Consensus Tracking of Multi-agent Systems with Switching Topologies (Emerging Methodologies and Applications in Modelling, Identification and Control) 0128183659, 9780128183656

Table of contents :
Cover
Consensus Tracking of Multiagent Systems with Switching Topologies
Copyright
Contents
About the authors
Acknowledgment
1 Introduction
1.1 A brief of multiagent systems
1.1.1 Related graph theory
1.1.2 Consensus control
1.1.3 Formation control
1.1.3.1 Position-based control
1.1.3.2 Displacement-based control
1.1.3.3 Distance-based control
1.1.4 Consensus tracking
1.2 Multiagent systems with time-varying topologies
1.2.1 Multiagent systems with switching topologies
1.2.2 Multiagent systems with jointly connected topologies
1.2.3 Multiagent systems with continuously time-varying topologies
1.3 Multiagent systems with node failures/actuator faults
1.4 Applications of multiagent systems
1.4.1 Application to cooperative robots
1.4.2 Application to multi-unmanned aerial vehicle
1.4.3 Application to intelligent train control
References
Further reading
2 Multiagent systems with continuously switching topologies
2.1 Introduction
2.2 First-order multiagent systems with continuously switching topologies
2.2.1 Main results
2.2.2 A numerical example
2.3 Second-order multiagent systems with continuously switching topologies
2.3.1 Consensus tracking with double-integrator dynamics
2.3.2 Consensus tracking with second-order nonlinear dynamics
2.3.3 Numerical examples
2.4 Conclusion
References
3 High-order multiagent systems with continuously switching topologies
3.1 Introduction
3.2 Problem description
3.3 High-order multiagent systems with continuously switching topologies
3.3.1 Stability analysis
3.3.2 Simulation results
3.4 High-order multiagent systems with continuously switching topologies based on polytopic model
3.4.1 Stability analysis
3.4.2 A numerical example
3.5 Conclusion
References
4 High-order multiagent systems with time delays and continuously switching topologies based on polytopic model
4.1 Introduction
4.2 Problem description
4.3 Main section
4.4 Numerical examples
4.5 Conclusion
References
5 Sliding mode control for multiagent systems with continuously switching topologies based on polytopic model
5.1 Introduction
5.2 Preliminaries and problem description
5.3 Sliding mode controller design and stability analysis
5.3.1 Sliding mode controller design
5.3.2 Stability analysis
5.3.3 Numerical examples
5.4 Sliding mode control for multiagent systems with disturbances
5.4.1 Sliding mode controller design for systems with disturbances
5.4.2 Stability analysis for systems with disturbances
5.4.3 Numerical examples
5.5 Conclusion
References
6 Cooperative relay tracking strategy for multiagent systems with assistance of Voronoi diagrams
6.1 Introduction
6.2 Voronoi diagrams
6.3 Relay tracking algorithm
6.4 Controller design and stability analysis
6.5 Numerical examples
6.6 Conclusion
References
Further reading
7 Stability of a class of multiagent relay tracking systems with unstable subsystems
7.1 Introduction
7.2 Related preliminaries
7.3 Problem formulation
7.4 Controller design and stability analysis
7.5 Disturbance attenuation analysis
7.6 Numerical examples
7.7 Conclusion
References
8 Multiagent relay tracking systems with damaged agents and time-varying number of agents
8.1 Introduction
8.2 Relay tracking systems with damaged agents
8.2.1 Problem formulation
8.2.2 Relay tracking algorithm
8.2.3 Controller design and stability analysis
8.2.4 A numerical simulation
8.3 Relay tracking systems with time-varying number of agents
8.3.1 Preliminaries and problem formulation
8.3.2 Main results of relay tracking systems with time-varying number of agents
8.3.3 A numerical simulation
8.4 Conclusion
References
9 Multiagent relay tracking systems with time-varying number of agents and time delays
9.1 Introduction
9.2 Linear relay tracking systems with time-varying number of agents and time delays
9.2.1 Problem formulation
9.2.2 Main results
9.2.3 Numerical simulations
9.3 Nonlinear relay tracking systems with time-varying number of agents and time delays
9.3.1 Related preliminaries
9.3.2 Stability analysis
9.3.3 Numerical simulations
9.4 Conclusion and discussions
References
10 Finite time stability analysis and coordination strategies of multiagent relay tracking systems
10.1 Introduction
10.1.1 Finite time boundedness
10.1.2 Finite time coordination relay strategies
10.2 Stability analysis of nonlinear multiagent relay tracking systems over a finite time interval
10.2.1 Relay tracking problem formulation
10.2.2 Stability analysis over a finite time interval
10.2.3 Stability analysis for systems subject to disturbances
10.2.4 Numerical examples
10.3 Finite-time coordination control of multiagent systems for target tracking with node failures and active replacements
10.3.1 Tracking problem description and fundamentals
10.3.2 Event-triggered coordination strategy
10.3.3 Nonsingular terminal sliding mode controller design
10.3.4 Finite time analysis with modified sliding mode control
10.3.5 Stability analysis of systems with agent replacements
10.3.6 Numerical simulations
10.4 Conclusion
References
Further reading
11 Conclusions and future research
11.1 Conclusions
11.1.1 Consensus tracking control of systems with continuously switching topologies
11.1.2 Relay tracking control of systems with agent replacements
11.2 Future research
11.2.1 General nonlinear multiagent systems
11.2.2 Cooperative control algorithms based on deep learning
References
Appendix A
A.1 Notations
A.2 Assumptions and lemmas
References
Index
Back Cover

Citation preview

Consensus Tracking of Multiagent Systems with Switching Topologies

Consensus Tracking of Multiagent Systems with Switching Topologies

Lijing Dong School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, Beijing, P.R. China

Sing Kiong Nguang Department of Electrical, Computer, and Software Engineering, University of Auckland, Auckland, New Zealand

Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1650, San Diego, CA 92101, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom Copyright © 2020 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN: 978-0-12-818365-6 For Information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Matthew Deans Acquisitions Editor: Brain Guerin Editorial Project Manager: Joshua Mearns Production Project Manager: R. Vijay Bharath Cover Designer: Victoria Pearson Typeset by MPS Limited, Chennai, India

Contents About the authors Acknowledgment

ix xi

1.

1

Introduction 1.1 A brief of multiagent systems 1.1.1 Related graph theory 1.1.2 Consensus control 1.1.3 Formation control 1.1.4 Consensus tracking 1.2 Multiagent systems with time-varying topologies 1.2.1 Multiagent systems with switching topologies 1.2.2 Multiagent systems with jointly connected topologies 1.2.3 Multiagent systems with continuously time-varying topologies 1.3 Multiagent systems with node failures/actuator faults 1.4 Applications of multiagent systems 1.4.1 Application to cooperative robots 1.4.2 Application to multi-unmanned aerial vehicle 1.4.3 Application to intelligent train control References Further reading

2.

Multiagent systems with continuously switching topologies 2.1 Introduction 2.2 First-order multiagent systems with continuously switching topologies 2.2.1 Main results 2.2.2 A numerical example 2.3 Second-order multiagent systems with continuously switching topologies 2.3.1 Consensus tracking with double-integrator dynamics 2.3.2 Consensus tracking with second-order nonlinear dynamics 2.3.3 Numerical examples 2.4 Conclusion References

2 4 5 6 8 10 10 11 13 14 17 17 18 19 20 27

29 29 30 30 34 36 37 40 42 45 46

v

vi

3.

4.

5.

Contents

High-order multiagent systems with continuously switching topologies

47

3.1 Introduction 3.2 Problem description 3.3 High-order multiagent systems with continuously switching topologies 3.3.1 Stability analysis 3.3.2 Simulation results 3.4 High-order multiagent systems with continuously switching topologies based on polytopic model 3.4.1 Stability analysis 3.4.2 A numerical example 3.5 Conclusion References

54 56 58 63 64

High-order multiagent systems with time delays and continuously switching topologies based on polytopic model

65

4.1 Introduction 4.2 Problem description 4.3 Main section 4.4 Numerical examples 4.5 Conclusion References

65 66 67 73 85 85

Sliding mode control for multiagent systems with continuously switching topologies based on polytopic model

87

47 48 49 50 53

5.1 Introduction 88 5.2 Preliminaries and problem description 89 5.3 Sliding mode controller design and stability analysis 89 5.3.1 Sliding mode controller design 89 5.3.2 Stability analysis 91 5.3.3 Numerical examples 94 5.4 Sliding mode control for multiagent systems with disturbances 98 5.4.1 Sliding mode controller design for systems with disturbances 98 5.4.2 Stability analysis for systems with disturbances 99 5.4.3 Numerical examples 102 5.5 Conclusion 102 References 105

Contents

6.

7.

8.

Cooperative relay tracking strategy for multiagent systems with assistance of Voronoi diagrams

107

6.1 Introduction 6.2 Voronoi diagrams 6.3 Relay tracking algorithm 6.4 Controller design and stability analysis 6.5 Numerical examples 6.6 Conclusion References Further reading

107 109 113 115 120 127 128 129

Stability of a class of multiagent relay tracking systems 131 with unstable subsystems 7.1 Introduction 7.2 Related preliminaries 7.3 Problem formulation 7.4 Controller design and stability analysis 7.5 Disturbance attenuation analysis 7.6 Numerical examples 7.7 Conclusion References

131 132 134 136 139 141 149 149

Multiagent relay tracking systems with damaged agents and time-varying number of agents

151

8.1 Introduction 8.2 Relay tracking systems with damaged agents 8.2.1 Problem formulation 8.2.2 Relay tracking algorithm 8.2.3 Controller design and stability analysis 8.2.4 A numerical simulation 8.3 Relay tracking systems with time-varying number of agents 8.3.1 Preliminaries and problem formulation 8.3.2 Main results of relay tracking systems with time-varying number of agents 8.3.3 A numerical simulation 8.4 Conclusion References

9.

vii

Multiagent relay tracking systems with time-varying number of agents and time delays 9.1 Introduction 9.2 Linear relay tracking systems with time-varying number of agents and time delays

151 152 153 154 155 161 163 164 168 171 172 174

177 177 178

viii

Contents

9.2.1 Problem formulation 9.2.2 Main results 9.2.3 Numerical simulations 9.3 Nonlinear relay tracking systems with time-varying number of agents and time delays 9.3.1 Related preliminaries 9.3.2 Stability analysis 9.3.3 Numerical simulations 9.4 Conclusion and discussions References

192 192 195 200 204 206

10. Finite time stability analysis and coordination strategies of multiagent relay tracking systems

207

10.1 Introduction 10.1.1 Finite time boundedness 10.1.2 Finite time coordination relay strategies 10.2 Stability analysis of nonlinear multiagent relay tracking systems over a finite time interval 10.2.1 Relay tracking problem formulation 10.2.2 Stability analysis over a finite time interval 10.2.3 Stability analysis for systems subject to disturbances 10.2.4 Numerical examples 10.3 Finite-time coordination control of multiagent systems for target tracking with node failures and active replacements 10.3.1 Tracking problem description and fundamentals 10.3.2 Event-triggered coordination strategy 10.3.3 Nonsingular terminal sliding mode controller design 10.3.4 Finite time analysis with modified sliding mode control 10.3.5 Stability analysis of systems with agent replacements 10.3.6 Numerical simulations 10.4 Conclusion References Further reading

11. Conclusions and future research

178 181 189

208 208 209 209 209 212 214 215 218 220 222 224 226 228 229 235 235 237 239

11.1 Conclusions 239 11.1.1 Consensus tracking control of systems with continuously switching topologies 239 11.1.2 Relay tracking control of systems with agent replacements 240 11.2 Future research 240 11.2.1 General nonlinear multiagent systems 241 11.2.2 Cooperative control algorithms based on deep learning 241 References 241 Appendix A Index

243 249

About the authors Lijing Dong Lijing Dong received the BE and the PhD degree from the School of Automation, Beijing Institute of Technology, Beijing, China, in 2010 and 2016, respectively. From 2013 to 2014, she was a visiting scholar with the Department of Electrical and Computer Engineering, University of Auckland. Currently, she is an Associate Professor at School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University. Her current research interests include multiagent systems and distributed control systems. She has obtained the authorization of multiple invention patents. In 2016 she got the first prizes of science and technology award from China Railway Society and innovation achievement award from China Industry-UniversityResearch Institute Collaboration Association. She has published more than 20 refereed journal and conference papers on multiagent systems and largescale complex systems and has/had served as reviewer of a number of international journals. Sing Kiong Nguang Sing Kiong Nguang received the BE (with first class honors) and the PhD degree from the Department of Electrical and Computer Engineering of the University of Newcastle, Callaghan, Australia, in 1992 and 1995, respectively. Currently, he is a Chair Professor at the Department of Electrical, Computer, and Software Engineering, University of Auckland, Auckland, New Zealand. He has published more than 300 refereed journal and conference papers on nonlinear control design, nonlinear control systems, nonlinear time-delay systems, nonlinear sampled-data systems, biomedical systems modeling, fuzzy modeling and control, biological systems modeling and control, and food and bioproduct processing. He has/had served on the editorial board of a number of international journals. He is the chief editor of the International Journal of Sensors, Wireless Communications and Control.

ix

Acknowledgment We gratefully acknowledge the support from the National Natural Science Foundation of China under Grant 61903022.

xi

Chapter 1

Introduction Chapter Outline 1.1 A brief of multiagent systems 2 1.1.1 Related graph theory 4 1.1.2 Consensus control 5 1.1.3 Formation control 6 1.1.4 Consensus tracking 8 1.2 Multiagent systems with time-varying topologies 10 1.2.1 Multiagent systems with switching topologies 10 1.2.2 Multiagent systems with jointly connected topologies 11 1.2.3 Multiagent systems with continuously time-varying topologies 13

1.3 Multiagent systems with node failures/ actuator faults 14 1.4 Applications of multiagent systems 17 1.4.1 Application to cooperative robots 17 1.4.2 Application to multi-unmanned aerial vehicle 18 1.4.3 Application to intelligent train control 19 References 20 Further reading 27

Multiagent systems (MASs) have become a hot research area in the recent two decades due to the wide applications to mobile robots, unmanned aerial vehicles (UAVs), autonomous underwater vehicles, and satellites. Consensus tracking problem is a typical issue of MASs. In practical the velocity of each maneuvering agent is time varying and the communication radius of each agent is finite. Therefore the communication topology between the agents may change from time to time, and tracking problem of MASs under timevarying topologies is of vital necessity. This chapter summarizes the recent development of MASs with different types of time-varying topologies, such as switching topologies, jointly connnected topologies, and continuously time-varying topologies. Then, systems with node failures or actuator faults are introduced and the concept of relay tracking is presented. Finally, the applications of multiagent systems are introduced.

Consensus Tracking of Multi-agent Systems with Switching Topologies. DOI: https://doi.org/10.1016/B978-0-12-818365-6.00001-X © 2020 Elsevier Inc. All rights reserved.

1

2

1.1

Consensus Tracking of Multi-agent Systems with Switching Topologies

A brief of multiagent systems

Inspired by the group activities of humans and collective behavior of insects or birds, researchers have done a lot of work to explain this kind of phenomena [1,2]. Over the last two decades, MASs have received much attention from researchers in various areas [3
12]. Numerous results have been achieved under various circumstances such as consensus tracking with switching topologies [13,14], the disturbance-rejection consensus [15], finitetime tracking control [16,17], pinning adaptive
impulsive control [18], and optimal coordination [19,20]. A MAS is formed by multiple autonomous agents that are capable of sensing the environment, moving, and information processing. An agent can be an UAV, or an unmanned ground vehicle (UGV), or a spacecraft, or an autonomous train, or a robot, etc. A MAS is of high efficiency, low cost, and reliability which make it an effective solution to solve complex tasks. The complex task is divided into multiple small tasks, each of which is assigned to a specific agent, namely, the complex task is accomplished through cooperation of individual agents, which is the salient feature of MASs. Currently, there are many different categories in the cooperative control of MASs. Consensus control, formation control, and tracking control are three of the most attractive ones. Since consensus of MASs is a fundamental problem in this research area, it has attracted increasing attention of researchers from various disciplines of engineering, biology, and science. In networks of agents, consensus means to reach an agreement regarding a certain quantity of interest that depends on the state of all agents. A consensus algorithm is an interaction rule that specifies the information exchange between an agent and all of its neighbors on the network. The consensus problems have been formulated as consensus of leaderless problems or leader-following problems [21
23]. For a cooperative MAS, leaderless consensus means that each agent updates its state based on local information of its neighbors such that all agents eventually reach an agreement on a common value, while leader-following consensus means that there exists a virtual leader that specifies an objective for all agents to follow. Formation control is another hot topic, where a group of interconnected agents is controlled to cooperatively move with a desired formation pattern. The desired formation could be time invariant [24,25] or time varying [26,27]. Specifically, Lu et al. [25] obtain sufficient conditions guaranteeing the exponentially converging speeds for both time-invariant and timevarying formation problems of MASs with directed graph interconnection topologies and time-varying coupling delays. Wang et al. [27] design a novel event-triggered integral sliding mode control strategy that makes sure the high-order agents achieve a time-varying formation.

Introduction Chapter | 1

3

Tracking control is a typical issue of MASs [7,9,11,13,28]. Many researchers have achieved significant results [13,29
32] on the tracking problem as it is an important topic in MASs’ research area. Consensus tracking of a target can be regarded as leader-following consensus problem. For example, Hajshirmohamadi et al. [33] propose unified event-triggered framework that requires the agents to transmit their information when the triggering condition is satisfied. In Ref. [34], two adaptive event-triggered communication schemes are presented for the consensus tracking control of MASs with stochastic actuator failures. Linear and dynamic-gain-based nonlinear observers are designed for solving the consensus tracking problem of second-order MASs with disturbance in Ref. [35]. Since agents are in various specific forms, consequently, the dynamics of agents are in different mathematical models, which can be generally categorized into linear and nonlinear dynamics. In the past few years, the MASs with integer dynamics [36,37] have been widely studied by many researchers due to its simple construction and convenience to analyze. Certainly, there are some researchers spend efforts on MAS with nonlinear dynamics [21,38] or switching topologies [22,39]. Single-integrator dynamics described by (1.1) and double-integrator dynamics described by (1.2) are basic forms of agents [35,40,41]. x_i ðtÞ 5 ui ðtÞ;

ð1:1Þ

where xi ðtÞ and ui ðtÞ are the state and the control input of the ith agent. x_i ðtÞ 5 vi ðtÞ; v_i ðtÞ 5 ui ðtÞ;

ð1:2Þ

where vi ðtÞ is the velocity state of the ith agent. As a direct extension of the study of the MASs with single-integrator or double-integrator dynamics, systems with general linear dynamics described by (1.3) are also studied recently [33]. x_i ðtÞ 5 Axi ðtÞ 1 Bui ðtÞ;

ð1:3Þ

where xi ðtÞAℝm is state vector of ith agent, AAℝm3m and BAℝm3p are constant matrices, ui ðtÞAℝp is the control input. As a further extension, MASs with nonlinear dynamics have attracted much attention of researchers. Nonlinear systems include first-, second-, and high-order dynamics, in which the second-order dynamics described by (1.4) is the most common form in the literature [42,43]. x_i ðtÞ 5 vi ðtÞ; v_i ðtÞ 5 f ðxi Þ 1 ui ðtÞ; where f ðxi Þ is the nonlinear dynamics of ith agent.

ð1:4Þ

4

Consensus Tracking of Multi-agent Systems with Switching Topologies

1.1.1

Related graph theory

For a MAS, agents and their relations are modeled using graph theory. In the view of graph theory, each agent can be treated as a node. Then the communication topology of tracking agents and can be treated as a
the target  dynamic graph. A weighted graph G :5 N ; E; A is denoted with a node set N 5 f1; 2;  . ..; N g, an edge set EDN 3 N , and a weighted adjacency matrix A 5 aij AℝN3N with nonnegative elements. The nodes within the communication range of
node i are called  the set of neighbors of node i, which is denoted by N i 5 jjjAN ; ðj; iÞAE . j= 2N i means node j is beyond the communication range of node i, aij 5 0; otherwise aij . 0: If aii 6¼ 0; we say that node has self-loop. In this book, it is assumed that no self-loop exists. Specifically, as illustrated in Fig. 1.1, the dash circles represent communication ranges of agents. Agent i and agent j are within each other’s communication range, thus, there is a connection between these two agents and aij 5 1: On the other hand, agent k is not within agent i’s communication range, there is no connection between these two agents and aik 5 0: In terms of consensus tracking problem, the target is involved. bi decides whether agent i is able to communicate the target or not. bi 5 0 means agent i cannot get the information of target, otherwise bi . 0: Denote ℬ 5 diag{b1, b2, . . ., bN}. The in-degree and out-degree of node i are, respectively, defined as degin fig 5

N P j51;j6¼i

aij; degout fig 5

N P j51;j6¼i

aji

FIGURE 1.1 Illustration of communication range and topology.

Introduction Chapter | 1

5

A digraph is called balanced if deg  in fig 5 degout fig; ’iAv. The Laplacian matrix ℒ 5 lij AℝN 3 N of graph G is defined as ℒ: 5 D 2 A where 8 9

xT 2xa1 1 yT 2ya1 5 dT1 > > < qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2 ð6:4Þ xT 2xa2 1 yT 2ya2 5 dT2 > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > 2
2 :
xT 2xa3 1 yT 2ya3 5 dT3 After simple calculation, coordination of the target is obtained. "

 #21   xT 2 xa1 2xa3 2 ya1 2ya3 
 5
yT 2 xa2 2xa3 2 ya2 2ya3 " # x2a1 2 x2a3 1 y2a1 2 y2a3 1 dT23 2 dT21

ð6:5Þ

x2a2 2 x2a3 1 y2a2 2 y2a3 1 dT23 2 dT22 It can be seen that at least three agents are required for localization, which means this monitored area should be 3-coverage. To achieve 3-coverage of a specific two-dimensional area, with Eq. (6.3), the minimum number of agents can be calculated. Then, virtual force
based approach (VFA) is adopted in the distributed deployment process. Each agent behaves as a source giving a force to others. This force may be either attractive force or repulsive force. If two agents are too close, they exert repulsive forces to separate each other; otherwise, attractive forces are exerted. The total force Fi exerted on agent i is determined by the summation of all forces contributed by its neighbors. Under the effect of force Fi agent i will move till the force is zero. In VFA for deployment, every agent will eventually converge to a steady state. For the stability analysis of the VFA, the readers can refer to Ref. [25]. For a target crossing a 3-coverage area, the tracking algorithm is triggered by the detection of a target. A target t is said to reside at agent ai if target t is closest to agent ai compared to all other agents in the monitored area. For instance, in Fig. 6.1, the target resides at Voronoi site agent a2 . Every agent sends its range measurement information to agent a2 , and then agent a2 estimates the location of t according to trilateration algorithm. The monitored area is divided into Voronoi cells with each agent responsible for calculating the location of targets within its own cell with trilateration algorithm. The corresponding Voronoi diagram is shown in Fig. 6.3. The agents in a 3-coverage area can monitor and estimate the locations of targets. However, if monitoring agents move to track a target, the 3coverage will be destroyed. In this situation, when another target breaks into this area, we will fail to estimate its location. Therefore redundant agents are introduced. Assume at most Nt targets may enter the monitored area and each target is supposed to be tracked by Nf tracking agents. Then the number

112

Consensus Tracking of Multi-agent Systems with Switching Topologies

FIGURE 6.3 Voronoi diagram for 3-coverage.

of redundant agents is Nt 3 Nf , and the whole number of agents in this area is Na 5 3Nm 1 Nt Nf : The Voronoi diagram with redundant agents deployed is shown in Fig. 6.4. Definition 6.1: An agent is called a monitoring agent if it is on the key position that assure the area is 3-overlapped. Definition 6.2: An agent is called a tracking agent if it is tracking a target. Definition 6.3: An agent is called a redundant agent if it is neither a monitoring nor tracking agent. Definition 6.4: A Voronoi site agent can be either a monitoring or redundant agent. Remark 6.1: Monitoring agents are deployed in specific locations to reach 3-coverage. However, redundant agents are initially randomly deployed in this area and do not participate in the virtual force
deployment approach. An agent is able to determine its own role through comparing the distances to its neighboring agents. If the distances between an agent and more than three of its neighbors are equal, it is a monitoring agent; otherwise, it is a

Cooperative relay tracking strategy Chapter | 6

113

FIGURE 6.4 Voronoi diagram with redundant agents.

redundant agent. In Fig. 6.3, all the agents are monitoring agents. In Fig. 6.4, except the monitoring agents, the others which are randomly deployed are called redundant agents. All the agents in Fig. 6.4 can be called Voronoi site agents.

6.3

Relay tracking algorithm

The structure of tracking algorithm is shown in Fig. 6.5. Initial deployment of agents based on virtual force approach has been addressed in Section 6.2. This section mainly focuses on relay tracking strategy. Remark 6.2: A significant feature in our model for tracking targets is that we do not seek consensus of all the agents. In this chapter, a large number ðNa Þ of agents are deployed on an interested area to carry out surveillance and tracking tasks. When a target enters this area, a distributed tracking application is activated. At each instant of time, Nf agents are assigned the task of capturing the maneuvering target, whereas all other agents in this region remain stationary. In other words, only Nf agents are supposed to be consensus with the target. The cooperative relay tracking algorithm is addressed in detail as follows:

114

Consensus Tracking of Multi-agent Systems with Switching Topologies

Tracking control

Tracking strategy design Distance measurement and processing

Voronoi diagram–based 3-coverage area

FIGURE 6.5 Tracking structure.

1. Calculate the required number of agents and initialize the deployment guaranteeing the monitored area is 3-overlapped. 2. When a target goes into the monitored area, the three neighboring agents ai ; 1 # i # 3; send their detected distances to the Voronoi site agent (local police station collects information about a thief from neighboring stations), where the target resides. 3. Using trilateration algorithm, the Voronoi site agent estimates the target’s location through three distance measurements (local station infers the thief’s position with collected information). This means during the tracking process only one agent is able to access the thief’s position. 4. Then the Voronoi site agent and Nf 2 1 other nearest agents start tracking the target and become tracking agents. 5. If tracking agent is a monitoring agent, in order to guarantee at least 3coverage of this area, the nearest redundant agents move to the monitoring agent’s previous location immediately. If the tracking agent is one of the redundant agents, the other agents stay at their original positions. Then recreate a new Voronoi diagram, in which the target and the tracking agents are exclusive. 6. If the target is not captured in the first Voronoi cell, it will go to the next Voronoi cell. The new corresponding Voronoi site agent will become a tracking agent; meanwhile, one of the original tracking agents will quit tracking based on the distance discipline. The one which is furthest from the target quits. Then repeat from procedure 5 again. 7. When a target is caught, it stops moving and releases the corresponding tracking agents to become redundant agents. This is called as release policy, with which there are more redundant agents in the domain. Repeat all the procedures until there are no more new and uncaptured targets. We note that the monitored area can tolerate moving of agents (addition or deletion of nodes for a local domain). Any changes in Voronoi diagram topology are solved in a local manner by means of well-known local

Cooperative relay tracking strategy Chapter | 6

115

algorithms [4], which means the moving of agents does not cause a global reassignment of Voronoi diagram.

6.4

Controller design and stability analysis

The agents in the monitored area are identical and described as x_i ðtÞ 5 f ðt; xi ðtÞÞ 1 ui ðtÞ;

iAf1; 2; . . .; Na g

ð6:6Þ

where xi ðtÞAℝ is position state of the ith agent. f ðt; xi ðtÞÞ is a nonlinear vector-valued continuous function to describe the self-dynamics of ith tracking agent. ui ðtÞAℝ2 is the control input of the ith agent. The maneuvering targets are assumed to possess the following dynamics. 2

x_tk ðtÞ 5 f ðt; xtk ðtÞÞ;

kAf1; 2; . . .; Nt g

ð6:7Þ

where xtk ðtÞAℝ is position state of the kth target. The objective of this chapter is to design a tracking strategy ensuring that the tracking agents effectively catch the targets in a two-dimensional space. In this work, we assume the nonlinear function f ðUÞ in (6.6) and (6.7) satisfies the Lipschitz Assumption A.1. The communication structure among agents may be static, but in many real-world applications, it may be time-varying. More recently, stochastic switching topologies have become a powerful tool to solve this problem. With the movement of the target, that is, the target enters a new Voronoi cell, the agents which are tracking the target may change. This may result in change of topology, which can be treated as switching topologies driven by Markov chain. Then the Laplacian matrix turns into ℒσðtÞ 1 BσðtÞ , where σðtÞ; t $ 0 is a right-continuous Markov chain on the probability space taking values in a finite state set S 5 f1; 2; . . .; Ns g; where
 Ns is the number of possible topologies. Corresponding generator Π 5 πij Ns 3 Ns is given by   πij h 1 oðhÞ; i 6¼ j; Pr σðt 1 hÞ 5 jjσðtÞ 5 i 5 1 1 πij h 1 oðhÞ; i 5 j; 2

in which h . P0; πij $ 0 is the transition rate from state i to j. When i 5 j; πii 5 2 j6¼i πij :   Denote graph by G 5 Gð1Þ; Gð2Þ; . . .; GðNs Þ ; where  the communication  GðkÞ 5 N GðkÞ; EGðkÞ; AGðkÞ is the communication graph. Denote the topology graph at time t as Gt ; then Gt 5 GðkÞ when σðtÞ 5 k: Piecewise constant signal σðtÞ changes only when the target enters into a new Voronoi cell. Remark 6.3: In this chapter the topology changes according to the movement of a target. When the target enters into a new Voronoi cell, the corresponding Voronoi site will become a tracking agent, meanwhile, one of the original tracking agents quits tracking. The one which is at the final layer of the topology quits. This means not only the topology changes but also the

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Consensus Tracking of Multi-agent Systems with Switching Topologies

nodes, which is reflected by N GðkÞ . This is quite different from the traditional switching systems, in which only the topology switches. Remark 6.4: It is possible that there exists a special case in which the target is captured in the first Voronoi cell. This means there is no longer switching in the tracking process. In this chapter, we consider the general case stated in Remark 6.3. Fig. 6.6 illustrates the rules of the switching of topologies. We take the situation that Nf 5 3 as an example. As analyzed in Section 6.3, only one agent can access the target’s position, which means in the topology of the multiagent system, only one of the bi ; i 5 1; . . .; Nf is nonzero. When three tracking agents are supposed to track one target, as shown in Fig. 6.6, there are two cases during the evolution of topology. In Fig. 6.6A, there are three layers in the topology, the target is at layer 1, agent 1 is at layer 2, and agents 2 and 3 are at layer 3. The final layer has two agents; therefore when the target moves into a new Voronoi cell, the corresponding Voronoi site will either replace agent 2 or agent 3. Meanwhile, since agent 1 is in target’s sense range, and target is now at the new Voronoi site’s sense range, then based on Rc $ 2Rs ; agent 1 can communicate with the new Voronoi site agent. That is why no matter which agent at layer 3 is replaced, the new agent, which is represented by noting the number in red, is always connected with agent 1. However, the new agent may connect or may not connect with the other(s) at layer 3. In Fig. 6.6B, there are four layers in the topology, and of course agent 3 is replaced. The switching process is stochastic and fits Markov property, that is, the switching process is a Markov process.

New agent

T

Layer 1

Layer 2

2

1 T

T

T

3

3 2

1 T

3

Layer 3

T

3

2

1

T 2

1 3

(A)

3

T

3 2

1

2

1

2

1

2

1 3

(B)

FIGURE 6.6 Rules of switching of topology. (A) Demonstration of switching rules for systems with three layers. (B) Demonstration of switching rules for systems with four layers.

Cooperative relay tracking strategy Chapter | 6

117

Definition 6.5: Under stochastic switching topologies, the tracking agents (6.6) are said to track the target (6.7) successfully in the mean square sense if. lim Eðjjxi ðtÞ 2 xt ðtÞjjÞ-0;

t-N

iANf :

ð6:8Þ

in which E denotes for expectation. Since we can only estimate the position of a target, we adopt the following format of control protocol for the ith tracking agent. 8 9 < X = u i ðt Þ 5 2 α aij ðσðtÞÞeij ðtÞ 1 bi ðσðtÞÞei ðtÞ ; ð6:9Þ : ; jAN i ðtÞ

where α . 0 is the control parameter to be designed. eij ðtÞ 5 xi ðtÞ 2 xj ðtÞ is the position disagreement vector between ith tracking agent and jth tracking agent. ei ðtÞ 5 xi ðtÞ 2 xt ðtÞ is the position disagreement vector between ith tracking agent and the target. Then, from Definition 6.5, the tracking problem can be interpreted as stability problem of the following overall tracking error system:  E_ ð
tÞ 5 Fðt; EðtÞÞ
2 αLσðtÞ EðtÞ;
t 6¼ σ ðtÞ; 
   ð6:10Þ 1 2 E σ t 5 E σ ðt Þ 2 ΔE σ ðtÞ ; t 5 σ ðtÞ; h iT where EðtÞ 5 eT1 ðtÞ; eT2 ðtÞ; . . .; eTNf ðtÞ is the collective position tracking error
 between tracking agents and the target, at time t. LσðtÞ 5 ℒσðtÞ 1 BσðtÞ  I2 is the associated  Laplacian matrix of  the dynamic graph including the target. satisfies Assumption A.1, where F ðt; EðtÞÞ 5 col f1 ðtÞ; f2 ðtÞ; . . .; fNf ðtÞ fi ðtÞ 5 f ðt; xi ðtÞÞ 2 f ðt; xt ðtÞÞ; iAN is the collective nonlinear self-dynamic   disagreement to the target at time t. σ
ðtÞ is the switching time. σ ðt2 Þ indi cates the time before switching and σ t1 represents the time after switch
  ing. ΔE σ ðtÞ is the jump of tracking error.

Remark 6.5: As mentioned in Remark 6.3, when the topology switches, the tracking agent which is furthest to the target is replaced by the new Voronoi site agent. This means the norm of tracking error EðtÞ decreases at every  switching time σ ðtÞ: The element in
EðtÞ corresponding to the replaced  tracking agent jumps, reflected by ΔE σ ðtÞ
: For ith
if the  tracking instance,   
 1 2 agent is replaced at time σ ð t Þ; jje σ t σ ð t Þ jj while jj # jje i i
 

 jjej
σ
t1 jj 5 jjej σ
ðt2 Þ jj; j 6¼ i; jAN , which implies
that    jjET σ t1 jj # jjET σ ðt2 Þ jj, that is, the jump of tracking error ΔE σ ðtÞ is nonnegative. Definition 6.6: [26] Assume that there exists a stochastic Lyapunov function V ðEðtÞ; i; tÞ for system (6.10) with Markov switching topologies such that

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Consensus Tracking of Multi-agent Systems with Switching Topologies

dV ðEÞ # 0; then the equilibrium solution of E 5 0 of the stochastic differential Eq. (6.10) is stochastically stable. The operator dV is defined as [27] dV ðEðtÞ; i; tÞ  1 5 lim EV ðEðt 1 hÞ; σðt 1 hÞjEðtÞ; σðtÞ 5 iÞ 2 V ðEðtÞ; σðtÞ 5 iÞ h-0 h 5 Vt ðEðtÞ; i; tÞ 1 VE ðEðtÞ; i; tÞE_ ðtÞ 1

Ns X

ð6:11Þ

πij V ðEðtÞ; i; tÞ;

j51

where @V ðEðtÞ; i; tÞ @t 0 1 @V ðEðtÞ; i; tÞ @V ðEðtÞ; i; tÞ @V ðEðtÞ; i; tÞA VE ðEðtÞ; i; tÞ 5 @ : ; ; . . .; @E1 @E2 @ENf Vt ðEðtÞ; i; tÞ 5

The following theorem provides a sufficient condition for successful tracking.

