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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: Wiley - IEEE
- Sprache: Englisch
A timely guide using iterative learning control (ILC) as a solution for multi-agent systems (MAS) challenges, showcasing recent advances and industrially relevant applications
- Explores the synergy between the important topics of iterative learning control (ILC) and multi-agent systems (MAS)
- Concisely summarizes recent advances and significant applications in ILC methods for power grids, sensor networks and control processes
- Covers basic theory, rigorous mathematics as well as engineering practice
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Seitenzahl: 337
Veröffentlichungsjahr: 2017
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Table of Contents
Cover
Title Page
Preface
1 Introduction
1.1 Introduction to Iterative Learning Control
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
2.3 Main Results
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
3.4 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
4.4 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
5.4 Extension to Iteration‐Varying Graph
5.5 Illustrative Examples
5.6 Conclusion
6 A HOIM‐Based Iterative Learning Control Scheme for Multi‐agent Formation
6.1 Introduction
6.2 Kinematic Model Formulation
6.3 HOIM‐Based ILC for Multi‐agent Formation
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
7.3 Convergence Properties with Lyapunov Stability Conditions
7.4 Convergence Properties in the Presence of Bounding Conditions
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
8.3 Controller Design for First‐Order Multi‐agent Systems
8.4 Extension to High‐Order Systems
8.5 Illustrative Example
8.6 Conclusion
9 Distributed Adaptive Iterative Learning Control for Nonlinear Multi‐agent Systems with State Constraints
9.1 Introduction
9.2 Problem Formulation
9.3 Main Results
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 Examples
10.6 Conclusion
11 Generalized Iterative Learning for Economic Dispatch Problem in a Smart Grid
11.1 Introduction
11.2 Preliminaries
11.3 Main Results
11.4 Learning Gain Design
11.5 Application Examples
11.6 Conclusion
12 Summary and Future Research Directions
12.1 Summary
12.2 Future Research Directions
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
End User License Agreement
List of Tables
Chapter 08
Table 8.1 Agent Parameters.
Chapter 11
Table 11.1 Generator Parameters.
Table 11.2 Initializations.
Chapter 12
Table 12.1 The topics covered in this book.
List of Illustrations
Chapter 01
Figure 1.1 The framework of ILC.
Figure 1.2 Example of a network.
Chapter 02
Figure 2.1 Communication topology among agents in the network.
Figure 2.2 Tracking errors of all agents at different iterations.
Figure 2.3 Maximum tracking error vs. iteration number.
Chapter 03
Figure 3.1 Communication topology among agents in the network.
Figure 3.2 Maximum norm of error vs. iteration number.
Chapter 04
Figure 4.1 Communication topology among agents in the network.
Figure 4.2 Output trajectories at the 150 th iteration under D‐type ILC learning rule.
Figure 4.3 Output trajectories at the 50 th iteration under PD‐type ILC learning rule.
Figure 4.4 Tracking error profiles at the 50 th iteration under PD‐type ILC learning rule.
Chapter 05
Figure 5.1 Iteration‐invariant communication topology among agents in the network.
Figure 5.2 Supremum norm of error in log scale vs. iteration number.
Figure 5.3 Iteration‐varying communication topology among agents in the network.
Figure 5.4 Supremum norm of error under iteration‐varying graph vs. iteration number.
Figure 5.5 Uniformly strongly connected communication graph for agents in the network.
Figure 5.6 Supremum norm of error under uniformly strongly connected graph vs. iteration number.
Chapter 06
Figure 6.1 Kinematics model.
Figure 6.2 Switching process between two different structure formations.
Figure 6.3 Multi‐Agent formation at 29th (odd) iteration.
Figure 6.4 Multi‐Agent formation at 30th (even) iteration.
Figure 6.5 Error convergence along the iteration axis.
Chapter 07
Figure 7.1 Tracking error profiles vs. iteration number for
and
.
Figure 7.2 Tracking error profiles vs. iteration number for system with bounded local Lipschitz term.
Figure 7.3 Desired torque profile.
Figure 7.4 Tracking error profiles vs. iteration number under control saturation.
