Second-Order Consensus of Continuous-Time Multi-Agent Systems [1 ed.] 032390131X, 9780323901314

Table of contents :
Front Matter
Copyright
Contents
List of figures
Preface
Acknowledgments
1 Second-order consensus seeking in directed networks of multi-agent dynamical systems via generalized linear local interaction protocols
1.1 Introduction
1.2 Preliminaries and model formulation
1.2.1 Graph theory
1.2.2 Notation
1.2.3 Consensus protocols
1.3 Main results
1.4 Illustrative examples
1.5 Conclusion
References
2 Robust finite-time leader-following consensus algorithms for second-order multi-agent systems with nonlinear dynamics
2.1 Introduction
2.2 Preliminaries
2.2.1 Notation
2.2.2 Graph theory
2.2.3 Supporting lemmas
2.3 Finite-time consensus analysis
2.3.1 Problem description
2.3.2 Main results
2.4 Illustrative examples
2.5 Conclusion
References
3 Second-order consensus of multi-agent systems with nonlinear dynamics over random switching directed networks
3.1 Introduction
3.2 Preliminaries
3.2.1 Notation
3.2.2 Random graph
3.2.3 Problem formulations
3.3 Main results
3.3.1 Orthogonal decomposition
3.3.2 The case of time-delay-free coupling
3.3.3 The case of time-delay coupling
3.4 Illustrative examples
3.5 Conclusion
References
4 Second-order locally dynamical consensus of multi-agent systems with arbitrarily fast switching directed topologies
4.1 Introduction
4.2 Preliminaries
4.2.1 Notation
4.2.2 Model of optimization problem
4.2.3 Communication network
4.3 Local stability under arbitrarily fast switchings
4.4 Illustrative examples
4.5 Conclusion
References
5 Second-order global consensus in multi-agent systems with random directional link failure
5.1 Introduction
5.2 Preliminary and problem formulation
5.2.1 Graph theory
5.2.2 Problem formulation
5.3 Main results
5.4 Illustrative examples
5.5 Conclusion
References
6 Algebraic criteria for second-order global consensus in multi-agent networks with intrinsic nonlinear dynamics and directed topologies
6.1 Introduction
6.2 Preliminaries
6.2.1 Notation
6.2.2 Graph theory
6.2.3 Supporting lemmas and definitions
6.3 Some delay-independent algebraic criteria for second-order global consensus
6.4 Illustrative examples
6.5 Conclusion
References
7 Event-triggering sampling-based leader-following consensus in second-order multi-agent systems
7.1 Introduction
7.2 Preliminaries
7.2.1 Notation
7.2.2 Algebraic graph theory
7.2.3 Model description and problem formulation
7.3 Main results
7.3.1 Fixed topology
7.3.2 Switching communication topologies
7.4 Illustrative examples
7.5 Conclusion
References
8 Consensus analysis of multi-agent systems with second-order nonlinear dynamics and general directed topology: an event-triggered scheme
8.1 Introduction
8.2 Preliminaries
8.3 Main results
8.4 Second-order consensus in networks containing a directed spanning tree
8.5 Illustrative examples
8.6 Conclusion
References
Index

Citation preview

SECOND-ORDER CONSENSUS OF CONTINUOUS-TIME MULTI-AGENT SYSTEMS

SECOND-ORDER CONSENSUS OF CONTINUOUS-TIME MULTI-AGENT SYSTEMS HUAQING LI DAWEN XIA QINGGUO LÜ ZHENG WANG XIANGZHAO WU HUIWEI WANG LIANGHAO JI

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 © 2021 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. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-323-90131-4 For information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals Publisher: Mara Conner Acquisitions Editor: Chris Katsaropoulos Editorial Project Manager: Gabriela D. Capille Production Project Manager: Omer Mukthar Designer: Miles Hitchen Typeset by VTeX

Contents

List of figures Preface Acknowledgments

1. Second-order consensus seeking in directed networks of multi-agent dynamical systems via generalized linear local interaction protocols 1.1. Introduction 1.2. Preliminaries and model formulation 1.2.1. Graph theory 1.2.2. Notation 1.2.3. Consensus protocols 1.3. Main results 1.4. Illustrative examples 1.5. Conclusion References

2. Robust finite-time leader-following consensus algorithms for second-order multi-agent systems with nonlinear dynamics 2.1. Introduction 2.2. Preliminaries 2.2.1. Notation 2.2.2. Graph theory 2.2.3. Supporting lemmas 2.3. Finite-time consensus analysis 2.3.1. Problem description 2.3.2. Main results 2.4. Illustrative examples 2.5. Conclusion References

3. Second-order consensus of multi-agent systems with nonlinear dynamics over random switching directed networks 3.1. Introduction 3.2. Preliminaries 3.2.1. Notation 3.2.2. Random graph 3.2.3. Problem formulations 3.3. Main results 3.3.1. Orthogonal decomposition

ix xi xiii

1 1 3 3 4 4 5 11 16 17

19 19 21 21 21 22 23 23 24 29 35 36

39 39 41 41 42 43 45 45

v

vi

Contents

3.3.2. The case of time-delay-free coupling 3.3.3. The case of time-delay coupling 3.4. Illustrative examples 3.5. Conclusion References

4. Second-order locally dynamical consensus of multi-agent systems with arbitrarily fast switching directed topologies 4.1. Introduction 4.2. Preliminaries 4.2.1. Notation 4.2.2. Model of optimization problem 4.2.3. Communication network 4.3. Local stability under arbitrarily fast switchings 4.4. Illustrative examples 4.5. Conclusion References

48 56 58 62 62

65 65 67 67 68 72 73 79 82 83

5. Second-order global consensus in multi-agent systems with random directional link failure 85 5.1. Introduction 5.2. Preliminary and problem formulation 5.2.1. Graph theory 5.2.2. Problem formulation 5.3. Main results 5.4. Illustrative examples 5.5. Conclusion References

6. Algebraic criteria for second-order global consensus in multi-agent networks with intrinsic nonlinear dynamics and directed topologies 6.1. Introduction 6.2. Preliminaries 6.2.1. Notation 6.2.2. Graph theory 6.2.3. Supporting lemmas and definitions 6.3. Some delay-independent algebraic criteria for second-order global consensus 6.4. Illustrative examples 6.5. Conclusion References

7. Event-triggering sampling-based leader-following consensus in second-order multi-agent systems 7.1. Introduction 7.2. Preliminaries

85 87 87 88 91 99 102 103

105 105 107 107 108 108 110 117 121 121

125 125 126

Contents

7.2.1. Notation 7.2.2. Algebraic graph theory 7.2.3. Model description and problem formulation 7.3. Main results 7.3.1. Fixed topology 7.3.2. Switching communication topologies 7.4. Illustrative examples 7.5. Conclusion References

8. Consensus analysis of multi-agent systems with second-order nonlinear dynamics and general directed topology: an event-triggered scheme 8.1. Introduction 8.2. Preliminaries 8.3. Main results 8.4. Second-order consensus in networks containing a directed spanning tree 8.5. Illustrative examples 8.6. Conclusion References Index

126 126 127 127 128 132 134 136 137

139 139 142 150 160 168 174 174 177

vii

List of figures

Figure 1.1 Figure 1.2 Figure 1.3 Figure 1.4 Figure 1.5 Figure 2.1 Figure 2.2 Figure 2.3

Figure 2.4 Figure 2.5 Figure 2.6 Figure 2.7 Figure 2.8

Figure 2.9 Figure 2.10 Figure 2.11 Figure 3.1

Figure 3.2

Figure 3.3

The directed interaction topology of five agents. Velocity and position states of five agents in a network under linear consensus protocols. Velocity and position states of five agents in a network under periodic consensus protocols. Velocity and position states of five agents in a network under periodic consensus protocols. Velocity and position states of five agents in a network under divergent consensus protocols. The directed interaction topology of five agents. Positions and velocities of all agents without control inputs. Results of simulation performed on a group of ten agents moving in the position space and velocity space under the influence of the control protocols (2.5). The evolutions of tracking errors eix (t) and eiv (t) and the sliding-mode variable si (t). Positions and velocities of all agents with control inputs. The evolutions of tracking errors eix (t) and eiv (t) and the sliding-mode variable si (t). Positions and velocities of all agents without control inputs. The results of simulation performed on a group of ten agents moving in the position and velocity spaces under the influence of the control protocols (2.5). The evolutions of tracking errors eix (t) and eiv (t) and the sliding-mode variable si (t). Positions and velocities of all agents without control inputs. The evolutions of tracking errors eix (t) and eiv (t) and the sliding-mode variable si (t). The responses of position and velocity states of all agents with time-delay-free couplings in the random switching network. (a) Position states of all agents. (b) Velocity states of all agents. The time evolution of algorithm with regard to the consensus position and velocity errors with time-delay-free couplings in the random switching network. (a) The time evolution of log (Ex (t)). (b) The time evolution of log (Ev (t)). The responses of position and velocity states of all agents with time-delay couplings in the random switching network. (a) Position states of all agents. (b) Velocity states of all agents.

12 13 14 15 16 29 30

30 31 32 32 33

33 34 35 35

60

60

61

ix

x

List of figures

Figure 3.4

Figure 4.1

Figure 4.2

Figure 4.3

Figure 4.4 Figure 5.1 Figure 5.2 Figure 5.3 Figure 6.1 Figure 6.2 Figure 7.1 Figure 7.2 Figure 7.3 Figure 8.1 Figure 8.2 Figure 8.3 Figure 8.4 Figure 8.5 Figure 8.6

The time evolution of logarithms with regard to the consensus position and velocity errors with time-delay couplings in the random switching network. (a) The time evolution of log (Ex (t)). (b) The time evolution of log (Ev (t)). The responses of position and velocity states of all agents in the designed arbitrarily fast switching network (4.33). (a) Position states of all agents. (b) Velocity states of all agents. Time evolution of consensus position and velocity errors in the designed arbitrarily fast switching network (4.33). (a) The time evolution of Ex (t); (b) The time evolution of Ev (t). Time evolution of consensus position and velocity errors in the designed arbitrarily fast switching network (4.33). (a) The time evolution of (Ex (t)). (b) The time evolution of (Ev (t)). Time-varying generalized matrix measure using agents’ states. Position and velocity states of five agents in dynamically switching network G(t). (a) Position states. (b) Velocity states. (a) The time evolution of the Euclidean norm of time-varying Laplacian matrix L (t). (b) The time evolution of Re (λ2 (L (t))). The time evolution of logarithm with regard to V (y(t), t). Time evolution of consensus error. The time evolution of states of networked system (6.19). The states of position and velocity of all agents. (a) The distributed control input ui1 (t) of all follower agents. (b) Event time instants of all follower agents. Time evolutions of y1i (t) and y2i (t), i = 1, 2, 3, 4. Structure of a small network (the vertices in the dotted boxes are in the same strongly connected components). The states of position and velocity of all agents. Control inputs. exv1 (t) and its threshold Th1 (t). The sampling time instants of all agents (ϕ2 = 0). The sampling time instants of all agents (ϕ2 = 0).

61

81

81

81 82 100 101 101 120 120 135 136 136 170 172 172 172 173 173

Preface

Coordination through local interaction appears frequently in nature systems such as synchronization flashing of fireflies, movement of a school of fish, the understanding of brain seizures, nonlinear optics, meteorology, and so on. These collective activities of creatures and objects have inspired compelling researches on coordination of multiagent systems, which help not only in better understanding the general mechanisms and interconnection rules of natural collective phenomena, but also benefit designs of artificial networked cyber-physical systems, including sensor networks, multi-robot systems, unmanned autonomous vehicles (UAV), complex networks, and so on. One of the most fundamental approaches to achieve coordination of multi-agent systems is consensus control that makes all agents reach an agreement on a common value of interest depending on the states of all agents, especially by negotiating (communicating) with their neighbors. It has witnessed considerable developments in many fields, including cooperative control of UAV, formation control of mobile robots, control of communication networks, management science and statistics, swarm-based computing, and so on. When the control input is added on the velocity term, agents can be modeled simply as first-order integrators. However, in many practical engineering applications, agents such as unmanned aerial vehicles and mobile robots are usually governed by second-order systems with both position and velocity terms. Hence, compared with its first-order counterparts, second-order consensus received more attentions in recent years. Although it has become a hot issue in scientific field and significant progresses have been made, second-order consensus of multi-agent systems that meets the actual applications comes across many issues. The extension of consensus algorithms for multi-agent systems from first-order to second-order is non-trivial; the obtained firstorder consensus criteria are usually failed to address consensus problem for second-order multi-agent systems. Besides, inherent nonlinearities of agents, complexity of physical system, and uncertainty of communication become the difficulties that protocols or algorithms of consensus control for second-order multi-agent systems need to conquer. Analysis and synthesis includes leader-following consensus, global consensus, unbalanced directed topology, random switching topology, nonlinear dynamics of agents, random directional link failures, continuous-time control, finite-time control, eventtriggered communication, which are all thoroughly studied. This book mainly investigates second-order consensus of continuous-time multi-agent systems. In general, the following problems are focused in this book: 1) Second-order consensus seeking in directed networks of multi-agent dynamical systems via generalized linear local interaction protocols; 2) Robust finite-time leader-following consensus algorithms for second-order multi-agent systems with nonlinear dynamics; 3) Second-order consen-

xi

xii

Preface

sus of multi-agent systems with nonlinear dynamics over random switching directed networks; 4) Second-order locally dynamical consensus of multi-agent systems with arbitrarily fast switching directed topologies; 5) Second-order global consensus in multiagent systems with random directional link failure; 6) Algebraic criteria for second-order global consensus in multi-agent networks with intrinsic nonlinear dynamics and directed topologies; 7) Event-triggering sampling-based leader-following consensus in second-order multi-agent systems; 8) Consensus analysis of multi-agent systems with second-order nonlinear dynamics and general directed topology, an event-triggered scheme. Among the topics, simulation results of examples practical applications are presented to illustrate the effectiveness and the practicability of the control protocols and algorithms proposed in the previous parts. This book is appropriate as a college course textbook for undergraduate and graduate students majoring in computer science, automation, artificial intelligence, and so on and as a reference material for researchers and technologists in related fields.

Southwest University, China Guizhou Minzu University, China Southwest University, China Southwest University, China Southwest University, China Southwest University, China Chongqing University of Posts and Telecommunications, China

Huaqing Li Dawen Xia Qingguo Lü Zheng Wang Xiangzhao Wu Huiwei Wang Lianghao Ji

Acknowledgments

This book was supported in part by the National Natural Science Foundation of China under Grants 61773321, 61762020, and 61876200, in part by the Fundamental Research Funds for the Central Universities under Grant XDJK2019AC001, and in part by the Innovation Support Program for Chongqing Overseas Returnees under Grant cx2019005. We would like to begin by acknowledging Zuqing Zheng, Wentao Ding, Enbing Su, Liang Ran, and Youcheng Niu, who have unselfishly given their valuable time in arranging raw materials. Their assistance has been invaluable to the completion of this book. The authors are especially grateful to their families for their encouragement and never ending support when it was most required. Finally, we would like to thank the editors at Elsevier for their professional and efficient handling of this book.

xiii

CHAPTER 1

Second-order consensus seeking in directed networks of multi-agent dynamical systems via generalized linear local interaction protocols 1.1. Introduction The recent literature has witnessed steadily increasing recognition and attention of coordinated motion of mobile agents across a broad range of disciplines. Applications can be found in many fields, including biology or ecology (aggregation behavior of animals), physics (collective motion of particles), computer science (distributed computation), and control engineering (formation control in robots) [1,2]. Research on multi-agent coordinated control problems not only helps in better understanding general mechanisms and interconnection rules of natural collective phenomena, but also benefits many practical applications of networked cyber-physical systems, such as the coordination and control of distributed sensor networks [3], formation control in multi-robots [4], unmanned autonomous vehicle (UAV) formations [5,6], flocking [7], complex networks [8,9,30–35], and so on [10,12]. A fundamental approach to achieve cooperative control is consensus. Roughly speaking, the consensus problem refers to how to make the states of multi-agent systems reach an agreement on a common value of interest, especially by negotiating with their neighbors. This common value might be the attitude in multispacecraft alignment, the heading direction in flocking behavior, or the average in the distributed computation [20]. When the velocity is introduced into the control input, each agent can be simply modeled as a first-order integrator. In this case the task of the consensus protocols is to ensure that position of all agents in the network converges to a constant value. Such consensus can be called the “stationary consensus”. Based on the algebraic graph theory, Olfati-Saber and Murray [11] presented a systematic framework to analyze the first-order consensus algorithms and showed that the consensus problem can be solved if the digraph (directed graph) is strongly connected. Ren and Beard [12] generalized the results of [11] and presented a more relaxed condition for the topology of directed networks, that is, the interaction graph has a directed spanning tree. Sun et al. [13] discussed the first-order average consensus problem of dynamic agents with multiple time-varying communication delays. Lu et al. [14] studied the first-order consensus problem over directed networks with arbitrary finite communication delays and nonlinear couplings. Second-Order Consensus of Continuous-Time Multi-Agent Systems https://doi.org/10.1016/B978-0-32-390131-4.00008-X

Copyright © 2021 Elsevier Inc. All rights reserved.

1

2

Second-Order Consensus of Continuous-Time Multi-Agent Systems

However, in the general case, where the driving force (acceleration) is considered as the control input, each agent should be modeled as a double integrator. The second-order consensus problem of multi-agent systems has received increasing attention; see [15–20] and the references therein. Unlike the first-order consensus, Ren and Atkins [7] showed that the existence of a directed spanning tree is only a necessary rather than a sufficient condition to reach a second-order consensus; for example, a second-order consensus may fail to be achieved in many cases even if the interaction topology contains a directed spanning tree. Therefore the extension of consensus algorithms from first-order to second-order is nontrivial [15], and the second-order consensus problem is more complicated and challenging than the first-order case. More surprisingly, Yu et al. [29] mentioned that consensus may no longer to be reachable within multi-agent systems by adding one connection between a chosen pair of agents, which has originally been able to reach a consensus. This is inconsistent with the intuition that more connections are helpful for reaching a consensus. Some special second-order consensus protocols were presented and some consensus conditions were obtained in [21–25]. For a class of simple linear protocols, Yu et al. [21] studied some necessary and sufficient conditions for second-order consensus multi-agent systems and revealed that both real and imaginary parts of the eigenvalues of the Laplacian matrix of the corresponding directed network play key roles in reaching a consensus. Ren [22] discussed the consensus problem of coupled secondorder linear harmonic oscillators with local interaction, which means that states of all agents converge to the same periodic function. Zhang and Tian [23] introduced the concept of consentability under two kinds of special second-order consensus protocols and obtained necessary and sufficient conditions for the discrete multi-agent systems. Zhu et al. [24] discussed a more general linear form of consensus protocols for multiagent systems with double-integrator dynamics. By choosing different consensus gains different dynamics including linear, periodic, and positive exponential dynamics can be achieved. Under the same general linear protocols, Zhu [25] further analyzed the maximum consensus speed and discovered that the largest and smallest nonzero eigenvalues of the Laplacian matrix of the interaction topology commonly determine the maximum consensus speed. However, the disadvantage of the existing literature lies in that the authors do not analytically investigate the final consensus convergence state of all agents in the networks and reveal the underlying relations among the final consensus state and the network topology, initial states of all agents, and the designed protocols. This is a relatively interesting issue, which deserves further studying in detail. In this chapter, for a class of generalized linear consensus protocols of the multi-agent systems with double-integrator dynamics, we analytically investigate some necessary and sufficient conditions for reaching a consensus. All agents in the fixed directed network topology are governed by double-integrator dynamics, and almost all existing linear consensus protocols can be considered as particular cases of the present chapter. Moreover, the final consensus convergence states of all agents are also analytically determined.

Second-order consensus seeking in directed networks of multi-agent dynamical systems

According to the obtained results, we find that both linear gains and eigenvalues of the Laplacian matrix associated with the directed network topology play key roles in reaching a consensus. Finally, we demonstrate the effectiveness and correctness of our theoretical findings by some numerical examples. The rest of this chapter is outlined as follows. In Section 1.2, we give some preliminaries on the graph theory and the model formulation. In Section 1.3, we establish the main results, including four corollaries. In Section 1.4, we simulate several numerical examples to verify the theoretical analysis. Finally, in Section 1.5, we draw conclusions.

1.2. Preliminaries and model formulation 1.2.1 Graph theory In this subsection, we introduce some basic concepts and results about algebraic graph theory. For more details about algebraic graph theory, we refer to [26]. Suppose that information exchange among agents in multi-agent systems can be modeled by an interaction digraph. By G = (V , ε, A) we denote a directed graph, where V = {1, 2, . . . , N } is a set of nodes, ε ⊆ V × V is a set of edges, and A = (aij )N ×N is an adjacency matrix. A directed edge εij in the network G is denoted by the ordered pair of nodes (i, j), where i is the head, and j is the tail, which means that node i can receive information from node j [27]. The elements of the adjacency matrix A are defined such that aij = 1 ⇔ εij ∈ ε, whereas aij = 0 ⇔ εij ∈/ ε. We always assume that there is no self-loop in network G, that is, aii = 0 for all i ∈ V . A weighted adjacency matrix A of the digraph G can be defined such that aij is a positive weight if εij ∈ ε, whereas aij = 0 if εij ∈/ ε. The set of neighbors of node i is denoted by Ni = {j ∈ V : (i, j) ∈ ε}. If there is a sequence of edges of the form (i, j1 ), (j1 , j2 ), . . . , (jm , j) ∈ ε composing a directed path beginning with i and ending with j in the digraph G with distinct nodes jk , k = 1, 2, . . . , m, then the node j is said to be reachable from node i. A digraph is strongly connected if for any distinct nodes i and j, there exists a directed path from node i to node j. A digraph has a directed spanning tree if there exists at least one node, called a root, that has a directed path to all the other nodes [28]. A digraph is balanced if n n j=1 aij = j=1 aji for all i. Let (generally, nonsymmetric) Laplacian matrix L = (lij )N ×N  associated with directed network G be defined as lii = N j=1, j=i aij and lij = −aij , where i = j [22]. Especially, for an undigraph, L is symmetric positive semi-definite. However, L for a digraph does not have this property. Remark 1.1. A network G is called undirected if there is a connection between two nodes i and j in G such that aij = aji > 0; otherwise, aij = aji = 0 (i = j, i, j = 1, 2, ..., N). Therefore undirected networks are particular cases of directed networks with aij = aji for all i, j = 1, 2, ..., N.

3

4

Second-Order Consensus of Continuous-Time Multi-Agent Systems

1.2.2 Notation For simplicity, we use some mathematical notations. We denote by In (On ) the identity (zero) matrix of dimension n; 1n (0n ) is the column vector with n elements being 1 (0); Re (·) and Im (·) represent the real part and imaginary part of a complex number, respectively; C n is the n-dimensional complex vector space.

1.2.3 Consensus protocols Consider N agents in the directed network G, labeled 1, 2, ..., N, moving in the twodimensional Euclidean space. Suppose that the ith agent in the directed network G is governed by double-integrator dynamics, that is, r˙i (t) = vi (t), v˙ i (t) = ui (t), i = 1, 2, ..., N ,

(1.1)

where ri (t) ∈ R is the position state of agent i, vi (t) ∈ R is its velocity state, and ui (t) ∈ R is the control input, which is designed based on local information exchange. For reaching a second-order consensus, we consider the most general linear protocols of the following form for i = 1, 2, ..., N: ui (t) = −α ri (t) − β vi (t) + γ



aij (rj (t) − ri (t)) + ξ

j∈Ni



aij (vj (t) − vi (t)),

(1.2)

j∈Ni

where α (β) denotes the position (velocity) damping gain, and γ (ξ ) represents the coupling strength of positions (velocities) between neighboring agents. Let x(t) = [r1 (t), r2 (t), ..., rN (t)]T and y(t) = [v1 (t), v2 (t), ..., vN (t)]T . With the generalized linear local interaction protocol (1.2), the closed-loop system consisting of (1.1) and (1.2) can be rewritten in the compact matrix form as 

x˙ (t) y˙ (t)



 = 

ON −α IN − γ L

IN 

−β IN − ξ L



x(t)



y(t)

 ,

(1.3)

Q

where L ∈ RN ×N is the (nonsymmetric) Laplacian matrix associated with digraph G. Definition 1.2. A second-order consensus in the multi-agent system (1.1) under control input (1.2) is said to be achieved if for any initial conditions ri (0) and vi (0), limt→∞ |ri (t) − rj (t)| = 0 and limt→∞ |vi (t) − vj (t)| = 0 foe all i, j = 1, 2, ..., N. Remark 1.3. Throughout this chapter, we only consider the particular case where ri (t), vi (t), ui (t) ∈ R1 . As for ri (t), vi (t), ui (t) ∈ Rm (m ≥ 2), the obtained results can be easily extended to this case by using the Kronecker product.

Second-order consensus seeking in directed networks of multi-agent dynamical systems

1.3. Main results Before stating our main results, we first give the following lemmas. Lemma 1.4. [12]. Let L be the (nonsymmetric) Laplacian matrix associated with a directed network G. Then L has a simple zero eigenvalue, and all the other eigenvalues have positive real parts if and only if G has a directed spanning tree. In addition, there exist 1N satisfying L1N = 0N and p ∈ RN satisfying p ≥ 0 (all the elements are non-negative), pT L = 0N , and pT 1N = 1. (That is, 1N and p are respectively the right and left eigenvectors of L associated with the zero eigenvalue.) For linear model (1.3), the eigenvalues of the matrix Q are very important in convergence analysis. In fact, a second-order consensus can be achieved if and only if all the eigenvalues of matrix Q associated with the non-zero eigenvalues of Laplacian matrix L have negative real parts. This will be further demonstrated in detail. Moreover, the right and left eigenvectors of L associated with the zero eigenvalue also play a significant role in determining the final consensus state. To analyze the consensus of model (1.3), we first investigate the relations of eigenvalues and eigenvectors of the matrix Q and Laplacian L. Therefore, the following results are helpful. Lemma 1.5. Let μi ∈ C be the ith eigenvalue of −L. Let χri ∈ C N and χli ∈ C N be, respectively, the right and left eigenvectors of −L associated with μi . Then the eigenvalues of Q defined in (1.4) are given by λi± =

−β+ξ μi ± (β−ξ μi )2 −4(α−γ μi )

associated with right eigenvectors

T

T  γ +λi± ξ T . ϕri± = χriT , λi± χriT and left eigenvectors ϕli± = χliT , γ (β+λ χ li )−αξ i± 2

Proof. Let λ be an eigenvalue of the matrix Q. Note that the characteristic polynomial of Q is  det (λI2N − Q) = det

λI N α IN + γ L

−I N λIN + β I N + ξ L



(1.4)

= det(λ2 IN + λ(β IN + ξ L ) + α IN + γ L ).

Letting μi be the ith eigenvalue of −L, we get det(λIN + L ) = that det (λI2N

N  i=1

(λ − μi ). Thus it follows

N   2 − Q) = λ + (β − ξ μi )λ + α − γ μi .

(1.5)

i=1

Therefore the roots of det (λI2N − Q) = 0 (i.e., the eigenvalues of Q) satisfy λ2 + (β − ξ μi ) λ + α − γ μi = 0.

(1.6)

5

6

Second-Order Consensus of Continuous-Time Multi-Agent Systems

We know that each eigenvalue μi of −L corresponds to two eigenvalues λi± of Q. Their relations can be described by λi ± =

−β + ξ μi ±



(β − ξ μi )2 − 4(α − γ μi )

2

, i = 1, 2, ..., N .

(1.7) 

Without loss of generality, let μ1 = 0. It follows from (1.7) that λ1± = −β± 2β −4α . Let  T T ∈ C 2N be an associated right eigenvector. λ be an eigenvalue of Q, and let ϕr = xT r , yr Then we have 

ON −α IN − γ L



IN −β IN − ξ L

xr yr



 =λ

xr yr

2



(1.8)

.

From (1.8) it follows that 

yr = λxr , −α xr − γ Lxr − β yr − ξ Lyr = λyr .

(1.9)

2

After some manipulations, we have −Lxr = α+βλ+λ xr (γ , ξ = 0). Suppose that μ is an γ +λξ eigenvalue of −L with an associated right eigenvector χr . Then we can obtain that α+βλ+λ2 = μ and xr = χr . Therefore it follows from (1.9) that the eigenvalues of Q are γ +λξ 

T

given by λi± (defined in (1.7)) associated with right eigenvectors ϕri± = χriT , λri± χriT . T  Similarly, let ϕl = xTl , yTl ∈ C 2N be a left eigenvector of Q associated with eigenvalue λ. Then we get that 

T

xTl , yl



ON −α IN − γ L

IN −β IN − ξ L



 T = λ xT l , yl .

(1.10)

From (1.10) we have ⎧ ⎪ ⎨ yT = l

γ + λξ xT , γ (β + λ) − αξ l

(1.11)

⎪ ⎩ γ xT − γ ξ yT L = γ (β + λ) yT . l l l

Solving (1.11) gives −xT l L=

α + βλ + λ2 T xl (γ , ξ = 0) . γ + λξ

(1.12)

A similar argument to that for the right eigenvectors shows that the left eigenvectors of

T

γ +λi± ξ Q associated with eigenvalue λi± are ϕli± = χliT , γ (β+λ χT . i± )−αξ li

Second-order consensus seeking in directed networks of multi-agent dynamical systems

Lemma 1.6. Under the assumption that the directed network G has a directed spanning tree, the real parts of all eigenvalues of matrix Q associated with non-zero eigenvalues of −L are negative if and only if  max −2αβ 2 + 4 Re(μi )αβξ − 2Re2 (μi )αξ 2 − 2 Re(μi )β 3 ξ + 2 Re(μi )β 2 γ 2≤i≤N  + Re2 (μi )β 2 ξ 2 − 2Re3 (μi )βξ 3 − 2 2Re2 (μi ) + Im2 (μi ) βγ ξ   +2 Re(μi ) Re2 (μi ) + Im2 (μi ) ξ 2 γ + 2Im2 (μi )γ 2 < 0,

(1.13)

where μi is the nonzero eigenvalue of −L, i = 2, 3, ..., N. Proof. From Lemma (1.4) we have that −L has a simple zero eigenvalue and all other eigenvalues have negative parts if and only if the directed network G has a directed spanning tree, that is, Re(λi± ) < 0, i = 2, ..., N. For convenience, let (β − ξ μi )2 − 4(α − γ μi ) = c + id, where c, d are real numbers, and i =

√

−1. It foli )±c lows from (1.7) that Re(λi± ) < 0 implies −β+ξ Re(μ < 0, i = 2 , ..., N. This equivalently 2 implies that the inequalities ξ Re(μi ) − β < c < −ξ Re(μi ) + β and Re(μi ) < 0 hold simultaneously. Therefore

c 2 < ξ 2 Re2 (μi ) + β 2 − 2ξβ Re(μi ), i = 2, 3, ..., N .

(1.14)

Also note that (β − ξ μi )2 − 4(α − γ μi ) = c 2 − d2 + i2cd. By some calculations and separating the real and imaginary parts we obtain β 2 + ξ 2 Re2 (μi ) − ξ 2 Im2 (μi ) − 2βξ Re(μi ) − 4α + 4γ Re(μi ) = c 2 − d2

(1.15)

and ξ 2 Re(μi ) Im(μi ) − βξ Im(μi ) + 2γ Im(μi ) = cd.

(1.16)

It follows from (1.15) and (1.16) that c 4 − Bc 2 − A = 0,

(1.17)

where B = β 2 + ξ 2 Re2 (μi ) − ξ 2 Im2 (μi ) − 2βξ Re(μi ) − 4α + 4γ Re(μi ),

(1.18)

and 

2

A = ξ 2 Re(μi ) Im(μi ) − βξ Im(μi ) + 2γ Im(μi ) .

(1.19)

7

8

Second-Order Consensus of Continuous-Time Multi-Agent Systems

Solving (1.17) and combining (1.14) yield √

B ± B2 + 4A < ξ 2 Re2 (μi ) + β 2 − 2ξβ Re(μi ), i = 2, 3, ..., N . c = 2 2

(1.20)

After some simplifications, we derive inequality (1.13). Relying on algebraic graph theory and matrix theory, we show some necessary and sufficient conditions for convergence of the generalized linear protocols in (1.2) under fixed interaction topology. Theorem 1.7. Let p and 1N be defined as in Lemma 1.4. Let μi , λi± , ϕri± , ϕli± be defined as in Lemma 1.5. Using the protocol defined in (1.2), a second-order consensus in multi-agent system (1.1) can be achieved if and only if the directed network contains a directed spanning tree and inequality (1.13) in Lemma 1.6 holds. Moreover, if a second-order consensus is reached as t → ∞, then the convergence state can be analytically described by the following formulas: ⎧ N ⎪ etλ1+ +etλ1−  p ⎪ ⎪ r ( t ) = i j rj (0) ⎪ 2 ⎪ ⎪ j=1 ⎪ ⎪ ⎪  tλ ⎪ ⎪ λ1+ ξ +γ e 1+ ⎪ + ⎪ 2 γ (β+λ1+ )−αξ + ⎨

λ1− ξ +γ e t λ1− 2 γ (β+λ1− )−αξ

N ⎪ λ1+ etλ1+ +λ1− etλ1−  p ⎪ ⎪ v ( t ) = i j rj (0) ⎪ 2 ⎪ ⎪ j=1 ⎪ ⎪ ⎪  tλ ⎪ t λ1− ⎪ 1+ 1+ (λ1+ ξ +γ ) ⎪ ⎪ + e 2 γλ(β+λ +e2 ⎩ 1+ )−αξ

λ1+ (λ1+ ξ +γ ) γ (β+λ1− )−αξ

 N j=1

pj vj (0),

 N j=1

(1.21)

pj vj (0).

Proof. (Sufficiency) Note that the directed network topology G contains a directed spanning tree. It follows from Lemma 1.4 that −L has a simple zero eigenvalue and all the other eigenvalues have negative real parts. Moreover, 1N and p ∈ RN are right and left eigenvectors of −L associated with zero eigenvalue, that is, L1N = 0N , pT L = 0N , and pT 1N = 1. Without loss of generality, let μ1 = 0, and then we get that Re(μi ) < 0, i = 2, 3, ..., N. Moreover, if inequality (1.3) in Lemma 1.6 is satisfied, then it follows that all eigenvalues of the matrix Q associated with non-zero eigenvalue of −L have negative real parts. Accordingly, it follows from Lemma 1.5 that thecorresponding right and left eigen2 vectors of Q associated with eigenvalues λ1± = −β± 2β −4α are given by   T T ϕr1± = 1T , λ 1 , ϕ = pT , ± N l1± N

γ + λ± ξ pT γ − (β + λ1± ) − αξ

Note that Q can be written in the Jordan canonical form as

T .

(1.22)

Second-order consensus seeking in directed networks of multi-agent dynamical systems

Q = PJP −1 ⎛ ⎜ = (ω1 , ω2 , ..., ω2N ) ⎝

⎞

λ1+

0

0

λ1−

0(2N −2)×1

0(2N −2)×1

⎛

01×(2N −2) ⎜ ⎜ ⎟⎜ 01×(2N −2) ⎠ ⎜ ⎜ ⎝ J˜

1T 2T

(1.23)

⎞

⎟ ⎟ ⎟ , .. ⎟ ⎟ . ⎠

T 2N

where ωj ∈ R2N and j ∈ R2N (j = 1, 2, ..., 2N ) can be chosen as the right and left eigenvectors or generalized eigenvectors of Q, respectively. Since PP −1 = I2N , ωi and i must be chosen such that ωiT i = 1 and ωiT k = 1 for i = k (i = 1, 2, ..., 2N ). Here 2 2 λ1+ = −β+ 2β −4α , λ1− = −β− 2β −4α , and J˜ is the upper diagonal Jordan block matrix

−β+ξ μi ± (β−ξ μi )2 −4(α−γ μi )

associated with the nonzero eigenvalues λi± = , i = 2, 3, ..., N. 2 Before choosing these eigenvectors, the following analysis is essential for setting their coefficients.

−β+ξ μi ± (β−ξ μi )2 −4(α−γ μi )

γ +λξ = A, where λ = , i = 2, ..., N. Then we Let λ γ (β+λ)−αξ 2 % % % know that λ2 + (γ ξ − Aγ ξ )λ − βγ A ξ + α A = 0. Combining it with λ2 + (β − ξ μ)λ + α − μγ = 0, we obtain



which yields A = 1. Similarly, let λ1 =

% % γ ξ − Aγ ξ = β − ξ μ, % % −βγ A ξ + α A = −βγ A ξ + α A,

−β+ξ μi ± (β−ξ μi )2 −4(α−γ μi )

(1.24)

−β+ξ μi ∓ (β−ξ μi )2 −4(α−γ μi )

, λ2 = , and 2 2 γ +λ2 ξ γ λ1 +ξ(α−μi ) λ1 γ (β+λ2 )−αξ = γ (β+λ2 )−αξ = B. Then it follows that γ λ1 + ξ α − ξ μi γ = −αξ B + γβ B + γ λ2 B. According to Vieta’s theorem, we have γ λ1 + γ λ2 = γ (ξ μi − β). Therefore (γ λ2 + γβ + ξ α)(1 + B) = 0, which yields B = −1. By the above analysis we can choose ω1 = ϕr1+ , ω2 = ϕr1− , 1 = 12 ϕl1+ , and  T T and ϕ 2 = 12 ϕl1− to make PP −1 = I2N hold, where ϕr1± = 1T l1± = N , λ1± 1N 1 2

T

γ +λ1±'ξ pT , γ &β+λ pT . Also note that lim eJ˜t = 0(2N −2)×(2N −2) . Then it follows that 1± −αξ

t→∞

 lim eQt = lim PeJt P −1 = (ω1 , ω2 )

t→+∞

t→+∞



etλ1+ 0

0 etλ1− T



1T 2T 



(1.25)

. T

By state transition theory from (1.3) it follows that x(t), y(t) = eQt x(0), y(0) . Then  ' T & T we have limt→∞ x(t), y(t) = limt→∞ eQt x(0), y(0) . After some calculations, we

9

10

Second-Order Consensus of Continuous-Time Multi-Agent Systems

obtain ⎧ etλ1+ + etλ1− ⎪ ⎪ 1N pT x(0) x ( t ) = ⎪ ⎪ ⎪ 2 ⎪ ) ( tλ1+ ⎪ ⎪ e λ1+ ξ + γ etλ1− λ1− ξ + γ ⎪ ⎪ ⎪ 1N pT y(0), + + ⎨ 2 γ (β + λ1+ ) − αξ 2 γ (β + λ1− ) − αξ ⎪ λ1+ etλ1+ + λ1− etλ1− ⎪ ⎪ y ( t ) = 1N pT x(0) ⎪ ⎪ 2 ⎪ ⎪ ) ( tλ1+ ⎪ ⎪ e λ1+ (λ1+ ξ + γ ) etλ1− λ1+ (λ1+ ξ + γ ) ⎪ ⎪ 1N pT y(0), + + ⎩ 2 γ (β + λ1+ ) − αξ 2 γ (β + λ1− ) − αξ

(1.26)

as t → ∞. This is equivalent to the results in (1.21). (Necessity) If the condition that matrix Q has exactly two eigenvalues λ1± =  −β± β 2 −4α and all the other eigenvalues have negative real parts is not satisfied, then 2 limt→∞ eQt has a rank greater than 2, which contradicts the assumption that a secondorder consensus is reached. (See [21] for a similar argument.) The proof is thus completed. Remark 1.8. For α = β = 0, it follows from (1.7) that Q has a zero eigenvalue with algebraic multiplicity two. Therefore, when Jordan canonical form is used to analyze the convergence results, the Jordan block matrix J in (1.23) should take the form ⎛ ⎜ ⎝

0 0

1 0

0(2N −2)×1

0(2N −2)×1

⎞

01×(2N −2) 01×(2N −2) ⎟ ⎠. ˜J

Moreover, ω1 , ω2 , 1 , and 2 take the generalized right and left eigenvectors. For more detail, we refer to [21]. Thus we obtain the following corollary. Corollary 1.9. For α = β = 0 and γ , ξ > 0, a second-order consensus in multi-agent system (1.3) can be achieved if and only if the network contains a directed spanning tree and ξ2 > max 2≤i≤N γ



* Im2 (μi )  , − Re(μi ) Re2 (μi ) + Im2 (μi )

(1.27)

where μi are nonzero eigenvalues of matrix −L, i = 2, 3, ... , N. In addition, if a second N order consensus is reached, then we have ri (t) → N j=1 pj rj (0) + j=1 pj vj (0)t and vi (t ) → N T j=1 pj vj (0) as t → ∞, where p = [p1 , p2 , ..., pN ] is the unique positive left eigenvector of −L associated with eigenvalue 0 satisfying pT 1N = 1. Yu et al. [21] studied the second-order consensus under the linear protocol ui (t) = γ

 j∈Ni

aij (rj (t) − ri (t)) + ξ

 j∈Ni

aij (vj (t) − vi (t)), i = 1, 2, ..., N ,

(1.28)

Second-order consensus seeking in directed networks of multi-agent dynamical systems

and obtained similar conclusions as Corollary 1.9. Therefore Theorem 1.7 can be viewed as an extension of the results in [21]. Corollary 1.10. For α > 0, β = 0 and γ = ξ = 1, a second-order consensus in multiagent system (1.3) can be achieved if and only if the network contains a directed spanning tree. √  In addition, if a second-order consensus is reached, we have ri (t) → cos( α t) N pj rj (0) + √ N √ √ N √ Nj=1 1 √ sin( α t ) j=1 pj vj (0) and vi (t ) → − α sin( α t ) j=1 pj rj (0) + cos( α t ) j=1 pj vj (0) as α t → ∞, where p = [p1 , p2 , ..., pN ]T is the unique positive left eigenvector of −L associated with eigenvalue 0 satisfying pT 1N = 1. Ren [22] investigated the second-order consensus with the linear protocol ui (t) = −α ri (t) +



aij (vj (t) − vi (t)), i = 1, 2, ..., N ,

(1.29)

j∈Ni

where α > 0. A stable periodic consensus can be achieved. For α > 0, β = 0, and γ = ξ = 1, after some calculations, the inequality in Theorem 1.7 can be simplified to −Re2 (μi )α < 0, which is satisfied for all α > 0. Therefore the result reported in [22] can be treated as a particular case of the results in this chapter. Corollary 1.11. For α > 0, β > 0, γ > 0 and ξ > 0, a second-order consensus in multiagent system (1.3) can be achieved if and only if the network contains a directed spanning tree and inequality (1.13) in Lemma 1.6 holds. In addition, if a second-order consensus is reached as t → ∞, the consensus is trivial, that is, ri (t) → 0 and vi (t) → 0 as t → ∞. Under the assumptions in Corollary 1.11, all eigenvalues of Q have negative parts, which implies that limt→+∞ eQt = 02N ×2N . Therefore the second-order consensus is trivial. Corollary 1.12. For α > 0, β < 0, γ > 0, and ξ > 0, a second-order consensus in multiagent system (1.3) can be achieved if and only if the network contains a directed spanning tree and inequality (1.13) in Lemma 1.6 holds. In addition, if a second-order consensus is reached as t → ∞, the consensus is divergent, and the final consensus state can be estimated analytically by formulas (1.21) in Theorem 1.7.  2 Under the assumptions of Corollary 1.12, the two eigenvalues of Q, λ1± = −β± 2β −4α , have positive real parts, whereas the real parts of all other eigenvalues are negative. Similarly to Theorem 1.7, we easily prove Corollary 1.12.