Theorem 6.1: Consider the multiagent system (6.10). Suppose that for given positive scalars α, l, there exist symmetric positive definite matrices Pi of appropriate dimensions such that

 P s 2αPi Li 2 αLTi Pi 1 Nj51 πij Pj 1 I lPi ,0 ð6:12Þ  2I for all i; j 5 1; 2; . . .; Ns : Then system (6.10) is stochastically stable, which means the tracking agents can track the target. Proof: Let us select the Lyapunov function as V ðEðtÞ; i; tÞ 5 ET ðtÞPi EðtÞ;



t 6¼ σ ðtÞ:

ð6:13Þ

The derivative of Lyapunov function (6.13) along the trajectory of system (6.10) is dV ðEðtÞ; i; tÞ

Ns X T 5 E_ ðtÞPi EðtÞ 1 ET ðtÞPi E_ ðtÞ 1 πij ET ðtÞPj EðtÞ j51

P Ns T T 5 E ðtÞ 2αPi Li 2 αLi Pi 1 j51 πij Pj EðtÞ 1 ET ðtÞPi F ðt; EðtÞÞ

1 F T ðt; EðtÞÞPi EðtÞ: Applying Lemma A.1, we have

Cooperative relay tracking strategy Chapter | 6

119

dV ðEðt Þ; i; tÞ

P T # ET ðtÞ 2αPi Li 2 αLTi Pi 1 Ns j51 πij Pj E ðtÞ 1 E ðtÞEðtÞ 1 F T ðt; EðtÞÞPi Pi F ðt; EðtÞÞ: In light of Assumption A.1 the above equation turns to be dV ðEðtÞ; i; tÞ # ET ðtÞ 2αPi Li 2 αLTi Pi 1

Ns X

! πij Pj 1 I 1 l2 Pi Pi EðtÞ:

j51

By inequality (6.12) and Lemma A.4, it is easy to see that  dV ðE
ðtÞ;
i; tÞ, 0 holds. At the switching times σ ðtÞ; since
   jjET σ t1 jj # jjET σ ðt2 Þ jj; we have




 
  ET σ t1 Pi E σ t1 2 ET σ ðt2 Þ Pi E σ ðt2 Þ # 0: Therefore the switching accelerates the degradation of tracking errors, which will be verified in Example 6.3. Then according to Definition 6.6, we can obtain that Theorem 6.1 ensures the stochastic stability of the disagreement system (6.10). The disagreement between the tracking agents and the target tends to 0 when t-N, that is, tracking agents finally track the target according to Definition 6.5. This completes the proof. & The above analysis is definitely suitable for the situation in which only one target is considered. If we consider multitargets, there could be four possible cases based on when and where targets enter the monitored area. Here are the detail descriptions and mathematical statements of the four cases. Case 6.1: At all the time, there is only one target in the monitored area, that is, no new target enter into this area until the existed target got caught. The mathematical statement of this case is
 Tint ðti Þ - Tint tj 5 [; i 6¼ jAf1; 2; . . .; Nt g; ð6:14Þ where Tint ðti Þ means the time interval from target ti enters into the monitored area till it is captured. In this case, proof of the successful tracking is the same with that in Theorem 6.1 due to which we just need to deal with each target separately. Case 6.2: There might be several targets in the monitored area at one time, but the targets are far-field distributed. They enter this area from various directions, are far away from each other, and there is no interaction between them. The mathematical statement of this case is

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Consensus Tracking of Multi-agent Systems with Switching Topologies


 T int ðti Þ -
T int  tj 6¼ [; Rðti Þ - R tj 5 [; i 6¼ jAf1; 2; . . .; N t g;

ð6:15Þ

where Rðti Þ means the route of target ti : In this case the analysis of capturing each target is independent, we can still prove the stability of tracking system using Theorem 6.1. Case 6.3: Several targets might enter the monitoring target along a same route over a time interval. The time interval makes sure that two targets are not likely to appear in one Voronoi cell at the same time. The mathematical statement of this case is
 Tint ðti Þ -
Tint tj 6¼ [; Rðti Þ - R tj 6¼
[; ð6:16Þ  i 6¼ jAf1; 2; . . .; Nt g; T Vak ðti Þ - TVak tj 5 [; kAf1; 2; . . .; Na g; where TVak ðti Þ is the time interval of target ti resides at Voronoi cell V ðak Þ: In this case, even though the targets share a route, they are passing Voronoi cells at different time; this means the approach in Theorem 6.1 for capturing each target is also applicable. Case 6.4: Several targets are in the monitored area at the same time. The targets are close to each other, a target could interact with another. The mathematical statement of this case is
 Tint ðti Þ - Tint t
j 6¼ [; i 6¼ jAf1; 2; . . .; Nt g; ð6:17Þ TVak ðti Þ - TVak tj 6¼ [; kAf1; 2; . . .; Na g; where TVak ðti Þ is the time interval of target ti resides at Voronoi cell V ðak Þ: In this case, when several targets are in a same Voronoi cell, the Voronoi site agent tracks the nearest target, the topology of other target tracking system stay static.

6.5

Numerical examples

In this section we present numerical examples to show the correctness and the effectiveness of the main results derived above. Consider a scenario where a group of three tracking agents tracking a maneuvering target in a two-dimensional space. Since the monitored area is 3-overlapped, when a target enters into this area, with the initialization principle addressed in Section 6.3, the three tracking agents can communicate with each other. Then the topology of the tracking system with three tracking agents is always as shown in the first graph in Fig. 6.7. According to the rules of evolution of a topology, there are nine possible topologies when the tracking number is set as Nf 5 3:

Cooperative relay tracking strategy Chapter | 6

T 1

T

1 2

3

1

2

1

T

2

3

1

T

4 2

T

T

7

3

3

2 3

T

8 2

1

6

1

3

2

1

T

5 2

1

3

3 2

3

T

121

9 2

1 3

FIGURE 6.7 Rules of switching of topology.

The nonlinear dynamics of the tracking agents and target are f ðt; xðtÞÞ 5 250 sinð0:0035xðtÞÞ 1 200 cos ð2:5tÞ 1 10 sin ð2:5tÞ:

ð6:18Þ

The control parameter is chosen as α 5 3:2: Consider there are at most five targets may enter a 1000 3 1000 m2 monitored area, the initial deployment of agents are shown in Fig. 6.8. The time of the first target breaks into this domain is considered as initial time. Coordination of the location where the first target enters this area is (0,100). It is easy to find that Cases 3 and 4 are more complicated than Cases 1 and 2. Actually, Cases 1 and 2 can be reflected in Cases 3 and 4. Therefore we give two examples under Cases 3 and 4, respectively. Example 6.1: This is an example for Case 3. At time steps of 50 and 150, two other new targets enter at locations (0,200) and (200,0) respectively. The third target is isolated to the other two targets, and the other two targets meet during the tracking. The tracking process is illustrated in Fig. 6.9. As can be seen clearly in this figure, the discontinuous trajectories represent switching of tracking agents. Comparing the two figures, we can find that in Fig. 6.9, at the end of the trajectories, there are more redundant agents monitoring this area. This is because we adopt the release policy, then when a target is caught, its

FIGURE 6.8 Initial deployment of agents.

FIGURE 6.9 Tracking trajectories.

Cooperative relay tracking strategy Chapter | 6

123

tracking agents are released to become redundant agents.Figs. 6.10
6.12 show tracking errors on x-axis, y-axis, and the norm of tracking errors, respectively. Figs. 6.10 and 6.11 obviously reflect the switching signal. Every switching of topology and tracking agents brings a decrease jump in tracking error. However, the decrease may not occur in both x-axis and yaxis. The norm of tracking errors is depicted in Fig. 6.12, in which the signal decreases every time a switching occurs. Example 6.2: This is an example for Case 4. At the initial time, targets 1 and 2 move into the monitored area simultaneously and share the same route. At time steps of 150, a new target enters this area from another direction. The third target is isolated to the other two targets. The tracking trajectories are illustrated in Fig. 6.13. When targets 1 and 2 move into this area at the same time and same location, three nearest agents join the tracking team of target 1, the initial tracking agents for target 2 is far-field distributed. Figs. 6.14
6.16 show tracking errors on x-axis, y-axis, and the norm of tracking errors, respectively. No matter the targets are isolated or not, they are captured eventually, which vividly shows the feasibility and effectiveness of our proposed tracking strategy.

FIGURE 6.10 Error trajectories of tracking on x-axis.

FIGURE 6.11 Error trajectories of tracking on y-axis.

FIGURE 6.12 Norm of tracking errors.

FIGURE 6.13 Tracking trajectories.

FIGURE 6.14 Error trajectories of tracking on x-axis.

FIGURE 6.15 Error trajectories of tracking on y-axis.

FIGURE 6.16 Norm of tracking errors.

Cooperative relay tracking strategy Chapter | 6

127

FIGURE 6.17 Norm of tracking errors comparison.

Example 6.3: To verify the advantage of proposed tracking strategy, we compare the tracking performance between the relay tracking strategy and the nonswitching tracking strategy. In the two tracking strategies the initial tracking agents are the same, and it is assumed the target enters this monitored area from position (0,100). However, in the nonswitching tracking strategy, the tracking agents do not switch when the target enters a new Voronoi cell. Fig. 6.17 shows norm of tracking errors with two tracking strategies. As can be seen from the figure, it is obvious that the target can be tracked successfully within a shorter time with proposed tracking strategy. The norms of tracking error of one tracking agent at steps 150 and 200 are marked. Here, only one tracking agent is marked, since all the three are at approximate magnitude. Specifically, at step 150, the norm of tracking error with proposed strategy is 405.3, which is overwhelmingly less than its counterpart value 1030 of the nonswitching strategy.

6.6

Conclusion

This chapter has proposed a cooperative relay tracking strategy to capture targets intruding a domain monitored by a large number of smart agents.

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Consensus Tracking of Multi-agent Systems with Switching Topologies

Trilateration algorithm is used to calculate the location of a target, the key information for designing a tracking controller. Since a target is tracked by several tracking agents, the tracking agents need to cooperate and communicate with each other. The domain is divided into many Voronoi cells with assistance of Voronoi diagram. In the tracking process the tracking agent which is able to access the location of a target is relayed by a series of Voronoi sites. When a target is moving into a new Voronoi cell, the Voronoi site replaces one of the tracking agents, which results in a switching topology. The switching problem of topologies is solved by modeling it as Markov chain process, and the relay of tracking agents is reflected by the jump of tracking errors. Simulation results prove the advantage of proposed relay tracking algorithm.

References [1] E. Bakolas, P. Tsiotras, Relay pursuit of a maneuvering target using dynamic Voronoi diagrams, Automatica 48 (9) (2012) 2213
2220. [2] E. Bakolas, P. Tsiotras, Optimal pursuit of moving targets using dynamic Voronoi diagrams, Proceedings of IEEE International Conference on Decision and Control, Hilton Atlanta Hotel, Atlanta, GA, 2010, pp. 7431
7436. [3] J.S. Li, H.C. Kao, Distributed K-coverage self-location estimation scheme based on Voronoi diagram, IET Commun. 4 (2) (2010) 167
177. [4] M.A. Mostafavi, C. Gold, M. Dakowicz, Delete and insert operations in Voronoi/ Delaunay methods and applications, Computers Geosci. 29 (4) (2003) 523
530. [5] H. Kawakami, T. Namerikawa, Cooperative target-capturing strategy for multi-vehicle systems with dynamic network topology, in: Proceedings of the American Control Conference, St Louis, MO, 2009, pp. 635
640. [6] M. Cao, C. Yu, B.D.O. Anderson, Formation control using range-only measurements, Automatica 47 (4) (2011) 776
781. [7] J. Yan, X. Guan, F. Tan, Target tracking and obstacle avoidance for multi-agent systems, Int. J. Autom. Comput. 7 (4) (2010) 550
556. [8] G. Wen, Z. Duan, G. Chen, W. Yu, Consensus tracking of multi-agent systems with Lipschitz-type node dynamics and switching topologies, IEEE Trans. Circuits Syst. I— Regul. Pap. 61 (2) (2014) 499
511. [9] G. Xu, Z. Guan, D. He, M. Chi, Y. Wu, Distributed tracking control of second-order multi-agent systems with sampled data, J. Frankl. Inst. 351 (10) (2014) 4786
4801. [10] L. Dong, S. Chai, B. Zhang, Tracking of a third-order maneuvering target under an arbitrary topology, Chin. Phys. B 23 (1) (2014) 010508. [11] L. Dong, S. Chai, B. Zhang, S.K. Nguang, Tracking problem under a time-varying topology, Chin. Phys. B 23 (6) (2014) 060502. [12] O. Hajek, Pursuit Games: An Introduction to the Theory and Applications of Differential Games of Pursuit and Evasion, Dover Publications, Mineola, NY, 2008. [13] S. Chae, S.K. Nguang, SOS based robust HN fuzzy dynamic output feedback control of nonlinear networked control systems, IEEE Trans. Cybern. 44 (7) (2014) 1204
1213. [14] S. Saat, S.K. Nguang, Nonlinear HN output feedback control with integrator for polynomial discrete-time systems, Int. J. Robust. Nonlinear Control. 25 (7) (2015) 1051
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[15] J. Zhang, P. Shi, J. Qiu, S.K. Nguang, A novel observer-based output feedback controller design for discrete-time fuzzy systems, IEEE Trans. Fuzzy Syst. 23 (1) (2015) 223
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5568. [17] Y. Xiao, Y. Zhang, Surveillance and tracking system with collaboration of robots, sensor nodes, and RFID tags, in: Proceedings of 18th International Conference on Computer Communications and Networks, San Francisco, CA, 2009, pp. 1
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2) (2013) 65
75. [22] K. You, Z. Li, L. Xie, Consensus condition for linear multi-agent systems over randomly switching topologies, Automatica 49 (10) (2013) 3125
3132. [23] A. Kulaib, R. Shubair, M. Al-Qutayri, J.W. Ng, An overview of localization techniques for wireless sensor networks, 2011 International Conference on Innovations in Information Technology, IEEE, Abu Dhabi, 2011, pp. 167
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67.

Further reading Y. Qian, X. Wu, J. Lu¨, J. Lu, Second-order consensus of multi-agent systems with nonlinear dynamics via impulsive control, Neurocomputing 125 (2014) 142
147. S. Boyd, L.E. Ghaou, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, vol. 15, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1994.

Chapter 7

Stability of a class of multiagent relay tracking systems with unstable subsystems Chapter Outline 7.1 7.2 7.3 7.4

Introduction Related preliminaries Problem formulation Controller design and stability analysis

131 132 134

7.5 Disturbance attenuation analysis 7.6 Numerical examples 7.7 Conclusion References

139 141 149 149

136

A class of nonlinear multiagent tracking problems with unstable subsystems is investigated. During the tracking process the topology and tracking agents switch that may lead the tracking system to be stable or unstable. The system switches either among consecutive stable subsystems and consecutive unstable subsystems or between stable and unstable subsystems. A tracking strategy guaranteeing overall successful tracking despite the existence of unstable subsystems is designed. Also, extended discussions are addressed on the case where the dynamics of agents are subject to disturbances and the disturbance attenuation level is achieved.

7.1

Introduction

When the target successively transits from one Voronoi cell to another, the tracking agent with furthest distance to the target is replaced by the corresponding Voronoi site agent. The switching of tracking agents thus results in tracking error jumps and involves switching of communication topologies, which is quite different from the traditional time-triggered switching topologies. The topology could be unconnected due to the fact that the topologies and tracking agents switch during the tracking process, which may lead to the instability of the corresponding system. The system switches not only among consecutive stable subsystems or consecutive unstable subsystems but also between stable and unstable subsystems, which is a well-known difficult

Consensus Tracking of Multi-agent Systems with Switching Topologies. DOI: https://doi.org/10.1016/B978-0-12-818365-6.00007-0 © 2020 Elsevier Inc. All rights reserved.

131

132

Consensus Tracking of Multi-agent Systems with Switching Topologies

issue to tackle [1,2]. Qin et al. [3] consider a setting that the weak connectivity of the interaction topologies is kept for some disconnected time intervals with short length. Qin et al. [4] also obtain some results on the case where the individual uncoupled system is allowed to be strictly unstable. In both of the two papers, the agents are assumed to be holding linear dynamics, and the agents do not switch during the moving process. The goal of this chapter is to design a tracking strategy guaranteeing successful tracking of a class of nonlinear tracking system with tolerance of unstable subsystems. We also investigate the situation in which the dynamics of agents are subject to external disturbances. From a practical point of view the external disturbances cannot be avoided in the presence of communication noises and environmental effects [5,6]. The effects of the disturbance on multiagent systems (MASs) have been investigated in Refs. [7
11]. In Ref. [9] a fuzzy HN controller is designed to achieve a guaranteed HN performance level for nonlinear MASs in which the nonlinear agents are described by Takagi
Sugeno (T
S) fuzzy models. Saboori and Khorasani [11] achieve the disturbance attenuation for MASs with switching topology networks with an average dwell time. Most of the existing literatures relevant to the disturbance attenuation issue for MASs are with fixed topology [7
10] or switching among stable subsystems [11]. The tracking problem for MASs with both unstable subsystems and disturbance has not been tackled in existing literatures, which motivates us to do this work. The main contributions are stated as follows. First, a new multiagent tracking problem frame is constructed to simulate a realistic area monitoring and target tracking situation, and a cooperative relay strategy is proposed to solve this problem. Second, we derive a new mathematical model for the tracking problem. Such a model can effectively solve the jumping problem of the tracking errors caused by the relay scheme. Third, a new switched technique is adopted to tackle the existence of unstable subsystems, which makes the analysis of the tracking problem more difficult. Furthermore, we also discuss the case that the agents are constrained by external disturbances because such disturbances cannot be avoided due to the presence of communication noises or environmental effects.

7.2

Related preliminaries

Since the dynamic change of the topology could happen when the target enters a new Voronoi cell, the Laplacian matrix is rewritten in the following format as ℒσðtÞ 1 BσðtÞ ; where σðtÞ 5 k; tA½tk ; tk11 Þ; kAℕ. At each event-triggered time the tracking system may switch from a stable subsystem to an unstable subsystem or a stable subsystem, and vice versa. We introduce a sequence pm ; mAℕ that belongs to the time sequence tk ; kAℕ; and the relationship between tk and pm satisfies (Fig. 7.1)

133

Stability of a class of multiagent relay tracking systems Chapter | 7

T (tk, tk+1)

S

t0

S

t1

Un S

UnS

t2

t3

UnS

t4

...

S

t5

t6

UnS

tk

S

...

S

tk+1 tk+2

tk+3

t

p0 = t0; s1 = 2; p1 = ts1 = t2; s2 = 5; p2 = ts2 = t5 … T – (p0, p2) = T (p0, p1);T + (p0, p2) = T ( p1, p2) FIGURE 7.1 Illustration of event-triggered switching times.

t0 5 p0 , t1 , ? , ts1 5 p1 , ts1 11 , ? , ts2 5 , p2 , . . .

ð7:1Þ

where sm ; mAℕ is the number of switching among consecutive stable sub½p ; p Þ 5 systems or consecutive unstable subsystems. , 1N


m50 m m11  ½t0 ; 1 NÞ; ½pm ; Pm11 Þ 5 tsm ; tsm 11 , tsm 11 ; tsm 12 , ? , tsm 11 2 1; tsm 11 . Then we suppose there exist a set of positive constants ηm ; ’mAℕ for each interval of ½pm ; pm11 Þ; and a positive constant η such that pm11 2 pm # ηm # η , 1 N;

’mAℕ:

ð7:2Þ

p0 5 t0 ; s1 5 2; p1 5 ts1 5 t2 ; s2 5 5; p2 5 ts2 5 t5 . . . T 2 ðp0 ; p2 Þ 5 T ðp0 ; p1 Þ; T 1 ðp0 ; p2 Þ 5 T ðp1 ; p2 Þ Fig. 7.1 can be interpreted as follows, tk ; kAℕ are all the switching moments, including both switching between stable and unstable subsystems and switching from stable to stable or unstable to unstable subsystems. On the other hand, pm ; mAℕ stand for the switching moments from a stable subsystem to an unstable subsystem or from an unstable subsystem to a stable subsystem. T ðtk ; tk11 Þ refers to the time interval ½tk ; tk11 Þ: For any time τ, less than time t; T 1 ðτ; tÞ and T 2 ðτ; tÞ refer to the total time length of unstable and stable from initial time τ till the present time t, respectively.  subsystems  lr 5 T 1 ðτ; tÞ =ðT 2 ðτ; tÞÞ is called length rate of unstable subsystems on time interval T ðτ; tÞ: Let N ðτ; tÞ denote the number of switching during ½τ; tÞ. Definition 7.1: [12] The system x_ðtÞ 5 f ðt; xðtÞÞ; t $ t0 ; xðt0 Þ 5 x0 ; t0 $ 0 is globally exponentially stable if there exist positive scalars c1 ; c2 . 0 such that jjxðtÞjj # c1 jjx0 jje2c2 ðt2t0 Þ holds for all t $ t0 .

ð7:3Þ

134

Consensus Tracking of Multi-agent Systems with Switching Topologies 1200

1000

800

600

400

200

0 0

200

400

600

800

1000

1200

FIGURE 7.2 Tracking trajectories.

7.3

Problem formulation

The dynamics of the ith agent deployed on the supervisory area is described as x_i ðtÞ 5 f ðt; xi ðtÞÞ 1 ui ðtÞ; xi ðt0 Þ 5 xi0 ; t0 $ 0

t $ t0

ð7:4Þ

where xi ðtÞAℝ2 is the position state of the ith agent, f ðt; xi ðtÞÞ is a nonlinear function, and ui ðtÞAℝ2 is the control input of the ith agent. The agents are expected to monitor this area and capture any intruded target. We consider the following dynamics of the target. x_t ðtÞ 5 f ðt; xt ðtÞÞ; t $ t0 xt ðt0 Þ 5 xt0 ; t0 $ 0

ð7:5Þ

where xt ðtÞAℝ2 is the position state of the target. f ðt; xt ðtÞÞ denotes the change of the force which is imposed on the target. In this chapter, we assume f ðUÞ satisfies the Lipschitz condition A.1. Definition 7.2: For first-order multiagent tracking problem, tracking agents are supposed to move along with the target, that is, the positions are required to be consensus. Mathematically, tracking problem boils down to the following expression. lim jjxi ðtÞ 2 xt ðtÞjj-0;

t-N

iANf :

ð7:6Þ

Stability of a class of multiagent relay tracking systems Chapter | 7

135

The purpose is to make sure positions of tracking agents and target finally to be the same. Thus the following triggering event is adopted, which ensures the tracking errors decrease along with every switching. fe5 jjexj ðtÞjj2 jjexi ðtÞjj # 0; j 6¼ i; i:e: d xt ðtÞ; xj ðtÞ 5 jjexj ðtÞjj # dðxt ðtÞ; xi ðtÞÞ 5 jjexi ðtÞjj

ð7:7Þ

where exi ðtÞ 5 xi ðtÞ 2 xt ðtÞ is the position tracking error between the ith tracking agent and the target. In this chapter, exi ðtÞ 5 ½exi1 ðtÞ; exi2 ðtÞ denotes the position tracking error, in which exi1 ðtÞ; exi2 ðtÞ are the position disagreement on x-axis and y-axis, respectively. Norm of the position tracking error between the ith tracking agent and the target is the same as the distance between ith agent and the tarpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi get, that is, jjexi ðtÞjj 5 dðxt ðtÞ; xi ðtÞÞ 5 e2xi1 ðtÞ 1 e2xi2 ðtÞ. Therefore the triggering event (7.7) means that distance between agent j and target is less than that of agent i. Then according to the definition of Voronoi diagram, this triggering event that the target moves from Voronoi cell V ðai Þ to  implies  Voronoi cell V aj . In this chapter the agents share the same relay tracking algorithm as that in Chapter 6, Cooperative relay tracking strategy for MASs with assistance of Voronoi diagrams. At each instant of time, only Nf pursuers are assigned the task of tracking the maneuvering target, whereas all other agents in this region will remain stationary. Remark 7.1: In this chapter, every agent is able to communicate in two modes: omni and directional. In the omni mode, each agent is capable of receiving and transmitting signals in all directions (360 degrees) with gain G. On the other hand, in the directional mode, an agent can point its beam toward a specific direction with gain Gd (where Gd . G). Because of the higher gain, agents in directional mode have a greater range in comparison with the omni mode [13]. At each time, it is assumed only the tracking agent, where the target resides, operates in directional mode while all the others operate in omni mode. When a target enters the monitoring area, the corresponding Voronoi site agent i operates in directional mode and rotates to send target’s position to neighboring agents. The neighboring agents calculate their relative distances with the target and send this information back. Then, the Voronoi site agent i decides the Nf 2 1 nearest agents and sends tracking commands to selected agents. Thereafter, Voronoi site agent stops rotating its communication direction and focuses on the direction of target. According to the abovementioned analysis, computation complexity of   the proposed relay pursuit tracking strategy is O 2Nf . It should be noted

136

Consensus Tracking of Multi-agent Systems with Switching Topologies

that this includes the tracking computation and dynamic Voronoi diagram computation.

7.4

Controller design and stability analysis

It is assumed that agent i can only receive the state information of its neighbors and target, then we adopt the following format of control protocol for the ith tracking agent. nX o a ð σ ð t Þ Þe ð t Þ 1 b ð σ ð t Þ Þe ð t Þ ; ð7:8Þ ui ð t Þ 5 2 α ij xij i xi jAN ðtÞ i

where α . 0 is the control parameter to be designed. exij ðtÞ 5 xi ðtÞ 2 xj ðtÞ is the position disagreement vector between the ith tracking agent and the jth tracking agent. Then, from Definition 7.2, the tracking problem can be interpreted as stability problem of the following overall disagreement system: E_ ðtÞ 5  F ðt;EðtÞÞ 2 αLk EðtÞ; t 6¼ tk ð7:9Þ E tk1 5 E tk2 2 ΔEðtk Þ; t 5 tk ; k 5 0; 1; 2; . . . h iT where EðtÞ 5 eTx1 ðtÞ; eTx2 ðtÞ; . . .; eTxNf ðtÞ :Lk 5 ðℒk 1 Bk Þ  I2 is the associated Laplacian matrix of the  dynamic graph, including the target in time interval  ½tk ; tk11 Þ: F ðt; EðtÞÞ 5 col f1 ðtÞ; f2 ðtÞ; . . .; fNf ðtÞ , where fi ðtÞ 5 f ðt; xi ðtÞÞ 2 f ðt; xt ðtÞÞ; iANf is the collective nonlinear self-dynamic disagreement to the target at time t.tk is the switching time, tk2 indicates the time before switching, and tk1 represents the time after switching. ΔEðtk Þ is the jump of tracking error at switching time tk . Remark 7.2: When the topology switches, the tracking agent which is furthest to the target is replaced by the new Voronoi site. This means the norm of tracking error E(t) decreases at every switching time tk . The element in E (t) corresponding to the replaced tracking agent jumps, and the jump value is reflected by ΔEðtk Þ: For instance, if the ith tracking agent     is replaced at time tk , then jjexi tk1 jj # jjexi tk2 jj while jjexj tk1 jj 5 jjexj tk2 jj; j 6¼ i; i; jANf : Tracking   error jumps  down at switching time tk ; which implies that jjE tk1 jj2 # μjjE tk2 jj2 ; 0 , μ # 1. Remark 7.3: In switched systems, sliding mode phenomenon arises when there are no impulse effects and the states do not jump at the switching events. However, in this chapter, the states (tracking errors) jump due to our proposed relay tracking strategy, in which a tracking agent is replaced by another one at the switching event. Therefore the sliding mode phenomenon on the switching instants will not occur in the switched system with the proposed relay tracking strategy.

Stability of a class of multiagent relay tracking systems Chapter | 7

137

Next, we will analyze the multiagent tracking system and figure out when it will be a stable subsystem or an unstable one. Let us select the Lyapunov candidate for the tracking system (7.9) during time interval ½tk ; tk11 Þ as V ðt Þ 5

1 T E ðtÞEðtÞ; 2

tA½tk ; tk11 Þ;

kAℕ:

ð7:10Þ

For any tA½tk ; tk11 Þ; the upper right-hand Dini derivative of V(t) along the trajectory (7.9) is D1 V ðtÞ 5

1 _T 1 E ðtÞEðtÞ 1 ET ðtÞE_ ðtÞ 2 2

5 2 αET ðtÞLTk EðtÞ 1 F ðt; EðtÞÞT EðtÞ

ð7:11Þ

# ð 2αλmin ðLk Þ 1 lÞjjEðtÞjj2 : Apparently, with an event-triggered switching, the communication topologies change along with time, the tracking system can be stable or unstable. As stated in Lemma A.5, the eigenvalues of Laplacian matrix are all nonnegative, as a result, there are two cases, that is, λmin ðLk Þ . 0 and λmin ðLk Þ . 0. The corresponding two cases are addressed as following. For some communication topologies, Lk is positive definite, we are always able to find appropriate positive control parameter α that enables 2αλmin ðLk Þ 1 l 5 2 γ 1 , 0, where γ 1 is a positive constant. In this case, we call the tracking system a stable subsystem. The corresponding Lyapunov function satisfies V ðtÞ 5 V1 ðtÞ # e2γ1 ðt2tk Þ V ðtk Þ:

ð7:12Þ

For some other communication topologies, λmin ðLk Þ 5 0, thus, 2αλmin ðLk Þ 1 l 5 γ 2 . 0, where γ 2 is a positive constant. In this case, we call the tracking system an unstable subsystem. The corresponding Lyapunov function satisfies V ðtÞ 5 V2 ðtÞ # eγ2 ðt2tk Þ V ðtk Þ:

ð7:13Þ

It should be noted that the control parameter α is a constant and is determined offline before the tracking process. According to the nonlinear dynamics of the agents, the Lipschitz coefficient can be calculated, and then it is able to obtain a conservative control parameter satisfying 2αλmin ðLk Þ 1 l 5 2 γ 1 , 0, where Lk is the worst possible case. Based on the above conditions, the overall stability of the tracking system is analyzed in this section.

138

Consensus Tracking of Multi-agent Systems with Switching Topologies

Theorem 7.1: Consider  the multiagent system (7.9), if there exist positive  numbers γ A 0; γ 1 and μ # 1 such that the length rate lr of unstable subsystems satisfies 

T 1 ðt0 ; t Þ γ 1 2 γ # ; ð7:14Þ T 2 ðt0 ; tÞ γ2 1 γ the tracking multiagent system (7.9) with event-triggered switching topologies is overall stable. Moreover, the overall tracking error satisfies lr 5

jjEðtÞjj2 # μN ðt0 ;tÞ jjEðt0 Þjj2 e2γ



ðt2t0 Þ

:

ð7:15Þ

Proof: Without loss of generality, we assume, at the first time interval,
 the subsystem is stable. Then, during the time interval, ½p0 ; p1 Þ 5 t0 ; ts1 , the Lyapunov function is   V1 t12 5 e2γ1 Tðt0 ;t1 Þ V1 ðt0 Þ     V1 t11 # μV1 t12 # μe2γ1 Tðt0 ;t1 Þ V1 ðt0 Þ     V1 t22 5 e2γ1 Tðt1 ;t2 Þ V1 t11     V1 t21 # μV1 t22 # μ2 e2γ1 Tðt0 ;t2 Þ V1 ðt0 Þ

ð7:16Þ

?   V1 ts21 5 e2γ1 T ðts121 ;ts1 Þ V1 ts11 21 : At time p1 5 ts1 , the tracking system switches from a stable subsystem to an unstable subsystem.   V2 tS11 # μV1 ts21 # μs1 e2γ1 T ðt0 ;ts1 Þ V1 ðt0 Þ   V2 ts21 11 5 eγ2 T ðts1 ;ts111 Þ V2 ts11  1  # μV2 ts21 11 V2 ts111 ð7:17Þ # μs1 11 eγ2 T ðts1 ;ts111 Þ2γ1 T ðt0 ;ts1 Þ V1 ðt0 Þ ?   V2 ts22 5 eγ2 T ðts221 ;ts2 Þ V2 ts12 21 : At time p2 5 ts2 , the tracking unstable subsystem to a stable subsystem.

system

switches

from

an

# μV2 ðts22 Þ # μs2 eγ2 Tðts1 ;ts2 Þ2γ1 Tðt0 ;ts1 Þ V1 ðt0 Þ 5 e2γ1 Tðts2 ;ts211 Þ V1 ðts12 Þ ð7:18Þ # μV1 ðtS22 11 Þ 2 # μs2 11 eγ2 T1ðt0 ;ts211 Þ2γ1 T ðt0 ;ts211 Þ V1 ðt0 Þ Then, in general case, at time tA½tk ; tk11 Þ, the Lyapunov function boils down to V1 ðts12 Þ V1 ðts22 11 Þ V1 ðts12 11 Þ

Stability of a class of multiagent relay tracking systems Chapter | 7

V ðtÞ # μk eγ2 T

1

ðt0 ;tÞ2γ 1 T2ðt0 ;tÞ

V ðt0 Þ:

From the condition (7.14) in Theorem 7.1, we can get     T 1 ðt0 ; tÞ γ 2 1 γ 2 T 2 ðt0 ; tÞ γ 1 2 γ # 0   γ 2 T 1 ðt0 ; tÞ 2 γ 1 T 2 ðt0 ; tÞ 1 γ T 1 ðt0 ; tÞ 1 T 2 ðt0 ; tÞ # 0  γ 2 T 1 ðt0 ; tÞ 2 γ 1 T 2 ðt0 ; tÞ # 2 γ ðt 2 t0 Þ

139

ð7:19Þ

ð7:20Þ

Associate (7.19) with (7.20), it is straightforward to obtain that 

ð7:21Þ V ðtÞ # μN ðt0 ;tÞ V ðt0 Þe2γ ðt2t0 Þ :   T     Since V ðtÞ 5 1=2 E ðtÞEðtÞ 5 1=2 jjEðtÞjj2 and V ðt0 Þ 5 1=2 jjEðt0 Þjj2 . Then (7.15) follows from (7.21). According to Definition 7.1, the overall error system is exponentially stable. The errors between the tracking agents and the target tend to 0 when t-N, that is, tracking agents finally track the target. Despite the system consists of unstable subsystems, the tracking agents still successfully track the target with the proposed relay tracking strategy. &

7.5

Disturbance attenuation analysis

In this section, we investigate the disturbance attenuation performance of our proposed tracking strategy for the MAS. The agents are subject to external disturbances, and the dynamics of the ith agent is then described as x_i ðtÞ 5 f ðt; xi ðtÞÞ 1 ui ðtÞ 1 ωi ðtÞ; xi ðt 0 Þ 5 xi0 ; t 0 $ 0

t $ t0

ð7:22Þ

where ωi ðtÞAℒ2 ½0; NÞ is the disturbance input of the ith agent. The dynamics of the target is then described as x_t ðtÞ 5 f ðt; xt ðtÞÞ 1 ωt ðtÞ; xt ðt0 Þ 5 xt0 ; t0 $ 0

t $ t0

ð7:23Þ

where ωt ðtÞAℒ2 ½0; NÞ is the disturbance input of the target. Then, from Definition 7.2, the tracking problem can be interpreted as stability problem of the following overall disagreement system: E_ ðtÞ 5  F ðt;EðtÞÞ 2 αLk EðtÞ 1 ωðtÞ; t 6¼ tk E tk1 5 E tk2 2 ΔEðtk Þ; t 5 tk ; k 5 0; 1; 2; . . . h  T iT where ωðtÞ 5 ðω1 ðtÞ2ωt ðtÞÞT ; ðω2 ðtÞ2ωt ðtÞÞT ; . . .; ωNf ðtÞ2ωt ðtÞ :

ð7:24Þ

For its similarity to the case without disturbance, some of the derivation is omitted. For stable subsystems, exists a positive scalar γ ω1 satisfies  22 there  γ

1 ω 2 # αλmin ðLk Þ 2 l 2 λ0 =2 , where λ0 . 0 is a scalar positively involved with the disturbance attenuation level. The piecewise Lyapunov function V(t) satisfies

140

Consensus Tracking of Multi-agent Systems with Switching Topologies

2γ ω1 ðt2tk Þ

V ðtÞ # e

λ2 V ðtk Þ 1 0 2

ðt

ω

e2γ1 ðt2τ Þ ωT ðτ Þωðτ Þdτ:

ð7:25Þ

tk

ω For unstable  ω subsystems,  22 λmin ðLk Þ 5 0, there exists a positive scalar γ 2 satisfying γ 2 =2 $ l 1 λ0 =2 , and then ð λ20 t γω ðt2τ Þ T γ ω2 ðt2tk Þ V ðtk Þ 1 e2 ω ðτ Þωðτ Þdτ: ð7:26Þ V ðt Þ # e 2 tk

Now, we are ready to give the main result on the robustness of the overall tracking system, which states the disturbance attenuation level from ω to the tracking errors E. Theorem 7.2: Considerthe disturbed multiagent system (7.24), if there exist   positive numbers γ ω A 0; γ ω1 and μ # 1 such that the length rate lωr of unstable subsystems satisfies 

lωr 5

T 1 ðt0 ; tÞ γ ω1 2 γ ω # ; T 2 ðt0 ; tÞ γ ω2 1 γ ω

ð7:27Þ

the tracking pffiffiffiffiffiffimultiagent system (7.24) achieves a disturbance attenuation level λ0 = γ ω . Proof: According to (7.16)
(7.19), (7.27) and similar to the derivation of (7.20), we obtain ð   λ2 t ð7:28Þ V ðtÞ # e2γω ðt2t0 Þ V ðt0 Þ 1 0 e2γω ðt2τ Þ ωT ðτ Þωðτ Þdτ: 2 t0 By integrating the abovementioned inequality (7.28) in terms of time from t0 to N, we have ðN ðN  V ðtÞdt # e2γω ðt2t0 Þ V ðt0 Þdt t0

t0

λ2 1 0 2 5

ðN ðt t0



e2γω ðt2τ Þ ωT ðτ Þωðτ Þdτdt

t0

1 λ20  V ðt0 Þ 1 γω 2γ ω

ðN

ð7:29Þ

ωT ðtÞωðtÞdt:

t0

  According to the definition of Lyapunov function V ðtÞ 5 1=2 ET ðtÞEðtÞ, the following inequality ðN ð 2 λ2 N T ET ðtÞEðtÞdt #  V ðt0 Þ 1 0 ω ðtÞωðtÞdt ð7:30Þ γω γ ω t0 t0

Stability of a class of multiagent relay tracking systems Chapter | 7

141

holds for any ωðtÞAℒ2 ½0; NÞ. This means that       jjEjj2ℒ2 # 2=γ ω V ðt0 Þ 1 λ20 =γ ω jjωjj2ℒ2 ; thus the disturbance attenuation pffiffiffiffiffiffi level λ0 = γ ω is achieved. Our proposed tracking strategy is also robust against external disturbances for the multiagent tracking systems. &

7.6

Numerical examples

In this section, we present numerical examples to show the correctness and the effectiveness of the main results derived from the earlier equation. The numerical examples include the two cases where a target is tracked by three agents, and the case in which the agents are subject to disturbances. Both of the cases are conducted on the following scenario. Consider a scenario where a group of three tracking agents track a maneuvering target in two-dimensional space. The nonlinear dynamics of the tracking agents and target are f ðt; xðtÞÞ 5 250sinð0:0035xðtÞÞ 1 200cosð2:5tÞ 1 20sinð2:5tÞ:

ð7:31Þ

Then, jjf ðt; xi Þ 2 f ðt; xt Þjj # 250 3 0:0035jjxi 2 xt jj 5 0:875jjxi 2 xt jj: Consider there are at most three targets may enter a 1200 3 1200 m2 monitoring area and the sensing radius is set as Rs 5 125 m. The time of the first target breaks into this domain is considered as initial time. The coordinate where the first target enters this area is (0,100). Example 7.1: The target is tracked by three agents. This example addresses the situation in which an intruded target is tracked by three agents. The tracking process is illustrated in Fig. 7.2. In this figure, the continuous line represents the trajectory of target. When the target enters a new Voronoi cell, the associated Voronoi site agent replaces one of the original tracking agents, which leads the tracking trajectories to be discontinuous. The discontinuity of different colored lines clearly shows which agent is replaced. Fig. 7.3 shows tracking errors on x-axis, y-axis, which obviously reflects the switching. Every switching of topology and tracking agents brings a jump in tracking error. Actually, each discontinuity of a line in Fig. 7.2 corresponds to a jump in Fig. 7.3. For instance, at time 1.28 s, agent 2 is replaced by a new Voronoi site agent and its error trajectory sees a significant jump. Correspondingly, there is an obvious discontinuity in the line as we can see in Figs. 7.2 and 7.3. Fig. 7.4 depicts the switching signal σðtÞ. The system switches five times over the entire tracking process, which is verified by the summation of norm of all tracking errors, that is, the value of Lyapunov function in Fig. 7.5. At

Error trajectories of x-axis 200 Agent 1 Agent 2 Agent 3

100

0

−100

0

1

2

3

4

5

6

7

8

Time (s) Error trajectories of y-axis 200

0 Agent 1 Agent 2 Agent 3

−200

−400

0

1

2

3

4

5

6

7

6

7

Time (s) FIGURE 7.3 Tracking errors of three tracking agents.

