Finite-Time Stability: An Input-Output Approach. Wiley Series in Dynamics and Control of Electromechanical Systems

  • ID: 4469593
  • Book
  • 184 Pages
  • John Wiley and Sons Ltd
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Systematically presents the input–output finite–time stability (IO–FTS) analysis of dynamical systems, covering issues of analysis, design and robustness

The interest in finite–time control has continuously grown in the last fifteen years. This book systematically presents the input–output finite–time stability (IO–FTS) analysis of dynamical systems, with specific reference to linear time–varying systems and hybrid systems. It discusses analysis, design and robustness issues, and includes applications to real world engineering problems.

While classical FTS has an important theoretical significance, IO–FTS is a more practical concept, which is more suitable for real engineering applications, the goal of the research on this topic in the coming years.

Key features:

  • Includes applications to real world engineering problems.
  • Input–output finite–time stability (IO–FTS) is a practical concept, useful to study the behavior of a dynamical system within a finite interval of time.
  • Computationally tractable conditions are provided that render the technique applicable to time–invariant as well as time varying and impulsive (i.e. switching) systems.
  • The LMIs formulation allows mixing the IO–FTS approach with existing control techniques (e. g. H control, optimal control, pole placement, etc.).
  • The book is equipped with a CD containing a software library for the solution of the finite–time control problems discussed in the book itself.

This book is essential reading for university researchers as well as post–graduate engineers practicing in the field of robust process control in research centers and industries. Topics dealt with in the book could also be taught at the level of advanced control courses for graduate students in the department of electrical and computer engineering, mechanical engineering, aeronautics and astronautics, and applied mathematics.

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1 Introduction 5

1.1 Finite–Time Stability (FTS) 5

1.2 Input–Output Finite–Time Stability 11

1.3 FTS and Finite–Time Convergence 15

1.4 Backgrounds 15

1.4.1 Vectors and signals 15

1.4.2 Impulsive dynamical linear systems 17

1.5 Book Organization 19

2 Linear Time–Varying Systems: IO–FTS Analysis 21

2.1 Problem Statement 21

2.2 IO–FTS forW2 Exogenous Inputs 22

2.2.1 Preliminaries 22

2.2.2 Necessary and sufficient conditions for IO–FTS forW2 exogenous inputs 29

2.2.3 Computational issues 33

2.3 A Sufficient Condition for IO–FTS forW1 Exogenous Inputs 34

2.4 Summary 38

3 Linear Time–Varying Systems: Design of IO Finite–Time Stabilizing Controllers 41

3.1 IO Finite–Time Stabilization via State Feedback 42

3.2 IO–Finite–Time Stabilization via Output Feedback 44

3.3 Summary 52

4 IO–FTS with Nonzero Initial Conditions 55

4.1 Preliminaries 55

4.2 Interpretation of the Norm of the Operator LSNZ 58

4.3 Sufficient Conditions for IO–FTS–NZIC 63

4.4 Design of IO Finite–Time Stabilizing Controllers NZIC 66

4.4.1 State feedback 68

4.4.2 Output feedback 69

4.5 Summary 72

5 IO–FTS with Constrained Control Inputs 73

5.1 Structured IO–FTS and Problem Statement 73

5.2 Structured IO–FTS Analysis 75

5.3 State Feedback Design 78

5.4 Design of an Active Suspension Control System Using Structured IO–FTS 80

5.5 Summary 84

6 Robustness Issues and the Mixed H1/FTS Control Problem 85

6.1 Preliminaries 87

6.1.1 System setting 87

6.1.2 IO–FTS with an H1 bound 87

6.2 Robust and Quadratic IO–FTS with an H1 Bound 92

6.2.1 Main result 93

6.2.2 A numerical example 96

6.3 State Feedback Design 96

6.3.1 Numerical example: Cont d 100

6.4 Case study: Quadratic IO–FTS with an H1 Bound of the Inverted Pendulum 101

6.5 Summary 102

7 Impulsive Dynamical Linear Systems: IO–FTS Analysis 105

7.1 Backgrounds 106

7.1.1 Preliminary results for theW2 case 106

7.2 Main Results: Necessary and Sufficient Conditions for IO–FTS in Presence of W2 Signals 108

7.3 Example and Computational Issues 113

7.4 Main Result: A Sufficient Condition for IO–FTS in Presence of W1 Signals 115

7.4.1 An illustrative example 117

7.5 Summary 117

8 Impulsive Dynamical Linear Systems: IO Finite–Time Stabilization via Dynamical Controllers 119

8.1 Problem Statement 119

8.2 IO Finite–Time Stabilization of IDLSs: W2 Signals 120

8.2.1 A numerical example 124

8.3 IO Finite–Time Stabilization of IDLSs: W1 Signals 126

8.3.1 Illustrative example: Cont d 127

8.4 Summary 127

9 Impulsive Dynamical Linear Systems with Uncertain Resetting Times 131

9.1 Arbitrary Switching 131

9.2 Uncertain Switching 133

9.3 Numerical Example 134

9.3.1 Known resetting times 135

9.3.2 Arbitrary switching 136

9.3.3 Uncertain switching 137

9.4 Summary 137

10 Hybrid Architecture for Deployment of Finite–Time Control Systems 139

10.1 Controller Architecture 139

10.2 Examples 142

10.2.1 Hybrid active suspension control 142

10.2.2 Lateral collision avoidance system 143

10.3 Summary 148

A Fundamentals on Linear Time–Varying Systems 151

A.1 Existence and Uniqueness 151

A.2 The State Transition Matrix 152

A.3 Lyapunov Stability of Linear Time–Varying Systems 155

A.4 Input to State and Input to Output Response 156

B Schur Complements 157

C Computation of Feasible Solutions to Optimizations Problems Involving DLMIs 159

C.1 Numerical Solution to a Feasibility Problem Constrained by a DLMI Coupled with LMIs 159

C.2 Numerical Solution to a Feasibility Problem Constrained by a D/DLMI Coupled with LMIs 161

D Solving Optimization Problems Involving DLMIs using MATLAB® 165

D.1 MATLAB® Script for the Solution of the Optimization Problem with DLMI/LMI Constraints Presented in Example 4 165

D.2 MATLAB® Script for the Solution of the D/DLMI/LMI Feasibility Problem Presented in Section 8.3.1 166

E Examples of Applications of IO–FTS Control Design to Real–World Systems 173

E.1 Building Subject to Earthquakes 173

E.2 Quarter car suspension model 176

E.3 Inverted Pendulum 179

E.4 Yaw and Lateral Motions of a Two–Wheel Vehicle 180

Index 183

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Francesco Amato
Gianmaria De Tommasi
Alfredo Pironti
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