless controllers independent of time delay, because the time
delayisvariableinpracticalsystemsanditisimpossibletoobtaintheexactvalues
of time delay. Hence, the controller design of nonlinear time-delay systems is
different from that of the nonlinear systems free of time delay, as how to system-
atically design controllers to overcome the effect of time delay is very difficult.
v
vi Preface
This book is devoted to report the latest results on nonlinear time-delay control
theory and applications, especially the robust control of time-delay systems with
nonlinearuncertainties. Thisbook collects some novel worksrelated to commonly
encountered nonlinear time-delay systems, such as the nonlinear systems in the
form of high-order polynomial dynamics, systems with nonlinear input, and tri-
angular nonlinear systems. Undoubtedly, the results in this book will enrich non-
linear system theory and time-delay system theory.
Our treatment is theoretically oriented, although some illustrative examples are
includedinthisbook.Thereaderisassumedtohavesomebackgroundinnonlinear
systems and control. Although this book is primarily intended for students and
practitionersofcontroltheory,itisalsoavaluablereferenceforthoseinfieldssuch
ascommunicationengineeringandeconomics.Moreover,webelievethatthisbook
should be suitable for certain advanced courses or seminars.
In Chap. 1, the background and some descriptions of nonlinear time-delay
system are provided.
Then, the rest of this book will be presented under the following parts:
PartI:Thefirstpartofthisbookisconcernedwithnonlineartime-delaysystems
with uncertainties in high-order polynomial form. In Chap. 2, based on the
Lyapunov–Krasovskii functional and Razumikhin lemma, two classes of memo-
rylesscontroldesignmethodsareproposedforsinglenonlineartime-delaysystem.
InChap.3,theresultsareextendedtoaclassoflarge-scaletime-delaysystemswith
interrelated N subsystems and the decentralized robust model reference adaptive
controller is constructed.
Part II: The second part of this book focuses on some new results on nonlinear
time-delay systems with general uncertainties. In Chap. 4, the decentralized
adaptive state feedback control strategy is proposed for interconnected systems. In
Chap. 5, the stabilization problem is investigated for a class of single uncertain
mismatched systems with multiple time delays and a memoryless state feedback
controllerisconstructed. InChap.6, theresult isextendedtoaclass oflarge-scale
systems and the solution of the resulting closed-loop system can exponentially
converge to a ball with adjustable radius. In Chap. 7, the control problem of
nonlinear time-delay systems is studied via the T-S fuzzy approach.
PartIII:Thethirdpartofthisbookisdedicatedtononlineartime-delaysystems
withtwokindsofnonlinearinputs.InChap.8,theadaptivetrackingcontrollawfor
nonlinear time-delay system with non-symmetric dead-zone input is presented. In
Chap. 9, based on T-S fuzzy approach, the decentralized networked control prob-
lem is investigated for large-scale time-delay systems with sector input.
Part IV: The last part of this book is devoted to controller design for nonlinear
time-delay systems with triangular structure. In Chap. 10, the robust control
problemisinvestigatedfornonlineartime-delaysystemswiththeformoftriangular
structureviaRazumikhinlemma.InChap.11,thestatefeedbackcontrolproblemis
addressed for a class of nonlinear time-delay systems via Lyapunov–Krasovskii
function.InChap.12,therobustoutputtrackingcontrolproblemisinvestigatedfor
a class of nonlinear time-delay systems with unmodeled dynamics. In Chap. 13,
decentralizeddynamicoutputfeedbackcontrolproblemisconsideredforaclassof
Preface vii
nonlinearinterconnectedsystemswithtimedelay.InChap.14,theoutputfeedback
problem is investigated for a class of uncertain nonlinear time-delay systems with
unknowncontrol direction.InChap.15,dynamicoutputfeedbacktrackingcontrol
strategy is presented for stochastic interconnected time-delay systems with pre-
scribed performance requirement.
ThesupportfromtheNationalNaturalScienceFoundationofChina(61673335,
61290322, 61322303, 61273222, 60974018, 60604004), Nature Science
Foundation of Hebei Province (F2016203467, F2014203267, F2011203110,
15967629D,GCC2014033),ProgramfortheOutstandingYoungInnovativeTalent
of China is gratefully acknowledged.
