Analysis and Synthesis for Discrete-Time Switched Systems: A Quasi-Time-Dependent Method

212 Pages·2020·4.963 MB·English

by  Zhongyang Fei, Shuang Shi, Peng Shi

Studies in Systems, Decision and Control 244
Zhongyang Fei
Shuang Shi
Peng Shi
Analysis and
Synthesis for
Discrete-Time
Switched Systems
A Quasi-Time-Dependent Method
Studies in Systems, Decision and Control
Volume 244
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Zhongyang Fei Shuang Shi Peng Shi
(cid:129) (cid:129)
Analysis and Synthesis
for Discrete-Time Switched
Systems
A Quasi-Time-Dependent Method
123
ZhongyangFei ShuangShi
Research Institute of Intelligent Research Institute of Intelligent
Control andSystems Control andSystems
Harbin Institute of Technology Harbin Institute of Technology
Harbin, Heilongjiang, China Harbin, Heilongjiang, China
PengShi
Schoolof Electrical andElectronic
Engineering
University of Adelaide
Adelaide, SA,Australia
ISSN 2198-4182 ISSN 2198-4190 (electronic)
Studies in Systems,DecisionandControl
ISBN978-3-030-25811-5 ISBN978-3-030-25812-2 (eBook)
https://doi.org/10.1007/978-3-030-25812-2
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Preface
Switched systems, which are powerful in modeling many man-made or physical
systems displaying switching properties, have attracted increasing attention in past
decades. Typical applications of switched systems include manipulator robots,
networked control systems, vehicle industry, flight control systems, biological
systems, and so on. Generally, a switched system contains a finite number of
subsystems and a switching signal governing the switching among them.
Ithasbeenwelldemonstratedthatswitchingsignalshavegreatinfluenceonthe
systemperformanceofswitchedsystems.Improperswitchingsignalmayevenlead
toinstabilitiesofthesystem.Astwofundamentaldeterministicswitchingrules,the
dwell-time(DT)switchingandaveragedwell-time(ADT)switchinghavereceived
extensiveattention.Inaddition,thereexistsanothermuchmorecommonswitching
signal,thepersistentdwell-time(PDT)switching.BothDTandADTswitchingcan
be termed as special forms of PDT switching. However, due to its complex
switching mechanism, the PDT switching is rarely concerned. On the other hand,
all of the aforementioned signals are mode-independent, which means that all
subsystems have to share common parameters, which leads to a certain conser-
vatism. By extending these signals to mode-dependent ones, more general and
flexible results can be obtained.
Furthermore,themultipleLyapunov-likefunctionmethodhasbecomeastandard
tool to analyze and synthesize switched systems. However, the traditional
Lyapunov-like function is time-independent. In this book, we introduce a time
scheduler to construct a time-scheduled multiple Lyapunov-like function for
discrete-timeswitchedsystems,whichisgenerallytermedasthequasi-time-dependent
(QTD)technique.BenefitedbytheQTDmethod,wecanachievebetterperformance
comparedwiththeresultsobtainedbytheconventionaltime-independentmethod.
In this book, we focus on the analysis and synthesis for discrete-time switched
systemsbyadoptingtheQTDtechnique.Twopartsareincluded:thefirstpartbased
onthemode-dependentaveragedwell-time(MDADT)switching,whilethesecond
part based on the mode-dependent persistent dwell-time (MDPDT) switching.
Thesemethodologiesprovideaframeworkforstabilityanddisturbanceattenuation
performanceanalysis,finite-timeperformanceanalysis,controllerandfilterdesign,
v
vi Preface
and reachable set estimation. Solutions to the design problems are presented in
terms of linear matrix inequalities.
Thecompendiousframeanddescriptionofthebookaregivenasfollows.Themain
contents of the first part include: Chap. 2 focuses on the stability analysis for
discrete-timeswitchedsystemswithMDADT;Chap.3studiesdisturbanceattenua-
tionperformanceinthesenseof‘ -gain;Chaps.4and5providetheH controland
2 1
filtering schemes for discrete-time switched systems with MDADT, respectively;
Chap.6studiestheasynchronouscontrolandfiltering;Chap.7givesareachableset
estimation; and Chap. 8 considers the finite-time performance. The main contents
ofthesecondpartinclude:Chap.9investigatethestabilityand‘ -gainperformance
2
withMDPDT;Chaps.10and11providecontrolandfilteringschemeswithMDPDT,
andChap.12studiesfinite-timeH controlandfilteringundertheMDPDTscheme.
1
To summarize, this book provides a framework to investigate the analysis and
synthesisissuesfordiscrete-timeswitchedsystems.Thebookisprimarilyintended
for researchers and engineers in the system and control community, especially for
