Table Of ContentStudies in Systems, Decision and Control 146
Dan Zhang · Li Yu
Analysis and
Synthesis of Switched
Time-Delay Systems:
The Average Dwell
Time Approach
Studies in Systems, Decision and Control
Volume 146
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Dan Zhang Li Yu
(cid:129)
Analysis and Synthesis
of Switched Time-Delay
Systems: The Average Dwell
Time Approach
123
Dan Zhang LiYu
Department ofAutomation ZhejiangUniversity of Technology
ZhejiangUniversity of Technology Hangzhou, Zhejiang, China
Hangzhou, Zhejiang, China
ISSN 2198-4182 ISSN 2198-4190 (electronic)
Studies in Systems,DecisionandControl
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Preface
Theswitchedsystemsarehybridsystemscontainingafinitenumberofsubsystems
andaswitchingsignalgoverningtheswitchingamongthem.Inthelastdecade,the
analysis and synthesis of switched systems have attracted increasing attention due
to the fact they are effectively in modeling of many physical systems such as
electronic circuits, process industry, robotic systems, and so on. The main features
of such kind of systems are that the dynamics of the overall systems are not only
dependent on the dynamics of each subsystem but also closely related to the
switching law. Therefore, the analysis and synthesis of such kind of complicated
systems are known to be challenging.
Ontheotherhand,thedynamicsofmanyrealsystemsarenotonlyrelatedtothe
current state but also be involved with the past state, say, e.g., biological systems,
circuit systems, chemical processes, manufacturing systems, and so on. Hence, it
would be better to use the time-delayed mathematical models to describe the
evolutionofthesesystems.Itiswellobservedthattheexistenceoftimedelaymay
degrade the system performance or even lead to the instability of systems. Since
1950s, the stability analysis of time-delay systems has become an active field in
control community, and a great number of results have been reported. Recent
research attention has been paid on the derivation of delay-dependent stability
conditions,whichcanestablishtherelationship between thedelayinformation and
the system performance.
With the increasing complexity of engineering systems, the time delay and the
parameter switching behaviors are more frequently encountered in one system,
whichgivesrisetotheswitchedtime-delaysystems.Comparedwiththeindividual
switched systems and the time-delay systems, the analysis and synthesis of swit-
ched time-delay systems are more difficult because the parameter switching and
delaymayinteractwitheachotherintheanalysisofsuchsystems.Furthermore,the
presence of nonlinear perturbation may also introduce difficulty. Therefore, due to
thegreatchallenges,thelastfewyearshavewitnessedtherapiddevelopmentonthe
robuststability,filteringandcontrolofsuchsystems,anddifferentapproacheshave
been proposed. This book addresses these challenging issues by using the average
dwell time approach. It presents new formulations, methods, and solutions for the
v
vi Preface
stability, filtering, and control of switched time-delay systems. Specifically, we
wish to give a timely, comprehensive, and self-contained coverage of the recent
advancesinasinglevolumeforeasyaccessbytheresearchersintherelatedareas.
Theresultstobepresentedinthisbookareoutlinedasfollows:Chap.1presents
theintroductionandsomebasicresultsontheswitchedtime-delaysystem.Chapter
2 gives some fundamentals on the stability analysis, controller synthesis, and H
1
filtering of linear time-invariant systems, time-delay systems, and switched linear
systems, respectively. The guideline of Matlab LMI Toolbox is also introduced.
Chapters 3 and 4 discuss the exponential stability of continuous-time and
discrete-time switched systems with mixed time delays and nonlinearities, respec-
tively. Chapters 5–10 address the robust filtering of switched time-delay systems,
which cover the phenomena such as the random measurement delays, stochastic
measurementdropouts,asynchronousswitching,andsoon.Finally,Chaps.11and
12 show two stabilization results, where the fault-tolerant control of switched
systems with actuator saturation was studied, and the finite-time stabilization
algorithm of switched time-delay systems with signal quantization was presented,
respectively.
This book would not have been possible without supports from our colleagues.
In particular, we are indebted to Dr. Ziyi Su and Dr. Haiyu Song for their fruitful
collaboration with us. Currently, they are working in Zhejiang Shuren University,
China, and Zhejiang University of Finance and Economics, China, respectively.
The support from the National Natural Science Foundation of China under Grant
61873237 andtheZhejiang ProvincialNatural Science FoundationofChinaunder
Grant LZ15F030003 are also gratefully acknowledged.
Hangzhou, China Dan Zhang
April 2018 Li Yu
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Switched Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Three Basic Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 Results on Problem 1 . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.2 Results on Problem 2 . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.3 Results on Problem 3 . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Switched Time-Delay Systems. . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Organization of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1 Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Linear Time-Invariant Systems . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.1 Stability Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.2 Stabilization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.3 H Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1
2.3 Time-Delay Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3.1 Stability Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4 Switched Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4.1 Stability Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.5 Matlab LMI Toolbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3 Exponential Stability of Continuous-Time Switched Systems
with Mixed Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3 Exponential Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 43
vii
viii Contents
3.4 Numerical Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.5 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4 Stability and Passivity Analysis of Discrete-Time Switched
Time-Delay Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2 Problem Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.4 A Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.5 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5 H‘ Filtering for Continuous-Time Switched Systems
with Mixed Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.2 Problem Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.3.1 H Performance Analysis. . . . . . . . . . . . . . . . . . . . . . 73
1
5.3.2 Filter Gain Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.4 A Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.4.1 Modeling of Switched Neural Networks . . . . . . . . . . . 80
5.4.2 State Estimation of Switched Neural Networks. . . . . . . 83
5.5 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6 H‘ Filtering for Continuous-Time Switched Stochastic
Time-Delay Systems with Delayed Measurement . . . . . . . . . . . . . . 87
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.2 Problem Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.4 A Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.4.1 Modeling of Switched Genetic Regulatory
Networks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.4.2 State Estimation of Switched Genetic Regulatory
Networks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.5 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
7 H‘ Filtering for Discrete-Time Switched Stochastic
Time-Delay Systems with Missing Measurement. . . . . . . . . . . . . . . 107
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.2 Problem Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Contents ix
7.3.1 H Filtering Analysis. . . . . . . . . . . . . . . . . . . . . . . . . 109
1
7.3.2 H Filter Gain Design . . . . . . . . . . . . . . . . . . . . . . . . 115
1
7.4 A Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.5 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
8 Asynchronous Filtering of Discrete-Time Switched Time-Delay
Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
8.2 Problem Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
8.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
8.4 The Simulation Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
8.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
9 H‘ Filtering of Discrete-Time Singular Switched Time-Delay
Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
9.2 Problem Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
9.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
9.3.1 Filtering Performance Analysis . . . . . . . . . . . . . . . . . . 137
9.3.2 Filter Gain Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
9.4 A Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
9.4.1 Modeling of Singular Switched I/O Systems . . . . . . . . 148
9.4.2 State Estimation of Singular Switched I/O Systems . . . 149
9.5 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
10 Fault Detection of Discrete-Time Switched Time-Delay Systems. . . 153
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
10.2 Problem Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
10.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
10.3.1 Exponential H Fault Detection Performance
1
Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
10.3.2 H Fault Detection Filter Design . . . . . . . . . . . . . . . . 159
1
10.4 A Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
10.5 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
11 Fault Tolerant Control for Discrete-Time Switched Time-Delay
Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
11.2 Problem Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
11.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