Figure 7.5 Communication topology among agents in the network.
Figure 7.6 Maximal tracking error profiles vs. iteration number for MAS with local Lipschitz uncertainties.
Chapter 08
Figure 8.1 Communication among agents in the network.
Figure 8.2 The trajectory profiles at the 1st and 50th iterations under
iic
.
Figure 8.3 Maximum tracking error vs. iteration number under
iic
.
Figure 8.4 The trajectory profiles at the 1st and 50th iterations under alignment condition.
Figure 8.5 Maximum tracking error vs. iteration number under alignment condition.
Figure 8.6 The trajectory profiles at the 1st iteration.
Figure 8.7 The trajectory profiles at the 50th iteration.
Figure 8.8 Maximum tracking errors vs. iteration number.
Figure 8.9 The trajectory profiles at the 1st iteration with initial rectifying action.
Figure 8.10 The trajectory profiles at the 20th iteration with initial rectifying action.
Chapter 09
Figure 9.1 Communication graph among agents in the network.
Figure 9.2 Trajectories of four agents at the 1st iteration and 20th iteration: (
) case.
Figure 9.3 Tracking performance at the 1st iteration, 10th iteration, and 20th iteration for each agent: (
) case.
Figure 9.4 Maximum tracking error along iteration axis: (
) case.
Figure 9.5 Input profiles at the 1st iteration, 10th iteration, and 20th iteration for each agent: (
) case.
Figure 9.6 Trajectories of four agents at the 1st iteration and 20th iteration: (
) case.
Figure 9.7 Maximum tracking error along iteration axis: (
) case.
Figure 9.8 Input profiles at the 1st iteration, 10th iteration, and 20th iteration for each agent: (
) case.
Figure 9.9 Trajectories of four agents at the 1st iteration and 20th iteration: (
) case.
Figure 9.10 Maximum tracking error along iteration axis: (
) case.
Figure 9.11 Input profiles at the 1st iteration, 10th iteration, and 20th iteration for each agent: (
) case.
Figure 9.12 Trajectories of four agents at the 1st iteration and 20th iteration: (
) case.
Figure 9.13 Tracking performance at the 1st iteration, 10th iteration, and 20th iteration for each agent: (
) case.
Figure 9.14 Maximum tracking error along iteration axis: (
) case.
Figure 9.15 Input profiles at the 1st iteration, 10th iteration, and 20th iteration for each agent: (
) case.
Figure 9.16 Trajectories of four agents at the 1st iteration and 20th iteration: (
) case.
Figure 9.17 Maximum tracking error along iteration axis: (
) case.
Figure 9.18 Input profiles at the 1st iteration, 10th iteration, and 20th iteration for each agent: (
) case.
Chapter 10
Figure 10.1 Directed acyclic graph for describing the communication among agents.
Figure 10.2 Trajectory profiles at the 1st iteration.
Figure 10.3 Trajectory profiles at the 70th iteration, all trajectories overlap with each other.
Figure 10.4 Maximum tracking error profile.
Figure 10.5 Control input profiles at the 1st iteration.
Figure 10.6 Control input profiles at the 70th iteration.
Chapter 11
Figure 11.1 Communication topology among generators and command vertex in the network.
Figure 11.2 Results obtained with generator constraints.
Figure 11.3 Robustness test when the command vertex is connected to generators 2 and 4.
Figure 11.4 Communication topology with the fifth generator.
Figure 11.5 Results obtained with the fifth generator.
Figure 11.6 Results obtained with time‐varying demand.
Figure 11.7 Results obtained with 100 generators.
Figure 11.8 Relation between convergence speed and learning gain.
Chapter 12
Figure 12.1 The schematic diagram of an agent under networked control.
y
i
,
ỹ
i
, and
y
d
are plant output, output signal received at the ILC side, reference output, respectively.
u
i
is is the control profile generated by the ILC mechanism.
Figure 12.2 Freeway traffic behaves as a multi‐agent system, where each vehicle has to coordinate its location, speed, acceleration, and orientation with neighboring vehicles.