1.4. Illustrative examples In this section, we present several simulation results validating the theoretical results in Section 1.3. We consider a group of five agents. The interaction topology between agents can be described by a directed network G shown in Fig. 1.1. It follows from

11

12

Second-Order Consensus of Continuous-Time Multi-Agent Systems

Figure 1.1 The directed interaction topology of five agents.

Fig. 1.1 that the adjacency matrix A and Laplacian matrix L of G are ⎡ ⎢ ⎢ ⎢ A=⎢ ⎢ ⎣

0 0 0 0 0

1 0 0 0 1

1 1 0 0 0

0 0 1 0 0

0 0 0 1 0

⎤

⎡

⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥,L = ⎢ ⎢ ⎥ ⎣ ⎦

2 0 0 0 0

−1

1 0 0 −1

−1 −1

1 0 0

0 0 −1 1 0

0 0 0 −1 1

⎤ ⎥ ⎥ ⎥ ⎥. ⎥ ⎦

(1.30)

Moreover, we can see that there exists a directed spanning tree in the directed network G. We first consider the linear consensus protocol in Corollary 1.9. By simple calculations we obtain that the eigenvalues of −L are μ1 = 0, μ2 = −1 + i, μ3 = −1 − i, μ4 = −2, and μ5 = −2. The left eigenvector of −L associated with zero eigenvalue satisfies p > 0 and pT 1N = 1, where p = [0.2, 0.2, 0.2, 0.2, 0.2]T . By computation we have that the right side of (1.27) in Corollary 1.9 equals 0.5. Let α = β = 0, γ = 0.45, and ξ = 0.5. Equality (1.27) in Corollary 1.9 is satisfied since 0.25/0.45 = 0.5556 > 0.5. Suppose that the initial state vector, which is composed of the initial position and veloc T T ity, is x (0) ; y (0) = [1, 0.86, 1.23, 0.23, 0.56; 2, 6, 5, 4, 0.58]T . Figs. 1.2(a) and 1.2(b) show the linear consensus of position and velocity states of five agents, and it takes about 400 s to reach consensus. When γ is assumed to be 0.3 and 0.2, respectively, and the other parameters are as before, the conditions of Corollary 1.9 hold. A second-order consensus can be also achieved, and the simulation results are shown in Figs. 1.2(c)–1.2(d) and 1.2(e)–1.2(f), respectively. In these cases, it takes about 60 s and 40 s, respectively, to reach a consensus. However, when γ = 0.51 (the conditions in Corollary 1.9 are not satisfied), a second-order consensus cannot be achieved, and the simulation results are shown in Figs. 1.2(g)–1.2(h). From Fig. 1.2 we also find that when a second-order consensus is achieved, the convergence speed is related positively 1 2 2 Im2 (μi ) . with ξγ − max − Re(μ )Re 2 (μ )+Im2 (μ ) 2≤i≤N

i

i

i

Secondly, we focus on the case of periodic consensus in Corollary 1.10. The linear gains in protocol (1.2) are assumed to be α = 0.25, β = 0, γ = 1, and ξ = 1, and the corresponding simulation results can be seen in Figs. 1.3(a)–1.3(b). When α = 0.49 and 1, the results are shown in Figs. 1.3(c)–1.3(d) and Figs. 1.3(e)–1.3(f), respectively.

Second-order consensus seeking in directed networks of multi-agent dynamical systems

Figure 1.2 Velocity and position states of five agents in a network under linear consensus protocols.

13

14

Second-Order Consensus of Continuous-Time Multi-Agent Systems

Figure 1.3 Velocity and position states of five agents in a network under periodic consensus protocols.

From Fig. 1.3 we can see that the period of consensus position and velocity states becomes smaller as α increases. Moreover, it takes more time to reach consensus as α increases. Next, we will validate Corollary 1.11. The linear gains in protocol (1.2) are assumed to be α = 0.1, β = 0.21, γ = 0.45, and ξ = 0.5. By some calculations we have that the inequality in Theorem 1.7 holds (−0.3962 < 0). The results of trivial second-order consensus can be reached, which are shown in Figs. 1.4(a)–1.4(b). When β = 0.1, the inequality in Theorem 1.7 also holds (−0.2325 < 0). The corresponding results can

Second-order consensus seeking in directed networks of multi-agent dynamical systems

Figure 1.4 Velocity and position states of five agents in a network under periodic consensus protocols.

be seen in Figs. 1.4(c)–1.4(d). However, for β = 0.3 and ξ = −0.25, the inequality in Theorem 1.7 does not hold any longer (0.3962 > 0), and the corresponding results are shown in Figs. 1.4(e)–1.4(f). Finally, we discuss Corollary 1.12. The linear gains in protocol (1.2) are assumed to be α = 5, β = −0.3, γ = 0.32, and ξ = 0.6. By some calculations we have that the inequality in Theorem 1.7 holds (−0.9976 < 0). The divergent consensus can also be reached, which is shown in Figs. 1.5(a)–1.5(b). However, for ξ = 0.2, we get by compu-

15

16

Second-Order Consensus of Continuous-Time Multi-Agent Systems

Figure 1.5 Velocity and position states of five agents in a network under divergent consensus protocols.

tation that the inequality in Theorem 1.7 does not hold (0.0992 > 0). Figs. 1.5(c)–1.5(d) display this result. From the above illustrative examples we can find that if a second-order consensus is reached, then the convergence (final consensus state) is determined commonly by the initial states of agents, the linear gains in the protocols, and the directed network topology (the left eigenvector p depends on the directed network topology).

1.5. Conclusion In this chapter, we studied the convergence of the second-order consensus of multiagent systems composed of coupled double-integrators dynamics. The considered protocol can be treated as extensions of most linear local interaction protocols in the existing literature. We have found that if a second-order consensus is reached, then the convergence is determined commonly by the initial states of agents, the linear gains in the protocols, and the directed network topology. Several simulation results further validated the effectiveness of theoretical analysis.

Second-order consensus seeking in directed networks of multi-agent dynamical systems

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[28] Q. Song, J. Cao, W. Yu, Second-order leader-following consensus of nonlinear multi-agent systems via pinning control, Systems & Control Letters 59 (2010) 553–562. [29] W. Yu, G. Chen, M. Cao, J. Kurths, Second-order consensus for multiagent with directed topologies and nonlinear dynamics, IEEE Transactions on Systems, Man and Cybernetics. Part B. Cybernetics 40 (2010) 881–891. [30] F. Ge, Z. Wei, Y. Lu, Y. Tian, L. Li, Decentralized coordination of autonomous swarms inspired by chaotic behavior of ants, Nonlinear Dynamics 70 (2012) 571–584. [31] J. Wang, H. Wu, Synchronization criteria for impulsive complex dynamical networks with timevarying delay, Nonlinear Dynamics 70 (2012) 13–14. [32] S. Pereira, A. Pages-Zamora, Mean square convergence of consensus algorithms in random WSNs, IEEE Transactions on Signal Processing 58 (2010) 2866–2874. [33] S. Pereira, A. Pages-Zamora, Consensus in correlated random wireless sensor networks, IEEE Transactions on Signal Processing 59 (2011) 6279–6284. [34] M. Porfiri, F. Fiorilli, Global pulse synchronization of chaotic oscillators through fast-switching: theory and experiments, Chaos, Solitons and Fractals 41 (2009) 245–262. [35] R. Amritkar, C. Hu, Synchronized state of coupled dynamics on time-varying networks, Chaos 16 (2006) 015117.

CHAPTER 2

Robust finite-time leader-following consensus algorithms for second-order multi-agent systems with nonlinear dynamics 2.1. Introduction In recent years, coordination of multi-agent systems has received compelling attention from scientific communities and emerged as a challenging new research area. The applications of multi-agent systems are diverse, such as satellite formation flying, cooperative control of unmanned air vehicles, scheduling of automated highway systems, air traffic control, congestion control of communication networks, distributed optimization of multiple mobile robotic systems, designation of sensor-network, flocking of social insects, swarm-based computing, and so on [1–5,7]. Research on multi-agent coordinated control problems not only helps in better understanding the general mechanisms and interconnection rules of natural collective phenomena, but also benefits many practical applications mentioned. A fundamental approach to achieve cooperative control is consensus analysis of multi-agent systems. Moreover, tools from algebraic graph theory have been introduced successfully to construct consensus conditions. Roughly speaking, consensus problem refers to how to design appropriate protocols and algorithms such that all agents can reach an agreement on a common value of interest that depends on the states of all agents, especially by negotiating with their neighbors (adjacent peers). This common value may be the attitude in multi-spacecraft alignment, the heading direction in flocking behavior, or the average in the distributed computation [6]. Most existing reports on the consensus problem focus on the case where the agents are governed by first-order dynamics when velocity is considered as the control input [1,4,7–9]. It is shown by Ren and Beard [4] that a first-order consensus can be achieved asymptotically if and only if the fixed directed network topology contains a directed spanning tree or the time-varying network topology contains a directed spanning tree frequently enough as the network evolves with time. Nevertheless, the second-order consensus problem has come to be recognized as an important issue when driving force (acceleration) is considered as the control input [5,10–12,14–17]. Here the second-order consensus problem is concerned with reaching an agreement among a group of agents governed by second-order dynamics. The insight into the secondorder consensus problem may lead to introducing more realistic dynamics into the Second-Order Consensus of Continuous-Time Multi-Agent Systems https://doi.org/10.1016/B978-0-32-390131-4.00009-1

Copyright © 2021 Elsevier Inc. All rights reserved.

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Second-Order Consensus of Continuous-Time Multi-Agent Systems

model of each individual agent based on the general framework of multi-agent systems, which is especially meaningful for the implementation of cooperative control strategies in engineering networked systems [13]. It has been shown that, in sharp contrast to the first-order consensus problem, a consensus may fail to be achieved for agents with second-order dynamics even if the network topology has a directed spanning tree [13,30]. More surprisingly, Yu et al. [10] mentioned that a consensus may no longer to be reachable within a multi-agent system by adding one connection between a chosen pair of agents, which has originally been able to reach a consensus. This is inconsistent with the intuition that more connections are helpful for reaching a consensus. Therefore the extension of consensus algorithms from first-order to second-order is non-trivial [31], and the second-order consensus problem is more complicated and challenging than the first-order case. Broadly specking, a complex network consists of a set of nonlinear oscillators with first-order dynamics for synchronization problems. However, in reality, some oscillators, for example, harmonic oscillators [14,21] and pendulums [18–20], are governed by second-order dynamics with the position and velocity terms. Hence, it is necessary to study the consensus problem of a multi-agent system composed of secondorder oscillators, in which the dynamics of each agent is not only determined by the interactions among agents, but also by its own more complicated dynamics, that is, intrinsic dynamics [11,21,22] (disturbances and unmodeled uncertainties of agent). In recent years, few consensus protocols consider the intrinsic dynamics of each individual agent in a multi-agent system with second-order dynamics. Note that Yu et al. [10] proposed a nonlinear multi-agent system by introducing complex dynamics into each agent and investigated the leaderless second-order consensus with fixed directed topology. Leader-following consensus means that there exists a leader agent that specifies an objective for all agents to follow. Song et al. [11], studied the second-order leaderfollowing consensus of nonlinear multi-agent systems by employing pinning control schemes. Through pinning a very small fraction of agents, all the agents in a large-scale multi-agent system can be effectively forced to track the leader agent with asymptotical convergence rate. In other words, the states of multi-agent systems cannot reach a consensus in finite time with their presented consensus algorithms. Also note that the convergence rate is an important topic in the study of the second-order consensus problem. Therefore finite-time consensus algorithms are more desirable. Besides a faster convergence rate, the closed-loop systems under finite-time control usually demonstrate better disturbance rejection properties. The main purpose of this chapter is studying the robust finite-time consensus problem for leader-following multi-agent systems with second-order nonlinear dynamics by using pinning control schemes. With the help of matrix theory, graph theory, and finite-time control technique we develop continuous distributed control algorithms in a quite unified way for each follower agent in the network. We also give a rigorous proof by using Lyapunov theory and show that the closed-loop systems are provided

Robust finite-time leader-following consensus algorithms for second-order multi-agent systems

with fast finite-time stability and strong robustness against uncertainties. Then we establish a sufficient condition ensuring that the robust leader-following consensus can be reached in a finite-time. From the obtained results, we find that the finite-time is determined by not only the parameters in the control algorithms but also by the initial states of agents in the network. The contributions of this chapter lie in two aspects. First, we introduce a new kind of terminal sliding-mode variable to establish our finite-time consensus protocols with faster convergence speed. Second, we use a saturation function with a small linear width to substitute the standard sign function, which can effectively remove the chattering coming from the high-frequency switching of discontinuous sign function. Compared with [11], we construct a sufficient condition ensuring that the states of agents globally exponentially approach the state of the leader rather than with asymptotic rate. The remainder of this chapter is organized as follows. In Section 2.2, we present some preliminaries including notations, graph theory, and supporting lemmas. Problem statements are provided, and main results are established in Section 2.3. Two numerical examples are simulated in Section 2.4 to verify the theoretical results. Finally, concluding remarks and future works are stated in Section 2.5.

2.2. Preliminaries 2.2.1 Notation The following mathematical notations will be used for simplicity throughout this chapter. We denote by 1n (0n ) the column vector with n elements being 1 (0) and by sign (x) the standard signum function; Rn denotes the set of all n-dimensional real column vectors; En represents the n-dimensional unit matrix; ⊗ denotes the Kronecker product; || · ||2 refers to the standard Euclidean norm for vectors; A−1 and AT denote the inverse and transpose matrices of a matrix A, respectively. For a vector x = [x1 , x2 , ... , xn ]T , we denote by diag{x} the diagonal matrix whose element in the ith row and ith column equals xi ; xα represents a column vector, that is, [xα1 , xα2 , ..., xαn ]T .

2.2.2 Graph theory Algebraic graph theory is widely used for representing multi-agent systems. We introduce some basic concepts and results about algebraic graph theory. For more detail, we refer to [23,24]. Suppose that information exchange among agents in multi-agent systems can be modeled by an interaction digraph (directed graph). Let G = (V , ε, A) be a weighted digraph of order N with the set of nodes V = {1, 2, ... , N }, a set of edges ε ⊆ V × V , and a weighted adjacency matrix A = (aij )N ×N with nonnegative elements. An edge of G is denoted by an ordered pair (i, j) that starts from node i and ends at node j. The element aij associated with the edge of the digraph is positive, that is, aij > 0 ⇔ (j, i) ∈ ε, and (i, j) ∈ ε means that node i can directly receive information

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Second-Order Consensus of Continuous-Time Multi-Agent Systems

from node j. Moreover, we assume that there is no self-loop in digraph G, that is, aii = 0 for all i ∈ V . The set of neighbors is denoted by Ni = {j ∈ V : (j, i) ∈ ε, i = j}, and the  in-degree and out-degree of node i are defined, respectively, as degin (i) = N j=1, j=i aij  and degout (i) = N a [11]. If there is a sequence of edges of the form (i, j1 ), j=1, j=i ji (j1 , j2 ), ..., (jm , j) ∈ ε composing a directed path beginning from i and ending at j in a directed graph G with distinct vertices jk , k = 1, 2, ..., m, then node j is said to be reachable from node i. A directed graph is strongly connected if there exists a directed path between any two distinct nodes. A directed graph has a directed spanning tree if there exists at least one node, called a root, that has a directed path to all other nodes. Moreover, if degin (i) = degout (i) for all i ∈ V , then the digraph is called balanced. Let  D = diag{d1 , d2 , ..., dN } be the diagonal matrix with diagonal elements di = N j=1 aij for all i ∈ V . The Laplacian matrix of a weighted digraph G is defined as L = D − A. To investigate the leader-following problem, we assume that besides the agents 1 ¯ ) to model the network to N, there exists a leader labeled by 0. We use g¯ = (V¯ , ε¯ , A ¯ is the adjacency matrix topology in this case, where V¯ = {0, 1, 2, ... , N }, ε ⊆ ε¯ , and A for agents 0, 1, 2, ..., N. To depict whether agents are connected to the leader in digraph g¯ , we define the leader adjacency matrix B = diag{b1 , b2 , ... , bN } associated with g¯ , where bi = ai0 > 0 if node 0 (the leader) is a neighbor of node i and bi = 0 ¯ otherwise. From the above  we can see that the relations between matrices A, A, and B1 T 0 0N ¯ = can be described as A , where B1 = [b1 , b2 , ... , bN ]T . Throughout this B1 A chapter, we assume that the weights of all edges are 1. Remark 2.1. A graph G is called undirected if there is a connection between two nodes i and j in G. Then aij = aji > 0; otherwise, aij = aji = 0 (i = j, i, j = 1, 2, ..., N ). Therefore undirected networks are a particular case of directed networks with aij = aji for all i, j = 1, 2, ..., N.

2.2.3 Supporting lemmas We give several necessary lemmas. Lemma 2.2. [25,26] Consider the system x˙ = f (x), f (0) = 0, x ∈ Rn . Suppose there exist a positive definite continuous function V (x) : U → R, real numbers c > 0 and α ∈ (0, 1), and an open neighborhood U0 ⊂ U of the origin such that V˙ (x) + c (V (x))a ≤ 0, x ∈ U0 \{0}. Then ))1−α V (x) will approach 0 in a finite time. In addition, the finite settling time T ≤ V (cx(1(0−α) . Lemma 2.3. [27] Let L be the (nonsymmetric) Laplacian matrix associated with a directed network G with N agents. Then L has a simple zero eigenvalue, and all other eigenvalues have positive real parts (rank(L ) = N − 1) if and only if G has a directed spanning tree. In addition, there exist 1N satisfying L1N = 0N and p ∈ RN satisfying p ≥ 0 (all the elements are nonnegative), pT L = 0N , and pT 1N = 1.

Robust finite-time leader-following consensus algorithms for second-order multi-agent systems

Lemma 2.4. [28] For xi ∈ R, i = 1, 2, ... , n, and 0 < p < 2, we have    ( ni=1 x2i )p/2 . In particular, ni=1 |xi | ≤ ( ni=1 x2i )1/2 for p = 1.

n

i=1 |xi |

p

≤

Lemma 2.5. [29] The fast terminal sliding mode can be described by the first-order dynamics s(t) = x˙ (t) + α x(t) + β(x(t))q/p , where x(t) ∈ R is a scalar variable, α, β > 0, p, q are the positive odd integers, and only the real solution is considered so that for any real number x(t), xq/p (t) is always a real number. When s = 0, we have x˙ (t) = −α x(t) − β(x(t))q/p . For properly chosen parameters, given an initial state  (x(t), x˙ (t)) converge to (0, 0) in a  x(0) = 0, the states p (p−q)/p α finite time T satisfying T ≤ α(p−q) ln 1 + β (x(0)) . Remark 2.6. The physical interpretation of Lemma 2.5 can be described as follows. When x(t) is far away from zero, the approximate dynamics become x˙ (t) = −α x(t), whose fast convergence is well understood. When x(t) is close to x = 0, the approximate dynamics become x˙ (t) = −β(x(t))q/p , which is a terminal attractor. Lemma 2.7. [22] Consider a leader-following system described by a directed network topology g¯ . If the directed network contains a directed spanning tree that takes the leader agent as the root node, then L + B is an invertible matrix, where L is the Laplacian matrix associated with follower agents, and B is the leader adjacency matrix.

2.3. Finite-time consensus analysis In this section, we present robust finite-time algorithms for nonlinear multi-agent systems and coupled double-integrator dynamics to achieve a second-order leaderfollowing consensus.

2.3.1 Problem description Consider a nonlinear multi-agent system consisting of coupled agents. Here the moving model of each agent in the group is given by 

x˙ i (t) = vi (t), v˙ i (t) = f (t, xi (t), vi (t)) + di (t) + ui (t),

(2.1)

where 1 ≤ i ≤ N, xi (t) = [xi1 (t), xi2 (t), ... , xin (t)]T and vi (t) = [vi1 (t) , vi2 (t) , · · · , vin (t)]T denote the position and velocity states of agent i, respectively, f (t, xi (t), vi (t)) = [f1 (t, xi (t), vi (t)), ... , fn (t, xi (t), vi (t))] is a nonlinear vector-valued continuous function describing the intrinsic dynamics of agent i, di (t) = [di1 (t), ... , din (t)] represents the corresponding uncertainties (including disturbances and unmodeled uncertainties), and ui (t) ∈ Rn is the control input for agent i. When f = 0 and di (t) = 0, the multi-agent system (2.1) reduces to the double-integrator dynamics [11].

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Second-Order Consensus of Continuous-Time Multi-Agent Systems

The motion model of the leader is assumed to be 

x˙ r (t) = vr (t), v˙ r (t) = f (t, xr (t), vr (t)) + dr (t),

(2.2)

where xr (t) ∈ Rn , vr (t) ∈ Rn , f (t, xr (t), vr (t)) ∈ Rn , and dr (t) ∈ Rn denote respectively the position, velocity, intrinsic dynamics, and uncertainties of the leader agent. Definition 2.8. The multi-agent system (2.1) is said to achieve a second-order leader-following consensus under the control input ui (t) if for any initial conditions xi (0), vi (0) ∈ Rn , we have limt→∞ xi (t) − xr (t) 2 = 0 and limt→∞ vi (t) − vr (t) 2 = 0 for all i = 1, 2, ... , N.

2.3.2 Main results Before moving on, we make the following assumption for the analysis. ¯ ¯ ¯ Assumption There exist   2.1.    positive  numbers   f and d such that ||f (t, x(tn), v(t))||2 ≤ f and max{d1 (t)2 , d2 (t)2 , ... , dN (t)2 , dr (t)2 } ≤ d¯ for all x(t), v(t) ∈ R .

When f = 0 and di (t) = 0, the reference velocity, that is, the velocity of the leader agent (2.2), is time-varying. Now we start to design consensus algorithms such that all agents in the nonlinear multi-agent system (2.1) can approach the leader agent (2.2) within a finite-time and have better disturbance rejection properties. Considering that not all the agents have access to the reference trajectory information, the tracking errors eix (t) and eiv (t) of agent i are defined as follows based on the neighboring information: ⎧ N   ⎪ x ⎪ e ( t ) = a ( x − x ) + b ( x − x ) = Lij xj + bi (xi − xr ), ⎪ ij i j i i r i ⎨ j ∈N j=1 i

N ⎪   ⎪ v ⎪ aij (vi − vj ) + bi (vi − vr ) = Lij vj + bi (vi − vr ). ⎩ ei (t) = j∈Ni

(2.3)

j=1

x T v T ] , Σv (t) = [e1v , e2v , ... , eN ] , D(t) = [d1 (t), d2 (t), ..., dN (t)]T , Letting Σx (t) = [e1x , e2x , ... , eN U (t) = [u1 (t), u2 (t), ..., uN (t)]T , and F (t, Σx (t), Σv (t)) = [f T (t, x1 (t), v1 (t)), ..., f T (t, xN (t), vN (t))]T , the closed-loop tracking error system can be rewritten in the compact matrix form as follows:

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

Σ˙ x (t) = Σv (t), Σ˙ v (t) = ((L + B) ⊗ En )(F (t, Σx (t), Σv (t)) + D(t)) −(B ⊗ En )(1N (f (t, xr (t), vr (t)) + dr (t))) +((L + B) ⊗ En )U (t).

(2.4)

Robust finite-time leader-following consensus algorithms for second-order multi-agent systems

Theorem 2.9. Suppose that Assumption 2.1 holds and a small fraction of agents in the directed network are pinned by the leader agent so that the whole network contains a directed spanning tree, which takes the leader agent as the root node. Suppose the sliding mode vector S(t) is defined as S(t) = [s1 (t), s2 (t), ... , sN (t)]T , where si (t) = eiv (t) + α eix (t) + β(eix (t))q/p , α, β > 0, and p and q are positive odd integers satisfying q < p < 2p. Then, for the leader-following system (2.1)–(2.2), there always exists a control input q U (t) = − ((L + B) ⊗ En )−1 [(α ENn + β diag{(Σx (t))q/p−1 })Σv (t) p ¯ ¯ + ((2N (f + d) + k)ENn + (B ⊗ En )(f¯ + d¯ ))sign(S(t))],

(2.5)

where k > 0, such that on the terminal sliding-mode surface, a second-order consensus can be achieved in a finite time. Proof. The first-order time derivative of sliding-mode variable si (t) (i = 1, 2, ... , N), is q q p q x p−q 1 v v v = e˙i (t) + α ei (t) + β (ei (t)) ei (t). p

˙si (t) = e˙iv (t) + α˙eix (t) + β (eix (t)) p−1 e˙ix (t)

(2.6)

Then the sliding-mode vector S(t) can be described by the first-order dynamics q q S˙ (t) = Σ˙ v + [α ENn + β diag{(Σx ) p−1 }]Σv . p

(2.7)

Selecting the Lyapunov function as V (t) = (1/2)ST (t)S(t), its first-order derivative with respect to time is V˙ (t) = ST (t)S˙ (t) q p

q

= ST (t)(Σ˙ v (t) + [α ENn + β diag{(Σx (t)) p−1 }]Σv (t)) = ST (t)(((L + B) ⊗ En )(F (t, Σx (t), Σv (t)) + D(t))

(2.8)

− (B ⊗ En )(1N (f (t, xr (t), vr (t)) + dr (t)))

q p

q

+ ((L + B) ⊗ En )U (t) + [α ENn + β diag{(Σx (t)) p−1 }]Σv (t)).

Note that 

S(t)

((L + B) ⊗ En )(F (t, Σx (t), Σv (t)) + D(t)) −(B ⊗ En )(1N (f (t, xr (t), vr (t)) + dr (t)))



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Second-Order Consensus of Continuous-Time Multi-Agent Systems

 = S(t) =

N

N

amj + bm ) ⊗ En )(f (t, xm (t), vm (t)) + dm (t))

j=1, j=m

N

m=1

−



−(B ⊗ En )(1N (f (t, xr (t), vr (t)) + dr (t)))]

sm (t)((

m=1

−

((D − A + B) ⊗ En )(F (t, Σx (t), Σv (t)) + D(t))

N

sm (t)[

N

(amj ) ⊗ En .(f (t, xj (t), vj (t)) + dj (t))]

j=1, j=m

sm (t)(bm ⊗ En )(f (t, xr (t), vr (t)) + dr (t))

m=1

≤

N

N sm (t) 2 (f¯ + d¯ ) +

m=1

+

N

N sm (t) 2 (f¯ + d¯ )

m=1

N

sm (t) 2 bm (f¯ + d¯ ).

(2.9)

m=1

Substituting the expressions of U (t) in the theorem and using Lemma 2.4, we have the following result: q q ST (t)(((L + B) ⊗ En )U (t) + [α ENn + β diag{(Σx (t)) p−1 }]Σv (t)) p = ST (t)[−((2N (f¯ + d¯ ) + k)ENn + (B ⊗ En )(f¯ + d¯ ))sign(S(t))]

=−

N

n

2N (f¯ + d¯ )(

m=1

−

j=1

N

n

bm (f¯ + d¯ )(

m=1

≤− −

N

−

m=1

n

k(

|smj (t)|)

j=1

|smj (t)|)

1/2 1/2 n N n

2 2 ¯ ¯ 2N (f + d)( |smj (t)| ) − k( |smj (t)| ) j=1

N

n

m=1

=−

N

j=1

m=1

N

|smj (t)|) −

bm (f¯ + d¯ )(

m=1

j=1

|smj (t)|2 )

j=1

2N (f¯ + d¯ )( sm (t) 2 ) −

m=1 N

m=1 1/2

N

k( sm (t) 2 )

m=1

bm (f¯ + d¯ )( sm (t) 2 ).

(2.10)

Robust finite-time leader-following consensus algorithms for second-order multi-agent systems

Combining (2.8), (2.9), and (2.10), we obtain V˙ (t) ≤ −k

N

1/2 N n

2 sm (t) 2 = −k ( smj (t))

m=1

m=1 j=1

n N

≤ −k (

s2mj (t))1/2

(2.11)

√

= − 2k(V (t))

1/2

.

m=1 j=1

It is clear that if V (0) = 0, then by Lemma 2.2 the sliding-mode vector S(t) will reach S = 0 within a finite time T1 , which satisfies √  N n

2  (smj (0))2 , T1 ≤

2k

(2.12)

m=1 j=1



v x x x where smj (0) = emj (0) + α emj (0) + β(emj (0))q/p , emj (0) = N l=1 aml (xmj (0) −xlj (0)), and  N v emj (0) = l=1 aml (vmj (0) − vlj (0)) + bm (vmj (0) − vrj (0)). After the sliding mode S = 0 is reached, we have e˙ix (t) + α eix (t) + β(eix (t))q/p = 0, i = 1, 2, ... , N. Using Lemma 2.5, we can see that all error states eix (t) and eiv (t) converge to zero within a finite-time T2 satisfying

p



α ln 1 + T2 ≤ α(p − q) β

 max

1≤m≤N , 1≤j≤n

v {emj (0)}

 p−p q  .

(2.13)

Then we have Σx = 0 and Σv = 0. Recalling the definition of Σx (t) and L1N = 0, after Σx = 0 is reached, we easily see that (L + B)[x1 (t), x2 (t), ... , xN (t)]T = (L + B)1N xr (t).

From Lemma 2.7 we know that after some agents are pinned, the whole network topology contains a directed spanning tree, and then L + B has an inverse matrix. Therefore [x1 (t), x2 (t), ... , xN (t)]T = 1N xr (t). Similarly, we can prove that [v1 (t), v2 (t), ..., vN (t)]T = 1N vr (t). We can see in the control input (2.5) that the second term containing diag{(Σx (t))q/p−1 }Σv (t) may cause the occurrence of singularity if Σv = 0 when Σx = 0. This situation does not occur in the ideal sliding mode since when S = 0, Σv = −αΣx − β(Σx )q/p , and hence as long as q < p < 2q, the term diag{(Σx )q/p−1 }Σv is equivalent to diag{−αΣxq/p − β(Σx )2q/p−1 }, which is non-singular. From all the above statements we can see that the second-order leader-following consensus can be reached in finite time T ≤ T1 + T2 .

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Second-Order Consensus of Continuous-Time Multi-Agent Systems

Remark 2.10. To eliminate chattering effectively, the sign function in the control input is usually replaced by the saturation function or sigmoid functions. The chattering phenomenon often appears in sliding-mode control, and it comes from the high-frequency switching of sign function due to its discontinuity at the origin. The chattering will weaken the performance of system when it is severe. Therefore, to remove the chattering to a large extent, in practice, it is necessary to use a saturation function instead of the sign function. The saturation function can be defined as sat(x) = 1, x > δ ; sat(x) = −1, x < −δ ; and sat(x) = x, x ≤ |δ|, where δ > 0 is called the linear width. Assuming that δ is a very small positive number, the saturation function can approximate the sign function with high accuracy. Remark 2.11. For the robust finite-time consensus of second-order systems with nonlinear dynamics, a more conservative condition is necessary, that is, all the following agents in the group must know the common upper bound of all the agents’ nonlinear functions. However, for the asymptotical case, this condition can be relaxed by Lipschitz conditions, which are shown in [10,11]. Remark 2.12. When there are many agents in the network, position and velocity information is only provided to a very small fraction of follower agents. Pinning control schemes can be used to guide a subgroup of agents such that all agents can approach the leader agents based on local information exchanged from neighboring agents. Therefore the selection of the pinned nodes is very important. From Proposition 1 in [11] we know that: (1) The agents with zero in-degrees must be pinned because their states are not influenced by any other agents; (2) The leader agent and all the followers should form a directed spanning tree, in which the leader node is the only root, such that the states of all agents can be directly on indirectly affected by the state of the leader. Remark 2.13. In the design of the controller, we need the upper bounds for nonlinear ¯ If the variables in the nonlinear function dynamics and uncertainties, that is, f¯ and d. are bounded, then the nonlinear function is also bounded, which can be easily satisfied for many systems, for example, the Lorenz system, Chen system, Chua circuit, and so on. In practice, we can select a sufficiently large number instead of f¯ + d¯ in the design of the controller. Remark 2.14. If (2.3) is designed as ⎧  x ⎪ aij (xi + Δi − xj − Δj ) + bi (xi + Δi − xr − Δ0 ), ⎨ ei (t) = j∈Ni  v ⎪ aij (vi + δi − vj − δj ) + bi (vi + δi − vr − δ0 ), i = 1, 2, ... , N , ⎩ ei (t) = j∈Ni

then [x1 (t), x2 (t), ... , xN (t)]T = 1N (xr (t) + Δ0 ) − [Δ1 , Δ2 , ... , ΔN ]T and [v1 (t), v2 (t), ... , vN (t)]T = 1N (vr (t) + δ0 )−[δ1 , δ2 , ... , δN ]T , that is, relative state deviations with a time-varying consensus reference state can be reached.

Robust finite-time leader-following consensus algorithms for second-order multi-agent systems

Figure 2.1 The directed interaction topology of five agents.

2.4. Illustrative examples In this section, we give illustrative examples to demonstrate the efficiency and applicability of the proposed method and validate the theoretical analysis. Example 1. In view of (2.1), we consider the multi-agent system consisting of ten agents described by 

x˙ i (t) = vi (t), v˙ i (t) = f (t, xi (t), vi (t)) + di (t) + ui (t), i = 1, 2, ... , 10,

(2.14)

where xi (t) and vi (t) ∈ R are the position and velocity states of agent i, respectively. The nonlinear function f is as follows: f (t, xi (t), vi (t)) = xi (t) − x3i (t) − avi (t) + b cos(ωt),

(2.15)

where a = 0.5, b = 0.4, and ω = 1. The second-order isolated system with nonlinear function (2.15) is chaotic with positive Lyapunov exponent 0.56. The uncertainties di (t) including external perturbations and unmodeled dynamics of agent i are assumed to be 0.01i sin(t), i = 1, 2, ... , 10, and the uncertainties dr (t) for the leader agent are taken 0.05 sin(t). Through simulation observation we obtain that f is bounded with   f  ≤ 1.5. According to Assumption 2.1, the parameters can be chosen as f¯ = 1.5 and 2 d¯ = 0.1, respectively. We assume that the interaction digraph of ten agents in the group is as shown in Fig. 2.1 with all the edge weights equal to 1. It is obvious that the directed topology in Fig. 2.1 does not have a directed spanning tree. By the statements of Remark 2.12 we can select agents 3 and 7 to be pinned by the leader agent so that the directed network including the leader agent contains a directed spanning tree that takes the leader agent as the root node. From the interaction topology of all agents in the group we can obtain A, ¯ respectively. To avoid occupying too much space, we omit their expressions D, L, B, A, here since it is very easy to write the expressions if needed. The initial positions and

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Second-Order Consensus of Continuous-Time Multi-Agent Systems

Figure 2.2 Positions and velocities of all agents without control inputs.

Figure 2.3 Results of simulation performed on a group of ten agents moving in the position space and velocity space under the influence of the control protocols (2.5).

velocities of the ten agents and the leader agent are chosen as [0.3965, 1.7360, 0.7149, 1.0310, 0.9725, −0.4311, 0.6219, −1.3153, 0.8242, −1.8727, 1.0621]T and [−0.8923, −1.8153, −1.6115, 1.2938, 0.7793, −0.7316, 1.8009, −1.8622, −0.2450, −0.4738, 1.1808]T , respectively. Without control input, the evolutions of positions and velocities of all agents are shown in Figs. 2.2(a) and 2.2(b), respectively. Choosing p = 5, q = 3, α = β = 0.5, and k = 0.8, Figs. 2.3(a) and 2.3(b) show some results of simulation performed on a group of ten agents moving in the position and velocity spaces under the influence of the control protocols (2.5) (here the sign function is replaced by a saturation function with small linear width δ = 0.15). Through computation by formulas (2.12) and (2.13) we have T1 = 11.5728 s and T2 = 4.6937 s. From Figs. 2.3(a) and 2.3(b) we can observe that all agents eventually arrive at the same position and velocity as the leader agent in a finite time T = 16.2665 s. The evolutions of tracking errors eix (t), eiv (t) and the sliding-

Robust finite-time leader-following consensus algorithms for second-order multi-agent systems

Figure 2.4 The evolutions of tracking errors exi (t) and evi (t) and the sliding-mode variable si (t).

mode variable si (t), i = 1, 2, ... , 10, are plotted in Figs. 2.4(a), 2.4(b), and 2.4(c). We can see that si (t) converges to zero within T2 = 4.6937 s and eix (t), eiv (t) converge to zeros within T = 16.2665 s. In the case of using saturation function, the corresponding control inputs for each agent are plotted in Fig. 2.4(d). From this we can also clearly see that neither singularity nor chattering occurs in the control input. Moreover, the controller has very good robustness to the uncertainties. A simulation is performed for comparison with the control inputs in which the sign function is used. The evolutions of positions and velocities of all agents under control input are shown in Figs. 2.5(a) and 2.5(b), respectively. The evolutions of tracking errors eix (t) and eiv (t) and the sliding-mode variable si (t), i = 1, 2, ... , 10, are plotted in Figs. 2.6(a), 2.6(b), and 2.6(c). We can see that the tracking performance is not very good. The corresponding control inputs are shown in Fig. 2.6(d). From observation it is clear that there exists chattering in the control inputs, which is caused by the discontinuity of the sign function (it can also be understood as the control inputs have high-frequency switching in the neighborhood of sliding mode S = 0). Therefore the controller equipped with saturation function with a very small linear width has better performance than that with sign function.

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Figure 2.5 Positions and velocities of all agents with control inputs.

Figure 2.6 The evolutions of tracking errors exi (t) and evi (t) and the sliding-mode variable si (t).

Example 2. When fi (t, xi (t), vi (t)) = 0, and di (t) = 0, each agent in the multi-agent system (2.1) reduces to the following double-integrator: 

x˙ i (t) = vi (t), v˙ i (t) = ui (t), i = 1, 2, ... , 10,

(2.16)

Robust finite-time leader-following consensus algorithms for second-order multi-agent systems

Figure 2.7 Positions and velocities of all agents without control inputs.

Figure 2.8 The results of simulation performed on a group of ten agents moving in the position and velocity spaces under the influence of the control protocols (2.5).

and the leader agent (2.2) is described by 

x˙ r (t) = vr (t), v˙ r (t) = 0,

(2.17)

which indicates that the reference velocity is a constant. The other parameters are assumed to be the same as in Example 1. Without control input, the evolutions of positions and velocities of all agents are shown in Figs. 2.7(a) and 2.7(b), respectively. Figs. 2.8(a) and 2.8(b) show some results of simulation performed on a group of ten agents moving in the position and velocity spaces under the influence of the control protocols (2.5) (here the sign function is replaced by a saturation function with a small linear width δ = 0.15). The evolutions of tracking errors eix (t) and eiv (t) and the sliding mode variable si (t), i = 1, 2, ... , 10, are plotted in Figs. 2.9(a), 2.9(b), and 2.9(c). We can see that si (t) converges to zero within

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Second-Order Consensus of Continuous-Time Multi-Agent Systems

Figure 2.9 The evolutions of tracking errors exi (t) and evi (t) and the sliding-mode variable si (t).

T2 = 4.6937 s, and eix (t) and eiv (t) converge to zeros within T = 16.2665 s. In the case of using a saturation function, the corresponding control inputs for each agent are plotted in Fig. 2.9(d). From this we can also clearly see that neither singularity nor chattering occurs in the control input. Moreover, the controller has very good robustness to the uncertainties. A simulation is also performed for the purpose of comparing with the control inputs in which the sign function is used. The evolutions of positions and velocities of all agents under control input are shown in Figs. 2.10(a) and 2.10(b), respectively. The evolutions of tracking errors eix (t) and eiv (t) and the sliding-mode variable si (t), i = 1, 2, ... , 10, are plotted in Figs. 2.11(a), 2.11(b), and 2.11(c). We can see that the tracking performance is not very good. The corresponding control inputs are shown in Fig. 2.11(d). From observation it is clear that there exists chattering in the control inputs, which is caused by the discontinuity of the sign function (it can also be understood as the control inputs have high-frequency switching in the neighborhood of sliding mode S = 0). Therefore the controller equipped with a saturation function with a very small linear width has better performance than that with the sign function.

Robust finite-time leader-following consensus algorithms for second-order multi-agent systems

Figure 2.10 Positions and velocities of all agents without control inputs.

Figure 2.11 The evolutions of tracking errors exi (t) and evi (t) and the sliding-mode variable si (t).

2.5. Conclusion In this chapter, we have investigated the robust finite-time leader-following consensus problem for multi-agent systems, in which all agents are governed by second-order nonlinear intrinsic dynamics. A kind of terminal sliding-mode variable is introduced to

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solve the problem. The presented algorithms can enable the elimination of the singularity problem associated with the conventional terminal sliding-mode control.

References [1] A. Jadbabaie, J. Lin, A.S. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Transactions on Automatic Control 48 (2003) 988–1001. [2] J.A. Fax, R.M. Murray, Information flow and cooperative control of vehicle formations, IEEE Transactions on Automatic Control 49 (2004) 1465–1476. [3] L. Moreau, Stability of multiagent systems with time-dependent communication links, IEEE Transactions on Automatic Control 50 (2005) 169–182. [4] W. Ren, R.W. Beard, Consensus seeking in multiagent systems under dynamically changing interaction topologies, IEEE Transactions on Automatic Control 50 (2005) 655–661. [5] Y.G. Hong, J.P. Hu, L.X. Gao, Tracking control for multi-agent consensus with an active leader and variable topology, Automatica 42 (2006) 1177–1182. [6] P. Lin, Y.M. Jia, Consensus of second-order discrete-time multi-agent systems with nonuniform timedelays and dynamically changing topologies, Automatica 45 (2009) 2145–2158. [7] R.O. Saber, R.M. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Transactions on Automatic Control 49 (2004) 1520–1533. [8] Y. Sun, L. Wang, G. Xie, Average consensus in networks of dynamic agents with switching topologies and multiple time-varying delays, Systems & Control Letters 57 (2008) 175–183. [9] J. Lu, D.W.C. Ho, J. Kurths, Consensus over directed static networks with arbitrary finite communication delays, Physical Review E 80 (2009) 066121. [10] W. Yu, G. Chen, M. Cao, J. Kurths, Second-order consensus for multiagent systems with directed topologies and nonlinear dynamics, IEEE Transactions on Systems, Man and Cybernetics. Part B. Cybernetics 40 (2010) 881–891. [11] Qiang Song, Jinde Cao, Wenwu Yu, Second-order leader-following consensus of nonlinear multiagent systems via pinning control, Systems & Control Letters 59 (2010) 553–562. [12] Y. Hong, G. Chen, L. Bushnell, Distributed observers design for leader-following control of multiagent networks, Automatica 44 (2008) 846–850. [13] W.W. Yu, G.R. Chen, M. Cao, Some necessary and sufficient conditions for second-order consensus in multi-agent dynamical systems, Automatica 46 (6) (2010) 1089–1095. [14] W. Ren, Synchronization of coupled harmonic oscillators with local interaction, Automatica 44 (12) (2008) 3195–3200. [15] Y. Zhang, Y.P. Tian, Consentability and protocol design of multi-agent systems with stochastic switching topology, Automatica 45 (5) (2009) 1195–1201. [16] J.D. Zhu, Y.P. Tian, J. Kuang, On the general consensus protocol of multi-agent systems with doubleintegrator dynamics, Linear Algebra and Its Applications 431 (5–7) (2009) 701–715. [17] J.D. Zhu, On the consensus speed of multi-agent systems with double-integrator dynamics, Linear Algebra and Its Applications 434 (5–7) (2011) 294–306. [18] D.D. Humieres, M.R. Beasley, B.A. Huberman, A. Libchaber, Chaotic states and routes to chaos in the forced pendulum, Physical Review A 26 (6) (1982) 3483–3496. [19] P. Amster, M.C. Mariani, Some results on the forced pendulum equation, Nonlinear Analysis – Theory Methods & Applications 68 (2008) 1874–1880. [20] H.K. Khalil, Nonlinear Systems, 3rd edition, Prentice Hall, 2002. [21] H. Su, X.F. Wang, Z. Lin, Synchronization of coupled harmonic oscillators in a dynamic proximity network, Automatica 45 (2009) 2286–2291. [22] Khoo Suiyang, Xie Lihua, Man Zhihong, Robust finite-time consensus tracking algorithm for multirobot systems, IEEE/ASME Transactions on Mechatronics 14 (2009) 219–228. [23] C. Godsil, G. Royle, Algebraic Graph Theory, Springer-Verlag, New York, 2001. [24] W. Zhu, D.Z. Cheng, Leader-following consensus of second-order agents with multiple time-varying delays, Automatica 46 (12) (2010) 1994–1999.