Switching signal s (t)

8 7 6 5 4 3 2 1 0

0

1

2

3

4 Time (s)

FIGURE 7.4 Switching signal of the tracking process.

5

8

143

Stability of a class of multiagent relay tracking systems Chapter | 7

10

Summation of norm of all tracking errors

x 104

5

0

0

1

2

3

4 Time (s)

5

6

7

8

7

8

Minimum eigenvalue of extended Laplacian matrix 0.4 0.3 0.2 0.1 0

0

1

2

3

4 Time (s)

5

6

FIGURE 7.5 Norm of errors and minimum eigenvalues.

every switching, the summation of norm of all tracking errors jumps down, satisfying V ðtk11 Þ # μV ðtk Þ. According to Fig. 7.5, during simulation time (0.1.28 s) and (2.32, 2.18 s), the minimum eigenvalues of extended Laplacian matrix is 0, that is, λmin ðLk Þ 5 0, then 2αλmin ðLk Þ 1 l 5 γ 2 5 0:875, the related subsystems are unstable subsystems and the corresponding Lyapunov function is increasing as can be clearly seen from Fig. 7.5. At the other time steps, λmin ðLk Þ 5 0:2679. To ensure the stability of the subsystems, it is required 2αλmin ðLk Þ 1 l , 0, then the designed control parameter is α . 3:2661: Under this design principle, we choose the control parameter as α 5 6, thus γ 1 5 0:7324. Thus the summation norm of tracking error holds a decreasing       = γ 5 0:4942; and then trend. Select γ 5 0:3, we have γ 2 γ 1 γ 2 1       lr 5 T 1 ðt0 ; tÞ =ðT 2 ðt0 ; tÞÞ 5 0:291 , γ1 2 γ = γ 2 1 γ . It is verified that the condition in Theorem 7.1 is satisfied. Example 7.2: The target is tracked by five agents. In order to show our proposed relay tracking strategy still works well when there are more tracking agents, we perform a simulation with five

144

Consensus Tracking of Multi-agent Systems with Switching Topologies 1200

1000

800

600

400

200

0

0

200

400

600

800

1000

1200

FIGURE 7.6 Tracking trajectories of five tracking agents.

tracking agents. All the other parameters are configured as the same in Example 7.1. Similar to the presentation form of tracking performance in Example 7.1, the tracking trajectories and tracking errors are depicted in Figs. 7.6 and 7.7, respectively. Also, we give the switching signals in Fig. 7.8. In terms of this case, the topology and tracking agents switch more than the case with three tracking agents. As illustrated in Fig. 7.9, for this case, there are two values of λmin ðLk Þ and both are smaller than the counterpart in Example 7.1. It requires a larger control parameter and we choose α 5 8 to make sure the stability of those subsystems. It is worth to note that for the two different minimum eigenvalues, the decreasing rate varies. Specifically, the bigger the minimum eigenvalue is, the more dramatic the tracking error falls. Example 7.3: The agents are subject to disturbances. A simulation of multiagent tracking system with disturbances is conducted to verify the theoretical result in Section 7.5. For simulation analysis, we assume a disturbance d(t) shown in Fig. 7.10, and the disturbances imposed on target and three tracking agents are ωt ðtÞ 5 d ðtÞ; ω1 ðtÞ 5 1:2d ðtÞ; ω2 ðtÞ 5 0:5d ðtÞ and ω3 ðtÞ 5 2 dðtÞ respectively. The rest of the simulation scenario is the same as that in Example 7.1. The tracking trajectories of the disturbed system are shown in Fig. 7.11.  Choose λ0 5 6, we obtain γ ω1 5 1:3; γ ω2 5 1:9. Select γ ω 5 0:2; we have   ðγ ω1 2 γ ω Þ=ðγ ω2 1 γ ω Þ 5 0:5238. In this example, the length rate of

Error trajectories of x-axis 400 Agent 1 Agent 2 Agent 3 Agent 4 Agent 5

200 0 −200 −400

0

1

2

3

4 Time (s)

5

6

7

Error trajectories of y-axis

200

Agent 1 Agent 2 Agent 3 Agent 4 Agent 5

0

−200

−400

8

0

1

2

3

4 Time (s)

5

6

7

8

FIGURE 7.7 Tracking errors of five tracking agents.

Switching signal s (t) 16 14 12 10 8 6 4 2 0

0

1

2

3

4 Time (s)

FIGURE 7.8 Switching signals with five tracking agents.

5

6

7

3

Summation of norm of all tracking errors

x 105

2

1

0 0

1

2

3

4 Time (s)

5

6

7

8

7

8

Minimum eigenvalue of extended Laplacian matrix 0.2

0.1

0

−0.1 0

1

2

3

4 Time (s)

5

6

FIGURE 7.9 Norm of errors and minimum eigenvalues with five tracking agents.

10 8 6

Disturbance

4 2 0 −2 −4 −6 −8 −10

0

1

2

3

4

5

6

Time (s) FIGURE 7.10 Illustrative description of disturbance signal d ðtÞ.

7

8

9

10

Stability of a class of multiagent relay tracking systems Chapter | 7

147

Tracking trajectories with disturbances 1200

1000

800

600

400

200

0

0

200

400

600

800

1000

1200

FIGURE 7.11 Tracking trajectories of the system subject to disturbances.

150

Ratio of the tracking error energy to the disturbance energy

X: 1.34 Y: 148.5

100

50

0

0

1

2

3

4

5

6

Time (s) FIGURE 7.12 Ratio of the tracking error energy to the disturbance energy.

7

8

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Consensus Tracking of Multi-agent Systems with Switching Topologies

x 104

Norm of tracking errors with relay strategy

8 6 4 X: 3 Y: 1.625e+04

2

X: 6 Y: 191.8

0

0

1

2

x 104

3

4

5 Time (s)

6

7

8

9

10

8

9

10

Norm of tracking errors without switching

8 X: 3 Y: 7.436e+04

6 4 2 0

X: 6 Y: 2414

0

1

2

3

4

5

6

7

Time (s) FIGURE 7.13 Norm of tracking errors comparison.

unstable subsystems is lr 5 0:4959; which apparently satisfies the condition in p Theorem 7.2. The disturbance attenuation level is designed as ffiffiffiffiffiffi error energy to the disturbance λ0 = γ ω 5 13:4164. The Ð N ratio of the tracking ÐN input energy, that is, t0 ET ðtÞEðtÞdt= t0 ωT ðtÞωðtÞdt is depicted in Fig. 7.12 that shows that the maximum value of the ratio is 148.5. Therefore the dispffiffiffiffiffiffiffiffiffiffiffi turbance attenuation level is 148:5 5 12:1861, which verifies that the pffiffiffiffiffiffi designed value λ0 = γ ω 5 13:4164 is achieved. Example 7.4: Comparison of tracking performance with nonrelay method. To verify the advantage of proposed tracking strategy, we compare the tracking performance between the relay pursuit strategy and the nonrelay pursuit strategy. In the two tracking strategies the initial tracking agents are the same. However, in the nonrelay pursuit strategy the tracking agents do not switch when the target enters a new Voronoi cell. Fig. 7.13 shows norms of tracking errors with two tracking strategies. As can be seen from the figure, it is obvious that the target can be tracked successfully within a shorter time with proposed tracking strategy. The norms of tracking errors at times 3 and 6 s are marked. Specifically, at time 6 s, the

Stability of a class of multiagent relay tracking systems Chapter | 7

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norm of tracking errors with proposed strategy is 191.8, which is overwhelmingly less than its counterpart value 2414 of the nonrelay strategy.

7.7

Conclusion

The tracking problem of a class of nonlinear MASs on a certain domain is investigated. This domain is monitored by a number of smart agents and divided into many Voronoi cells with assistance of Voronoi diagram. At the same time, the smart agents are set to track any intruded target agent with the proposed tracking strategy. Several tracking agents cooperate and communicate with each other when they are supposed to track the same target. During the tracking process the tracking agents are relayed by a series of Voronoi sites. The switching of topologies and relay of tracking agents are triggered when a target is moving into a new Voronoi cell and this makes the tracking system either stable or unstable. The system not only switches among consecutive stable subsystems or consecutive unstable subsystems but also between stable and unstable subsystems. Our proposed tracking strategy guarantees overall successful tracking with tolerance of unstable subsystems and disturbances. One issue that has not been addressed in this chapter, and is worth pursuing in the future work, is whether this cooperative relay tracking strategy can be used to capture a target under the network-based environment with time delays and packet dropouts, which is known to be an important issue in networked systems [14,15]. In this case the subsystems are more likely to be unstable, and it will be more difficult to deal with. Another possible extension is to consider a situation where the agents and targets are described by Takagi
Sugeno (T
S) fuzzy dynamic models [15
18]. In terms of more complex nonlinear systems, when the state variable is not measurable, observer-based control [19,20] could be adopted.

References [1] X. Sun, W. Wang, Integral input-to-state stability for hybrid delayed systems with unstable continuous dynamics, Automatica 48 (9) (2012) 2359
2364. [2] X. Sun, S. Du, P. Shi, W. Wang, L. Wang, Input-to-state stability for nonlinear systems with large delay periods based on switching techniques, IEEE Trans. Circuits Syst. 61 (6) (2014) 1789
1800. [3] J. Qin, C. Yu, H. Gao, Coordination for linear multiagent systems with dynamic interaction topology in the leader-following framework, IEEE Trans. Ind. Electron. 61 (5) (2014) 2412
2422. [4] J. Qin, H. Gao, C. Yu, On discrete-time convergence for general linear multi-agent systems under dynamic topology, IEEE Trans. Autom. Control 59 (4) (2014) 1054
1059. [5] S. Chae, S.K. Nguang, SOS based robust HN fuzzy dynamic output feedback control of nonlinear networked control systems, IEEE Trans. Cybern. 44 (7) (2014) 1204
1213.

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[6] S. Saat, S.K. Nguang, Nonlinear HN output feedback control with integrator for polynomial discrete-time systems, Int. J. Robust. Nonlinear Control. 25 (7) (2015) 1051
1065. [7] X. Wang, S. Li, P. Shi, Distributed finite-time containment control for double-integrator multiagent systems, IEEE Trans. Cybern. 44 (9) (2014) 1518
1528. [8] Y. Liu, J. Lunze, Leader-follower synchronisation of autonomous agents with external disturbances, Int. J. Control. 87 (9) (2014) 1914
1925. [9] Y. Zhao, B. Li, J. Qin, H. Gao, H.R. Karimi, HN consensus and synchronization of nonlinear systems based on a novel fuzzy model, IEEE Trans. Cybern. 43 (6) (2013) 2157
2169. [10] D. Meng, K. Moore, Studies on resilient control through multiagent consensus networks subject to disturbances, IEEE Trans. Cybern. 44 (11) (2014) 2050
2064. [11] I. Saboori, K. Khorasani, HN consensus achievement of multi-agent systems with directed and switching topology networks, IEEE Trans. Autom. Control. 59 (11) (2014) 3104
3109. [12] P.H.A. Ngoc, New criteria for exponential stability of nonlinear time-varying differential systems, Int. J. Robust. Nonlinear Control. 24 (2) (2014) 264
275. [13] V. Friderikos, K. Papadaki, M. Dohler, A. Gkelias, H. Agvhami, Linked waters [marine communication], Commun. Eng. 3 (2) (2005) 24
27. [14] H. Li, C. Wu, P. Shi, Y. Gao, Control of nonlinear networked systems with packet dropouts: Interval type-2 fuzzy model-based approach, IEEE Trans. Cybern. 45 (11) (2014) 2378
2389. [15] W. Tong, H. Gao, J. Qiu, A combined adaptive neural network and nonlinear model predictive control for multirate networked industrial process control, IEEE Trans. Neural Netw. Learn. Syst. 27 (99) (2015) 416
425. [16] J. Qiu, S. Ding, H. Gao, S. Yin, Fuzzy-model-based reliable static output feedback HN control of nonlinear hyperbolic PDE systems, IEEE Trans. Fuzzy Syst. 24 (2) (2016) 388
400. [17] J. Qiu, G. Feng, H. Gao, Static-output-feedback HN control of continuous-time T-S fuzzy affine systems via piecewise Lyapunov functions, IEEE Trans. Fuzzy Syst. 21 (2) (2013) 245
261. [18] J. Qiu, H. Tian, Q. Lu, H. Gao, Nonsynchronized robust filtering design for continuoustime T-S fuzzy affine dynamic systems based on piecewise Lyapunov functions, IEEE Trans. Cybern. 43 (6) (2013) 1755
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1245.

Chapter 8

Multiagent relay tracking systems with damaged agents and time-varying number of agents Chapter Outline 8.1 Introduction 8.2 Relay tracking systems with damaged agents 8.2.1 Problem formulation 8.2.2 Relay tracking algorithm 8.2.3 Controller design and stability analysis 8.2.4 A numerical simulation 8.3 Relay tracking systems with time-varying number of agents

151 152 153 154 155 161

8.3.1 Preliminaries and problem formulation 164 8.3.2 Main results of relay tracking systems with time-varying number of agents 168 8.3.3 A numerical simulation 171 8.4 Conclusion 172 References 174

163

It is practical that some of the tracking agents are possible to be out of order over the tracking course. Thus we propose a cooperative relay tracking strategy to ensure the successful tracking with the existence of damaged agents. In this case, tracking errors jump and the dimension of Laplacian matrix of the multiagent system (MAS) changes, which increases the difficulty of analysis. In order to solve this issue, a new type of average Lyapunov function is constructed to compensate the unmatched dimension of communication topology. Thereafter, conditions guaranteeing the stability of each subsystem and overall stability of the switched tracking system are obtained.

8.1

Introduction

We have presented some results on cooperative relay tracking for first-order MASs in previous chapters. In these two chapters the monitored domain is divided into many Voronoi cells with the assistance of knowledge of Voronoi diagrams. Then, when a target enters a new Voronoi cell, the Consensus Tracking of Multi-agent Systems with Switching Topologies. DOI: https://doi.org/10.1016/B978-0-12-818365-6.00008-2 © 2020 Elsevier Inc. All rights reserved.

151

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Consensus Tracking of Multi-agent Systems with Switching Topologies

Voronoi site agent replaces one of the original tracking agents. Furthermore, the overall stability conditions for a tracking system being unstable over part of time intervals are obtained. The stability conditions are established on the 2 basis that the norm
of tracking errors
2  2jjEðtÞjj decreases at every switching 2 1 time tk , that is, jjE tk jj # μjjE tk jj ; 0 , μ , 1. The condition 0 , μ , 1 is guaranteed by the switching law that one of the original tracking agents is replaced by the Voronoi site agent when the target enters its Voronoi cell. However, the condition 0 , μ , 1 is not easy to be always satisfied, such as in applications for protecting sensitive areas against offensive intrusion [1] and occasionally tracking targets in a typical military, environmental, or habitat monitoring areas [2]. In practical applications, along with the moving of the target, one or some of the tracking agents may be damaged due to different causes. For example, the tracking agents could be attacked, out of power, or in disorder. When one or some of the tracking agents are damaged, the equal numbers of the other deployed agents are supposed to join the tracking. Under this circumstance the tracking error is likely to jump up at the switching times, that is, μ . 1, differing from the basis condition 0 , μ , 1 in previous chapters. Therefore the theorems achieved in Chapter 6, Cooperative relay tracking strategy for multiagent systems with assistance of Voronoi diagrams, and Chapter 7, Stability of a class of multiagent relay tracking systems with unstable subsystems, are inapplicable for this situation. The replacement of damaged tracking agents triggers the change of dynamic tracking topology and possibly causes jump up of the tracking error. On the other hand, although there are numerous results on MASs with switching topologies, most of the results in the literature are based on the assumption that the tracking agents are fixed. Replacement of tracking agents is not considered, and the number of tracking agents is always the same as that at the initial time. However, the tracking agents may not be the same during the whole tracking process, and it is possible that the number of tracking agents changes because of various reasons. Some of the tracking agents may be called to implement other tasks, without new agents joining the tracking, causing jump of the summation norm of tracking errors. Therefore it is no longer appropriate to analyze the system with the norm of all the tracking errors. The existing results on cooperative tracking of MASs with fixed or switching topologies are not applicable to the tracking problem with variable number of tracking agents. The practical situation addressed here motivates us to explore the tracking problem of MASs with damaged agents and time-varying number of tracking agents in this chapter.

8.2

Relay tracking systems with damaged agents

First, compared with the previous works, a more general case of tracking problem is explored, and the practical problem is modeled in mathematical

Multiagent relay tracking systems Chapter | 8

153

form, precisely describing the switch of tracking agents and the arise of jump errors. Second, a new selection rule of the relay agent considering moving direction of the target is proposed for the second-order MASs. Finally, the stability for the relay tracking system with jump errors at switching instants ðμ . 0Þ is analyzed.

8.2.1

Problem formulation

The dynamics of the ith agent deployed on the monitored area is described as x_i ðtÞ 5 vi ðtÞ v_i ðtÞ 5 f ðt; xi ðtÞ; vi ðtÞÞ 1 ui ðtÞ; t $ t0 xi ðt0 Þ 5 xi0 ; vi ðt0 Þ 5 vi0 ; t0 $ 0;

ð8:1Þ

where xi ðtÞAℝ2 is the position state of the ith agent, vi ðtÞAℝ2 is the velocity state of the ith agent, f ðt; xi ðtÞ; vi ðtÞÞ is a nonlinear function, and ui ðtÞAℝ2 is the control input of the ith agent. i 5 1; 2; . . .; Nf and Nf is the number of tracking agents. The agents are expected to monitor this area and track any intruded target. We consider the following dynamics of the target. x_t ðtÞ 5 vt ðtÞ v_t ðtÞ 5 f ðt; xt ðtÞ; vt ðtÞÞ; t $ t0 ; xt ðt0 Þ 5 xt0 ; vt ðt0 Þ 5 vt0 ; t0 $ 0

ð8:2Þ

where xt ðtÞAℝ2 and vt ðtÞAℝ2 are position and velocity state of the target, respectively. f ðt; xt ðtÞ; vt ðtÞÞ denotes the nonlinear dynamics of the target. In this chapter, we assume f ðUÞ satisfies the Lipschitz Assumption A.1. The following definition is employed to obtain the main results. Definition 8.1: For second-order multiagent tracking problems, not only the position but also the velocity of tracking agents are required to be consensus with the target. Then, it is said the tracking agents successfully track the target if the following conditions satisfy. lim jjxi ðtÞ 2 xt ðtÞjj 5 0;

t-N

lim jjvi ðtÞ 2 vt ðtÞjj 5 0;

t-N

i 5 1; 2; . . .; Nf :

ð8:3Þ

It should be noted that for the first-order MASs, success of tracking and capture share the same meaning, requiring limt-N jjxi ðtÞ 2 xt ðtÞjj 5 0. However, for the second-order MASs, the definitions of tracking and capture are different. limt-N jjxi ðtÞ 2 xt ðtÞjj 5 0 guarantees the success of capture without any requirement on the velocities of agents, whereas the success of

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Consensus Tracking of Multi-agent Systems with Switching Topologies

tracking for second-order systems needs both the positions and velocities to be the same eventually.

8.2.2

Relay tracking algorithm

To tackle the issue of multiagent tracking systems with damaged agents, we propose a relay tracking algorithm. The proposed relay tracking algorithm is addressed as follows: 1. A number of agents capable of sensing and communicating are randomly deployed on a designed area. This area could be any interested place in need of monitoring and tracking the intruded targets. 2. Once a target intrudes this area, the nearest Nf agents are set to track the intruded target. It should be noted that at each instant of time, only Nf agents are assigned the task of tracking the target, whereas all the other agents in this region remain stationary. 3. During the tracking process, one or some of the tracking agents may run out of power or be attacked. The relay is triggered by the occurrence of damaged agents. In order to guarantee successful tracking of the target, equivalent number of agents (relay agents) are set to replace the original tracking agents and join the tracking task. 4. The selection of relay agents is subject to the guidance rule illustrated in Fig. 8.1. The one (agent 2) on the direction, toward which the target is moving to, is selected to replace the original agent, even though it is farther away from the target than agent 1. Intuitively, this guidance rule is in favor of reducing the tracking time. Also, it may affect the average dwell time, and some detailed explanations are given in Remark 8.3. 5. Repeat procedure 3 until the target is caught.

FIGURE 8.1 Guidance rule for selecting the relay agent. (A) Illustration of original tracking agents and topology. (B) Illustration of relay tracking agents and switching topology.

Multiagent relay tracking systems Chapter | 8

155

Remark 8.1: The stationary agent itself determines whether it replaces the damaged tracking agent or not. When one of the tracking agents is damaged, its neighboring tracking agents cannot receive its information anymore. Then, the neighboring tracking agents would know that a damaged agent occurs, and they broadcast to agents nearby. This is realizable with communication technique by using an identification label or other methods. In this chapter, both position and velocity information are able to be sensed. Thus the nearby agents can easily calculate the angle between its location and the moving direction of the target. Then the nearby agents broadcast the angle to others; after comparison, the original stationary agent with minimum angle to the moving direction of the target joins tracking. Let N ðt0 ; tÞ denote the switching times during the time interval ½t0 ; tÞ: It satisfies the average dwell time property (8.4). N ðt0 ; tÞ # N0 1

t 2 t0 ; τa

ð8:4Þ

in which N0 is a positive number and τ a . 0 represents the average dwell time.

8.2.3

Controller design and stability analysis

It is assumed that agent i can only receive the state information (position and velocity) of its neighbors and target, then we adopt the following format of control protocol for the ith tracking agent. X 
  ui ð t Þ 5 2 α aij ðσðtÞÞ exij ðtÞ 1 evij ðtÞ 1 bi ðσðtÞÞðexi ðtÞ 1 evi ðtÞÞ ; ð8:5Þ jAN i ðtÞ

where α . 0 is the control parameter to be designed. exij ðtÞ 5 xi ðtÞ 2 xj ðtÞ and evij ðtÞ 5 vi ðtÞ 2 vj ðtÞ are the position and velocity disagreement vectors between the ith tracking agent and the jth tracking agent. exi ðtÞ 5 xi ðtÞ 2 xt ðtÞ and evi ðtÞ 5 vi ðtÞ 2 vt ðtÞ are the position and velocity disagreement vectors between the ith tracking agent and target. When the original tracking agents quit due to malfunction, a new agent joins tracking according to the relay tracking algorithm. Since the positions of the new agent and the original agent are different and due to the limitation of communication range, the topology switches as described in solid line in Fig. 8.1. Correspondingly, the Laplacian matrix of the topology changes. At the same time, the norm of tracking error E(t) decreases or increases at every switching time tk . The element in E(t) corresponding to the replaced tracking agent jumps.
For  instance,
2 if the
ith1 tracking
agent  1 2 is replaced at time t , then jje t jj 6 ¼ jje k xi xi k
1
2
1
2tk jj;jjevi tk jj 6¼ jjevi tk jj 6 i; i; jAN f . Tracking while jjexj tk jj 5 jjexj tk jj, jjevj tk jj 5 jjevj tk jj; j ¼ error jumps up or jumps down at switching time tk , which implies that

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Consensus Tracking of Multi-agent Systems with Switching Topologies

V(t)

0

t1

t2

t3

t4

t5

t

FIGURE 8.2 Illustration of tracking error jumps.






  jjEx tk1 jj2 1 jjEv tk1 jj2 # μk jjEx tk2 jj2 1 jjEv tk2 jj2 ; μk . 0. As depicted in Fig. 8.2, it is more realistic that both cases exist during a whole tracking course. Then, from Definition 8.1, the relay tracking problem can be interpreted as stability problem of the following switching system with jump errors. E_ x ðtÞ 5 Ev ðtÞ E_ v ðtÞ 5 F ðt; Ex ðtÞ; Ev ðtÞÞ 2 αLk ðEx ðtÞ 1 Ev ðtÞÞ; t 6¼ tk




  jjEx tk1 jj2 1 jjEv tk1 jj2 # μk jjEx tk2 jj2 1 jjEv tk2 jj2 ;

ð8:6Þ

μk . 0; t 5 tk ; kAℕ: h iT h iT where Ex ðtÞ 5 eTx1 ðtÞ; eTx2 ðtÞ; . . .; eTxNf ðtÞ ; Ev ðtÞ 5 eTv1 ðtÞ; eTv2 ðtÞ; . . .; eTvNf ðtÞ : Lk 5 ðℒk 1 Bk Þ  I2 is the associated Laplacian matrix of the dynamic graph,  including the target in time interval ½tk ; tk11 Þ.F ðt; Ex ðtÞ; Ev ðtÞÞ 5 where fi ðtÞ 5 f ðt; xi ðtÞ; vi ðtÞÞ 2 f ðt; xt ðtÞ; vt ðtÞÞ; col f1 ðtÞ; f2 ðtÞ; . . .; fNf ðtÞ i 5 1; 2; . . .; N f is the collective nonlinear self-dynamic disagreement to the target at time t. tk is the switching time, tk2 indicates the time before switching, and tk1 represents the time after switching. Remark 8.2: Theoretical results achieved in this chapter can be applied but not limited to the practical situation with damaged agents. The situation with damaged agents is given to illustratively explain the motivation of this chapter. The theoretical results are able to be applied to any situation fitting the features of replacing tracking agents and jumps of tracking errors. From the abovementioned analysis the relay tracking system is organized as stability problem of a switching system with jump errors. To solve the relay tracking problem, two steps are involved: first step is to prove the stability of consensus tracking problem for a nonlinear dynamic and fixed network graph in each interval of ½tk ; tk11 Þ. The second step is to prove the

Multiagent relay tracking systems Chapter | 8

157

overall stability problem of switched MAS with tracking error jumps at each switching instant. Some techniques solving consensus tracking problem for a nonlinear dynamic and fixed network graph in each interval of ½tk ; tk11 Þ are employed [37]. In Refs. [35] the authors investigate consensus tracking for MASs with nonlinear dynamics under fixed topology. This is only equivalent to the first step in our manuscript. Specifically, in Ref. [3], Schur complement lemma and linear matrix inequality technique are adopted in the proof of theorems. The tracking problem with switching topologies worth further exploring is also claimed in the future work by Song et al. [4]. In Refs. [6,7] the stability problem for second-order nonlinear MASs with switching topologies is explored. However, it is assumed there is no replacement of tracking agents, and the jump of tracking errors at each switching instant are not considered. To some extent, this chapter makes more contribution than the existing literature. Before analyzing the stability of the multiagent tracking system, definitions of two kinds of stabilities are given. Definition 8.2: [8] Let x 5 0 be an equilibrium point for x_ 5 f ðt; xÞ. Let V be a positive and differentiable function such that V_ # 2 λV, where λ is a positive constant. Then, x 5 0 is exponentially stable. If the assumptions hold globally, x 5 0 is globally and exponentially stable. Definition 8.3: [9] The switched system x_ 5 fσ ðt; xÞ is said to be uniformly asymptotically stable, if there exists a class Kℒ function fKℒ such that for all switching signals σ the solutions of x_ 5 fσ ðt; xÞ satisfy the inequality jjxðtÞjj # fKℒ ðjjxðt0 Þ; tjjÞ. If there exists a positive constant μ such that Vp ðxÞ # μVq ðxÞ; p; qAℕ holds globally in the state space, the globally asymptotic stability can be established. It should be noted that the term “uniform” is used here to describe uniformity with respect  to switching T signals. Set EðtÞ 5 ExT ðtÞ; EvT ðtÞ . Then, the system (8.6) can be rewritten as follows: E_ ðtÞ 5 M1 EðtÞ 1 M2 F ðt; Ex ðtÞ; Ev ðtÞÞ where

 M1 5

0 2αLk

ð8:7Þ

 I 0 ; ; M2 5 I 2αLk

with 0 and I in appropriate dimensions. The following theorem gives sufficient conditions ensuring the globally exponential stability for the subsystem of the tracking system (8.7) during the time interval ½tk ; tk11 Þ.

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Consensus Tracking of Multi-agent Systems with Switching Topologies

Theorem 8.1: Consider the multiagent system (8.7) over time interval ½tk ; tk11 Þ; kAℕ. If its communication topology is connected and at least one tracking agent has access to the target’s information, the corresponding subsystem is exponentially stable if inequality (8.8) holds.   2β 2 5 sup λmax fΩk g 1 β 1 , 0; kAℕ; ð8:8Þ where



22αLk Ωk 5 2αLk 1 I

2αLk 1 I ; 22αLk

β 1 5 maxf3lx 1 lv ; 3lv 1 lx g, and I is an identity matrix with appropriate dimensions. Proof: Let us select the Lyapunov candidate for the tracking system (8.7) during the time interval ½tk ; tk11 Þ as Vk ðtÞ 5 ET ðtÞPk EðtÞ; tA½tk ; tk11 Þ; where

 Pk 5

ð8:9Þ

kAℕ;

αLk 1 I I : I I

In order to guarantee the positive definitiveness of Pk , according to Schur complement, it is required that αLk 1 I 2 I . 0: Straightforwardly, Pk . 0 requires Lk . 0: As it is assumed that the communication topology is connected and at least one tracking agent has access to the target’s information, the Laplacian matrix Lk . 0 as analyzed in Lemma A.5. Then, it has λ1 jjEðtÞjj2 # Vk ðtÞ # λ2 jjEðtÞjj2 ; tA½tk ; tk11 Þ;

kAℕ;

ð8:10Þ

where λ1 5 inf fλmin fPk gg; λ2 5 supfλmax fPk gg; kAℕ. For any tA½tk ; tk11 Þ the upper right-hand Dini derivative of V ðtÞ along the trajectory (8.7) is D1 Vk ðtÞ

T 5 ET ðtÞPk E_ ðtÞ 1 E_ ðtÞPk EðtÞ T 5 E ðtÞPk ½M1 EðtÞ 1 M2 F ðt; Ex ðtÞÞ 1 T ½M1 EðtÞ1M  2 F ðt; Ex TðtÞ; E  v ðtÞÞ PkTEðtÞ T 5 E ðtÞ Pk M1 1 M1 Pk EðtÞ 1 E ðtÞPk M2 F ðt; Ex ðtÞ; Ev ðtÞÞ 1 F T ðt; Ex ðtÞ; Ev ðtÞÞM2T Pk EðtÞ ð8:11Þ

Observe Ωk

5 Pk M1 1 M1T Pk 2αLk 1 I 22αLk 5 2αLk 1 I 22αLk

ð8:12Þ

Multiagent relay tracking systems Chapter | 8

Then, it has

  ET ðtÞ Pk M1 1 M1T Pk EðtÞ # λmax fΩk gjjEðtÞjj2 :

159

ð8:13Þ

The second part of (8.11) can be derived as T ET ðtÞPk M2 F ðt; Ex ðtÞ; Ev ðtÞÞ 1 F Tðt; E x ðtÞ; Ev ðtÞÞM2 Pk EðtÞ
 αLk 1 I I 0 F ðt; Ex ðtÞ; Ev ðtÞÞ 1 5 ExT ðtÞEvT ðtÞ I I  I 
 αLk 1 I I Ex ð t Þ T F ðt; Ex ðtÞ; Ev ðtÞÞ 0 I I I Ev ð t Þ T T 5 Ex ðtÞF ðt; Ex ðtÞ; Ev ðtÞÞ 1 Ev ðtÞF ðt; Ex ðtÞ; Ev ðtÞÞ 1 F T ðt; Ex ðtÞ; Ev ðtÞÞEx ðtÞ 1 F T ðt; Ex ðtÞ; Ev ðtÞÞEv ðtÞ

ð8:14Þ

In light of Assumption A.1 and noticing β 1 5 maxf3lx 1 lv ; 3lv 1 lx g, it has ET ðtÞPk M2 F ðt; Ex ðtÞ; Ev ðtÞÞ 1 F T ðt; Ex ðtÞ; Ev ðtÞÞM2T Pk EðtÞ # 2lx jjEx ðtÞjj2 1 2lv jjEv ðtÞjj2 1 ð2lx 1 2lv ÞjjEx ðtÞjj jjEv ðtÞjj # ð3lx 1 lv ÞjjEx ðtÞjj2 1 ð3lv 1 lx ÞjjEv ðtÞjj2 # β 1 jjEðtÞjj2

ð8:15Þ

To obtain the globally exponentially stable condition of the subsystems, it is further derived as the following:
 D1 Vk ðtÞ # λmax fΩk g 1 β 1 jjEðtÞjj2 ð8:16Þ # 2 β 2 jjEðtÞjj2 Considering condition (8.10), it is straightforward that D1 Vk ðtÞ # 2 βVk ðtÞ with β 5 β 2 =λ2 .

ð8:17Þ ’

Theorem 8.1 provides globally exponentially stable conditions for individual subsystems, which is the basis of Theorem 8.2. The overall asymptotic stability of the switched system (8.6) could not be achieved if the individual subsystems do not satisfy the conditions in Theorem 8.1. According to the inequality (8.10) and the third inequality in (8.6), at switching time tk , the Lyapunov candidate complies
 λ2
 Vk11 tk1 # μk Vk tk2 : λ1

ð8:18Þ

Theorem 8.2: Consider the switched multiagent tracking system (8.6) satisfying the conditions in Theorem 8.1. The tracking agents will asymptotically track the target if inequality (8.18) holds and the average dwell time τ a satisfies (8.19).

