Iterative Learning Control for Multi-agent Systems Coordination

Shiping Yang, Jian-Xin Xu, and Xuefang LiNational University of Singapore Dong ShenBeijing University of Chemical Technology, P.R. China

• Preface ix 1 Introduction 1 1.1 Introduction to Iterative Learning Control 1 1.1.1 Contraction-Mapping Approach 3 1.1.2 Composite Energy Function Approach 4 1.2 Introduction to MAS Coordination 5 1.3 Motivation and Overview 7 1.4 Common Notations in This Book 9 2 Optimal Iterative Learning Control for Multi-agent Consensus Tracking 11 2.1 Introduction 11 2.2 Preliminaries and Problem Description 12 2.2.1 Preliminaries 12 2.2.2 Problem Description 13 2.3 Main Results 15 2.3.1 Controller Design for Homogeneous Agents 15 2.3.2 Controller Design for Heterogeneous Agents 20 2.4 Optimal Learning Gain Design 21 2.5 Illustrative Example 23 2.6 Conclusion 26 3 Iterative Learning Control for Multi-agent Coordination Under Iteration-Varying Graph 27 3.1 Introduction 27 3.2 Problem Description 28 3.3 Main Results 29 3.3.1 Fixed Strongly Connected Graph 29 3.3.2 Iteration-Varying Strongly Connected Graph 32 3.3.3 Uniformly Strongly Connected Graph 37 3.4 Illustrative Example 38 3.5 Conclusion 40 4 Iterative Learning Control for Multi-agent Coordination with Initial State Error 41 4.1 Introduction 41 4.2 Problem Description 42 4.3 Main Results 43 4.3.1 Distributed D-type Updating Rule 43 4.3.2 Distributed PD-type Updating Rule 48 4.4 Illustrative Examples 49 4.5 Conclusion 50 5 Multi-agent Consensus Tracking with Input Sharing by Iterative Learning Control 53 5.1 Introduction 53 5.2 Problem Formulation 54 5.3 Controller Design and Convergence Analysis 54 5.3.1 Controller Design Without Leader’s Input Sharing 55 5.3.2 Optimal Design Without Leader’s Input Sharing 58 5.3.3 Controller Design with Leader’s Input Sharing 59 5.4 Extension to Iteration-Varying Graph 60 5.4.1 Iteration-Varying Graph with Spanning Trees 60 5.4.2 Iteration-Varying Strongly Connected Graph 60 5.4.3 Uniformly Strongly Connected Graph 62 5.5 Illustrative Examples 63 5.5.1 Example 1: Iteration-Invariant Communication Graph 63 5.5.2 Example 2: Iteration-Varying Communication Graph 64 5.5.3 Example 3: Uniformly Strongly Connected Graph 66 5.6 Conclusion 68 6 A HOIM-Based Iterative Learning Control Scheme for Multi-agent Formation 69 6.1 Introduction 69 6.2 Kinematic Model Formulation 70 6.3 HOIM-Based ILC for Multi-agent Formation 71 6.3.1 Control Law for Agent 1 72 6.3.2 Control Law for Agent 2 74 6.3.3 Control Law for Agent 3 75 6.3.4 Switching Between Two Structures 78 6.4 Illustrative Example 78 6.5 Conclusion 80 7 P-type Iterative Learning for Non-parameterized Systems with Uncertain Local Lipschitz Terms 81 7.1 Introduction 81 7.2 Motivation and Problem Description 82 7.2.1 Motivation 82 7.2.2 Problem Description 83 7.3 Convergence Properties with Lyapunov Stability Conditions 84 7.3.1 Preliminary Results 84 7.3.2 Lyapunov Stable Systems 86 7.3.3 Systems with Stable Local Lipschitz Terms but Unstable Global Lipschitz Factors 90 7.4 Convergence Properties in the Presence of Bounding Conditions 92 7.4.1 Systems with Bounded Drift Term 92 7.4.2 Systems with Bounded Control Input 94 7.5 Application of P-type Rule in MAS with Local Lipschitz Uncertainties 97 7.6 Conclusion 99 8 Synchronization for Nonlinear Multi-agent Systems by Adaptive Iterative Learning Control 101 8.1 Introduction 101 8.2 Preliminaries and Problem Description 102 8.2.1 Preliminaries 102 8.2.2 Problem Description for First-Order Systems 102 8.3 Controller Design for First-Order Multi-agent Systems 105 8.3.1 Main Results 105 8.3.2 Extension to Alignment Condition 107 8.4 Extension to High-Order Systems 108 8.5 Illustrative Example 113 8.5.1 First-Order Agents 114 8.5.2 High-Order Agents 115 8.6 Conclusion 118 9 Distributed Adaptive Iterative Learning Control for Nonlinear Multi-agent Systems with State Constraints 123 9.1 Introduction 123 9.2 Problem Formulation 124 9.3 Main Results 127 9.3.1 Original Algorithms 127 9.3.2 Projection Based Algorithms 135 9.3.3 Smooth Function Based Algorithms 138 9.3.4 Alternative Smooth Function Based Algorithms 141 9.3.5 Practical Dead-Zone Based Algorithms 145 9.4 Illustrative Example 147 9.5 Conclusion 174 10 Synchronization for Networked Lagrangian Systems under Directed Graphs 175 10.1 Introduction 175 10.2 Problem Description 176 10.3 Controller Design and Performance Analysis 177 10.4 Extension to Alignment Condition 183 10.5 Illustrative Example 184 10.6 Conclusion 188 11 Generalized Iterative Learning for Economic Dispatch Problem in a Smart Grid 189 11.1 Introduction 189 11.2 Preliminaries 190 11.2.1 In-Neighbor and Out-Neighbor 190 11.2.2 Discrete-Time Consensus Algorithm 191 11.2.3 Analytic Solution to EDP with Loss Calculation 192 11.3 Main Results 193 11.3.1 Upper Level: Estimating the Power Loss 194 11.3.2 Lower Level: Solving Economic Dispatch Distributively 194 11.3.3 Generalization to the Constrained Case 197 11.4 Learning Gain Design 198 11.5 Application Examples 200 11.5.1 Case Study 1: Convergence Test 201 11.5.2 Case Study 2: Robustness of Command Node Connections 202 11.5.3 Case Study 3: Plug and Play Test 203 11.5.4 Case Study 4: Time-Varying Demand 205 11.5.5 Case Study 5: Application in Large Networks 207 11.5.6 Case Study 6: Relation Between Convergence Speed and Learning Gain 207 11.6 Conclusion 208 12 Summary and Future Research Directions 209 12.1 Summary 209 12.2 Future Research Directions 210 12.2.1 Open Issues in MAS Control 210 12.2.2 Applications 214 Appendix A Graph Theory Revisit 223 Appendix B Detailed Proofs 225 B.1 HOIM Constraints Derivation 225 B.2 Proof of Proposition 2.1 226 B.3 Proof of Lemma 2.1 227 B.4 Proof of Theorem 8.1 229 B.5 Proof of Corollary 8.1 230 Bibliography 233 Index 245