Figure 12.3 Layout of a road network in central business district (CBD). The circle, squares and triangles represent agents, and each controls the traffic signals at an intersection.
Figure 12.4 UAVs enter a CBD region to perform exploration, searching, surveillance, and tracking tasks. UAV‐1 is fixed wing, UAV‐2 and UAV‐3 are rotary wing.
Figure 12.5 Formations of live agents.
Figure 12.6 Bacteria quorum sensing with either individual or group behaviors, where the ellipses represent bacteria and triangles represent autoinducers.
Figure 12.7 The receptors and associated ligands on T cell, dendritic cell, tissue macrophage cell, and tumor cell. Receptors and ligands binding together will generate positive/negative stimuli, and accordingly activate/inhibit the T cell.
Appendix B
Figure B.1 The boundary of complex parameter
.
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Iterative Learning Control for Multi‐agent Systems Coordination
Shiping Yang
Jian‐Xin Xu
Xuefang Li
National University of Singapore
Dong Shen
Beijing University of Chemical Technology, P.R. China
This edition first published 2017© 2017 John Wiley & Sons Singapore Pte. Ltd
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Names: Yang, Shiping, 1987– author. | Xu, Jian‐Xin, author. | Li, Xuefang, 1985– author. | Shen, Dong, 1982– author.Title: Iterative learning control for multi‐agent systems coordination / by Shiping Yang, Jian‐Xin Xu, Xuefang Li, Dong Shen.Description: Singapore : John Wiley & Sons, Inc., 2017. | Includes bibliographical references and index.Identifiers: LCCN 2016052027 (print) | LCCN 2016056133 (ebook) | ISBN 9781119189046 (hardback) | ISBN 9781119189060 (pdf) | ISBN 9781119189077 (epub)Subjects: LCSH: Intelligent control systems. | Multiagent systems. | Machine learning. | Iterative methods (Mathematics) | BISAC: TECHNOLOGY & ENGINEERING / Robotics.Classification: LCC TJ217.5 .Y36 2017 (print) | LCC TJ217.5 (ebook) | DDC 629.8/9–dc23LC record available at https://lccn.loc.gov/2016052027
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Preface
The coordination and control problems of multi‐agent systems (MAS) have been extensively studied by the control community due to the broad practical applications, for example, the formation control problem, search and rescue by multiple aerial vehicles, synchronization, sensor fusion, distributed optimization, the economic dispatch problem in power systems, and so on. Meanwhile, many industry processes require both repetitive executions and coordination among several independent entities. This observation motivates the research of multi‐agent coordination from an iterative learning control (ILC) perspective. This book is dedicated to the application of iterative learning control to multi‐agent coordination problems.
In order to study multi‐agent coordination by ILC, an extra dimension, the iteration domain, is introduced into the problem. A challenging issue in controlling multi‐agent systems by ILC is the non‐perfect repeating characteristics of MAS. The inherent nature of MAS such as heterogeneity, information sharing, sparse and intermittent communication, imperfect initial conditions, and inconsistent target tracking trajectories increases the complexity of the problem. Due to these factors, controller design becomes a challenging problem. This book provides detailed guidelines for the design of learning controllers under various coordination conditions, in a systematic manner. The main content can be classified into two parts following the two main frameworks of ILC, namely, contraction‐mapping (CM) and composite energy function (CEF) approaches. Chapters 2–7 apply the CM approach, while Chapters 8–10 apply the CEF approach. Each chapter studies the coordination problem under certain conditions. For example, Chapter 2 assumes a fixed communication topology, Chapter 3 assumes a switching topology, and Chapter 4 addresses the initial state error problem in multi‐agent coordination. In a sense, each chapter addresses a unique coordination control problem for MAS. Chapters 2–10 discuss continuous‐time systems. In Chapter 11 we present a generalized iterative learning algorithm to solve an optimal power dispatch problem in a smart grid by utilizing discrete‐time consensus algorithms.
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. Two minimal 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.