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Consensus Tracking of Multi-agent Systems with Switching Topologies

 1=k where μ 5 Πki51 μi .


 log λ2 μ=λ1 ; τa . β

Proof: Without loss of generality, let t0 5 0, then it has
 V1 t12 5 e2βt1 V ð0Þ

ð8:19Þ

ð8:20Þ

Considering inequality (8.18), at switching times t1 and t2 , it is derived that
 λ2 μ1
2  λ2 μ1 2βt V2 t11 # V1 t 1 # e 1 V ð 0Þ λ1 λ1  2
 λ2 μ2
2  λ2 V3 t21 # V2 t 2 # μ1 μ2 e2βt2 V ð0Þ: λ1 λ1

ð8:21Þ

Iterating this inequality till tA½tk11 ; tk12 Þ; we have k λ2 μ λ2 μk 2βðt2tk Þ
2  i 2βt e Vk tk # Π e V ð0Þ: ð8:22Þ i51 λ1 λ1  1=k Recall that μ 5 Πki51 μi , then this together with (8.22) implies that  λ2 μ k 2βt Vk11 ðtÞ # e V ð0Þ: ð8:23Þ λ1

Vk11 ðtÞ #

In the light of the dwell time property inequality (8.4), Eq. (8.23) is able to be rewritten as Vk11 ðtÞ # e2βt1ðN0 1ðt=τ a ÞÞlogðλ2 μ=λ1 Þ V ð0Þ 5 eN0 logðλ2 μ=λ1 Þ eððlogðλ2 μ=λ1 Þ=τ a Þ2β Þt V ð0Þ:

ð8:24Þ

From the condition (8.19) in Theorem 8.2, it is straightforward to obtain that Vk11 ðtÞ converges to zero exponentially as t-N. Using the fact that a positive definite function V ðxÞ satisfies fK1 ðjjxjjÞ # V ðxÞ # fK2 ðjjxjjÞ, where fK1 and fK2 are class K functions, we have !  λ2 μ N0 ððlogðλ2 μ=λ1 Þ=τ a Þ2β Þt 21 e fK2 ðjjEð0ÞjjÞ : jjEðtÞjj # fK1 λ1 According to Definition 8.3, the overall tracking error system is asymptotically stable, implying that the tracking error converges to zero. Then from the definition of successful tracking, it is straightforward to conclude that the tracking agents successfully track the target with the proposed tracking strategy. &

Multiagent relay tracking systems Chapter | 8

161



 Remark 8.3: When λ2 μ=λ1 . 1; log λ2 μ=λ1 =β is always greater than 0, suggesting
there is always a minimum value constraint on τ a . However, when 0 , λ2 μ=λ1 # 1, the condition (8.19) is fulfilled without any restrict constraint on τ a . In addition, the selection rule of the relay agents affects the value of μ directly, since different new agents result in different jump errors. As addressed in the relay tracking algorithm, if an original agent is replaced by a new agent on the target’s moving direction with farther distance to the target than another deployed agent on the opposite direction, the value of λ2 μ=λ1 increases while the tracking time decreases. In light of the fact that τ a is determined by β and λ2 μ=λ1 , some optimization work could be performed to balance the tracking time and average dwell time, which is a possible extension of this chapter in the future work.

8.2.4

A numerical simulation

In this section a numerical simulation is carried out to verify the correctness and the effectiveness of the main results derived earlier. The simulation is conducted on the following scenario. For a 1500 m 3 1500 m twodimensional space, Nm agents are randomly dispensed. The sensing radius is set as Rs 5 100 m. When a target intrudes, a group of five agents starts tracking the target. The nonlinear dynamics of the agents and target are f ðt; xðtÞ; vðtÞÞ 5 3:5sinð0:01xðtÞÞ 1 0:2cosð0:02vðtÞÞ 1 0:2sinð2tÞ:

ð8:25Þ

Hence, jjf ðt; xi ; vi Þ 2 f ðt; xt ; vt Þjj # 0:035jjxi 2 xt jj 1 0:004jjvi 2 vt jj. Time of the target breaks into this domain is considered as initial time. The coordinate where the first target enters this area is (0,100). During the tracking process, there is a worst possible case in terms of Lk . It is straightforward that if the control parameter meets condition (8.8) in Theorem 8.1 for the worst case, condition (8.8) satisfies for all other cases. The minimum value of control parameter is thus determined. In this simulation, to guarantee the stability of all subsystems, it is required α . 0:12. Under this design principle, we choose the control parameter as α 5 0:25; thus λ1 5 0:4458; λ2 5 5:7122; β 2 5 2:1410. As illustrated in Fig. 8.3, this area is monitored by a large number of agents (hollow circles). When a target (asterisk) breaks into this area, five agents (solid circles) start tracking the target. When an original agent is out of order, another monitoring agent replaces it to continue tracking, causing the tracking trajectories to be discontinuous. The discontinuity of different lines clearly shows which agent is replaced. The trajectories of tracking agents and the target are described in Fig. 8.3. Fig. 8.4 depicts position and velocity tracking errors. Every switching of topology and tracking agents brings a jump in tracking error. Actually, each discontinuity of a line in Fig. 8.3 corresponds to a jump in Fig. 8.4. Fig. 8.5

Tracking trajectories 1500 Agents Target Agent 1 Agent 2 Agent 3 Agent 4 Agent 5

(m)

1000

500

0

0

500

1000

1500

(m) FIGURE 8.3 Tracking trajectories.

Error trajectories of position

Agent 1 Agent 2 Agent 3 Agent 4 Agent 5

200

(m)

100

0

−100

0

10

20

30 40 t (s) Error trajectories of velocity

50

20

50

40 20 (m/s)

60 Agent 1 Agent 2 Agent 3 Agent 4 Agent 5

0 −20 −40

0

10

FIGURE 8.4 Tracking errors.

30 t (s)

40

60

Multiagent relay tracking systems Chapter | 8

163

Switching signal s(t)

8 7 6 5 4 3 2 1 0

0

10

20

30 t (s)

40

50

60

FIGURE 8.5 Norm of errors and minimum eigenvalues.

TABLE 8.1 Relationship between average dwell time τ a and control parameter α. α

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

τa

12.15

9.27

8.74

9.05

9.57

10.30

10.99

11.19

depicts switching signal σðtÞ. According to the simulation data, it is able to obtain that μ 5 2:0642. To guarantee the overall stability the minimum average dwell time is τ a 5 8:7387 seconds. From the simulation results the tracking agents relay six times over 56 seconds. It is clear that the average dwell time is greater than τ a , which verifies the correctness of Theorem 8.2. Furthermore, the relationship between average dwell time τ a and control parameter α is investigated and tabulated in Table. 8.1. It can be seen that the allowable average dwell time first decreases then increases along with control parameter.

8.3 Relay tracking systems with time-varying number of agents The main contributions of this section are addressed as follows. First, we create a brand new but more realistic tracking scenario where a number of tracking agents changes and establishes its mathematical model. Second, a novel type of Lyapunov function using the average norm of tracking errors

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Consensus Tracking of Multi-agent Systems with Switching Topologies

is designed to analyze the stability of each subsystem. Finally, switched technique is employed to solve the jump of average tracking errors and obtain the stability condition for the overall tracking system.

8.3.1

Preliminaries and problem formulation

Assume that a target holding nonlinear dynamics (8.26) moves in the threedimension space. x_t ðtÞ 5 vt ðtÞ; v_t ðtÞ 5 at ðtÞ; a_t ðtÞ 5 f ðt; xt ðtÞ; vt ðtÞ; at ðtÞÞ; t $ t0 ; xt ðt0 Þ 5 xt0 ; vt ðt0 Þ 5 vt0 ; at ðt0 Þ 5 at0 ; t0 $ 0;

ð8:26Þ

where xt ðtÞAℝ3 , vt ðtÞAℝ3 , and at ðtÞAℝ3 are position, velocity, and acceleration states of the target, respectively. f ðt; xt ðtÞ; vt ðtÞ; at ðtÞÞ decides the nonlinear behavior of the target, satisfying Lipchitz Assumption A.1. Variable agents are committing the task of tracking the target. The dynamics of the ith agent is described as x_i ðtÞ 5 vi ðtÞ; v_i ðtÞ 5 ai ðtÞ; a_i ðtÞ 5 f ðt; xi ðtÞ; vi ðtÞ; ai ðtÞÞ 1 ui ðtÞ; xi ðt0 Þ 5 xi0 ; vi ðt0 Þ 5 vi0 ; ai ðt0 Þ 5 ai0 ;

t $ t0 ; t0 $ 0;

ð8:27Þ

where xi ðtÞAℝ3 is position state of the ith agent, vi ðtÞAℝ3 is velocity state of the ith agent, and ai ðtÞAℝ3 is acceleration state of the ith agent. ui ðtÞAℝ3 is the control input of the ith agent. f ðt; xi ðtÞ; vi ðtÞ; ai ðtÞÞ is the nonlinear behavior of ith agent, sharing Assumption A.1. The relay tracking process with dynamically changing number of agents is illustrated in Fig. 8.6. The target is represented by the diamond, and the tracking agents are described by solid circles. The evolution of the relay tracking process is indicated by lines with arrows. In order to clearly explain the switching process, the sequences are marked. It is assumed there are four different tracking numbers. As shown in Fig. 8.6, six agents are tracking the target initially, then two agents quit tracking. Thus the number of agents apparently turns to be four and so on. After a round of evolution the tracking number turns to be six again. However, the communication topology and the tracking errors are not the same as the initial. This process continues till the target is successfully tracked. In this chapter, the number of tracking agents is variable, implying the communication topology, and node set are not fixed either. Thus the Laplacian matrix is rewritten as ℒσðtÞ 1 BσðtÞ , and the node set turns into NσðtÞ , where σðtÞ 5 tk ; kAℕ. The topology and node set are piecewise constant between switching times t0 ; t1 ; . . .; tk . Denote Nσðtk Þ as the number of tracking agents over time interval ½tk ; tk11 Þ.

Multiagent relay tracking systems Chapter | 8

165

FIGURE 8.6 Illustration of a relay tracking multiagent system with dynamic agents.

It is assumed that agent i is limited to access the state information of its neighbors and target. Then, we adopt the following format of control protocol for the ith tracking agent. ui ðtÞ 5 2 αbi ðexi ðtÞ 1 evi ðtÞ 1 eai ðtÞÞ 2 X 
 α aij exij ðtÞ 1 evij ðtÞ 1 eaij ðtÞ ;

ð8:28Þ

jAN σðtÞ

where α . 0 is the control parameter to be designed. exij ðtÞ 5 xi ðtÞ 2 xj ðtÞ; evij ðtÞ 5 vi ðtÞ 2 vj ðtÞ; eaij ðtÞ 5 ai ðtÞ 2 aj ðtÞ are the position, velocity, and acceleration disagreement vectors between the ith and the jth tracking agent. exi ðtÞ 5 xi ðtÞ 2 xt ðtÞ; evi ðtÞ 5 vi ðtÞ 2 vt ðtÞ; eai ðtÞ 5 ai ðtÞ 2 at ðtÞ are the position, velocity, and acceleration disagreement vectors between the ith tracking agent and the target. Then, the tracking problem can be interpreted as stability problem of the collective tracking error system. For time interval tA½tk ; tk11 Þ; k 5 0; 1; 2; . . .; the collective error system is E_ x ðtÞ 5 Ev ðtÞ; ð8:29Þ E_ v ðtÞ 5 Ea ðtÞ; E_ a ðtÞ 5 F ðt; Ex ðtÞ; Ev ðtÞ; Ea ðtÞÞ 2 αLk ðEx ðtÞ 1 Ev ðtÞ 1 Ea ðtÞÞ; h iT h iT where Ex ðtÞ5 eTx1 ðtÞ;eTx2 ðtÞ;...;eTxNσ t ðtÞ , Ev ðtÞ5 eTv1 ðtÞ;eTv2 ðtÞ;...;eTvNσ t ðtÞ , ð kÞ ð kÞ h iT T T T and Ea ðtÞ5 ea1 ðtÞ;ea2 ðtÞ;...;exNσ t ðtÞ are collective position error, velocity ð kÞ

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Consensus Tracking of Multi-agent Systems with Switching Topologies

error, and acceleration error, respectively. Lk 5 ðℒk 1Bk ÞI3 is the extended Laplacian matrix of the dynamic graph, including the target, in which h iT n o Bk 5diag b1 ;b2 ;...;bNσðt Þ . F ðt;Ex ðtÞ;Ev ðtÞ; Ea ðtÞÞ5 f1 ðtÞ;f2 ðtÞ;...;fNσðt Þ ðtÞ , k

k

where fi ðtÞ5f ðt;xi ðtÞ;vi ðtÞ;ai ðtÞÞ2f ðt;xt ðtÞ; vt ðtÞ;at ðtÞÞ is the collective nonlinear self-dynamic disagreement to the target at time t. During the whole tracking process, part of the tracking agents are likely to quit, while some new agents are possible to join the tracking task. The instant when new agents join and the instant when the original agents quit are not always the same time. This means the number of tracking agents is variable, implying the rank of Laplacian matrix of the multiagent topology is no longer constant. Also, since the number of tracking agents is not constant during the whole tracking course, the dimension of E is not the same over different time intervals. At the switching time tk the average norm of tracking errors satisfies
 2
1 2
1 2
 2
2 2
2 2 jjEx t1 jjEx t2 k jj 1jjEv tk jj 1jjEa tk jj k jj 1jjEv tk jj 1jjEa tk jj #μk Nσðtk Þ Nσðtk21 Þ ð8:30Þ where tk2 indicates the time before switching, tk1 represents the time after switching, and μk is a positive scalar indicating the jump of tracking errors. Next, we introduce some definitions and lemmas, which will play an important role in the proof of our main theorems. Definition 8.4: For the third-order multiagent tracking problem (8.29) with variable number of tracking agents, it is said the tracking agents successfully track the target if there exists a sufficient small scalar δ . 0 such that lim

t-N

NσðtÞ X


 jjexi ðtÞjj2 1 jjevi ðtÞjj2 1 jjeai ðtÞjj2 # δ:

ð8:31Þ

i51

Lemma 8.1: [10] Suppose the system x_ 5 fp ðxÞ switches among m modes, and their associated Lyapunov functions are Vp ðxÞ; p 5 1; 2; . . .; m. The switching times are denoted as t0 ; t1 ; . . .; ti ; . . .. For switching times tj and ti , when the system switches into mode p, if the Lyapunov function Vp satisfies

 ð8:32Þ Vp x tj # Vp ðxðti ÞÞ; where ti , tj , then the switched system x_ 5 fp ðxÞ; p 5 1; 2; . . .; m is stable in the sense of Lyapunov. Remark 8.4: Fig. 8.7 is an illustrative explanation of Lemma 8.1, which implies that a cyclically switched system is stable if the initial Lyapunov

Multiagent relay tracking systems Chapter | 8

167

FIGURE 8.7 Illustrative explanation of Lemma 8.1.

FIGURE 8.8 Illustration of the cyclic switching.

function values for same modes fall. As can be seen in Fig. 8.7, this lemma relaxes the constraints on Lyapunov function values at switching times. Lemma 8.1 is applicable for the case where the modes are cyclically changing rather than stochastically switching. As shown in Fig. 8.8, the modes of switched system are switching cyclically. However, the periods are not required to be identical. In terms of the tracking system, this means the number of tracking agents is different over time intervals but is required to be changing cyclically. As the number of tracking agents changes during the whole tracking course, it is no longer appropriate to measure the tracking performance with the summation of all tracking errors. Therefore for each subsystem, we design the Lyapunov function as the average tracking error in quadratic form: V ðt Þ 5

1 ET ðtÞEðtÞ; 2NσðtÞ

 T where EðtÞ 5 ExT ðtÞ; EvT ðtÞ; EaT ðtÞ .

ð8:33Þ

168

Consensus Tracking of Multi-agent Systems with Switching Topologies

Remark 8.5: As claimed in Ref. [10], Lemma 8.1 can be used for the stability analysis of heterogeneous systems, and the analysis can be based upon the existence of any type of Lyapunov function. In this chapter the dimension of the state variable is piecewise constant but time-varying over different time intervals. The subsystems with different dimensions of the state variable essentially are heterogeneous subsystems. Namely, the overall system is a switched system between heterogeneous subsystems. Also, the Lyapunov function exists since we design an average norm of tracking errors as Lyapunov function. Therefore Lemma 8.1 is applicable to the switched system where dimension of the state variable is time-varying.

8.3.2 Main results of relay tracking systems with time-varying number of agents The following theorem gives sufficient condition ensuring the stability for the subsystem of the tracking system (8.29) during the time interval ½tk ; tk11 Þ. Theorem 8.3: Consider the multiagent system (8.29) with connected undirected topology over the time interval ½tk ; tk11 Þ; kAℕ. If the control parameter α satisfies α $ supfmaxfΦk1 ; Φk2 ; Φk3 gg; ð8:34Þ 
 in which Φk1 5 3Nσðtk Þ 1 lx =λmin ðLk Þ; Φk2 5 6Nσðtk Þ 1 lv =λmin ðLk Þ; and
Φk3 5 3Nσðtk Þ 1 lx 1 lv 1 2la =4λmin ðLk Þ; the corresponding subsystem is stable.


Proof: Let us select the Lyapunov candidate for the tracking system (8.29) during the time interval ½tk ; tk11 Þ as V ðt Þ

1 ET ðtÞEðtÞ 2Nσðtk Þ  1
T 5 Ex ðtÞEx ðtÞ 1 EvT ðtÞEv ðtÞ 1 EaT ðtÞEa ðtÞ ; 2Nσðtk Þ tA½tk ; tk11 Þ; kAℕ: 5

ð8:35Þ

For any tA½tk ; tk11 Þ the upper right-hand Dini derivative of V ðtÞ along the trajectory (8.29) is D1 V ðtÞ

5

1 

 ExT ðtÞEv ðtÞ 1 EvT ðtÞEa ðtÞ 1 EaT ðtÞE_ a ðtÞ

Nσðtk Þ 1  T 5 E ðtÞEv ðtÞ 1 EvT ðtÞEa ðtÞ 1 Nσðtk Þ x

 EaT ðtÞ½F ðt; Ex ðtÞ; Ev ðtÞ; Ea ðtÞÞ 2 αℒk ðEx ðtÞ 1 Ev ðtÞ 1 Ea ðtÞÞ : ð8:36Þ

169

Multiagent relay tracking systems Chapter | 8

In light of Assumption A.1, we have D1 V ðtÞ 1  # jjEx ðtÞjj jjEv ðtÞjj 1 jjEv ðtÞjj jjEa ðtÞjj 1 Nσðtk Þ

ð8:37Þ

ðlx 2 αλmin ðℒk ÞÞjjEa ðtÞjj jjEx ðtÞjj 1  ðlv 2 αλmin ðℒk ÞÞjjEa ðtÞjj jjEv ðtÞjj 1 ðla 2 αλmin ðℒk ÞÞjjEa ðtÞjj2 : From condition (8.34), it is apparent that lx 2 αλmin ðLk Þ , 0; lv 2 αλmin ðLk Þ , 0; la 2 αλmin ðLk Þ , 0: Considering the fact that for any pffiffiffi vector XAℝn , it has jjXjjN # jjXjj # njjXjjN ; in this chapter, 3N Ex ; Ev ; Ea Aℝ σðtk Þ , subsequently, the abovementioned equation is able to be derived as
 1 
3Nσðtk Þ 1 lx 2 αλmin ðLk Þ jjEx ðtÞjj2N 1 6Nσðtk Þ 2Nσðtk Þ 
 1 lv 2 αλmin ðLk ÞÞjjEv ðtÞjj2N 1 3Nσðtk Þ 1 lx 1 lv 1 2la 2 4αλmin ðLk Þ jjEa ðtÞjj2N :

D1V ðtÞ #

ð8:38Þ From condition (8.34), it is straightforward to conclude that D1 V ðtÞ is nonpositive, which leads to the stability of subsystem over time interval ½tk ; tk11 Þ. & Denote

  2 β k 9sup max 3Nσðtk Þ 1 lx 2 αλmin ðLk Þ;

6Nσðtk Þ 1 lv 2 αλmin ðLk Þ;  3Nσðtk Þ 1 lx 1 lv 1 2la 2 4αλmin ðLk Þ ; ð8:39Þ

then we have D1 V ðtÞ # 2 β k V ðtÞ:

ð8:40Þ

V ðtÞ # 2 V ðtk Þe2β k t :

ð8:41Þ

Thus for tA½tk ; tk11 Þ,

Here, we are ready to analyze the overall stability of the switched tracking system. Without the loss of generality and for simplicity, it is assumed there are two modes switch and V1 ðt0 Þ 5 V ðt0 Þ: Theorem 8.4: Based on the condition in Theorem 8.3, suppose that τ 1 β 1 1 τ 2 β 2 $ lnμ1 μ2 ð8:42Þ     Hold, where μ1 5 max μ2k21 ; μ2 5 max μ2k , τ 1 5 minfτ 2k21 g; τ 2 5 minfτ 2k g; k 5 1; 2; . . . with τ 2k21 5 t2k21 2 t2ðk21Þ; τ 2k 5 t2k 2 t2k21 : Then, the tracking agents described in (8.27) are able to successfully track the target described in (8.26).

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Consensus Tracking of Multi-agent Systems with Switching Topologies

Remark 8.6: In this chapter, we consider the case where μ2k21 and μ2k are bounded with known constant which is a reasonable assumption in  bounds,  reality. We use μ1 5 max μ2k21 to denote the bound of μ2k21 , that is, μ2k21 # μ1 for all kAℕ1 . Correspondingly, μ2 5 max μ2k denotes the bound of μ2k , that is,μ2k # μ2 for all kAℕ1 . Therefore the condition in Theorem 8.4 is independent of k. Proof: Without loss of generality, let t0 5 0, then it has V1 ðt2k Þ # μ2k V2 ðt2k Þ 5 μ2k V2 ðt2k21 Þe2τ 2k β 2 # μ2k21 μ2k V1 ð
t2k21 Þe2τ 2k β2 5 μ2k21 μ2k V1 t2ðk21Þ e2ðτ 2k β 2 12τ 2k21 β1 Þ :

ð8:43Þ

According to Lemma 8.1, to ensure the stability of the switched multiagent tracking system, it requires μ2k21 μ2k e2ðτ 2k β2 1τ 2k21 β 1 Þ # 1:

ð8:44Þ

After mathematical derivation, it follows from (9.70) that τ 2k β 2 1 τ 2k21 β 1 $ 1nμ2k21 μ2k :

ð8:45Þ

For V2 we have V2 ðt2k11 Þ

# μ2k11 V1 ðt2k11 Þ 5 μ2k11 V1 ðt2k Þe2τ 2k11 β1 # μ2k μ2k11 V2 ðt2k Þe2τ 2k11 β1 5 μ2k μ2k11 V2 ðt2k21 Þe2ðτ 2k β2 1τ 2k11 β 1 Þ :

ð8:46Þ

From Lemma 8.1, V2 needs to meet the same requirement that V2 ðt2k11 Þ # V2 ðt2k21 Þ. After similar derivation, it is straightforward to obtain τ 2k β 2 1 τ 2k11 β 1 $ lnμ2k μ2k11 : ð8:47Þ    Since μ1 5 max μ2k21 ; μ2 5 max μ2k ; τ 1 5 minfτ 2k21 g; τ 2 5 minfτ 2k g; k 5 1; 2; . . .; conditions (8.45) and (8.47) can be organized in a consolidated form as (8.42). Then, given Lemma 8.1, it can be concluded that the overall error system (8.29) is stable in the sense of Lyapunov, implying the tracking agents (8.27) successfully track the target (8.26). This completes the proof. & 

Remark 8.7: It is worth noting that the number of V ðtÞ is not limited to two but can be any finite number m. However, the modes must be cyclic as illustrated in Fig. 8.8 rather than stochastically switching. As shown in Fig. 8.8, modes are switching cyclically, but each mode is not necessary to last for identical periods. In this case, condition (8.42) in Theorem 8.4 turns into

Multiagent relay tracking systems Chapter | 8 m X



m

171



τ p β p $ ln Π μi

p51

p51

n o   with τ p 5 min τ mk2ðm2pÞ ; μp 5 max μmk2ðm2pÞ . In fact, the number of V ðtÞ is ought to be determined according to the actual situation. This is not less conservative than that with two modes. However, the results obtained in this chapter are less conservative than those in which a common Lyapunov function is involved for a switched system. The derivation is similar to that of two modes. Therefore in this chapter, it is assumed there are two modes switch for the sake of simplicity. Remark 8.8: The results in this chapter can be easily extended to the nonlinear systems with unknown but bounded disturbances as considered in Ref. [11]. However, since the results achieved in this chapter depend on the Lipschitz constants of nonlinear dynamics, it is not able to extend the results to systems with unknown nonlinear dynamics as considered in Ref. [12]. Remark 8.9: According to (8.39), it is obvious that the value of β k is closely related to the designed parameter α. With satisfying condition (8.34) in Theorem 8.3, the increase of designed parameter α leads to the increase of β k . Then according to condition (8.42) in Theorem 8.4, for fixed μ1 ; μ2 , which are independent of control parameter α, the increase of β relaxes the constraint on dwell time τ. Therefore it is able to conclude that designed parameter α affects the minimum allowable dwell time of the switched system. With a sufficiently large α the system is allowed to switch quickly with smaller dwell times. However, it should be noted that, in practical applications, the control input is always subject to saturation due to the constraint of actuators. Thus the designed control parameter also needs to compromise with the specific application constraints.

8.3.3

A numerical simulation

A numerical simulation is carried out to illustrate the previous developments. In the simulation, it is assumed that during the tracking process, number of tracking agents switch between three and five. Initially, three agents start tracking the target. The initial position, velocity, and acceleration of the target in three-dimensional space are xt ð0Þ 5 ½0; 0; 0T ; vt ð0Þ 5 ½0:25; 0:8; 0:5T ; at ð0Þ 5 ½20:2; 0:3; 0:4T , respectively. The initial positions of three tracking agents in three-dimensional space are x1 ð0Þ 5 ½3; 22:8; 1:5T ; x2 ð0Þ 5 ½22:5; 24:6; 4T ; x3 ð0Þ 5 ½5; 20:5; 23T , respectively. The initial velocities of three tracking agents in three-dimensional space are v1 ð0Þ 5 ½20:5; 1:5; 21:2T ; v2 ð0Þ 5 ½1:6; 0:5; 20:7T ; v3 ð0Þ 5 ½21:2; 20:6; 0:3T , respectively. The initial accelerations of tracking agents are 0.

172

Consensus Tracking of Multi-agent Systems with Switching Topologies

The nonlinear dynamics imposing on the target and agents is a jerk system with chaotic behavior. The nonlinear function is specified as [13] f ðUÞ 5 2 0:6aðtÞ 2 vðtÞ 1 jxðtÞj 2 1:

ð8:48Þ

The function f ðUÞ satisfies Assumption A.1 with lx 5 1; lv 5 1; la 5 0:6. In this numerical example λmin ðLk Þ 5 2: Then, based on the condition (8.34) in Theorem 8.3, the restricted condition on control parameter α is α $ 15:5. We choose α 5 18 to make sure the stability of each subsystem. The summation norm of tracking errors and switch of tracking number are depicted in Fig. 8.9. In this figure, stars and circles are instants when three and five agents start tracking the target correspondingly matching the switch of tracking number. As clearly illustrated in Fig. 8.9, the starting values of mode 1 where the tracking number is 3 (i.e., the values where the stars stand) decrease, so do the starting values of mode 2 where the tracking number is 5 (i.e., the values where the circles stand). However, the starting values of modes 1 and 2 are not necessarily to decrease. To further prove the effectiveness and feasibility of proposed control strategy, the tracking errors of position, velocity, and acceleration are also given in Figs. 8.10, 8.11, and 8.12, respectively.

8.4

Conclusion

This chapter provides a relay tracking solution for successful tracking of an intruded target while damaged agents occur. Several tracking agents

Norm of error

60 40 20 0

0

5

10

15

20

25

15

20

25

Number of tracking agents

Time (s) 6

4

2

0

0

5

10 Time (s)

FIGURE 8.9 Summation of norm of tracking errors.

Multiagent relay tracking systems Chapter | 8

173

Error trajectories of position x-Axis (m)

5 0 −5

0

5

10

15

20

25

Time (s) y-Axis (m)

10 0 −10

0

5

10

15 Time (s)

z-Axis (m)

10

Agent 1 20 Agent 2 Agent 3 Agent 4 Agent 5

25

0 −10

0

5

10

15

20

25

15

20

25

15

20

15

20

Time (s) FIGURE 8.10 Tracking errors of position.

x-Axis (m/s)

Error trajectories of velocity 5 0 −5

0

5

10

y-Axis (m/s)

Time (s) 5 0 −5

0

5

10

z-Axis (m/s)

Time (s) 5

Agent 1 Agent 2 25 Agent 3 Agent 4 Agent 5

0 −5

0

5

10 Time (s)

FIGURE 8.11 Tracking errors of velocity.

25

174

Consensus Tracking of Multi-agent Systems with Switching Topologies

x-Axis (m/s2)

Error trajectories of acceleration 5 0 −5

0

5

10

15

20

25

y-Axis (m/s2)

Time (s) 10 0 −10

0

5

10

15

z-Axis (m/s2)

Time (s) 10

Agent 1 Agent 2 20 Agent 3 Agent 4 Agent 5

25

0 −10 0

5

10

15

20

25

Time (s) FIGURE 8.12 Tracking errors of acceleration.

cooperate and communicate with each other when they are assigned to track the same target. The switching of topologies and relay of tracking agents are triggered when a tracking agent is damaged. Our proposed tracking strategy guarantees overall successful tracking with the tolerance of decreases and arises of tracking errors. One possible extension of this work is to perform some optimization work to balance the tracking time and average dwell time. Furthermore, a class of multiagent tracking systems with variable number of agents is investigated. As analyzed, the stability of the switched system with variable tracking number is related to the balance of dwell time and jump of average norm of tracking errors.

References [1] Y. Zhang, Y. Meng, A decentralized multi-robot system for intruder detection in security defense, in: IEEE/RSJ International Conference on Intelligent Robots and Systems, Taipei, 2010, pp. 55635568. [2] Y. Xiao, Y. Zhang, Surveillance and tracking system with collaboration of robots, sensor nodes, and RFID tags, in: Proceedings of 18th International Conference on Computer Communications and Networks, San Francisco, CA, 2009, pp. 16. [3] Y. Zhao, Z. Li, Z. Duan, Distributed consensus tracking of multi-agent systems with nonlinear dynamics under a reference leader, Int. J. Control 86 (10) (2013) 18591869. [4] Q. Song, J. Cao, W. Yu, Second-order leader-following consensus of nonlinear multi-agent systems via pinning control, Syst. Control Lett. 59 (9) (2010) 553562.

Multiagent relay tracking systems Chapter | 8

175

[5] R. Dong, Consensus tracking for multiagent systems with nonlinear dynamics, Sci. World J. 2014 (2014) 110. [6] G. Wen, Y. Yu, Z. Peng, A. Rahmani, Consensus tracking for second-order nonlinear multi-agent systems with switching topologies and a time-varying reference state, Int. J. Control 89 (10) (2016) 20962106. [7] W. Liu, S. Zhou, Y. Qi, X. Wu, Leaderless consensus of multi-agent systems with Lipschitz nonlinear dynamics and switching topologies, Neurocomputing 173 (2016) 13221329. [8] H.K. Khalil, Nonlinear Systems, third ed., Publishing House of Electronics Industry, Beijing, 2012. [9] D. Liberzon, Switching in Systems and Control, Springer Science 1 Business Media, New York, 2003. [10] R. Shorten, F. Wirth, O. Mason, K. Wulff, C. King, Stability criteria for switched and hybrid systems, Siam Rev. 49 (4) (2007) 545592. [11] X. Wang, D. Xu, H. Ji, Robust almost output consensus in networks of nonlinear agents with external disturbances, Automatica 70 (C) (2016) 303311. [12] H. Zhang, F.L. Lewis, Adaptive cooperative tracking control of high-order nonlinear systems with unknown dynamics, Automatica 48 (7) (2012) 14321439. [13] S.J. Linz, J.C. Sprott, Elementary chaotic flow, Phys. Lett. A 259 (3) (1999) 240245.

Chapter 9

Multiagent relay tracking systems with time-varying number of agents and time delays Chapter Outline 9.1 Introduction 177 9.2 Linear relay tracking systems with time-varying number of agents and time delays 178 9.2.1 Problem formulation 178 9.2.2 Main results 181 9.2.3 Numerical simulations 189

9.3 Nonlinear relay tracking systems with time-varying number of agents and time delays 9.3.1 Related preliminaries 9.3.2 Stability analysis 9.3.3 Numerical simulations 9.4 Conclusion and discussions References

192 192 195 200 204 206

Time-varying number of tracking agents consequently results in jump of tracking error and variable dimension of Laplacian matrix of the multiagent systems (MASs), which involves significant difficulty on stability analysis especially with the effect of communication time delays. This chapter is a further extension of the relay tracking systems introduced in Chapter 8, Multiagent relay tracking systems with damaged agents and time-varying number of agents. In the current framework the tracking agents are dynamically changing with a varying number of tracking agents over different time intervals. Another focus of this work is to develop novel stability conditions for relay tracking systems with time delays for both linear and nonlinear MASs.

9.1

Introduction

Communication time delays between agents are not considered in Chapter 8. However, this issue cannot be ignored in real applications. It is worth further investigating how time delays affect the performance of relay tracking multi-agent systems with switching number of agents. However, due to the Consensus Tracking of Multi-agent Systems with Switching Topologies. DOI: https://doi.org/10.1016/B978-0-12-818365-6.00009-4 © 2020 Elsevier Inc. All rights reserved.

177

178

Consensus Tracking of Multi-agent Systems with Switching Topologies

distinct features of relay tracking systems with variable number of agents, the existing results on switched time-delay multi-agent systems in the literature cannot be directly applied to analyze the cooperative relay tracking system with dynamically changing number of agents and time delays. Therefore this chapter investigates both linear and nonlinear multi-agent relay tracking systems with time-varying number of agents under the effect of communication time delays. First, the linear relay tracking systems with time-varying number of agents and time delays are investigated in Section 9.2. The contributions of this section are threefold: (1) discontinuity of tracking errors at the switching times is analyzed for different cases with a varying number of agents. (2) A novel impulse-time-dependent average Lyapunov function is proposed for analyzing the stability of the complicated multiagent relay tracking system. (3) Sufficient conditions guaranteeing successful tracking of targets under a varying number of agents and time delays are given in terms of switching time intervals, control parameter, and time delay bound. Then, we further discuss this issue where agents are with nonlinear dynamics in Section 9.3. A new type of average Lyapunov function is constructed to compensate the unmatched dimensions of communication topologies over different time intervals. Generalized reciprocally convex Lemma and a more relaxed switched technique are employed to achieve a less conservative switched stability condition for the MAS with variable tracking number and time delays. Finally, through a series of numerical simulations, the effectiveness and feasibility of derived results are verified. The relationship between maximum allowable communication time delays and various control parameters is obtained in a quantitative way.