those who are interested in switched systems and their applications.
Harbin, China Zhongyang Fei
Harbin, China Shuang Shi
Adelaide, Australia Peng Shi
April 2019
Acknowledgements
Herearenumerousindividuals, withoutwhoseusefulsuggestionsandconstructive
comments,themonographcouldnothavebeenfinished.Wewouldliketoexpress
our sincere thanks to Profs. Wei Xing Zheng (University of Western Sydney,
Australia), Xudong Zhao (Dalian Institute of Technology, China), Choon Ki Ahn
(Korea University, Korea), Huijun Gao (Harbin Institute of Technology, China),
Ligang Wu(HarbinInstitute ofTechnology,China), Ming Liu(Harbin Institute of
Technology, China), Yang Shi (University of Victoria, Canada), and also to our
fellow colleagues who have offered invaluable support and encouragement
throughoutthisresearch.Wewouldliketoacknowledgethecontributionsfromour
studentsChaoxuGuan,LiuYang,ZhengpengShi,andYujunrongMa.Theauthors
areespeciallygratefultotheirfamiliesfortheirencouragementandendlesssupport.
The publishing of this book was supported in part by the National Natural
ScienceFoundationofChina(61503094,61873310,61790564and61773131),the
Natural Science Foundation of Heilongjiang Province (F2018015), the
Fundamental Research Funds for the Central Universities and the Australian
Research Council (DP170102644).
vii
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Switched Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Background and Motivation . . . . . . . . . . . . . . . . . . . . . 2
1.2 Switching Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Classification of Switching Signals . . . . . . . . . . . . . . . . 4
1.2.2 Time-Dependent Switching Signals. . . . . . . . . . . . . . . . 5
1.3 Overview of Switched Systems . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Quasi-Time-Dependent Technique for Switched Systems . . . . . 9
1.5 Organization of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Part I Analysis and Synthesis for Discrete-Time Switched Systems
with Mode-Dependent Average Dwell-Time
2 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1 Extended Mode-Dependent Average Dwell-Time . . . . . . . . . . . 21
2.2 Revised Multiple Lyapunov-Like Functions . . . . . . . . . . . . . . . 22
2.3 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.1 Stability Analysis for Discrete-Time Switched
Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.2 Stability Analysis for Discrete-Time Switched
Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
ix
x Contents
3 Weighted ‘2-Gain Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1 Weighted ‘ -Gain Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2
3.1.1 Weighted ‘ -Gain Analysis for Discrete-Time
2
Switched Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1.2 Weighted ‘ -Gain Analysis for Discrete-Time
2
Switched Linear Systems . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4 H Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
1
4.1 State Feedback Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 Output Feedback Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.3 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5 H Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
1
5.1 H Filter Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
1
5.2 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6 Asynchronous Control and Filtering. . . . . . . . . . . . . . . . . . . . . . . . 75
6.1 Stability and ‘ -Gain Analysis with Asynchronous
2
Switching. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.1.1 Revised Multiple Lyapunov-Like Functions. . . . . . . . . . 75
6.1.2 Stability and ‘ -Gain Analysis . . . . . . . . . . . . . . . . . . . 77
2
6.2 Asynchronous Control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.3 Asynchronous Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.4 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7 Reachable Set Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.1 Reachable Set Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.2 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
8 Finite-Time Control and Filtering. . . . . . . . . . . . . . . . . . . . . . . . . . 111
8.1 Finite-Time Performance Analysis . . . . . . . . . . . . . . . . . . . . . . 111
8.2 Finite-Time Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
8.3 Finite-Time Filtering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
8.4 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130