Limits of Stability and Stabilization of Time-Delay Systems: A Small-Gain Approach (Advances in Delays and Dynamics Book 8)

"This authored monograph presents a study on fundamental limits and robustness of stability and stabilization of time-delay systems, with an emphasis on time-varying delay, robust stabilization, and newly emerged areas such as networked control and multi-agent systems. The authors systematically develop an operator-theoretic approach that departs from both the traditional algebraic approach and the currently pervasive LMI solution methods. This approach is built on the classical small-gain theorem, which enables the author to draw upon powerful tools and techniques from robust control theory. The book contains motivating examples and presents mathematical key facts that are required in the subsequent sections. The target audience primarily comprises researchers and professionals in the field of control theory, but the book may also be beneficial for graduate students alike" -- Proporcionat per l'editor Preface 7 Contents 10 Symbols and Abbreviations 13 1 Introduction 15 1.1 Examples of Time-Delay Systems 15 1.1.1 A Gyroscopic System 15 1.1.2 A Classical Economic Model 16 1.1.3 Communication Networks 18 1.1.4 Biological Systems 20 1.1.5 The Dujiangyan Irrigation Project: A Showcase of Early Mankind's Ingenuity 21 1.2 Linear Time-Delay Systems 23 1.2.1 Stability of Linear Time-Delay Systems 25 1.2.2 Stabilization of Linear Time-Delay Systems 27 1.3 Delays in Networked Control Systems 29 1.4 Delays in Multi-agent Systems 31 1.5 Summary 33 2 Mathematical Background 34 2.1 Signal and System Norms 34 2.1.1 mathcalL2 and mathcalH2 Norms 35 2.1.2 mathcalLp-Induced System Norms 37 2.2 Small-Gain Theorems 38 2.2.1 mathcalL2 Small-Gain Theorem 39 2.2.2 mathcalLinfty Small-Gain Theorem 40 2.2.3 Mean-Square Small-Gain Theorem 41 2.3 mathcalHinfty and mathcalH2 Optimal Control 43 2.3.1 Youla–Kuera Parameterization 44 2.3.2 Inner–Outer Factorization 46 2.4 Robust Stability Margin Problems 49 2.4.1 System Poles and Zeros 49 2.4.2 Analytic Function Interpolation 51 2.4.3 Gain and Phase Margin Problems 52 2.5 LMI and GEVP 54 2.6 Summary 55 3 Small-Gain Stability Conditions 56 3.1 Introduction 56 3.2 Uncertainty Modeling and Model Transformation 57 3.3 mathcalL2-Stability Conditions 61 3.3.1 Range-Independent Conditions 61 3.3.2 Range-Dependent Conditions 63 3.3.3 Extension to Noncentric Delay Intervals 69 3.3.4 Extension to Transformed Nominal Systems 72 3.4 mathcalLinfty-Stability Conditions 74 3.4.1 Delay-Independent Conditions 75 3.4.2 Delay-Dependent Conditions 79 3.5 Summary 82 4 Delay Margin 84 4.1 Introduction 84 4.2 Upper Bounds on the Delay Margin 86 4.3 Lower Bounds on the Delay Margin 92 4.4 Summary 105 5 Stabilization of Linear Systems with Time-Varying Delays 107 5.1 Introduction 107 5.2 Rate-Independent Bound 109 5.3 Rate-Dependent Conditions 110 5.4 Summary 119 6 Delay Margin Achievable by PID Controllers 120 6.1 Introduction 120 6.2 First-Order Unstable Systems 123 6.3 Second-Order Unstable Plants 127 6.3.1 Real Poles 127 6.3.2 Complex Conjugate Poles 134 6.3.3 Oscillatory Poles 137 6.4 Second-Order Examples 139 6.5 Effect of Nonminimum Phase Zeros 143 6.6 Unstable Plants with Time-Varying Delay 146 6.7 Summary 154 7 Delay Radius of MIMO Delay Systems 0 7.1 Introduction 155 7.2 Region of Stability 157 7.3 Uniform Delays 164 7.4 Region of Instability 165 7.5 Summary 175 8 Stabilization of Networked Delay Systems 176 8.1 Introduction 176 8.2 Mean-Square Stabilizability 178 8.3 SISO Systems 179 8.4 MIMO Systems 186 8.5 Summary 195 9 Consensus of Multi-agent Systems Under Delay 196 9.1 Introduction 196 9.2 Consensus over Undirected Graph 198 9.3 Consensus over Directed Graph 211 9.4 Summary 217 Appendix References 218 218 Index 226 Front Matter ....Pages i-xvi Introduction (Jing Zhu, Tian Qi, Dan Ma, Jie Chen)....Pages 1-19 Mathematical Background (Jing Zhu, Tian Qi, Dan Ma, Jie Chen)....Pages 21-42 Small-Gain Stability Conditions (Jing Zhu, Tian Qi, Dan Ma, Jie Chen)....Pages 43-70 Delay Margin (Jing Zhu, Tian Qi, Dan Ma, Jie Chen)....Pages 71-93 Stabilization of Linear Systems with Time-Varying Delays (Jing Zhu, Tian Qi, Dan Ma, Jie Chen)....Pages 95-107 Delay Margin Achievable by PID Controllers (Jing Zhu, Tian Qi, Dan Ma, Jie Chen)....Pages 109-143 Delay Radius of MIMO Delay Systems (Jing Zhu, Tian Qi, Dan Ma, Jie Chen)....Pages 145-165 Stabilization of Networked Delay Systems (Jing Zhu, Tian Qi, Dan Ma, Jie Chen)....Pages 167-186 Consensus of Multi-agent Systems Under Delay (Jing Zhu, Tian Qi, Dan Ma, Jie Chen)....Pages 187-208 Back Matter ....Pages 209-219