9.2 Linear relay tracking systems with time-varying number of agents and time delays 9.2.1

Problem formulation

Consider a relay tracking system consisting of multiple agents. The dynamics of agent i is given by x_i ðtÞ 5 Axi ðtÞ 1 Bui ðtÞ; xi ðt0 Þ 5 xi0 ; t0 $ 0

t $ t0

ð9:1Þ

where xi ðtÞAℝm is state vector of ith agent, AAℝm 3 m and BAℝm 3 p are constant matrices, ui ðtÞAℝp is the control input, xi0 is the initial condition of ith agent. The agents are randomly deployed in a certain area to monitor and track any intruded target, the dynamics of which is

Multiagent relay tracking systems with time-varying Chapter | 9

x_t ðtÞ 5 Axt ðtÞ; t $ t0 xt ðt0 Þ 5 xt0 ; t0 $ 0

179

ð9:2Þ

where xt ðtÞAℝm is state vector of the target and xt0 is the initial condition of the target. The communication topology of tracking agents and the target is piecewise constant between switching times. Denote the corresponding number of
tracking agents as nðkÞA 1; 2; . . .; Np over time interval ½tk ; tk11 Þ; where among Np is the maximum number of tracking agents. The communication
 tracking agents forms a weighted graph GðσðkÞÞ, where σðkÞA 1; 2; . . .; Mp and Mp $ Np is the maximum number of possible communication topologies. aij ðσðkÞÞ represents the connection status between ith agent and jth agent at time tA½tk ; tk11 Þ; aij ðσðkÞÞ 5 aji ðσðkÞÞ . 0 implies agents i and j can access each other’s information; otherwise, aij ðσðkÞÞ 5 aji ðσðkÞÞ 5 0. ÞÞ is LσðkÞ 5  The corresponding Laplacian matrix of graph GðσðkP ðk Þ lij ðσðkÞÞ nðkÞ 3 nðkÞ ; where lij ðσðkÞÞ 5 2 aij ðσðkÞÞ; i 6¼ j; lii ðσðkÞÞ 5 nj51 aij ðσðkÞÞ: In a relay tracking problem, since a target is involved, bi ðσðkÞÞ is introduced to represent the connect status of ith agent and the target. If the tracking agent i is able to access the target’s information
at time tA½tk ; tk11 Þ; bi ðσðkÞÞ 5 1; otherwise, bi ðσðkÞÞ50. Denote BσðkÞ 5diag b1 ðσðkÞÞ; b2 ðσðkÞÞ; ...;bnðkÞ ðσðkÞÞg: Denote ei ðtÞ 5 xi ðtÞ 2 xt ðtÞ as the position disagreement vector between the ith tracking agent and the target. Denote eij ðtÞ 5 xi ðtÞ 2 xj ðtÞ as the disagreement vector between ith tracking agent and jth tracking agent. The definition of successful tracking is addressed as following. Definition 9.1: For the multiagent tracking problem with a varying number of tracking agents, it is said the tracking agents successfully track the target if there exists a sufficient small scalar δ . 0 such that lim

t-N

nðkÞ X

jjei ðtÞjj2 # δ:

ð9:3Þ

i51

When communication time delays among agents are considered, state information received from neighboring agents is not real time values. Thus at time tA½tk ; tk11 Þ; the control protocol for ith tracking agent is ( ui ðtÞ 5 2 K1 bi ðσðkÞÞ ei ðt 2 τ i ðtÞÞ 1

n ðk Þ X

) K2 aij ðσðkÞÞ eij ðt 2 τ i ðtÞÞ ;

ð9:4Þ

j51

where ei ðt 2 τ i ðtÞÞ 5 xi ðt 2 τ i ðtÞÞ 2 xt ðt 2 τ i ðtÞÞ is the position disagreement vector between the ith tracking agent and the target. eij ðt 2 τ i ðtÞÞ 5 xi ðt 2 τ i ðtÞÞ 2 xj ðt 2 τ i ðtÞÞ is the disagreement vector between ith tracking agent and jth tracking agent. τ i ðtÞ represents the maximum communication

180

Consensus Tracking of Multi-agent Systems with Switching Topologies

time delay between agent i and its neighbors. From the viewpoint of each agent the delays are time-varying and may be different for each agent, but we assume the delays share a common upper bound τ. Substituting the controller (9.4) into (9.1), the tracking problem is then interpreted as a stability problem of the following tracking error system with the time delay.   E_ nðkÞ ðtÞ 5 AnðkÞ EnðkÞ ðtÞ 2 BσðkÞ  BK1 1 LσðkÞ  BK2 EnðkÞ ðt 2 τ Þ: ð9:5Þ h iT T is the collective tracking error. where EnðkÞ ðtÞ 5 e T1 ðtÞ; e T2 ðtÞ; . . .; e nðkÞ AnðkÞ 5 InðkÞ  A. Since the multiagent tracking system suffers from a varying number of agents, the dimension of EðtÞ is not the  same  over different time intervals.   At the switching time tk , let dim Enðk21Þ tk2 5 m 3 nðk 2 1Þ; dim EnðkÞ tk1 5 m 3 nðkÞ; where tk2 indicates the time before switching, tk1 represents the time after switching. Two assumptions are made in the relay tracking strategy: (1) No new agents are willing to join the tracking task when the absolute value of the overall tracking errors is less than δ1 and (2) a new agent is willing to join the tracking task only if the absolute value of its tracking error is less than δ2 . During the whole tracking process, there are three possible cases at switching times in terms of the number of tracking agents and tracking errors. Here are the detail descriptions and mathematical statements of the cases at the switching time tk . Case 1. Some of the original agents quit tracking without new agents join the task. In this case the number of tracking agents decreases, that is, nðkÞ , nðk 2 1Þ; and the absolute value of the overall tracking errors is   2   2 :EnðkÞ tk1 : # γ 1k :Enðk21Þ tk2 : ; 0 , γ 1k # 1: ð9:6Þ Case 2. Some new agents join the task without original agents quit the tracking. In this case the number of tracking agents increases, that is, nðkÞ . nðk 2 1Þ: Denote   2   2 :EnðkÞ tk1 : # γ 2k :Enðk21Þ tk2 : ; γ 2k . 1: ð9:7Þ According to Assumption (2), it has   2   2 :EnðkÞ tk1 : # :Enðk21Þ tk2 : 1 δ2 :

ð9:8Þ

Combining (9.7), (9.8), and in light of Assumption (1), γ 2k #

δ2 1 1: δ1

ð9:9Þ

Multiagent relay tracking systems with time-varying Chapter | 9

181

Case 3. New agents join tracking at the same time when some agents quit tracking. The communication topology could and the tracking   change  2 2 2 error E suffers discontinuity. In this case, :EnðkÞ tk1 : # γ 2k :Enðk21Þ  : P tk : Fig. 9.1 illustrates the switching of different cases. tα1 5 Τ p1 ðUÞ =p1 denotes dwell time of Case 1, p1 denotes the number of Case 1. Pthe average  tα2 5 Tp2 ðUÞ =p2 denotes the average dwell time of Cases 2 and 3, p2 denotes the number of Cases 2 and 3. Let the time ratio between Case 1 and Cases 2/3, κ as κ5

9.2.2

p1 tα1 : p2 tα2

ð9:10Þ

Main results

In this section, we construct an impulse-time-dependent Lyapunov function, which is the basis of stability analysis. First, several auxiliary functions associated with the impulse time sequence are introduced. For given time sequences tk , piecewise linear functions g0 ðtÞ; g1 ðtÞ:½t0 ; NÞ-ℝ1 are defined as g0 ð t Þ 5

1 ; g1 ðtÞ 5 ðt 2 tk Þg0 ðtÞ: tk11 2 tk

ð9:11Þ

exist  positive constants β 1 ; β 2 such that g0 ðtÞ satisfies  There   1=β 1 # g0 ðtÞ # 1=β 2 for all t . 0: From the definition of g0 ðtÞ, it can be seen that β 2 represents the minimum interval of switching, which significantly influences the stability of the relay tracking system. This will be analyzed in detail given next. Define impulse-time-dependent function as ψðtÞ 5 μg1 ðtÞ ; μ . 1. According  to (9.11), it is easy to obtain that ψ tk1 5 1; ψ tk2 5 μ: Based on Bessel-Legendre Inequality introduced in Lemma A.7, we define 2 3 E ðt Þ 6 Ð t L0 ðsÞEðsÞds 7 6 7 ^ EðtÞ 5 6 t2τ 7: 4 5 ^ Ðt t2τ LN21 ðsÞE ðsÞds For the sake of simplicity, we let N 5 2, that is, 2 3 E ðt Þ Ð 6 t 7 E^ ðtÞ 5 4 t2τ L0 ðsÞEðsÞds 5: Ðt t2τ L1 ðsÞE ðsÞds

ð9:12Þ

Consensus Tracking of Multi-agent Systems with Switching Topologies

FIGURE 9.1 Illustration of switching signals between different cases.

182

Multiagent relay tracking systems with time-varying Chapter | 9

183

We then define 2

E ðt Þ Eðt 2 τ Þ

3

7 6 ð 7 61 t 7 6 L ð s ÞE ð s Þds 7 6 0 E^ ζ ðtÞ 5 6 τ t2τ 7: 7 6 ð 7 61 t 5 4 L1 ðsÞEðsÞds τ t2τ

ð9:13Þ

The relationship between E^ ðtÞ and E^ ζ ðtÞ is E^ ðtÞ 5 M1 E^ ζ ðtÞ; E_^ ðtÞ 5 M2 E^ ζ ðtÞ, h iT   where M2 5 HnσððkkÞÞΤ ; ΓT0 ; ΓT1 ; and HnσððkkÞÞ 5 AnðkÞ ; 2 LσðkÞ  BK; 0mnðkÞ ; 0mnðkÞ ; 2 3 ImnðkÞ 0mnðkÞ 0mnðkÞ 0mnðkÞ 6 7 M1 5 4 0mnðkÞ 0mnðkÞ τImnðkÞ 0mnðkÞ 5: 0mnðkÞ 0mnðkÞ 0mnðkÞ τImnðkÞ For the multiagent relay tracking system with varying number of agents and communication time delays (9.5) over time interval ½tk ; tk11 Þ; the impulse-time-dependent average Lyapunov function is defined as V ðt Þ 5

 1
V1k ðtÞ 1 V2k ðtÞ 1 V3k ðtÞ : nð k Þ

ð9:14Þ

in which V1k ðtÞ 5 ψðtÞ E^ nðkÞ ðtÞ Q~ 1 E^ nðkÞ ðtÞ T

V2k ðtÞ 5

ðt t2τ

V3k ðtÞ 5 τ

ðt t2τ

ðt θ

  EnΤðkÞ ðsÞ ψðtÞ eλðs2tÞ InðkÞ  Q2 EnðkÞ ðsÞ ds

  T E_ nðkÞ ðsÞ ψðtÞ eλðs2tÞ InðkÞ  Q11 E_ nðkÞ ðsÞ ds dθ

ð9:15Þ ð9:16Þ ð9:17Þ


where λ . 0 is a positive constant and Q~ 1 5 diag InðkÞ  Q11 ; InðkÞ  Q12 ; InðkÞ Q13 g with Q11 ; Q12 ; Q13 Aℝm 3 m ; Q2 Aℝm 3 m are positive definite symmetric matrices. The following theorem gives sufficient conditions ensuring the successful tracking of a target by agents with dynamically changing number and communication time delays. Theorem 9.1: Consider the multiagent relay tracking system (9.5) with varying number of agents and communication time delays under Assumptions (1) and (2). For given positive constants λ; μ . 1 and the time ratio κ, if there exist positive definite symmetric matrices Q11 ; Q12 ; Q13 ; Q2 , matrices @κ1 1 Aℝp 3 m ; @κ1 2 Aℝp 3 m ; positive constants α1 ; α2 , and α3 such that

184

Consensus Tracking of Multi-agent Systems with Switching Topologies



 (9.18),  (9.19), and (9.20) hold for iA 1; 2; . . .; Np ; jA RA1; 2; . . .; Mp j rank LR 5 ig: " #  Ξ1ij 1 Ξ2i 2 e2λτ Ξ32i , 0; ð9:18Þ τΞW 2Ii  Q11 31ij 2

α1 ln μ 2 λ , 0;

ð9:19Þ 3

K α3 5 α2 1 lnμ 1 α1 lnμ 2 4 ð1 1 KÞ ð1 1 KÞ

ð9:20Þ

  α3 ln δ2 =δ1 1 1 2 λ , 0; ð1 1 KÞ where

2

Ξ11 1ij

6 6 Ξ21 1ij Ξ1ij 5 6 6 4 τIi  Q12 τIi  Q13







0





-τIi  Q12 τIi  Q13

λτ 2 Ii  Q12 -2τIi  Q13

λτ Ii  Q13



3 7 7 7 7 5

2

where κ1 κ2 21 Ξ11 1ij 5 λIi  Q11 1 He ðIi  AQ11 Þ; Ξ1ij 5 -Bj  B@1 -Lj  B@1 :
 Ξ2i 5 diag Ii  Q2 ; 2 e2λτ ðIi  Q2 Þ; 0mi ; 0mi ;

2

3Τ 2 32 3 Γ0 Ii  Q11 0mi 0mi Γ0 Ξ32i 5 4 Γ1 5 4 0mi 3Ii  Q11 0mi 54 Γ1 5; Γ2 0mi 0mi 5Ii  Q11 Γ2     κ1 κ2 ΞW 31ij 5 Ii  AQ11 2 Bj  B@1 1 Lj  B@1 0mi 0mi : Then, the relay tracking multiagent system (9.5) with time delay τ and κ2 21 controller (9.4) is stable with K1 5 @κ1 1 Q21 11 ; K2 5 @1 Q11 ; the average times tα1 5 1=α2 ; tα2 5 1=α3 and the minimum dwelling time β 2 5 1=α1 : Proof: Over the time interval ½tk ; tk11 Þ; kAℕ1 ; the derivative of Lyapunov function (9.15) is  T T T _ _ 1 _ ~ ~ ~ ^ ^ ^ ^ ^ ^ D V1k ðtÞ 5 ψðtÞEnðkÞ ðtÞQ1 EnðkÞ ðtÞ 1 ψðtÞ EnðkÞ ðtÞQ1 EnðkÞ ðtÞ 1 EnðkÞ ðtÞQ1 EnðkÞ ðtÞ : ð9:21Þ

Multiagent relay tracking systems with time-varying Chapter | 9

185

Denote @κ1 1 5 K1 Q11 ; @κ1 2 5 K2 Q11 ; α1 5 1=β 2 : Then, considering the property of impulse-time-dependent function ψðtÞ and by noting that E^ ðtÞ 5 M1 E^ ζ ðtÞ; E_^ ðtÞ 5 M2 E^ ζ ðtÞ; (9.21) can be reorganized as Τ ^ D1 V1k ðtÞ # ðα1 ln μ 2 λÞV1k ðtÞ 1 ψ ðtÞ E^ζn ðkÞ ðtÞ Ξ lnðkÞσðkÞ EζnðkÞ ðtÞ:

The derivative of Lyapunov function (9.16) is     Ðt D1 V2k ðtÞ # ψ_ ðtÞ 2 λψðtÞ e2λt t2τ EnTðkÞ ðsÞ eλs InðkÞ  Q2 EnðkÞ ðsÞ ds h   1 ψðtÞ EnTðkÞ ðtÞ InðkÞ  Q2 EnðkÞ ðtÞ i   2 EnΤðkÞ ðt 2 τ Þe2λτ InðkÞ  Q2 EnðkÞ ðt 2 τ Þ

ð9:22Þ

ð9:23Þ

Τ # ðα1 lnμ 2 λÞV2k ðtÞ 1 ψðtÞE^ ζnðkÞ ðtÞΞ2nðkÞ E^ ζnðkÞ ðtÞ:

The derivative of Lyapunov function (9.17) is 1

D V3k ðtÞ # ðα1 lnμ 2 λÞV3k ðtÞ 1 ψðtÞe2λt τ ðt h i     T T E_ nðkÞ ðtÞeλt InðkÞ  Q11 E_ nðkÞ ðtÞ 2 E_ nðkÞ ðθÞeλθ InðkÞ  Q11 E_ nðkÞ ðθÞ dθ t2τ

  T # ðα1 lnμ 2 λÞV3k ðtÞ 1 τ 2 ψðtÞE_ nðkÞ ðtÞ InðkÞ  Q11 E_ nðkÞ ðtÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 2 τψðtÞe2λt 3

V31k

ðt t2τ

T E_ nðkÞ ðθÞe

 λθ

 InðkÞ  Q11 E_ nðkÞ ðθÞdθ:

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} V32k

ð9:24Þ Substituting dynamics of overall tracking error system (9.5) into (9.24), the following equality holds:  T   V31k 5 τ 2 ψðtÞ AnðkÞ EnðkÞ ðtÞ2 BσðkÞ  BK1 1LσðkÞ  BK2 EnðkÞ ðt2τ Þ      InðkÞ  Q11 AnðkÞ EnðkÞ ðtÞ 2 BσðkÞ  BK1 1 LσðkÞ  BK2 EnðkÞ ðt 2 τ Þ : ð9:25Þ Recalling

@κ1 1

5 K1 Q11 ; (9.25) can be reformed as T V31k 5 τ 2 ψðtÞ E^ ζnðkÞ ðtÞ Ξ31nðkÞσðkÞ E^ ζnðkÞ ðtÞ:

where

2

  AΤnðkÞ InðkÞ  Q11 AnðkÞ

6 21 6Ξ 31nðkÞσðkÞ Ξ31nðkÞσðkÞ 5 6 6 4 0mi 0mi

*

*

Ξ22 31nðkÞσðkÞ

*

0mi 0mi

0mi 0mi

ð9:26Þ

*

3

7 * 7 7; 7 * 5 0mi

186

Consensus Tracking of Multi-agent Systems with Switching Topologies

 T   with Ξ21 InðkÞ  Q11 Ai ; Ξ22 31nðkÞσðkÞ 5 2 BσðkÞ  BK1 1LσðkÞ  BK2 31nðkÞσðkÞ 5  T    BσðkÞ  BK1 1LσðkÞ  BK2 InðkÞ  Q11 BσðkÞ  BK1 1 LσðkÞ  BK2 : Since eλθ is monotonically increasing, then 2 τψ ðtÞe2λt V32k # 2 τψðtÞe2λτ 3 ðt   T E_ nðkÞ ðθÞ InðkÞ  Q11 E_ nðkÞ ðθÞdθ:

ð9:27Þ

t2τ

Recalling yields

@κ1 2

5 K2 Q11 and applying BesselLegendre Lemma A.7, it

2τψ ðtÞe2λt V32k # 2 ψðtÞe2λτ E^ ζnðkÞ ðtÞ Ξ32nðkÞ E^ ζnðkÞðtÞ: T

ð9:28Þ

Combining the abovementioned derivations, we have D1 V ðtÞ # ðα1 lnμ 2 λÞV ðtÞ 1

1 Τ ψðtÞ E^ ζnðkÞ ðtÞ ΞnðkÞσðkÞ E^ ζnðkÞ ðtÞ; nðkÞ

ð9:29Þ

where ΞnðkÞσðkÞ 5 Ξ1nðkÞσðkÞ 1 Ξ2nðkÞ 1 τ 2 Ξ31nðkÞσðkÞ 2 e2λτ Ξ32nðkÞ :

ð9:30Þ

According to Schur complement Lemma, it is easy to conclude that ΞnðkÞσðkÞ , 0 equals to  * Ξ1nðkÞσðkÞ 1 Ξ2nðkÞ 2 e2λτ Ξ32nðkÞ ΞsnðkÞσðkÞ 5 ,0 ð9:31Þ τΞ031nðkÞσðkÞ 2InðkÞ  Q21 11 where

    Ξ031nðkÞσðkÞ 5 AnðkÞ 2 BσðkÞ  BK1 1 LσðkÞ  BK2 0 0
 Denote W 5 diag ImnðkÞ ; ImnðkÞ ; ImnðkÞ ; ImnðkÞ ; InðkÞ  Q11 : After congruent transformation, inequality (9.31) is equivalent to W ΞsnðkÞσðkÞ W T , 0;

ð9:32Þ

which can be reorganized as (9.18). According to (9.18) in Theorem 9.1, we obtain D1 V ðtÞ # 2 ðλ 2 α1 lnμÞ V ðtÞ:

ð9:33Þ

For multiagent relay tracking error system (9.5) with time delays derivations and conditions (9.18) and (9.19) prove that the tracking error decays over time intervals. Next, constraint conditions on the switching properties of the time delay multiagent relay tracking system are to be derived. At the switching time tk the Lyapunov function of multiagent relay tracking system with time delays (9.5) satisfies

Multiagent relay tracking systems with time-varying Chapter | 9

187

  1  1  ^Τ  1  ^T  1  ψ t k E n ðk Þ t k E n ðk Þ t k V tk1 5 nð k Þ ð1   1 tk 1 1 EnTðkÞ ðsÞψ tk1 eðλ1α1 ln μÞðs2tk Þ EnðkÞ ðsÞ ds ð9:34Þ nðkÞ t1k 2τ ð t1 ð t1 k k   τ 1 T 1 E_ nðkÞ ðsÞψ tk1 eðλ1α1 ln μÞðs2tk Þ E_ nðkÞ ðsÞ ds dθ: 1 nðkÞ tk 2τ θ For Cases 2 and 3, since the new joined agent stays static before they start tracking, the third term in (9.34) is not larger than that before switching.   Thus  1  there is2 a γ 2k such that 1 , γ 2k , γ 2k : In light of the fact that ψ tk 5 1; ψ tk 5 μ; and tracking error jumps at switching times, the average impulse-time-dependent Lyapunov function satisfies the following inequality at switching time tk :    nð k 2 1Þ     γ 1k or γ 2k V tk2 : V tk1 # μ21 nð k Þ For tA½tk ; tk11 Þ, it has   V ðtÞ 5 V tk1 e2ðλ2α1 lnμÞðt2tk Þ    nð k 2 1Þ  γ 1k or γ 2k V tk2 e2ðλ2α1 ln μÞðt2tk Þ # μ21 nðkÞ   2  2ðλ2α ln μÞðt2t Þ nð k 2 1Þ  1 k21 γ 1k or γ 2k V tk21 e # μ21 nðkÞ    2  2ðλ2α nð k 2 2Þ  1 γ 1k or γ 2k γ 1k21 or γ 2k21 V tk21 # μ22 e nðkÞ

ð9:35Þ

ln μÞðt2tk21 Þ

:

ð9:36Þ Through manipulating Eq. (9.36), it can be obtained that k     nð0Þ 3 L γ 1i or γ 2i V t01 e2ðλ2α1 lnμÞðt2t0 Þ : ð9:37Þ nð k Þ i51  p1 1=p1  p2 1=p2 Denote γ 1 5 Li51 γ 1i ; γ 2 5 Li51 γ 2i : Assume t0 5 0; (9.37) is able to be reorganized as   nð0Þ γ 1 p1 γ 2 p2 V ðt Þ # V ð0Þe2ðλ2α1 lnμÞt : ð9:38Þ nð k Þ μ μ

V ðtÞ # μ2p1 2p2

Noticing the fact that the average dwell time p1 tα1 1 p2 tα2 5 t; and in light of (9.10), n o     K K ln γ =μ 1 ln γ =μ 2 ð λ2α lnμ Þ t nð 0Þ 1 1 2 ð11K Þtα2 V ð0Þ 3 e ð11K Þtα1 V ðt Þ # : nð k Þ ð9:39Þ

188

Consensus Tracking of Multi-agent Systems with Switching Topologies

The worst situation for Case 1 is γ 1 5 1. Recalling the upper bound condition (9.9) on γ 2 , the condition (9.20) guarantees that the relay tracking error system is exponentially stable, indicating the tracking errors converge to zero exponentially. Then, it can be concluded that the relay tracking multiagent system (9.5) with time delays and varying number of agents and jump in tracking errors is stable in sense of Lyapunov stability. It implies that the tracking agents (9.1) successfully track the target (9.2) cooperatively. This completes the proof.& Remark 9.1: From (9.18) and (9.19) in Theorem 9.1, for a given control matrix, the tracking performance of the multiagent system with a varying number of agents is closely related to the minimum time interval β 2 . The stability of tracking system partly depends on the minimum time interval β 2 and also relies on the average dwell time tα1 and tα2. Remark 9.2: In Theorem 9.1, (9.20) is the sufficient condition on switching κ 1 frequency of the relay tracking system. Since ln , 0, it can be ð1 1 κÞtα1 μ concluded that the occurrence of Case 1 facilitates Cases 2 and 3. Namely, the relay tracking system can tolerate more new joined agents if some agents quit during the tracking process. Remark 9.3: Theorem 9.1 provides sufficient conditions for controller designing of the relay tracking system with time-varying number of agents. In real applications with different given variables, the conditions in Theorem 9.1 can be applied in different ways, which mainly classified into three cases. Application 9.1: For a relay tracking multiagent system with given λ, the controller guarantees convergence over time intervals is then determined by (9.18). Then, the theorem can be used for stabilization. The coupling constraints for the minimum dwell time β 2 , average dwell times tα1 ; tα2 can be estimated with various time ratio κ. Application 9.2: The characteristics of switching signals are given, that is, the time ratio κ, minimum dwell time β 2 , and the average dwell times tα1 ; tα2 are known previously. According to conditions (9.19) and (9.20), the convergence rate λ can be obtained. Then, the controller matrices are able to be calculated with (9.18). Application 9.3: If the characteristics of switching signals are not known, then, in order to implement designing forcontroller,condition (9.20) is dis carded. Instead, we choose μ to ensure ln δ2 =δ1 1 1 =μ , 0: Thereafter, by adjusting the minimum dwell time β 2 from (9.19), the convergence rate λ can be estimated and make sure (9.18) has feasible solutions at the same time.

Multiagent relay tracking systems with time-varying Chapter | 9

9.2.3

189

Numerical simulations

In order to verify the feasibility of developed theoretical results, simulations are carried out. In the simulations, agents are randomly deployed in a square area. Initially, five agents are assigned to track a target, then, during the whole tracking process, the number of tracking agents changes between three, four, and five. It is set that no new agents are willing to join the tracking task when the current norm of overall tracking errors is not greater than δ1 5 4. A new agent is allowed to join the tracking task only if the norm of its tracking error is not greater than δ2 5 15: Consider a MAS with   0:1 0:8 0:5 A5 ;B5 : ð9:40Þ 21:5 20:5 0:5 Then, according to condition (9.18) in Theorem 9.1, the maximum allowable communication time delays with different convergence rate λ are listed in Table 9.1. As can be seen in the table, the maximum allowable communication time delay τ changes in inverse proportion to the convergence rate λ. The tradeoff between these two key indexes should be made for specific application scenarios. Simulations of a multiagent relay tracking system with a varying number of agents under different switching frequencies are conducted in order to verify the correctness of derived theoretical results. In the following specific simulations, we choose λ 5 0:4; μ 5 1:1, and communication time delay is set as τ 5 0:15 second.

TABLE 9.1 Allowable communication time delays with different convergence rate λ. λ

τ ðs Þ

K1

K2

0.4

0.16

[0.6778, 0.3873]

[0.4382, 0.2505]

0.3

0.31

[0.5056, 0.2824]

[0.3239, 0.1812]

0.2

0.42

[0.1967, 0.1359]

[0.1256, 0.0868]

0.1

0.511

[0.1232, 0.1005]

[0.0772, 0.0631]

0.08

0.528

[0.1006, 0.0861]

[0.0635, 0.0543]

0.06

0.544

[0.0967, 0.0853]

[0.0608, 0.0537]

0.04

0.56

[0.0801, 0.0740]

[0.0507, 0.0468]

0.02

0.575

[0.0837, 0.0792]

[0.0526, 0.0498]

0.01

0.583

[0.0687, 0.0672]

[0.0436, 0.0427]

190

Consensus Tracking of Multi-agent Systems with Switching Topologies

Under this configuration the feasible solutions are  3:6558 21:3710 ; Q11 5 21:3710 7:0546  0:0229 20:0158 ; Q12 5 20:0158 0:0289  15:1462 220:3614 ; Q13 5 220:3614 39:0278  0:0038 20:0026 Q2 5 ; 20:0026 0:0048     @κ1 1 5 1:9470 1:8032 ; @κ1 2 5 1:2585 1:1663 : Thus the control matrices are obtained. K1 5 ½0:6778 0:3873; K2 5 ½0:4382 0:2505: Example 9.1: Simulation of relay tracking under slow switching This section conducts a simulation where the agent replacements occur with slow frequency, as shown in Fig. 9.2. In this simulation the time ratio between Case 1 and Cases 2/3 is κ 5 0:6387. According to conditions (9.19) and (9.20) in Theorem 9.1, the feasible solutions for the allowable minimum dwell time β 2 and average dwell times tα1 ; tα2 are obtained with the math in Table 9.2 with tα1 19 2 534tα2 *β 2 = software “Maple” and are listed  4000*tα2 *β 2 2 953*tα2 2 6439*β 2 ; tα2 1 5 2 534tα2 *β 2 = 4000 * tα2 * β 2 2 953*tα2 2 6439*β 2 ÞÞ. From Fig. 9.2, it can be calculated that the allowable minimum dwell time β 2 5 0:719 second; the average dwell times tα1 5 1:559 seconds; tα2 5 3:6615 seconds; which satisfy the second situation in Table 9.2. The relay tracking trajectories are illustrated in Fig. 9.3. As can be seen in the figure, with proposed relay tracking strategy, a time-varying number of agents successfully track the target even under the effect of communication time delay τ 5 0:15 second: Example 9.2: Simulation of relay tracking under fast switching This section conducts a simulation where the agent replacements occur with high frequency. In this simulation the time ratio between Case 1 and Cases 2/3 is κ 5 0:8388; with which the feasible solutions for the allowable minimum dwell time β 2 and average dwell times tα1 ; tα2 are obtained as listed in   0 Table 9.3 with tα1 1 9 2 435tα2 * β 2 = 4000 tα2 * β 2 2 953 * tα2 2 7955 * β 2 ; 0 tα2 1 9 2 435tα2 * β 2 = 4000 * tα2 * β 2 2 953 * tα2 2 7955 * β 2 . From Fig. 9.4, it can be calculated that the allowable minimum dwell time β 2 5 0:349 second; the average dwell times tα1 5 0:4562 second; tα2 5 0:4079 second.

191

Multiagent relay tracking systems with time-varying Chapter | 9 6

Number of agents

5

4

3

2

1

0

0

2

4

6 Time (s)

8

10

12

FIGURE 9.2 Number evolution of tracking agents (slow switching).

TABLE 9.2 Feasible solutions of dwell times β 2 ; tα1 , and tα2 ðκ 5 0:6387Þ. β2

tα1

tα2

0:238 , β 2 , 1:715

β 2 , tα1 , tα11

6439 *β 2 6439 *β 2 , tα2 , 4000 * β 2 2 419 4000 * β 2 2 953

0:238 , β 2 , 1:715

tα1 . β 2

1:715 , β 2 , 1:848

β 2 , tα1 , tα21

1:715 , β 2 , 1:848

tα1 . β 2

β 2 . 1:848

tα1 . β 2

tα2 .

6439 * β 2 4000 * β 2 2 953

β 2 , tα2 , tα2 .

6439 * β 2 4000 * β 2 2 953

6439 * β 2 4000 * β 2 2 953

tα2 . β 2

According to Table 9.3, when minimum dwell time β 2 5 0:349 second, it is required that the average dwell time tα2 . 3:1621 seconds, which is clearly not satisfied. Namely, the successful track of a target cannot be achieved if the number of agents switches in a high frequency, as depicted in Fig. 9.5.

192

Consensus Tracking of Multi-agent Systems with Switching Topologies

100 80 60 40 20 0 −20 −40 −60 −80 −100 −100

−80

−60

−40

−20

0

20

40

60

80

100

FIGURE 9.3 Tracking trajectories (slow switching).

TABLE 9.3 Feasible solutions of dwell times β 2 ; tα1 , and tα2 ðκ 5 0:8388Þ. β2

tα1

tα2 0

0:238 , β 2 , 2:118

β 2 , tα1 , tα11

0:238 , β 2 , 2:118

tα1 . β 2

2:118 , β 2 , 2:227

β 2 , tα1 , tα21

2:118 , β 2 , 2:227

tα1 . β 2

β 2 . 2:227

tα1 . β 2

3977:5 * β 2 7955 * β 2 , tα2 , 2000 * β 2 2 259 4000 * β 2 2 953 tα2 ,

7955 * β 2 4000 * β 2 2 953

0

β 2 , tα2 , tα2 ,

7955 * β 2 4000 * β 2 2 953

7955 * β 2 4000 * β 2 2 953

tα2 . β 2

9.3 Nonlinear relay tracking systems with time-varying number of agents and time delays 9.3.1

Related preliminaries

This section investigates a relay tracking MAS in which the dynamics of ith tracking agent is described as

193

Multiagent relay tracking systems with time-varying Chapter | 9 6

Number of agents

5

4

3

2

1

0

0

2

4

6 Time (s)

8

10

12

FIGURE 9.4 Number evolution of tracking agents (fast switching).

x_i ðtÞ 5 f ðt; xi ðtÞÞ 1 ui ðtÞ; xi ðt0 Þ 5 xi0 ; t0 $ 0

t $ t0

ð9:41Þ

where xi ðtÞAℝ2 denotes the position states on x- and y-axes, f ðt; xi ðtÞÞ represents the nonlinear dynamics, and ui ðtÞAℝ2 is the control input of ith agent. It should be noted that in the considered model (9.41), the controller is imposed on each control channel. This is a commonly adopted model for MASs [1,2] and is reasonable for some practical systems, such as the spacecraft [3] and the wheeled mobile robot [4]. The agents are supposed to track any intruded target with dynamics: x_t ðtÞ 5 f ðt; xt ðtÞÞ; t $ t0 xt ðt0 Þ 5 xt0 ; t0 $ 0

ð9:42Þ

where xt ðtÞAℝ2 denotes the position state and f ðt; xt ðtÞÞ represents the nonlinear dynamics of the target. In this chapter, it is assumed the nonlinear function f ðUÞ meets Lipschitz assumption. Since all the agents can only receive information of agents and target within its communication range, and there exist communication time delays, the position values from the neighbors taking account of possible delays are employed in designing control protocol for ith tracking agent.

194

Consensus Tracking of Multi-agent Systems with Switching Topologies 100 80 60 40 20 0 −20 −40 −60 −80

−100 −100

−80

−60

−40

−20

0

20

40

60

80

100

FIGURE 9.5 Tracking trajectories (fast switching).

( ui ðtÞ 5 2 γ bi ðσðtÞÞei ðt 2 τ i ðtÞÞ 1

NX ðσ ðt ÞÞ

) aij ðσðtÞÞeij ðt 2 τ i ðtÞÞ ;

ð9:43Þ

j51

where γ . 0 is the control parameter. ei ðt 2 τ i ðtÞÞ 5 xi ðt 2 τ i ðtÞÞ 2 xt ðt 2 τ i ðtÞÞ is the position tracking error of the ith tracking agent. eij ðt 2 τ i ðtÞÞ 5 xi ðt 2 τ i ðtÞÞ 2 xj ðt 2 τ i ðtÞÞ denotes the position disagreement between tracking agents i and j. τ i ðtÞ represents the maximum communication time delay between agent i and its neighbors. From the viewpoint of each agent the communication time delays are not the same for each agent, but it is reasonable to assume that they share a common upper bound τ ðtÞ # τ M and its derivative satisfies τ_ ðtÞ # τ d , 1 at time t. Therefore substituting controller (9.43) into the dynamics of tracking agents (9.41), straightforwardly, the successful tracking of a target is equivalent to the stability of the following error system at 0. E_ σðtÞ ðtÞ 5 2 γ 1 LσðtÞ EσðtÞ ðt 2 τðtÞÞ 1 Fðt; EσðtÞ ðtÞÞ; t 6¼ tk ð9:44Þ h iT is the collective position error. where EσðtÞ ðtÞ 5 eT1 ðtÞ; eT2 ðtÞ; . . .; eTNσðtÞ ðtÞ    T F t; EσðtÞ ðtÞ 5 f1 ðtÞ; f2 ðtÞ; . . .; fNσðtÞ ðtÞ ; with fi ðtÞ 5 f ðt; xi ðtÞÞ 2 f ðt; xt ðtÞÞ; is the collective nonlinear self-dynamic disagreement. LσðtÞ 5 ℒσðtÞ 1 BσðtÞ is the extended Laplacian matrix including the target. Due to the change of number of tracking agents over different time intervals, the ranks of

Multiagent relay tracking systems with time-varying Chapter | 9

195

collective tracking error EσðtÞ ðtÞ and collective nonlinear dynamics   F t; EσðtÞ ðtÞ are time-varying. In this chapter the tracking position error ei ðtÞ is composed of ei1 ðtÞ and ei2 ðtÞ, which denotes the tracking error on x-axis and y-axis, respectively. Its norm equals to the distance between agent i and the target, that is, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jjei ðtÞjj 5 dðxt ðtÞ; xi ðtÞÞ 5 e2i1 ðtÞ 1 e2i2 ðtÞ: In the complicated scenario where the tracking agents relay during the whole tracking course and number of agents is time-varying, the dimension of EðtÞ switches over different time intervals. The jump of tracking errors before and after switching time tk is described as   jjEk21 tk2 jj2 jjEk ðtk Þjj2 # Δk ; t 5 tk ; kAℕ1 ð9:45Þ Nk Nk21 where tk2 and tk represent the instant before and after switching time tk , and Δk . 0 indicates the jump caused by dynamic tracking agents at tk .

9.3.2

Stability analysis

The tracking system is divided into many subsystems over different time intervals ½tk21 ; tk Þ; where tk ; k 5 1; 2; 3. . . are the instants when the number of tracking agents switches. Specifically, the error system (9.44) is a piecewise system over time intervals ½tk21 ; tk Þ and the characteristics of the jumps at switching times are depicted in (9.45). Therefore the stability analysis of the tracking system includes two parts, that is, the decay of tracking errors in time intervals and the constraints on switching times. In this section, sufficient conditions on the convergence of the cooperative relay tracking MAS (9.44) with time delays during time interval ½tk21 ; tk Þ are derived. Since the relay tracking system involves dynamic number of tracking agents, the summation of all tracking errors cannot reflect the tracking performance any more. Thus it is not reasonable to design the Lyapunov function with the summation of all tracking errors. Then, for system over time interval ½tk21 ; tk Þ; an average Lyapunov function for the relay tracking MAS with communication time delays is designed as V k ðt Þ 5

 1
V1 ðtÞ 1 V2 ðtÞ 1 V3 ðtÞ 1 V4 ðtÞ : 2Nk

V1 ðtÞ 5 EkT ðtÞEk ðtÞ Ðt V2 ðtÞ 5 t2τ M EkΤ ðsÞe2αðs2tÞ Ek ðsÞds Ðt V3 ðtÞ 5 t2τ ðtÞ EkT ðsÞe2αðs2tÞ Ek ðsÞds Ð0 Ðt T V4 ðtÞ 5 τ M 2τ M t1θ E_ k ðsÞe2αðs2tÞ E_ k ðsÞdsdθ in which α is a positive constant.