Qinhuangdao, China Changchun Hua
Qinhuangdao, China Liuliu Zhang
Shanghai, China Xinping Guan
January 2017
Contents
1 Introduction... .... .... ..... .... .... .... .... .... ..... .... 1
1.1 Background .. .... ..... .... .... .... .... .... ..... .... 1
1.2 Description of Nonlinear Time-Delay Systems. .... ..... .... 3
1.2.1 Quasi Nonlinear Time-Delay Systems. .... ..... .... 3
1.2.2 Pure Nonlinear Time-Delay Systems.. .... ..... .... 4
1.2.3 Interconnected Nonlinear Time-Delay Systems... .... 5
1.3 Problems Studied in This Book .... .... .... .... ..... .... 6
1.3.1 High-Order Polynomial Uncertainties . .... ..... .... 6
1.3.2 General Uncertainties . .... .... .... .... ..... .... 7
1.3.3 Nonlinear Input.. .... .... .... .... .... ..... .... 7
1.3.4 System with Lower Triangular Structure... ..... .... 8
1.4 Summary .... .... ..... .... .... .... .... .... ..... .... 9
Part I High-Order Polynomial Nonlinear Uncertainties
2 Robust Stabilization of Single Nonlinear Time-Delay System . .... 13
2.1 Introduction .. .... ..... .... .... .... .... .... ..... .... 13
2.2 System Formulation and Preliminaries ... .... .... ..... .... 14
2.3 Adaptive Robust State Feedback Controller ... .... ..... .... 16
2.4 Novel Nonlinear Feedback Controller.... .... .... ..... .... 19
2.5 Simulations... .... ..... .... .... .... .... .... ..... .... 22
2.6 Conclusion ... .... ..... .... .... .... .... .... ..... .... 26
3 Robust Model Reference Adaptive Control for Interconnected
Time-Delay Systems .... ..... .... .... .... .... .... ..... .... 27
3.1 Introduction .. .... ..... .... .... .... .... .... ..... .... 27
3.2 System Formulation and Preliminaries ... .... .... ..... .... 28
3.3 Main Results.. .... ..... .... .... .... .... .... ..... .... 30
3.4 Numerical Example ..... .... .... .... .... .... ..... .... 35
3.5 Conclusion ... .... ..... .... .... .... .... .... ..... .... 39
ix
x Contents
Part II General Nonlinear Uncertainties
4 Decentralized Adaptive Control for Interconnected
Time-Delay Systems .... ..... .... .... .... .... .... ..... .... 43
4.1 Introduction .. .... ..... .... .... .... .... .... ..... .... 43
4.2 System Formulation and Preliminaries ... .... .... ..... .... 44
4.3 Decentralized Feedback Control.... .... .... .... ..... .... 46
4.4 Application to Decentralized Control for a Class
of Interconnected Systems .... .... .... .... .... ..... .... 50
4.5 Illustrative Examples..... .... .... .... .... .... ..... .... 54
4.6 Conclusion ... .... ..... .... .... .... .... .... ..... .... 59
5 Memoryless State Feedback Control for Uncertain
Nonlinear Time-Delay System . .... .... .... .... .... ..... .... 61
5.1 Introduction .. .... ..... .... .... .... .... .... ..... .... 61
5.2 System Description. ..... .... .... .... .... .... ..... .... 62
5.3 Controller Design.. ..... .... .... .... .... .... ..... .... 63
5.4 Simulation Example ..... .... .... .... .... .... ..... .... 71
5.5 Conclusion ... .... ..... .... .... .... .... .... ..... .... 73
6 Exponential Stabilization for Interconnected Time-Delay
Systems... .... .... .... ..... .... .... .... .... .... ..... .... 75
6.1 Introduction .. .... ..... .... .... .... .... .... ..... .... 75
6.2 System Formulation and Preliminaries ... .... .... ..... .... 76
6.3 Controller Design.. ..... .... .... .... .... .... ..... .... 78
6.4 Numerical Example ..... .... .... .... .... .... ..... .... 88
6.5 Conclusion ... .... ..... .... .... .... .... .... ..... .... 90
7 Robust Adaptive Control for Time-Delay System
via T-S Fuzzy Approach. ..... .... .... .... .... .... ..... .... 93
7.1 Introduction .. .... ..... .... .... .... .... .... ..... .... 93
7.2 System Formulation and Assumptions ... .... .... ..... .... 94
7.3 Virtual Control Design... .... .... .... .... .... ..... .... 98
7.4 Controller Design.. ..... .... .... .... .... .... ..... .... 101