1Introduction
1.1 Introduction to Iterative Learning Control
Iterative learning control (ILC), as an effective control strategy, is designed to improve current control performance for unpredictable systems by fully utilizing past control experience. Specifically, ILC is designed for systems that complete tasks over a fixed time interval and perform them repeatedly. The underlying philosophy mimics the human learning process that “practice makes perfect.” By synthesizing control inputs from previous control inputs and tracking errors, the controller is able to learn from past experience and improve current tracking performance. ILC was initially developed by Arimoto et al. (1984), and has been widely explored by the control community since then (Moore, 1993; Bien and Xu, 1998; Chen and Wen, 1999; Longman, 2000; Norrlof and Gunnarsson, 2002; Xu and Tan, 2003; Bristow et al. 2006; Moore et al. 2006; Ahn et al. 2007a; Rogers et al. 2007; Ahn et al. 2007b; Xu et al. 2008; Wang et al. 2009, 2014).
Figure 1.1 shows the schematic diagram of an ILC system, where the subscript i denotes the iteration index and yd denotes the reference trajectory. Based on the input signal, ui, at the i th iteration, as well as the tracking error , the input for the next iteration, namely the th iteration, is constructed. Meanwhile, the input signal will also be stored into memory for use in the th iteration. It is important to note that in Figure 1.1, a closed feedback loop is formed in the iteration domain rather than the time domain. Compared to other control methods such as proportional‐integral‐derivative (PID) control and sliding mode control, there are a number of distinctive features about ILC. First, ILC is designed to handle repetitive control tasks, while other control techniques don’t typically take advantage of task repetition—under a repeatable control environment, repeating the same feedback would yield the same control performance. In contrast, by incorporating learning, ILC is able to improve the control performance iteratively. Second, the control objective is different. ILC aims at achieving perfect tracking over the whole operational interval. Most control methods aim to achieve asymptotic convergence in tracking accuracy over time. Third, ILC is a feedforward control method if viewed in the time domain. The plant shown in Figure 1.1 is a generalized plant, that is, it can actually include a feedback loop. ILC can be used to further improve the performance of the generalized plant. As such, the generalized plant could be made stable in the time domain, which is helpful in guaranteeing transient response while learning takes place. Last but not least, ILC is a partially model‐free control method. As long as an appropriate learning gain is chosen, perfect tracking can be achieved without using a perfect plant model.
Figure 1.1 The framework of ILC.
Generally speaking, there are two main frameworks for ILC, namely contraction‐mapping (CM)‐based and composite energy function (CEF)‐based approaches. A CM‐based iterative learning controller has a very simple structure and is easy to implement. A correction term in the controller is constructed from the output tracking error; to ensure convergence, an appropriate learning gain is selected based on system gradient information in place of an accurate dynamic model. As a partially model‐free control method, CM‐based ILC is applicable to non‐affine‐in‐input systems. These features are highly desirable in practice as there are plenty of data available in industry processes but there is a shortage of accurate system models. CM‐based ILC has been adopted in many applications, for example X‐Y tables, chemical batch reactors, laser cutting systems, motor control, water heating systems, freeway traffic control, wafer manufacturing, and so on (Ahn et al., 2007a). A limitation of CM‐based ILC is that it is only applicable to global Lipschitz continuous (GLC) systems. The GLC condition is required by ILC in order to form a contractive mapping, and rule out the finite escape time phenomenon. In comparison, CEF‐based ILC, a complementary approach to CM‐based ILC, applies a Lyapunov‐like method to design learning rules. CEF is an effective method to handle locally Lipschitz continuous (LLC) systems, because system dynamics is used in the design of learning and feedback mechanisms. It is, however, worthwhile pointing out that in CM‐based ILC, the learning mechanism only requires output signals, while in CEF‐based ILC, full state information is usually required. CEF‐based ILC has been applied in satellite trajectory keeping (Ahn et al. 2010) and robotic manipulator control (Tayebi, 2004; Tayebi and Islam, 2006; Sun et al., 2006).
This book follows the two main frameworks and investigates the multi‐agent coordination problem using ILC. To illustrate the underlying idea and properties of ILC, we start with a simple ILC system.