Robust finite-time leader-following consensus algorithms for second-order multi-agent systems

[25] S.P. Bhat, D.S. Bernstein, Finite-time stability of continuous autonomous systems, SIAM Journal on Control and Optimization 38 (3) (2000) 751–766. [26] Shihua Li, Haibo Du, Xiangze Lin, Finite-time consensus algorithm for multi-agent systems with double-integrator dynamics, Automatica 47 (2011) 1706–1712. [27] W. Ren, R.W. Beard, Consensus seeking in multi-agent systems under dynamically changing interaction topologies, IEEE Transactions on Automatic Control 50 (5) (2005) 655–661. [28] Mohammad Pourmahmood Aghababa, Sohrab Khanmohammadi, Ghassem Alizadeh, Finite-time synchronization of two different chaotic systems with unknown parameters via sliding mode technique, Applied Mathematical Modelling 35 (2011) 3080–3091. [29] Shuanghe Yu, Xinghuo Yu, Zhihong Man, A fuzzy neural network approximator with fast terminal sliding mode and its applications, in: International Conference on Neural Information Processing, vol. 148, 2004, pp. 469–486. [30] W. Ren, E. Atkins, Second-order consensus protocols in multiple vehicle systems with local interactions, in: AIAA Guidance, Navigation, and Control Conference and Exhibit, San Francisco, California, 2005. [31] W. Ren, E. Atkins, Distributed multi-vehicle coordinated control via local information exchange, International Journal of Robust and Nonlinear Control 17 (10–11) (2007) 1002–1033.

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CHAPTER 3

Second-order consensus of multi-agent systems with nonlinear dynamics over random switching directed networks 3.1. Introduction In recent years, there has been an increasing interest in the study of the interplay between communication and control in networks [1–23]. In particular, coordination control of multi-agent systems has received compelling attention from scientific communities and emerged as a challenging new research field [1–6]. Coordination control means that local communication and cooperation among individual agents in the network may lead to certain desirable global behaviors. The local interaction mechanism also frequently appears in nature such as synchronization flashing of fireflies [35], movement of a school of fish [35], descriptions of the heart [36], the understanding of brain seizures [37], nonlinear optics [38], meteorology [39], and so on. These collective activities of creatures have inspired the designs of many practical engineering applications [8], including (but not limited to) the formation control of multi-robots [40] and unmanned autonomous vehicles (UAV) [41] in control engineer, the distributed computation [42] and the coordination control of distributed sensor networks [43] in computer science, swarming or flocking [44], complex networks [6,10–21], to name a few. A fundamental approach to make the states of multi-agent systems reach an agreement on a common value of interest is consensus analysis, because it not only helps in better understanding the general mechanisms and interconnection rules of natural collective phenomena, but also benefits many practical applications of networked cyber-physical systems. When the control input is added to the velocity term, each agent can be modeled simply as a first-order integrator. Significant progresses have been made toward the consensus problem of multi-agent systems for this case, see [1,2,4], to name a few. Also note that almost reported works can be treated as a particular case of the synchronization problem of complex dynamical networks [45], which has been widely studied in the past decades [6,10–21]. However, as pointed out in [46], the extension of consensus algorithms for agents from first-order dynamics to second-order dynamics (when the control input is added to the driving force/acceleration term) is non-trivial. That is, the obtained first-order consensus criteria usually failed to handle with consensus problem for second-order multi-agent systems [47]. Very recently, the second-order consensus of multi-agent systems has attracted more and more attention under various assumptions, such as communication time delays [3,48,49], switching topology [7,9,49,50], nonlinSecond-Order Consensus of Continuous-Time Multi-Agent Systems https://doi.org/10.1016/B978-0-32-390131-4.00010-8

Copyright © 2021 Elsevier Inc. All rights reserved.

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ear dynamics [45,47,51,52] or nonlinear coupling [51], leader [45,49,50] or leaderless [3,5,7,9,47,51,52], and communication noises [3,49]. Usually, in reality, some oscillators, for example, harmonic oscillators [51,53] and pendulums [54], are governed by second-order systems with both position and velocity terms. Hence it is necessary to investigate the consensus problem of multi-agent systems composed of second-order oscillators, in which the dynamics of each agent are not only determined by the interactions among agents, but also by their own complicated dynamics, that is, intrinsic dynamics [45,47,51,52,55] (disturbances and unmodeled uncertainties of agent). Protocols or algorithms dealing with the second-order consensus of multi-agent systems with nonlinear dynamics have not been emphasized until the works [45,47,51]. In [47] a kind of measurement for directed strongly connected graph, that is, general algebraic connectivity, was first defined by Yu et al. The authors built the bridge between the general algebraic connectivity and the performance of reaching an agreement for second-order multi-agent systems with nonlinear dynamics. A similar problem has also been paid attention by Song et al. [45] via using pinning control technique, and it is worth mentioning that the above approaches have overcome the restriction in [47] that the interaction network is strongly connected. Based on local adaptive strategies, Su et al. [51] have found that if one agent has access to the information of the virtual leader, then all agents in the group can synchronize with the virtual leader. However, there is a common drawback in the previous works: The network topology is deterministic or static, and the inner coupling matrix is constant in time. In real-world applications such as the case of terrestrial planet finder (TPF) mission and other similar mission scenarios, the sensing and inter-spacecraft communication topology often changes over time due to the dynamic nature of each state of a spacecraft (e.g., range limitations on relative sensing, shadowing scenario, etc.) [1]. Hence the study on the consensus problem of multi-agent systems with switching topology under the removal of old links and/or the addition of new links with mobile nodes is not only important but also necessary [52]. Generally speaking, the consensus of multi-agent systems with switching topologies can be divided into the following cases: arbitrary switching [50], Markov switching [7], controlled switching [49,52], and random switching [1,2,19,23]. Random switching means that communication among agents over a network depends on a time-varying topology, which may vary randomly based on a pre-given probability matrix. Moreover, the other switching modes may be regarded as particular cases of random switching in which some special switching sequences may take place. In addition, problems on directed graphs are theoretically more challenging than those on undirected graphs due to algebraic properties mostly known for undirected graphs [4]. Also, many important real networks have directed edges [4]. Moreover, with increasingly strict requirements for control speed and system performance, the unavoidable time-delays in both controllers and actuators have also become a serious problem. For instance, all digital controllers, analogue of

Second-order consensus of multi-agent systems with nonlinear dynamics over random switching directed networks

anti-aliasing and reconstruction filters, have also exhibited a certain time-delay during operation, and the hydraulic actuators and human being interaction usually show even more significant time-delays. To the best of the authors’ knowledge, few authors have considered the second-order dynamic consensus problem for multi-agent systems with time-delay(-free) coupling over random switching directed networks (that communicate via a stochastic information network) thus far. Motivated by the previous discussions, in this chapter, we investigate this challenging scenario. Communication among agents is modeled as a randomly directed periodically switching graph with different edge weights. The existence of any edge is probabilistic and independent of the existence of any other edge. We further allow each edge to be weighted differently. By applying the orthogonal decomposition method the system state vector can be decomposed as two transversal components, one of which evolves along the consensus manifold, and the other evolves transversally with the consensus manifold. Several sufficient criteria for asymptotically almost sure consensus are derived for the cases of time-delay-free coupling and time-delay coupling, respectively. For the first case, we find that if there exists one directed spanning tree in the network that corresponds to the fixed timeaverage topology and the switching rate of the dynamic network is sufficiently fast, then a second-order dynamic consensus can be guaranteed by the choice of suitable parameters. For the second case, we also propose three criteria based on algebraic inequality for reaching a second-order dynamic consensus. The obtained results are quite powerful and can be further used to solve various switching cases for complex dynamical networks. The rest of this chapter is structured as follows. In Section 3.2, we introduce some basic concepts of a random graph and formulate the problem under investigation. In Section 3.3, we derive several sufficient conditions for reaching a second-order dynamic consensus over random networks for the cases of time-delay-free coupling and timedelay coupling. We illustrate our main results by numerical simulations in Section 3.4. Concluding remarks and future research topics are drawn in Section 3.5.

3.2. Preliminaries 3.2.1 Notation Unless otherwise stated, the vectors in this chapter are assumed to be columns. By Z + we denote the set of positive integers; Rn and Rn×m denote the n-dimensional Eumatrices, respectively. For a vector u ∈ Rn , clidean space and the set of all n × m real  T 1/2 its Euclidean norm is defined as u = u u ; In and On denote the identity matrix and zero matrix of order n, respectively; λmin (A) and λmax (A) denote the minimum and maximum eigenvalue of a square matrix A, respectively; λi (A) denotes the ith eigenvalue of a square matrix A. Let e = [1, 1, . . . , 1]T . The Kronecker product, denoted by ⊗, facilitates the manipulation of matrices of appropriate dimensions by the follow-

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Second-Order Consensus of Continuous-Time Multi-Agent Systems

ing properties: (A ⊗ B) (C ⊗ D) = AC ⊗ BD and (A⊗ B)T = AT ⊗ BT . The symmetric part of a matrix B ∈ Rm×m is defined as sym(B) = 1/2 B + BT ; Re (λ) and Im (λ) denote the real and imaginary parts of a complex number λ, respectively.

3.2.2 Random graph Graph theory is the study of objects, naturally called graphs, consisting of a set of vertices, each pair of which is endowed with an incidence relation represented by an edge. In variety of emerging applications, including mobile ad hoc networks [24], opinion dynamics [25], cooperative control [19,26,27], mathematical epidemiology [28], information exchange among agents in multi-agent networked systems can be modeled by interaction random graphs. Mathematically, a dynamically switching directed/undirected random network can be described by a sequence of weighted directed/undirected random graphs G = (V , E, We ), where V = {1, 2, ..., N } is a vertex set whose elements denote the agents in the networks, E ⊆ V × V is an edge set whose elements denote the directed/undirected communication links between agents. A directed edge Eij ∈ E   in a graph G is represented by an ordered pair of vertices i, j , where i is the head, and j is the tail, which also means that vertex i can receive information from vertex j   for i, j ∈ V . We assume that the existence of an edge from vertex i to vertex j j = i in a graph G (t) is determined randomly and is also independent of other edges with probability pij , 0 ≤ pij ≤ 1. An information link is referred to as a potential link when the associated edge probability pij > 0. The probabilities pij are collected in the probability   matrix P = pij . We also assume that the random graph ha no self-loops, that is, no single edge starts and ends at the same vertex. Thus we have pii = 0 for i ∈ V . Then we define N (N − 1) independent Bernoulli random variables δij , i, j ∈ V , i = j, as fol1 − pij , where each random lows: δij = 1 with probability pij and δij= 0 with probability   variable δij is associated with the edge i, j . We = wij is a weight matrix with all diagonal elements equal to 0, and each element wij denotes the weight associated with edge   i, j . The weight denotes how each agent evaluates the information collected from its neighboring agents to update the consensus algorithm. The matrix W is assumed to be symmetric for undirected graphs, whereas it can be asymmetric for directed graphs. Moreover, a directed graph has a directed spanning tree if there exists at least one vertex, called root, that has a directed path to all the other vertices. Algebraically, a weighted directed random graph G (t) is represented by the adja cency matrix A = aij and the Laplacian matrix L = Lij defined as follows: aij = 0 if i = j and aij = wij δij if i = j, where wij is the corresponding entry of the weight matrix  We , and Lij = N k=1 aik if i = j and Lij = −aij if i = j [4,19]. Both the adjacency and Laplacian matrices are essentially random. The Laplacian matrix L is a zero row-sum matrix, and therefore e = [1, 1, ..., 1]T is an eigenvalue vector of L associated with the eigenvalue 0. In addition, the rank of L equals to N − 1 if and only if any undirected graph G is connected; any directed graph G has a directed spanning tree. In one

Second-order consensus of multi-agent systems with nonlinear dynamics over random switching directed networks

of the circumstances, the spectrum of L can be ordered: 0 = Reλ1 (L ) < Reλ2 (L ) ≤ · · · ≤ ReλN −1 (L ) ≤ λN (L ) [29]. As in [19], the authors also consider a class of random   graphs G (t) that keeps unchanged in the interval t ∈ kΔ, (k + 1)Δ and switches at a   series of time instants kΔ, k ∈ Z + , where Δ is called the fixed period or switching rate. The finite sample space of the random directed graph is indicted by G , and the  j elementary events (possible graphs) are indicted by G , j = 1, 2, ... , |G |, where |G | represents the cardinality. The Laplacian matrix corresponding to graph G j is denoted  as L j . In this sense, a multi-agent system corresponding to a random switching network can be viewed as a set of nonlinear stochastic switched systems. It follows from the switching mechanisms described above that the graph edges are independent random variables. The fixed time-average topology of the random graph Laplacian matrix,      written as E [L ] = E Lij , may be computed entrywise by E Lij = −pij wij for i = j    and E Lij = N k=1 pik wik for i = j. The fixed time-average topology, that is, E [L ], corresponds to a weighted directed graph, which does not necessarily belong to G. We refer to this graph as the average graph, denoted by E [G] as in [19].

3.2.3 Problem formulations In this chapter, we consider a dynamical network G (t) composed of N identical agents with second-order nonlinear dynamics. Suppose G (t) is interconnected pairwise via a random, weighted, directional time-delay states information interaction, in which each agent is an n-dimensional dynamical unit. The model of each agent can be described by x˙ i (t) = vi (t) , v˙ i (t) = f (xi (t) , vi (t) , t) − α

N

Lij (t/Δ) B (t) xj (t − τ )

j=1

−β

N

(3.1)

Lij (t/Δ) B (t) vj (t − τ ), i = 1, 2, . . . , N ,

j=1

where xi (t) = (xi1 (t) , . . . , xin (t))T ∈ Rn and vi (t) = (vi1 (t) , . . . , vin (t))T ∈ Rn are the position and velocity vectors of the ith agent, respectively, f : Rn × Rn × R+ → Rn is a continuously differentiable vector-valued function, α > 0 and β > 0 stand for position and velocity coupling strengths between any two agents in the network, B (t) ∈ Rn×n is a semi-positive definite diagonal matrix modeling the time-varying inner coupling   agents, L (t/Δ) = Lij (t/Δ) N ×N is the Laplacian matrix representing the topological structure of the random network G (t) at time t, and τ ≥ 0 specifies the coupling delay between agents.

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For system (3.1), if xi (t) = e ⊗ s1 (t), vi (t) = e ⊗ s2 (t), i = 1, 2, . . . , N, and some  s (t) = sT1 (t) , sT2 (t) ∈ R2n is a solution of the individual subsystem

˙s1 (t) = s2 (t) , ˙s2 (t) = f (s1 (t) , s2 (t) , t),

(3.2)

then we can see that the second-order dynamic consensus can be achieved. Here s (t) is called a consensus manifold. Generally, s (t) can be an equilibrium point, a nontrivial periodic orbit, or even a chaotic attractor defined for infinite-dimensional systems [30]. Also note that the consensus we discussed in this chapter refers to the almost sure consensus over random switching directed network G (t). An almost sure consensus is also called a consensus with probability one [4]. The problem of second-order consensus for multi-agent systems with nonlinear dynamics that communicate via a stochastic information network corresponds to stability  T analysis of a consensus manifold s (t) = sT1 (t) , sT2 (t) in the randomly coupled dynamic networks (3.1). To this end, subtracting (3.2) from (3.1) and noticing that the row sums of L (t/Δ) equal one, we obtain the following error dynamical system: x˙ˆ i (t) = vˆ i (t) , 

v˙ˆ i (t) = f xi (t) , vi (t) , t) − f (s1 (t) , s2 (t) , t −α

N



Lij (t/Δ) B (t) xˆ j (t − τ )

(3.3)

j=1

−β

N

Lij (t/Δ) B (t) vˆ j (t − τ ),

j=1

where xˆ i (t) = xi (t) − s1 (t) and vˆ i (t) = vi (t) − s2 (t), i = 1, 2, . . . , N. Linearizing (3.3) around the consensus manifold s (t) leads to x˙ˆ i (t) = vˆ i (t) , v˙ˆ i (t) = Dx f (s1 , s2 , t) xˆ i (t) + Dy f (s1 , s2 , t) vˆ i (t) −α

N

Lij (t/Δ) B (t) xˆ j (t − τ )

(3.4)

j=1

−β

N

Lij (t/Δ) B (t) vˆ j (t − τ )

j=1

    + Oi xˆ i (t) , t + Oi vˆ i (t) , t , i = 1, 2, . . . , N , 



where Dx f (s1 , s2 , t) and Dy f (s1 , s2 , t) denote the Jacobian matrices of f x, y, t toward the state vectors x and y on s (t) = sT1 (t) , sT2 (t) , respectively. In addition, the high-order

Second-order consensus of multi-agent systems with nonlinear dynamics over random switching directed networks





 







          terms  satisfy limxˆ i (t)→0 ( Oi xˆ i (t) , t / xˆ i (t) ) = 0 and limvˆ i (t)→0 Oi (ˆvi (t), t) / vˆ i (t) = 0. Equivalently, (3.4) can be rewritten in the following compact vector form:



y˙ (t) = (IN ⊗ F (s1 , s2 , t)) y (t) − L˜ (t/Δ) ⊗ B (t)

(3.5)

  × y (t − τ ) + O y (t) , t ,



T



T



where y (t) = xˆ T (t) , vˆ T (t) , xˆ (t) = xˆ T1 (t) , xˆ T2 (t) , ... , xˆ TN (t) , vˆ (t) = vˆ 1T (t) , vˆ 2T (t) ,  T (t ) T , ... , vˆ N 







On In ON ON F (s1 , s2 , t) = , and L˜ (t/Δ) = . Dx f (s1 , s2 , t) Dy f (s1 , s2 , t) α L (t/Δ) β L (t/Δ) 

  

Similarly, the high-order term also satisfies limy→0 O(y, t)/y = 0. Now we introduce the following lemma. Lemma 3.1. [14] Consider a nonlinear system of the form of x˙ (t) = A (t) x (t) + B (t) x (t −τ ) + O (x, t), where O (x, 0) = 0. The state x (t) is uniformly asymptotically stabilized at the origin if the following conditions are satisfied: i) For arbitrary t, limx→0 (O (x, t)/x) = 0; ii) A (t) and B (t) are bounded for arbitrary time t; iii) Its linear part x˙ (t) = A (t) x (t) + B (t) × x (t − τ ) is uniformly asymptotically stabilized at the origin. It follows from Lemma 3.1 that we analyze the asymptotic stability of the linear part of randomly switching system (3.5) at the origin, that is, we investigate the second-order dynamic consensus of all agents in the random switching network.

3.3. Main results 3.3.1 Orthogonal decomposition In this section, we give some results regarding the case of time-delay-free case (τ = 0) and the time-delay case (τ > 0), respectively. In general, to investigate the second-order dynamic consensus of all agents in the random switching network, we need to consider the local stability of error system (3.5) along the consensus manifold s (t). To begin with,   we first decompose the state vector of (3.5) (neglecting the high-order term O y(t), t ) into two components that are orthogonal to each other [9,13,23]. One component evolves along the consensus manifold s (t), and the other evolves transversely to the consensus manifold. Since e ∈ RN , we denote its spanned subspace by A. On the other hand, each subspace of RN has only one orthogonal complementary subspace, so the orthogonal subspace of A uniquely exists. Suppose that this orthogonal complementary space A⊥ is the column space of a matrix W ∈ RN ×(N −1) that satisfies W T e = 0 and

45

46

Second-Order Consensus of Continuous-Time Multi-Agent Systems

W T W = IN −1 , where W = (w1 , w2 , ..., wN −1 ) consists of an array of N − 1 basis vectors in RN . In the following section, we will discuss the existence of a matrix W . Obviously, RN = A ⊕ A⊥ . Similarly, we can also expand this decomposition into RnN , that is, RnN = B ⊕ B⊥ , where B is the subspace spanned by e ⊗ In , and B⊥ represents the orthogonal complement space spanned by W ⊗ In . Note that the consensus state e ⊗ s(t) = 0 in the range of e ⊗ In = 0 and then in the null space of W T ⊗ In [13]. So the state variable y(t) ∈ R2nN can be decomposed into a component in the subspace B spanned by e ⊗ In = 0 and a component in the subspace B⊥ spanned by W T ⊗ In as follows: y (t) = e ⊗ y¯ (t) + (W ⊗ I2n ) η (t) ,

(3.6)

where y¯ (t) = N1 (e ⊗ I2n )T y (t) ∈ R2n and η (t) = (W ⊗ I2n )T y (t) ∈ R2n(N −1) . Note that the average of all the components in y (t) and the two components are orthogonal to each other, that is,      [(W ⊗ I2n )η (t)]T e ⊗ y¯ (t) = ηT (t) W T ⊗ I2n e ⊗ y¯ (t)   = ηT (t) W T e ⊗ y¯ (t)

(3.7)

= 0.

Using the state transformation 

y¯ (t) η (t)



 =

T 1 N (e ⊗ I2n ) (W ⊗ I2n )T



y (t) ,

(3.8)

the linear part of (3.5) can be partitioned into two dynamical coupled subsystems. The first subsystem can be described as 

 1 (e ⊗ I2n )T (IN ⊗ F (s1 , s2 , t)) y (t)− L˜ (t/Δ) ⊗ B (t) y (t − τ ) N

 1 T 1 e ⊗ F (s1 , s2 , t) y (t) − (eT ⊗ I2n ) L˜ (t/Δ) ⊗ B (t) y (t − τ ) = N N

  T 1 1 T e ⊗ I2n L˜ (t/Δ) ⊗ B (t) = (IN ⊗ F (s1 , s2 , t)) e ⊗ I2n y (t) − N N   × e ⊗ y¯ (t − τ ) + (W ⊗ I2n ) η (t − τ )

y˙¯ (t) =

= (IN ⊗ F (s1 , s2 , t)) y¯ (t) − Δ1 − Δ2 ,

(3.9)

Second-order consensus of multi-agent systems with nonlinear dynamics over random switching directed networks

where

   1 T e ⊗ I2n L˜ (t/Δ) ⊗ B (t) e ⊗ y¯ (t − τ ) N     1 T ON ON = 2 e ⊗ I2n ⊗ B (t ) × α L (t/Δ) β L (t/Δ) N

Δ1 =









e ⊗ eT ⊗ I2n y (t − τ )

 1  = 2 eT ⊗ I2n N 

=



ON ON α L (t/Δ) β L (t/Δ)

⊗ B(t) ×

 









e ⊗ eT ⊗ In  xˆ (t − τ ) e ⊗ eT ⊗ In vˆ (t − τ )

 1 T e ⊗ I2n 2 N



(3.10) 0

nN    α L (t/Δ) e ⊗ B (t) eT ⊗ In xˆ (t − τ ) 

0nN  +   +β L (t/Δ) e ⊗ B (t) eT ⊗ In vˆ (t − τ )    0nN 1 T = 2 e ⊗ I2n 0nN

N

=02n

and Δ2 =

 1 T e ⊗ I2n L˜ (t/Δ) ⊗ B (t) (W ⊗ I2n ) η (t − τ ) . N

(3.11)

 1 T e ⊗ I2n × N

L˜ (t/Δ) ⊗ B (t) (W ⊗ I2n ) η (t − τ ) .

(3.12)

Therefore y˙¯ (t) = (IN ⊗ F (s1 , s2 , t)) y¯ (t) −

Similarly, the second subsystem can be expressed as η˙ (t) =(W ⊗ I2n )T y˙ (t)

   = W T ⊗ I2n (IN ⊗ F (s1 , s2 , t)) y (t) − L˜ (t/Δ) ⊗ B (t) y (t − τ )

    = W T ⊗ F (s1 , s2 , t) y (t) − W T ⊗ I2n L˜ (t/Δ) ⊗ B (t) y (t − τ )    = W T ⊗ F (s1 , s2 , t) e ⊗ y¯ (t) + (W ⊗ I2n )η (t) − Δ3   =W T e ⊗ F (s1 , s2 , t) y¯ (t) + W T W ⊗ F (s1 , s2 , t) η (t) − Δ3 = (IN −1 ⊗ F (s1 , s2 , t)) η (t) − Δ3 ,

(3.13)

47

48

Second-Order Consensus of Continuous-Time Multi-Agent Systems

where

  Δ3 = W T T ⊗ I2n L˜ (t/Δ) ⊗ B (t) y (t − τ )

    = W T ⊗ I2n L˜ (t/Δ) ⊗ B (t) e ⊗ y¯ (t − τ ) + (W ⊗ I2n ) η (t − τ )

  = W T ⊗ I2n L˜ (t/Δ) ⊗ B (t) (e ⊗ y¯ (t − τ ))

  + W T ⊗ I2n L˜ (t/Δ) ⊗ B (t) (W ⊗ I2n ) η (t − τ )  

  0nN   T + W T ⊗ I2n L˜ (t/Δ) ⊗ B (t) (W ⊗ I2n ) η (t − τ ) = W ⊗ I2n  = W T ⊗ I2n



0nN

(3.14)

L˜ (t/Δ) ⊗ B (t) (W ⊗ I2n ) η (t − τ ) .

Combining (3.12) and (3.13) yields the following two coupled subsystems:  1 T e ⊗ I2n × N

L˜ (t/Δ) ⊗ B (t) (W ⊗ I2n ) η (t − τ ) ,

y˙¯ (t) = (IN ⊗ F (s1 , s2 , t)) y¯ (t) −

(3.15)

  η˙ (t) = (IN −1 ⊗ F (s1 , s2 , t)) η (t) − W T ⊗ I2n ×

L˜ (t/Δ) ⊗ B (t) (W ⊗ I2n ) η (t − τ ) .

In the following, we will discuss two cases, τ = 0 and τ > 0. First, we will give two useful lemmas for deriving our main results regarding the delay-free case (τ = 0). We establish a relation between the asymptotic stability of the second differential equation in (3.15) and that of a derived sampled-data system (at the switching instants Δk for all k ∈ Z + ). Second, we establish a criterion that guarantees the asymptotical stability of the system of associated fixed time-average topology corresponding to the second differential equation of (3.15).

3.3.2 The case of time-delay-free coupling Lemma 3.2. For system (3.15) with τ = 0, suppose that F (s1 , s2 , t), L˜ (t/Δ), and B (t) defined in (3.5) are bounded and piecewise continuous functions for all t ≥ 0 and that L˜ (t/Δ)       switches at time instants Δk for all k ∈ Z + . For any t ∈ kΔ, k + 1 Δ , k ∈ Z + , if η kΔ almost surely converges to zero, then η (t) decays to zero. 



 





Proof. For any t ∈ kΔ, k + 1 Δ , k ∈ Z + , η (t) can be computed by η (t) = φη t, kΔ ×   η kΔ , where φη (t, τ ) denotes

the transition matrix of η˙ (t) = (IN −1 ⊗ F (s1 , s2 , t)) η (t)   T ˜ − W ⊗ I2n L (t/Δ) ⊗ B (t) (W ⊗ I2n ) η (t). Since F (s1 , s2 , t), L˜ (t/Δ), and B (t) are bounded and piecewise continuous functions, there exist positive constants m, λ, and β¯

Second-order consensus of multi-agent systems with nonlinear dynamics over random switching directed networks

 

 

such that for any t ≥ 0, F (s1 , s2 , t)) ≤ m, L˜ (t/Δ) ≤ λ, and B (t) ≤ β¯ . In addition,   from the definition of the matrix W we obtain W  = W T  = 1. According to the Gronwall–Belmann inequality [see [19], proof of Theorem 1], η (t) can be bounded by 

   (IN −1 ⊗ F (s1 , s2 , t)) − W T ⊗ I2n × kΔ 

 L˜ (t/Δ) ⊗ B (t) (W ⊗ I2n ) dτ.

η (t) ≤ ηk  exp





t

(3.16)

 

Therefore, for any t ∈ kΔ, k + 1 Δ , k ∈ Z + , we obtain that  η (t) ≤ ηk  exp

  (IN −1 ⊗ F (s1 , s2 , t)) − W T ⊗ I2n ×

(k+1)Δ 

kΔ

  L˜ (t/Δ) ⊗ B (t) (W ⊗ I2n ) dτ.



(3.17)

Thus 

(k+1)Δ 

η (t) ≤ ηk  exp







m + λβ¯ dτ = ηk  m + λβ¯ Δ,

(3.18)

kΔ

and the claim follows immediately. For the convenience of description, we write the deterministic dynamic expectation system corresponding to (3.15) as follows:  1 T e ⊗ I2n × N

E[L˜ ] ⊗ B (t) (W ⊗ I2n ) ξ (t) ,

y˙¯ 1 (t) = (IN ⊗ F (s1 , s2 , t)) y¯ 1 (t) −

  ξ˙ (t) = (IN −1 ⊗ F (s1 , s2 , t)) ξ (t) − W T ⊗ I2n ×

E[L˜ ] ⊗ B (t) (W ⊗ I2n ) ξ (t) ,

(3.19)

where E[L˜ ] denotes the expected value of the extended Laplacian matrix L˜ (t/Δ). Lemma 3.3. Assume that F (s1 , s2 , t) ≤ m, where m is a positive constant. If there exists a positive definite symmetric matrix Q ∈ R2n×2n such that m Q − 2λmin



WT ⊗ Q

then ξ (t) asymptotically converges to zero.





E[L˜ ] ⊗ B (t) (W ⊗ I2n ) < 0,

(3.20)

49

50

Second-Order Consensus of Continuous-Time Multi-Agent Systems

Proof. Define the candidate Lyapunov function V (ξ (t)) = ξ T (t) (IN −1 ⊗ Q) ξ (t) ,

(3.21)

where Q ∈ R2n×2n is a positive definite symmetric matrix. The time derivative of V (ξ (t)) along the solution of (3.19) for t > 0 can be given by dV (ξ (t)) =2ξ T (t) (IN −1 ⊗ Q) ξ˙ (t) dt =2ξ T (t) (IN −1 ⊗ Q) [(IN −1 ⊗ F (s1 , s2 , t)) ξ (t)

   − W T ⊗ I2n E[L˜ ] ⊗ B (t) (W ⊗ I2n )ξ (t)

(3.22)

=2ξ T (t) [IN −1 ⊗ (QF (s1 , s2 , t))] ξ (t)

  − 2ξ T (t) W T ⊗ Q E[L˜ ] ⊗ B (t) (W ⊗ I2n ) ξ (t) =2ξ T (t) H (ξ (t) , t) ξ (t) ,

where H (ξ (t) , t) = [IN −1 ⊗ (QF (s1 , s2 , t))]

  − 2 W T ⊗ Q E[L˜ ] ⊗ B (t) (W ⊗ I2n )

=2IN −1 ⊗ sym (QF (s1 , s2 , t))

  − 2 W T ⊗ Q E[L˜ ] ⊗ B (t) (W ⊗ I2n ) .

(3.23)

The largest eigenvalue of H (ξ (t) , t) can be estimated by using Weyl’s inequality [31]:   λmax (H (ξ (t) , t)) ≤λmax 2IN −1 ⊗ sym (QF (s1 , s2 , t)) 

 + λmax −2 W T ⊗ Q E[L˜ ] ⊗ B (t) (W ⊗ I2n ) .

(3.24)

The first summand in the right-hand side of (3.24) can be bounded as   λmax 2IN −1 ⊗ sym (QF (s1 , s2 , t))   ≤ max |λi sym (QF (s1 , s2 , t)) | 1≤i≤2n   ≤sym (QF (s1 , s2 , t))

(3.25)

2

≤Q2 F (s1 , s2 , t)2 ,

and



 λmax −2 W T ⊗ Q E[L˜ ] ⊗ B(t) (W ⊗ I2n )

  = − 2λmin W T ⊗ Q E[L˜ ] ⊗ B (t) (W ⊗ I2n ) .

(3.26)

Second-order consensus of multi-agent systems with nonlinear dynamics over random switching directed networks

Let μ = Q2 F (s1 , s2 , t)2 − 2λmin



WT ⊗ Q





E[L˜ ] ⊗ B (t) (W ⊗ I2n ) < 0.

(3.27)

If mQ2 − 2λmin



WT ⊗ Q





E[L˜ ] ⊗ B (t) (W ⊗ I2n ) < 0,

(3.28)

then dV (ξ (t))/dt < 0. This completes the proof. Based on the previous lemmas, the main results for the time-delay-free case can be stated as follows. Theorem 3.4. Suppose that there exists a directed spanning tree in the time-average graph ¯ corresponding to the fixed time-average topology of a random Laplacian matrix E [L˜ ] G and that F (s1 , s2 , t), L˜ (t/Δ)  , and B (t) are piecewise continuous functions bounded by   F (s1 , s2 , t) ≤ m, L˜ (t/Δ) ≤ λ, and B (t) ≤ β¯ for all t ≥ 0. Also, suppose that there exists a positive definite symmetric matrix Q ∈ R2n×2n such that the inequality (3.20) holds. Then the second-order dynamic consensus over directed random switching network G (t) can be achieved if the random network switches at a sufficiently fast rate. 



 

Proof. By (3.2) it is clear that for any t ∈ kΔ, k + 1 Δ , k ∈ Z + , if we can prove the    stochastic sequence η kΔ , k ∈ Z + converges to zero for sufficiently large k ∈ Z + , then the second-order dynamic consensus over directed random switching network G(t) can be guaranteed as t → ∞. For the convenience of description, we define M (t) = (IN −1 ⊗ F (s1 , s2 , t)) η (t)

  − W T ⊗ I2n L˜ (t/tΔ) ⊗ B (t) (W ⊗ I2n ) and



(3.29)



¯ (t) = (IN −1 ⊗ F (s1 , s2 , t)) ξ (t) − W T ⊗ I2n × M

(3.30)

E[L˜ ] ⊗ B (t) (W ⊗ I2n ) ξ (t) . 







¯ (t) ξ (t). Let ηk = η kΔ and ξk = ξ kΔ . Then we have η˙ (t) = M (t) η (t) and ξ˙ (t) = M Suppose that φη (t, t0 ) and φξ (t, t0 ) are the transition matrices of systems (3.29) and    + (3.30), respectively. Therefore for all k ∈ Z , ηk+1 = φη k + 1 Δ, kΔ ηk and ξk+1 =    φξ k + 1 Δ, kΔ ξk . First, based on the conditions in the theorem, we know that system (3.30) is uniformly asymptotically stable at the origin, because the uniform asymptotical stability of solutions is equivalent to the exponential stability of linear systems [13,19,33]. By the definition of exponential stability we have that there exist a positive definite symmetric

51

52

Second-Order Consensus of Continuous-Time Multi-Agent Systems

time-varying matrix Q (t) and positive scalars q1 , q2 , and q3 such that the Lyapunov function 1 V (ξ (t) , t) = ξ T (t) Q (t) ξ (t) 2

(3.31)

q2 q1 ξ (t)2 ≤ V (ξ (t) , t) ≤ ξ (t)2 2 2

(3.32)

q3 d V (ξ (t) , t) ≤ − ξ (t)2 dt 2

(3.33)

satisfies

and



 



for all t ∈ kΔ, k + 1 Δ , k ∈ Z + (see [34], proof of Theorem 7.4). Second, we will show that if the random network G (t) switches at a sufficiently fast rate and V (η (t) , t) is also a Lyapunov function for system (3.29), then the uniform exponential stability of (3.29) can be still guaranteed. This claim  is achieved by proving that for sufficiently    ΔV = V k + 1 Δ, ξk+1 − V kΔ, ξk is negative small values of Δ, the difference   define for all t ∈ kΔ, k + 1 Δ , k ∈ Z + . Moreover, from the proof of Theorem 7.4 in [32] we can get the inequality 



V k + 1 Δ, ξk+1













q1 q3 − V kΔ, ξk ≤ − 1 − exp − Δ 2 q2 



ξk 2 .

(3.34)

 

This also implies that for all t ∈ kΔ, k + 1 Δ , k ∈ Z + ,            ξkT φξT k + 1 Δ, kΔ Q k + 1 Δ φξ k + 1 Δ, kΔ ξk − ξkT Q kΔ ξk

≤ −q1 1 − exp − qq32 Δ ξk 2 .

(3.35)

In addition, we use the Peano–Baker series representation for the transition matrix φη ((k + 1)Δ, kΔ) (see [19,23]) to define φη









k + 1 Δ, kΔ =I2(N −1)n + +

∞ 

i=2





M (σ1 )dσ1 kΔ



(k+1)Δ

kΔ

M (σi )dσi . . . dσ1 . t





(3.36)

σi−1

... t





σ1

M (σ1 )





(k+1)Δ





Let H k + 1 Δ, kΔ = φη k + 1 Δ, kΔ − φξ k + 1 Δ, kΔ . Then we obtain that 





H k + 1 Δ, kΔ =



(k+1)Δ 

kΔ



¯ (σ1 ) dσ1 + εk , M (σ1 ) − M

(3.37)

Second-order consensus of multi-agent systems with nonlinear dynamics over random switching directed networks

where εk =

∞ 

i=2

−



(k+1)Δ

kΔ (k+1)Δ

τi−1

M (τi )dτi . . . dτ1

... t

∞ 

i=2



τ1

M (τ1 ) ¯ (τ1 ) M

t



kΔ



τ1

τi−1

... t

(3.38) ¯ (τi )dτi . . . dτ1 . M

t

¯ (t) are uniformly bounded. In general, M (t) and M      ¯  . By the hypothesis (k+1)Δ M (σ )dσ = ΔM ¯ (σ ) Define γ = sup max M (t) , M kΔ t≥0    we can compute an upper bound for H k + 1 Δ, kΔ :     H k + 1 Δ, kΔ  ≤ 2 (exp (Δγ ) − 1 − Δγ ) .

(3.39)

Moreover, we have     φη k + 1 Δ, kΔ  ≤ exp



(k+1)Δ 

 M ¯ (t)dt ≤ exp (Δγ ) .

(3.40)

kΔ









Note that φη = φξ + H. Then ΔV η, k + 1 Δ, kΔ can be expressed by     ΔV η, k + 1 Δ, kΔ      1 1 = ηkT+1 Q k + 1 Δ ηk+1 − ηkT Q kΔ ηk

2 2         1 T T  = ηk φη k + 1 Δ, kΔ Q k + 1 Δ φη k + 1 Δ, kΔ ηk 2   1 − ηkT Q kΔ ηk 2      1   = ηkT φξT k + 1 Δ, kΔ + H T k + 1 Δ, kΔ × 2        Q((k + 1)Δ) φξ k + 1 Δ, kΔ + H k + 1 Δ, kΔ ηk   1 − ηkT Q kΔ ηk 2           1 T  T  = ηk φξ k + 1 Δ, kΔ Q k + 1 Δ φξ k + 1 Δ, kΔ − Q kΔ ηk 2         ⎛ T  ⎞ φξ k + 1 Δ, kΔ Q k + 1 Δ H k + 1 Δ, kΔ +          ⎟ 1 ⎜ + ηkT ⎝ H T k + 1 Δ, kΔ Q k + 1 Δ φξ k + 1 Δ, kΔ + ⎠ ηk . 2          H T k + 1 Δ, kΔ Q k + 1 Δ H k + 1 Δ, kΔ 





(3.41)



The task is now to compute an upper bound for ΔV η, k + 1 Δ, kΔ and show that this bound is negative if the switching rate is sufficiently small. We use several relations: Q (t) ≤ q2 ,

(3.42)

53

54

Second-Order Consensus of Continuous-Time Multi-Agent Systems

       φξ k + 1 Δ, kΔ  ≤ q2 exp − q3 Δ ,

q1





V k + 1 Δ, ξ

(3.43)

q2

       q3 k + 1 Δ ≤ exp − Δ V kΔ, ξ kΔ ,



q2

(3.44)

for all k ∈ Z + . Recalling inequalities (3.39), (3.40), (3.42), (3.43), and (3.43), the upper     bound for ΔV η, k + 1 Δ, kΔ can be estimated by        q1 q3 ΔV η, k + 1 Δ, kΔ ≤ − 1 − exp − Δ ηk 2

2

+ 2q2



q2

  q2 q3 exp − Δ (exp (Δγ ) − 1 − Δγ ) ηk 2

q1

(3.45)

q2

+ 4q2 (exp (Δγ ) − 1 − Δγ )2 ηk 2 .

Define        q1 q3 q2 q3 1 − exp − Δ + 2q2 exp − Δ × g (Δ, ηk ) = −

2

q2

q1

q2



(3.46)

(exp (Δγ ) − 1 − Δγ ) + 4q2 (exp (Δγ ) − 1 − Δγ ) ηk  . 2

2

∂ g (0, ηk ) < 0 when ηk = 0. Thus since It can be shown that g (0, ηk ) = 0 and ∂Δ g(Δ, ηk ) → +∞ as Δ → +∞ for ηk = 0, there exists Δ∗ > 0 such that g (Δ∗ , η) = and     g (Δ, η) < 0 for all Δ ∈ (0, Δ∗ ). Therefore ΔV η, k + 1 Δ, kΔ is negative for all Δ ∈ (0, Δ∗ ) when ηk = 0. From the above analysis we get that there exists a positive real number r1 such that

  2     ΔV η, k + 1 Δ, kΔ ≤ −r1 η kΔ  





(3.47)



when η kΔ = 0. This also implies that η kΔ converges to zero for sufficiently large k. It follows from Lemma 3.2 that η (t) also decays to zero as t → ∞. Therefore, if all the conditions in the theorem are satisfied, then the second-order dynamic consensus over directed random switching network G(t) will be reached. The proof is thus completed. Remark 3.5. The effectiveness of the orthogonal decomposition strictly depends on the existence of the matrix W ∈ RN ×(N −1) that satisfies W T e = 0 and W T W = IN −1 . For a directed graph G with a directed spanning tree, the eigenvalues corresponding to its Laplacian matrix L can be ordered as 0 = Reλ1 (L ) < Reλ2 (L ) ≤ · · · ≤ ReλN (L ) [56]. Moreover, e is an eigenvector of L associated with the eigenvalue for an  0. Therefore,

T undirected connected graph G , e is also an eigenvector of 1/2 L  + L  , where L 

Second-order consensus of multi-agent systems with nonlinear dynamics over random switching directed networks

is the Laplacian matrix of G . Thus we can choose N − 1 eigenvectors, denoted by W1 , W2 , . . . , WN −1 , associated with the non-zero eigenvalues of the Laplacian matrix  T corresponding to 1/2 We + We . Therefore the condition can guarantee the existence of the decomposition matrix W . 