7.5 Simulation ... .... ..... .... .... .... .... .... ..... .... 107
7.6 Conclusion ... .... ..... .... .... .... .... .... ..... .... 112
Part III Nonlinear Input
8 Adaptive Tracking of Time-Delay System with Unknown
Dead-Zone Input... .... ..... .... .... .... .... .... ..... .... 115
8.1 Introduction .. .... ..... .... .... .... .... .... ..... .... 115
8.2 System Description. ..... .... .... .... .... .... ..... .... 116
8.3 Controller Design.. ..... .... .... .... .... .... ..... .... 118
8.4 Simulation Examples .... .... .... .... .... .... ..... .... 125
8.5 Conclusion ... .... ..... .... .... .... .... .... ..... .... 128
Contents xi
9 Decentralized Fuzzy Networked Control Systems Design
with Sector Input... .... ..... .... .... .... .... .... ..... .... 129
9.1 Introduction .. .... ..... .... .... .... .... .... ..... .... 129
9.2 System Formulation and Assumptions ... .... .... ..... .... 130
9.3 Virtual Control Design... .... .... .... .... .... ..... .... 133
9.4 Controller Design.. ..... .... .... .... .... .... ..... .... 136
9.4.1 Parameters Known Case ... .... .... .... ..... .... 137
9.4.2 Parameters Unknown Case . .... .... .... ..... .... 144
9.5 Simulations... .... ..... .... .... .... .... .... ..... .... 147
9.6 Conclusion ... .... ..... .... .... .... .... .... ..... .... 155
Part IV Time-Delay System with Lower Triangular Structure
10 Robust Control for a Class of Time-Delay System
via Razumikhin Lemma . ..... .... .... .... .... .... ..... .... 159
10.1 Introduction .. .... ..... .... .... .... .... .... ..... .... 159
10.2 Problem Formulation and Preliminaries .. .... .... ..... .... 160
10.3 Main Results.. .... ..... .... .... .... .... .... ..... .... 162
10.4 Conclusion ... .... ..... .... .... .... .... .... ..... .... 171
11 Backstepping Control for Nonlinear Time-Delay System
via L-K Function... .... ..... .... .... .... .... .... ..... .... 173
11.1 Introduction .. .... ..... .... .... .... .... .... ..... .... 173
11.2 System Description and Preliminaries.... .... .... ..... .... 174
11.3 Controller Design for the Second-Order System.... ..... .... 176
11.4 Extension to the nth-Order System.. .... .... .... ..... .... 180
11.5 Application to Chemical Reactor Systems .... .... ..... .... 185
11.6 Simulations... .... ..... .... .... .... .... .... ..... .... 187
11.7 Conclusion ... .... ..... .... .... .... .... .... ..... .... 191
12 NN-Based Output Feedback Tracking of Nonlinear Time-Delay
System ... .... .... .... ..... .... .... .... .... .... ..... .... 193
12.1 Introduction .. .... ..... .... .... .... .... .... ..... .... 193
12.2 System Formulation and Some Assumptions .. .... ..... .... 194
12.3 Preliminary Knowledge .. .... .... .... .... .... ..... .... 197
12.4 Observer Design... ..... .... .... .... .... .... ..... .... 198
12.5 Controller Design.. ..... .... .... .... .... .... ..... .... 200
12.6 Choosing Proper Functions.... .... .... .... .... ..... .... 208
12.7 Simulation Example ..... .... .... .... .... .... ..... .... 211
12.8 Conclusion ... .... ..... .... .... .... .... .... ..... .... 214
13 Output Feedback Stabilization for Interconnected Time-Delay
Systems... .... .... .... ..... .... .... .... .... .... ..... .... 215
13.1 Introduction .. .... ..... .... .... .... .... .... ..... .... 215
13.2 System Formulation ..... .... .... .... .... .... ..... .... 216
xii Contents
13.3 Robust Controller Design. .... .... .... .... .... ..... .... 220
13.3.1 Observer Design . .... .... .... .... .... ..... .... 220
13.3.2 Controller Design .... .... .... .... .... ..... .... 222
13.4 Adaptive Neural Network Control .. .... .... .... ..... .... 232
13.5 Simulation Investigation.. .... .... .... .... .... ..... .... 240
13.6 Conclusion ... .... ..... .... .... .... .... .... ..... .... 246
14 Robust Control of Time-Delay System with Unknown Control