Consider the following linear time‐invariant dynamics:
where i is the iteration index, a is an unknown constant parameter, and T is the trial length. Let the target trajectory be xd(t), which is generated by
with ud(t) is the desired control signal. The control objective is to tune ui(t) such that without any prior knowledge about the parameter a, the tracking error can converge to zero as the iteration number increases, that is, for .
We perform the ILC controller design and convergence analysis for this simple control problem under the frameworks of both CM‐based and CEF‐based approaches, in order to illustrate the basic concepts in ILC and analysis techniques. To restrict our discussion, the following assumptions are imposed on the dynamical system (1.1).
Assumption 1.1
The identical initialization condition holds for all iterations, that is, , .
Assumption 1.2
For , , there exists a ud(t), such that implies , .
1.1.1 Contraction‐Mapping Approach
Under the framework of CM‐based methodology, we apply the following D‐type updating law to solve the trajectory tracking problem:
where is the learning gain to be determined. Our objective is to show that the ILC law (1.3) can converge to the desired ud, which implies the convergence of the tracking error ei(t), as i increases.
Define . First we can derive the relation
Furthermore, the state error dynamics is given by
Combining (1.4) and (1.5) gives:
Integrating both sides of the state error dynamics and using Assumption 1.1 yields
Then, substituting (1.7) into (1.6), we obtain
Taking λ ‐norm on both sides of (1.8) gives
where , and the λ ‐norm is defined as
The λ ‐norm is just a time weighted norm and is used to simplify the derivation. It will be formally defined in Section 1.4.
If in (1.9), it is possible to choose a sufficiently large such that . Therefore, (1.9) implies that , namely .
1.1.2 Composite Energy Function Approach
In this subsection, the ILC controller will be developed and analyzed under the framework of CEF‐based approach. First of all, the error dynamics of the system (1.1) can be expressed as follows:
where xd is the target trajectory.
Let k be a positive constant. By applying the control law
and the parametric updating law ,
we can obtain the convergence of the tracking error ei as i tends to infinity.
In order to facilitate the convergence analysis of the proposed ILC scheme, we introduce the following CEF:
where is the estimation error of the unknown parameter a.
The difference of Ei is
By using the identical initialization condition as in Assumption 1.1, the error dynamics (1.10), and the control law (1.11), the first term on the right hand side of (1.14) can be calculated as
In addition, the second term on the right hand side of (1.14) can be expressed as
where the updating law (1.12) is applied. Clearly, ϕixiei appears in (1.15) and (1.16) with opposite signs. Combining (1.14), (1.15), and (1.16) yields
The function Ei is a monotonically decreasing sequence, hence is bounded if E0 is bounded.
Now, let us show the boundedness of E0. For the linear plant (1.1) or in general GLC plants, there will be no finite escape time, thus E0 is bounded. For local Lipschitz continuous plants, ILC designed under CEF guarantees there is no finite escape time (see Xu and Tan, 2003, chap. 7), thus E0 is bounded. Hence, the boundedness of E0(t) over [0, T] is obtained.
Consider a finite sum of ΔEi,
and apply the inequality (1.17); we have:
Because of the positiveness of Ei and boundedness of E0, ei(t) converges to zero in a pointwise fashion as i tends to infinity.
1.2 Introduction to MAS Coordination
In the past several decades, MAS coordination and control problems have attracted considerable attention from many researchers of various backgrounds due to their potential applications and cross‐disciplinary nature. Consensus in particular is an important class of MAS coordination and control problems (Cao et al., 2013). According to Olfati‐Saber et al. (2007), in networks of agents (or dynamic systems), consensus means to reach an agreement regarding certain quantities of interest that are associated with all agents. Depending on the specific application, these quantities could be velocity, position, temperature, orientation, and so on. In a consensus realization, the control action of an agent is generated based on the information received or measured from its neighborhood. Since the control law is a kind of distributed algorithm, it is more robust and scalable compared to centralized control algorithms.
The three main components in MAS coordination are the agent model, the information sharing topology, and the control algorithm or consensus algorithm.