T

On the other hand, we decompose the state vector as ξ T (t) = ξ1T (t) , ξ2T (t) , where ξ1 (t) , ξ2 (t) ∈ Rn(N −1) . Then the second differential equation in (3.19) can be equivalently transformed as 

ξ˙1 (t) ξ˙2 (t)



 =

0

In(N −1) A1 (t) A2 (t)



ξ1 (t) ξ2 (t)

 ,

(3.48)

where 







A1 = IN −1 ⊗ Dx f (s1 , s2 , t) − α W T E[L ]W ⊗ B (t) , and A2 = IN −1 ⊗ Dy f (s1 , s2 , t) − β W T E[L ]W ⊗ B (t) .  

Definition 3.6. The matrix measure of a complex square matrix C = cij ∈ C n×n is defined as follows [57,58]: μθ (C ) = limε→0+ In +εCε θ −1 , where ·θ is the  induced matrix norm. When the matrix norms C 1 = maxj ni=1 |cij |, C 2 =  [λmax (C T C )]1/2 , and C ∞ = maxi nj=1 |cij |, we can obtain the matrix measures

 

[57,58] μ1 (C ) = maxj Re cjj +

maxj Re (cii ) +

n  j=1, j=i

n  i=1, i=j

|cij | , μ2 (C ) = 12 λmax (C ∗ + C ), and μ∞ (C ) =

|cij | , respectively, where C ∗ is the complex conjugate transpose

of a complex matrix. To this end, we can immediately obtain the following theorem for the time-delayfree case. ¯ Theorem 3.7. Suppose that there exists a directed spanning tree in the time-average graph G ˜ corresponding to the fixed time-average topology of random Laplacian matrix E[L ]. Moreover, let F (s1 , s2 , t), L˜ (t/Δ), and B (t) be piecewise continuous functions uniformly bounded by   F (s1 , s2 , t) ≤ m, L˜ (t/Δ) ≤ λ, and B (t) ≤ β¯ for all t ≥ 0. If 



+∞

μθ 0

0

In(N −1) A1 (t) A2 (t)



dt = −∞, θ ∈ {1, 2, ∞} ,

then the second-order dynamic consensus over directed random switching network will be reached if the random network G (t) switches at a sufficiently fast rate.

55

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Second-Order Consensus of Continuous-Time Multi-Agent Systems

Proof. The main proof procedures are similar to those of Theorem 3.4. We only need to    +∞ 0 In(N −1) prove that under the condition 0 μθ dt = −∞, ξ (t) will converge A1 (t) A2 (t) asymptotically to zero. From (3.48) we have  d|ξ (t)|θ dt

− μθ

In(N −1) A1 ( t ) A2 ( t ) 

= lim+

|ξ (t+ε)|θ −|ξ (t)|θ ε

= lim+

1

ε→0

 ε→0



0

− μθ

|ξ (t)|θ

0

In(N −1) A1 (t) A2 (t) 

|ξ (t + ε)|θ − |ξ (t)|θ − εμθ

ε

 |ξ (t)|θ

0

In(N −1) A1 (t) A2 (t)



 |ξ (t)|θ

!  !   ! ! 0 In(N −1) ! ! ξ (t)! ≤ lim+ ε |ξ (t + ε)|θ − |ξ (t)|θ − ε !μθ ! ! A1 (t) A2 (t) ε→0 θ ! !     ! ! 0 In(N −1) ! ! ξ(t)! ≤ lim+ 1ε |ξ (t + ε)|θ − ! I + εμθ ! ! A1 (t) A2 (t) ε→0 θ !  !   ! ! 0 In(N −1) ! ! ≤ lim+ 1ε !ξ (t + ε) − ξ (t) − εμθ ξ (t)! ! ! A1 (t) A2 (t) ε→0 

1

(3.49)

θ

= 0.

Thus  |ξ (t)|θ ≤ |ξ (0)|θ exp



t

μθ 0

0

In(N −1) A1 (t) A2 (t)



dτ.

(3.50)

Therefore ξ (t) will asymptotically tend to zero as t → ∞. The remaining proof is similar to that of Theorem 3.4 and is omitted here. The proof is thus completed.

3.3.3 The case of time-delay coupling Note that for linear systems, the uniform asymptotic stability of solutions is equivalent to the uniform exponential stability [13,23]. Thus if the second differential equation in (3.15) is uniformly asymptotically stable, then by the definition of the exponential stability, for all ε > 0, there exist two constants κ > 0 and μ (ε) > 0 such that η (t0 ) < μ and η (t) < ε exp (−κ t) η (t0 ) for all t0 ≥ 0. This also implies that η (t − τ ) < ε exp (−κ (t − τ )) η (t0 ). Therefore η (t, t0 ) and η (t − τ, t0 ) will simultaneously converge to zero as t → ∞. Then the transversal components will disappear, and the second-order dynamic consensus over random switching network with time-delay

Second-order consensus of multi-agent systems with nonlinear dynamics over random switching directed networks

couplings can be achieved. Before we give a sufficient condition for system (3.1) to reach a consensus, we state the following fundamental lemmas. Consider the linear time-delay dynamical system x˙ (t) = Ax (t) + Bx (t − τ ) ,

(3.51)

where x (t) ∈ Rn , A, B ∈ Rn×n , and τ > 0 is time-delay. We summarize the fundamental results that give conditions for the asymptotic stability of the this dynamical system in the following lemmas. Lemma 3.8. If μθ (A) + Bθ < 0, then the zero solution of system (3.51) is asymptotically stable, where θ ∈ {1, 2, ∞}. Lemma 3.8 is a delay-independent criterion, and the stability is guaranteed under this condition for any value of time-delay τ . Lemma 3.9. Let l1 = μθ (A) + Bθ ≥ 0 (otherwise, system (3.51) is stable because of   Lemma 3.8) and l2 = μθ −jA + Bθ . Stability of the zero solution of system (3.51) is achieved if the following conditions are satisfied: Reλi (A + B exp(−τ s)) < 0, i = 1, 2, ... , N, s = jω, 0 ≤ ω ≤ l2 , s = l1 + jω, s = r + jl2 , 0 ≤ r ≤ l1 , where j2 = −1, and μθ (X ) is the matrix measure for X ∈ C N ×N defined by Definition 3.6. Lemma 3.9, which includes the information of the delay, is referred to as the delaydependent stability criterion. In this chapter, as an example, we only take matrix measure μ2 (X ), and the other two measures can be mimicked similarly. 



Theorem 3.10. If 12 λmax F T (s1 , s2 , t) + F (s1 , s2 , t) + max {α, β } × L (t/Δ)2 B (t)2 < 0 for all t, then the second-order dynamic consensus for system (3.1) over random switching network G (t) can be realized for all time-delays τ ≥ 0. Proof. Consider the following time-delay system:   η˙ (t) = (IN −1 ⊗ F (s1 , s2 , t)) η (t) − W T ⊗ I2n ×

L˜ (t/Δ) ⊗ B (t) (W ⊗ I2n ) η (t − τ ) .

(3.52)

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Second-Order Consensus of Continuous-Time Multi-Agent Systems

Under the conditions of Theorem 3.10, inspired by Lemma 3.8, we can show the convergence of (3.52) as follows:   

   μ2 (IN −1 ⊗ F (s1 , s2 , t)) + − W T ⊗ I2n L˜ (t/Δ) ⊗ B (t) (W ⊗ I2n ) 2   T     ≤ μ2 (F (s1 , s2 , t)) +  W ⊗ I2n 2 L˜ (t/Δ) ⊗ B (t) (W ⊗ I2n )2 2   T  ˜    ≤ μ2 (F (s1 , s2 , t)) + W 2 L (t/Δ) B (t)2 W 2 2   ˜  ≤ μ2 (F (s1 , s2 , t)) + L (t/Δ) B (t)2

(3.53)

2

≤ μ2 (F (s1 , s2 , t)) + max {α, β} L (t/Δ)2 B (t)2 = 12 λmax (F T (s1 , s2 , t) + F (s1 , s2 , t)) + max {α, β} L (t/Δ)2 B (t)2 < 0,

where the last equality comes from F (s1 , s2 , t) ∈ R2n×2n . The claim immediately follows from Lemma 3.8 and the proof thus is completed.  

 

Theorem 3.11. If l1 = μ2 (F (s1 , s2 , t)) + (W T ⊗ I2n )(L˜ (t/Δ) ⊗ B(t)) × (W ⊗ I2n ) 







for some t and Reλmax IN −1 ⊗ F (s1 , s2 , t) − W T ⊗ I2n × L˜ (t/Δ) ⊗ B (t) (W ⊗ I2n ) ×  exp (−τ s) < 0 for all t, then the second-order dynamic consensus can be realized, where s takes the value in the range given by s = jω, 0 ≤ ω ≤ l2 , s = l1 + jω, 0 ≤ ω ≤ l2 , s= jω, 0 ≤ ω ≤ l2 , 





with l1 = μ2 −jF (s1 , s2 , t) +  W T ⊗ I2n





L˜ (t/Δ) ⊗ B (t) (W ⊗ I2n ) . 2

Proof. The theorem can be immediately proved by means of Lemma 3.9.  

Corollary 3.12. If l1 = μ2 (F (s1 , s2 , t)) +  W T ⊗ I2n



 

L˜ (t/Δ) ⊗ B (t) × (W ⊗ I2n )

2

  ≥ 0 for some t and μ2 (IN −1 ⊗ F (s1 , s2 , t) − W T ⊗ I2n × L˜ (t/Δ) ⊗ B (t) (W ⊗ I2n ) × exp (−τ s)) for all t, then the second-order dynamic consensus can be realized.

Proof. For any matrix μ2 (X ), we have Re λmax ≤ μθ (X ), where θ ∈ {1, 2, ∞} [59]. The proof is thus completed.

3.4. Illustrative examples In this section, we perform some numerical simulations to illustrate the feasibility and effectiveness of the theoretical results presented in the previous sections. For the convenience of representation, we assume that there are four agents in the random switching network. Each agent is modeled as the second-order system with nonlinear dynamics

x˙ i (t) = vi (t) , v˙ i (t) = f (vi (t)) ,

(3.54)

Second-order consensus of multi-agent systems with nonlinear dynamics over random switching directed networks



for i = 1, 2, 3, 4, where xi = (xi1 , xi2 , xi3 )T , vi = (vi1 , vi2 , vi3 )T , f (vi ) = vi2 , vi3 , −cvi1  2 T , and a, b, c are constants. When a = 0.44, b = 1.1, and c = 1, the − bvi2 − avi3 + vi3 second-order oscillator produces a chaotic $ By computation we obtain that # behavior [60]. "





the norm F 2 = λmax F T F = max

√

"

√

2, 1 + 4vi12 = 2 is a bounded constant. We

assume that the link weights among agents are ⎡ ⎢ ⎢ ⎣

We = ⎢

0 0.2319 0.4614 0.1209

0.3301 0 0.2384 0.2344

0.3530 0.1199 0 0.4326

0.2055 0.2124 0.4771 0

⎤ ⎥ ⎥ ⎥ ⎦

(3.55)

and the potential link probabilities among agents are ⎡ ⎢ ⎢ ⎣

P=⎢

0 0.1527 0.4147 0.4853

0.1501 0 0.2204 0.0031

0.1459 0.3459 0 0.0417

0.0979 0.4887 0.1831 0

⎤ ⎥ ⎥ ⎥. ⎦

(3.56)

According to Remark 3.5, we choose the decomposition matrix as W = [W2 , W3 , W4 ], where W2 = [−0.0423, −0.7987, 0.3630, 0.4780]T , W3 = [0.7615, 0.3203, 0.1113, −0.5524]T , and W4 = [−0.4103, 0.0971, 0.7784, −0.4651]T . For the case of time-delay-free coupling, the other parameters are selected as √ B = diag {1, 1, 1}, m = 2, Q = 0.1E6 , α = 30, and β = 30. By calculation we know that the left-hand side of inequality (3.28) is −0.4164, and thus the conditions of Theorem 3.4 are satisfied. From Theorem 3.7 we can calculate that, under the selected   0 In(N −1) is always a negative number, that is, the conditions parameters, μθ A1 (t) A2 (t) in Theorem 3.7 are also satisfied. In our experiments, we use the Runge–Kutta method to solve the differential equations by letting the step size h = 0.005, the switching rate Δ = 0.1, and the initial position and velocity conditions both selected randomly from the interval [0, 0.5]. The total consensus of position! and velocity errors of the net! 4 ! 4 ! ! ! ! work (3.1) are defined by Ex (t) = j=2 xj (t) − x1 (t) and Ev (t) = j=2 vj (t) − v1 (t)!, respectively. Figs. 3.1(a) and 3.2(b) show the evolution process of position states and velocity states of all agents with time-delay-free couplings over the random switching network, respectively. Figs. 3.2(a) and 3.2(b) show the time evolution of algorithms with regard to the consensus position and velocity errors with time-delay-free couplings, respectively. It is easy to see that the second-order dynamic consensus over the designed random switching network is achieved. For the case of time-delay coupling, time-delay is chosen as τ = 0.1, other parameters are selected the same as those for the time-delay coupling case. We can easily

59

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Second-Order Consensus of Continuous-Time Multi-Agent Systems

Figure 3.1 The responses of position and velocity states of all agents with time-delay-free couplings in the random switching network. (a) Position states of all agents. (b) Velocity states of all agents.

Figure 3.2 The time evolution of algorithm with regard to the consensus position and velocity errors with time-delay-free couplings in the random switching network. (a) The time evolution of log (Ex (t)). (b) The time evolution of log (Ev (t)).

check that the inequalities in Theorem 3.11 and Corollary 3.12 are satisfied due to relatively large values of coupling strengths, for example, α = 30 and β = 30. Figs. 3.3(a) and 3.2(b) show the evolution process of position and velocity states of all agents with time-delay couplings over the random switching network, respectively. Figs. 3.4(a) and 3.2(b) show the time evolution of logarithms with regard to the consensus position and velocity errors with time-delay couplings, respectively. It is easy to see that the second-order dynamic consensus over the designed random switching network is achieved. Remark 3.13. The weight denotes how each agent updates the consensus algorithm. In general, we do not require We to have nonnegative elements. A negative weight may imply deteriorated communication channels or natural disagreement of the child node

Second-order consensus of multi-agent systems with nonlinear dynamics over random switching directed networks

Figure 3.3 The responses of position and velocity states of all agents with time-delay couplings in the random switching network. (a) Position states of all agents. (b) Velocity states of all agents.

Figure 3.4 The time evolution of logarithms with regard to the consensus position and velocity errors with time-delay couplings in the random switching network. (a) The time evolution of log (Ex (t)). (b) The time evolution of log (Ev (t)).

over the information obtained from its parent node. Note that in our experiments, we have considered the case in which some communication links have negative weights, for example, we23 = −0.5 and we42 = −0.37. We find that the second-order dynamic consensus over random switching network can also be achieved. Therefore the obtained results in this chapter are more practical in applications of engineering. Remark 3.14. The results obtained in this chapter have provided new insights about the requirements for second-order dynamic consensus when the network topology is random switching. The results also show that even if the network is not always connected instantaneously, then sufficient information is propagated through the network to allow almost sure consensus as long as the expected value of the network is connected and the switching rate is sufficiently fast.

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3.5. Conclusion In this chapter, we studied the problem of second-order dynamic consensus problem over random switching networks in detail. To simplify the theoretical analysis, we used the orthogonal decomposition method. The obtained results are quite powerful and can be further used to solve various switching cases for complex dynamical networks.

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CHAPTER 4

Second-order locally dynamical consensus of multi-agent systems with arbitrarily fast switching directed topologies 4.1. Introduction In recent years the collective behaviors of multi-agent systems have attracted extensive attention due to the growing interest in animal group behaviors [1]. Numerous issues have also been addressed such as the consensus problem [2–4,30,31,33–35], formation control [5], and so on. A fundamental approach to make the states of multi-agent systems reach an agreement on a common value of interest is consensus analysis. Roughly speaking, the consensus problem generally means how to design network interaction protocols (called algorithms) based on local information of node dynamics such that all the agents asymptotically or even exponentially reach an agreement on their states. At present the consensus problem has attracted extensive attention in different research fields due to its broad applications in cooperative control of unmanned air vehicles [11], formation control of mobile robots [5,32], control of communication networks [7], management science and statistics [8], system and control [9,29], flocking of social insects [10], swarm-based computing [6], and so on. In the past decade, numerous studies had been conducted on the consensus problem for multi-agent networks with first-order linear dynamics, [2,3,5,11–14]. A framework of consensus problem in networked dynamical agents was established by Olfati-Saber and Murray [12]. Jadbabaie et al. [13] further discussed the linearized Vicsek model and obtained that consensus can always be reached as long as the switching topologies are periodically jointly connected or ultimately connected. Ren et al. [15] extended the results of [12,13] and presented some more less restrictive conditions for reaching consensus of agents under dynamically varying interaction topologies. However, when the driving force (acceleration) is introduced as the control input in practical systems, each agent should be modeled by a double-integrator dynamical system. Moreover, in reality a broad class of harmonic oscillators [16] and pendulums [17] are second-order dynamical models, which are governed by both position and velocity terms. In contrast to the first-order consensus problem, it has been shown that consensus may fail to be achieved for agents with the second-order dynamics even if the network topology has a directed spanning tree [18]. The consensus problem of double-integrator agent systems Second-Order Consensus of Continuous-Time Multi-Agent Systems https://doi.org/10.1016/B978-0-32-390131-4.00011-X

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has also been studied extensively in [4,6,16,18], to name a few. By the existing literature we know that consensus of the double-integrator agent systems depends not only on the topology of the network but also on the coupling strengths of relative position and velocity states between neighboring agents. In real communication environment, dynamical behaviors of agents in networks are determined by the interactions among them and their intrinsic dynamics (disturbances and unmodeled uncertainties of agent) [19,20]. Protocols dealing with second-order consensus of multi-agent systems with nonlinear dynamics have not been involved until the recent works [19,20]. In [20] a kind of measurement for directed strongly connected graphs, that is, general algebraic connectivity, was first introduced by Yu et al. The authors built the bridge between the general algebraic connectivity and the performance of reaching an agreement of the second-order multi-agent systems with nonlinear dynamics. A similar problem has also been studied by Song et al. [19] through the use of pinning control technique, and it is worth mentioning that the restriction on the aforementioned approach in which the interaction network is strongly connected is removed in [20]. There exists a common drawback in the literature that the network topology under investigation is fixed. However, practical networks are usually in uncertain communication environments [2–4,14,15], so in the real world the information flow between any pair of agents may be subject to failure with a certain probability. In such cases the network topology is constant in some time intervals and randomly jumps to another topology when certain occasional events occur at some random moments. Therefore it is important to study the system with random network topology, the existence of whose edge is probabilistic. Moreover, the dwell time of each topology is unknown in advance, and the weighted adjacent matrix is not necessarily nonnegative due to the probable existence of deteriorated communication channels in reality. Mathematically, dynamics of agents in such kind of networks can be modeled as a kind of arbitrarily fast switching nonlinear systems. As described previously, the study on the second-order dynamical consensus in multi-agent systems with arbitrarily fast switching topologies is complicated and challenging, particularly when there exist isolated agents in some topologies in the switching sequences. Motivated by the aforementioned discussions, in this chapter, we focus on the second-order locally dynamical consensus of multi-agent systems with nonlinear dynamics in the directed networks with arbitrarily fast switching topologies. In our designed framework, each link in the network that represents the information flow between any pair of agents can be subject to failure with certain probability. Moreover, we assume that the dwell time of each topology is unknown in advance and the corresponding adjacency weighted matrix is not necessarily nonnegative due to the deteriorated communication channels. By the orthogonal decomposition method we can further decompose the state vector of the resultant error dynamical system into two transversal components, one of which evolves along the consensus manifold, and the

Second-order locally dynamical consensus of multi-agent systems with arbitrarily fast switching directed topologies

other evolves transversally with the consensus manifold. Thus the problem to be solved is simplified as analyzing the locally asymptotical stability of a fast switching dynamical system with respect to the component that evolves transversally with the consensus manifold. The difficulty using Lyapunov theory is that there is no general method on how to construct a suitable Lyapunov function for such a kind of fast switching systems. In what follows, by introducing the generalized matrix measure [25,26] we instead use contraction and circle analysis methods [27,28] to study in detail the second-order locally dynamical consensus of multi-agent systems with arbitrarily fast switching directed topologies. We also derive several easily verified criteria derived for reaching the second-order dynamical consensus. From our main results, to verify whether the locally dynamical consensus in a multi-agent network with arbitrarily fast switching directed topologies can be achieved, we only need to validate the obtained criteria along each cycle that consists of some specific topologies. If all the values are negative, then the consensus can be guaranteed. Moreover, we also find that only if the position and velocity coupling strengths are sufficiently large, then the arbitrarily fast switching can be tolerated. Finally, we provide a numerical simulation illustrating the feasibility and effectiveness of our theoretical results. Moreover, the obtained results are quite powerful and can be further used to explore various switching cases for complex dynamical networks. The rest of this chapter is organized as follows. We give some necessary concepts and knowledge with respect to random graph theory and state problems in Section 4.2. In Section 4.3, we provide the main results. In Section 4.4, we provide a numerical example illustrating the effectiveness of the obtained theoretical results. Finally, we conclude this chapter in Section 4.5.

4.2. Preliminaries 4.2.1 Notation Unless otherwise stated, the vectors in this chapter are assumed to be columns. By R+ we denote the set of positive real numbers and by Rn and Rn×m the n-dimensional Euclidean space and the set of all n × m real matrices, respectively. For a vector u ∈ Rn ,  1/2 its Euclidean norm (or called 2-norm) is defined as u = uT u ; In and On denote the n-dimensional identity and zero matrices, respectively. Let e = [1, 1, . . . , 1]T . For a symmetric matrix A ∈ Rm×m , we denote by λmin (A) and λmax (A) the minimum and maximum eigenvalues, respectively. In addition, we order the algebraic spectrum {λi (A)}m i=1 of A to satisfy λmin (A) = λ1 (A) ≤ λ2 (A) ≤ · · · ≤ λm−1 (A) ≤ λm (A) = λmax (A). The Kronecker product, denoted by ⊗, facilitates the manipulation of matrices of appropriate dimensions by the following properties: (A ⊗ B) (C ⊗ D) = AC ⊗ BD and (A ⊗ B)T = AT ⊗ BT . For a set M, by |M | we denote its cardinality.

67

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Second-Order Consensus of Continuous-Time Multi-Agent Systems

4.2.2 Model of optimization problem Suppose that there are N agents in the dynamical network G (t) and each one is an ndimensional dynamical unit with second-order nonlinear dynamics. At time t, all agents are pairwise interconnected using the weighted and directional position and velocity state information based on the topology of G (t). The model of each agent through coupling can be described by x˙ i (t) =vi (t), v˙ i (t) = f (xi (t), vi (t), t) − α

N 

Lij (m(t))B(t)xj (t)

j=1

−β

N 

(4.1)

Lij (m(t))B(t)vj (t),

j=1

i = 1, 2, ... , N, where xi (t) = (xi1 (t), ... , xin (t))T ∈ Rn and vi (t) = (vi1 (t), ... , vin (t))T ∈ Rn are the position and velocity state vectors of the ith agent, respectively, f : Rn × Rn × R+ → Rn is a continuously differentiable vector-valued function, which represents the inner nonlinear dynamics of uncoupled agent i, α > 0 and β > 0 stand for position and velocity coupling strengths between any two agents in the network, and B(t) ∈ Rn×n is a semi-positive definite diagonal matrix modeling the time-varying inner coupling agents. At time t, L (m(t)) = (Lij (m(t)))N ×N is the Laplacian matrix of the dynamical network, which switches arbitrarily fast. Definition 4.1. The second-order nonlinear consensus in multi-agent systems (4.1) is   said to be local if for any ε > 0, there exist δ(ε) > 0 and T > 0 such that xi (0) − xj (0) ≤       δ(ε) and vi (0) − vj (0) ≤ δ(ε) imply xi (t) − xj (t) ≤ ε and vi (t) − vj (t) ≤ ε for any t > T and i, j = 1, ..., N. If a second-order dynamical consensus of (4.1) is achieved in G(t), then there exist two n-dimensional time-varying vectors s1 (t) and s2 (t) such that for i = 1, 2, ... , N, xi (t) = e ⊗ s1 (t), e ⊗ s2 (t) and 

˙s1 (t) = s2 (t), ˙s2 (t) = f (s1 (t), s2 (t), t).



T

(4.2)

Here s (t) = sT1 (t) , sT2 (t) ∈ R2n is called the consensus manifold. Generally, s(t) can be a nontrivial periodic orbit or even a chaotic attractor defined for finite-dimensional systems.

Second-order locally dynamical consensus of multi-agent systems with arbitrarily fast switching directed topologies

Subtracting (4.2) from (4.1) and noticing that all the row sums of Laplacian matrix L (m(t)) equal zero, we get the following error dynamical system: ⎧ ⎪ x˙ˆ i (t) = vˆ i (t), ⎪ ⎪ ⎪ ⎪ ⎪ v˙ˆ i (t) = f (xi (t), vi (t), t) − f (s1 (t), s2 (t), t) ⎪ ⎪ ⎨ N

−α Lij (m(t))B(t)ˆxj (t) ⎪ j=1 ⎪ ⎪ ⎪ N ⎪

⎪ ⎪ −β Lij (m(t))B(t)ˆvj (t), ⎪ ⎩

(4.3)

j=1

where xˆ i (t) = xi (t) − s1 (t) and vˆ i (t) = vi (t) − s2 (t), i = 1, 2, . . . , N. Then we get that the second-order locally nonlinear consensus of all agents in the network G(t) with arbitrarily fast switching topologies is equivalent to the locally asymptotical stability of switched nonlinear system (4.3). Thus linearizing (4.3) around the consensus manifold s(t) leads to x˙ˆ i (t) =ˆvi (t), v˙ˆ i (t) =D1 f (s1 , s2 , t)ˆxi (t) + D2 f (s1 , s2 , t)ˆvi (t) −α

N 

Lij (m(t))B(t)ˆxj (t)

(4.4)

j=1

−β

N 

Lij (m(t))B(t)ˆvj (t)

j=1

+ Oi (ˆxi (t), t) + Oi (ˆvi (t), t), i = 1, 2, ... , N , 



where D1 f (s1 , s2 , t) and D2 f (s1 , s2 , t) denote the Jacobian matrices of f x, y, t toward  T the highthe state vectors x and y on s (t) = sT1 (t) , sT2 (t) , respectively. In addition,         / , ˆ order terms satisfy lim (|| O (ˆ x ( t ), t )||/||(ˆ x ( t )||) = 0 and lim O v t t ( ) xˆ i (t)→0 vˆ i (t)→0 i i i i i   vˆ i (t) = 0. Equivalently, (4.4) can be rewritten in the following compact vector form: 

y˙ (t) = (IN ⊗ F (s1 , s2 , t)) y (t) − L˜ (m (t)) ⊗ B (t)   × y (t) + O y (t) , t ,

where 

T

y (t) = xˆ T (x) , vˆ T (t) , 

T

xˆ (x) = xˆ T1 (x) , xˆ T2 (x) , . . . , xˆ TN (x) , 

T

T vˆ (t) = vˆ 1T (t) , vˆ 2T (t) , . . . , vˆ N (t) ,



(4.5)

69

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Second-Order Consensus of Continuous-Time Multi-Agent Systems





On In , , F ( s1 , s2 , t ) = D1 f (s1 , s2 , t) D2 f (s1 , s2 , t) and 

ON L˜ (m(t)) = α L (m(t))



ON . β L (m(t)) 

  

Similarly, the high-order term also satisfies limy→0 O(y, t)/y = 0. We first   decompose the state vector y (t) of (4.5) (neglecting the high-order term O y (t) , t ) into two components orthogonal to each other. One component evolves along the consensus manifold s (t), and the other evolves transversally to the consensus manifold s (t). Since e ∈ RN , we denote its spanned subspace by A. On the other hand, each subspace of RN has only one orthogonal complementary subspace, so the orthogonal subspace of A uniquely exists. Suppose that this orthogonal complementary space A⊥ is the column space of matrix V ∈ RN ×(N −1) that satisfies V T e = 0 and V T V = IN −1 , where V = (V1 , V2 , ... , VN −1 ) consists of an array of N − 1 basis vectors in RN . Obviously, RN = A ⊕ A⊥ . Similarly, we also can expand this decomposition into RnN , that is, RnN = B ⊕ B⊥ , where B is the subspace spanned by e ⊗ In , and B⊥ represents the orthogonal complement space spanned by V ⊗ In . Note that the consensus state e ⊗ s (t) = 0 in the range of e ⊗ In = 0 and then in the null space of V T ⊗ In [21,22]. So the state variable y (t) ∈ R2nN can be decomposed into a component in the subspace B spanned by e ⊗ In = 0 and a component in the subspace B⊥ spanned by V T ⊗ In , which are described as follows: y (t) = e ⊗ y¯ (t) + (V ⊗ I2n ) η (t) ,

(4.6)

where y¯ (t) = (1/N ) (e ⊗ I2n )T y (t) ∈ R2n and η (t) = (V ⊗ I2n )T y (t) ∈ R2n(N −1) . Note that y¯ (t) is the average of all the components in y (t), and the two components are orthogonal to each other, that is,      [(V ⊗ I2n ) η (t)]T e ⊗ y¯ (t) = ηT (t) V T ⊗ I2n e ⊗ y¯ (t)   = ηT (t) V T e ⊗ y¯ (t)

(4.7)

= 0.

Using the state transformation 







1 y¯ (t) (e ⊗ I2n )T y (t) , = N η (t) (V ⊗ I2n )T

(4.8)

Second-order locally dynamical consensus of multi-agent systems with arbitrarily fast switching directed topologies

the linear part of (4.5) can be partitioned into two dynamical coupled subsystems. The first subsystem can be described as     1 (e ⊗ I2n )T (IN ⊗ F (s1 , s2 , t)) y (t) − L˜ (m (t) ⊗ B (t)) y (t) N    1 T 1 T e ⊗ F (s1 , s2 , t) y (t) − e ⊗ I2n L˜ (m (t) ⊗ B (t)) y (t) = N N    1 1 T e ⊗ I2n = (IN ⊗ F (s1 , s2 , t)) eT ⊗ I2n y (t) − N N   × L˜ (m (t)) ⊗ B (t) e ⊗ y¯ (t) + (V ⊗ I2n ) η (t)

y˙¯ (t) =

(4.9)

= (IN ⊗ F (s1 , s2 , t)) y¯ (t) − Δ1 − Δ2 ,

where Δ1 =

   1 T e ⊗ I2n L˜ (m (t)) ⊗ B (t) e ⊗ y¯ (t) N

 1  = 2 eT ⊗ I2n N



    × e ⊗ eT ⊗ I2n y (t)   1 T ON = 2 e ⊗ I2n α L (m (t)) N   T   e ⊗ e ⊗ I2n xˆ (t)    × e ⊗ eT ⊗ I2n vˆ (t) =









ON ON ⊗ B (t ) α L (m (t)) β L (m (t))

ON ⊗ B (t ) β L (m (t))

(4.10)

    1 T e ⊗ I2n 0nN α L (m (t)) e ⊗ B (t) eT ⊗ In xˆ (t) + 0nN 2 N    +β L (m (t)) e ⊗ B (t) eT ⊗ In vˆ (t) 

 0nN 1  = 2 eT ⊗ I2n 0nN N



=02n

and Δ2 =

  1 T e ⊗ I2n L˜ (m (t)) ⊗ B (t) (V ⊗ I2n ) η (t) . N

(4.11)

Therefore y˙¯ (t) = (IN ⊗ F (s1 , s2 , t)) y¯ (t) − × (V ⊗ I2n ) η (t) .

  1 T e ⊗ I2n L˜ ((m (t) ⊗ B (t)) N

(4.12)

71

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Second-Order Consensus of Continuous-Time Multi-Agent Systems

Similarly, the second subsystem can be expressed as    η˙ (t) = (IN −1 ⊗ F (s1 , s2 , t)) η (t) − V T ⊗ I2n L˜ (m (t)) ⊗ B (t) × (V ⊗ I2n ) η (t) .

(4.13)

If η(t) → 0 as t → +∞, then from (4.12) we know that y¯ (t) will also approach zero along the consensus manifold s(t). Thus the second-order locally dynamical consensus in multi-agent systems with arbitrarily fast switching topologies can be achieved. The great difficulty with Lyapunov theory is that there is no general method to construct a Lyapunov function for such a kind of arbitrarily fast switching dynamical systems. In the following section, we will use the methods of contraction and circle analysis to study under what conditions the state η(t) of arbitrarily fast switchings system (4.13) locally and asymptotically decays to zero. Remark 4.2. Our results are based on the orthogonal decomposition method. Therefore the existence of the decomposition matrix V ∈ RN ×(N −1) satisfying V T e = 0 and V T V = IN −1 is very essential. If an underlying undirected graph associated with the weighted matrix 0.5 W + W T contains a directed spanning tree, then the N − 1 eigenvectors associated with N − 1 nonzero eigenvalues of the Laplacian matrix L of the obtained connected graph are real vectors. Then we can use the N − 1 normalized real eigenvectors to construct the decomposition matrix V . We give some definitions for convenience of analysis.

 

Definition 4.3. [23] The matrix measure of a complex square matrix C = cij ∈ C n×n is defined as μθ (C ) = limε→0+ (In + εC θ − 1)/ε, where · is the induced matrix norm. θ  1/2

, and C ∞ = For the matrix norms C 1 = maxj ni=1 |cij |, C 2 = λmax CT C  

maxi nj=1 |cij |, we obtain the matrix measures μ1 (C ) = maxj Re cjj + ni=1, i =j |cij | ,

 

μ2 (C ) = (1/2) λmax (C ∗ + C ), and μ∞ (C ) = maxi Re(cii ) + nj=1, j =i |cij | , respectively, where C ∗ is the complex conjugate transpose of a complex matrix, and Re(C ) denotes

the real part of a complex C. Definition 4.4. [25] For any continuously differentiable nonlinear function f (x), the   generalized matrix measure of f (x), μ˜ θ f (x) , CS , in a compact set CS is of the  form       μ˜ θ f (x) , CS =supx∈CS μθ ∂ f /∂ x , and the generalized matrix norm f (x)θ, CS =   supx∈CS ∂ f /∂ xθ , where ·θ is the induced matrix norm as that in Definition 4.1, and θ = {1, 2, ∞}.

4.2.3 Communication network A graph consists of a set of vertices, each pair of which is endowed with an incidence relation represented by an edge. The set of vertices and edges of the graph G are denoted by V and E, respectively; in this case, we write G = (V , E). In our designed

Second-order locally dynamical consensus of multi-agent systems with arbitrarily fast switching directed topologies

framework, for a random graph on N vertices, the existence of an edge between a pair of vertices in the set V = {1, 2, . . . , N } is determined randomly and independent of other edges with probability p = 0.5. The sample space (the set of all possible topologies) of such a random graph is defined by M. Let m (t) be a topology indicator, that is, m : R+ → M. Let tk be the switching moments appearing in the running process of the time-varying dynamical network G (t), where tk ∈ R+ , k = 1, 2, . . .. Thus m (t) is a piecewise left continuous function, which keeps unchanged when t ∈ [tk , tk+1 ) and randomly jumps to another topology in M at t = tk+1 . Since there are totally 2CN2 potential directed edges among N vertices, the cardinality |M | of M is bounded, and its 2 upper bound is 22CN . In this chapter, we use the tool of random graph to describe a multi-agent network whose directed topology switches arbitrarily fast. Especially, the existence (nonexistence) of a directed edge Eij ∈ E in the random graph G (t) is rep  resented by an ordered pair of vertices i, j , which practically means that agent j can (cannot) receive the position and/or velocity information from agent i in the network. For a dynamical network G (t) with N vertices, we assign it an adjacency weighted ma  trix W = wij N ×N . Negative weights may also exist in W , which means the probable existence of deteriorated communication channels or natural disagreement of the child node over the information obtained from its parent node. If the one-way information channel between agent i and agent j is established successfully when t ∈ [tk , tk+1 ), then the edge Eij exists and is naturally assigned a weight wij when t ∈ [tk , tk+1 ). Meanwhile, the effect on agent i by the information received from agent j is a positive correlation   with wij when t ∈ [tk , tk+1 ). Supposing that W = w ij N ×N is the weighted matrix of   a random graph G (t) at time t, we use L (m (t)) = Lij m (t) N ×N to denote the corre sponding Laplacian matrix of G (t), where Lij = −wIJ if i = j and Lij = N k=1 w ik if i = j. The Laplacian matrix L is a zero row sums matrix; therefore e = [1, 1, . . . , 1]T is an eigenvector of L associated with the zero eigenvalue [15]. In addition, the rank of L is equal to N − 1 if and only if for an undirected graph, it is connected; for a directed graph, it contains a directed spanning tree [15]. Let tsk,i and tfk,i denote, respectively, the kth starting time  and the kth ending  timeof topology i in M. Furthermore, suppose that k k 0 < Δ1,i = infk tf ,i − ts,i ≤ supk tfk,i − tsk,i Δ2,i < +∞, that is, the dwell time of topology   i is in the interval Δ1,i , Δ2,i .

4.3. Local stability under arbitrarily fast switchings Before giving the main results, we present the following lemmas. A logical path in the arbitrarily fast switched system (4.13) can be described by a           topology sequence m tk1 , m tk2 , ... , m tk1+n . A finite logical path m tk1 , m tk2 , ... ,       m tk1+n is closed if m tk1 = m tk1+n . A closed path CP = m(tk1 ), m(tk2 ), . . . , m(tk1+n ) in which no state appears more than once except for the one that is the first and the last is

73

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Second-Order Consensus of Continuous-Time Multi-Agent Systems

a cycle [27]. We can find all types of cycles by using the knowledge from graph theory. We state the following lemmas. Lemma 4.5. Every closed path is composed of some cycles. 











Lemma 4.6. Suppose that n + 1 states m tk1 , m tk2 , ... , m tk1+n belong to a set composed   t of different states m ki , 1 ≤ i ≤ n. Then there exists at least one cycle in the logical path      m tk1 , m tk2 , ... , m tk1+n . In the following, we will derive some sufficient conditions by studying the convergence of ηT (t) η (t) using the contraction and circle analysis [27,28] instead of constructing a Lyapunov function. From (4.13) we have   dηT (t) η (t) = ηT (t) AT (s1 , s2 , m (t)) + A (s1 , s2 , m (t)) η (t) , dt

(4.14)

where 

A (s1 , s2 , m (t)) =IN −1 ⊗ F (s1 , s2 , t) − W T ⊗ I2n



  × L˜ (m (t)) ⊗ B (t) (W ⊗ I2n ) .

For each topology m (t) ∈ M, let CS (m (t)) denote the set CS (m (t)) ={η (t) |A (s1 , s2 , m (t)) + AT (s1 , s2 , m (t)) is negative definite},

(4.15)

and let UCS = ∪ CS (m (t)) . m(t)∈M

(4.16)

We are in a position to state our main results. Theorem 4.7. The second-order locally dynamical consensus of multi-agent systems (4.1) with arbitrarily fast switching directed topologies can be achieved if there exists a compact set  contains η∗ = 0 and for each cycle LC j 1 ≤ j ≤ θ in the switchSCS ⊆ CS such that SCS  ing sequences, we have α˜ j = i∈LCj μ˜ 2 (A (s1 , s2 , i) SCS) , Δ∗i < 0, where Δ∗i = Δ2, i for i ∈ N1 , and Δ∗i = Δ1, i for i ∈ N2 with N1 = {i|μ˜ 2 (A (s1 , s2 , i) , SCS) ≥ 0} and N2 = {i|μ˜ 2 (A (s1 , s2 , i) , SCS) < 0}. Proof. Without loss of generality, we assume that the switching sequences can be partitioned as m1 , m2 , ... , mp , where m1 = m(t0 ), ml+q = m(t), p ≤ |M |, and q ≤ |M |. It is worth mentioning that the partition is not unique. According to Lemma 4.5, the closed path CP is composed of some cycles, denoted by LC (1) , LC (2) , ... , LC (θ ), where

Second-order locally dynamical consensus of multi-agent systems with arbitrarily fast switching directed topologies

    LC j  ≤ M, and there are no subcycles in LC j , j ∈ {1, 2, ... , θ }. Let ts, CP and tf , CP ,

respectively, represent the start and ending times of the closed path CP. Then we get  that mp is the former topology of m ts, CP and ml+1 is the latter topology of m tf , CP . By (4.14) and the definition of a generalized matrix measure we have              η tfk, i  ≤ exp μ˜ 2 (A (s1 , s2 , i) , SCS) tfk, i − tsk, i η tsk, i  .

(4.17)

The |M | potential topologies can be divided into the following two sets: N1 = {i|μ˜ 2 (A (s1 , s2 , i) , SCS) ≥ 0} , N2 = {i|μ˜ 2 (A (s1 , s2 , i) , SCS) ≥ 0} . Therefore (4.17) is equivalent to the following two inequalities:            η tfk, i  ≤ exp μ˜ 2 (A (s1 , s2 , i) , SCS) Δ2, i η tsk, i  , i ∈ N1 ,

(4.18)

           η tfk, i  ≤ exp μ˜ 2 (A (s1 , s2 , i) , SCS) Δ1, i η tsk, i  , i ∈ N2 .

(4.19)

Define Δ∗i = Δ2, i if i ∈ N1 and Δ∗i = Δ1, i if i ∈ N2 . Then we get            η tfk, i  ≤ exp μ˜ 2 (A (s1 , s2 , i) , SCS) Δ∗i η tsk, i  .