ð9:46Þ

ð9:47Þ

196

Consensus Tracking of Multi-agent Systems with Switching Topologies

Due to the switching rules and the inequality of tracking error’s norm at switching times (9.45), the average Lyapunov function satisfies the following inequality with μk . 0.   V k11 ðtk Þ # μk V k tk2 ð9:48Þ Theorem 9.2: Consider the relay tracking multiagent system (9.44) with communication time delays and connected undirected topology over time interval ½tk21 ; tk Þ; kAℕ1 : The corresponding subsystem is stable if there exists a positive constant, such that condition (9.49) holds. Φk 5 Φ1k 1 Φ2k 1 Φ3k 1 τ 2M Φ41k 2 e22ατ M Φ42k # 0

ð9:49Þ

in which

2 3 2 2 3  0 Ik 0 l 1 1 1 2α Ik 2 γLk 0 5; 0 Φ1k 5 4 2 γLk 0 0 5; Φ2k 5 4 0 0 22ατ M 0 0 2 e I 0 0 0 k 2 3 2 2 3 Ik 0 0 2l Ik 0 0 Φ3k 4 0 2 ð1 2 τ d Þe22ατ M Ik 0 5; Φ41k 5 4 0 2γ 2 Lk Lk 0 5; 0 0 0 0 0 0  1 2 12 Φ42k 5 ½11 2 12 ; 12 2 13  ΦðλÞ 1 ; 12 2 13

where

" # " # zfflfflfflffl}|fflfflfflffl{2Nk zfflfflfflffl}|fflfflfflffl{ 2Nk zfflfflfflffl}|fflfflfflffl{ 2Nk zfflfflfflffl}|fflfflfflffl{ 2Nk zfflfflfflffl}|fflfflfflffl{ 2Nk zfflfflfflffl}|fflfflfflffl{2Nk ; 12 5 ½0; . . .; 0; 1; . . .; 1; 0; . . .; 0; ; 11 5 ½1; . . .; 1; 0; . . .; 0; 0; . . .; 0;

"

# zfflfflfflffl}|fflfflfflffl{2Nk zfflfflfflffl}|fflfflfflffl{ 2Nk zfflfflfflffl}|fflfflfflffl{2Nk 13 5 ½0; . . .; 0; 0; . . .; 0; 1; . . .; 1; ;

ΦðλÞ is a delay dependent matrix defined

in (A.5) with R 5 I; λA½0; 1: Proof: Let us select the Lyapunov candidate function for relay tracking system (9.44) over time interval ½tk21 ; tk Þ as (9.46). The derivative of Lyapunov function (9.46) is D1 V1 ðtÞ 5 E_ k ðtÞEk ðtÞ 1 EkT ðtÞE_ k ðtÞ: T

ð9:50Þ

Substituting the overall tracking error system (9.44) into (9.50), and with simple derivations, it has D1 V1 ðtÞ 5 ½ 2γLk Ek ðt2τ ðtÞÞ; F ðt2EðtÞÞT Ek ðtÞ 1 EkΤ ðtÞ½ 2 γLk Ek ðt 2 τ ðtÞÞ; F ðt 2 Ek ðtÞÞ 5 2 γEkΤ ðt 2 τ ðtÞÞ; LTk Ek ðtÞ 2 γEkT ðtÞLk Ek ðt 2 τ ðtÞÞ 1 2ðF ðt2Ek ðtÞÞÞT Ek ðtÞ:

ð9:51Þ

197

Multiagent relay tracking systems with time-varying Chapter | 9

 T Denote E k ðtÞ 5 EkT ðtÞ; EkT ðt2τ ðtÞÞ; EkT ðt2τ M Þ ; from Assumption A.1 and Lemma A.1, the following inequality can be deduced. D1 V1 ðtÞ # 2 γEkT ðt 2 τ ðtÞÞLTk Ek ðtÞ 22 γEkT ðtÞLk Eðt 2 τ ðtÞÞ   1 EkT ðtÞ l2 1 1 Ek ðtÞ 5 2 2αV1 ðtÞ 1 E Tk ðtÞΦ1k E k ðtÞ

ð9:52Þ

Since 0 # τ ðtÞ # τ M ; the derivative of second part in Lyapunov function (9.46) is D1 V2 ðtÞ # 2 2αV2 ðtÞ 1 EkT ðtÞEk ðtÞ 2 EkT ðt 2 τ M Þe22ατ M Ek ðt 2 τ M Þ 5 2 2αV2 ðtÞ 1 E Tk ðtÞΦ2k E k ðtÞ:

ð9:53Þ

For τ_ ðtÞ # τd , 1, it has D1 V3 ðtÞ # 2 2αV3 ðtÞ 1 EkT ðtÞEk ðtÞ 2 ð1 2 τ d ÞEkT ðt 2 τ ðtÞÞe22ατ M Ek ðt 2 τ ðtÞÞ 5 2 2αV3 ðtÞ 1 E Tk ðtÞΦ3k E k ðtÞ: ð9:54Þ Τ D1 V4 ðtÞ # 2 2αV4 ðtÞ 1 τ 2M E_ k ðtÞE_ k ðtÞ 2 τ M

ðt t2τ M

T E_ k ðsÞe22ατ M E_ k ðsÞds:

ð9:55Þ Substituting dynamics of overall tracking error system (9.44) into (9.55), the second term in (9.55) is derived as τ 2M E_ ðtÞE_ ðtÞ 5 τ 2M ½2γLk Ek ðt2τ ðtÞÞ1F ðt; Ek ðtÞÞT ½ 2γLk Ek ðt 2 τ ðtÞÞ 1 F ðt; Ek ðtÞÞ
5 τ 2M F T ðt; Ek ðtÞÞF ðt; Ek ðtÞÞ 2 γF T ðt; Ek ðtÞÞLk Ek ðt 2 τ ðtÞ    2 γEkT t 2 τ ðtÞLk F ðt; Ek ðtÞÞ 1 γ 2 EkT t 2 τ ðtÞLTk Lk Ek ðt 2 τ ðtÞÞ : T

ð9:56Þ Applying Assumption A.1 and Lemma A.1, inequality (9.57) follows.
  T τ 2M E_ ðtÞ E_ ðtÞ # τ 2M 2l2 EkT ðtÞ Ek ðtÞ 1 2γ 2 EkT t 2 τ ðtÞ LTk Lk Ek ðt 2 τ ðtÞÞ 5 τ 2M E Tk ðtÞ Φ41k E k ðtÞ:

ð9:57Þ

The third term in (9.55) can be rewritten as ðt T E_ k ðsÞe22ατ M E_ k ðsÞds 2 τM t2τ M

22ατ M

# 2e

τM

ðt

t2τ ðtÞ

T E_ ðsÞE_ k ðsÞds 2 τ M

ð t2τ ðtÞ t2τ M

T E_ k ðsÞE_ k ðsÞds:

ð9:58Þ

198

Consensus Tracking of Multi-agent Systems with Switching Topologies

In light of Lemma A.3, it has Ðt T 2 τ M t2τ M E_ k ðsÞe22ατ M E_ k ðsÞ ds ð t T ð t T Τ 22ατ M τ M _ _ E ðsÞ ds Ek ðsÞ ds # 2e τ ðtÞ t2τ M k t2τ M ð t2τ ðtÞ T ð t2τ ðtÞ τM Τ Τ 22ατ M _ _ E ðsÞ ds Ek ðsÞ ds 2e τ ðtÞ 2 τ M t2τ M k t2τ M τM ½Ek ðtÞ2Ek ðt2τ ðtÞÞT ½Ek ðtÞ 2 Ek ðt 2 τ ðtÞÞ 5 2 e22ατ M τ ðt Þ τM ½Ek ðt2τ ðtÞÞ2Ek ðt2τ M ÞT ½Ek ðt 2 τ ðtÞÞ 2 Ek ðt 2 τ M Þ: 2 e22ατ M τ ðtÞ 2 τ M ð9:59Þ Applying generalized reciprocally convex Lemma A.6, inequality (9.60) follows. ðt T E_ k ðsÞe22ατ M E_ k ðsÞ ds 2 τM t2τ M

22ατ M

# 2e



Ek ðtÞ2Ek ðt2τ ðtÞÞ Ek ðt2τ ðtÞÞ2Ek ðt2τ M Þ

T

 ΦðλÞ



EðtÞ 2 Eðt 2 τ ðtÞÞ Eðt 2 τ ðtÞÞ 2 Eðt 2 τ M Þ

5 2 e22ατ M E Tk ðtÞ Φ42k E k ðtÞ: ð9:60Þ where ΦðλÞ is defined in (A.5) with R 5 I; λ 5 τ=τ M : Combining (9.55)(9.60), it can be obtained that   D1 V4 ðtÞ # 2 2αV4 ðtÞ 1 E Tk ðtÞ τ 2M Φ41k 2 e22ατ M Φ42k E k ðtÞ:

ð9:61Þ

Combining the abovementioned derivations, then D1 V k ðtÞ # 2 2αV k ðtÞ 1

1 T E ðtÞ Φk E k ðtÞ: 2Nk k

ð9:62Þ

According to condition (9.49) in Theorem 9.2, it is straightforward to conclude that D1 V k ðtÞ # 2 2αV k ðtÞ:

ð9:63Þ

This indicates that the tracking errors of the time-delay multiagent system decay exponentially over time interval ½tk21 ; tk Þ. Thus for tA½tk21 ; tk Þ; V ðtÞ # V ðtk21 Þ e22αðt2tk21 Þ :

ð9:64Þ

With the derived results in the abovementioned section, this section analyzes the overall stability and gives constraints on the switching of the relay tracking multiagent system.&

199

Multiagent relay tracking systems with time-varying Chapter | 9

Theorem holds, the ics (9.44) times will

9.3: On the basis of condition (9.49) in Theorem 9.2, if (9.65) relay tracking multiagent system composed of time delay dynamover time intervals and jump characteristics (9.45) at switching achieve stability. m

m X

p51

p51

ln L μp #

ð9:65Þ

2αΤ p


 
 with Τ p 5 min T tmðk21Þ1p21 ; tmðk21Þ 1 p ; μp 5 max μmk 2 ðm 2 pÞ : Proof: Without loss of generality, let V 1 ðt0 Þ 5 V ðt0 Þ: Then,   V 2 ðt1 Þ # μ1 V 1 t12 5 μ1 e22αT ðt0 ;t1 Þ V 1 ðt0 Þ ^ 2  V m ðtm21 Þ # μm21 V m21 tm21 5 μm21 ?μ2 μ1 e22αT ðt0 ;tm21 Þ V 1 ðt0 Þ   V 1 ðtm Þ # μm V m tm2

ð9:66Þ

m

5 L μp e22αT ðt0 ;tm Þ V 1 ðt0 Þ: p51

  V2 ðt1 Þ # μ1 V1 t12

ð9:67Þ

5 μ1 e22αT ðt0 ;t1 Þ V1 ðt0 Þ; ^

2  Vm ðtm21 Þ # μm21 Vm21 tm21 22αT ðt0 ;tm21 Þ

5 μm21 ?μ2 μ1 e   V1 ðtm Þ # μm Vm tm2 m

5 L μp e22αT ðt0 ;tm Þ V1 ðt0 Þ:

ð9:68Þ V1 ðt0 Þ: ð9:69Þ

p51

According to Lemma 8.1, it is required that V 1 ðtm Þ # V 1 ðt0 Þ in order to guarantee the stability of the switched multiagent tracking system. Therefore m

L μp e22αT ðt0 ;tm Þ # 1:

ð9:70Þ

p51

Applying logarithm on both sides of (9.70), it has m

ln L μp # 2αT ðt0 ; tm Þ:

ð9:71Þ

p51

With similar derivations the stable condition in a consolidated form (9.65) could be obtained. Then, according to Lemma 8.1, the relay tracking

200

Consensus Tracking of Multi-agent Systems with Switching Topologies

multiagent system (9.44) with variable number of agents and jump of tracking errors achieves stability in the sense of Lyapunov theory. This means the relay tracking agents (9.41) are able to successfully track the target (9.42) cooperatively. This completes the proof.& Remark 9.4: The proposed method and achieved theoretical results for solving multiagent relay tracking problems with the variable number of tracking agents can be directly applied to solve the problems with bounded nonuniform delays as in Ref. [5] and are possible to extend to systems with nonuniform delays without a bound as in Ref. [6]. The difference is that the delay related parts V2 ; V3 ; V4 in the Lyapunov function could be more complicated as that in Ref. [6]. Remark 9.5: Generalized reciprocally convex Lemma is adopted to deal with the double integral part in the Lyapunov function when analyzing the stability of subsystems with time delays. First, this part is divided into two subparts and then the generalized reciprocally convex combination method is adopted. This leads to a less conservative result comparing with those which adopt Jensen’s inequality or other integral inequality in dealing with the double integral part in the Lyapunov function [7,8]. In addition, as demonstrated in Ref. [9], the generalized reciprocally convex combination Lemma is less restrictive than the original reciprocally convex combination Lemma developed in Ref. [10]. Furthermore, multiple Lyapunov functions and a more relaxed switched technique are applied in analyzing the overall stability of the multiagent system with variable tracking number. This relaxes the constraints on values of the Lyapunov functions at switching times especially comparing to the results with common Lyapunov functions [11]. Remark 9.6: There are two disadvantages associated with the developed method and results: (1) the number of tracking agents is different over time intervals but is required to be changing cyclically and (2) in order to choose a Lyapunov function for the multiagent relay tracking subsystem in each time interval, the subsystems must be individually stable. Further investigation on developing more general results applicable for systems with stochastically switching number of tracking agents is worthwhile. Relaxation on the stability of each subsystem over a time interval is another effective way to improve the feasibility and practicability of the results.

9.3.3

Numerical simulations

Some numerical simulations are implemented to demonstrate the previous theoretical developments. The nonlinear dynamics of the target and agents are specified as

Multiagent relay tracking systems with time-varying Chapter | 9

f ðUÞ 5 16sinð0:028xðtÞ 1 0:12Þ 1 20cosð2tÞ:

201

ð9:72Þ

The function f ðUÞ satisfies Assumption A.1 with l 5 0:448: The agents are initially deployed on a 100 m 3 100 m area and the communication radius of each agent is set as 15 m. The initial position of target is (5,25). Example 9.3: Numerical simulation with two different tracking numbers. This section conducts a simulation in which the number of tracking agents switches between four and five during the tracking process. At the initial time, five agents are assigned to track the target, after a while, an agent quits, and this situation sustains until another agent   joins the  task. The time delay function is assumed to be τ ðtÞ 5 ð1 1 eÞ= e 1 1 esint τ M : In this situation the maximum allowable τ M for different control parameters γ are listed in Table 9.4. As shown in Table 9.4, the maximum allowable time delay and control parameter are in an inverse relationship. Specifically, with the increase of control parameter, the maximum allowable time delay becomes smaller. In order to illustratively display the tracking performance, a simulation with control parameter designed as γ 5 1.6 and maximum communication time delay τ M 5 0:044 second is carried out. In this simulation the maximum value of μ1 and the minimum value of dwell time T1 for mode 1 (with five tracking agents) are 1.1538 and 0.444 second, respectively. The maximum value of μ2 and the minimum value of dwell time T2 for mode 2 (with four tracking agents) are 1.5403 and 0.44 seconds, respectively. If we choose parameter α . 0.3252, the stability of the whole tracking system can be reached according to condition (9.65) in Theorem 9.3. The tracking trajectories, the dynamic changing number of tracking agents, and average norm of tracking errors are illustrated in Figs. 9.69.8, respectively. As shown in the figures, the trajectories of agents are discontinuous. The discontinuities occur because tracking agents drop out and join in during the whole tracking process. However, the trajectories of agents and the target are finally consensus implying the success of tracking. Namely, under the effect of proposed switched technique and achieved results, the tracking agents with variable numbers and communication time delays could successfully track the target. The number of agents switches between four and five as depicted in Fig. 9.7. Correspondingly, at the instant time when tracking number switches, the average norm of tracking errors jumps as illustrated in Fig. 9.8.

TABLE 9.4 The maximum values of τ M against different γ. γ

1.6

1.8

2.0

2.2

2.4

2.6

τ M ðsÞ

0.195

0.16

0.116

0.096

0.088

0.08

202

Consensus Tracking of Multi-agent Systems with Switching Topologies 100 90 80 70

(m)

60 50 40 30 20 10 0

0

10

20

30

40

50 (m)

60

70

80

90

100

FIGURE 9.6 Trajectories of agents with two different numbers.

6

Tracking number

5.5

5

4.5

4

3.5

3

0

1

2

3

4 Time (s)

5

FIGURE 9.7 Change of tracking number between four and five.

6

7

8

203

Multiagent relay tracking systems with time-varying Chapter | 9 2.5

Tracking errors

2

1.5

1

0.5

0

0

1

2

3

4 Time (s)

5

6

7

8

FIGURE 9.8 Average norm of tracking errors with two different numbers.

TABLE 9.5 The maximum values of τ M against different γ. γ

1.2

1.4

1.6

1.8

2.0

2.2

τ M (s)

0.076

0.54

0.132

0.12

0.106

0.09

The initial values of average tracking errors with different numbers constantly decrease. However, there are no compulsory requirements on the values before and after each switching time between different modes. The number of agents switches six times, which is mutually verified by Figs. 9.7 and 9.8. Example 9.4: Numerical simulation with three different tracking numbers. Furthermore, a more complicated simulation in which the number of tracking agents switches between three, four, and five is carried out in the purpose of verifying the feasibility of developed results. In this situation the maximum allowable τ M for different control parameters γ are listed in Table 9.5. By comparing Tables 9.4 and 9.5, it is obvious that when the tracking number changes substantially in a multiagent system, the maximum time delay it could endure becomes smaller. Choose control parameter as γ 5 1.6 with maximum communication time delay τ M 5 0:044 second: In this simulation, there are three modes during the

204

Consensus Tracking of Multi-agent Systems with Switching Topologies

whole tracking process. The maximum value of μ1 and the minimum value of dwell time T1 for mode 1 (with five tracking agents) are 1.3319 and 0.598, respectively. The maximum value of μ2 and the minimum value of dwell time T2 for mode 2 (with three tracking agents) are 0.8382 and 0.426 seconds, respectively. The maximum value of μ3 and the minimum value of dwell time T3 for mode 3 (with four tracking agents) are 2.1833 and 0.46 seconds, respectively. If we choose parameter α . 0:3002; the condition (9.65) in Theorem 9.3 is satisfied, which guarantees the stability of the relay tracking system. The tracking trajectories are described in Fig. 9.9. The change of related tracking number is illustrated in Fig. 9.10. As illustrated, the number of agents changes nine times during the whole tracking process, and the corresponding discontinuities of the average norm of tracking errors are described in Fig. 9.11. It can be seen that the tracking agents are still able to successfully track the target, even if the number of agents changes more frequently, which verifies the flexibility of theoretical results.

9.4

Conclusion and discussions

This chapter intensively explores a complicated relay tracking MASs that suffers from dynamically changing number of agents and communication time delays at the same time. For linear systems the constructed 100 90 80 70

(m)

60 50 40 30 20 10 0

0

10

20

30

40

50 (m)

60

70

FIGURE 9.9 Trajectories of agents with three different numbers.

80

90

100

Multiagent relay tracking systems with time-varying Chapter | 9

205

6 5.5

Tracking number

5 4.5 4 3.5 3 2.5 2

0

1

2

3

4 Time (s)

5

6

7

8

7

8

FIGURE 9.10 Change of tracking number between three, four, and five.

2.5

Tracking errors

2

1.5

1

0.5

0

0

1

2

3

4 Time (s)

5

6

FIGURE 9.11 Average norm of tracking errors with three different numbers.

206

Consensus Tracking of Multi-agent Systems with Switching Topologies

impulse-time-dependent average Lyapunov function fully considers the features of multiagent relay tracking system at the switching times. This makes it possible to compare the tracking performance, and precisely indicate the jump of tracking errors at the same time in the stability analysis. Stability criteria of relay tracking systems with a varying number of agents is given in the form of relationship between switching time intervals, control parameter, and time delays. In the stability analysis of nonlinear relay tracking systems, reciprocally convex Lemma is employed to solve the inequality scaling issue in the derivation of stability condition. As analyzed, maximum allowable communication time delays are closely related to the switching of tracking numbers and are in an inverse relationship with the control parameter.

References [1] Y. Zhang, Y. Yang, Y. Zhao, G. Wen, Distributed finite-time tracking control for nonlinear multi-agent systems subject to external disturbances, Int. J. Control 86 (1) (2013) 2940. [2] Y. Cui, Y. Liu, E. Alsaadi, W. Zhang, Event-based consensus for a class of nonlinear multi-agent systems with sequentially connected topology, IEEE Trans. Circuits Syst. I: Regul. Pap. 65 (10) (2018) 35063518. [3] C. Pukdeboon, Second-order sliding mode controllers for spacecraft relative translation, Appl. Math. Sci. 6 (97100) (2012) 49654979. [4] S.I. Han, Prescribed consensus and formation error constrained finite-time sliding mode control for multi-agent mobile robot systems, IET Control Theory Appl. 12 (2) (2018) 282290. [5] Z.J. Tang, T.Z. Huang, J.L. Shao, J.P. Hu, Consensus of second-order multi-agent systems with nonuniform time-varying delays, Neurocomputing 97 (1) (2012) 410414. [6] D. Meng, Y. Jia, J. Du, Consensus seeking via iterative learning for multi-agent systems with switching topologies and communication time-delays, Int. J. Robust Nonlinear Control 26 (17) (2016) 37723790. [7] D. Xie, S. Xu, Y. Chu, Y. Zou, Event-triggered average consensus for multi-agent systems with nonlinear dynamics and switching topology, J. Franklin Inst. 352 (3) (2015) 10801098. [8] G. Wen, Y. Yu, Z. Peng, A. Rahmani, Consensus tracking for second-order nonlinear multi-agent systems with switching topologies and a time-varying reference state, Int. J. Control 89 (10) (2016) 20962106. [9] A. Seuret, K. Liu, F. Gouaisbaut, Generalized reciprocally convex combination lemmas and its application to time-delay systems, Automatica 95 (2018) 488493. [10] P.G. Park, J.W. Ko, C. Jeong, Reciprocally convex approach to stability of systems with time-varying delays, Automatica 47 (2011) 235238. [11] J. Qin, H. Gao, W. Zheng, Second-order consensus for multi-agent systems with switching topology and communication delay, Syst. Control Lett. 60 (6) (2011) 390397.

Chapter 10

Finite time stability analysis and coordination strategies of multiagent relay tracking systems Chapter Outline 10.1 Introduction 208 10.1.1 Finite time boundedness 208 10.1.2 Finite time coordination relay strategies 209 10.2 Stability analysis of nonlinear multiagent relay tracking systems over a finite time interval 209 10.2.1 Relay tracking problem formulation 209 10.2.2 Stability analysis over a finite time interval 212 10.2.3 Stability analysis for systems subject to disturbances 214 10.2.4 Numerical examples 215 10.3 Finite-time coordination control of multiagent systems for target tracking with node failures and active replacements 218

10.3.1 Tracking problem description and fundamentals 220 10.3.2 Event-triggered coordination strategy 222 10.3.3 Nonsingular terminal sliding mode controller design 224 10.3.4 Finite time analysis with modified sliding mode control 226 10.3.5 Stability analysis of systems with agent replacements 228 10.3.6 Numerical simulations 229 10.4 Conclusion 235 References 235 Further reading 237

This chapter explores the stability of a class of multiagent systems (MASs) with proposed relay tracking strategy over a finite time interval. Agents are deployed in an area to monitor and track the intruded targets. According to proposed relay tracking strategy, the tracking agents and their communication topologies switch during the whole tracking process. This results in impulsive effects on the overall tracking errors. The relationship of finite time interval against desired overall tracking errors, and control parameter is derived quantitatively for the multiagent relay tracking system. Consensus Tracking of Multi-agent Systems with Switching Topologies. DOI: https://doi.org/10.1016/B978-0-12-818365-6.00010-0 © 2020 Elsevier Inc. All rights reserved.

207

208

Consensus Tracking of Multi-agent Systems with Switching Topologies

Then, stability conditions for the system with disturbances and impulsive effects are obtained. Moreover, finite time coordination strategies of second-order multiagent relay tracking systems are presented in this chapter. Coordination strategies are designed for three typical events: initial tracking agents determination for multiple targets, node failures, and active replacements. Modified nonsingular terminal sliding mode (NTSM) controller is designed for tracking agents. We propose a novel continuous function in the controller to eliminate the singularity. With modified NTSM control method, it is able to estimate the finite tracking time. Based on this, generalized Voronoi diagram considering the velocity of second-order agents and targets is presented. The event that the target enters a new generalized Voronoi cell triggers active replacement. Comparison lemma is utilized to deal with the agent switching issue caused by node failures or active replacements. The coordination strategies and designed NTSM controller lead to successful tracking of multiple targets.

10.1 Introduction Most of existing works in the literature only guarantee asymptotic stability of consensus problem. This means that network consensus is realized when time goes to infinity. Although these works are theoretically of tremendous value, it is desired to reach network consensus in a finite time from a practical point of view.

10.1.1 Finite time boundedness In real applications, we usually care about the tracking performance in a finite length of time rather than infinity time. The finite time interval that the tracking agents need to successfully track a target is a critical index. Finite time stability/boundedness is a special stability concept, assuming that the system states lie within specific bounds over a finite time interval [1,2]. Recently, some interesting results on the finite-time multiagent problems have been presented, such as Refs. [3
8]. Specifically, in Ref. [7], two finite-time sliding mode controllers are proposed and fast terminal sliding mode (TSM) control algorithms are developed to guarantee finite-time reachability of a given desired tracking motion of robot manipulators. As for Ref. [8], robust finite-time tracking problems of second-order nonlinear MASs with directed topologies are explored. Another important finite time control method is iterative learning control. This method is initially proposed to deal with tracking problem for systems that operate repetitively and now is extended to nonrepetitive uncertain systems [9]. Some results on consensus/tracking problem of MASs with nonlinear dynamics over a finite time interval without requiring the dynamics to satisfy the global Lipschitz assumption have also been achieved (see Ref. [10] for more detailed explanations on this method).

Finite time stability analysis and coordination strategies Chapter | 10

209

10.1.2 Finite time coordination relay strategies The objective of Ref. [11] is to determine an optimal pursuit strategy that minimizes the time of target pursuit. The boundaries of Voronoi diagram are partitioned with considering the tracking time. On the other hand, the area is divided into Voronoi cells only based on the position information in other related papers [12
15]. The new agent actively replaces one of the original tracking agents when the target is closer to the new agent. For first-order systems the active replacement strategy according to positions of agents could achieve minimum tracking time. In practical systems such as coordinated control of the robots and unmanned aircrafts, agents are governed by both position and velocity states; the problem of second-order MASs is more challengeable and complex than the first-order ones [16,17]. However, for second-order systems, this active strategy is effective but not optimal for reducing the tracking time. It is not reasonable to directly use the summation of position and velocity to determine the boundaries of Voronoi cells since the tracking time is influenced by complex factors. Therefore we design a generalized dynamic Voronoi diagram, in which the boundaries of Voronoi cells are determined by the finite time that required to achieve successful tracking. Furthermore, a novel NTSM control method is proposed to estimate the finite tracking times for second-order agents. In the current literature the results concerning stability of MASs over finite time intervals are based on the assumptions that there are no agent replacements and the tracking errors are continuous during the tracking process under switching topologies. However, the relay tracking system suffers from impulsive effects and dynamic changing topologies even with unconnected graphs. Taking these issues into account, the stability analysis over finite time intervals will be more challenging [18]. Thus the existing results are not applicable to analyze the nonlinear multiagent relay tracking system over a finite time, which elaborates the motivation of this chapter.

10.2 Stability analysis of nonlinear multiagent relay tracking systems over a finite time interval 10.2.1 Relay tracking problem formulation In this section the relay tracking problem is formulated in a mathematical form. The dynamics of ith agent deployed on the supervisory area is described as x_i ðtÞ 5 f ðt; xi ðtÞÞ 1 ui ðtÞ; t $ t0 xi ðt0 Þ 5 xi0 ; t0 $ 0

ð10:1Þ

where xi ðtÞAℝ2 is position state of ith agent, f ðt; xi ðtÞÞ is a nonlinear function, ui ðtÞAℝ2 is the control input of ith agent.

210

Consensus Tracking of Multi-agent Systems with Switching Topologies

The agents are expected to monitor this area and capture any intruded target, the dynamics of which is x_t ðtÞ 5 f ðt; xt ðtÞÞ; t $ t0 xt ðt0 Þ 5 xt0 ; t0 $ 0

ð10:2Þ

where xt ðtÞAℝ2 is position state of the target, f ðt; xt ðtÞÞ denotes the change of the force imposed on the target. In this chapter, we assume f ðUÞ satisfies Assumption A.1. According to the relay tracking algorithm, the event that the target escapes to a neighbor Voronoi cell triggers relay of the tracking agents and consequently triggers switching of communication topology of the agents. When the triggering event happens, Voronoi site agent aj joins the tracking task and an original tracking agent quits at the mean time. This could cause dynamic change of topology, the connection between agent i and j becomes aij ðσðtÞÞ; and the connection between agent i and the target turns to be bi ðσðtÞÞ; where σðtÞ 5 k; tA½tk ; tk11 Þ; kAℕ. Correspondingly, the Laplacian matrix is rewritten in the following format as ℒσðtÞ 1 BσðtÞ . In this chapter, it is assumed that the replacement of tracking agents and dynamic change of topology happens only when the target enters a different Voronoi region. The case where the topology switches when the target is in the same Voronoi region but gets closer to some agent as opposed to another agent is not considered. The switching topologies are event-triggered and then the topologies are piecewise constant between the event times t0 ; t1 ; . . .; tk . Based on the above analysis, at time t, control protocol for the ith tracking agent is designed with received neighboring agent’s information as (10.3). ( ) X aij ðσðtÞÞexij ðtÞ ; ð10:3Þ ui ðtÞ 5 2 α bi ðσðtÞÞexi ðtÞ 1 jANi ðtÞ

where α . 0 is the  control parameter to be designed. exi ðtÞ 5 xi ðtÞ 2 xt ðtÞ 5 exi1 ðtÞ; exi2 ðtÞ is the position disagreement vector between the ith tracking agent and the target. exij ðtÞ 5 xi ðtÞ 2 xj ðtÞ; jAN i ðtÞ is the position disagreement vector between the ith tracking agent and the jth tracking agent. Considering that there are Nf tracking agents, the collective tracking error vector between tracking agents and the target at time t is EðtÞ 5 ½eTx1 ðtÞ; eTx2 ðtÞ; . . .; eTxNf ðtÞT . The associated Laplacian matrix of the dynamic graph including the target over time interval ½tk ; tk11Þ is Lk5 ðℒk 1 Bk Þ  I2: . Fðt; EðtÞÞ 5 colff1 ðtÞ; f2 ðtÞ; . . .; fNf ðtÞg, where fi ðtÞ 5 f t; xi ðtÞ 2 f ðt; xt ðtÞÞ; iAN is the collective nonlinear self-dynamic disagreement to the target at time t. Remark 10.1: It should be noted that the dynamic graph addressed in this chapter means the communication topology switches when relay of the tracking agents occurs. There have been some results on typically switching topologies in the literature (see, e.g., Ref. [19]). A class of dynamic signed graphs, which involves entries in the form of transfer functions, has also

Finite time stability analysis and coordination strategies Chapter | 10

211

been investigated (see Ref. [20] for detailed explanations). The main difference of the dynamic graph addressed in this chapter is that both the tracking agents and the communication graph are dynamically changing. According to the proposed relay tracking algorithm, the collective tracking error at switching time tk follows Eðtk Þ 5 Δk Eðtk2 Þ, where Δk is a matrix with appropriate dimension. The norm of exi ðtk Þ is always smaller than that of exi ðtk2 Þ due to the triggering event. In terms of norm of the collective tracking error at switching time tk , it has jjEðtk Þjj # ηk jjEðtk2 Þjj;

0 , ηk # 1:

ð10:4Þ

In (10.4), ηk indicates the jump of tracking error at switching time tk . The goal of this chapter is to analyze the tracking performance over a finite time interval. If the tracking errors between all tracking agents and the target are decreased into a sufficiently small value E in finite time interval T, it is said the tracking system achieves stability. Straightforwardly, the tracking problem can be interpreted as stability problem of the overall tracking error system. Substituting controller (10.3) into ith tracking agent’s dynamic Eq. (10.1), then subtracting the target’s state Eq. (10.2), and considering the Laplacian matrix of topology, it is able to obtain the overall tracking error system in terms of EðtÞ. E_ðtÞ 5 2 αLσðtÞ EðtÞ 1 Fðt; EðtÞÞ; Eðtk Þ 5 Δk Eðtk2 Þ; jjEðtk Þjj # ηk jjEðtk2 Þjj;

t 5 tk ;

t 6¼ tk k 5 0; 1; 2; . . .

ð10:5Þ

Eðt0 Þ 5 E0 : The notations will be used throughout this chapter: For any time τ less than time t; Ts ðτ; tÞ refers to the total time  length of stable subsystems from initial time τ till the present time t. ls 5 ðTs ðτ; tÞÞ=ðTðτ; tÞÞ is called length rate of stable subsystems on time interval Tðτ; tÞ. The following definition and lemmas is needed to be reviewed in this chapter. Definition 10.1: [18] The switched system e_ðtÞ 5 Ai eðtÞ 1 fi ðt; eðtÞÞ; t $ t0 ; eðt0 Þ 5 e0 ; t0 $ 0 is said to be stable with respect to ðt0 ; R1 ; R2 ; T; ls Þ if jjeðt0 Þjj # R1 .jjeðtÞjj # R2 ;

’t . T;

ð10:6Þ

where ls is the length rate of stable subsystems on time interval ½t0 ; T Þ. Lemma 10.1: (Gronwall
Bellman inequality) Let x and f be real continuous functions which are defined on time  interval  c . 0; d . 0 Ð t ½t0 ; tÞ; t . 0, and are positive constants. If jjx ð t Þjj # c 1 1 d jjx ð τ Þjj f ð τ Þdτ , then jjxðtÞjj # t0 Ðt cecd t0 f ðτ Þdτ.

