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Finite-Time Stability


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Table of Contents

Preface xi

List of Acronyms xiii

1. Introduction 1

1.1 Finite-Time Stability (FTS) 1

1.2 Input-Output Finite-Time Stability 6

1.3 FTS and Finite-Time Convergence 10

1.4 Background 10

1.4.1 Vectors and signals 10

1.4.2 Impulsive dynamical linear systems 12

1.5 Book Organization 13

2. Linear Time-Varying Systems: IO-FTS Analysis 15

2.1 Problem Statement 15

2.2 IO-FTS for W2 Exogenous Inputs 16

2.2.1 Preliminaries 16

2.2.2 Necessary and sufficient conditions for IO-FTS for W2exogenous inputs 22

2.2.3 Computational issues 25

2.3 A Sufficient Condition for IO-FTS for W Inputs 26

2.4 Summary 29

3. Linear Time-Varying Systems: Design of IO Finite-Time Stabilizing Controllers 33

3.1 IO Finite-Time Stabilization via State Feedback 34

3.2 IO-Finite-Time Stabilization via Output Feedback 36

3.3 Summary 42

4. IO-FTS with Nonzero Initial Conditions 45

4.1 Preliminaries 45

4.2 Interpretation of the Norm of the Operator LSNZ 48

4.3 Sufficient Conditions for IO-FTS-NZIC 52

4.4 Design of IO Finite-Time Stabilizing Controllers NZIC 55

4.4.1 State feedback 56

4.4.2 Output feedback 57

4.5 Summary 58

5. IO-FTS with Constrained Control Inputs 61

5.1 Structured IO-FTS and Problem Statement 61

5.2 Structured IO-FTS Analysis 63

5.3 State Feedback Design 65

5.4 Design of an Active Suspension Control System Using Structured IO-FTS 67

5.5 Summary 70

6. Robustness Issues and the Mixed H /FTS Control Problem 71

6.1 Preliminaries 72

6.1.1 System setting 72

6.1.2 IO-FTS with an H bound 73

6.2 Robust and Quadratic IO-FTS with an H Bound 77

6.2.1 Main result 78

6.2.2 A numerical example 80

6.3 State Feedback Design 82

6.3.1 Numerical example: Cont'd 85

6.4 Case study: Quadratic IO-FTS with an H Bound of the Inverted Pendulum 86

6.5 Summary 88

7. Impulsive Dynamical Linear Systems: IO-FTS Analysis 89

7.1 Background 90

7.1.1 Preliminary results for the W2 case 90

7.2 Main Results: Necessary and Sufficient Conditions for IO-FTS in Presence of W2 Signals 91

7.3 Example and Computational Issues 96

7.4 Main Result: A Sufficient Condition for IO-FTS in Presence of W Signals 98

7.4.1 An illustrative example 99

7.5 Summary 100

8. Impulsive Dynamical Linear Systems: IO Finite-Time Stabilization via Dynamical Controllers 103

8.1 Problem Statement 103

8.2 IO Finite-Time Stabilization of IDLSs: W2 Signals 104

8.2.1 A numerical example 107

8.3 IO Finite-Time Stabilization of IDLSs: W Signals 108

8.3.1 Illustrative example: Cont'd 110

8.4 Summary 111

9. Impulsive Dynamical Linear Systems with Uncertain Resetting Times 113

9.1 Arbitrary Switching 113

9.2 Uncertain Switching 114

9.3 Numerical Example 116

9.3.1 Known resetting times 117

9.3.2 Arbitrary switching 118

9.3.3 Uncertain switching 118

9.4 Summary 119

10. Hybrid Architecture for Deployment of Finite-Time Control Systems 121

10.1 Controller Architecture 121

10.2 Examples 123

10.2.1 Hybrid active suspension control 123

10.2.2 Lateral collision avoidance system 124

10.3 Summary 129

A. Fundamentals on Linear Time-Varying Systems 131

B. Schur Complements 137

C. Computation of Feasible Solutions to Optimizations Problems

Involving DLMIs 139

D. Solving Optimization Problems Involving DLMIs using MATLAB (R) 145

E. Examples of Applications of IO-FTS Control Design to Real-World Systems 151

References 159

Index 167

About the Author

FRANCESCO AMATO is Professor of Bioengineering, Dean of the School of Computer and Biomedical Engineering and the Coordinator of the Doctorate School in Biomedical and Computer Engineering at the University of Catanzaro, Italy. The scientific activity of Francesco Amato has developed in the fields of systems and control theory; robust control, finite-time stability and control, control of nonlinear quadratic systems with applications to the contexts of aircraft control, computational biology and bioengineering. He has published around 250 papers in international journals and conference proceedings and two monographs with Springer Verlag entitled "Robust Control of Linear Systems subject to Uncertain Time-Varying Parameters" and "Finite-Time Stability and Control". GIANMARIA DE TOMMASI is Associate Professor with the Department of Electrical Engineering and Information Technology, University of Naples Federico II, Italy. Since 2002, he has been a Visiting Researcher with the Joint European Torus (JET) Tokamak, Oxfordshire, U.K., where he has participated in various projects connected to the JET plasma current and shape control system. He has authored more than 100 journal and conference papers, and is a co-author of the monograph "Finite-Time Stability and Control" (Springer). His current research interests include control of nuclear fusion devices, fault detection for discrete event systems, identification of discrete event systems modeled with Petri nets, and stability on finite-time horizon of hybrid systems. Dr. De Tommasi is a member of the IEEE Control System Society Conference Editorial Board, and has been Guest Editor of the Fusion Engineering and Design special issue titled "Design and Implementation of Real-Time Systems for Magnetic Confined Fusion Devices". ALFREDO PIRONTI is a Full Professor of System and Control Theory in the Department of Electrical and Information Technology Engineering, University of Naples Federico II, Italy. He spent several periods as visiting researcher at the Max Planck Institute for Plasma Physics in Garching (Germany), the Center for Control Engineering and Computation (University of California at Santa Barbara), the ITER Joint Work Site of Naka (Japan), and the EFDA-JET site of Culham (UK). His research interests include application of feedback control to nuclear fusion problems, robust control of uncertain systems, and differential games theory. In 2005 he was guest editor for the IEEE Control Systems Magazine journal, where he contributed to two special issues focused on the control of plasmas in tokamak machines. He is the author of more than 200 papers published in international journals, books, and conference proceedings.

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