(4.20)

Let tj represent all the switching moments in the closed path CP. By (4.17)–(4.20) we know that 

   η tj  ≤



i∈ST ts, CP ; tj





    exp μ˜ 2 (A (s1 , s2 , i) , SCS) Δ∗i η ts, CP  ,





(4.21) 

where ST ts, CP ; tj = {i|i is the node (topology) in the path from m ts, CP to m(tj )} . We consider two cases.     Case 1. There is no appearance of a cycle in the path from m ts, CP to m tj . Then by (4.21) the following inequalities hold: 

  η(tj ) ≤

i∈N1 ∩ST (ts, CP ; tj )



exp{μ˜ 2 (A(s1 , s2 , i), SCS)Δ∗i }×   exp{μ˜ 2 (A(s1 , s2 , i), SCS)Δ∗i } η(ts, CP )

i∈N2 ∩ST (ts, CP ; tj )

≤



i∈N1

exp{μ˜ 2 (A(s1 , s2 , i), SCS)Δ∗i }

(4.22)

75

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Second-Order Consensus of Continuous-Time Multi-Agent Systems

×

 i∈N2

≤



  exp{μ˜ 2 (A(s1 , s2 , i), SCS)Δ∗i } η(ts, CP )   exp{μ˜ 2 (A(s1 , s2 , i), SCS)Δ∗i } η(ts, CP ) .

i∈N1

 

 

 



Case 2. There exist cycles LC 1 , LC 2 , . . . , LC θ in the path from m ts, CP   to m tj . Using (4.21), we have ⎧ ⎪ ⎪ ⎪ ⎪ ⎨





⎪  ⎪ ⎪ ⎪ ⎩i∈ ST (ts, CP ; tj )−

∪



  k0 ∈ 1 , 2 , ... , θ

ST (ts, LC(k0 ) ; tf , LC(k0 )

 

   exp{μ˜ 2 (A(s1 , s2 , i), SCS)Δ∗i } × η(ts, CP ) ⎛ ⎞   < exp ⎝ α( ˆ k0 )⎠ exp{μ˜ 2 (A(s1 , s2 , i), SCS)Δ∗i } 

k0 ∈{1 , 2 , ... , θ }

(4.23)

i∈N1

  × η(ts, CP )    exp{μ˜ 2 (A(s1 , s2 , i), SCS)Δ∗i } η(ts, CP ) ≤α θ i∈N1


R. If η (t0 ) ∈ SCS, then there exists t∗∗ such that η (t∗∗ ) = R and η (t) ≤ R, t ∈ [t0 , t∗∗ ]. From (4.13) and the definition of a generalized matrix measure we know that A (s1 , s2 , m (t0 )) + AT (s1 , s2 , m (t0 )) ≤μ˜ 2 (A (s1 , s2 , m (t0 )) , SCS) × I2n(N −1)

(4.28)

for all t ∈ [t0 , t∗∗ ]. By (4.28) we have   ∗∗     η t  ≤ exp μ˜ 2 (A (s1 , s2 , m (t0 )) , SCS) t∗∗ − t0 η (t0 ) < R,

(4.29)

which contradicts with η (t∗∗ ) = R. Thus η (t) ∈ SCS for t0 ≤ t ≤ t1 . Moreover, if R0 =

R max (exp (μ˜ 2 (A (s1 , s2 , i) , SCS) Δ∗ )) i∈N1

×

1

!

exp 2 max (μ˜ 2 (A (s1 , s2 , i) , SCS)) (|M | + 1) Δ∗ i∈N1

"

(4.30)

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and η (t0 ) ≤ R0 , we get η (t1 ) ≤ exp (μ˜ 2 (A (s1 , s2 , m (t0 )) , SCS) (t1 − t0 )) η (t0 )    ≤ max exp μ˜ 2 (A (s1 , s2 , i) , SCS) Δ∗ η (t0 ) i∈N1    ≤ max exp μ˜ 2 (A (s1 , s2 , i) , SCS) Δ∗ R0 i∈N1

R

!

=

(4.31) "

exp 2 max (μ˜ 2 (A (s1 , s2 , i) , SCS) Δ∗ ) (|M | + 1) Δ∗ i∈N1

< R.

Supposing that η (t) ∈ SCS for t0 ≤ t ≤ tN , we will show that also η (t) ∈ SCS for tN ≤ t ≤ tN +1 . From (4.27) we obtain η (tN +1 ) ≤α



NC t0 ; tN +1



i∈N1

× η (t0 ) ≤α NC

" !  ∗  exp max (μ˜ 2 (A (s1 , s2 , i) , SCS)) p + q + 2 Δ



t0 ; tN +1



" ! exp 2 max (μ˜ 2 (A (s1 , s2 , i) , SCS)) (|M | + 1) Δ∗ i∈N1

× η (t0 ) ≤

(4.32)

R max (exp (μ˜ 2 (A (s1 , s2 , i) , SCS))) i∈N1

t0 , that is, η (t) ∈ SCS. Note that NC (t0 ; t) → +∞ as t → +∞. Thus we can see that if the initial condition satisfies η (t0 ) ≤ R0 , then limt→∞ η (t) = 0. We obtain from (4.12) that y¯ (t) will also approach zero along the consensus manifold s (t). Thus we can achieve the second-order locally dynamical consensus of multi-agent systems with arbitrarily fast switching topologies. The proof is thus completed. Based on Theorem 4.7, we can immediately derive the following corollaries, and we omit the proofs. Corollary 4.8. The second-order locally dynamical consensus of multi-agent systems (4.1) with arbitrarily fast switching directed topologies can be achieved if there exists a compact set SCS ⊆ UCS such that SCS contains η∗ = 0 and for all m (t) ∈ M, αˆ (m (t)) = μ˜ 2 (A(s1 , s2 , m (t)), SCS) < 0.

Second-order locally dynamical consensus of multi-agent systems with arbitrarily fast switching directed topologies

Remark 4.9. According to the expression of A (s1 , s2 , m (t)), for all potential topologies m (t) ∈ M, we can see that αˆ (m (t)) < 0 can be easily guaranteed by selecting sufficiently large coupling strengths α and β . That is, the arbitrarily fast switching directed topologies can be effectively tolerated, and the consensus can be achieved for all agents in the network. In this chapter, we always assume that the states of all agents in the network are bounded. In fact, the network after coupling may be divergent even if the isolated agent is bounded. However, the boundedness of the whole coupling network is not the main focus of this chapter. The fixed topology can be considered as a particular case of switching topologies, and therefore we get the following: Corollary 4.10. The second-order locally dynamical consensus of multi-agent systems (4.1) with fixed topology can be achieved if there exists a compact set SCS ⊆ UCS such that SCS contains η∗ = 0 and αˆ (m) = μ˜ 2 (A (s1 , s2 , m) , SCS) < 0.

4.4. Illustrative examples In this section, we present a numerical simulation example illustrating the correctness of the obtained theoretical results. For convenience, we assume that there are totally five agents in the dynamical network G (t) and the dwell time of each topology in the switching sequences is randomly distributed in the interval. Each agent is equipped with a second-order oscillator with nonlinear dynamics. At time t, all agents are coupled by position and velocity states with other agents according to the topology of the network G (t). Suppose that the agents’ inner coupling matrix B = I3 . Through coupling, the nonlinear dynamics of agent i can be described by x˙ i (t) =vi (t) , v˙ i (t) =f (vi (t)) − α

5 

Lij (m (t)) xj (t) (4.33)

j=1

−β

5 

Lij (m (t)) vj (t),

j=1



for i = 1, 2, ... , 5, where xi = (xi1 , xi2 , xi3 )T , vi = (vi1 , vi2 , vi3 )T , f (vi ) = vi2 , vi3 , −cvi1 − T bvi2 − avi3 + vi32 , and a, b, c are constants. Especially, when a = 0.44, b = 1.1, and c = 1, the second-order isolated oscillator (α = β = 0) depicts a chaotic attractor [24].

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We assume that the link weights among agents are ⎡ ⎢ ⎢ ⎢ W =⎢ ⎢ ⎣

0 0.0683 0.1956 0.1776 0.3161

0.0997 0 0.2845 0.1737 0.0747

0.0965 0.4799 0 0.2172 0.0507

0.3701 0.3416 0.4514 0 0.3319

0.4459 0.3368 0.0013 0.1888 0

⎤ ⎥ ⎥ ⎥ ⎥. ⎥ ⎦

(4.34)

Note that W is asymmetric and the underlying time-varying dynamical network G (t) is directed. According to Remark 4.2, we can choose the decomposition matrix as V = [V2 , V3 , V4 , V5 ], where V2 =[0.4845, −0.4021, −0.5630, −0.0522, 0.5328]T , V3 =[−0.5331, 0.5548, −0.3687, −0.1520, 0.4990]T , V4 =[0.5283, 0.5361, −0.1873, −0.5423, 0.3303]T , V3 =[−0.0444, −0.2188, 0.5586, −0.6929, 0.3976]T . Due to the boundedness of the chaotic trajectory, the generalized matrix measure αˆ (m (t)) = μ˜ 2 (A (s1 , s2 , m (t)) , SCS) < 0) for each topology m (t) in the switching sequences can be easily guaranteed by selecting relatively large positive real numbers α and β . Here SCS denotes a small neighborhood of η∗ = 0. In our experiments, we

use the Runge–Kutta method to solve the differential equations by letting the step size h = 0.005, and the initial position and velocity conditions are selected randomly from the interval [0, 0.3] and [0, 0.35], respectively. The coupling strengths among agents are selected as α = 20 and β = 20. The total consensus position and velocity  er  rors of the network (4.33) are respectively defined by Ex (t) = 5j=2 xj (t) − x1 (t) and 

 Ev (t) = 5j=2 vj (t) − v1 (t). Figs. 4.1(a) and 4.1(b) show the evolution process of position states and velocity states of five agents in the dynamical network G (t), whose directed topology arbitrarily fast switches. Figs. 4.2(a) and 4.2(b) show the time evolution of consensus position and velocity errors, respectively. Figs. 4.3(a) and 4.3(b) show the time evolution of logarithms with regard to the consensus position and velocity errors, respectively. It is easy to see that by choosing suitable coupling strengths the second-order locally dynamical consensus over the arbitrarily fast switching network is achieved. Before the dynamic T T T consensus is achieved, the consensus state vector s (t) = s1 (t) , s2 (t) is unknown to us. Therefore the generalized matrix measure αˆ (m (t)) = μ˜ 2 (A (s1 , s2 , m (t)) , SCS) for those topologies m (t) in which the consensus is reached cannot be precisely computed. However, we still can use one of the agents’ state vectors to substitute the consensus state vector s (t) to approximately estimate the index αˆ (m (t)). Using the states of all agents to replace s (t), we show the time-varying generalized matrix measure in Fig. 4.4. From

Second-order locally dynamical consensus of multi-agent systems with arbitrarily fast switching directed topologies

Figure 4.1 The responses of position and velocity states of all agents in the designed arbitrarily fast switching network (4.33). (a) Position states of all agents. (b) Velocity states of all agents.

Figure 4.2 Time evolution of consensus position and velocity errors in the designed arbitrarily fast switching network (4.33). (a) The time evolution of Ex (t); (b) The time evolution of Ev (t).

Figure 4.3 Time evolution of consensus position and velocity errors in the designed arbitrarily fast switching network (4.33). (a) The time evolution of (Ex (t)). (b) The time evolution of (Ev (t)).

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Figure 4.4 Time-varying generalized matrix measure using agents’ states.

Fig. 4.4 we can see that the generalized matrix measures are always negative and near each other by selecting suitable coupling strengths α = 20 and β = 20. In addition, the link weight between two agents denotes how to update the designed consensus algorithms. In general, we do not require the link weights to be nonnegative. However, in practice, negative weights may usually imply the existence of deteriorated communication channels or natural disagreement of the child node over the information obtained from its parent node. Note that in our experiments, we have considered the case in which some communication links have negative weights, for example, W23 = −0.4799. In the simulation, other parameters keep unchanged. We find that the secondorder locally dynamical consensus over the fast switching network G (t) can be also achieved. We do not present the simulation results because of limited space. Therefore the obtained results are more practical in the applications of engineering.

4.5. Conclusion The problem of second-order consensus of multi-agent systems with nonlinear dynamics, especially with random switching directed topologies, is very difficult. There are still few works involving the second-order nonlinear consensus of multi-agent systems with arbitrarily fast switching topologies. The traditional Lyapunov stability theory may fail to analyze the stability of such kinds of fast switching nonlinear systems. Using local linearization technique, generalized matrix measure, contraction analysis, and cycle analysis methods, we settle the second-order locally dynamical consensus. From our theoretical results and simulations we find that under sufficiently large coupling strengths, the arbitrarily fast switching can be effectively tolerated. It is worth mentioning that the global consensus of second-order agents with nonlinear dynamics over random networks by time-delay couplings is still an open problem.

Second-order locally dynamical consensus of multi-agent systems with arbitrarily fast switching directed topologies

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CHAPTER 5

Second-order global consensus in multi-agent systems with random directional link failure 5.1. Introduction Distributed coordination of multi-agent systems has been intensively studied in recent years. The consensus problem as one of the most fundamental research topics in the field of coordination control of multi-agent systems has attracted considerable attention over the past few years due to its extensive applications in cooperative control of mobile autonomous robots, design of distributed sensor networks, spacecraft formation control, and other areas [1–3]. Consensus means that the states of all agents reach an agreement on a common value of interest by using local information of each agent’s neighbors. The consensus problem for agents with first-order dynamics has been investigated from various perspectives [4–10,28,29]. In many practical situations, agents such as unmanned aerial vehicles and mobile robots can be controlled directly by their accelerations rather than by their velocities [1,11]. Hence there has been an increasing research interest on the consensus problems of second-order multi-agent systems, where the agents are governed by both position and velocity states. Ren and Atkins [12] proposed several control algorithms for secondorder consensus under directed graphs. It has been shown that the systems described by second-order integrators may not achieve consensus even if the directed graph has a directed spanning tree. Yu et al. [13], Zhu et al. [14], and Li et al. [15] presented some necessary and sufficient conditions ensuring the second-order consensus under a directed graph. It is known that in most cases, second-order consensus can be reached in multi-agent systems if the coupling control gains and the spectra of the Laplacian matrix satisfy some additional conditions, which are somewhat different from those in multiagent systems with first-order dynamics [6–8]. Moreover, in reality the agents may be governed by some nonlinear terms, and the agents usually have time-varying intrinsic velocities rather than constant ones, even after a velocity consensus has been reached. In this case the agents are not only affected by the interaction among neighboring agents, but also by their own intrinsic nonlinear dynamics [16–19]. In [16] the authors studied the leaderless consensus problem for second-order multi-agent systems with intrinsic nonlinear dynamics under directed graphs. The work is extended by Song et al. [17] to the leader-following tracking case, by Su et al. [18] to a connectivity-preserving control algorithm, and by Li et al. [19] to the robust leader-following consensus in a finite time. Second-Order Consensus of Continuous-Time Multi-Agent Systems https://doi.org/10.1016/B978-0-32-390131-4.00012-1

Copyright © 2021 Elsevier Inc. All rights reserved.

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The connectivity of a graph plays a key role in the behavior of multi-agent systems. However, in practice the connectivity of a graph may vary over time, and their interaction topology may also be changing dynamically with time. Therefore the problem of second-order consensus of multi-agent systems with nonlinear dynamics and switching topology deserves more attention. Generally, the extension of consensus algorithms for multi-agent systems from first-order to second-order is non-trivial except in the case where all topologies in the switching sequence have directed spanning trees. Guo et al. [20] investigated the flocking problem of leader-following multi-agent systems in directed graphs with switching topologies. From the obtained results it was found that when the multi-agent networks run on topologies with isolated agents, the proposed techniques in [20] need to add several directed edges between the leader agent and the corresponding isolated agents to constantly guarantee the connectivity of the underlying interaction topology. Thus the results essentially require that all directed topologies in the switching sequence have directed spanning trees. To this end, Li et al. [21] considered the second-order consensus multi-agent systems with nonlinear dynamics and random switching directed topology. Using the orthogonal decomposition technique and local linearization method, we derive two criteria for almost surely second-order nonlinear consensus for the non-time-delay coupling and time-delay coupling. Meanwhile, Li et al. in [22] further discussed the second-order consensus of multi-agent systems with nonlinear dynamics and arbitrarily fast switching directed topologies by using the generalized matrix measure and tools from contraction and circle analysis. The theoretical results in [21,22] require that the inherent nonlinear term of all individual agents are continuously differentiable to meet the need of local linearization and thus are limited to second-order local rather than global consensus. Inspired by the studies mentioned, in this chapter, we address the problem of secondorder globally nonlinear consensus in multi-agent networks with directed topology and random switching interconnections, in which there may exist some isolated agents in certain topology during the random switching sequence. This topic is not only significant but also fundamental and was not fully addressed so far. This is because in such a case the methods for analyzing the stability of switched systems, for example, the common Lyapunov function method or the multiple Lyapunov functions method, fail to deal with the stability of the resultant error dynamical system associated with the random switching network. We resort to the Lyapunov function designed for the corresponding time-average network to further capture the convergence characteristics of the multi-agent systems associated with the random switching directed network. First, by constructing a suitable Lyapunov function for the time-average network we derive a criterion for the second-order global consensus. Then by associating the solution of random switching nonlinear system with the constructed Lyapunov function we analytically investigate the exponential stability of the resultant error system corresponding to the random switching nonlinear system. We establish a sufficient condition

Second-order global consensus in multi-agent systems with random directional link failure

for second-order globally nonlinear consensus in a multi-agent network with random directed interconnections. The obtained results require that the second-order consensus can be achieved in the time-average network and the designed Lyapunov function decreases along the solution of the random switching nonlinear system at an infinite subsequence of the switching moments. From the results we find that even though the network is not always connected instantaneously in time, sufficient information can be propagated through the dynamical network to guarantee the second-order globally nonlinear consensus. Finally, we present a numerical example to show the effectiveness of the designed distributed interaction protocols and the correctness of the theoretical analysis. This chapter proceeds as follows. We give some concepts and knowledge of graph theory in Section 5.2 and formulate the problem to be solved. In Section 5.3, we systematically investigate the criteria for reaching the second-order globally nonlinear consensus of multi-agent networks with the time-average directed topology and the random switching directed topologies. In Section 5.4, we present an illustrative example. Section 5.5 draws the conclusions of this chapter.

5.2. Preliminary and problem formulation 5.2.1 Graph theory The notations used throughout this chapter are quite standard. Let R+ be the set of positive real numbers. The symmetric part of a matrix C ∈ Rm×m is denoted by sym(C ) = 1/2(C + C T ); N refers to the set of all nonnegative integers. For x ∈ R+ , x is the largest nonnegative integer that is smaller than x. Let G = (V , E, We ) be a weighted directed graph of order N with the set of nodes V = {v1 , v2 , . . . , vN }, the set of edges E ⊆ V × V , and a weighted adjacent matrix We = (Weij )N ×N . A directed edge in G is denoted by eij = (vi , vj ). If there is a directed edge from node vi to node vj , we say that node vi can reach node vj , and we assign it an edge weight Weji > 0; otherwise, Weji = 0. We assume that there are no self-loops or multiple edges in G, that is, Weii = 0 for i = 1, 2, . . . , N. A graph G has a directed spanning tree if there exists at least one node, called the root, that has a directed path to all other nodes in G. The Laplacian matrix denoted by L = (Lij )N ×N of G is defined by Lij = −Weij for  i = j and Lii = N j=1,j=i Weij , where i, j = 1, 2, . . . , N. To represent a random switching directed graph G (t) = (V , E, W (t)), we allocate an edge probability matrix P = (pij )N ×N for G. The existence of a directed edge eij in G(t) between a pair of nodes vi and vj is randomly determined and is independent of other edges with probability pij satisfying 0 ≤ pij < 1. An information link is referred to as a potential link when the associated edge probability is positive. We define N (N − 1) independent Bernoulli random variables δij , i, j = 1, 2, . . . ; , N , i = j, as follows: δij = 1 with probability pij and δij = 0 with probability 1 − pij , where each random variable

87

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Second-Order Consensus of Continuous-Time Multi-Agent Systems

δij is associated with the edge eij . Therefore, at time t, the weighted adjacency ma  trix W (t) = Wij (t) N ×N of G(t) can be determined as Wij (t) = Weij δij for i = j and Wii (t) = 0, where i, j = 1, 2, . . . , N. Correspondingly, denote by L (t) = (Lij (t))N ×N the time-varying Laplacian matrix associated with the random directed graph G(t). The sampling space (all possible topologies) of all such random directed graphs in this framework is represented by the set M. Let tk , k ∈ N, be the random switching moments appearing in the running process of the dynamical network G(t) during t ∈ [0, +∞). Suppose that the random switching instants satisfy t0 < t1 < . . . < tk−1 < tk < . . ., limk→∞ tk = +∞ and 0 < Δ1 = infk {tk − tk−1 } ≤ supk {tk − tk−1 } = Δ2 < +∞. Define Δtk = tk − tk−1 , k ∈ N. In this chapter, we assume that the sequence {Δtk }∞ k=1 consists of independent and identically distributed random variables. Moreover, the time and topology switchings are independent with each other. ¯ as the time-average graph of G(t). The Laplacian matrix of the timeDenote G   average graph, denoted as L¯ = L¯ ij N ×N , can be computed in elements by L¯ ij = −pij Weij  for i = j and L¯ ii = N j=1 pij Weij for i, j = 1, 2, . . . , N.

5.2.2 Problem formulation Suppose that the multi-agent network under consideration consists of N agents, which dynamically update their states based on local information exchange. Each isolated agent is governed by second-order nonlinear dynamics. At time t, all agents are randomly interconnected pairwise according to the edge probability matrix by using the weighted directional position and velocity information based on the dynamical topology of the directed random switching network G(t). Therefore a more general version of multiagent networks of linearly coupled second-order dynamical systems can be formulated as follows: ⎧ x˙ i (t) = vi (t), ⎪ ⎪ ⎪ ⎪ N ⎪ ⎨ v˙ (t) = f (x (t), v (t), t) + α  W (t)B x (t) − x (t) i i i ij j i j=1 ⎪ ⎪ N    ⎪ ⎪ ⎪ + β Wij (t)B vj (t) − vi (t) , i = 1, 2, ..., N , ⎩

(5.1)

j=1

where xi (t) = (x1i (t), ..., xni (t))T ∈ Rn and vi (t) = (vi1 (t), ..., vin (t))T ∈ Rn are the position and velocity variable vectors of agent i, respectively, f (xi (t), vi (t), t) = (f1 (xi (t), vi (t), t), ..., fn (xi (t), vi (t), t))T : Rn × Rn × R+ → Rn represents a continuous but not necessarily differentiable vector-valued function, which models the inherent nonlinear dynamics of the uncoupled agent i, α > 0 and β > 0 represent the position and velocity coupling strengths between any two agents in G(t), and B ∈ Rn×n denotes the inner coupling configuration between the agents. Definition 5.1. The second-order globally nonlinear consensus in the multi-agent dynamical network (5.1) with random switching directed topology is considered to be

Second-order global consensus in multi-agent systems with random directional link failure





achieved if, for any initial conditions xi (t0 ), vi (t0 ) ∈ RN , limt→∞ xi (t) − xj (t) = 0 and limt→∞ vi (t) − vj (t) = 0, i = j, ∀ i, j = 1, 2, ..., N. In what follows, for i = 2, 3, ..., N, let Xi1 (t) = xi (t) − x1 (t) and Vi1 (t) = vi (t) − v1 (t) be the position and velocity differences between agent i and agent 1, respectively, in  T the multi-agent dynamical network G(t). For convenience, we define X¯ (t) = X21 (t),   T   T (t ) T , V T (t ) T , and F (X ¯ (t) = V21 ¯ (t), V¯ (t), t) = f T (x2 (t), v2 (t), t) − (t) , ... , VN1 ..., XN1 T f T (x1 (t), v1 (t), t), . . . , f T (xN (t), vN (t), t) − f T (x1 (t), v1 (t), t) . Then from (5.1) we derive the following error dynamical system in compact vector form: 





X˙¯ (t) O(N −1)n = ˙ ¯ α S V (t) W (t ) ⊗ B

I(N −1)n β SW (t) ⊗ B









X¯ (t) 0(N −1)n + , V¯ (t) F (X¯ (t), V¯ (t), t)

(5.2)

where SW (σ (t))  ⎡ −W12 (t) − W2j (t)

⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎣

j=2

W32 (t) − W12 (t)

W23 (t) − W13 (t) −W13 (t) −

.. . WN2 (t) − W12 (t)

 j=3

W3j (t)

.. . WN3 (t) − W13 (t)

...

W2N (t) − W1N (t)

...

W3N (t) − W1N (t)

..

.. .  −W1N (t) − WNj (t)

. ...

⎤ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎦

j=N

(5.3) Correspondingly, for a multi-agent network associated with the time-average di¯ we obtain the following error rected topology corresponding to the Laplacian matrix L, dynamical system:





X˙¯ E (t) O(N −1)n = α S¯ W ⊗ B V˙¯ E (t)

I(N −1)n β S¯ W ⊗ B









X¯ E (t) 0(N −1)n + , V¯ E (t) F (X¯ E (t), V¯ E (t), t)

(5.4)

where ⎡

S¯ W

⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎣ 

¯ 12 − −W

 j=2

¯ 2j W

¯ 32 − W ¯ 12 W .. . ¯ ¯ 12 WN2 − W 

¯ 23 − W ¯ 13 W ¯ 13 − −W

 j=3

¯ 3j W

.. . ¯ ¯ 13 WN3 − W

...

¯ 2N − W ¯ 1N W

...

¯ 3N − W ¯ 1N W

..

. ...

¯ 1N −W

.. .  ¯ Nj − W

⎤ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦

(5.5)

j=N



T (t ), . . . , ¯ = W ¯ ij = Weij Pij , i, j = 1, 2, . . . , N. Moreover, X¯ E (t) = XE21 ¯ ij ,W with W N ×N    T T T (t ) T (t ), . . . , V T (t ) , where X (t ) = x (t ) − x (t ) and XEN1 and V¯ E (t) = VE21 Ei1 Ei E1 EN1

89

90

Second-Order Consensus of Continuous-Time Multi-Agent Systems



VEi1 (t) = vEi (t) − vE1 (t) for i = 2, 3, . . . , N, F (X¯ E (t), V¯ E (t), t) = f T (xE2 (t), vE2 (t), t) − T f T (xE1 (t), vE1 (t), t), . . . , f T (xEN (t), vEN (t), t) − f T (xE1 (t), vE1 (t), t) . We further present some assumptions and lemmas, which will be used to derive our main results. Assumption 5.1. [17] For the nonlinear function f in (5.1), there exist two constant matrices     M 1 = wij n×n and M 2 = mij n×n with wij ≥ 0 and mij ≥ 0 such that n       f (x, v, t) − f (y, z, t) ≤ wij xj − yj  + mij vj − zj 

(5.6)

j=1

for all i = 1, 2, . . . , n, x, y, v, z ∈ Rn , and t ≥ 0. Remark 5.2. Note that Assumption 5.1 is a Lipschitz-type condition satisfied by many well-known nonlinear dynamical systems. By the classical method of differential equation theory, Assumption 5.1 guarantees the existence and uniqueness of the solution for the random switching nonlinear system (5.2) and the time-average system (5.4). Moreover, f (0, 0, t) = 0 for all t ≥ t0 , which makes second-order consensus possible. Lemma 5.3. [1] The Laplacian matrix L has a simple eigenvalue (zero) and all the other eigenvalues have positive real parts if and only if the underlying network G has a directed spanning tree. ¯ associated with time-average Laplacian matrix Lemma 5.4. [23] Suppose that the network G ¯ L has a directed spanning tree. Then all the eigenvalues of thematrix −S¯ W are the same as those    of L¯ except for the zero eigenvalue, that is, λi −S¯ W = λi+1 L¯ for i = 1, 2, . . . , N − 1.

Lemma 5.5. [24] Let k ∈ R, and let X, Y , P, Q be matrices of appropriate dimensions. Then we have: (i) (kX ) ⊗ Y = X ⊗ (kY ); (ii) (X + Y ) ⊗ P = X ⊗ P + Y ⊗ P; (iii) (X ⊗ Y )(P ⊗ Q) = (XP ) ⊗ (YQ); (iv) (X ⊗ Y )T = X T ⊗ Y T . Let C = diag{c1 , c2 , . . . , cN −1 } and D = diag{d1 , d2 , . . . , dN −1 }, where ci and di , i = 1, 2, . . . , N − 1, are positive real numbers. Using two matrices M 1 and M 2 as in Assumption 5.1, we define the following parameters:  ρ1 = max

1≤i≤N −1

ρ2 = max

1≤i≤N −1

ρ3 = max

1≤i≤N −1

 max di

1≤j≤n



n   k=1

wjk2ε

+ m2jkε

+ wkj2(1−ε)

 n    , max di m2kj(1−ε)

1≤j≤n





max ci

1≤j≤n

k=1

n   k=1

wkj2(1−ε)

 ,

 ,

Second-order global consensus in multi-agent systems with random directional link failure

and  ρ4 = max

1≤i≤N −1

 max ci

1≤j≤n

n  

wjk2ε

+ m2jkε

+ m2kj(1−ε)



 ,

k=1

with ε ∈ [0, 1]. To this end, we further make the following assumption: Assumption 5.2. There exist two positive constants α and β and two positive definite diagonal matrices C and Dsuch that the following conditions hold:    ¯ (a) A = −β sym (D ⊗ In ) (SW ⊗ B) − α sym (C ⊗ In ) (S¯ W ⊗ B) > 0; (b) C ⊗ In − (D ⊗ In ) A−1 (D ⊗ In ) > 0; 3 (c) α sym (D ⊗ In ) (S¯ W ⊗ B) + ρ1 +ρ 2 I(N −1)n < 0; 4 (d) D ⊗ In + β sym (C ⊗ In ) (S¯ W ⊗ B) + ρ2 +ρ 2 I(N −1)n < 0.

5.3. Main results We are now in the position to state our main results with regard to the second-order global consensus in multi-agent systems with nonlinear dynamics and directed fixed topology. Theorem 5.6. If there exist two positive definite diagonal matrices C and D and two suitable positive coupling strengths α and β such that Assumption 5.2 holds, then the zero solution of the second-order nonlinear system (5.4) is globally asymptotically stable. Equivalently, the second-order globally nonlinear consensus in a multi-agent network with fixed directed topology corresponding to the time-average Laplacian matrix L¯ can be achieved asymptotically. Proof. Construct the following Lyapunov function:



 1 X¯ (t) , V (X¯ E (t), V¯ E (t), t) = X¯ ET (t), V¯ ET (t) Ω ¯ E VE (t) 2

where Ω =

A

(5.7)



D ⊗ In with matrices A, C, and D the same as those defined in C ⊗ In

D ⊗ In Assumption 5.2. First, we show that the Lyapunov function (5.7) is valid, which implies Ω > 0. By the Schur complement theorem [17] the positive definiteness of matrix Ω can be guaranteed by Assumption 5.2(a,b). The time derivative of (5.7) along the solution of (5.4) is V˙ (X¯ E (t), V¯ E (t), t) = X¯ ET (t)AV¯ E (t) + V¯ ET (t) (D ⊗ In ) V¯ E (t)

91

92

Second-Order Consensus of Continuous-Time Multi-Agent Systems

  + α X¯ ET (t) (D ⊗ In ) (S¯ W ⊗ B) X¯ E (t)   + β X¯ ET (t) (D ⊗ In ) (S¯ W ⊗ B) V¯ E (t)   + α V¯ ET (t) (C ⊗ In ) (S¯ W ⊗ B) X¯ E (t)   + β V¯ ET (t) (C ⊗ In ) (S¯ W ⊗ B) V¯ E (t) + X¯ ET (t) (D ⊗ In ) F (X¯ E (t), V¯ E (t), t)

(5.8)

+ V¯ ET (t) (C ⊗ In ) F (X¯ E (t), V¯ E (t), t)   = α X¯ ET (t) (D ⊗ In ) (S¯ W ⊗ B) X¯ E (t) + V¯ ET (t) (D ⊗ In ) V¯ E (t)   + β V¯ ET (t) (C ⊗ In ) (S¯ W ⊗ B) V¯ E (t) + X¯ ET (t) (D ⊗ In ) F (X¯ E (t), V¯ E (t), t) + V¯ ET (t) (C ⊗ In ) F (X¯ E (t), V¯ E (t), t).

By Assumption 5.1 and the algebraic inequality 2μ|xy| ≤ μ2ε x2 + μ2(1−ε) y2 for μ ≥ 0, x, y ∈ R, ε ∈ [0, 1] (see, e.g., [17]) we derived that X¯ ET (t) (D ⊗ In ) F (X¯ E (t), V¯ E (t), t) = =

N −1 



di x(i+1)e (t) − x1e (t)

i=1 n N −1  



T 



f (x(i+1)e (t), v(i+1)e (t), t) − f (x1e (t), v1e (t), t) 

di xj(i+1)e (t) − xj1e (t) fj (x(i+1)e (t), v(i+1)e (t), t) − fj (x1e (t), v1e (t), t)



i=1 j=1

≤

n  n N −1  

 

 

 

 

 

 

k di xj(i+1)e (t) − xj1e (t) wjk xk(i+1)e (t) − xk1e (t) + mjk v(ki+1)e (t) − v1e (t)

i=1 j=1 k=1

≤

n N −1 n  2 1     2ε j 2 di wjk (x(i+1)e (t) − xj1e (t)) + wjk2(1−ε) (xk(i+1)e (t) − xk1e (t)) 2 i=1 j=1 k=1 n N −1 n  2 1     2ε j 2 k di mjk (x(i+1)e (t) − xj1e (t)) + m2jk(1−ε) (v(ki+1)e (t) − v1e (t)) + 2 i=1 j=1 k=1

≤

n N −1 n 2 1    2ε j di wjk (x(i+1)e (t) − xj1e (t)) 2 i=1 j=1 k=1

+

n N −1 n 2 1    2ε j di mjk (x(i+1)e (t) − xj1e (t)) 2 i=1 j=1 k=1

(5.9)

Second-order global consensus in multi-agent systems with random directional link failure

+

n N −1 n 2 1    2(1−ε) j j di wkj (x(i+1)e (t) − x1e (t)) 2 i=1 j=1 k=1

+

n N −1 n 2 1    2(1−ε) j j di mkj (v(i+1)e (t) − v1e (t)) 2 i=1 j=1 k=1

⎧

⎫

n N −1 n  2 ⎬ 1  ⎨    2ε ≤ di wjk + m2jkε + wkj2(1−ε) xj(i+1)e (t) − xj1e (t) ⎭ 2 i=1 ⎩ j=1 k=1

⎧

⎫

n N −1 n 2 ⎬ 1  ⎨   2(1−ε)  j j + di mkj v(i+1)e (t) − v1e (t) ⎭ 2 i=1 ⎩ j=1 k=1

≤

ρ1

2

2

X¯ E (t) +

ρ2

2

2

V¯ E (t) .

By a derivation process similar to that for obtaining inequality (5.9) and the definitions of ρ3 and ρ4 we obtain 2 ρ 4 2 ρ3 V¯ ET (t) (C ⊗ In ) F (X¯ E (t), V¯ E (t), t) ≤ X¯ E (t) + V¯ E (t) . 2 2

(5.10)

Substituting inequalities (5.9) and (5.10) into (5.8), after some manipulation, we obtain V˙ (X¯ E (t), V¯ E (t), t)

   ρ1 + ρ3 T ¯ I(N −1)n X¯ E (t) ≤XE (t) α sym (D ⊗ In ) (S¯ W ⊗ B) + 2   ρ2 + ρ4  T ¯ I(N −1)n V¯ E (t). + VE (t) D ⊗ In + β sym (C ⊗ In ) (S¯ W ⊗ B) +

(5.11)

2

By Assumption 5.2(c,d) we have V˙ ≤ 0 for all X¯ E (t) and! V¯ E (t). Moreover, V˙ = 0  T  if and only if X¯ E (t) = 0 and V¯ E (t) = 0. Therefore the set M = X¯ ET (t), V¯ ET (t) X¯ E (t) = "

!

T 

"

V¯ E (t) = 0 is the largest invariant set contained in the set D = X¯ ET (t), V¯ ET (t) V˙ = 0 for the nonlinear dynamical system (5.4). According to LaSalle’s invariance principle [17], all trajectories of system (5.4) starting from any initial condition will finally approach the set M as t → ∞, that is, X¯ E (t) → 0 and V¯ E (t) → 0 as t → ∞. This means that the second-order globally nonlinear consensus in a multi-agent network associated with the underlying interaction topology corresponding to the time-average Laplacian matrix L¯ can be achieved asymptotically. This completes the proof. Remark 5.7. According to Lemma 5.4 and the algebraic graph theory [25], L¯ has a zero ¯ aseigenvalue with algebraic multiplicity m if and only if the fixed topology of graph G sociated with the Laplacian matrix L¯ has m connectivity branches. This also means that the algebraic multiplicity of eigenvalue zero of the matrix S¯ W equals to m − 1 (when

93

94

Second-Order Consensus of Continuous-Time Multi-Agent Systems

the algebraic multiplicity of eigenvalue zero of matrix S¯ W equals to 0, we claim that S¯ W does not have zero eigenvalue). In fact, conditions (c) and (d) in Assumption 5.2 imply that the matrix S¯ W does not have zero eigenvalue. Otherwise, the matrices   β sym (D ⊗ In ) (S¯ W ⊗ B) and α sym (C ⊗ In ) (S¯ W ⊗ B) have zero eigenvalues, which further indicate that Assumption 5.2(c,d) does not hold. Furthermore, Assumption 5.2 implies that there exists a directed spanning tree in the underlying fixed time-average ¯ Therefore the connectivity of a graph topology associated with the Laplacian matrix L. is a necessary condition for consensus in multi-agent systems. In addition, in Theorem 5.6, as well as in the following Theorem 5.8, we omit the proof of existence and uniqueness of the solution to nonlinear systems (5.2) and (5.4). Under the Lipschitztype condition as described in Assumption 5.1, this can be proved using the classic differential equation theory. In what follows, we study the second-order global consensus in multi-agent networks with nonlinear dynamics and random switching directed interconnections. For the case where all topologies of random dynamical network G(t) contain directed spanning trees (e.g., the connectivity of network can be maintained constantly), the results can be obtained trivially by mimicking the analysis procedure for fixed topology. However, when there are some isolated agents in certain topologies during the random switching process, these isolated agents fail to receive neighbors’ state information to further update their current states. Thus it is possible that these isolated agents may deviate the consensus manifold as time goes by. From the theoretical perspective this can also be viewed as that the coefficient matrix of the resultant error dynamical system (5.2) is a singular matrix when the associated topology does not contain a directed spanning tree. Thus the classical methods for dealing with the switched system, that is, common Lyapunov function method or multi-Lyapunov functions method, are no longer valid. However, in general, the time-average network of the random switching dynamical network G(t) may contain a directed spanning tree. It is natural to think of using the Lyapunov function constructed for the nonlinear system associated with time-average network to further characterize the convergence properties of the corresponding random switching nonlinear system. Therefore by associating it with the solution of the random switching dynamical network G(t) we can use the constructed Lyapunov function for the time-average network to analyze the second-order global consensus in G(t). For convenience, let y(t) = (X¯ T (t), V¯ T (t))T and V (y(t), t) = 1/2yT (t)Ω y(t), where Ω is as in (5.7). We further make the following assumption. Assumption 5.3. There exists an infinite subsequence composed of some fixed moments in the random switching sequence, denoted by t0 = t1∗ < t2∗ < . . . < tk∗−1 < tk∗ < . . ., satisfying limk→+∞ tk∗ = +∞. Moreover, there exists a finite positive integer q such that ti∗+1 − ti∗ ≤ qΔ2 2 and V (y(ti∗+1 ), ti∗+1 ) − V (y(ti∗ ), ti∗ ) ≤ −p y(ti∗ ) , where i = 1, 2, . . ., and p > 0.

Second-order global consensus in multi-agent systems with random directional link failure

We are now in the position to state our main results for achieving the second-order globally nonlinear consensus in the multi-agent network (5.1) with directed topology and random switching interconnections. Theorem 5.8. Suppose that Assumptions 5.2 and 5.3 hold. Then the globally exponential stability of zero solution of random switching nonlinear system (5.2) can be guaranteed. Equivalently, the second-order globally nonlinear consensus in the multi-agent network (5.1) with directed topology and random switching interconnections can be reached. Proof. It follows from Assumption 5.2 that Theorem 5.6 holds and the second-order globally nonlinear consensus in time-average network of random switching network G(t) can be achieved. Moreover, if it is associated with the solution of the random switching dynamical network G(t) with (5.7), then the positive definiteness of Lyapunov function V (y(t), t) can also be ensured. In the following, we will use the properties in Assumption 5.3 of V (y(t), t) to further guarantee the exponential stability of zero solution of random switching nonlinear system (5.2). According to algebraic theory, we 2 2 obtain 1/2λmin (Ω) y(t) ≤ V (y(t), t) ≤ 1/2λmax (Ω) y(t) . From Assumption 5.3 we have that there exists k0 ∈ N such that tk∗0 − t0 < qΔ2 . For simplicity, let

O(N −1)n Fˆ (X¯ (t), V¯ (t), t) = α SW (σ (t)) ⊗ B



I(N −1)n y(t) β SW (σ (t)) ⊗ B

(5.12)

+ F˜ (X¯ (t), V¯ (t), t),

where F˜ (X¯ (t), V¯ (t), t) =

0(N −1)n ¯ F (X (t), V¯ (t), t)



.

For t ∈ [t0 , tk∗0 ], we have #

y(t) = y(t0 ) +

t

Fˆ (X¯ (s), V¯ (s), s)ds.

(5.13)

t0

For s ∈ [t0 , t], by computation we have ˆ ¯ F (X (s), V¯ (s), s) ≤ K y(s) ,

(5.14)

where K = K1 + K2 with $

K1 = (max {α, β})2 A1 + 1

(5.15)

and %

K2 = 2 max {B1 , C1 },

(5.16)

95

96

Second-Order Consensus of Continuous-Time Multi-Agent Systems

in which 







T A1 = λmax SW (σ (t))SW (σ (t)) λmax BT B ,



!

B1 = max max wjk2 1≤j≤n

"

,

1≤k≤n

and



!

C1 = max max m2jk 1≤j≤n

" .

1≤k≤n

By some computation steps we have

O(N −1)n α SW (σ (t)) ⊗ B

I(N −1)n β SW (σ (t)) ⊗ B



T (σ (t ))S (σ (t )) ⊗ BT B α 2 SW W = T (σ (t ))S (σ (t )) ⊗ BT B αβ SW W

T (σ (t ))S (σ (t )) ⊗ BT B α 2 SW W ≤ T (σ (t ))S (σ (t )) ⊗ BT B αβ SW W

T

O(N −1)n α SW (σ (t)) ⊗ B

I(N −1)n β SW (σ (t)) ⊗ B

 T (σ (t ))S (σ (t )) ⊗ BT B αβ SW W T (σ (t ))S (σ (t )) ⊗ BT B I + β 2 SW W 

T (σ (t ))S (σ (t )) ⊗ BT B αβ SW I W + T 2 T β SW (σ (t))SW (σ (t)) ⊗ B B 0



(5.17) 

0 . I

The maximum eigenvalue of this matrix is bounded by $











T (σ (t ))S (σ (t )) λ T K1 = (max {α, β})2 λmax SW W max B B + 1.

(5.18)

Moreover, we have the following inequality: F (X¯ E (t), V¯ E (t), t) 2 N −1 f (x(i+1)e (t), v(i+1)e (t), t) − f (x1e (t), v1e (t), t) 2 = = ≤ =

i=1 N −1  n 

 fj (x(i+1)e (t), v(i+1)e (t), t) − fj (x1e (t), v1e (t), t)2

i=1 j=1 N −1  n  n

i=1 j=1 k=1 N −1  n  n

i=1 j=1 k=1 N −1  n  n  i=1 j=1 k=1







2

2



k wjk xk(i+1)e (t) − xk1e (t) + mjk v(ki+1)e (t) − v1e (t)

i=1 j=1 k=1 N −1  n  n 

+ ≤



2 



k wjk2 xk(i+1)e (t) − xk1e (t) + m2jk v(ki+1)e (t) − v1e (t)

    k (t) +2wjk mjk xk(i+1)e (t) − xk1e (t) v(ki+1)e (t) − v1e 

2



2 

k 2wjk2 xk(i+1)e (t) − xk1e (t) + 2m2jk v(ki+1)e (t) − v1e (t)

 2 2  2 ≤ K22 X¯ E (t) + V¯ E (t) = K22 y(t) ,

(5.19)

Second-order global consensus in multi-agent systems with random directional link failure

!

!

!

where K2 = 2 max max1≤j≤n max1≤k≤n wjk2

""

! ! """ . Therefore , max1≤j≤n max1≤k≤n m2jk

||Fˆ (X¯ (s), V¯ (s), s)|| ≤ K y(s) , where K = K1 + K2 .

