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Distributed Artificial Intelligence Laboratory, ERC-FCDE, MoE
School of Mathematics, Renmin University of China
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Book:SILC
 Brief Introduction

This book is self contained and intensive. Prior knowledge of ILC and MAS is not required. Chapter 1 provides a rudimentary introduction to the two areas. Twominimal examples of ILC are presented in Chapter 1, and a short review of some terminologies in graph theory is provided in Appendix A. Readers can skip the preliminary parts if they are familiar with the domain. We present detailed convergence proofs for each controller as we believe that understanding the theoretical derivations can benefit readers in two ways. On the one hand, it helps readers appreciate the controller design. On the other hand, the control design and analysis techniques can be transferred to other domains to facilitate further exploration in various control applications. Specifically for industrial experts and practitioners, we provide detailed illustrative examples in each chapter to show how those control algorithms are implemented.The examples demonstrate the effectiveness of the learning controllers and can be modified to handle practical problems.

Contents

Preface

1 Introduction
1.1 Introduction to Iterative Learning Control
1.1.1 Contraction-Mapping Approach
1.1.2 Composite Energy Function Approach
1.2 Introduction to MAS Coordination
1.3 Motivation and Overview
1.4 Common Notations in This Book

2 Optimal Iterative Learning Control for Multi-agent Consensus Tracking
2.1 Introduction
2.2 Preliminaries and Problem Description
2.2.1 Preliminaries
2.2.2 Problem Description
2.3 Main Results
2.3.1 Controller Design for Homogeneous Agents
2.3.2 Controller Design for Heterogeneous Agents
2.4 Optimal Learning Gain Design
2.5 Illustrative Example
2.6 Conclusion

3 Iterative Learning Control for Multi-agent Coordination Under Iteration-Varying Graph
3.1 Introduction
3.2 Problem Description
3.3 Main Results
3.3.1 Fixed Strongly Connected Graph
3.3.2 Iteration-Varying Strongly Connected Graph
3.3.3 Uniformly Strongly Connected Graph
3.4 Illustrative Example
3.5 Conclusion

4 Iterative Learning Control for Multi-agent Coordination with Initial State Error
4.1 Introduction
4.2 Problem Description
4.3 Main Results
4.3.1 Distributed D-type Updating Rule
4.3.2 Distributed PD-type Updating Rule
4.4 Illustrative Examples
4.5 Conclusion

5 Multi-agent Consensus Tracking with Input Sharing by Iterative Learning Control
5.1 Introduction
5.2 Problem Formulation
5.3 Controller Design and Convergence Analysis
5.3.1 Controller DesignWithout Leader’s Input Sharing
5.3.2 Optimal DesignWithout Leader’s Input Sharing
5.3.3 Controller Design with Leader’s Input Sharing
5.4 Extension to Iteration-Varying Graph
5.4.1 Iteration-Varying Graph with Spanning Trees
5.4.2 Iteration-Varying Strongly Connected Graph
5.4.3 Uniformly Strongly Connected Graph
5.5 Illustrative Examples
5.5.1 Example 1: Iteration-Invariant Communication Graph
5.5.2 Example 2: Iteration-Varying Communication Graph
5.5.3 Example 3: Uniformly Strongly Connected Graph
5.6 Conclusion

6 A HOIM-Based Iterative Learning Control Scheme for Multi-agent Formation
6.1 Introduction
6.2 KinematicModel Formulation
6.3 HOIM-Based ILC for Multi-agent Formation
6.3.1 Control Law for Agent 1
6.3.2 Control Law for Agent 2
6.3.3 Control Law for Agent 3
6.3.4 Switching Between Two Structures
6.4 Illustrative Example
6.5 Conclusion

7 P-type Iterative Learning for Non-parameterized Systems with Uncertain Local Lipschitz Terms
7.1 Introduction
7.2 Motivation and Problem Description
7.2.1 Motivation
7.2.2 Problem Description
7.3 Convergence Properties with Lyapunov Stability Conditions
7.3.1 Preliminary Results
7.3.2 Lyapunov Stable Systems
7.3.3 Systems with Stable Local Lipschitz Terms but Unstable Global Lipschitz Factors
7.4 Convergence Properties in the Presence of Bounding Conditions
7.4.1 Systems with Bounded Drift Term
7.4.2 Systems with Bounded Control Input
7.5 Application of P-type Rule in MAS with Local Lipschitz Uncertainties
7.6 Conclusion

8 Synchronization for Nonlinear Multi-agent Systems by Adaptive Iterative Learning Control
8.1 Introduction
8.2 Preliminaries and Problem Description
8.2.1 Preliminaries
8.2.2 Problem Description for First-Order Systems
8.3 Controller Design for First-OrderMulti-agent Systems
8.3.1 Main Results
8.3.2 Extension to Alignment Condition
8.4 Extension to High-Order Systems
8.5 Illustrative Example
8.5.1 First-Order Agents
8.5.2 High-Order Agents
8.6 Conclusion

9 Distributed Adaptive Iterative Learning Control for Nonlinear Multi-agent Systems with State
9.1 Introduction
9.2 Problem Formulation
9.3 Main Results
9.3.1 Original Algorithms
9.3.2 Projection Based Algorithms
9.3.3 Smooth Function Based Algorithms
9.3.4 Alternative Smooth Function Based Algorithms
9.3.5 Practical Dead-Zone Based Algorithms
9.4 Illustrative Example
9.5 Conclusion

10 Synchronization for Networked Lagrangian Systems under Directed Graphs
10.1 Introduction
10.2 Problem Description
10.3 Controller Design and Performance Analysis
10.4 Extension to Alignment Condition
10.5 Illustrative Example
10.6 Conclusion

11 Generalized Iterative Learning for Economic Dispatch Problem in a Smart Grid
11.1 Introduction
11.2 Preliminaries
11.2.1 In-Neighbor and Out-Neighbor
11.2.2 Discrete-Time Consensus Algorithm
11.2.3 Analytic Solution to EDP with Loss Calculation
11.3 Main Results
11.3.1 Upper Level: Estimating the Power Loss
11.3.2 Lower Level: Solving Economic Dispatch Distributively
11.3.3 Generalization to the Constrained Case
11.4 Learning Gain Design
11.5 Application Examples
11.5.1 Case Study 1: Convergence Test
11.5.2 Case Study 2: Robustness of Command Node Connections
11.5.3 Case Study 3: Plug and Play Test
11.5.4 Case Study 4: Time-Varying Demand
11.5.5 Case Study 5: Application in Large Networks
11.5.6 Case Study 6: Relation Between Convergence Speed and Learning Gain
11.6 Conclusion

12 Summary and Future Research Directions
12.1 Summary
12.2 Future Research Directions
12.2.1 Open Issues in MAS Control
12.2.2 Applications

Appendix A Graph Theory Revisit

Appendix B Detailed Proofs
B.1 HOIM Constraints Derivation
B.2 Proof of Proposition 2.1
B.3 Proof of Lemma 2.1
B.4 Proof of Theorem 8.1
B.5 Proof of Corollary 8.1

Bibliography
Index

Sample Chapter

Chapter 1: Introduction
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