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10.2.2 Stability analysis over a finite time interval Theorem Given positive constants R1 . 0; R2 . 0; T . 0. Denote  10.1:  η 5 max ηi ; i 5 1; 2; . . .; k;  and assume λ is the minimum eigenvalue of  λmin ðLk Þ . 0; i:e:; λ9inf λmin ðLk Þjλmin ðLk Þ . 0 : Assume the initial tracking error of multiagent system (10.5) under an undirected communication topology satisfies jjEðt0 Þjj , R1 ; if there exists a constant 0 , γ , 1 such that 

R2 5 ηk e2αλls T1γ lT R1 ;

ð10:7Þ

then the tracking problem is said to be stable under the effect of proposed relay tracking algorithm with respect to ðt0 ; R1 ; R2 ; T; ls Þ. The tracking error shrinks to R2 within finite time interval [0, T] with T#

ln ηk R1 2 ln R2 :  s 2 γl αλl

ð10:8Þ

Proof: Without loss of generality, we assume t0 5 0. For tA½0; t1 Þ; we have ðt EðtÞ 5 e2αL0 t Eðt0 Þ 1 e2αL0ðt2τÞ F ðτ; Eðτ ÞÞdτ: ð10:9Þ t0

Then, at switching time t1 ;   2 E t12 5 e2αL0 t1 Eðt0 Þ 1

ð t2 1

e2αL0 ðt1 2τ Þ F ðτ; Eðτ ÞÞdτ: 2

ð10:10Þ

t0

The tracking   error at switching times subjects to impulsive effect Eðtk Þ 5 Δk E tk2 ; thus for tA½t1 ; t2 Þ; it follows that ðt 2αL1 ðt2t1 Þ EðtÞ 5 e Eðt1 Þ 1 e2αL1 ðt2τ Þ F ðτ; Eðτ ÞÞdτ t1

 ð t1 2αL0 ðt1 2τ Þ 2αL1 ðt2t1 Þ 2αL0 t1 5e Δ1 e Eðt0 Þ 1 e F ðτ; Eðτ ÞÞdτ 1

ðt

t0

e2αL1 ðt2τ Þ F ðτ; Eðτ ÞÞdτ

t1 2αL1 ðt2t1 Þ2αL0 t1

5 Δ1 e

1 Δ1 e2αL1 ðt2t1 Þ 1

ðt t1

ð t1

ð10:11Þ

Eðt0 Þ e2αL0 ðt1 2τ Þ F ðτ; Eðτ ÞÞdτ

t0

e2αL1 ðt2τ Þ F ðτ; Eðτ ÞÞdτ:

  At switching time t2 ; Eðt2 Þ 5 Δ2 E t22 : Repeating this procedure, for tA½tk ; tk11 Þ; we obtain

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213

k

EðtÞ 5 L Δj e2αLk ðt2tk Þ...2αL1 ðt2 2t1 Þ2αL0 t1 Eðt0 Þ j51 k

2αLk ðt2tk Þ...2αL1 ðt2 2t1 Þ

1 L Δj e j51

2αLk ðt2tk Þ

1 ? 1 Δk e 1

ðt

ð t1

e2αL0 ðt1 2τ Þ F ðτ; Eðτ ÞÞdτ ð10:12Þ

t0

ð tk

2αLk21 ðtk 2τ Þ

e

F ðτ; Eðτ ÞÞdτ

tk21

e2αLk ðt2τ Þ F ðτ; Eðτ ÞÞdτ:

tk

Considering (10.4) and taking norm operation of both sides of inequality (10.12), we have jjEðtÞjj # η1 η2 ?ηk e2αðλk ðt2tk Þ 1?1 λ1 ðt2 2t1 Þ1λ0 t1 Þ jjEðt0 Þjj ð t1 2αðλk ðt2tk Þ 1?1 λ1 ðt2 2t1 ÞÞ e2αλ0 ðt1 2τ Þ ljjEðτ Þjjdτ 1 η1 η2 ?ηk e 1 ? 1 ηk e2αλk ðt2tk Þ 1

ðt

ð tk

t0

e2αλk21 ðtk 2τ Þ ljjEðτ Þjjdτ

ð10:13Þ

tk21

e2αλk ðt2τ Þ ljjEðτ Þjjdτ:

tk

Apparently, with an event-triggered switching, the communication topologies change along with time. This causes the undirected communication graphs to be connected or unconnected. As stated in Lemma A.5, the eigenvalues of Laplacian matrix are all nonnegative; as a result, there are two cases, that is, λmin ðLk Þ . 0 and λmin ðLk Þ 5 0. 

jjEðtÞjj # η1 η2 ?ηk e2αλTs ð0;tÞ jjEðt0 Þjj ð t1  1 η1 η2 ?ηk e2αλTs ðt2τ Þ ljjEðτ Þjjdτ 1 ? 1 ηk 1

ðt

ð tk

t0 

e2αλTs ðt2τ Þ ljjEðτ Þjjdτ

ð10:14Þ

tk21 

e2αλTs ðt2τ Þ ljjEðτ Þjjdτ:

tk

 Since η 5 max ηi ; i 5 1; 2; . . .; k; and it is obvious that there exists a constant 0 , γ , 1 such that  s ð0;tÞ k 2αλT

jjEðtÞjj # η e

jjEðt0 Þjj 1 γ

ðt t0



e2αλTs ðt2τ Þ ljjEðτ Þjjdτ:

ð10:15Þ

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Consensus Tracking of Multi-agent Systems with Switching Topologies

Recalling ls 5 ðTs ðτ; tÞÞ=ðT ðτ; tÞÞ; then 



jjEðtÞjj # e2αλls t ηk jjEðt0 Þjj 1 γe2αλls t

ðt



leαλls τ jjEðτ Þjjdτ:

ð10:16Þ

Through reforming the above equation, we get ðt   jjEðtÞjjeαλls t # ηk jjEðt0 Þjj 1 γl eαλls τ jjEðτ Þjjdτ:

ð10:17Þ

t0

t0

Applying Gronwall
Bellman inequality on (10.17) yields 

jjEðtÞjjeαλls t # ηk jjEðt0 Þjjeγlt :

ð10:18Þ

Then in light of (10.7), at time T, the norm of the tracking error satisfies jjEðT Þjj # R2 :

ð10:19Þ

As can be seen clearly, tracking error shrinks to a smaller value R2 within finite time interval T, and the final tracking error jjEðT Þjj mainly depends on the control parameter and the length rate of stable subsystems. Applying natural logarithm on both sides of (10.18), then after simple mathematical derivations, finite time interval T is able to be expressed in an explicit form (10.8). This completes the proof. &

10.2.3 Stability analysis for systems subject to disturbances In this section, we investigate the control performance of our proposed tracking strategy over a finite time interval for the MAS with external disturbances. The agents are subject to disturbances, and the dynamics of ith agent is then described as x_i ðtÞ 5 f ðt; xi ðtÞÞ 1 ui ðtÞ 1 ωi ðtÞ; xi ðt0 Þ 5 xi0 ; t0 $ 0

t $ t0

ð10:20Þ

where ωi ðtÞ is the disturbance input of ith agent. The dynamics of the target is then described as x_t ðtÞ 5 f ðt; xt ðtÞÞ 1 ωt ðtÞ; xt ðt0 Þ 5 xt0 ; t0 $ 0

t $ t0

ð10:21Þ

where ωt ðtÞ is the disturbance input of the target. Then, the tracking problem can be interpreted as stability problem of the following overall disagreement system: E_ ðtÞ 5 2 αLk EðtÞ 1 F ðt; EðtÞÞ 1 ΩðtÞ; t 6¼ tk   Eðtk Þ 5 Δk E tk2 ;   jjEðtk Þjj # ηk jjE tk2 jj; t 5 tk ; k 5 0; 1; 2; . . . E ð t 0 Þ 5 E0 :

ð10:22Þ

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215

h  T iT where ΩðtÞ5 ðω1 ðtÞ2ωt ðtÞÞT ; ðω2 ðtÞ2ωt ðtÞÞT ;.. .; ωNf ðtÞ2ωt ðtÞ : jjΩðtÞjj # Ð T λτ ωðtÞ; which is a Lebesgue function satisfying 0 e ωðτ Þdτ , N: Here, based on the above results, we are ready to analyze tracking performance of MAS (10.22) with disturbances over a finite time interval. Theorem 10.2: Assume the initial tracking error of multiagent tracking system with disturbances (10.22) under undirected topology satisfies jjEðt0 Þjj , R1 ; then, within finite time interval [0,T], the tracking error of multiagent system (10.22) is attenuated to jjEðT Þjj # eðγl2αλls ÞT ηk jjEðt0 Þjj 1 Mω ; ÐT   with Mω 5 eðγl2αλls ÞT 0 e2αλls τ ωðτ Þdτ: 

ð10:23Þ

Proof: In terms of the overall tracking system with disturbance (10.22), for tA½0; t1 Þ; we have ðt  EðtÞ 5 e2αL0 t Eðt0 Þ 1 e2αL0 ðt2τ Þ F ðτ; Eðτ ÞÞ 1 Ωðτ Þ dτ: ð10:24Þ t0

After similar derivation as that from (10.10) to (10.15), for tA½tk ; tk11 Þ; we obtain 

jjEðtÞjj # ηk e2αλTs ð0;tÞ jjEðt0 Þjj  Ðt  1 γ t0 e2αλTs ðt2τ Þ ljjEðτ Þjj 1 ωðτ Þ dτ: Recalling ls 5 ðTs ðτ; tÞÞ=ðT ðτ; tÞÞ; it has h i Ðt   jjEðtÞjj # e2αλls t ηk jjEðt0 Þjj 1 γ t0 e2αλls τ ωðτ Þdτ  Ðt  1 γe2αλls t t0 leαλls τ jjEðτ Þjjdτ:

ð10:25Þ

ð10:26Þ

Applying Gronwall
Bellman inequality and reorganizing the inequality, we get

ðt  s Þt  sτ γl2αλl k 2αλl ð jjEðtÞjj # e η jjEðt0 Þjj 1 e ωðτ Þdτ : ð10:27Þ t0

At time T, norm of the tracking error satisfies (10.23). This completes the proof. &

10.2.4 Numerical examples In this section, we present numerical examples to show the effectiveness and correctness of the main results derived previously. All the numerical simulations are conducted on the following scenario.

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Consensus Tracking of Multi-agent Systems with Switching Topologies

Consider a scenario where a group of three tracking agents track a maneuvering target in two-dimensional space. The nonlinear dynamics of the tracking agents and target is f ðt; xðtÞÞ 5 0:28 sin ðxðtÞ 1 5:2Þ 1 50 cos ðtÞ:

ð10:28Þ

Then, jjf ðt; xi Þ 2 f ðt; xt Þjj # 0:28jjxi 2 xt jj. The Lipschitz constant l 5 0.28. The instant time when target breaks into this domain is considered as initial time. The coordinate where the first target enters this area is (0, 20). Example 10.1: The target is tracked by three agents. This example addresses the situation in which an intruded target is tracked by three agents. Fig. 10.1 illustrates the evolution of tracking errors along with time in a 3-dimension graph. The brown circles denote the maximum value of tracking error at each time. Clearly, with proposed relay control strategy, the tracking errors shrink sharply. Switching of tracking errors leads to the discontinuity of change of brown circles, which accelerates the tracking process. The tracking errors on x-axis and y-axis and the summation norm of all tracking errors are illustrated in Fig. 10.2. In this simulation,

3 Agent 1 Agent 2 Agent 3

2.5

Time (s)

2 1.5 1 0.5 0 10

15 10

0

5 0 –5

–10 Tracking error on y-axis (m)

–10 –15

Tracking error on x -axis (m)

FIGURE 10.1 Evolution of tracking errors in three-dimension ðα 5 8Þ.

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y-axis (m)

x-axis (m)

Tracking error 10 Agent 1 Agent 2 Agent 3

0 –10

0

0.5

1

1.5 2 Time (s)

2.5

3

3.5

10 Agent 1 Agent 2 Agent 3

0 –10

0

0.5

1

1.5 2 Time (s)

2.5

3

3.5

3

3.5

Summation norm of all tracking errors 40 X: 0.84 Y: 0.9691

20 0

0

0.5

1

1.5 2 Time (s)

2.5

FIGURE 10.2 Tracking errors under relay tracking algorithm ðα 5 8Þ.

jjEðt0 Þjj 5 28:3; λ 5 0:2679; η 5 0:8007; ls 5 0:9533; then if the tracking error is assumed to achieve a smaller value R2 5 EðT Þ , 1 with designed control parameter α 5 8; the up boundary of mandatory finite    time T 5 1:3313s  s 2 γl in Theorem 10.1. according to T # lnηk jjEðt0 Þjj 2 lnjjEðT Þjj = αλl As illustrated in Fig. 10.2, at time t 5 0:84 , 1:3313s; the tracking error becomes smaller than 1. Example 10.2: Simulation results subject to disturbances This simulation is conducted to verify the effectiveness of proposed relay tracking strategy for the situation where agents are subject to disturbances. Except the agents are subject to disturbances, the other simulation conditions are same as those in Example 10.1. Fig. 10.3 exploits the evolution of tracking errors in three-dimension. The tracking error shrinks to a smaller value within finite time. Compared with Fig. 10.1, the tracking error vibrates within the certain strip due to the effect of disturbances. Fig. 10.4 depicts the summation norm of tracking errors. For this case the tracking error enters a smaller region R2 , 1 within finite time T 5 0:94s. The tracking system requires more time than the system without disturbances to successfully track the target.

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Consensus Tracking of Multi-agent Systems with Switching Topologies

3

Time (s)

2.5

Agent 1 Agent 2 Agent 3

2 1.5 1 0.5 0 10 0 –10

–15

Tracking error on y-axis (m)

–10

–5

0

10

5

15

Tracking error on x-axis (m)

FIGURE 10.3 Evolution of tracking errors subject to disturbances in three-dimension ðα 5 8Þ.

Summation norm of all tracking errors 40

X: 0.94 Y: 0.9885

20 0

0

0.5

1

1.5 2 Time (s)

2.5

3

3.5

FIGURE 10.4 Tracking errors subject to disturbances ðα 5 8Þ.

10.3 Finite-time coordination control of multiagent systems for target tracking with node failures and active replacements Finite-time consensus problem of MASs with sliding mode control methods has been widely studied in many cases [6]. These methods also suffer from chattering, as the control input is directly influenced by the switching function. Thus it can easily damage the actuator. Among many chattering mitigation techniques, the well-known high/second-order sliding mode control is developed with higher robustness and accuracy [21]. A chattering alleviation protocol for finite-time nonlinear multiagent tracking systems is proposed based on full order sliding surface and super twisting algorithm [22]. TSM is a special nonlinear switching manifold with finite time mechanism. The conventional TSM may have a singular problem. This leads to infinite control values, which is impossible to implement in practical systems.

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To solve this problem, NTSM is proposed. Traditionally, the nonsingular sliding mode surface is designed as s 5x 1 βx_p=q ; where β . 0; p; q ðp . qÞ are positive odds. Then, it has s_s # 1=β p=q x_p=q21 ηjsj 5 η0 jsj: Since x_ 5 0; x ¼ 6 0 is not a stable equilibrium, the sliding mode can be achieved in finite time. However, the settling time cannot be estimated accurately because η0 contains the time-varying term x_p=q21 . In addition, when x_ tends to be a sufficiently small value, the convergence rate becomes smaller. Recently, some researchers propose a modified sliding surface and design a continuous sinusoidal function in the nonlinear control law for fixed time consensus of the MASs [16,23,24]. The state space is divided into two different areas, and the settling time to reach the sliding surface contains the time of traveling the second area. The proposed algorithm successfully eliminates the singularity. However, as stated in Ref. [23], the traveling time across the second area cannot be estimated precisely. For target tracking applications the priority is to minimize the tracking time. In order to achieve this object the key issues include accurate estimation of tracking time, determination of Voronoi diagrams, and selection of active replaced agents. Based on the existing results and motivated by the above observation, this section is devoted to developing new algorithms for second-order systems with node failures and active replacements. Compared with the previous works, the contributions of this section are threefold: 1. A generalized dynamic Voronoi diagram based on tracking time with consideration of velocity states is presented. For second-order tracking systems the agent closest to target may not be the optimal one that can successfully track it in shortest time. A target resides in the Voronoi cell where the corresponding Voronoi site agent is available to track it in a shortest time. Different from the traditional distance-based Voronoi diagram adopted in [12
15], the generalized dynamic boundaries of Voronoi cells are determined according to the time needed for successful tracking. This is the foundation for the following event-triggered coordination control strategy. 2. Event-triggered coordination control strategy is proposed for different kinds of events such as initial tracking agents’ determination for multiple targets or node failures. Normally, the term “event-triggered control” is used to describe the mechanism where agents broadcast their states or update controller only when the condition is triggered. Then, the agents take the sample of information or controllers take the corresponding action for the sake of energy saving [25
27]. In this chapter, “event-triggered coordination control strategy” means that when events like node failures occur, the coordination control strategy is triggered. When targets enter this monitored area, nearby agents are triggered to track the target. Corresponding coordination strategy is that nearby Voronoi site agents distributively estimate their tracking time and decide the joining agents.

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Consensus Tracking of Multi-agent Systems with Switching Topologies

If multiple targets occur in a same local area, the strategy plays a significant role in coordinating initial tracking agents for each target. Then, if the target moves into a new Voronoi cell, the active replacement strategy is triggered to initiatively replace one of the original tracking agents. Meanwhile, when node failures occur, the coordination strategy decides which agent joins in. 3. An efficient modified nonsingular TSM control method is proposed to accurately estimate the tracking time, which is the key basis for generalized Voronoi diagram and proposed coordination control strategy. Recently, some researchers propose a newly designed nonsingular terminal sliding surface and design a continuous sinusoidal function in the nonlinear control law for fixed time consensus of the MASs [16,23,24]. The state space is divided into two different areas and the settling time to reach sliding surface contains the time of traveling the second area. The proposed algorithm successfully eliminated the singularity. Nevertheless, as stated in Ref. [23], the traveling time across the second area cannot be estimated precisely since the convergence rate contained a time-varying term. This section designs a novel continuous function in the sliding mode controller to eliminate the singularity and the time-varying term in the convergence rate at the same time. This guarantees the target tracking of the MAS to be achieved in finite time, which is able to be estimated precisely.

10.3.1 Tracking problem description and fundamentals Consider a multiagent tracking system , in which agent i has dynamics given by x_i ðtÞ 5 vi ðtÞ; v_i ðtÞ 5 f ðxi ; vi ; tÞ 1 ui ðtÞ;

ð10:29Þ

where xi ðtÞ; vi ðtÞAℝm are position and velocity state vectors of ith agent, ui ðtÞAℝm is the control input, f ðxi ; vi ; tÞ is the nonlinear dynamics of ith agent. The agents are expected to track a target with dynamics: x_0 ðtÞ 5 v0 ðtÞ; v_0 ðtÞ 5 f ðx0 ; v0 ; tÞ;

ð10:30Þ

where x0 ðtÞ; v0 ðtÞAℝm are position and velocity state vectors of the target. f ðx0 ; v0 ; tÞ is the dynamics of the target. Assumption 10.1: Agents’ communication topology graph G contains a spanning tree and at least one tracking agent can access the target’s information. Before addressing the main results, the following definition and lemmas are introduced.

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Definition 10.2: For the multiagent tracking problem, successful tracking is said to be achieved in finite time T if Nf X   xi ðtÞ 2 x0 ðtÞ2 5 0; i51 Nf X

  vi ðtÞ 2 v0 ðtÞ2 5 0;

’t $ T; ð10:31Þ ’t $ T:

i51

Lemma 10.2: [1,28] Consider a system x_ 5 f ðxÞ with f ð0Þ 5 0 and suppose there exists a continuous function V:D-ℝ such that the following conditions hold (1) V is positive definite and (2) there exist real numbers c . 0 and αAð0; 1Þ such that V_ ðxÞ 1 cV α ðxÞ # 0; xAD. Then the origin is a finitetime-stable equilibrium. Moreover, the settling time depending on the initial state xð0Þ 5 x0 satisfies T ðxÞ # 1=ðcð1 2 αÞÞV 12α ðx0 Þ: Lemma 10.3: (Comparison Lemma) [29] Let f(t, u) be continuous on an open (t, u)—set E and u 5 u0 ðtÞ the maximal solution of u_ 5 f ðt; uÞ; uðt0 Þ 5 u0 . Let ½t0 ; t0 1 T Þ be the maximal interval of existence of the solution u(t) and suppose uðtÞAE; ’tA½t0 ; t0 1 T Þ. Let vðtÞ be a continuous function whose upper right-hand derivative satisfies D1 vðtÞ # f ðt; vðtÞÞ. Then, on a common interval existence of u0 ðtÞ and vðtÞ; vðtÞ # u0 ðtÞ. Generalized dynamic Voronoi diagram: With traditional dynamic Voronoi diagram, an area is partitioned into Voronoi cells in which agents are set as Voronoi sites on the basis of distance. For target tracking applications, velocity is involved. Our purpose is to track the target within finite time as short as possible. Therefore a generalized dynamic Voronoi diagram is designed. n o   Vτ ðai Þ 5 xAXjT ðx; ai Þ # T x; aj for all j 6¼ i : ð10:32Þ where T ðx; ai Þ denotes the finite time for site agent ai to successfully track point x. Initially, deployed agents are static, namely, the initial velocities are zero. Hence, at the time before any target intrudes, the generalized Voronoi diagram is the same as basic Voronoi diagram. Once a target is detected, the nearby agents estimate tracking times T ðx0 ; xi Þ and reside Voronoi cell is determined according to (10.32). Apparently, it is critical to accurately estimate the tracking time. This is realized by designing a modified nonsingular TSM controller, which we will introduce in the following sections. It should be noted that the number of tracking agents Nf is less than the number of agents deployed in this area Na : Assume there are mostNt targets  would intrude this monitored area, then the constraint is Nt # floor Na =Nf :

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Consensus Tracking of Multi-agent Systems with Switching Topologies

10.3.2 Event-triggered coordination strategy This section considers multiple targets tracking scenario where a number of smart agents are initially deployed in an area. There are mainly three major events during the entire tracking process. The event-triggered coordination strategy includes the following cases: Coordination of initial tracking agents: Once a target is detected, the Voronoi cells are repartitioned based on generalized dynamic Voronoi diagram. The site agent i where the target resides and nearby site agents is set to track the target. If a site agent detects multiple targets at the same time, it determines which target to track based on estimated finite tacking time. For example, there are two targets that invade this area and are set to be tracked by four agents, which form two clusters as illustrated in Fig. 10.5. The three connected agents are triggered to recruit a new agent to join tracking. As shown in Fig. 10.5, agent 4 is able to receive the recruit information from both of two clusters. Agent 4 estimates its finite time needed to be consensus tracking for two clusters. The coordination strategy is that agent 4 joins the cluster that costs less finite tracking time. This is different from the traditional distance
based coordination strategy, in which the one with shortest distance joins tracking. If velocity is considered, the one with shortest distance is not always the one with shortest tracking time. Specifically, in Fig. 10.5, agent 4 is at the same velocity direction of agent 3 in cluster 2 but is at the opposite velocity direction of agent 3 in cluster 1. It could cost less time for agent 4 to be consensus tracking with cluster 2. Thus agent 4 may join cluster 2 rather than cluster 1 even though it is closer to cluster 1 with regard to distance. Coordination for active agent replacements: During the tracking process the Voronoi diagram dynamically changes in a local way as depicted in Fig. 10.6. When the target moves into a new Voronoi cell, the new corresponding Voronoi site agent replaces one of the original tracking agents. This process is called active replacement. With active agent replacements the tracking time could be reduced according to the principle of Voronoi cells. In Fig. 10.6 the solid line FIGURE 10.5 Illustration of coordination of initial tracking agents.

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223

FIGURE 10.6 Illustration of active replacement.

represents the initial Voronoi diagram and dash line represents the updated Voronoi diagram with agent 1 at new position according to traditional partition method. If velocity of the target is considered, the boundary of generalized Voronoi diagram between agent 1 and agent 2 could be the dotted line. From the perspective of tracking time, target has entered into Voronoi cell V ða2 Þ; which implies agent 2 can reach successful tracking of target in a shorter time than agent 1. In order to achieve faster successful tracking, agent 2 initiatively replaces one of the original tracking agents. Coordination for node failures: During the tracking process, some tracking agents might be damaged or out of power, etc. Then, the corresponding tracking team is triggered to recruit new agents. The agents that receive the recruit information estimate their finite times to reach successful consensus tracking and send feedback to the recruit agent. The recruit agent selects the one with shortest tracking time to replace the damaged one. This process is called passive replacement. For example, a target is tracked by four agents initially as illustrated in Fig. 10.7. If agent 4 occurs failure, its connected agent is triggered to recruit a new agent. As shown in Fig. 10.7, nearby agents 5 and 6 can both receive the recruit information. They estimate the finite time needed to be consensus tracking and interact with agent 3. Then, agent 3 determines which one joins the tracking. Geographically, agent 5 is closer to agent 3. However, agent 6 is on the moving direction of agent 3, while agent 5 is on the opposite moving direction of agent 3. Therefore for second-order tracking systems, agent 6 may require less time to achieve consensus with the original tracking system.

224

Consensus Tracking of Multi-agent Systems with Switching Topologies FIGURE 10.7 Illustration of coordination for node failures.

Remark 10.2: In this chapter, it is assumed that node failures only occur in the terminal tracking agents. For example, in Fig. 10.7, agent 4 is a terminal agent and is possible to occur failures. Meanwhile, other agents are not allowed to be out of order. Otherwise, the original tracking agents connected to it will be isolated. If agent 2 occurs failure, agent 3 and agent 4 will be isolated. This can also be solved by adopting jointly connected topology method. However, this is out of the scope of this chapter since we focus on designing event-triggered coordination strategies.

10.3.3 Nonsingular terminal sliding mode controller design From the above analysis, it is clear that the accurate estimation of finite tracking time is of great significance. In this section a novel nonsingular TSM controller is designed. With proposed controller, successful tracking can be achieved in finite time, which is possible to be estimated accurately. Denote the comprehensive tracking error of ith agent with target and other agents as Nf X   Exi ðtÞ 5 aij xi ðtÞ 2 xj ðtÞ 1 bi ðxi ðtÞ 2 x0 ðtÞÞ; j51 Nf X   Evi ðtÞ 5 aij vi ðtÞ 2 vj ðtÞ 1 bi ðvi ðtÞ 2 v0 ðtÞÞ:

ð10:33Þ

j51

Then, the dynamics of tracking error system is E_ xi ðtÞ 5 Evi ðtÞ; Nf X    E_ vi ðtÞ 5 aij ui ðtÞ 2 uj ðtÞ 1 f ðxi Þ 2 f xj 1 bi ðf ðxi Þ 2 f ðx0 ÞÞ: j51

ð10:34Þ

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Denote the overall tracking errors in terms of position and velocity as h iT T T T ðtÞ; Ex2 ðtÞ; . . .; ExN ð t Þ ; Ex ðtÞ 5 Ex1 f ð10:35Þ h iT T T T ðtÞ; Ev2 ðtÞ; . . .; EvN ð t Þ : Ev ðtÞ 5 Ev1 f Then, the error dynamics of the interconnected tracking system are governed by E_ x ðtÞ 5 Ev ðtÞ; ð10:36Þ E_ v ðtÞ 5 LðU ðtÞ 1 ΔF ðtÞÞ;   where Lk 5 ðℒ 1 BÞ  Im: , U ðtÞ 5 uT1 ðtÞ; uT2 ðtÞ; . . .; uTN ðtÞ and ΔF ðtÞ 5 h i    T ðf ðx1 Þ2f ðx0 ÞÞT ; ðf ðx2 Þ2f ðx0 ÞÞT ; . . .; f xNf 2f ðx0 Þ : The states of ith tracking agent xi ðtÞ; vi ðtÞ track the states of target x0 ðtÞ; v0 ðtÞ; that is; xi ðtÞ-x0 ðtÞ; vi ðtÞ-v0 ðtÞ in finite time if the overall tracking errors Ex ðtÞ and Ev ðtÞ converge to zero in finite time. The detailed proof is omitted in this chapter, interested readers can refer to Ref. [30]. It is apparent that the convergence of overall tracking errors requires the elements Exi ðtÞ and Evi ðtÞ converge to zero in finite time. Thus we analyze the finite time stability for tracking agent i in the following. The sliding manifold surface is designed as p=q

si 5 Exi ðtÞ 1 γEvi ðtÞ;

ð10:37Þ

where γ is a positive parameter, p and q are positive odds satisfying 1 , p=q , 2: During the whole tracking process, when triggered events happen at tk ; kAℕ; agents in the system and/or communication topologies change. The tracking system is divided into subsystems at time intervals ½tk ; tk11 Þ: At each time interval, agent i could receive information from neighboring agents. Thus at time tA½tk ; tk11 Þ; the nonsingular sliding mode controller for ith tracking agent is designed as following. !21 Nf X ui ðtÞ 5 aij 1bi "

j51

2

Nf X j51

# Nf X   aij f ðxi Þ 2 f xj 2 bi ðf ðxi Þ 2 f ðx0 ÞÞ 1 aij uj ðtÞ 1 uip ðtÞ ; 

j51

ð10:38Þ where uip ðtÞ is a new modified finite time nonlinear control law designed based on the novel continuous function ϑðEvi Þ as presented in (10.40).

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q 22p=q 12p=q Evi ðtÞ 1 η sign ðsi ÞϑðEvi ÞEvi ðt Þ ; γp   8 12p=q > π=2 2 arctan E ðtÞ > vi <   ; if jEvi ðtÞj # 1 p=q21 ϑðEvi Þ 5 ðtÞ sin 0:9033Evi > > : 1; otherwise: uip ðtÞ 5 2

ð10:39Þ

ð10:40Þ

It should be noted that the number 0.9033 in ϑðEvi Þ is a specific designed number, and it must be lessthan 1. It guarantees the continuity of ϑðEvi Þ.  When jEvi ðtÞj 5 1; ϑðEvi Þ 5 π=2 2 arctanð1Þ =ðsinð0:9033ÞÞ 5 1. Therefore the controller is continuous. At the same time, in the designed controller p=q21 (10.38), singularity is eliminated. When jEvi ðtÞj 5 0; Evi ðtÞ 5 0; 12p=q Evi 5 N: Since 0.9033 , 1, according to L’Hospital’s rule, it has   p=q21 p=q21 Evi ðtÞ 3 1= sin 0:9033 Evi ðtÞ 5 1=0:9033 $ 1 as Evi ðtÞ-0; which is an important  characteristic  in thederivation of finite time. Also, when 12p=q 12p=q Evi ðtÞ-0; π=2 2 arctan Evi ðtÞ 3 Evi ðtÞ 5 1, through which the singularity is eliminated. The modified nonsingular TSM controller with designed continuous function ϑðEvi Þ eliminates the singularity and makes it possible to accurately estimate the finite tracking time at the same time. This is a significant advantage compared to existing literatures on nonsingular TSM controllers, such as Refs. [16,23,24].

10.3.4 Finite time analysis with modified sliding mode control This section analyzes the tracking performance of the MAS with proposed event-triggered coordination control strategy. Estimation of tracking time with designed nonsingular sliding mode controller is presented. Theorem 10.3: Consider the multiagent tracking system (10.34) with communication topology satisfying Assumption 10.1. With proposed NTSM control law (10.38) and (10.39), the finite time that needed for agent i to achieve successful tracking of the target is jjsi ð0Þjj : Ti #  η γp=q

ð10:41Þ

Proof: Choose the Lyapunov function for agent i as Vi ð t Þ 5

1 T s si : 2 i

ð10:42Þ

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The derivative of Lyapunov function (10.42) along (10.37) is V_ i ðtÞ 5 sTi s_i   γp p=q21 _ Evi 5 sTi Evi ðtÞ 1 ðtÞEvi ðtÞ q   γp p=q21 q 22p=q Evi Evi 5 sTi ðt Þ ðtÞ 1 E_ vi ðtÞ : q γp

ð10:43Þ

Substituting controller (10.38) and (10.39) into tracking error system (10.34), we obtain

q 22p=q 12p=q E ðtÞ 1 ηsign ðsi ÞϑðEvi ÞEvi ðt Þ : ð10:44Þ E_vi ðtÞ 5 2 γp vi When jEvi ðtÞj . 1; ϑðEvi Þ 5 1: Substituting (10.44) and (10.39) into (10.43), the derivative of Lyapunov function satisfies the inequality (10.45). γp V_i ðtÞ 5 2 η sTi sign ðsi Þ q ð10:45Þ γp pffiffiffi 1=2 52η 2 Vi ð t Þ q When 0 , jEvi ðtÞj # 1, it can be obtained that ϑðEvi Þ $ 1 from (10.40). Then, V_i ðtÞ 5 2 #2

N X γp η sTi sign ðsi ÞϑðEvi Þ q i51 N X γp η sTi sign ðsi Þ q i51

ð10:46Þ

1

γp pffiffiffi 2 2Vi ðtÞ: 52η q p=q21 12p=q tÞj 50; E ðtÞ 5 0; Evi ðtÞ 5 N. Recalling the fact jEvi ð vi  p=q21 p=q21 5 1=0:9033 $ 1, π=2 2 arctan Evi ðtÞ 3 1= sin 0:9033 Evi ðtÞ

When that

12p=q

12p=q

ðtÞ 5 1 as Evi ðtÞ-0. It follows γp pffiffiffi 1=2 V_ i ðtÞ # 2 η 2Vi ðtÞ: ð10:47Þ q pffiffiffi 1=2 2Vi ðtÞ exists for all cases. Overall, the inequality V_ i ðtÞ # 2 η γp q According pffiffiffi to Lemma    10.2,the tracking time  Tiof agent i is obtained as 1=2 Ti # 2Vi ð0Þ = η γp=q 5 jjsi ð0Þjj= η γp=q . & ðEvi

ðtÞÞ 3 Evi

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Consensus Tracking of Multi-agent Systems with Switching Topologies

Remark 10.3: From the above analysis, it can be seen that the finite tracking time Ti for agent i can be estimated with initial values of slidingp mode surffiffiffi face si . The main advantage is that the convergence rate η γp=q 2 is constant rather than time-varying, which makes it able to estimate the finite time accurately. The sliding mode surface si defined in (10.37) contains overall position and velocity errors, which is available to agent i in a distributed way. Therefore each agent could estimate the finite tracking time distributively. Then, the proposed event-triggered coordination strategy in Section 10.3.2 is applicable.

10.3.5 Stability analysis of systems with agent replacements When agent replacement occurs in agent i, according to the coordination strategy introduced in Section 10.3.2, the tracking times of damaged agent and new joined agent are different. This leads jump in the tracking time, which is directly determined by sTi ðtk Þsi ðtk Þ. Namely,     sTi tk1 si tk1 5 sTi ðtk Þsi ðtk Þ 1 ΔsTi ðtk Þ si ðtk Þ: ð10:48Þ In terms of node failures, ΔsTi ðtk Þsi ðtk Þ could be positive or negative. With regard to active replacements, ΔsTi ðtk Þsi ðtk Þ is negative since the tracking time decreases. Since the multiagent tracking system (10.34) is divided into subsystems over time intervals ½tk ; tk11 Þ; it turns to be a switched system associated with impulsive effect. In the stability analysis for switched systems, the following definition is required: Let xðtÞ be a solution of x_ðtÞ 5 f ðt; xðtÞÞ; t $ t0 ; xðt0 Þ 5 x0 ; t0 $ 0 and V ðt; xðtÞÞ be a given function. Denote D1 V ðt; xðtÞÞ as the upper right-hand Dini derivative of V ðt; xðtÞÞ; that is, V ðt 1 h; xðt 1 hÞÞ 2 V ðt; xðtÞÞ : h h-01

D1 V ðt; xðtÞÞ 5 lim sup

Theorem 10.4: Consider the multiagent tracking system (10.34) with agent replacements and jumps (10.48) at switching times. The agents are able to track the target infinite time (10.49) under the effect of proposed modified nonsingular terminal sliding control law (10.38) and (10.39). ( ) k X   q  ð10:49Þ ðjjsi ð0ÞÞjj 1 Δjjsi tj jj : T # sup ηγp j51 Proof: Without loss of generality, suppose the successful tracking is achieved in time interval ½tk ; tk11 Þ Choose the Lyapunov function for

Finite time stability analysis and coordination strategies Chapter | 10

229

agent i over time interval ½tk ; tk11 Þ as the same form (10.42). Then, the derivative of Lyapunov function satisfies the inequality (10.50). γp pffiffiffi 1=2 2Vi ðtÞ ð10:50Þ DVi1 ðtÞ # 2 η q Based on comparison Lemma 10.3 and the inequality (10.50), when tA½tk ; tk11 Þ ηγp 1=2 1=2 ð10:51Þ Vi ðtÞ # Vi ðtk Þ 2 pffiffiffi ðt 2 tk Þ 2q At switching times tk ; kAℕ1 , agent replacement occurs, resulting in jumping of the Lyapunov function. ηγp 1=2 1=2   1=2 Vi ðtÞ # Vi tk2 2 ΔVi ðtk Þ 2 pffiffiffi ðt 2 tk Þ 2q ð10:52Þ ηγp 1=2  1  1=2 # Vi tk21 2 ΔVi ðtk Þ 2 pffiffiffi ðt 2 tk21 Þ 2q After iterations along with tk , we get 1=2

1=2

Vi ð t Þ # Vi ð t 0 Þ 1

k X j51

1=2 

ΔVi

 ηγp tj 2 pffiffiffi ðt 2 t0 Þ 2q

ð10:53Þ

Without loss of generality, we assume the initial time t0 5 0.  It can be pffiffiffi  1=2 seen from (10.53) that there exists a time Ti # 2q=ηγp Vi ð0Þ 1  Pk  1=2   tj such that Vi ðtÞ 5 0; ’t $ Ti : For the whole system, the j51 ΔVj    successful tracking can be achieved in T 5 sup Ti . & Remark 10.4: If node failures occur, the Lyapunov function could jump up, that is, ΔVi ðtk Þ might be greater than zero. According to the estimation of  Ti , this leads to a larger tracking time. On the other hand, in light of the fact that ΔVi ðtk Þ , 0 if agent i is actively replaced, it is apparent that proposed active replacement strategy facilitates decreasing of tracking time.

10.3.6 Numerical simulations Example 10.3: Simulation of coordination strategy for system with active replacements In order to verify the feasibility of developed theoretical results, simulations are carried out. In this section, we present numerical examples to show the effectiveness and correctness of the main results derived previously. Consider a scenario where a group of three agents track a maneuvering target

230

Consensus Tracking of Multi-agent Systems with Switching Topologies

in a two-dimensional space. Initially, 40 agents are deployed in a 1500 3 1500 m2 area. The number of tracking agents is set as Nf 5 3. The nonlinear dynamics of the tracking agents and target are f ðt; xðtÞÞ 5 3:5 sin ð0:01xðtÞÞ 1 0:2 cos ð0:02vðtÞÞ 1 0:2 sin ð2tÞ:

ð10:54Þ

The time of the target breaks into this domain is considered as initial time. Coordinate of the location where the target enters this area is x0 ð0Þ 5 ð0; 100Þ. The velocity of the target is v0 ð0Þ 5 ð30; 20Þ. The control parameter is chosen as η 5 60; γ 5 0:3; p 5 5; q 5 3. From Fig. 10.8, it can be seen that the agents finally successfully track the target. The trajectories suffer from some discontinuities, which reflect the active agent replacements. Furthermore, jjsjj of the system with and without active replacement strategy is depicted in Fig. 10.9. Apparently, under the same initial condition, the proposed active replacement strategy facilitates convergence of the tracking error. For this simulation the initial value of jjsjj is 606. According to the condition (10.41) derived in Theorem 10.4, the theoretical convergence time of system without agent replacement should be T # 20:2s. With proposed active agent replacement strategy the value of jjsjj shrinks about 275 initiatively. According to the condition (10.49) derived in Theorem 10.4, the theoretical convergence time of system turns  to be T # 11s. The actual tracking times with and without using active replacement strategy are 7.22 and 15.42s, respectively. This evidently Tracking trajectories 1500 Target Agent 1 Agent 2 Agent 3

y (m)

1000

500

0

0

500

1000 x (m)

FIGURE 10.8 Tracking trajectories with active replacements.