Direction . .... .... .... ..... .... .... .... .... .... ..... .... 247
14.1 Introduction .. .... ..... .... .... .... .... .... ..... .... 247
14.2 Problem Formulation .... .... .... .... .... .... ..... .... 248
14.3 Preliminaries.. .... ..... .... .... .... .... .... ..... .... 249
14.4 Observer Design... ..... .... .... .... .... .... ..... .... 251
14.5 Controller Design: Known Bound Functions .. .... ..... .... 253
14.6 Controller Design: Unknown Bound Functions .... ..... .... 260
14.7 Simulation Example ..... .... .... .... .... .... ..... .... 264
14.8 Conclusion ... .... ..... .... .... .... .... .... ..... .... 270
15 Decentralized Prescribed Performance Tracking of Stochastic
Time-Delay System . .... ..... .... .... .... .... .... ..... .... 271
15.1 Introduction .. .... ..... .... .... .... .... .... ..... .... 271
15.2 System Formulation and Preliminaries ... .... .... ..... .... 272
15.2.1 Problem Formulation.. .... .... .... .... ..... .... 272
15.2.2 Basic Knowledge on Stochastic System ... ..... .... 275
15.3 Controller Design.. ..... .... .... .... .... .... ..... .... 276
15.3.1 Reduced-Order Observer Design. .... .... ..... .... 276
15.3.2 Prescribed Performance Transformation.... ..... .... 278
15.3.3 Adaptive Neural Network Controller Design..... .... 278
15.4 Simulation Example ..... .... .... .... .... .... ..... .... 287
15.5 Conclusion ... .... ..... .... .... .... .... .... ..... .... 290
References.... .... .... .... ..... .... .... .... .... .... ..... .... 291
Chapter 1
Introduction
1.1 Background
Timedelayisaninherentcharacteristicofphysicalsystemwhenmaterialsorenergyis
transmittingthroughacertainroute.Thephenomenonoftimedelayexistsinvarious
engineeringsystemssuchaschemicalprocess,powersystem,rollingsystems,long
transmissionlinesinpneumaticsystems,andsystemscontrolledbycommunication
networks.Theexistenceoftimedelaymayleadtodeteriorationoftheclosed-loop
performance,andevendestabilizethesystems.Therefore,thestabilityanalysisand
controldesignoftime-delaysystemsaresignificantforpracticalengineeringappli-
cations.
Itiswellknownthatmostofthesystemsencounteredinengineeringprocesses
arenonlinearinessence.Forexample,inamechatronicsystem,theactuatorscan-
notincreaseitspowerinfinitely,andtherealwaysexistsaturationnonlinearities.In
general,linearmodelisanapproximationofarealnonlinearsystemandmodeling
errorsalwaysexist.Earlystudiesofcontroltheorymainlyfocusedonlinearsystem
models, because the system is relatively simple and high accuracy performance is
not required. However, with the rapid development of science and technology, the
industrialapplicationsystemsarebecomingmuchmorecomplexandtheaccuracy
requirementisalsoincreasing.Itisnoteasytoachieveexpectedcontrolobjectives
basedonlinearizedmodelsofindustrialprocesses,becausetheglobalstabilitycould
not be achieved as the linearized model is only locally feasible. Therefore, direct
researchworkonpracticalnonlinearsystemmodelsshouldbelaunchedinorderto
obtaineffectivenonlinearcontrollers.
In addition, the uncertainties in practical systems cannot be avoided due to the
inaccuracy of modeling, external disturbances, change of working conditions, and
component aging. The uncertainties could affect systems’ performance and even
destabilize the systems. It is well known that the development of modern control
theory based on state-space model has made a big breakthrough in 1960–1970s;
however,ithasbeenprovedthatthesetheoriesareseriouslydependentonaccurate
mathematicalmodelsthatresultintheinvalidnessofmoderncontroltheorywhenit
isappliedforpracticalengineeringapplications.Thus,thestudyofuncertaintiesin
systemsisofimportanceinboththeoreticalresearchandpracticalapplications.
©SpringerNatureSingaporePteLtd.2018 1
C.Huaetal.,RobustControlforNonlinearTime-DelaySystems,
DOI10.1007/978-981-10-5131-9_1