Thus, based on the Bellman–Growall formula (see, e.g., [26]), we obtain the following results from (5.13): y(t) ≤ y(t0 ) eK (t−t0 ) , t ∈ [t0 , t∗ ], k0

(5.20)

where tk∗0 is a switching instant in the subsequence defined in Assumption 5.3. 2



Since V (y(tk∗0 ), tk∗0 ) ≤ 12 λmax (Ω) y(tk∗0 )

and V (y(tk∗0 +1 ), tk∗0 +1 ) − V (y(tk∗0 ), tk∗0 ) ≤

2 −p y(tk∗0 ) < 0, we derive that ∗

&

∗

V (y(tk0 +1 ), tk0 +1 ) ≤ 1 −

'

2p λmax (Ω)

V (y(tk∗0 ), tk∗0 )

(5.21)

for t > tk∗0 . Since 0 ≤ V (y(tk∗0 +1 ), tk∗0 +1 ) < V (y(tk∗0 ), tk∗0 ), we have 0 ≤ 1 − 2p/λmax (Ω) < 1. Repeating this computation procedure m∗ times (as determined in (5.23)), we have &

V (y(tk∗0 +m∗ ), tk∗0 +m∗ ) ≤ 1 −

' m∗

2p λmax (Ω)

V (y(tk∗0 ), tk∗0 ).

(5.22)

Let φ = λmax (Ω) /λmin (Ω) and choose ') & '* ( & 2p λmin (Ω) m = ln ln 1 − . λmax (Ω) λmax (Ω) ∗

(5.23)

From (5.22) we derive that & 2 ) ≤ 1− k0 +m∗

∗ y(t

' m∗

2p λmax (Ω)

2 φ y(tk∗0 ) ,

(5.24)

and from Assumption 5.3 that &

1− Define λ=

'm∗

2p λmax (Ω)

 ln 1 −

2p

φ < 1.

(5.25)



λmax (Ω)

−q Δ 2

(5.26)

> 0.

By some computations we easily establish the following relation: &

1−

2p λmax (Ω)

' m∗ =e

−λm∗ qΔ2

≤e

 −λ tk∗

0 +m

∗ ∗ −tk

0



.

(5.27)

97

98

Second-Order Consensus of Continuous-Time Multi-Agent Systems

Combining (5.24)–(5.27), we derive that ∗ y(t

k0

+m∗

  − λ t∗ −t∗ ) ≤ φ e 2 k0 +m∗ k0 y(tk∗0 ) .

(5.28)

Then for all n ∈ N, by a similar argument as that leading to (5.24), we get ∗ y(t

k0

+nm∗

& 2 ) ≤ 1 −

'nm∗

2p λmax (Ω)

2 φ y(tk∗0 ) .

(5.29)

By the definitions of (5.23) and (5.26) we further have the inequality &

1−

'nm∗

2p λmax (Ω)

= e−λnm

∗ qΔ

2

≤e

 −λ tk∗

0 +nm

∗ ∗ −tk



0

.

(5.30)

Therefore, for all n ∈ N, we have the inequality % − λ t∗ ∗ −t∗  ∗ 2 k0 +nm k0 y(tk0 ) . k0 +nm∗ ) ≤ φ e

∗ y(t

(5.31)

On the other hand, for all t ≥ tk∗0 , there exists n0 ∈ N such that for t ∈ [tk∗0 +n0 m∗ , tk0 +(n0 +1)m∗ ), we have ∗

y(t) ≤ y(t∗

k0 +n0

m∗

) +

≤ y(tk∗0 +n0 m∗ ) + 

≤e

K

t−t∗



k0 +n0 m∗

 − λ2 tk∗

#

t

Fˆ (X¯ (s), V¯ (s), s)ds

tk∗

∗ 0 +n 0 m

#

t



tk∗

0 +n 0 m

∗

k0 +n0 m∗ )

∗ y(t 



K y(s) ds



(5.32) 

% ∗ φ y(tk0 )   % ∗ − λ t∗ −t∗ ≤e 2 k0 +n0 m∗ k0 eKm qΔ2 φ y(tk∗0 ) , ≤e

0 +n 0 m

∗ ∗ −tk

0

e

K t−tk∗

0 +n 0 m

∗

since t − tk∗0 +n0 m∗ ≤ m∗ qΔ2 . For t ∈ [tk∗0 +n0 m∗ , tk∗0 +(n0 +1)m∗ ), we have tk∗0 +n0 m∗ − tk∗0 ≥ t − m∗ qΔ2 − tk∗0 . Thus we have the inequality   % λ ∗ y(t) ≤ e− 2 t−tk0 e λ2 m∗ qΔ2 eKm∗ qΔ2 φ y(t∗ ) . k0

(5.33)

For any t > tk∗0 , since tk∗0 − t0 < k0 qΔ2 , we have t − tk∗0 + t0 > t − k0 qΔ2 . Combining this







with y(tk∗0 ) ≤ y(t0 ) eKk0 qΔ2 , it follows from (5.33) that

+    ∗ , % λ ∗ y(t) ≤ e− λ2 (t−t0 ) e 2 m +k0 +K m +k0 qΔ2 φ y(t0 ) , t ≥ t∗ . k0

(5.34)

Second-order global consensus in multi-agent systems with random directional link failure

For t ∈ [t0 , tk∗0 ), λ

e− 2 (t−t0 ) e

+  λ 2

,

m∗ +k0 +Km∗ qΔ2 % 

λ

φ ≥ 1 by (5.20) and e− 2 (t−t0 ) e

+  λ 2

,

m∗ +k0 +Km∗ qΔ2 % 

φ ≥ 1,

and we have +    ∗ , % λ ∗ y(t) ≤ e− λ2 (t−t0 ) e 2 m +1 +K m +1 qΔ2 φ y(t0 ) , t0 ≤ t < t∗ . k0

(5.35)

Therefore, under Assumptions 5.2 and 5.3, we can be establish the exponential stability of the zero solution of random switching nonlinear system (5.2) with convergence rate λ/2 > 0. This implies that we can achieve the second-order globally nonlinear consensus in the multi-agent dynamical network (5.1) with directed topology and random switching interconnections exponentially. The proof is thus completed.

5.4. Illustrative examples In this section, we use a simulation example to demonstrate the correctness of our theoretical results. We assume that there are five agents flying in a three-dimensional space. All the agents share their position and velocity states over a dynamically random switching directed time-varying network G(t). Moreover, the framework of the random switching directed network G(t) is determined by ⎡ ⎢ ⎢ ⎢ We = ⎢ ⎢ ⎣

and

⎡ ⎢ ⎢ ⎢ P=⎢ ⎢ ⎣

0 0.2800 0.1103 0.0976 0.1557

0 0.7235 0.6992 0.6714 0.7145

0.3264 0 0.4127 0.2423 0.4345

0.6080 0 0.6335 0.7108 0.6272

0.3044 0.4230 0 0.2180 0.4466

0.7193 0.7399 0 0.6521 0.7392

0.1514 0.1687 0.2496 0 0.0974

0.6894 0.6630 0.7042 0 0.7137

0.3194 0.3233 0.0151 0.2939 0

0.6644 0.7035 0.6768 0.6794 0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(5.36)

⎤ ⎥ ⎥ ⎥ ⎥. ⎥ ⎦

(5.37)

After the linear couplings, the dynamical behavior of agent i can be determined by ⎧ x˙ i (t) = vi (t), ⎪ ⎪ ⎪ ⎪ N ⎪ ⎨ v˙ (t) = f (v (t)) + α  W (m(t))B x (t) − x (t) i i eij j i j=1 ⎪ ⎪ N    ⎪ ⎪ ⎪ +β Weij (m(t))B vj (t) − vi (t) , ⎩ j=1

(5.38)

99

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Second-Order Consensus of Continuous-Time Multi-Agent Systems

Figure 5.1 Position and velocity states of five agents in dynamically switching network G(t). (a) Position states. (b) Velocity states.

where i = 1, 2, . . . , 5, xi (t) = (xi1 (t), xi2 (t), xi3 (t))T represents the ith agent’s position, and in a three-dimensional space. Thus vi (t) = (vi1 (t), vi2 (t), vi3 (t))T represents its velocity T   f (vi ) = a(vi2 − g(vi1 )), vi1 − vi2 + vi3 , −bvi2 with g(vi1 ) = cvi1 + 0.5 d − c (|vi1 + 1| − |vi1 − 1|). In particular, the isolated second-order oscillator (α = β = 0) exhibits a chaotic attractor at the parameters a = 10, b = 18, c = 1/4, and d = −1/3 (see [27]). All the agents’ inner coupling constant matrix B is assumed to be I3 . Let the coupling strengths be α = 25 and β = 50. Suppose that the dwell time of each topology in the random switching sequences are randomly and independently distributed over the interval [0.1, 0.2], that is, Δ1 = 0.1 and Δ2 = 0.2. In addition, we take ⎛ D = C = I4 and ⎞ ε = 0.5. By numerical computations we have 25/3 10 0 ⎜ ⎟ M 1 = O3 , M 2 = ⎝ 1 1 1⎠, ρ1 = ρ2 = max {55/3, 3, 18} = 18.3333, ρ3 = 0, ρ4 = 0 18 0   max{88/3, 32, 19} = 32, λmin (A) = 51.1097, λmin C ⊗ I3 − (D ⊗ I3 ) A−1 (D ⊗ I3 ) =  ρ1 +ρ3    0.9804, λmax α sym (D ⊗ I3 ) (S¯ W ⊗ B) + 2 I12 = −7.8699, and λmax (D ⊗ I3 +   4 β sym (C ⊗ I3 ) (S¯ W ⊗ B) + ρ2 +ρ 2 I12 ) = −7.9065. This implies that the positive definiteness of matrix Ω as defined in (5.7) can be guaranteed, and thus the conditions in Assumption 5.2 are satisfied. In the simulation, we use the Runge–Kutta method to solve the differential equations by taking the step size 0.0025. The initial position and velocity states of all agents are randomly selected from the intervals [0, 8] and [0, 5], respectively. The time evolutions of position and velocity states of all the agents are shown in Figs. 5.1 (a) and (b), respectively. As shown in the figures, the second-order nonlinear consensus in the random switching multi-agent directed network G(t) is achieved. Even though the network is not always connected instantaneously, all agents can still fly in a three-dimensional space with the same position and velocity states after the second-order consensus has been achieved. To illustrate the change of network

Second-order global consensus in multi-agent systems with random directional link failure

Figure 5.2 (a) The time evolution of the Euclidean norm of time-varying Laplacian matrix L(t). (b) The time evolution of Re (λ2 (L(t))).

Figure 5.3 The time evolution of logarithm with regard to V (y(t), t).

topology structure in the random switching sequence, we depict the time evolution of the Euclidean norm of the time-varying Laplacian matrix L (t) corresponding to G(t), which is shown in Fig. 5.2(a). Moreover, we also provide the time evolution of Re (λ2 (L (t))) (the real part of the second minimum eigenvalue of L (t)), which is depicted in Fig. 5.2(b). From Fig. 5.2(b), we can observe that in some time intervals, the value Re (λ2 (L (t))) is equal to zero, which indicates that there are no directed spanning trees in topologies running in these time intervals, i.e., there exist some isolated agents in these topologies. Fig. 5.3 shows the time evolution of logarithm with regard to V (y(t), t) = 1/2yT (t)Ω y(t), where y(t) = (X¯ T (t), V¯ T (t))T . It can be observed that the time derivative of V (y(t), t) has positive and negative values. This is because in the random switching sequences the random switching multi-agent network (5.1) undergoes some topologies, which do not contain directed spanning trees. In this case, as described before, the exponential stability associated with the resultant error dynamical system of (5.1) can also be established.

101

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From the simulation results we find that even though the time-varying network topology is not always connected instantaneously in time, sufficient information can also be propagated through the dynamical network to guarantee the second-order globally nonlinear consensus. The underlying mechanism that makes consensus is still possible even though some links between agents are broken within a period of time, which is useful in particular domains such as swarms, autonomous vehicles formation, or attitude adjustment of man-made satellites. Remark 5.9. Different from the results in [21,22], the inherent nonlinear function of the agent under study only needs to be continuous but not necessarily differentiable. Thus the results are in the sense of global consensus. A sufficient condition for reaching the second-order globally nonlinear consensus in a multi-agent network both with directed topology and random switching interconnections is obtained, and the consensus can also be achieved at an exponential convergence rate, which is estimated analytically. It is required that the designed Lyapunov function decreases along the solution of the nonlinear random switching system at an infinite subsequence of switching moments. To reach the second-order nonlinear consensus in networks of multiple agents with directed topologies and random switching connections, a relatively conservative condition, that is, real-time measurements of position and velocity states of all agents, is needed. This condition can effectively guide us to regulate the coupling strengths to overcome the irregular switching and finally achieves consensus. Remark 5.10. For a multi-agent network, the dynamical behavior of the coupled agents may be different from that of the isolated ones upon strong linear couplings. For example, the whole coupled network may be divergent even if the states of the isolated agent are bounded. This may be resulted from the sufficiently large coupling strengths or undesirable switching topologies. It is worth mentioning that even if the network dynamics change qualitatively due to these factors, the second-order nonlinear consensus in a multi-agent network with directed topologies and random switching connections can also be achieved. Therefore the results obtained in this chapter are more general. Here we omit the simulation results due to the limited space.

5.5. Conclusion We investigated the second-order consensus problem in a multi-agent network with inherent nonlinear dynamics and random switching directed interconnections. We found that the second-order global consensus of multi-agent systems with random switching topologies can be achieved as long as the consensus can be realized in the corresponding time-average network and the designed Lyapunov function decreases along the solution of the nonlinear random switching system at an infinite subsequence of switching moments. Although the obtained result seems to be slightly conservative, it can effectively

Second-order global consensus in multi-agent systems with random directional link failure

instruct us to regulate the coupling strengths to overcome the adverse effect caused by the random switching. The results of this chapter can also provide some insights on how the consensus is achieved even though the potential network topology is disconnected simultaneously. There are still a number of related interesting problems deserving further investigation. For example, it is desirable to study: i) consensus of agents with different nonlinear dynamics; ii) consensus of agents with time-varying delay couplings; iii) cluster consensus; and iv) consensus with the communication constraints such as packet loss, channel noise, and limited bandwidth. Some of them will be investigated in the near future.

References [1] W. Ren, R. Beard, E. Atkins, Information consensus in multivehicle cooperative control: collective group behavior through local interaction, IEEE Control Systems Magazine 27 (2007) 71–82. [2] W. Yu, G. Chen, Z. Wang, W. Yang, Distributed consensus filtering in sensor networks, IEEE Transactions on Systems, Man and Cybernetics. Part B. Cybernetics 39 (2009) 1568–1577. [3] S. Pereira, A. Pages-Zamora, Consensus in correlated random wireless sensor networks, IEEE Transactions on Signal Processing 59 (2011) 6279–6284. [4] J. Lu, J. Kurths, J. Cao, N. Mahdavi, C. Huang, Synchronization control for nonlinear stochastic dynamical networks: pinning impulsive strategy, IEEE Transactions on Neural Networks 23 (2012) 285–292. [5] Y. Tang, W. Wong, Distributed synchronization of coupled neural networks via randomly occurring control, IEEE Transactions on Neural Networks and Learning Systems 24 (2013) 435–447. [6] M. Cao, A. Morse, B. Anderson, Reaching a consensus in a dynamically changing environment: convergence rates, measurement delays, and asynchronous events, SIAM Journal on Control and Optimization 47 (2008) 601–623. [7] A. Jadbabaie, J. Lin, S.A. Morse, Coordination of groups of mobile agents using nearest neighbor rules, IEEE Transactions on Automatic Control 48 (2003) 988–1001. [8] M. Porfiri, D. Stilwell, Consensus seeking over random weighted directed graphs, IEEE Transactions on Automatic Control 52 (2007) 1767–1773. [9] X. Yang, J. Cao, J. Lu, Synchronization of Markovian coupled neural networks with nonidentical node-delays and random coupling strengths, IEEE Transactions on Neural Networks 23 (2012) 60–71. [10] D. Aeyels, J. Peuteman, On exponential stability of nonlinear time-varying differential equations, Automatica 35 (1999) 1091–1100. [11] H. Su, X. Wang, Z. Lin, Synchronization of coupled harmonic oscillators in a dynamic proximity network, Automatica 45 (2009) 2286–2291. [12] W. Ren, E. Atkins, Distributed multi-vehicle coordinated control via local information exchange, International Journal of Robust and Nonlinear Control 17 (2006) 1002–1033. [13] W. Yu, G. Chen, M. Cao, Some necessary and sufficient conditions for second-order consensus in multi-agent dynamical systems, Automatica 46 (2010) 1089–1095. [14] J. Zhu, Y. Tian, J. Kuang, On the general consensus protocol of multi-agent systems with doubleintegrator dynamics, Linear Algebra and Its Applications 431 (2009) 701–715. [15] H. Li, X. Liao, T. Dong, L. Xiao, Second-order consensus seeking in directed networks of multi-agent dynamical systems via generalized linear local interaction protocols, Nonlinear Dynamics 70 (2012) 2213–2226. [16] W. Yu, G. Chen, M. Cao, J. Kurths, Second-order consensus for multi-agent systems with directed topologies and nonlinear dynamics, IEEE Transactions on Systems, Man and Cybernetics. Part B. Cybernetics 40 (2010) 881–891. [17] Q. Song, J. Cao, W. Yu, Second-order leader-following consensus of nonlinear multi-agent systems via pinning control, Systems & Control Letters 59 (2010) 553–562.

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[18] H. Su, G. Chen, X. Wang, Z. Lin, Adaptive second-order consensus of networked mobile agents with nonlinear dynamics, Automatica 47 (2011) 368–375. [19] H. Li, X. Liao, G. Chen, Finite-time leader-following consensus for second-order multi-agent systems with nonlinear dynamics, International Journal of Systems Science 11 (2013) 422–426. [20] W. Guo, J. Lu, S. Chen, X. Yu, Second-order tracking control for leader-following multi-agent flocking in directed graphs with switching topology, Systems & Control Letters 60 (2011) 1051–1058. [21] H. Li, X. Liao, X. Lei, T. Huang, W. Zhu, Second-order consensus seeking in multi-agent systems with nonlinear dynamics over random switching directed networks, IEEE Transactions on Circuits and Systems. I, Regular Papers 60 (2013) 1595–1607. [22] H. Li, X. Liao, T. Huang, Second-order dynamic consensus of multi-agent systems with arbitrarily fast switching topologies, IEEE Transactions on Circuits and Systems. I, Regular Papers 43 (2013) 1343–1353. [23] W. Yu, Multi-agent collective behaviors analysis and applications in complex networks and systems, PhD thesis, City University of Hong Kong, Sep. 2009. [24] R. Horn, C. Johnson, Topics in Matrix Analysis, Cambridge Univ. Press, Cambridge, UK, 1985. [25] C. Godsil, G. Royle, Algebraic Graph Theory, Springer-Verlag, New York, 2001. [26] D. Aeyels, J. Peuteman, A new asymptotic criterion for nonlinear time-variant differential equations, IEEE Transactions on Automatic Control 43 (1998) 968–971. [27] L.O. Chua, M. Komuro, T. Matsumoto, The double scroll family, IEEE Transactions on Circuits and Systems 33 (1986) 1073–1118. [28] S. Wen, Z. Zeng, T. Huang, H∞ filtering for neutral systems with mixed delays and multiplicative noises, IEEE Transactions on Circuits and Systems. II, Express Briefs 59 (2012) 820–824. [29] S. Wen, Z. Zeng, T. Huang, Exponential stability analysis of memristor-based recurrent neural networks with time-varying delays, Neurocomputing 97 (2012) 233–240.

CHAPTER 6

Algebraic criteria for second-order global consensus in multi-agent networks with intrinsic nonlinear dynamics and directed topologies 6.1. Introduction In recent years, there has been an increasing research interest in the dynamical behaviors of isolated nonlinear systems [10–12,17,18] or networked control systems [49–52,54] with applications [16]. Especially, the design of distributed control algorithms based on agents’ local interaction information in multi-agent networks, such as rendezvous control of multi-nonholonomic agents [8], formation control [2,15], and flocking attitude alignment [4,23], has drawn much attention from researchers. The formation control, flocking, and rendezvous can be unified in the general framework of consensus setting. Consensus, a typical collective behavior in networked systems with a group of autonomous mobile agents, has recently received considerable attention due to its broad applications in biological systems, sensor networks, unmanned air vehicle (UAV) formations, robotic teams, underwater vehicles, and so on. The basic idea for information consensus is that each agent shares information only with its neighboring agents under a distributed protocol, whereas the whole group of agents can coordinate so as to achieve a certain global behavior of common interest [5]. It is worth mentioning that different methods to deal with the fuzzy shortest path problems, fuzzy shortest path in a network by Bellman dynamic programming approach, and multi-objective linear programming technique have been presented [9,39,43,55]. In recent years the consensus problem in the cooperative control community has been extensively studied [6,7,19–22,24–27,29–38,40–42,44,47,48,53], to name a few. Jadbabaie et al. [19] provided a theoretical explanation for the consensus behavior of the Vicsek model by using the graph and matrix theories. Under the assumption that the dynamics of each agent was a scalar continuous-time integrator, Olfati-Saber and Murray [33] further solved the average consensus problem for directed balanced networks. Ren and Beard [37] extended the results in [19,33] by providing more relaxed conditions. Moreau [30] used a set-valued Lyapunov approach to address the consensus problem with unidirectional time-dependent communication links. Moore and Lucarelli [29] extended the results for single consensus variables to include the cases of forced consensus. Carli et al. [6] discussed the quantized average consensus problem. Hui et al. [15] developed the robust Second-Order Consensus of Continuous-Time Multi-Agent Systems https://doi.org/10.1016/B978-0-32-390131-4.00013-3

Copyright © 2021 Elsevier Inc. All rights reserved.

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analysis results for nonlinear network consensus protocols. Ballal and Lewis [1] proposed a continuous-time and discrete-time bilinear trust update schemes for trust consensus. More profound theoretical results have been established for distributed consensus of networked dynamic systems [8,35,38], which are very important in a wide range of practical applications [7,32,44], whereas the scenario for networks of agents with timevarying asymptotic velocity exists ubiquitously in the study of synchronization [45,46]. When acceleration is considered as the control input, each agent should be modeled as a double integrator dynamics. In this case the consensus problem becomes more challenging. In many practical applications (e.g., autonomous underwater vehicles, unmanned aerial vehicles), the actuators can affect only the acceleration through the agents’ inertias. Particularly, Lee and Spong [23] addressed the stable flocking of multiple agents that had significant inertias and evolved on a balanced information graph. As presented in [23], the agents’ inertial effect can even cause unstable group behavior. However, as pointed out in [36], the extension of consensus algorithms for agents from first-order to second-order dynamics (when the control input is added to the driving force/acceleration term) is non-trivial. Protocols or algorithms dealing with the second-order consensus of multi-agent systems with nonlinear dynamics have not been emphasized until the works [24–26,40,42,53]. In [53] a kind of measurement for directed strongly connected graph, that is, general algebraic connectivity, was first defined. The authors built the bridge between the general algebraic connectivity and the performance of reaching an agreement for second-order multi-agent systems with nonlinear dynamics. But the relation between the generalized algebraic connectivity and the eigenvalues of the Laplacian matrix was not direct. The directed graph containing a directed spanning tree have to be divided into strongly connected components, and the generalized algebraic connectivity of each strongly connected component should be calculated to give sufficient conditions ensuring consensus. The work of [53] was extended to the leader-following case via pinning control in Song et al. [40] by using pinning control technique, and it is worth mentioning that the above approaches have overcome the restriction, that is, the interactive network is strongly connected in [53]. Based on the local adaptive strategies, Su et al., [42] found that if one agent has access to the information of the virtual leader, then all agents in the group can synchronize with the virtual leader. In [24] the finite-time second-order robust consensus problem of multi-agent networks with inherent nonlinear dynamics was considered, and the convergence time was obtained. In [25] the authors studied the final secondorder consensus convergence state of a multi-agent directed network by using a kind of generalized linear local interaction protocols. In [26], by introducing the generalized matrix measure and by applying the tools of contraction and circle analysis the secondorder locally dynamical consensus of multi-agent systems with arbitrarily fast switching directed topologies was theoretically investigated in detail, and some easily verified sufficient conditions were also presented. In [27] the authors discussed the second-order

Algebraic criteria for second-order global consensus in multi-agent networks

local consensus problem for multi-agent systems with nonlinear dynamics over dynamically switching random directed networks. By applying the orthogonal decomposition method the state vector of the resulted error dynamical system can be decomposed into two transversal components, one of which evolves along the consensus manifold, and the other evolves transversally with the consensus manifold. Several sufficient conditions for reaching almost surely second-order local consensus are derived for the cases of time-delay-free coupling and time-delay coupling, respectively. In the above results the time-delay coupling was not considered. If non-time-delay coupling and time-delay coupling terms exist at the same time, which play a more important role, then the second-order consensus problem of multi-agent systems is still an open problem. In this chapter, we systematically study the second-order global consensus problems in general multi-agent directed networks with both non-time-delay and time-delay coupling terms. Many consensus mathematical models under fixed topology can be seen as particular cases of our models. The Lyapunov directed method [46] is used to derive some delay-independent algebraic criteria for reaching the second-order global consensus. These criteria deeply reveal the underlying relations among the network interactive topologies, inner coupling matrices, and coupling gains to obtain the secondorder nonlinear consensus. Finally, we also provide a numerical simulation example to illustrate the feasibility and effectiveness of our theoretical results. The rest of this chapter is organized as follows. In Section 6.2, we provide some preliminaries. In Section 6.3, we present some delay-independent algebraic criteria for second-order global consensus. In Section 6.4, we give a numerical example to validate the theoretical analysis. Finally, in Section 6.5, we state some concluding remarks.

6.2. Preliminaries In this section, we provide some mathematical preliminaries, supporting lemmas, and definitions to derive the main results of this chapter.

6.2.1 Notation In this chapter, we use standard notations. Let I be the identity matrix of appropriate dimensions, and Rn and Rm×n denote, respectively, the n-dimensional real Euclidean space and the set of all m × n real matrices. We denote by e ∈ Rn the vector with all elements equal to one. For a real matrix A, by AT we denote its transpose. For a square matrix A, we denote its inverse by A−1 , and by λi (A) its ith eigenvalue in ascending order of the real part. For a real symmetric matrix X, we denote by λmin (X ) and λmax (X ) its minimum and maximum eigenvalues, respectively, and we write X > 0 (≥) or X < 0 (≤) if X is positive (semi-positive) or negative (semi-negative) definite. X  denotes the Euclidean norm of a vector X ∈ Rn or the corresponding induced norm for a matrix X ∈ Rm×n ; ⊗ denotes the Kronecker product [10]. For a real vector ω ∈ Rn , diag (ω) is

107

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Second-Order Consensus of Continuous-Time Multi-Agent Systems

the diagonal matrix whose element on the ith row and ith column is the ith component of ω. For two vectors x and y ∈ Rn , x⊥y represents their inner product, and x⊥ denotes the orthogonal complement space of all vectors orthogonal to x.

6.2.2 Graph theory The information exchange among the nodes in a complex network can be described by an interactive directed graph. Let G = {V , E, A} be a digraph in which V = {1, 2, . . . , N }   is the set of nodes, E ⊆ V × V is the set of edges, and A = aij N ×N with nonnegative   elements is called the adjacency matrix. A directed edge denoted by j, i means that node i can directly receive the information from node j. The elements of the adjacency matrix A are defined as follows: If there is a directed link from node j to node i = j, then aij > 0; otherwise aij = 0. In this chapter, we always assume that aii = 0 for all i ∈ V ,   that is, there are no loops in a digraph. The Laplacian matrix L = lij N ×N associated with the underlying interactive digraph or the adjacency matrix A is defined by lij =  N −aij ≤ 0 for i = j and lii = N j=1,j =i aij , which ensures that j=1 lii = 0. Generally speaking, the Laplacian matrix of a digraph is asymmetric. Note that an undirected graph is a particular case of digraph with aij = aji for all i, j = 1, 2, . . . , N. A directed path from   node j to node i is a sequence of edges j, i1 , (i1 , i2 ) , . . . , (im , i) in the digraph G with distinct nodes ik , k = 1, . . . m. A digraph is strongly connected if for any two distinct nodes j and i, there always exists a directed path from node j to node i. A digraph is a directed tree if there is a directed path from the root to every other vertex [41].

6.2.3 Supporting lemmas and definitions Lemma 6.1. [37] The Laplacian matrix L has a simple zero eigenvalue, and all the other eigenvalues have positive real parts if and only if the digraph associated with L has a directed spanning tree. Lemma 6.2. [14] For matrices A, B, C, and D of appropriate dimensions, we have the following conditions: (1) (γ A) ⊗ B = A ⊗ (γ B), where γ is a constant; (2) (A ⊗ B)T = AT ⊗ BT ; (3) (A ⊗ B) ⊗ C = A ⊗ C + B ⊗ C; (4) (A ⊗ B) (C ⊗ D) = (AC ) ⊗ (BD). Lemma 6.3. [13] Let A ∈ Rn×n be a symmetric matrix. We have λmin (A) xT x ≤ xT Ax ≤ λmax (A) xT x for all x ∈ Rn . Lemma 6.4. [46] For two real vectors x, y and a symmetric positive definite matrix K of suitable dimensions, we have xT y ≤ 1/2xT Kx + 1/2yT K −1 y.

Algebraic criteria for second-order global consensus in multi-agent networks

Lemma 6.5. [3] (Schur complement) The linear matrix inequality (LMI) 

Q (x) S (x) ST (x) R (x)

 > 0,

where Q (x) = QT (x) and R (x) = RT (x), is equivalent to each of the following conditions: (1) Q (x) > 0, R (x) − ST (x) Q−1 (x) S (x) > 0; (2) R (x) > 0, Q (x) − S (x) R−1 (x) ST (x) > 0. Definition 6.6. [46] W is the set of all real matrices with zero row sums and nonpositive off-diagonal elements; Ws is the set of all irreducible symmetric matrices in W . Definition 6.7. [46] For a digraph G, its Laplacian matrix is denoted by L. Define a (L ) = minx=1,x⊥e xT Lx. Remark 6.8. Under the conditions that the underlying interactive topology of L contains a directed spanning tree, e is an eigenvector associated with the zero eigenvalue of L, and all other nonzero eigenvalues have positive real parts [37]. We can construct an orthonormal basis K = [K1 , K2 , ..., KN ] for the space e⊥ , in which each vector  is orthogonal to e. Any vector π in e⊥ can be expressed as π = iN=−11 yi Ki , where yi , i = 1, 2, ..., N − 1, are real numbers. Moreover, π = 1 implies y = 1, where a(L ) = minx=1, x⊥e xT Lx = y = [y1 , y2 , ... , yN ]T . In this sense, we write π = Ky. Then  minKy=1 yT K T LKy = minKy=1 1/2yT K T LK + K T L T K y = 1/2λmin (K T LK + K T L T K ). Moreover, if the underlying interactive topology is undirected, then L is symmetric, and   a(L ) is equal to the smallest nonzero eigenvalue of 1/2 L + L T [46]. Definition 6.9. [46] For a strongly connected graph G, its Laplacian matrix L is an irreducible square matrix [53]. Let ω be the unique positive vector such that ωT L = 0 and maxi (ωi ) = 1 ,i ∈ V . The vector ω exists by Perron–Frobenius theory [28]. Let W = diag(ω). Then we define a3 (L ) = min

x =0, x⊥e



xT WLx T

.

ωω xT W −  x ωi i

Remark 6.10. Similarly, for an irreducible Laplacian matrix L, we can also construct an orthonormal basis K for the space e⊥ , in which each vector is orthogonal to e. Then a3 (L ) can be determined as follows: a3 (L ) = min

x =0, x⊥e



xT WLx T



ωω xT W −  x ωi i

= min

x =0, x⊥e

+ L T W )x ωωT W− x ωi

1 T 2 x (WL

xT

i

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Second-Order Consensus of Continuous-Time Multi-Agent Systems

= min

Ky =0

+ L T W )Ky . ωωT W− Ky ωi

1 T T 2 y K (WL

yT K T

i

Thus a3 (L ) can be solved by the following LMI:

a3 (L ) =

max δ

subject to

1 T 2 K (WL

 + L T W )K − δ K T W −

ωωT  i ωi



K ≥ 0.

6.3. Some delay-independent algebraic criteria for second-order global consensus Consider a linearly coupled complex network composed of N identical nodes, in which each node is a second-order n-dimensional dynamical system described by ⎧ ⎪ x˙ i (t) = vi (t), ⎪ ⎪ ⎪ N ⎪  ⎪ ⎪ v˙ i (t) = f (xi (t), vi (t), t) + α W1ij B1 (xj (t) − xi (t)) ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎨ N N   + α W1ij B1 (vj (t) − vi (t)) + β W2ij B2 (xj (t − τ ) − xi (t − τ )) ⎪ j=1 j=1 ⎪ ⎪ ⎪ N ⎪  ⎪ ⎪ + β W2ij B2 (vj (t − τ ) − vi (t − τ )), ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎩ i = 1, 2, ..., N , 

T



T

(6.1)

where xi (t) = x1i (t), x2i (t), ..., xni (t) ∈ Rn and vi (t) = vi1 (t), vi2 (t), ..., vin (t) ∈ Rn are the position and velocity state vectorsof the ith agent, respectively, f (xi (t), vi (t), t) =  T f1 (xi (t), vi (t), t), . . . , fn (xi (t), vi (t), t) : Rn × Rn × R+ → Rn represents a continuous but not necessarily differentiable vector-valued function, which models the inner     nonlinear dynamics of the ith agent, W1 = W1ij ∈ RN ×N and W2 = W2ij ∈ RN ×N are non-time-delay coupling and time-delay coupling adjacency matrices, respectively, B1 ∈ Rn×n and B2 ∈ Rn×n represent the inner non-time-delay coupling and time-delay coupling matrices, respectively, α > 0 and β > 0, respectively, stand for the position and velocity coupling strengths between any two agents in the network, and τ ≥ 0 specifies the coupling delay between agents resulted from the energy in the systems that propagates with finite speed. Note that in network (6.1) the non-time-delay coupling and time-delay coupling adjacency matrices W1 and W2 describe two classes of coupling topologies, whereas the matrices B1 and B2 describe the specific couplings between any two agents.   According to the graph theory, we can define the Laplacian matrices L1 = L1ij N ×N   and L2 = L2ij N ×N associated with the non-time-delay coupling and time-delay coupling adjacency matrices W1 and W2 , respectively. Then we equivalently get the fol-

Algebraic criteria for second-order global consensus in multi-agent networks

lowing coupled network: ⎧ x˙ i (t) = vi (t), ⎪ ⎪ ⎪ N ⎪  ⎪ ⎪ ⎪ v ( t ) = f ( x ( t ), v ( t ), t ) − α L1ij B1 (xj (t) + vj (t)) ˙ i i i ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

−β

N  j=1

j=1

(6.2)

L2ij B2 (xj (t − τ ) + vj (t − τ )),

i = 1, 2, ..., N .

Let x(t) = (x1 (t), x2 (t), ..., xN (t))T ∈ RN , v(t) = (v1 (t), v2 (t), ..., vN (t))T ∈ RN , and I ⊗ f (xi (t), vi (t), t) = f T (x1 (t), v1 (t), t), . . . , f T (xN (t), vN (t), t))T . The multi-agent network (6.2) can be recast in the following compact vector form: ⎧ ⎪ ⎨ x˙ (t) = v(t), v˙ (t) = I ⊗ f (xi (t), vi (t), t) − α(L1 ⊗ B1 )(x(t) + v(t)) ⎪ ⎩ − β(L2 ⊗ B2 )(x(t − τ ) + v(t − τ )).

(6.3)

We state our main results as follows. Theorem 6.11. The second-order nonlinear consensus in multi-agent network (6.3) can  xi (t) − xj (t) → 0 and be achieved globally in the sense that for all i, j ∈ V , i = j,   vi (t) − vj (t) → 0 as t → ∞ if there exist a matrix U ∈ Ws , a time-varying matrix P (t), three symmetric positive definite matrices satisfying A − BC −1 B > 0, and a matrix K > 0 of appropriate dimensions such that the following conditions are satisfied:  (i) xT (t)(U ⊗ B) I ⊗ f (xi (t), vi (t), t) + (I ⊗ P (t))x(t) < 0,   (ii) vT (t)(U ⊗ C ) I ⊗ f (xi (t), vi (t), t) + (I ⊗ P (t))v(t) < 0, (iii) R1 (t) = β 2 (UL2 ⊗ BB2 )T K −1 (UL2 ⊗ BB2 ) − α(U ⊗ B)(L1 ⊗ B1 ) −(U ⊗ B) (I ⊗ P (t)) + 2 (Q ⊗ I ) K (Q ⊗ I ) < 0, (iv) R2 (t) = −α(U ⊗ C )(L1 ⊗ B1 ) − (U ⊗ C ) (I ⊗ P (t))     T + 14 −α (UL1 ) ⊗ (BB1 ) + U ⊗ A − α L1T U ⊗ B1T C K −1      × −α (UL1 ) ⊗ (BB1 ) + U ⊗ A − α L1T U ⊗ B1T C +β 2 ((UL2 ) ⊗ (CB2 ))T K −1 ((UL2 ) ⊗ (CB2 )) +U ⊗ B + (Q ⊗ I ) K (Q ⊗ I ) < 0, where Q = I − 1/N (eeT ).

Proof. Construct the following Lyapunov function: V (x(t), v(t), t) = V1 (x(t), v(t), t) + V2 (x(t), v(t), t)  1 = xT (t), vT (t) Ω 2



x(t) v(t)



111

112

Second-Order Consensus of Continuous-Time Multi-Agent Systems

 +

t

t−τ



xT (s), vT (s)





Aτ 0

0 Cτ



x(s) v(s)



ds,

(6.4)





A B , U ∈ Ws , and A, B, and C are three symmetric positive where Ω = U ⊗ B C definite matrices of appropriate dimensions; two symmetric positive definite matrices Aτ and Cτ are will be determined later. By Schur complement (Lemma 6.5), the positive definiteness of the matrix Ω can be guaranteed by the conditions A > 0 and A − BC −1 B > 0. Calculating the time derivative of V1 (t) along the solutions of system (6.3) produces the following: V˙ 1 (x (t) , v (t) , t) = xT (t) (U ⊗ A) v (t) + vT (t) (U ⊗ B) v (t) + xT (t) (U ⊗ B) v˙ (t) + vT (t) (U ⊗ C ) v˙ (t) ,

(6.5)

where xT (t)(U ⊗ B)˙v(t) 

= x (t)(U ⊗ B) T

I ⊗ f (xi (t), vi (t), t) − α(L1 ⊗ B1 )(x(t) + v(t)) −β(L2 ⊗ B2 )(x(t − τ ) + v(t − τ ))



  = xT (t)(U ⊗ B) I ⊗ f (xi (t), vi (t), t)

− α xT (t)(U ⊗ B)(L1 ⊗ B1 )(x(t) + v(t)) − β xT (t)(U ⊗ B)(L2 ⊗ B2 )(x(t − τ ) + v(t − τ ))   = xT (t)(U ⊗ B) I ⊗ f (xi (t), vi (t), t) + (I ⊗ P (t))x(t) + xT (t) [−α(U ⊗ B)(L1 ⊗ B1 ) − (U ⊗ B) (I ⊗ P (t))] x(t) + xT (t) [−α(U ⊗ B)(L1 ⊗ B1 )] v(t) + xT (t) [−β(U ⊗ B)(L2 ⊗ B2 )] x(t − τ ) + xT (t) [−β(U ⊗ B)(L2 ⊗ B2 )] v(t − τ ),

(6.6)

and vT (t)(U ⊗ C )˙v(t) 

= v (t)(U ⊗ C ) T

I ⊗ f (xi (t), vi (t), t) − α(L1 ⊗ B1 )(x(t) + v(t)) −β(L2 ⊗ B2 )(x(t − τ ) + v(t − τ ))



  = vT (t)(U ⊗ C ) I ⊗ f (xi (t), vi (t), t) + (I ⊗ P (t))v(t) ) − vT (t) [(U ⊗ C )(IN ⊗ P (t))] v(t)

− α vT (t)(U ⊗ C )(L1 ⊗ B1 )x(t) − α vT (t)(U ⊗ C )(L1 ⊗ B1 )v(t) − β vT (t)(U ⊗ C )(L2 ⊗ B2 )x(t − τ ) − β vT (t)(U ⊗ C )(L2 ⊗ B2 )v(t − τ ).

(6.7)

Algebraic criteria for second-order global consensus in multi-agent networks

By Lemma 6.4 and the matrix factorizations − β(U ⊗ B)(L2 ⊗ B2 ) = (Q ⊗ I ) (−β(U ⊗ B)(L2 ⊗ B2 )) = H1 H2 , − β(U ⊗ C )(L2 ⊗ B2 ) = (Q ⊗ I ) (−β(U ⊗ C )(L2 ⊗ B2 )) = M1 M2 ,

(6.8a) (6.8b)

and   − α(U ⊗ B)(L1 ⊗ B1 ) + U ⊗ A − α L1T ⊗ B1T (U ⊗ C )    T   T  1 = 2 (Q ⊗ I ) −α (UL1 ) ⊗ (BB1 ) + U ⊗ A − α L1 U ⊗ B1 C

2

= 2J1 J2

(6.8c)

we can derive the following inequalities: 1 1 xT (t)H1 H2 x(t − τ ) ≤ xT (t)H1 KH1T x(t) + xT (t − τ )H2T K −1 H2 x(t − τ ), 2 2 1 T 1 T T T x (t)H1 H2 v(t − τ ) ≤ x (t)H1 KH1 x(t) + v (t − τ )H2T K −1 H2 v(t − τ ), 2 2 1 T 1 T T v (t)M1 M2 x(t − τ ) ≤ v (t)M1 KM1 v(t) + xT (t − τ )M2T K −1 M2 x(t − τ ), 2 2 1 T 1 T T T v (t)M1 M2 v(t − τ ) ≤ v (t)M1 KM1 v(t) + v (t − τ )M2T K −1 M2 v(t − τ ), 2 2 1 1 xT (t)J1 J2 v(t) ≤ xT (t)J1 KJ1T x(t) + vT (t)J2T K −1 J2 v(t). 2 2

(6.9a) (6.9b) (6.9c) (6.9d) (6.9e)

Moreover, the time derivative of V2 (t) is V˙ 2 (x(t), v(t), t) = xT (t)Aτ x(t) − xT (t − τ )Aτ x(t − τ ) + vT (t)Cτ v(t) − vT (t − τ )Cτ v(t − τ ).

(6.10)

If we choose Aτ = H2T K −1 H2 and Cτ = M2T K −1 M2 , then by some mathematical manipulations we get V˙ (x(t), v(t), t) ≤ xT (t)R1 x(t) + vT (t)R2 v(t).