1500

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700 Without agent replacement Active agent replacement

600

500

s

400

300

200

100

0 0

5

10

15

20

25

30

35

40

45

50

Time (s) FIGURE 10.9 Comparison of jjsjj for systems with and without active replacements.

verifies the correctness of conditions in Theorem 10.4 and the effectiveness of proposed active replacement strategy. Example 10.4: Simulation of coordination strategy for system with node failures and active replacements. This section illustrates the control performance of tracking system with active and passive agent replacements at the same time. The simulations are conducted under the same initial condition with that in Section 10.3. The tracking trajectories of multiagent system with node failures and active agent replacements are presented in Fig. 10.10, while the trajectories without active agent replacements are shown in Fig. 10.11. Comparison of jjsjj under the effect of node failures with and without active replacement strategy is depicted in Fig. 10.12. With effect of active replacements, jjsjj shrinks. Meanwhile, when node failures occur, jjsjj jumps up as shown in Fig. 10.12. In order to clearly show the effectiveness of proposed strategy, node failures are set to occur at same times in the two simulations. Although the performance of system without active replacements is unsatisfactory, the agents successfully track the target for both cases. Under the effect of designed NTSM controller, the agents are able to track the target even with node failures.

232

Consensus Tracking of Multi-agent Systems with Switching Topologies Tracking trajectories 1500 Target Agent 1 Agent 2 Agent 3

y (m)

1000

500

0

0

500

1000

1500

x (m) FIGURE 10.10 Tracking trajectories with node failures and active agent replacements.

Tracking trajectories 1500 Target Agent 1 Agent 2 Agent 3

y (m)

1000

500

0

0

500

1000

1500

x (m) FIGURE 10.11 Tracking trajectories with node failures and without active agent replacements.

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800 Node failures and active replacements Node failures

700 600

s

500 400 300 200 100 0 0

5

10

15

20

25

30

35

40

45

50

Time (s) FIGURE 10.12 Comparison of jjsjj for systems with and without active replacements suffers from node failures.

As seen from Fig. 10.12, the convergence time of the case with node failures and active replacements is generally the same as that without agent replacements in Section 10.3. This is because that the passive increase of jjsjj caused by node failures is compensated by the active decrease generated by active replacements. However, if the system suffers from node failures without active replacements, it needs more time to track the target. For this simulation, it costs almost double time to achieve successful tracking. Example 10.5: Simulation of coordination strategy for multiple targets. This section conducts simulation of coordination strategy for multiple targets. The time of the first target breaks into this domain is considered as initial time. The first target enters this area at position (0,100) with velocity (30,20). Then after 3s, another target intrudes this area at position (0,500) with velocity (20,10). The control parameter is chosen as η 5 40; γ 5 0:32; p 5 5; q 5 3: The control performance and jjsjj of tracking system with multiple targets are shown in Figs. 10.13 and 10.14, respectively. As can be seen from the results, the event-triggered coordination strategy is effective for tracking of multiple targets.

234

Consensus Tracking of Multi-agent Systems with Switching Topologies 1500 Target Agent 1 Agent 2 Agent 3

y (m)

1000

500

0

0

500

1000

1500

x (m) FIGURE 10.13 Tracking trajectories for multiple targets.

1400 Target 1 Target 2

1200

1000

s

800

600

400

200

0 0

5

10

15

20

25 Time (s)

FIGURE 10.14 jjsjj for two clusters.

30

35

40

45

50

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10.4 Conclusion A new switching tracking scheme for MASs is presented and its performance over a finite time interval is studied in this section. It is proved that the relay tracking strategy is in favor of reducing tracking time since at each switching time the tracking error witnesses an abrupt decreasing. A quantitative conclusion on tracking time interval, closely related to the control parameter and length rate of stable subsystems, is drawn for the relay tracking system. Given the threshold that the tracking error is expected to lie in, it is able to estimate the required time interval for the relay tracking system. Examples are provided to demonstrate the relative accuracy of estimated time interval and the effectiveness of proposed relay tracking strategy with or without external disturbances. In addition, we have proposed a modified NTSM control scheme for multiagent tracking systems. With designed continuous function in the control law, the singularity is eliminated and the convergence finite time can be estimated. This makes it realizable to establish generalized Voronoi diagram considering velocity of second-order agents. Event-triggered coordination strategies are presented for threat defense applications. Simulation results show the correctness of finite tracking time estimation with designed controller. The active replacement strategy effectively decreases the tracking time. The coordination strategies are feasible for multiple threat defense with agent failures and active replacements.

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766. [2] F. Amato, G.D. Tommasi, A. Pironti, Necessary and sufficient conditions for finite-time stability of impulsive dynamical linear systems, Automatica 49 (8) (2013) 2546
2550. [3] Y. Zhang, Y. Yang, Y. Zhao, G. Wen, Distributed finite-time tracking control for nonlinear multi-agent systems subject to external disturbances, Int. J. Control 86 (1) (2013) 29
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2206. [5] L. Lee, K.I. Kou, W. Zhang, J. Liang, Y. Liu, Robust finite-time boundedness of multiagent systems subject to parametric uncertainties and disturbances, Int. J. Syst. Sci. 47 (10) (2016) 2466
2474. [6] M. Ghasemi, S.G. Nersesov, Finite-time coordination in multiagent systems using sliding mode control approach, Automatica 50 (4) (2014) 1209
1216. [7] C. Suttirak, C. Pukdeboon, Finite-time convergent sliding mode controllers for robot manipulators, Appl. Math. Sci. 7 (63) (2013) 3141
3154. [8] H. Li, X. Liao, G. Chen, Leader-following finite-time consensus in second-order multiagent networks with nonlinear dynamics, Int. J. Control Autom. Syst. 11 (2) (2013) 422
426. [9] D. Meng, K.L. Moore, Robust iterative learning control for nonrepetitive uncertain systems, IEEE Trans. Autom. Control 62 (2) (2017) 907
913.

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[10] D. Meng, K.L. Moore, Robust cooperative learning control for directed networks with nonlinear dynamics, Automatica 75 (2017) 172
181. [11] E. Bakolas, P. Tsiotras, Relay pursuit of a maneuvering target using dynamic Voronoi diagrams, Automatica 48 (9) (2012) 2213
2220. [12] L. Dong, S. Chai, B. Zhang, S.K. Nguang, X. Li, Cooperative relay tracking strategy for multi-agent systems with assistance of Voronoi diagrams, J. Franklin Inst. 353 (17) (2016) 4422
4441. [13] L. Dong, S. Chai, B. Zhang, S.K. Nguang, A. Savvaris, Stability of a class of multiagent tracking systems with unstable subsystems, IEEE Trans. Cybern. 47 (8) (2017) 2193
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2326. [21] A. Levant, Universal single-input-single-output (SISO) sliding-mode controllers with finite-time convergence, IEEE Trans. Autom. Control 46 (9) (2001) 1447
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2909. [25] Z.G. Wu, X. Yong, R. Lu, Y. Wu, T. Huang, Event-triggered control for consensus of multiagent systems with fixed/switching topologies, IEEE Trans. Syst. Man Cybern. Syst. 48 (10) (2018) 1736
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[29] X. Lin, Y. Zheng, Finite-time consensus of switched multiagent systems, IEEE Trans. Syst. Man Cybern. Syst. 47 (7) (2017) 1535
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Further reading A. Saberi, E. Peymani, H.F. Grip, Homogeneous networks of non-introspective agents under external disturbances
HN almost synchronization, Automatica 52 (2015) 363
372. S. Saat, S.K. Nguang, Nonlinear HN output feedback control with integrator for polynomial discrete-time systems, Int. J. Robust Nonlinear Control 25 (7) (2015) 1051
1065. S. Chae, S.K. Nguang, SOS based robust HN fuzzy dynamic output feedback control of nonlinear networked control systems, IEEE Trans. Cybern. 44 (7) (2014) 1204
1213.

Chapter 11

Conclusions and future research Chapter Outline 11.1 Conclusions 239 11.1.1 Consensus tracking control of systems with continuously switching topologies 239 11.1.2 Relay tracking control of systems with agent replacements 240

11.2 Future research 11.2.1 General nonlinear multiagent systems 11.2.2 Cooperative control algorithms based on deep learning References

240 241

241 241

In practical applications of multiagent tracking systems, communication topologies between agents are time-varying and the original tracking agents could be replaced by others. This book mainly deals with these two issues and some significant results have been presented. However, there are still some open issues to be solved in the future research. In this chapter, concluding remarks are given and suggestions for future research work are discussed.

11.1 Conclusions 11.1.1 Consensus tracking control of systems with continuously switching topologies Continuously switching topologies means that the communication topology of the multiagent systems (MASs) shifts continuously rather than switches among several different structures. We derive necessary and sufficient conditions associated with eigenvalues and controller design principles, which ensure successful tracking of the nonlinear for first-order, second-order, and high-order MASs with continuously switching topologies. Then, we model the continuously switching topologies by a finite number of constant Laplacian matrices together with their corresponding scheduling functions, which is called polytopic model. With proposed polytopic model, sufficient conditions for the existence of a tracking strategy of a high-order tracking system have been expressed in terms of linear matrix inequalities. Consensus Tracking of Multi-agent Systems with Switching Topologies. DOI: https://doi.org/10.1016/B978-0-12-818365-6.00011-2 © 2020 Elsevier Inc. All rights reserved.

239

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Consensus Tracking of Multi-agent Systems with Switching Topologies

Then, tracking control of systems with time delays is investigated and the relationship between control parameter and the maximum allowable communication delays is tabulated. Furthermore, sliding mode control method is applied to solve this nonlinear tracking problem under a time-varying topology. It is proved that the proposed polytopic model structure makes it able to apply the consensus tracking scheme in a wider range.

11.1.2 Relay tracking control of systems with agent replacements This book presents a relay tracking control strategy, where some initial tracking agents are replaced by new agents. Namely, during the tracking process, not only the topology but also the tracking agents switch, which is significantly different from the traditional switching topologies. Moreover, the replacements of agents and switching topologies may lead the tracking system to be unstable. We derive some sufficient conditions that guarantee overall successful tracking despite the existence of unstable subsystems and/or for systems subject to disturbances. What is more, it is practical that some of the tracking agents are damaged without new agents join the tracking immediately. In this case the number of agents is time-varying and the dimension of Laplacian matrix of the MAS changes. In order to solve this problem, a new type of average Lyapunov function is constructed to compensate the unmatched dimension of communication topology. Then, a novel impulse-time-dependent average Lyapunov function is designed and sufficient conditions based on Bessel Legendre inequality guaranteeing successful tracking of linear systems under a varying number of agents and time delays are obtained. In order to tackle nonlinear tracking systems with variable number of agents and time delays, reciprocally convex Lemma and a more relaxable switched technique are employed to achieve a less conservative switched stability condition. Finally, this book explores finite time stability analysis and coordination strategies of multiagent relay tracking systems. Stability conditions for a class of MASs with proposed relay tracking strategy over a finite time interval are achieved despite impulsive effects on the overall tracking errors. The relationship of finite time interval against desired overall tracking errors and control parameter is derived quantitatively for the multiagent relay tracking system. Then, we propose a novel nonsingular terminal sliding mode control method to accurately estimate the tracking time, based on which finite time coordination strategies of second-order multiagent relay tracking systems are presented.

11.2 Future research There exist many challenging open questions in the topic of MASs. In this section, we discuss some natural extensions of the work presented in this book.

Conclusions and future research Chapter | 11

241

11.2.1 General nonlinear multiagent systems The nonlinear systems investigated in this book are assumed to satisfy the Lipschitz condition. However, the study of general nonlinear systems that do not necessarily satisfy the Lipschitz condition is more valuable. Based on the finite time Lyapunov stability theorem and matrix theory, Hua et al. [1] prove that the non-Lipschitz continuous control law guarantees the finite time consistency of high-order uncertain nonlinear MASs. Dai et al. [2] consider nonlinear stochastic fractional integral differential equations under nonLipschitz conditions, which are general and include many random (fractional) integral differential equations discussed in the literature. For nonlinear systems, Kelly and Lord [3] introduce a class of adaptive time-stepping strategies for stochastic differential equations with non-Lipschitz drift coefficients.

11.2.2 Cooperative control algorithms based on deep learning Distributed control algorithms are particularly important for the research of MASs. Most of the research in this book is focused on classic distributed control algorithms. In addition, first-order sliding mode control algorithms, terminal sliding mode control, and high-order sliding mode control algorithms are used to design distributed control protocols. However, more intelligent distributed control protocols, such as deep learning and reinforcement learning algorithms, are worth further exploring. Jiang et al. [4] propose a new method based on multiagent deep reinforcement learning for multitarget tracking to solve problems in existing tracking methods, such as different numbers of targets, noncausality and nonreal-time. Gupta et al. [5] propose human-learning strategies to adjust multiagent behavior in high-dimensional environments. Then, the behavior of the agent can be accurately predicted. In the field of multiagent collaborative control algorithms, deep learning and machine learning will be the future development trend.

References [1] C.C. Hua, X. You, X.P. Guan, Leader-following consensus for a class of high-order nonlinear multi-agent systems, Automatica 73 (2016) 138 144. [2] X. Dai, W. Bu, A. Xiao, Well-posedness and em approximations for non-Lipschitz stochastic fractional integro-differential equations, J. Comput. Appl. Math. 356 (2019) 377 390. [3] C. Kelly, G.J. Lord, Adaptive timestepping strategies for nonlinear stochastic systems IMa J Numer Anal. 38 (3) (2018) 1523 1549. [4] M. Jiang, T. Hai, Z. Pan, H. Wang, Y. Jia, C. Deng, Multi-agent deep reinforcement learning for multi-object tracker, IEEE Access. 7 (2019) 32400 32407. [5] J.K. Gupta, M. Egorov, M. Kochenderfer, Cooperative multi-agent control using deep reinforcement learning, in: International Conference on Autonomous Agents Multiagent Systems, 2017.

Appendix A A.1 Notations Throughout this book, the following notations are used. C , hx,yi A . 0 (A is a symmetric matrix) ’ A $ 0 (A is a symmetric matrix) N eA A 2 = diag (x) sign 1p 0p Im 0m 3 n max min ℝm ℝm 3 n xT jxj xγ jjxjj xi A21 jjAjj Aij colðUÞ λi ðAÞ L2 ½0; NÞ jjvjjL2

Subset Union Inner product of vectors x, y A is positive definite For all A is positive semidefinite Infinity Matrix exponential of square matrix A Element of Not element of Diagonal matrix with the vector x on its diagonal Sign function P 3 1 vector of all ones P 3 1 vector of all zeros m-by-m identity matrix m-by-n matrix whose entries are all zero Maximum value Minimum value The set of all m-dimensional real column vectors The set of all m-by-n dimensional real vectors Transpose of vector x 1-Norm of the vector x
γ γ  x1 ; x2 ; . . .; xnγ 2-Norm of vector x ith entry of vector x Inverse of matrix A 2-Norm of matrix A Entry of matrix A on ith row and jth column Column vector ith eigenvalue of A; A is symmetric and its eigenvalues are ordered from least to greatest value Space of square integrable vector functions over [0,N) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ÐN T L2 norm of ν, jjvjjL2 5 0 v ðt Þv ðt Þdt

243

244

Appendix A

σ ðA Þ AT rank(A) λmin ðAÞ λmax ðAÞ He fAg fK fK ℒ 

Spectral radius of matrix A Transpose of matrix A Rank of matrix A Minimum eigenvalue of matrix A Maximum eigenvalue of matrix A AT 1 A Class K functions Class K ℒ functions Kronecker product

A.2 Assumptions and lemmas Assumption A.1: There exists a nonnegative constant l $ 0 such that jj f ðt; xi Þ 2 f ðt; xt Þjj # ljjxi 2 xt jj: There exist finite constants lx $ 0; lv $ 0 such that jj f ðt; xi ; vi Þ 2 f ðt; xt ; vt Þjj # lx jjxi 2 xt jj 1 lv jjvi 2 vt jj: There exist finite constants lx $ 0; lv $ 0; la $ 0 such that jj f ðt; xi ; vi ; ai Þ 2 f ðt; xt ; vt ; at Þjj # lx jjxi 2 xt jj 1 lv jjvi 2 vt jj 1 la jjai 2 at jj: Assumption A.1 is a Lipschitz-type condition, satisfied by many wellknown systems, including Lorenz system, Chen system, Lu¨ system, and Chua’s circuit [1]. To analyze the tracking problem, the following lemmas are essential to introduce. Lemma A.1: [2] For vectors x; yAℝn with appropriate dimensions, inequality (A.1) holds. 6 2xT y # xT Mx 1 yT M 21 y;

ðA:1Þ

where M is a positive definite symmetric matrix. Proof. For scalars x, y, according to Young’s inequality, it has 2xy # x2 1 y2 : For vectors x; yAℝn , and a symmetric positive definite matrix A, 6 2xT y 5 6 2xT AA21 y # 2jjðAxÞT jjjjA21 yjj According to Young’s inequality, 2jjAxT jjjjA21 yjj # jjðAxÞT jj2 1 jjA21 yjj2  T 5 ðAxÞT Ax 1 A21 y A21 y  T 5 xT AT Ax 1 yT A21 A21 y;

Appendix A |

245

 T  21 21 21 Then, based on the fact that A21 5 AT ; A B 5 ðABÞ21 ;  21  T xT AT Ax 1 yT A21 A21 y 5 xT AT Ax 1 yT AT A y: Denote M 5 AT A; it has 6 2xT y # xT Mx 1 yT M 21 y. This completes the proof.

 Lemma A.2: [3] For matrices A 5 aij ARm 3 n and B 5 bij ARp 3 q , the Kronecker product is defined as 3 2 a11 B a12 B ? a1n B 7 6 6 a21 B a22 B ? a2n B 7 ðm1pÞ 3 ðn1qÞ 7AR A  B56 6 ^ ^ & ^ 7 5 4 am1 B am2 B ? amn B Consider matrices U; XARm 3 m ; V; YARm 3 n , the following properties exist: G G G G

G G

G

ðU 1 X Þ  V 5 U  V 1 X  V; ðU  V ÞðX  Y Þ 5 UX  VY; ðU  V ÞT 5 U T  V T ; if U and V are invertible, UV is invertible and ðU  V Þ21 5 U 21  V 21 ; if U and V are symmetric, UV is symmetric, if U and V are symmetric positive (semipositive) definite, UV is symmetric positive (semipositive) definite, suppose that U and V are square matrices of size n and m, respectively. Let λ1 ; . . .; λn be the eigenvalues of U and β 1 ; . . .; β m be those of V. Then the eigenvalues of UV are λi β j ; i 5 1; . . .; n; j 5 1; . . .; m.

Lemma A.3: [4] For a positive definite matrix Q, constants h1 and h2 satisfying 0 , h1 , h2, and a vector function wðtÞ:½h1 ; h2 /ℝn such that the integrals concerned as well defined, inequality (A.2) holds. !ð  ð T ð t2h1

t2h1

wðsÞdsÞ Q t2h2

wðsÞds # ðh2 2 h1 Þ

t2h2

t2h1

wT ðsÞQwðsÞds:

ðA:2Þ

t2h2

 Lemma A.4: [5] For a given symmetric matrix S 5 ing inequalities are equivalent:

S11

ð1ÞS , 0 ð2ÞS11 , 0; S22 2 ST12 S21 11 S12 , 0; T ð3ÞS22 , 0; S11 2 S12 S21 11 S12 , 0:



 S12 , the followS22

246

Appendix A

Definition A.1: [6] Let xðtÞ be a solution of x_ðtÞ 5 f ðt; xðtÞÞ; t $ t0 ; xðt0 Þ 5 x0 ; t0 $ 0 and V ðt; xðtÞÞ be a given function. Denote D1 V ðt; xðtÞÞ as the upper right-hand Dini derivative of V ðt; xðtÞÞ; that is, V ðt 1 h; xðt 1 hÞÞ 2 V ðt; xðtÞÞ : h h-01

D1 V ðt; xðtÞÞ 5 lim sup

Lemma A.5: [7,8] Given a graph G, then the Laplacian matrix ℒ, associated with the graph has at least one zero eigenvalue and all of the nonzero eigenvalues are in the open right half plane. For undirected graphs, the Laplacian matrix is symmetric with real eigenvalues. Therefore the set of eigenvalues of ℒ can be ordered sequentially in an ascending order as 0 5 λ1 ðℒÞ # λ2 ðℒÞ # . . .

ðA:3Þ

The eigenvalues of extended Laplacian matrix ℒ 1 ℬ considering the target agent are all nonnegative. The mathematical statement is 0 # λ1 ðℒ 1 ℬÞ;

iAℕ:

ðA:4Þ

Lemma A.6: [9] Consider a parameter dependent symmetric matrix ΦðλÞ in ℝm 3 m ;    ð2 2 λÞR 1 2ð1 2 λÞY1 ΦðλÞ 5 ðA:5Þ ð1 2 λÞX1T 1 λX2T ð1 2 λÞR 1 2λY2 where two symmetric matrices Y1 ; Y2 Aℝn 3 n two matrices X1 ; X2 Aℝn 3 n , such that the convex inequality (A.6) holds for all λA½0; 1: ΦðλÞ # ð1 2 λÞΦð0Þ 1 λΦð1Þ

ðA:6Þ

Then, there exists a symmetric matrix RAℝn 3 n such that the following reciprocally convex matrix inequality (A.7) holds. 0 1 1 R 0 Bλ C B C ðA:7Þ B C $ ΦðλÞ 1 @ 0 RA 12λ Lemma A.7: [10,11] BesselLegendre inequality: Consider Legendre polyk  l P nomials over interval [ 2 h,0] as Lk ðuÞ 5 ð21Þk pkl ðu1hÞ=h ; ’kAℕ; with l50

pkl

5 ð21Þ

l

k k11 Ll Ll ;

where

k Ll

Δ

5 k!=ððk 2 1Þ!l!Þis the coefficient and

Appendix A |

ð0 2h

Lk ðuÞLl ðuÞ 5

8 < 0;

h : 2k 1 1;

247

k 6¼ l; k 5 l:

ðA:8Þ

Based on the Legendre polynomials, let ωAC; R . 0 and h . 0, then the following inequality holds for all NAℕ. " # ðo 2 1 T X T T _ ζ ð2k 1 1ÞΓk RΓk ζ ðA:9Þ ω_ ðuÞRωðuÞdu h 2h k50 where

2

3 1 1 ΩT ; ΩT 5; ζ 5 4ωT ð0Þ; ωT ð2 hÞ; h 0 h 1
 Γk 5 I; ð21Þk11 ϒ k0 I; ϒ 1k I ;

with Ð0 Ωk 5 2h Lk ðuÞωðuÞdu; k 5 0; . . .; N; if i # k 2 ð2i 1 1Þð1 2 ð21Þk1i Þ; ϒ ik 5 0; if i . k:

References [1] Y. Qian, X. Wu, J. Lit, J. Lu¨, Second-order consensus of multi-agent systems with nonlinear dynamics via impulsive control, Neurocomputing 125 (2014) 142147. [2] S. Xu, J. Lam, A survey of linear matrix inequality techniques in stability analysis of delay systems, Int. J. Syst. Sci. 39 (12) (2008) 10951113. [3] A. Laub, Matrix Analysis for Scientists and Engineers, SIAM, Philadelphia, PA, 2005. [4] K. Gu, An integral inequality in the stability problem of time-delay systems, in: Proceedings of the IEEE Conference on Decision and Control, 2000, pp. 28052810. [5] S. Boyd, L.E. Ghaou, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, vol. 15, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1994. [6] N.M. Linh, V.N. Phat, Exponential stability of nonlinear time-varying differential equations and applications, Electron. J. Differ. Equ. 2001 (34) (2001) 113. [7] W. Ren, R.W. Beard, Consensus seeking in multiagent systems under dynamically changing interaction topologies, IEEE Trans. Autom. Control 50 (5) (2005) 655661. [8] R. Olfati-Saber, J. Fax, R. Murray, Consensus and cooperation in networked multi-agent systems, Proc. IEEE 95 (1) (2007) 215233. [9] A. Seuret, K. Liu, F. Gouaisbaut, Generalized reciprocally convex combination lemmas and its application to time-delay systems, Automatica 95 (2018) 488493. [10] A. Seuret, F. Gouaisbaut, Hierarchy of LMI conditions for the stability analysis of timedelay systems, Syst. Control Lett. 81 (2015) 17. [11] K. Liu, A. Seuret, Y. Xia, Stability analysis of systems with time-varying delays via the second-order Bessel-Legendre inequality, Automatica 76 (2017) 138142.

Index Note: Page numbers followed by “f” and “t” refer to figures and tables, respectively.

A Active replacement strategy, 209, 222 223, 223f, 235 coordination for, 222 Actuator faults, MAS with, 14 16 Adaptive control algorithm, 8 Adaptive distributed fixed-time consensus protocol, 88 Adaptive SMC law, 88, 102 Adaptive time-stepping strategies, 241 Agent replacements, stability analysis with, 228 229 Agent-based modeling, 19 Automatic train control (ATC), 19 Automatic train protection (ATP), 19 Automatic train supervision (ATS), 19

B Backstepping method, 88 Bellman Gronwall inequality, 7 Bolzano Weierstrass theorem, 7

C Collective consensus tracking error, 31 Consensus algorithm, 2 control, 5 6, 5f recovery method, 14 Consensus tracking, 8 10, 9f control of systems, 239 240 with double-integrator dynamics, 37 40, 43f velocity errors of MASs, 44f problem, 1 with second-order nonlinear dynamics, 40 42 Continuously switching topologies, 239 consensus tracking control of systems with, 239 240

continuously switching topologies based on polytopic model, 65, 67 73 problem description, 66 67 first-order MASs with, 30 35 high-order MAS with, 49 54 based on polytopic model, 54 63 numerical example, 58 63 simulation results, 53 54 stability analysis, 50 53, 56 58 trajectories of agents and maneuvering target, 54f numerical examples, 73 84 acceleration error trajectories, 79f, 84f global time-varying topology, 75f nonlinear group traces, 77f position error trajectories, 77f, 83f time-invariant tracking strategy, 82f, 83f tracking error trajectories, 80f, 81f trajectories of tracking agents, 76f velocity error trajectories, 78f, 84f second-order MASs with, 30 35 Continuously time-varying topologies, MASs with, 13 14, 13f Controller design, 115 120 Cooperative control algorithms, 241 Cooperative relay tracking algorithm, 113 114 controller design and stability analysis, 115 120 rules of switching of topology, 116f, 121f numerical examples, 120 127 error trajectories of tracking, 123f, 125f, 126f initial deployment of agents, 122f norm of tracking errors, 124f, 126f, 127f relay tracking algorithm, 113 115, 114f Voronoi diagrams, 109 113 with redundant agents, 113f for 3-coverage, 112f

249

250

Index

Cooperative relay tracking algorithm (Continued) Voronoi sites, cells, and distances, 110f Cooperative robots, MASs’ applications to, 17 18, 17f Coordination strategies, 208 for active agent replacements, 222 of initial tracking agents, 222 for node failures, 223

D Deep learning, 241 Digraph, 5 Directional mode, 135 Displacement-based control, 7 8 Distance-based control, 8 Distributed Bayesian information fusion, 67 68 Distributed control algorithms, 13, 241 Distributed linear tracking control algorithm, 9 Disturbance attenuation analysis, 139 141 Double-integrator dynamics, 3 Dynamic surface control technique, 88 Dynamics of tracking error system, 224 225

E Efficient modified nonsingular TSM control method, 220 Euclidean distance, 109 110 Euler Lagrange systems, 6 Event-triggered coordination control strategy, 219 220, 222 224 active replacement, 223f coordination for node failures, 224f coordination of initial tracking agents, 222f Event-triggered switching times, 133f

F Fast TSM control algorithms, 208 Finite-time boundedness, 208 control method, 208 coordination relay strategies, 208 209 interval, 207 208 stability analysis of nonlinear multiagent relay tracking systems, 209 217 multiagent problems, 208 sliding mode controllers, 208 stability/boundedness, 208 Finite time analysis with modified sliding mode control, 226 228 Finite-time coordination control of MASs, 218 234

event-triggered coordination strategy, 222 224 finite time analysis with modified sliding mode control, 226 228 nonsingular terminal sliding mode controller design, 224 226 numerical simulations, 229 234 tracking trajectories for multiple targets, 234f tracking trajectories with active replacements, 230f tracking trajectories with node failures, 232f stability analysis of systems with agent replacements, 228 229 tracking errors, 218f subject to disturbances, 218f tracking problem description and fundamentals, 220 221 First-order sliding mode control algorithms, 241 Formation control, 2, 6 8, 7f consensus tracking, 8 10 displacement-based control, 7 8 distance-based control, 8 position-based control, 7

G General nonlinear multiagent systems, 241 Generalized dynamic Voronoi diagram, 208 209, 219 221, 235 Global time-varying Laplacian matrix, 90 Graph theory, 4, 11 Gronwall Bellman inequality, 211, 215

H High-order MAS with continuously switching topologies, 49 54 problem description, 48 49 High-order sliding mode control algorithms, 241 High-order tracking agents, 67 Human-learning strategies, 241

I Impulse-time-dependent average Lyapunov function, 178, 181, 183, 185, 187, 195, 240 Intelligent distributed control protocols, 241 Intelligent train control, MASs’ applications to, 19 20

Index

J Jacobian matrix, 34 Jerk system, 74 Jointly connected topologies, MASs with, 11 13, 12f Jumping problem in tracking errors, 108, 132

K k-Coverage, 109, 110f Kalman filters, 67 68 Krasovsky’s theorem, 92

L Laplacian matrix, 5, 8, 30, 42, 48 49, 66, 68, 115, 132, 136 137, 164, 210 211, 239 240 eigenvalues of, 137 Linear relay tracking systems, 178 191. See also Nonlinear relay tracking systems main results, 181 188 numerical simulations, 189 191 problem formulation, 178 181 switching signals between different cases, 182f Linear time-varying systems, 31 32 Lipschitz condition, 66, 134 135 Lyapunov candidate for tracking system, 158 161, 168 Lyapunov function, 6, 38 42, 50 51, 56, 67, 118, 137 138, 140 141, 163 164, 166, 168, 185, 200, 227 229 of multiagent relay tracking system with time delays, 186 187 Lyapunov matrix function, 92, 100 Lyapunov Krasovskii function, 69 73

M Maneuvering target, 47 49, 53, 66, 89, 107, 115 Markov chain Monte Carlo based tracking methods, 67 68 Markov process, 116 117 MASs. See Multiagent systems (MASs) Maximum allowable communication time delays, 189 Mean Subsequence Reduced-type algorithms, 65 Modified nonsingular terminal sliding control law, 228 Modified NTSM control method, 208, 235 Modified sliding mode control, finite time analysis with, 226 228

251

Multi-unmanned aerial vehicle, MASs’ applications to, 18 19 Multiagent relay tracking system, 183 184. See also Nonlinear multiagent relay tracking systems class stability of, 131 controller design and stability analysis, 136 139 description of disturbance signal, 146f disturbance attenuation analysis, 139 141 event-triggered switching times, 133f norm of errors and minimum eigenvalues, 143f, 146f problem formulation, 134 136 related preliminaries, 132 133 switching signal, 142f, 145f tracking error energy, 147f tracking errors, 142f, 145f, 148f tracking trajectories, 134f, 144f Multiagent systems (MASs), 1 10, 29, 47 48, 65, 87, 132, 151, 177, 207 208, 239 applications of, 17 20 automatic multitrain control system, 19f to cooperative robots, 17 18, 17f to intelligent train control, 19 20 to multi-unmanned aerial vehicle, 18 19 relay tracking with time-varying number, 17f communication range and topology, 4f consensus control, 5 6 with continuously time-varying topologies, 13 14, 13f finite-time coordination control for target tracking, 218 234 first-order MASs with continuously switching topologies, 30 35 main results, 30 34 numerical example, 34 35 formation control, 6 8 with jointly connected topologies, 11 13, 12f with node failures/actuator faults, 14 16 related graph theory, 4 5 second-order MASs with continuously switching topologies, 36 45 consensus tracking with double-integrator dynamics, 37 40 consensus tracking with second-order nonlinear dynamics, 40 42 numerical examples, 42 45

252

Index

Multiagent systems (MASs) (Continued) tracking errors of agents with target, 36f SMC with disturbances, 98 102 with switching topologies, 10 11, 11f with time-varying topologies, 10 14, 11f

N Node failures, 10 MASs with node failures/actuator faults, 14 16 relay tracking, 16f Voronoi diagram, 15f Non-Lipschitz continuous control law, 241 Nonlinear continuous function, 36 37, 42 Nonlinear dynamics, 141, 161, 172 Nonlinear function, 53, 172 Nonlinear multiagent relay tracking systems, 241 numerical examples, 215 217 tracking errors evolution in threedimension, 216f relay tracking problem formulation, 209 211 stability analysis over finite time interval, 212 214 for systems subject to disturbances, 214 215 Nonlinear real matrix, 34 35 Nonlinear relay tracking systems, 192 204. See also Linear relay tracking systems numerical simulations, 200 204 related preliminaries, 192 195 stability analysis, 195 200 Nonlinear stochastic fractional integral differential equations, 241 Nonlinear systems, 3, 67, 89 Nonlinear tracking agents, 76 79 Nonsingular terminal sliding mode (NTSM), 208, 218 219 control method, 209, 240 controller design, 224 226 Numerical simulations, 229 234 of linear relay tracking systems, 189 191 of nonlinear relay tracking systems, 200 204

O Omni mode, 135

P Passive replacement, 223

Piecewise-constant switching topology signal, 10 11 Polytopic disagreement system, 99 Polytopic model, 55, 87 88, 239 240 high-order MASs with continuously switching topologies based on, 54 63 time delays and continuously switching topologies, 65, 67 73 of time-varying topology, 13 14 Position and velocity tracking errors of ith tracking agent, 89 90 Position-based control, 7

R Reinforcement learning (RL), 18, 241 Relay tracking. See also Consensus tracking algorithm, 113 115, 154 155, 210 control of systems with agent replacements, 240 error system, 188 multiagent system, 184, 199 problem formulation, 209 211 simulation under fast switching, 190 under slow switching, 190 strategy, 207 208 Relay tracking systems, 177 with damaged agents, 152 163, 165f controller design and stability analysis, 155 161 guidance rule for selecting relay agent, 154f numerical simulation, 161 163 problem formulation, 153 154 relay tracking algorithm, 154 155 tracking error jumps, 156f tracking trajectories, 162f linear relay tracking systems with timevarying number of agents and time delays, 178 191 with time-varying number of agents, 163 172 average dwell time and control parameter, 163t cyclic switching, 167f main results of, 168 171 norm of errors and minimum eigenvalues, 163f numerical simulation, 171 172 preliminaries and problem formulation, 164 168, 167f tracking errors of acceleration, 174f

Index

S Sensing radius, 161 Single-integrator dynamics, 3 Sliding mode control method (SMC method), 87 88, 91, 100, 239 240 arbitrary topology, 96f control performance comparison of control strategies, 98t controller design, 89 91 decomposed topologies, 96f for multiagent systems with disturbances, 98 102 controller design for systems with disturbances, 98 99 numerical examples, 102 stability analysis for systems with disturbances, 99 102 numerical examples, 94 98 position error trajectories, 97f preliminaries and problem description, 89 stability analysis, 91 94 velocity error trajectories, 97f SMC method. See Sliding mode control method (SMC method) Soccer robot, 18, 18f Stability analysis, 115 120 of nonlinear multiagent relay tracking systems, 209 217 of nonlinear relay tracking systems, 195 200 of SMC method, 91 94 of systems with agent replacements, 228 229 for systems with disturbances, 99 102 State transition matrix, 35 Switched system, 211 sliding mode phenomenon, 136 stability analysis for, 228 theory, 11 Switching topologies, MASs with, 10 11, 11f

T Takagi Sugeno fuzzy model (T S fuzzy model), 132 Terminal sliding mode (TSM), 208, 218 219, 241 Three-dimensional space, 53, 58 59, 74 Time-varying communication topology, 53, 78 Time-varying formation control protocol, 8 Time-varying matrix, 42

253

Time-varying number of tracking agents and time delays, 177 linear relay tracking systems with, 178 191 nonlinear relay tracking systems with, 192 204 Time-varying topologies, 13 14, 66, 90, 95 MASs with, 10 14, 11f Tracking. See also Consensus tracking; Relay tracking agents, 89, 98, 131 132, 134 137, 159 160, 164, 177 number evolution, 191f, 193f error, 186, 240 of multiagent tracking system, 215 system, 158, 162f problem, 8 9, 49, 68, 132, 136 137 process, 96, 141, 223 time, 75 76, 76t, 98 effect of initial distribution on, 80t trajectories, 82 83, 122f, 123, 125f agents with control method, 85f Tracking control protocol, 3, 49, 54 55, 67 68, 91 for ith tracking agent, 99 Trilateration algorithm, 109 110, 127 128 T S fuzzy model. See Takagi Sugeno fuzzy model (T S fuzzy model) TSM. See Terminal sliding mode (TSM)

U Uncertain communication environments, 13 14, 47 topology, 9 10, 65 Unmanned aerial vehicles (UAVs), 1 Unmanned ground vehicle (UGV), 2

V Variable agents, 164 Velocity disagreement vector, 155 in three-dimensional, 42 Virtual force based approach (VFA), 111 Voronoi cell, 131 132, 135, 141, 151 152 Voronoi diagram, 14 15, 15f, 107, 109 113, 135, 209, 219 220 for 3-coverage, 112f generalized dynamic, 219, 221 with redundant agents, 113f

W Weighted graph, 4, 179