Agent models range from simple single integrator model to complex nonlinear models. Consensus results on single integrators are reported by Jadbabaie et al. (2003), Olfati‐Saber and Murray (2004), Moreau (2005), Ren et al. (2007), and Olfati‐Saber et al. (2007). Double integrators are investigated in Xie and Wang (2005), Hong et al. (2006), Ren (2008a), and Zhang and Tian (2009). Results on linear agent models can be found in Xiang et al. (2009), Ma and Zhang (2010), Li et al. (2010), Huang (2011), and Wieland et al. (2011). Since the Lagrangian system can be used to model many practical systems, consensus has been extensively studied by means of the Lagrangian system. Some representative works are reported by Hou et al. (2009), Chen and Lewis (2011), Mei et al. (2011), and Zhang et al. (2014).
Information sharing among agents is one of the indispensable components for consensus seeking. Information sharing can be realized by direct measurement from on board sensors or communication through wireless networks. The information sharing mechanism is usually modeled by a graph. For simplicity in the early stages of consensus algorithm development, the communication graph is assumed to be fixed. However, a consensus algorithm that is robust or adaptive to topology variations is more desirable, since many practical conditions can be modeled as time‐varying communications, for example asynchronous updating, or communication link failures and creations. As communication among agents is an important topic in the MAS literature, various communication assumptions and consensus results have been investigated by researchers (Moreau, 2005; Hatano and Mesbahi, 2005; Tahbaz‐Salehi and Jadbabaie, 2008; Zhang and Tian, 2009). An excellent survey can be found in Fang and Antsaklis (2006). Since graph theory is seldom used in control theory and applications, a brief introduction to the topic is given in Appendix A.
A consensus algorithm is a very simple local coordination rule which can result in very complex and useful behaviors at the group level. For instance, it is widely observed that by adopting such a strategy, a school of fish can improve the chance of survival under the sea (Moyle and Cech, 2003). Many interesting coordination problems have been formulated and solved under the framework of consensus, for example distributed sensor fusion (Olfati‐Saber et al. 2007), satellite alignment (Ren and Beard, 2008), multi‐agent formation (Ren et al., 2007), synchronization of coupled oscillators (Ren, 2008), and optimal dispatch in power systems (Yang et al. 2013). The consensus problem is usually studied in the infinite time horizon, that is, the consensus is reached as time tends to infinity. However, some finite‐time convergence algorithms are available (Cortex, 2006; Wang and Hong, 2008; Khoo et al., 2009; Wang and Xiao, 2010; Li et al., 2011). In the existing literature, most consensus algorithms are model based. By incorporating ILC into consensus algorithms, the prior information requirement from a plant model can be significantly reduced. This advantage will be shown throughout this book.
1.3 Motivation and Overview
In practice, there are many tasks requiring both repetitive executions and coordination among several independent entities. For example, it is useful for a group of satellites to orbit the earth in formation for positioning or monitoring purposes (Ahn et al., 2010). Each satellite orbiting the earth is a repeated task, and the formation task fits perfectly in the ILC framework. Another example is the cooperative transportation of a heavy load by multiple mobile robots (Bai and Wen, 2010; Yufka et al., 2010). In such kinds of task implementation, the robots have to maneuver in formation from the very beginning to the destination. The economic dispatch problem in power systems (Xu and Yang, 2013; Yang et al., 2013) and formation control for ground vehicles with nonholonomic constraints (Xu et al., 2011) also fall in this category. These observations motivate the study of multi‐agent coordination control from the perspective of ILC.