(6.11)

If R1 < 0 and R2 < 0, then the trajectories (x(t), v(t)) will asymptotically approach the  set (x(t), v(t))  V˙ (t, x(t), v(t)) = 0 ; also, xT (t)R1 x(t) = 0

(6.12)

vT (t)R2 v(t) = 0

(6.13)

and

113

114

Second-Order Consensus of Continuous-Time Multi-Agent Systems

for all t > 0. Owing to the existence and uniqueness of solution of system (6.3), we know that x(t) = e ⊗ s1 (t) and v(t) = e ⊗ s2 (t) are the solutions of (6.12) and (6.13), respectively, because L1 e = 0, L2 e = 0, Qe = 0, and Ue = 0. Moreover, by substituting x(t) = e ⊗ s1 (t) and v(t) = e ⊗ s2 (t) into system (6.3) we also see that (s1 (t), s2 (t)) is a solution of an individual oscillator. The proof is thus completed. Theorem 6.11 is a little ambiguous and thus cannot provide us some clear underlying relations of the network topological structures L1 and L2 , inner coupling matrices B1 and B2 , the dynamics of an individual oscillator f (xi (t), vi (t), t), and the coupling gains α and β to reach the second-order global nonlinear consensus. We further will establish some easily manageable analytical criteria, which can clearly reveal the underlying relations of the network topological structures, inner coupling matrices, the dynamics of an individual oscillator, and the coupling gains for reaching the secondorder global consensus. Choose U = Q, P (t) = ηB1 , K = γ I, where η > 0 and γ > 0. Then Theorem 6.11 reduces to the following: Theorem 6.12. The second-order nonlinear consensus in multi-agent network (6.3) can be        achieved globally in the sense that xi (t) − xj (t) → 0 and vi (t) − vj (t) → 0 as t → ∞ if there exist three symmetric positive definite matrices A, B, C satisfying A − BC −1 B > 0 such that the following conditions are satisfied:   (i) xT (t)(Q ⊗ B) I ⊗ f (xi (t), vi (t), t) + η(I ⊗ B1 )x(t) < 0,   (ii) vT (t)(Q ⊗ C ) I ⊗ f (xi (t), vi (t), t) + η(I ⊗ B1 )v(t) < 0, 2 (iii) αηa(L1 ) + η > 3βQLBB2 +BBB TB  , λmin

(iv) αηa(L1 ) + η >

1

2

1

    B+3 max 12 −α QL1 ⊗BB1 +Q⊗A−α L1T Q⊗B1T C ,βQL2 CB2    . CB1 +B1T C λmin 2

Proof. The position solution space x(t) ∈ RN ×n of system (6.3) can be orthogonally decomposed into two components. One, denoted by xe (t), evolves along the position consensus manifold e ⊗ s1 (t), and the other, denoted by xe⊥ (t), evolves transversally with the position consensus manifold. Similarly, the velocity solution space v(t) ∈ RN ×n of system (6.3) can be also orthogonally decomposed into two components. One, denoted by ve (t), evolves along the velocity consensus manifold e ⊗ s2 (t), and the other, denoted by ve⊥ (t), evolves transversally with the velocity consensus manifold. Here s1 (t) ∈ Rn and s2 (t) ∈ Rn satisfy ˙s1 (t) = s2 (t), ˙s2 (t) = f (s1 (t), s2 (t), t). It is worth mentioning that once the T T   state vector xT (t), vT (t) of system (6.3) runs on a consensus manifold xTe (t), veT (t) at a certain time, the state of system (6.3) will not deviate the space generated by all the consensus manifold since then, because L1 e = 0 and L2 e = 0. Therefore we only need T  to find some analytical criteria under which all state vectors xT (t), vT (t) in space xe⊥ (t) × ve⊥ (t) will asymptotically approach zero.

Algebraic criteria for second-order global consensus in multi-agent networks

Consider U = Q, P = ηB1 , and K = γ I, where η > 0 and γ > 0. Denote by y a unit norm vector orthogonal to e and any unit norm vector z. We can derive the following: 







yT ⊗ zT ((−α QL1 − Q) ⊗ ηBB1 ) y ⊗ z

   = yT (−α QL1 − Q) y zT ηBB1 z    = −α yT QL1 y − yT Qy zT ηBB1 z    = −α yT L1 y − yT y zT ηBB1 z     BB1 + B1T B T T ≤ −α y L1 y − y y ηλmin 2  T    BB 1 + B1 B = −α yT a(L1 )y − yT y ηλmin 2   T BB1 + B1 B . ≤ [−α a(L1 ) − 1] ηλmin

(6.14)

2

In this derivation, we have used the equality Qy = y since y is an eigenvector associated with the eigenvalue one of matrix Q. In addition, we have 



















yT ⊗ zT (2γ Q ⊗ I ) y ⊗ z ≤ 2γ yT Qy zT z = 2γ yT y zT z = 2γ .

(6.15)

Furthermore,     β 2 γ −1 yT ⊗ zT (QL2 ⊗ BB2 )T (QL2 ⊗ BB2 ) y ⊗ z      = β 2 γ −1 yT ⊗ zT L2T Q ⊗ B2T B (QL2 ⊗ BB2 ) y ⊗ z    = β 2 γ −1 yT L2T Q2 L2 y zT B2T B2 B2 z ≤ β 2 γ −1 QL2 2 BB2 2 .

(6.16)

If we choose γ ≥ β QL2  BB2 , then we have that R1 is negative definite if αηa(L1 ) + η >

3β QL2  BB2  

λmin

BB1 +B1T B 2

.

(6.17)

By a similar proof as before we have that R2 is also negative definite if ⎫ ⎧ ⎡ ⎤   ⎪ ⎪ ⎬ ⎨  −α QL1 ⊗ BB1  ⎥ 1 ⎢ T T B + 3 max 2 ⎣ −α L1 Q ⊗ B1 C ⎦ , β QL2  CB2  ⎪ ⎪  ⎭ ⎩   +Q ⊗ A   . αηa(L1 ) + η > CB1 +B1T C λmin 2

This completes the proof.

(6.18)

115

116

Second-Order Consensus of Continuous-Time Multi-Agent Systems

Furthermore, if we select B = C = I and A = (1 + ε)I, where ε > 0, we get the following: Corollary 6.13. The second-order consensus in multi-agent network (6.3) can be achieved     globally in the sense that xi (t) − xj (t) → 0 and vi (t) − vj (t) → 0 as t → ∞ if for some ε > 0,   (i) xT (t)(Q ⊗ I ) I ⊗ f (xi (t), vi (t), t) + η(I ⊗ B1 )x(t) < 0,   (ii) vT (t)(Q ⊗ I ) I ⊗ f (xi (t), vi (t), t) + η(I ⊗ B1 ))v(t) < 0,  (iii) αηa(L1 ) + η >

1+3 max





1 −αηQL ⊗B +(1+ε)Q⊗I −αηL T Q⊗BT ,βQL B  1 1 2 2 1 1 2   B1 +B1T λmin 2

.

If the underlying network topologies associated with Laplacian matrix L1 and L2 are undirected, then QL1 = QL2 = 0, and a(L1 ) = λ2 (L1 ) (see [30]). We have the following: Corollary 6.14. The second-order consensus in multi-agent network (6.3) can be achieved     globally in the sense that xi (t) − xj (t) → 0 and vi (t) − vj (t) → 0 as t → ∞ if for some ε > 0,   (i) xT (t)(Q ⊗ I ) I ⊗ f (xi (t), vi (t), t) + η(I ⊗ B1 )x(t) < 0,   v ( t ), t ) + η( I ⊗ B )) v ( t ) < 0,  (ii) vT (t)(Q ⊗ I ) I ⊗ f (xi (t), i 1  (iii) αηλ2 (L1 ) + η >

1+3 max





1 −αηL ⊗B +BT +(1+ε)Q⊗I , βL B  1 1 2 2 1 2   B1 +B1T λmin 2

.

If the underlying network topology associated with Laplacian matrix L1 is strongly connected, by a proof similar to that of Corollary 6.14 we get the following main results. Theorem 6.15. The second-order consensus in multi-agent network (6.3) can be achieved   globally in the sense that xi (t) − xj (t) → 0 and vi (t) − vj (t) → 0 as t → ∞ if there exist three symmetric positive definite matrices A, B, C satisfying A − BC −1 B > 0 and the following conditions are satisfied:   (i) xT (t)(Q ⊗ B) I ⊗ f (xi (t), vi (t), t) + η(I ⊗ B1 )x(t) < 0,   (ii) vT (t)(Q ⊗ C ) I ⊗ f (xi (t), vi (t), t) + η(I ⊗ B1 )v(t) < 0, BB2   , (iii) αηa3 (L1 ) + η > 3βQL2  BB +BT B (iv) αηa3 (L1 ) + η > 

 

1 1 λ2 (U )λmin 2     1  B+3 max 2 −αηUL1 ⊗BB1 +U ⊗A−αηL1T U ⊗B1T C ,βUL2 CB2    , CB1 +B1T C λ2 (U )λmin 2

where U =

W − ωωT / i ωi and W = diag(ω) with the unique positive vector ω such that ωT L1 = 0 and maxi (ωi ) = 1, i ∈ V . Remark 6.16. Wu [46] studied synchronization in an array of coupled nonlinear systems with delay and nonreciprocal time-varying coupling. It is shown that the array synchronizes when the non-time-delay coupling term is cooperative and large enough. The

Algebraic criteria for second-order global consensus in multi-agent networks

obtained results in this chapter show that the results of [46] are no longer effective to reach the second-order global consensus. Therefore the extension of consensus algorithms for agents from first-order dynamics to second-order (when the control input is added to the driving force/acceleration term) is non-trivial.

6.4. Illustrative examples In this section, we give a simulation example demonstrating the potentials of our theoretical analysis. We investigate the following second-order nonlinear consensus of a multi-agent network composed of five coupled Lorenz systems: ⎧ x˙ i (t) = vi (t), ⎪ ⎪ ⎪ N ⎪  ⎪ ⎪ ⎪ ⎨ v˙ i (t) = f (vi (t), t) − α L1ij B1 (xj (t) + vj (t)) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

−β

N  j=1

j=1

(6.19)

L2ij B2 (xj (t − τ ) + vj (t − τ )),

i = 1, 2, ..., 5,



T



T

where xi (t) = x1i (t), x2i (t), x3i (t) , vi (t) = vi1 (t), vi2 (t), vi3 (t) the ith node. Furthermore, ⎛

⎞

σ (vi2 (t) − vi1 (t))

⎜

is the state variable of

⎟

f (vi (t), t) = ⎝ rvi1 (t) − vi1 (t)vi3 (t) − vi2 (t) ⎠ , vi1 (t)vi2 (t) − bvi3 (t)

(6.20)

where σ = 10, r = 28, and b = 8/3. We further validate the conditions of Theorem 6.12. We assume that the state variables of the coupled network still evolve in the attractive region as in an isolated Lorenz system. According to [41], we have that for i = 1, 2, ..., 5, 

2



vi2 (t) + vi3 (t) − r

2

≤

b2 r 2 . 4(b − 1)

(6.21)

Since 

vT (t)(Q ⊗ I ) I ⊗ f ( vi (t), t) + η(I ⊗ B1 )v(t) =−

( i 0. This is be0 l cause whenever agent i samples its state value or receives a new measurement state value from one of its neighbors, the controller with the latter scheme needs updating immediately, whereas the controller with the former version does not need to update the control input even though one of its neighbors completes the update. Our main purpose is designing a distributed event-triggering rule for agent i that analytically determines only on its local information when agent i has to be triggered to sample its current state and request new state measurements from its neighbors simultaneously such that the second-order leader-following consensus can be guaranteed. Before consensus is reached, for t ∈ [tki , tki +1 ), i = 1, . . . , N, we define the measurement errors exi (t), evi (t), exij (t) and evij (t) as exi (t) = xi (tki ) − xi (t), evi (t) = vi (tki ) − vi (t), and exij (t) = xj (tki ) − xj (t) if j ∈ Ni and exij (t) = 0 otherwise; evij (t) = vj (tki ) − vj (t) if j ∈ Ni and evij (t) = 0 otherwise, where i, j = 1, 2, . . . , N. We design the distributed event-triggering function for agent i as Ei (t) =(bi + di )exi (t) + evi (t) + Ai ˜exi (t) + e˜vi (t) + bi exi0 (t) + evi0 (t) − βi Hi (t) = 0, √

(7.4)



where Hi (t) = Hi1 (t) + Hi2 (t) with Hi1 (t) = j∈Ni \{0} [aij (xj (t) − xi (t))2 + aij (vj (t) − vi (t))2 ] and Hi2 (t) = bi [x0 (t) − xi (t)2 + v0 (t) − vi (t)2 ], βi > 0. Here Ai denotes the  T (t ), eT (t ), . . . , ith row of weighted adjacency matrix A, di = j∈Ni \{0} aij , e˜xi (t) = [exi1 xi2 T (t )]T , and e˜ (t ) = [eT (t ), eT (t ), . . . , eT (t )]T , i = 1, 2, . . . , N. exiN vi viN vi1 vi2

7.3.1 Fixed topology We are in the position to state our main results. Theorem 7.2. Consider the second-order leader-following system (7.1)–(7.2) with the distributed sampling control strategy (7.3) and the event-triggering sampling condition Ei (t) = 0. If λmin ((L + L T )/2 + B − IN ) > 2kβ h, then for any bounded initial condition xi (t0 ), vi (t0 ), ¯ the second-order consensus can be reached globally if and only if the communication topology G contains a directed spanning tree with the leader at the root. Moreover, the closed-loop system does not exhibit the Zeno behavior. Proof. Let ξi (t) = xi (t) − x0 (t), and ηi (t) = vi (t) − v0 (t), i = 1, 2, . . . , N. Define Aˆ = 2 diag{A1 , A2 , . . . , AN } ∈ RN ×N , D = diag{d1 , d2 , . . . , dN } ∈ RN , and T η(t) = [η1T (t), η2T (t), . . . , ηN (t)]T ∈ RN ×n ,

Event-triggering sampling-based leader-following consensus in second-order multi-agent systems

ξ(t) = [ξ1T (t), ξ2T (t), . . . , ξNT (t)]T ∈ RN ×n , T T T e˜x (t) = [˜ex1 (t), e˜x2 (t), . . . , e˜xN (t)]T ∈ RN

e˜v (t) ex (t) ev (t) ex0 (t) ev0 (t)

2 ×n

,

2 T T T = [˜ev1 (t), e˜v2 (t), . . . , e˜vN (t)]T ∈ RN ×n , T T T = [ex1 (t), ex2 (t), . . . , exN (t)]T ∈ RN ×n , T T T = [ev1 (t), ev2 (t), . . . , evN (t)]T ∈ RN ×n , T T T = [ex10 (t), ex20 (t), . . . , exN0 (t)]T ∈ RN ×n , T T T = [ev10 (t), ev20 (t), . . . , evN0 (t)]T ∈ RN ×n .

We have the following error dynamical system with compact matrix-vector form: ξ˙ (t) =η(t), η( ˙ t) = − [(L + B) ⊗ In ]ξ(t) − [(L + B) ⊗ In ]η(t) + (Aˆ ⊗ In ) × (˜ex (t) + e˜v (t)) − [(D + B) ⊗ In ](ex (t) + ev (t)) + (B ⊗ In )(ex0 (t) + ev0 (t)).

(7.5)

(Sufficiency) Suppose agent i samples the states of position and velocity at time instants t = tki , k ∈ Z0+ . Then the corresponding measurement errors exi (t), evi (t), exij (t), and evij (t) will be automatically reset to zeros. The distributed event-triggering sampling condition (7.4) thus enforces the following inequalities: (bi + di )exi (t) + evi (t) ≤ αi1 (t)βi Hi (t),

(7.6)

Ai ˜exi (t) + e˜vi (t) ≤ αi2 (t)βi Hi (t),

(7.7)

bi exi0 (t) + evi0 (t) ≤ αi3 (t)βi Hi (t)

(7.8)

and

for all t ≥ t0 , where αi1 (t), αi2 (t), αi3 (t) > 0, and αi1 (t) + αi2 (t) + αi3 (t) = 1. From the norm inequality p − q2 ≤ 2p2 + 2q2 for p, q ∈ Rn we have  2 (t )β 2 H 2 (t ) ≤ α 2 (t )β 2 [2a2 2 2 2 2 αi1 i i i iM i1 j∈Ni \{0} (ξj (t ) + ξi (t ) + ηj (t ) + ηi (t ) ) + 2 2 bM (ξi (t) +ηi (t) )], where aiM = max1≤j≤N {aij } and bM = max1≤i≤N {bi } = max1≤i≤N {bi }. N  2 2 2 2 ∗ 2 Therefore (bi + di )2m N i=1 exi (t )+ evi (t ) ≤ i=1 (bi + di ) exi (t )+ evi (t ) ≤ β [2(a ) (N − ∗ 2 2 N − 1) + bM ](ξ(t) + η(t) ), where (bi + di )m = min1≤i≤N {bi + di , bi + di > 0}, a∗ = max1≤i≤N {aiM }, and N ∗ = max1≤i≤N {|Ni |}. Thus we can derive the inequality B +Dex (t) + ev (t) ≤ B + D/(bi + di )m β h(ξ(t) + η(t)) for all t ≥ t0 , where h = 2(a∗ )2 N + 2(a∗ )2 (N ∗ − 1) + bM . By a similar analysis procedure, letting Am  = min1≤i≤N {Ai , Ai  > 0} and bm = min1≤i≤N {bi , bi > 0}, we get Aˆ ˜ex (t) + e˜v (t) ≤ Aˆ /Am β h(ξ(t) + η(t)) and Bex0 (t) + ev0 (t) ≤ bM /bm β h(ξ(t) + η(t)),

129

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Second-Order Consensus of Continuous-Time Multi-Agent Systems

where β = min1≤i≤N {βi }. Letting k = max{B + D/(bi + di )m , Aˆ /Am , bM /bm }, we get B + Dex (t) + ev (t) + Aˆ ˜ex (t) + e˜v (t) + Bex0 (t) + ev0 (t) ≤ k1 (ξ(t) + η(t)) for all t ≥ t0 , where k1 = kβ h. To analyze the asymptotical stability of the second-order closed-loop system (7.5), we construct the following Lyapunov function: 1 V (t) = yT (t) 2



(L T + L + 2B) ⊗ In

I N ×n

I N ×n I N ×n



y(t),

(7.9)

where y(t) = (ξ T (t), ηT (t))T . For y(t) = 0, the positive definiteness of V (t) can be guaranteed by λmin (L + L T + 2B − IN ) > 0. Calculating the time derivative of V (t) along the solution of (7.5), we have V˙ (t) = − ηT (t)[(L + B − IN ) ⊗ In ]η(t) − ξ T (t)[(L + B) ⊗ In ], × ξ(t) + ξ T (t)Q(t) + ηT (t)Q(t),

(7.10)

where Q(t) = (Aˆ ⊗ In )(˜ex (t) + e˜v (t)) − [(D + B) ⊗ In ](ex (t) + ev (t)) + (B ⊗ In )(ex0 (t) + ev0 (t)). Since ξ T (t)Q(t) ≤ 3k1 /2ξ(t)2 + k1 /2η(t)2 and η(t)T (t)Q(t) ≤ k1 /2ξ(t)2 + 3k1 /2η(t)2 , it follows from (7.10) that V˙ (t) ≤ − [λmin ((L + L T )/2 + B − IN ) − 2k1 ]η(t)2 − [λmin ((L + L T )/2 + B) − 2k1 ]ξ(t)2 .

(7.11)

If λmin ((L + L T )/2 + B − IN ) > 2k1 , then V˙ (t) ≤ 0 for all t ≥ t0 , that is, V˙ (t) ≤ 0 and V˙ (t) = 0 if and only if ξ(t) = 0 and η(t) = 0. The set M = {(ξ T (t), ηT (t))T |ξ(t) = η(t) = 0} is the largest invariant set contained in the set D = {(ξ T (t), ηT (t))T |V˙ (t) = 0} for (7.5). According to the LaSalle invariance principle [10], starting from any bounded initial condition in Rn , every solution of system (7.5) will approach the set M as t → ∞. Therefore the second-order leader-following consensus can be achieved under the distributed event-triggering sampling controller (7.3). (Necessity) Since the second-order leader-following consensus of multi-agent system (7.1)–(7.2) is achieved with the distributed control strategy (7.3), we have xi (t) → x0 (t) and vi (t) → v0 (t) as t → ∞ for i = 1, 2, . . . , N. So, by triggering condition (7.4) we get exi (t) → 0, evi (t) → 0, e˜xi (t) → 0, e˜vi (t) → 0, exi0 (t) → 0, and evi0 (t) → 0 as t → ∞. Hence (Aˆ ⊗ In )(˜ex (t) + e˜v (t)) − [(D + B) ⊗ In ](ex (t) + ev (t)) + (B ⊗ In )(  ex0 (t) + ev0 (t)) → 0 as ON IN ˜ t → ∞, that is, the real parts of all eigenvalues of the matrix L = −L − B −L − B are negative.  ˜ The characteristic equation of L˜ is 1≤i≤N (λ2 + λμi + Let λ be an eigenvalue of L. μi ) = 0, where μi are the eigenvalues of L + B, i = 1, 2, . . . , N. Hence λi1 = 12 (−μi +

Event-triggering sampling-based leader-following consensus in second-order multi-agent systems

  μ2i − 4μi ) and λi2 = 12 (−μi − μ2i − 4μi ). Let μ2i − 4μi = c + id, where c and d are real. Denote μi = Re(μi ) + iIm(μi ). It follows from Re(λ) < 0 that c 2 < Re2 (μi ). Moreover, by separating the real and imaginary parts we get Re2 (μi ) − Im2 (μi ) − 4Re(μi ) = c 2 − d2 and Re(μi )Im(μi ) − 2Im(μi ) = cd. By simple calculations we obtain c 4 − [Re2 (μi ) − Im2 (μi ) − 4Re(μi )]c 2 − [Re2 (μi ) × Im2 (μi ) + 4Im2 (μi ) − 4Re(μi )Im2 (μi )] = 0. Consequently, c 2 < Re2 (μi ) implies Re(μi )[2Re2 (μi ) + 3Im2 (μi )] > 2Re2 (μi ) + 2Im2 (μi ). So, Re(μi ) > 0, i = 1, 2, . . . , N. Namely, all eigenvalues of matrix L + B have strictly positive 

real parts. By Lemma 2 in [1] this implies that the underlying communication topology has a directed spanning tree with the leader at the root. It remains to show that the difference of inter-event time instants for all agents in G is lower bounded by a strictly positive constant. Assume that agent i triggers at time i i i i instants {tki }∞ k=0 , that is, exi (tk ) = 0, evi (tk ) = 0, exij (tk ) = 0, and evij (tk ) = 0. Suppose that ¯ the velocity and acceleration of all agents in the network G are respectively bounded by Mv > 0 and Mv˙ > 0. In each interval t ∈ [tki , tki +1 ), for agent i, we have exi (t) = t t t t  ti e˙xi (s)ds ≤ ti ˙exi (s)ds ≤ ti ˙xi (s)ds ≤ ti vi (s)ds ≤ Mv (t − tki ) for all t ∈ [tki , tki +1 ). k k k k Similarly, we get evi (t) ≤ Mv˙ (t − tki ), exij (t) ≤ Mv (t − tki ), and evij (t) ≤ Mv˙ (t − tki ) for all t ∈ [tki , tki +1 ), j ∈ Ni . Then we can derive that exi (t) + evi (t) ≤ (Mv + Mv˙ )(t − tki ), √ ˜exi (t) + e˜vi (t) ≤ |Ni |(Mv + Mv˙ )(t − tki ) and exi0 (t) + evi0 (t) ≤ (Mv + Mv˙ )(t − tki ) for all t ∈ [tki , tki +1 ). Consequently, (bi + di )exi (t) + evi (t) + Ai ˜exi (t) + e˜vi (t) + bi exi0 (t) + √ evi0 (t) ≤ (2bi + di + Ni Ai )(Mv + Mv˙ )(t − tki ). According to the event-triggering function (7.4), the next event will not be triggered until the trigger function Ei (t) = 0, that is, for agent i, the next sampling time instant t = tki +1 happens to be the moment when (bi + di )exi (t) + evi (t) + Ai ˜exi (t) + e˜vi (t) + bi ˜exi0 (t) + e˜vi0 (t) = βi Hi (t). Assume that before consensus is reached, there exists a positive constant M1 > 0 such i i i ∞ that βi Hi (t) ≥ M1 > 0 for some t ∈ {tli }∞ l=0 . Otherwise, Hi (tk ) = 0 for some tk ∈ {tl }l=0 , i and thus at the event time tk , the consensus has been achieved, and the event will not √ be necessarily triggered. Then we have (2bi + di + Ni Ai )(Mv + Mv˙ )(tki +1 − tki ) > M1 . Thus a lower bound on the inter-event intervals is given by τki = tki +1 − tki > M1 /{(2bi + √ di + Ni Ai )(Mv + Mv˙ )} > 0. This completes the proof of Theorem 7.2. Remark 7.3. In fact, the function of the event-triggering condition (7.4) is to guarantee that (bi + di )exi (t) + evi (t) + Ai ˜exi (t) + e˜vi (t) + bi exi0 (t) + evi0 (t) ≤ βi Hi (t) for all t ≥ t0 . For agent i, when the next sampling instant tki +1 comes, we have Ei (tki +1 ) = 0. The defined measurement errors will be automatically set to zeros. That is, for agent i, the event-triggering rule Ei (t) = 0 can guarantee the above inequality for all t ≥ t0 theoretically. However, due to the existence of numerical computation error, Ei (t) > 0 will appear with a certain probability, which may result in performance degradation of the designed distributed controller (7.3). In real applications, we may modify the sampling rule Ei (t) = 0 to be Ei (t) ≥ 0.

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The obtained results can be extended to the case where each agent is governed by a second-order nonlinear dynamics. Consider the following second-order nonlinear system of N agents: x˙ i (t) = vi (t), v˙ i (t) = f (xi (t), vi (t), t) + ui (t),

(7.12)

where i = 1, 2, . . . , N, and xi (t), vi (t), ui (t) ∈ Rn denote the position, velocity, and control input of agent i, respectively. The dynamics of the leader is expressed as x˙ 0 (t) = v0 (t), v˙ 0 (t) = f (x0 (t), v0 (t), t),

(7.13)

where x0 (t), v0 (t) ∈ Rn denote the position and velocity of the leader, respectively. Assumption 7.1. The nonlinear function f (xi (t), vi (t), t) = (f1 (xi (t), vi (t), t), f2 (xi (t), vi (t), t), . . . , fn (xi (t), vi (t), t)T ∈ Rn satisfies the Lipschitz conditions, that is, there exist constants kl > 0, l = 1, 2, . . . , n, such that fl (xi (t), vi (t), t) − fl (xj (t), vj (t), t) ≤ kl (xi (t) − xj (t) + vi (t) − vj (t)) for any xi (t), xj (t), vi (t), and vj (t) ∈ Rn , t ≥ t0 . Denote the diagonal matrix K = diag(k1 , k2 , . . . , kn ). We immediately have the following conclusion. Theorem 7.4. Consider a second-order leader-following nonlinear system (7.12)–(7.13) with distributed control strategy (7.3) and the event-triggering condition Ei (t) = 0. Assume that the nonlinear function satisfies Assumption 7.1 and the directed g¯ contains a directed spanning tree that takes the leader as the root. If λmin ((L + L T )/2 + B − IN ) > 2kβ h + K , then for any bounded initial conditions xi (t0 ), vi (t0 ) ∈ Rn , the second-order nonlinear leader-following consensus can be reached globally. Furthermore, the closed-loop system does not exhibit the Zeno behavior. Proof. The proof procedure is similar to that of Theorem 7.6 and is omitted here.

7.3.2 Switching communication topologies In this section, we investigate time-varying communication topologies. In particular, the topologies are allowed to switch at predefined instances of time. Denote the set of all possible graphs by Λ = {g1 , g2 , . . . , gm } and define the index set J = {1, 2, . . . , m}. The switches are described by the piecewise constant switching signal σ (t) : [t0 , +∞) → J, which defines the series of switching times t0 < t1 < t2 < · · · . Assume that the time interval between any two consecutive switches, called the dwell time, is lower bounded by a positive constant δ , that is, tl+1 − tl > δ for all l = 0, 1, . . .. Therefore the consensus error dynamics for the leader-following multi-agent system (7.1)–(7.2) with switching topologies is given by ξ˙ (t) =η(t),

Event-triggering sampling-based leader-following consensus in second-order multi-agent systems

η( ˙ t) = − [(L σ (t) + Bσ (t) ) ⊗ In ]ξ(t) − [(L σ (t) + Bσ (t) ) ⊗ In ] ˆ σ (t) ⊗ In )(˜ex (t) + e˜v (t)) − [(Dσ (t) + Bσ (t) ) × η(t) + (A ⊗ In ](ex (t) + ev (t)) + (Bσ (t) ⊗ In )(˜ex0 (t) + e˜v0 (t)).

(7.14)

The switching topology case poses some challenges for the event-triggering control strategy. First of all, note that each agent has to keep a copy of the latest measurement values, received from its neighbors, in its memory. To evaluate the distributed control input (7.4), each agent has to know its current neighbor set Niσ (t) , which depends on the current topology Gσ (t) . If the topology changes and agent i gets new neighbors, then it has no valid copy of these agents latest broadcasted values in its memory. To avoid such inconsistencies, it is necessary that each agent resets all stored measurement errors to zero at all switching instants, and accordingly, the corresponding measurement values have to be reset to the actual state values. Then the triggering condition has to be evaluated, and given it is fulfilled, new measurements have to be broadcasted. ¯ σ (t) uniformly jointly contains a directed spanDefinition 7.5. A time-varying graph G ning tree that takes the leader as the root if there exists a time horizon T > 0 such that ¯ σ (s) contains a directed spanning tree for all t ≥ t0 . the joint directed graph ∪s∈[t,t+T ) G

In this chapter, without explicit mention, we claim that a time-varying graph is jointly connected, which implies that it uniformly jointly contains a directed spanning tree. To see the asymptotical stability of the switching system (7.14), we consider an infinite sequence of nonempty bounded and contiguous time intervals [Tl , Tl+1 ), l = 0, 1, . . ., with T0 = t0 , Tl+1 − Tl ≥ T, where T is the same as that in Definition 7.5. Suppose that in each such interval [Tl , Tl+1 ), there is a sequence of non-overlapping subintervals [tl0 , tl1 ), . . ., [tlj , tlj+1 ), . . . , [tlml −1 , tlml ) satisfying tlj+1 − tlj ≥ δ , 0 ≤ j ≤ ml − 1. Here tlj , 0 ≤ j ≤ ml − 1, are all switching instants in the interval [Tl , Tl+1 ), l = 0, 1, . . .. Assumption 7.2. The directed graph ∪s∈[Tl ,Tl+1 ) g¯σ (s) uniformly jointly contains a directed spanning tree that takes the leader as the root across each interval [Tl , Tl+1 ), l = 0, 1, . . .. Theorem 7.6. Consider the second-order leader-following system (7.1)–(7.2) with distributed control strategy (7.3) and the event-triggering condition Ei (t) = 0, both in switching forms. Moreover, each agent in G samples at all switching instants even if the event-triggering condition is not satisfied. Assume that the switching signal σ (t) satisfies Assumption 7.2. Suppose that the parameter configuration satisfies ϑσ (s) = minl∈Z λmin (L˜ (Tl , Tl+1 )) > 2kβ h + 1, where L˜ (Tl , Tl+1 ) = ∪s∈[Tl ,Tl+1 ) {(L σ (s) + (L σ (s) )T )/2 + Bσ (s) }, l = 0, 1, . . .. Then for any bounded initial conditions xi (t0 ), vi (t0 ) ∈ Rn , the second-order leader-following consensus can be reached globally. Proof. Let δmin = minl∈Z {ϑσ (s) }. By a similar procedure as that for (7.9) we have V˙ (t) ≤ −δmin y(t) ≤ 0 for t ≥ t0 . Therefore from the criteria of limit we get that limt→∞ V (t)

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exists. Consider the infinite sequences V (Tl ), l = 0, 1, . . .. Using the Cauchy convergence criteria [2], we have that for any > 0, there exists a positive number M Tl +1 such that for all l ≥ M , |V (Tl+1 ) − V (Tl )| < , that is, Tl V˙ (t)dt < . This integral can be rewritten as the sum of integrals For each integral, we get

Tli+1 Tli

Tl1 Tl0

[−V˙ (t)]dt ≥ δmin

T 0 +δ > δmin [ T 0l y(t)dt + . . . + l

m −1 Tl l +δ m −1 Tl l

[−V˙ (t)]dt + . . . +

Tli+1 Tli

y(t)dt

Tlml

m −1

[−V˙ (t)]dt < .

Tl l Tli +δ ≥ δmin T i y(t)dt. l

Thus

y(t)dt]. Note that there are finite switches in

the interval [Tl , Tl+1 ) and the number ml is finite for each l = 0, 1, . . .. This implies that limt→∞ tt+δ [Mk y(t)]dt = 0 with Mk being a bounded positive constant. Invoking Barlabat’s lemma [2], we conclude that limt→∞ y(t) = 0. Thus, under the jointly connected switching topology, the second-order leader following consensus can be achieved under the distributed sampling controller (7.4). Consequently, there is no lower bound for the inter-event intervals, since the topology switching can lead to events at any time. However, since there is a positive lower bound δ on the dwell time, there can be no accumulation points in the sequences of event-times as long as there exists a positive lower bound on the inter-event times between the topology switching times. This completes the proof. For the second-order nonlinear case, we have the following results. Theorem 7.7. Consider the second-order leader-following system (7.12)–(7.13) with distributed control strategy (7.3) and the event-triggering condition Ei (t) = 0, both in switching forms. Moreover, each agent in G samples at all switching instants even if the eventtriggering condition is not satisfied. Assume that the nonlinear function satisfies Assumption 7.1 and the switching signal σ (t) satisfies Assumption 7.2. Suppose that the parameter configuration satisfies ϑσ (s) = minl∈Z λmin (L˜ (Tl , Tl+1 )) > 2kβ h + 1 + K , where L˜ (Tl , Tl+1 ) = ∪s∈[Tl ,Tl+1 ) {(L σ (s) + (L σ (s) )T )/2 + Bσ (s) }, l = 0, 1, . . .. Then for any bounded initial conditions xi (t0 ), vi (t0 ) ∈ Rn , the second-order leader-following nonlinear consensus can be reached globally. Proof. The proof procedure is similar to that of Theorem 7.6 and is omitted here.

7.4. Illustrative examples In this section, we give an example illustrating the validity of the theoretical results. Assume that there are five agents including a leader and four agent in the multi-agent system. The inherent nonlinear dynamics of each agent are f (vi (t), t) = 0.2 cos(t)vi (t), i = 0, 1, .⎛ . . , 4. We set the weighted adjacency matrix and leader adjacency matrix as ⎞ 0 0.27 0 0.22 ⎜ 0.44 0 0.2 0.42 ⎟ ⎜ ⎟ A=5×⎜ ⎟ and B = diag{0.45 0 0 0.475}. By computation 0 0 0.45 ⎠ ⎝ 0.45 0.32 0.25 0.16 0

Event-triggering sampling-based leader-following consensus in second-order multi-agent systems

Figure 7.1 The states of position and velocity of all agents.

we have bM = 4.75, bm = 4.5, a∗ = 2.25, (bi + di )m = 4.5, N ∗ = 3, h = 8.09, Am  = 1.74, k = 1.87, K  = 0.2, λmin ((L + L T )/2 + B − I4 ) = 2.15, and β = 0.034. Choosing βi = 0.03, i = 1, 2, 3, 4, the conditions in Theorem 7.2 are satisfied. The initial states of position and velocity are selected as ⎡

1.0 ⎢ [x0 (0) x1 (0) x2 (0) x3 (0) x4 (0)] = ⎣ 0.46 0.09

0.98 1.96 1.86

1.46 0.14 0.91

1.47 1.58 0.75

⎤

1.29 ⎥ 1.02 ⎦ 0.06

and ⎡

0.33 ⎢ [v0 (0) v1 (0) v2 (0) v3 (0) v4 (0)] = ⎣ 0.60 0.74

0.99 0.14 0.02

0.73 0.80 0.21

0.58 0.08 0.20

⎤

0.12 ⎥ 0.11 ⎦ . 0.05

We can see from Figs. 7.1(a) and 7.1(b) that the second-order consensus is achieved. Fig. 7.2(a) shows the distributed control input for each agent, which is piecewise constant, and Fig. 7.2(b) shows the individual event time instants. Let y1i (t) = (bi + di )exi (t) + evi (t) + Ai ˜exi (t) + e˜vi (t) + bi exi0 (t) + evi0 (t) and y2i (t) = βi Hi (t), i = 1, 2, 3, 4. For agents i = 1, 2, 3, 4, the two representative time response curves, which indicate the occurrence of sampling events, are shown in Figs. 7.3(a), 7.3(b), 7.3(c), and 7.3(d), respectively.

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Figure 7.2 (a) The distributed control input ui1 (t) of all follower agents. (b) Event time instants of all follower agents.

Figure 7.3 Time evolutions of y1i (t) and y2i (t), i = 1, 2, 3, 4.

7.5. Conclusion In this chapter, we studied the problem of the event-triggering asynchronous sampling based leader-following second-order consensus in multi-agent systems. Under the designed distributed event-triggering scheme, the asymptotical stability of the whole

Event-triggering sampling-based leader-following consensus in second-order multi-agent systems

networked system can be assured. The obtained results should be of great interest to practical control applications involving nodes with limited computation capability, limited capability of communication and actuation, and limited onboard energy source. The theoretical results were supported by simulated examples.

References [1] Q. Song, J. Cao, W. Yu, Second-order leader-following consensus of nonlinear multi-agent systems via pinning control, Systems & Control Letters 59 (2010) 553–562. [2] N. Wei, D. Cheng, Leader-following consensus of multi-agent systems under fixed and switching topologies, Systems & Control Letters 59 (2010) 209–217. [3] D. Dimarogonas, E. Frazzoli, K. Johansson, Distributed event-triggered control for multi-agent systems, IEEE Transactions on Automatic Control 57 (2012) 1291–1297. [4] Y. Fan, G. Feng, Y. Wand, C. Cheng, Distributed event-triggered control of multi-agent systems with combinational measurements, Automatica 42 (2013) 671–675. [5] W. Zhu, Z. Jiang, G. Feng, Event-based consensus of multi-agent systems with general linear models, Automatica 50 (2014) 552–558. [6] B. Shen, Z. Wang, X. Liu, Sampled-data synchronization control of complex dynamical networks with stochastic sampling, IEEE Transactions on Automatic Control 57 (2012) 2644–2650. [7] M. Mazo Jr., P. Tabuada, Decentralized event-triggered control over wireless sensor/actuator networks, IEEE Transactions on Automatic Control 56 (2011) 2456–2461. [8] A. Anta, P. Tabuada, To sample or not to sample: self-triggered control for nonlinear systems, IEEE Transactions on Automatic Control 55 (2010) 2030–2042. [9] C. Godsil, G. Royle, Algebraic Graph Theory, Springer, 2001. [10] H. Khalil, Nonlinear Systems, 3rd edition, Prentice Hall, 2002.

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CHAPTER 8

Consensus analysis of multi-agent systems with second-order nonlinear dynamics and general directed topology: an event-triggered scheme 8.1. Introduction Over the past few years, the distributed control of multi-agent systems has attracted great attention from various scientific communities due to its higher robustness, less communication cost, greater efficiency, and so on. Multi-agent systems can be found in many application areas, such as swarming and flocking [30,39], biological systems [36], teaming of multi-robotics [34,35], and control engineering [5,13,20–26,31,43,49–51]. An interesting and important issue arising from the distributed control of multi-agent systems is designing distributed protocols based only on the local and relative information so that the states of all agents reaching an agreement can be guaranteed. This is known as the distributed consensus problem. Consensus with a long history in computer science [5], especially in the field of distributed computing, is widely encountered in real-world applications and serves as a foundation for the study of collective behavior of multi-agent systems. The graph model is naturally applied, in which each agent is expressed as a vertex, and each communication is expressed as an edge between the corresponding agents (vertices). The consensus problem for agents with the first-order dynamics has recently been investigated from various perspectives [34,35]. Olfati-Saber et al. [31] presented a systematic framework to analyze the first-order consensus algorithms and showed that the consensus problem can be solved if a diagraph is strongly connected. Ren et al. [35] further proved that the first-order consensus can be achieved if the union of the dynamically changing interaction graphs has a directed spanning tree frequently enough as the system evolves. However, the second-order dynamics has recently received increasing attention due to many real-world applications, where mobile agents are governed by both position and velocity states [20,22–26,33,39,49]. Differently from the first-order consensus, Ren et al. [45] illustrated that the existence of a directed spanning tree is just a necessary condition rather than a sufficient one to reach the second-order consensus. In other words, the second-order consensus may not be achievable even if the interaction topology has a directed spanning tree. Therefore the extension of consensus Second-Order Consensus of Continuous-Time Multi-Agent Systems https://doi.org/10.1016/B978-0-32-390131-4.00015-7

Copyright © 2021 Elsevier Inc. All rights reserved.

