Finite-Time Stability: An Input-Output Approach

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 su?cient conditions for IO-FTS for W2 exogenous inputs 22 2.2.3 Computational issues 25 2.3 A Su?cient 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 Su?cient 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 Su?cient Conditions for IO-FTS in Presence of W2 Signals 91 7.3 Example and Computational Issues 96 7.4 Main Result: A Su?cient 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

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