As discussed in the previous subsection, the consensus tracking problem is an important multi‐agent coordination problem, and many other coordination problems can be formulated and solved in this framework, such as the formation, cooperative search, area coverage, and synchronization problems. We chose consensus tracking as the main topic in this book. Here we briefly describe a prototype consensus tracking problem and illustrate the concepts of distributed tracking error which are used throughout the book. In the problem formulation, there is a single leader that follows a prescribed trajectory, and the leader’s behavior is not affected by others in the network. There are many followers, and they can communicate with each other and with the leader agent. However, they may not know which one the leader is. Due to communication limitations, a follower is only able to communicate with its near neighbors. The control task is to design an appropriate local controller such that all the followers can track the leader’s trajectory. A local controller means that an agent is only allowed to utilize local information. To illustrate these concepts, Figure 1.2 shows an example of a communication network. (Please see Appendix A for a revision of graph theory.) Each node in the graph represents an agent (agents will be modeled by dynamic systems in later chapters). Edges in the graph show the information flow. For instance, there is an edge starting from agent 2 and ending at agent 1, which means agent 1 is able to obtain information from agent 2. In this example there are two edges ending at agent 1. This implies that agent 1 can utilize the information received from agents 0 and 2. Let xi denote the variable of interest for the i th agent, for instance, position, velocity, orientation, temperature, pressure, and so on. The distributed error ξ1 for agent 1 is defined as
Figure 1.2 Example of a network.
The distributed error ξ1 will be used to construct the distributed learning rule.
With this problem in mind, the main content of the book is summarized below.
In
Chapter 2
, a general consensus tracking problem is formulated for a group of global Lipschitz continuous systems. It is assumed that the communication is fixed and connected, and the perfect identical initialization condition (
iic
) constraint is satisfied as well. A D‐type ILC rule is proposed for the systems to achieve perfect consensus tracking. By adoption of a graph dependent matrix norm, a local convergence condition is devised at the agent level. In addition, optimal learning gain design methods are developed for both directed and undirected graphs such that the
λ
‐norm of tracking error converges at the fastest rate.
In
Chapter 3
, we investigate the robustness of the D‐type learning rule against communication variations. It turns out that the controller is insensitive to iteration‐varying topology. In the most general case, the learning controller is still convergent when the communication topology is uniformly strongly connected over the iteration domain.
In
Chapter 4
, the PD‐type learning rule is proposed to deal with imperfect initialization conditions as it is difficult to ensure perfect initial conditions for all agents due to sparse information communication—hence only a few of the follower agents know the desired initial state. The new learning rule offers two main features. On the one hand, it can ensure controller convergence. On the other hand, the learning gain can be used to tune the final tracking performance.
In
Chapter 5
, a novel input sharing learning controller is developed. In the existing literature, when designing the learning controller, only the tracking error is incorporated in the control signal generation. However, if the follower agents can share their experience gained during the process, this may accelerate the learning speed. Using this idea, the new controller is developed for each agent by sharing its learned control input with its neighbors.
In
Chapter 6
, we apply the learning controller to a formation problem. The formation contains two geometric configurations. The two configurations are related by a high‐order internal model (HOIM). Usually the ILC control task is fixed. The most challenging part of this class of problem is how to handle changes in configuration. By incorporating the HOIM into the learning controller, it is shown that, surprisingly, the agents are still able to learn from different tasks.
In
Chapter 7
, by combining the Lyapunov analysis method and contraction‐mapping analysis, we explore the applicability of the P‐type learning rule to several classes of local Lipschitz systems. Several sufficient convergence conditions in terms of Lyapunov criteria are derived. In particular, the P‐type learning rule can be applied to a Lyapunov stable system with quadratic Lyapunov functions, an exponentially stable system, a system with bounded drift terms, and a uniformly bounded energy bounded state system under control saturation. The results greatly complement the existing literature. By using the results of this chapter, we can immediately extend the results in
Chapters 2
–
5
to more general nonlinear systems.
In
Chapter 8
, the composite energy function method is utilized to design an adaptive learning rule to deal with local Lipschitz systems that can be modeled by system dynamics that are linear in parameters. With the help of a special parameterization method, the leader’s trajectory can be treated as an iteration‐invariant parameter that all the followers can learn from local measurements. In addition, the initial rectifying action is applied to reduce the effect of imperfect initialization conditions. The method works for high‐order systems as well.
Chapter 9
addresses the consensus problem of nonlinear multi‐agent system (MAS) with state constraints. A novel type of barrier Lyapunov function (BLF) is adopted to deal with the bounded constraints. An ILC strategy is introduced to estimate the unknown parameter