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algorithms from the first-order to the second-order is non-trivial, and the second-order consensus problem is more complicated and challenging. Note that most of the mentioned works are concerned with the case where the agents are governed by integrator-type dynamics, that is, an agent without inherent nonlinear dynamics. Since all physical systems are nonlinear in nature, it is necessary to study the consensus problem of a multi-agent system having second-order oscillators, in which the dynamics of each agent is not only determined by the interactions among agents, but also by its own dynamics, that is, self-dynamics. For example, in reality, some oscillators, for example, harmonic oscillators [33,40] and pendulums [17], are governed by the second-order dynamics with the position and velocity terms. From this perspective, Yu et al. [46] investigated the second-order consensus problem in multi-agent systems with nonlinear dynamics and directed topology by using tools from algebraic graph theory and Lyapunov control approach. Some sufficient conditions were derived for reaching the second-order consensus with time-varying consensus velocities. It is important to note that most of the aforementioned works on the secondorder consensus problems in multi-agent systems are derived based on the assumption that information is transmitted continuously among multi-agents. The designed control law requires real-time updates, which promotes nodes to be equipped with high-performance processors and needs communication channels with high-speed data transmission capabilities. Therefore strategies based on continuous-time control will undoubtedly consume much more energy and largely restrict practical application with limited computing resources and network bandwidth. In practice, autonomous nodes such as mobile robots are often equipped with digital microprocessors that coordinate the data acquisition, communication with other nodes, and control actuation. Thus it is necessary to implement control laws on a digital platform. In other words, control laws can only be updated at discrete times. A commonly used approach in the present literature is time-scheduled (periodic) control. It may be conservative in terms of the number of control update, since the constant sampling period has to guarantee the stability and convergence of the resulting error closed-loop system in the worst-case scenario. Other limitations of time-scheduled control methods also include: (i) It is a detailed and restrictive design process, that is, the whole design process and its time specification must be known in advance; (ii) The communication and task scheduling on control units have to be synchronized during operation to ensure the strict time specification in the system design [1], therefore an efficient implementation for some special cases is impossible, and a typical example is multi-rate sampling [2]; (iii) In terms of flexibility and scalability, there exists deficiency in the designed architecture: A small change in one subsystem may generally imply an entire new system design, thus the system design itself becomes very complicated, and there is still a lack of adequate tool for design process [18]; (iv) Time-scheduled system also leads to lower resource utilization. In such systems, all schedules are fixed and planned in the specified operating mode. The timeslot assigned to a specific task must be at least satisfying the maximum execution time

Consensus analysis of multi-agent systems with second-order nonlinear dynamics and general directed topology

of this task. Therefore, even if many different operating modes have been considered, there are still substantial not fully utilized idle resources [18]. Differently from time-triggered implementation, in event-triggered control the information for measurement is not transmitted periodically in time, but triggered by the occurrence of certain event. Event-triggered control mechanism has many advantages in comparison with time-scheduled control methods. These advantages can be summarized as follows: (i) Event-triggered systems have the ability to rapidly react to asynchronous external events that are unknown in advance [1]; (ii) Event-triggered systems can easily modify an operative task to an existing node, since all scheduling and synchronization decisions are deferred activation of this task at run time, and thus event-triggered systems possess a higher flexibility and extensibility, which in many cases allow the adaptation to the actual demand without redesigning the complete system; (iii) In event-triggered systems, only those tasks that are activated under the actual circumstances have to be scheduled. Since the scheduling decision is made dynamically, the CPU will be available again after the actual (and not the maximum) task execution time. Therefore, if load conditions are low or average, then the resource utilization of an event-triggered system will be much better than that of the corresponding time-scheduled system; (iv) Event-triggered systems have a better implementation under actual circumstances. In [6], based on the deterministic event-triggered strategy introduced in [41], distributed consensus algorithms for the first-order multi-agent systems were put forward, and a lower bound for the inter-event time was provided to ensure that there was no Zeno behavior. Also, event-triggered control was addressed in networked control systems and wireless sensors/actuators networks in [28,42]. Fan et al. [8] proposed a basic event-triggered control algorithm for the distributed rendezvous problem of single-integrators using combinational measurements. Peng et al. [32] proposed an event-triggered strategy and control co-design for sampled-data control systems to determine whether or not the sampled data were transmitted. Many results with regard to event-triggered control were reported, such as H∞ filtering [11,47], consensus [7,12], and output tracking control for T-S fuzzy systems [48]. In [29], average consensus problems were developed based on distributed event-based algorithms with sampled-data event detection. The highlight of this paper is that the minimum inter-event time was naturally low bounded by the synchronous sampling period. In [50] the event-based consensus problem of general linear multi-agent systems was considered. Two sufficient conditions with or without continuous communication between neighboring agents were presented to guarantee the consensus. Event-triggered tracking control for heterogeneous multi-agent systems with Markov communication delays was discussed in [45]. Along with the same design framework given in [6], pertinent works on event-triggered cooperative consensus for multi-agent systems include [9,19,38,44]. Seyboth et al. [37] proposed a new event-triggered strategy in which each agent broadcasts its actual state

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to neighbors only when its own trigger condition was violated, and the threshold was a function of time. This approach has the decentralized feature and requires no continuous information of neighbors. However, nearly all the existing literature does not consider the inherent nonlinear dynamics. Thus current methods may not deal with the consensus of multi-agent systems with nonlinear dynamics. The objective of this chapter is investigating the consensus algorithms for networks of second-order agents with inherent nonlinear dynamics and directed topologies by means of event-triggered control strategies. The contributions of this chapter are three-fold. Firstly, taking inherent nonlinear dynamics and directed topology into consideration, we present a general model of second-order multi-agent systems with decentralized event-triggered control strategy. Unlike the time-triggered control, the updates of control input of all agents are asynchronous. Secondly, we design a distributed state-dependent sampling event for analytically determining the sampling time instant sequence. In addition, the continuous communication between agents is avoided. In the proposed framework, continuous-time feedback control can be seen as our particular case under the condition that the parameters in the event are set to zeros. The Zeno behavior can be excluded efficiently, and high-frequency sampling is avoided, thus reducing communication among the network and computing cost of the controllers. Thirdly, we perform a rigorously theoretical analysis for achieving consensus on a network, which is strongly connected or contains a directed spanning tree consisting of some strongly connected components, and derive some criteria for consensus. The event-based control technique proposed in this chapter can save energy and reduce the network load. The rest of the chapter is organized as follows. In Section 8.2, we briefly outline some preliminaries about model formulation and graph theory. In Sections 8.3–8.4, we give some consensus criteria on strongly connected networks, networks with a directed spanning tree. In Section 8.5, we simulated a representative network to illustrate the theoretical analysis. Conclusions are drawn in Section 8.6.

8.2. Preliminaries In this section, we introduced some basic notions and properties from graph theory. Let G = (V , E , G) be a weighted directed network of order N with the set of nodes (vertices) V = {v1 , v2 , . .. , vN }, the set of directed edges E ⊆ V × V , and a weighted adjacency matrix G = G . A directed edge in graph G is denoted by the ordered pairs  ij N ×N of nodes Eij = vi , vj , where vi and vj are called the terminal and initial nodes. If there is an edge from node vj to vi , then we say that node vj can reach node vi (node vi can Eij ; otherreceive information from node vj ). Gij > 0 is the weight associated with edge    wise, Gij = 0. Denote by Ni the set of neighbors of agent i, that is, Ni = j Gij > 0 . As usual, we assume that there are no self-loops or parallel edges in G . A graph G is strongly

Consensus analysis of multi-agent systems with second-order nonlinear dynamics and general directed topology

connected if for any two distinct nodes vi and vj in G , there exists a directed path from node vi to vj . A graph G is said to have a directed spanning tree if there is a node that can reach other nodes following the edge directions in graph G . In this chapter, we will use the following notations. For a vector ξ = (ξ1 , ξ2 , . . . , ξN )T ∈ RN , we denote by diag {ξ } the real diagonal matrix with element ξi in the ith row and ith column; · and ·1 represent the Euclidean norm and one-norm for a vector or a matrix, respectively. For a matrix A, AT and A−1 denote its transpose and inverse, respectively; In and On are the identity and zero matrices of dimension n, respectively; O is the zero matrix of appropriate dimension; 1N is the N-dimensional column vector with all elements being one; N is the set of nonnegative integers;. Rm×n represents the set of all m × n-dimensional real matrices; D+ stands for the Dini (upper ˜ and B˜ of the same order, A ˜ > B˜ (A ˜ ≥ B˜ ) means that right) derivative. For matrices A N × N ˜ − B˜ is a positive (semi-)definite matrix. A matrix G ∈ R A is nonnegative if ev  N ery entry Gij ≥ 0 1 ≤ i ≤ N , 1 ≤ j ≤ N , and a vector x ∈ R is positive if every entry xi ≥ 0 (1 ≤ i ≤ N ). The maximum and minimum eigenvalues of a matrix A are denoted by λmax (A) and λmin (A), respectively. For a complex number λ, Re (λ) denotes its real part. For two sets A and B, A ∩ B is the intersection set, and |A| is its cardinality (the number of its elements). For a vector x, xp denotes its norm, where p = 1, 2, ∞. Unless otherwise stated, x represents the Euclidean norm (p = 2). Definition 8.1. ([15]) The weighted adjacency matrix G in a directed (undirected) network G is reducible if there are a permutation matrix P ∈ RN ×N and an integer m with 1 ≤ m ≤ N − 1 such that 

P GP = T

˜ 11 G ˜ G21

O ˜ 22 G

 ,

˜ 11 ∈ Rm×n , G ˜ 21 ∈ R(N −m)×m , and G ˜ 22 ∈ R(N −m)×(N −m) . Otherwise, G is called an where G irreducible matrix.

Obviously, a nonzero matrix G of order 1 is irreducible. The next lemma shows a relation between an irreducible matrix and the corresponding strong connectivity in a network. Lemma 8.2. ([4], [15]) A matrix G is irreducible if and only if the corresponding its network G is strongly connected.

Lemma 8.2 is very easy to understand. For a strongly connected network, G ca not be permutated in the form of the matrix detailed in Definition 8.1, where the matrix O denotes that in the network there are no connections from one component to another. Therefore G is irreducible if the network is strongly connected. It is not difficult to justify the strongly connected networks.

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Second-Order Consensus of Continuous-Time Multi-Agent Systems

The commonly studied second-order protocol is described as follows [4,13,34,46]: x˙ i (t) = vi (t) , v˙ i (t) = α

N



N



Gij xj (t) − xi (t) + β

j=1.j =i





Gij vj (t) − vi (t) ,

(8.1)

j=1.j =i

where xi ∈ Rn and vi ∈ Rn are the position and velocity states of the ith agent, α > 0 and β > 0 are the coupling strengths for the position and velocity, and G = (Gij )N ×N is the coupling configuration matrix representing the topological structure of the network. The Laplacian matrix L = (Lij )N ×N is defined by Lii = −

N

Lij ,

(8.2)

j=1,j =i

Lij = −Gij ,

which ensures the diffusion property N j=1 Lij = 0. For an undirected network, its Laplacian matrix is positive semi-definite. However, in general, it may not be true for a directed network. When the network reaches the second-order consensus in (8.1), the

velocities of all agents converge to a constant vector described by N j=1 ξj vj (0), which depends only on the initial velocities of the agents, where ξ = (ξ1 , ξ2 , . . . , ξN )T is the nonnegative left eigenvector of L associated with eigenvalue zero satisfying ξ T 1N = 1 [34]. However, in most of the applications of multi-agent formulations, the velocity of each agent is generally not a constant but a time-varying variable. Therefore Yu et al. [46] considered the following second-order consensus protocol with time-varying velocities: x˙ i (t) =vi (t) , v˙ i (t) =f (xi (t) , vi (t) , t) + α

N



j=1,j =i

+β

N





Gij xj (t) − xi (t)

(8.3)



Gij vj (t) − vi (t) ,

j=1,j =i

where f : Rn × Rn × R+ → Rn is a continuously but not necessarily differentiable vector-valued function. Herein, if f is a differentiable function, then we can take f = −∇ U (x, v), where U (x, v) is a potential function, and then the multi-agent system (8.3) includes many popular swarming and flocking models [10,30] as particular cases.

Consensus analysis of multi-agent systems with second-order nonlinear dynamics and general directed topology

Definition 8.3. The multi-agent system (8.3) is said to achieve second-order consensus if for any initial conditions,   lim xi (t) − xj (t) = 0, t→∞   lim vi (t) − vj (t) = 0, i, j = 1, 2, . . . , N . t→∞

When the network reaches second-order consensus in (8.3), the positions and velocities of all agents globally converge to the weighted average of their current states, respectively, in the sense that ⎧ N

⎪ ⎪ ⎪ lim xi (t) = ξj xj (t), ⎪ ⎨ t→∞ j=1

(8.4)

N ⎪

⎪ ⎪ lim = v t ξj vj (t) , i = 1, 2, . . . , N . ( ) ⎪ i ⎩ t→∞ j=1

In this chapter, we consider an event-triggered protocol for second-order multiagent systems with nonlinear dynamics described as follows: x˙ i (t) = vi (t) , v˙ i (t) = f (xi (t) , vi (t) , t) + ui (t) = f (xi (t) , vi (t) , t) + α

N

 



j=1,j =i

+β

N

 

 

Gij xj tkj (t) − xi tki



(8.5)

 i 

Gij vj tkj (t) − vi tk ,

j=1,j =i





t ∈ [tki , tki +1 ), k (t) = arg min j t − tlj , k ∈ N. Δ

l∈N:t≥tl

Generally, the objective of this chapter is to find out whether the positions and velocities of all agents are still able to globally asymptotically converge to the same weighted average of their current states as described in (8.4) with less communication cost. If yes, how can we design the distributed sampling event with high performance for each agent? The following lemmas, assumptions, and definitions are very helpful for the main results. Lemma 8.4. ([14]) The Laplacian matrix L has a simple eigenvalue zero, and all the other eigenvalues have positive real parts if and only if the corresponding directed network G has a directed spanning tree. In other words, the eigenvalues of L satisfy 0 =λ1 (L ) < Re (λ2 (L )) ≤ · · · ≤ Re (λN (L )) .

145

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Second-Order Consensus of Continuous-Time Multi-Agent Systems

Lemma 8.5. ([35]) Suppose that L is irreducible. Then L1N = 0, and there is a positive vector ξ = (ξ1 , ξ1 , . . . , ξN )T such that ξ T L = 0. In addition, there exists a positive-definite diagonal matrix Ξ = diag(ξ1 , ξ1 , . . . , ξN ) such that Lˆ = (1/2)(Ξ L + L T Ξ ) is symmetric, and

N

N ˆ ˆ j=1 Lij = j=1 Lji = 0 for all i = 1, 2, . . . , N. For an undirected or balanced network, it is well known that 1N is not only the left eigenvector but also the right eigenvector of G associated with eigenvalue 0, that is, G1N = 0 and 1TN G = 0. However, for a general directed network, 1N is not the left eigenvector of G associated with eigenvalue 0. The left eigenvector ξ is very important in determining the final asymptotic states. Lemma 8.6. ([16]) Let γ ∈ R, and let A, B, C, D be matrices of appropriate dimensions. Then, for the Kronecker product ⊗, the following statements hold: (i) (γ A) ⊗ B = A ⊗ (γ B); (ii) (A + B) ⊗ C = A ⊗ C + B ⊗ C; (iii) (A ⊗ B) (C ⊗ D) = (AC ) ⊗ (BD); (iv) (A ⊗ B)T = AT ⊗ BT . Lemma 8.7. (Schur complement [3]) The linear matrix inequality 

Q (x) S (x)

S (x) R (x)

 > 0,

where Q (x) = Q(x)T and R (x) = R(x)T , is equivalent to one of the following conditions: (i) Q (x) > 0, R (x) − S(x)T Q(x)−1 S (x) > 0, (ii) R (x) > 0, Q (x) − S (x) R(x)−1 S(x)T > 0. Lemma 8.8. For a continuously differentiable vector x(t) ∈ Rn , we have the following inequality: d x (t)p ≤ x˙ (t)p dt for p ∈ {1, 2, ∞}. Proof. Using the properties of norms, we have: x (t + Δ)p − x (t)p d x (t)p = lim Δ→0 dt Δ x (t + Δ)p − x (t)p

≤ lim Δ→0 |Δ|    x t + Δ) − x (t)  (   = x˙ (t)p . =  lim  Δ→0 Δ p

Consensus analysis of multi-agent systems with second-order nonlinear dynamics and general directed topology

Assumption 8.1. ([46]) There exist nonnegative constants ρ1 and ρ2 such that for all x, y, z ∈ Rn and t ≥ 0,      f (x, v, t) − f y, z, t  ≤ ρ1 x − y + ρ2 v − z , f (0, 0, t) = 0.

Note that Assumption 8.1 is a Lipschitz condition satisfied by many well-known systems. Definition 8.9. System (8.5) does not exhibit the Zeno behavior if infk {tki +1 − tki } > 0 for all i. In other words, there is no trajectory of the system with infinite number of events in finite time. For all agents, define the position and velocity measurement errors:  

exi (t) = xi tki − xi (t) ,  

evi (t) = vi tki − vi (t) , t

(8.6)

∈ [tki , tki +1 ), i = 1, 2, . . . , N .

Thus system (8.5) can be rewritten as x˙ i (t) = vi (t) , v˙ i (t) = f (xi (t) , vi (t) , t) + α +β

t

N





N

j=1,j =i



Gij exj (t) + xj (t) − exi (t) − xi (t)

(8.7)



Gij evj (t) + vj (t) − evi (t) − vi (t) ,

j=1,j =i i i ∈ [tk , tk+1 ), k ∈ N.

By the definition of Laplacian matrix L, we further get x˙ i (t) = vi (t) , v˙ i (t) = f (xi (t) , vi (t) , t) + α

N

j=1





Lij exj (t) + xj (t) − β

N

j=1





Lij evj (t) + vj (t) ,

(8.8)

t ∈ [tki , tki +1 ), k ∈ N.

Define x¯ (t) = N ¯ (t) = k=1 ξk xk (t ) and v vi (t) − vˆ (t). Then we have

N

k=1 ξk vk (t ).

Let xˆ i (t) = xi (t) − x¯ (t), vˆ i (t) =

⎧ x˙ i (t) = vi (t) , ⎪ ⎪ ⎪ N ⎪

⎪ ⎪ v˙ i (t) = f (xi (t) , vi (t) , t) − ξk f (xk (t) , vk (t) , t) ⎪ ⎪ ⎪ j=1 ⎨ N N N N





−α Lij xj (t) − β Lij vj (t) − α Lij exj (t) − β Lij evj (t) ⎪ ⎪ ⎪ j=1 j=1 j=1 j=1 ⎪ ⎪ ⎪ N N N N ⎪    





⎪ ⎪ +α ξk Lkj exj (t) + xj (t) + β ξk Lkj evj (t) + vj (t) . ⎩ j=1

j=1

j=1

j=1

(8.9)

147

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Second-Order Consensus of Continuous-Time Multi-Agent Systems

Note that N

ξk

j=1

N





















Lkj exj (t) + xj (t) = ξ T L ⊗ In (ex (t) + x (t)) = 0

(8.10)

j=1

and N j=1

ξk

N

Lkj evj (t) + vj (t) = ξ T L ⊗ In (ev (t) + v (t)) = 0,

(8.11)

j=1



T





T (t ) , eT (t ) , . . . , eT (t ) , e (t ) = eT (t ) , eT (t ) , . . . , eT (t ) , x (t ) = where ex (t) = ex1 v xN xN v1 v2   T x2T  T (t ) . It follows from (8.9) x1 (t) , xT2 (t) , . . . , xTN (t) , and v (t) = v1T (t) , v2T (t) , . . . , vN that

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

x˙ˆ i (t) = vˆ i (t) , v˙ˆ i (t) = f (xi (t) , vi (t) , t) −

N

j=1

ξk f (xk (t) , vk (t) , t)

N N N N





−α Lij xj (t) − β Lij vj (t) − α Lij exj (t) − β Lij evj (t) ⎪ ⎪ ⎪ j=1 j=1 j=1 j=1 ⎪ ⎪ ⎪ N N N N ⎪    





⎪ ⎪ ⎩ +α ξk Lkj exj (t) + xj (t) + β ξk Lkj evj (t) + xvj (t) . j=1

j=1

j=1



j=1

Since N j=1 Lij = 0 for all i = 1, 2, . . . , N, we have Therefore we have ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

x˙ˆ i (t) = vˆ i (t) , v˙ˆ i (t) = f (xi (t) , vi (t) , t) −

N

j=1

(8.12)

N

¯ (t ) = 0 j=1 Lij x

and

N

¯ (t) = 0. j=1 Lij v

ξk f (xk (t) , vk (t) , t)

N N N N





− β − α − β L x ˆ t L v ˆ t L e t Lij evj (t) −α ( ) ( ) ( ) ⎪ ij j ij j ij xj ⎪ ⎪ j=1 j=1 j=1 j=1 ⎪ ⎪ ⎪ N N N N ⎪    





⎪ ⎪ ⎩ +α ξk Lkj exj (t) + xj (t) + β ξk Lkj evj (t) + xvj (t) . j=1



j=1

j=1

(8.13)

j=1

T





T (t ) , f ∗ (x (t ) , Letting xˆ (t) = xˆ T1 (t) , xˆ T2 (t) , . . . , xˆ TN (t) , vˆ (t) = vˆ 1T (t) , vˆ 2T (t) , . . . , vˆ N   T T v (t) , t)= f (x1 (t) , v1 (t) , t) , f T (x2 (t) , v2 (t) , t) , . . . , f T (xN (t) , vN (t) , t) , yˆ (t) = xˆ (t) ,    T T vˆ T (t) , and e (t) = exT (t) , evT (t) , we have









y˙ˆ (t) = F (x (t) , v (t) , t) + L˜ ⊗ In yˆ (t) + L ⊗ In e (t) ,

(8.14)

Consensus analysis of multi-agent systems with second-order nonlinear dynamics and general directed topology

where



F (x(t), v(t), t) =  L =

ON

ON

−α L

−β L



0 ((IN − 1N ξ T ) ⊗ In )f ∗ (x, v, t) 

 , L˜ =

ON −α L

IN −β L

 ,

.

Definition 8.10. ([46]) For a strongly connected network with Laplacian matrix L, the general algebraic connectivity is defined as aξ (L ) =

ˆ xT Lx , xT ξ =0, x =0 xT Ξ x min

where Lˆ = (Ξ L + L T Ξ )/2, Ξ = diag(ξ1 , ξ2 , . . . , ξN ), ξ = (ξ1 , ξ2 , . . . , ξN )T > 0, and

ξ T L = 0, N i=1 ξi = 1. Lemma 8.11. ([46]) The general connectivity aξ (L ) of a strongly connected network can be computed by the following optimization problem: 

aξ (L ) = 

ˆT

where Q = IN −1 , − ξξN

T

max δ,

subject to QT (Lˆ − δΞ )Q ≥ 0,

∈ RN ×(N −1) and ξˆ = (ξ1 , ξ2 , . . . , ξN −1 )T .

Remark 8.12. If Ξ = 1η IN and the network is undirected, then aξ (L ) = λ2 (L ). Definition 8.13. For a strongly connected network with Laplacian matrix L ∈ RN ×N and positive-definite matrix Γ ∈ Rn×n , the extended general algebraic connectivity is defined as a˜ ξ (L , Γ ) =

yT (Lˆ ⊗ Γ )y , yT (ξ ⊗1n )=0, y =0 yT (Ξ ⊗ Γ )y min

where Lˆ = (Ξ L + L T Ξ )/2, Ξ = diag(ξ1 , ξ2 , . . . , ξN ), ξ = (ξ1 , ξ2 , . . . , ξN )T > 0, ξ T L = 0,

and N i=1 ξi = 1. Lemma 8.14. For a strongly connected network with Laplacian matrix L ∈ RN ×N and positive definite matrix Γ ∈ Rn×n , we have a˜ ξ (L , Γ ) = aξ (L ). 

T

Proof. Letting Q = IN −1 −ξˆ T /ξN ∈ RN ×(N −1) and ξˆ T = (ξ1 , ξ2 , . . . , ξN −1 )T , note ∗ that Q = Q ⊗ In constructs a group of base vectors for the orthogonal subspace associated with the space spanned by the vector ξ ⊗ 1n . Thus by letting y = Q∗ z we have the

149

150

Second-Order Consensus of Continuous-Time Multi-Agent Systems

following formula: yT (Lˆ ⊗ Γ )y yT (ξ ⊗1n )=0, y =0 yT (Ξ ⊗ Γ )y

a˜ ξ (L , Γ ) =

min

= min z =0

= min

(Q∗ z)T (Lˆ ⊗ Γ )Q∗ z (Q∗ z)T (Ξ ⊗ Γ )Q∗ z zT (QT ⊗ In )(Lˆ ⊗ Γ )(Q ⊗ In )z

zT (QT ⊗ In )(Ξ ⊗ Γ )(Q ⊗ In )z ˆ ⊗ Γ )z zT (QT LQ = min T T . z =0 z (Q Ξ Q ⊗ Γ )z z =0

Since for any vector z ∈ RNn , z = 0, we have the inequality a˜ ξ (L , Γ ) ≤ which is equivalent to the following optimization problem: 

a˜ ξ (L , Γ ) =

max δ,



ˆ ⊗Γ )z zT (QT LQ , zT (QT Ξ Q⊗Γ )z



subject to QT (Lˆ − δΞ )Q ⊗ Γ ≥ 0.

Since Γ > 0, we have 

a˜ ξ (L , Γ ) =

max δ,

subject to QT (Lˆ − δΞ )Q ≥ 0.

From Lemma 8.11 we have a˜ ξ (L , Γ ) = aξ (L ) for all positive-definite matrices Γ . Lemma 8.15. ([46]) If the Laplacian matrix L is irreducible, then aξ (L ) > 0. Lemma 8.16. For two vectors x(t) = [xT1 (t), xT2 (t), . . . , xTN (t)]T and y(t) = [yT1 (t), yT2 (t), . . . , yT (t)]T ∈ RN ×n , we have the following properties: N

N x (t) = x(t)1 ; (i)    

iN=1 iN 1     (ii) i=1 j=1 xi (t )1 yj (t ) 1 = x(t )1 y(t ) 1 ; (iii) x(t)1 x(t)1 ≤ Nnx(t)2 = NnxT (t)x(t). Proof. These conclusions can be drawn from the definitions of norms.

8.3. Main results Theorem 8.17. Suppose that the network G of multi-agent system (8.5) is strongly connected and Assumption 8.1 holds. For agents i = 1, 2, . . . , N in the network, its next sampling time instant tki +1 is analytically determined by the following individual event: 



tki +1 = inf t > tki , Ei (x(tki ), v(tki ), xi (t), vi (t), t) = 0 ,

(8.15)

Consensus analysis of multi-agent systems with second-order nonlinear dynamics and general directed topology

where Ei (x(tki ), v(tki ), xi (t), vi (t), t) = exi (t)1 + evi (t)1 − ϕ1



       i   i j j x ( t ) − x ( t ) + v ( t ) − v ( t )    i k j k (t) i k j k (t)  j∈Ni 1

1

− ϕ2 exp(−γ (t − t0 ))

with t0i = t0 ≥ 0, i = 1, 2, . . . , N. Then second-order consensus in system (8.5) is achieved if 

aξ (L ) > max



2αρ1 + αρ2 + βρ1 χ1 α 2βρ2 + αρ2 + βρ1 χ2 + 2, 2 + + 2 , 2α 2 α β 2β 2 β

(8.16)

where  χ1 = α max{α, β}



+

 √ 2  Πij M φ2 Nn 2κ

 

λmin (Ξ ) 

χ2 = β max{α, β}



3Πij M φ1 Nn 2





3Πij M φ1 Nn 2

+

 

 √ 2  Πij M φ2 Nn 2κ

λmin (Ξ )

    Πij  = max (Ξ L )ij , γ > 0, φ1 = M 1≤i, j≤N

0 ≤ ϕ1
0, Ξ = diag(ξ1 , ξ2 , . . . , ξN ), 1≤i≤N NM N

ξ = (ξ1 , ξ2 , . . . , ξN )T > 0, ξ T L = 0 and

N

i=1 ξi

= 1. Moreover, the Zeno behavior can

be excluded. Remark 8.18. We can see from the sampling event (8.15) that the threshold of the event is independent on the agents’ neighbors’ position and velocity states xj (t) and vj (t), j ∈ Ni . This implies that the continuous communication for exchanging the agents’ information in the network can be effectively avoided. In addition, the main function of the exponential decay term ϕ2 exp(−γ (t − t0 )) is guaranteeing that there are no accumulated points for all agents in the network, which will be shown as follows. Proof. Consider the following Lyapunov function candidate: 1 V (t) = yˆ T (t) (Ω ⊗ In ) yˆ (t), 2

(8.17)

151

152

Second-Order Consensus of Continuous-Time Multi-Agent Systems









A B αβΞ L + αβ L T Ξ αΞ = where Ω = . We will show that V (t) ≥ 0 αΞ βΞ B C with V (t) = 0 if and only if yˆ (t) = 0. From the definition of aξ (L ) we have   1 V (t) = xˆ T (t) (2αβ Lˆ ) ⊗ In xˆ (t) + xˆ T (t)(αΞ ⊗ In )ˆv(t) 2 1 + vˆ T (t)(βΞ ⊗ In )ˆv(t) 2  1 T  ≥ xˆ (t) (2αβ aξ (L )Ξ ) ⊗ In xˆ (t) + xˆ T (t)(αΞ ⊗ In )ˆv(t) 2 1 + vˆ T (t)(βΞ ⊗ In )ˆv(t) 2    1 T 2αβ aξ (L )Ξ αΞ ⊗ In y(t) = y (t) αΞ βΞ 2   1 ˆ ⊗ In y(t). = yT (t) Q 2

(8.18)

ˆ > 0 is equivalent to βΞ > 0 and 2αβ aξ (L )Ξ − By Schur complement Lemma 8.7, Q −1 ˆ > 0. Consequently, αΞ (βΞ ) αΞ > 0. From (8.15) we have aξ (L ) > 2αβ 2 , and thus Q V (t) ≥ 0, and V (t) = 0 if and only if yˆ (t) = 0. Taking the time derivative of V (t) along the trajectories of (8.13) yields

1 V˙ (t) = 2



2xˆ T (t)(A ⊗ In )x˙ˆ (t) + 2x˙ˆ (t)(B ⊗ In )ˆv(t) +2xˆ T (t)(B ⊗ In )v˙ˆ (t) + 2vˆ T (t)(C ⊗ In )v˙ˆ (t) T



= xˆ T (t) [−α(B ⊗ In )(L ⊗ In )] xˆ (t) + xˆ T (t) [A ⊗ In − β(B ⊗ In )(L ⊗ In )] vˆ (t)   − xˆ T (t) α(L T ⊗ In )(C ⊗ In ) vˆ (t) + vˆ T (t) [(B ⊗ In ) − β(C ⊗ In )(L ⊗ In )] vˆ (t)   + xˆ T (t)(B ⊗ In ) ((IN − 1N ξ T ) ⊗ In )f (x(t), v(t), t)   + vˆ T (t)(C ⊗ In ) ((IN − 1N ξ T ) ⊗ In )f (x(t), v(t), t) − α xˆ T (t)(B ⊗ In )(L ⊗ In )ex (t) − β xˆ T (t)(B ⊗ In )(L ⊗ In )ev (t) − α vˆ T (t)(C ⊗ In )(L ⊗ In )ex (t) − β vˆ T (t)(C ⊗ In )(L ⊗ In )ev (t).

(8.19) In view of B = αΞ , we have 



xˆ T (t)(B ⊗ In ) ((IN − 1N ξ T ) ⊗ In )f (x(t), v(t), t)

  = α xˆ T (t) (Ξ ⊗ In ) f (x(t), v(t), t) − 1N ⊗ f (¯x(t), v¯ (t), t)     + α xˆ T (t) (Ξ ⊗ In ) 1N ⊗ f (¯x(t), v¯ (t), t) − (1N ξ T ) ⊗ In f (x(t), v(t), t) .

(8.20)

Consensus analysis of multi-agent systems with second-order nonlinear dynamics and general directed topology





Since 1TN Ξ = ξ T , 1TN ξ = 1, and xˆ (t) = (IN − 1N ξ T ) ⊗ In x(t), we have 

xˆ T (t) (Ξ ⊗ In ) 1N ⊗ f (¯x(t), v¯ (t), t)



    T = 1T x(t), v¯ (t), t) (Ξ ⊗ In ) (IN − 1N ξ T ) ⊗ In x(t) N ⊗ f (¯   = (ξ T − ξ T ) ⊗ f T (¯x(t), v¯ (t), t) x(t)

(8.21)

= 0. 



Due to Ξ 1N = ξ , ξ T 1N = 1, and xˆ (t) = (IN − 1N ξ T ) ⊗ In x(t), we get 



xˆ T (t) (Ξ ⊗ In ) (1N ξ T ) ⊗ In f (x(t), v(t), t) = 0.

(8.22)

From (8.20), (8.21), and (8.22) it follows that 



xˆ T (t)(B ⊗ In ) (( IN − 1N ξ T ) ⊗ In )f (x(t), v(t), t)  f ( x ( t ), v ( t ), t ) − 1 ⊗ f (¯ x ( t ), v ¯ ( t ), t )+ N   = α xˆ T (t) (Ξ ⊗ In ) 1N ⊗ f (¯x(t), v¯ (t), t) − (1N ξ T ) ⊗ In f (x(t), v(t), t)   = α xˆ T (t) (Ξ ⊗ In ) f (x(t), v(t), t) − 1N ⊗ f (¯x(t), v¯ (t), t)  

N = α i=1 (xi (t) − x¯ (t))T ξi f (xi (t), vi (t), t) − f (¯x(t), v¯ (t), t)  2 2   

ρ2  ≤α N xˆ i (t) + ρ22 vˆ i (t) . i=1 ξi ρ1 + 2

(8.23)

Similarly, we have 

vˆ T (t)(C ⊗ In ) ((IN − 1N ξ T ) ⊗ In )f (x(t), v(t), t) 

= β vˆ T (t)(Ξ

⊗ In )



f (x(t), v(t), t) − 1N ⊗ f (¯x(t), v¯ (t), t)+ 1N ⊗ f (¯x(t), v¯ (t), t) − (1N ξ T ) ⊗ In f (x(t), v(t), t)

 .

(8.24)

By the same method used to derive (8.21) and (8.22) we have 



vˆ T (t) (Ξ ⊗ In ) 1N ⊗ f (¯x(t), v¯ (t), t) = 0

(8.25)

and 



vˆ T (t) (Ξ ⊗ In ) (1N ξ T ) ⊗ In f (x(t), v(t), t) = 0.

(8.26)

By these two equalities it follows from (8.24) that 

vˆ T (t)(C ⊗ In ) ((IN − 1N ξ T ) ⊗ In )f (x(t), v(t), t) ≤β

  N 2  2  

ξi ρ21 xˆ i (t) + ρ21 + ρ2 vˆ i (t) .

i=1



(8.27)

153

154

Second-Order Consensus of Continuous-Time Multi-Agent Systems

From (8.19)–(8.27) we get the following formula: V˙ (t) ≤ xˆ T (t) [−α(B ⊗ In )(L ⊗ In )] xˆ (t) + xˆ T (t) [A ⊗ In − β(B ⊗ In )(L ⊗ In )] vˆ (t) −α xˆ T (t)(L T ⊗ In )(C ⊗ In )ˆv(t) + vˆ T (t) [(B ⊗ In ) − β(C ⊗ In )(L ⊗ In )] vˆ (t)  2 ρ  2  

ρ2  2   ξ + x ( t ) + v ( t ) ˆ +α N ˆ ρ 1 i i i=1 i 2 2  2  ρ    

ρ1   + 1 + ρ2 vˆ i (t)2 +β N ξ x ( t ) ˆ i i i=1 2 2 −α xˆ T (t)(B ⊗ In )(L ⊗ In )ex (t) − β xˆ T (t)(B ⊗ In )(L ⊗ In )ev (t) −αˆvT (t)(C ⊗ In )(L ⊗ In )ex (t) − β vˆ T (t)(C ⊗ In )(L ⊗ In )ev (t)      2 +βρ1 Ξ ⊗ I xˆ (t) = xˆ T (t) −α 2 (Ξ L ) + 2αρ1 +αρ n 2   +xˆ T (t) (A − αβΞ L − αβ L T Ξ ) ⊗ In vˆ (t)     2 +βρ1 Ξ ) ⊗ In vˆ (t) +ˆvT (t) (αΞ − β 2 Ξ L + 2βρ2 +αρ 2 −α 2 xˆ T (t) [(Ξ L ) ⊗ In ] ex (t) −αβ xˆ T (t) [(Ξ L ) ⊗ In ] ev (t) − αβ vˆ T (t) [(Ξ L ) ⊗ In ] ex (t) −β 2 vˆ T (t) [(Ξ L ) ⊗ In ] ev (t).

(8.28) Since A = αβΞ L + αβ L T Ξ , from (8.28) we have

  Ξ ⊗ In xˆ (t)     2 +βρ1 Ξ ) ⊗ I vˆ (t) +ˆvT (t) (αΞ − β 2 Ξ L + 2βρ2 +αρ n 2

V˙ (t) ≤ xˆ T (t)



−α 2 (Ξ L ) +

−α 2 xˆ T (t) [(Ξ L ) ⊗ I



2αρ1 +αρ2 +βρ1 2

n ] ex



(t) − αβ xˆ T (t) [(Ξ L ) ⊗ I

(8.29)

n ] ev (t )

−αβ vˆ T (t) [(Ξ L ) ⊗ In ] ex (t) − β 2 vˆ T (t) [(Ξ L ) ⊗ In ] ev (t).

Note the following equality: xˆ T (t)(ξ ⊗ 1n ) =

N

i=1

ξi xT i (t )1n −

N

i=1

ξi

N

k=1

ξk xT k (t )1n

(8.30)

= 0.

According to (8.13), if xˆ (t) = 0 and xˆ T (t)(ξ ⊗ 1n ) = 0, then for any positive definite matrix Γ , we have a˜ ξ (L , Γ ) ≤

xˆ T (t)(Lˆ ⊗ Γ )ˆx(t) , xˆ T (t)(Ξ ⊗ Γ )ˆx(t)

(8.31)

Consensus analysis of multi-agent systems with second-order nonlinear dynamics and general directed topology

that is, −xˆ T (t)(Lˆ ⊗ Γ )ˆx(t) ≤ −˜aξ (L , Γ )ˆxT (t)(Ξ ⊗ Γ )ˆx(t).

(8.32)

Since a˜ ξ (L , Γ ) = aξ (L ), by taking Γ = In we have −xˆ T (t)(Lˆ ⊗ In )ˆx(t) ≤ −aξ (L )ˆxT (t)(Ξ ⊗ In )ˆx(t).

(8.33)

Similarly, we can derive −ˆvT (t)(Lˆ ⊗ In )ˆv(t) ≤ −aξ (L )ˆvT (t)(Ξ ⊗ In )ˆv(t).

(8.34)

Using inequalities (8.33) and (8.34), from (8.29) we have    2 +βρ1 Ξ ⊗ ˆ (t) −α 2 aξ (L ) + 2αρ1 +αρ I n x 2    2 +βρ1 Ξ ⊗ I vˆ (t) +ˆvT (t) α − β 2 aξ (L ) + 2βρ2 +αρ n 2

V˙ (t) ≤ xˆ T (t)

(8.35)

−α 2 xˆ T (t) [(Ξ L ) ⊗ In ] ex (t) − αβ xˆ T (t) [(Ξ L ) ⊗ In ] ev (t) −αβ vˆ T (t) [(Ξ L ) ⊗ In ] ex (t) − β 2 vˆ T (t) [(Ξ L ) ⊗ In ] ev (t).

Considering the distributed sampling event (8.16), the position and velocity measurement errors exi (t) = xi (tki ) − xi (t) and evi (t) = vi (tki ) − vi (t) play an important role in designing the sampling condition. For t ∈ [tki , tki +1 ), k ∈ N, from the definitions of measurement errors exi (t) and evi (t) we can see that exi (tki ) and evi (tki ) will be reset to zeros at each sampling time instant tki . For each agent i in the network, the next sampling event will not be triggered until t → (tki +1 )− , that is, for agent i, the next sampling time instant tki +1 is just the moment when the term exi (t)1 + evi (t)1 touches the time-varying threshold ϕ1

         i j j xi (tk ) − xj (tk (t) ) + vi (tki ) − vj (tk (t) ) + ϕ2 exp(−γ ((tki +1 )− − t0 )). j∈Ni

1

1

For agent i, at the time instant t = (tki +1 )+ the measurement errors exi (t) and evi (t) would be set to zeros once again. Therefore the term will not exceed the time-varying thresh

old ϕ1 j∈Ni [||xi (tki ) − xj (tkj (t) )||1 + ||vi (tki )−vj (tkj (t) )||1 ] + ϕ2 exp(−γ ((tki +1 )− − t0 ) thereafter, that is, for all t ≥ t0 , we have exi (t)1 + evi (t)1     −

 i     i j j ≤ ϕ1 xi (tk ) − xj (tk (t) ) + vi (tki ) − vj (tk (t) ) + ϕ2 e−γ ((tk+1 ) −t0 ) j∈Ni

1

1

     ≤ ϕ1 NM (ex (t)1 + ev (t)1 ) + ϕ1 NM xˆ (t)1 + vˆ (t)1 + ϕ2 e−γ (t−t0 ) .

(8.36)

155

156

Second-Order Consensus of Continuous-Time Multi-Agent Systems

Summing from 1 to N, from (8.36) we get that ex (t)1 + ev (t)1 ≤ ϕ1 NM N (ex (t)1 + ev (t)1 )      + ϕ1 NM N xˆ (t)1 + vˆ (t)1 + ϕ2 Ne−γ (t−t0 ) ,

(8.37)

ex (t)1 + ev (t)1 ≤ ϕ1 NM N (ex (t)1 + ev (t)1 )      + ϕ1 NM N xˆ (t)1 + vˆ (t)1 + ϕ2 Ne−γ (t−t0 ) .

(8.38)

that is,

Using ex (t)1 ≤ ex (t) and ev (t)1 ≤ ev (t), we get     ϕ2 N ϕ1 NM N  xˆ (t) + vˆ (t) + e−γ (t−t0 ) 1 1 1 − ϕ1 NM N 1 − ϕ1 NM N      Δ = φ1 xˆ (t) + vˆ (t) + φ2 e−γ (t−t0 ) .

ex (t) + ev (t) ≤

1

(8.39)

1

Note that

  2 T −α xˆ (t)Ξ Lex (t) − αβ xˆ T (t)Ξ Lev (t) − αβ vˆ T (t)Ξ Lex (t) − β 2 vˆ T (t)Ξ Lev (t)    

N    N

≤ α max{α, β}  Πij xˆ T i (t ) exj (t ) + evj (t )   i=1 j=1   

N   N

T +β max{α, β}  Π vˆ (t) exj (t) + evj (t)  . i=1 j=1 ij i 

(8.40)

Furthermore, we have the following result:    

N    N

 Πij xˆ T i (t ) exj (t ) + evj (t )   i=1 j=1 ⎡ ⎣

≤

 

 

 

  Πij 

3Πij  φ1 Nn M + 2

φ

√

M 2 2κ

λmin (Ξ )   Πij  φ1 Nn M vˆ T (t)(Ξ + 2λmin (Ξ )

Nn

2 ⎤ ⎦

(8.41) xˆ T (t)(Ξ

⊗ In )ˆx(t)

⊗ In )ˆv(t) + κ 2 e−2γ (t−t0 ) .

Similarly, we can derive the following formula:    

N

N     Πij vˆ iT (t) exj (t) + evj (t)    i=1 j=1 ⎡ ⎣

≤

 

 

 

3Πij  φ1 Nn M + 2

  Πij 

φ

M 2 2κ

λmin (Ξ )   Πij  φ1 Nn M xˆ T (t)(Ξ + 2λmin (Ξ )

√

Nn

2 ⎤ ⎦

(8.42) vˆ T (t)(Ξ

⊗ In )ˆv(t)

⊗ In )ˆx(t) + κ 2 e−2γ (t−t0 ) .

Consensus analysis of multi-agent systems with second-order nonlinear dynamics and general directed topology

Therefore from (8.40)–(8.42) we have   2 T −α xˆ (t)Ξ Lex (t) − αβ xˆ T (t)Ξ Lev (t) − αβ vˆ T (t)Ξ Lex (t) − β 2 vˆ T (t)Ξ Lev (t) ⎡   ⎫ ⎧ 2 ⎤    √   3Π  φ Nn Πij  φ2 Nn ⎪ ⎪ M ⎣ ij M 1 + ⎦ ⎪ ⎪   ⎬ ⎨ 2 2κ Πij  φ1 Nn M + β max{α, β} ≤ α max{α, β} λmin (Ξ ) 2λmin (Ξ ) ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ ׈xT (t)(Ξ ⊗ In ⎡)ˆx(t) ⎫ ⎧   2 ⎤    √   3Π  φ Nn Πij  φ2 Nn ⎪ ⎪ M ⎣ ij M 1 + ⎦ ⎪ ⎪   ⎬ ⎨ 2 2κ Πij  φ1 Nn M + α max{α, β} + β max{α, β} λmin (Ξ ) 2λmin (Ξ ) ⎪ ⎪ ⎪ ⎪ ⎭ ⎩

(8.43)

׈vT (t)(Ξ ⊗ In )ˆv(t) + [max{α, β}]2 κ 2 e−2γ (t−t0 ) Δ

= χ1 xˆ T (t)(Ξ ⊗ In )ˆx(t) + χ2 vˆ T (t)(Ξ ⊗ In )ˆv(t) + ηe−2γ (t−t0 ) .

Therefore from (8.28) and (8.45) we have '

&



2αρ1 + αρ2 + βρ1 Ξ ⊗ In xˆ (t) −α aξ (L ) + 2 '  & 2βρ2 + αρ2 + βρ1 T 2 + vˆ (t) α − β aξ (L ) + Ξ ⊗ In vˆ (t) 2 + χ1 xˆ T (t)(Ξ ⊗ In )ˆx(t) + χ2 vˆ T (t)(Ξ ⊗ In )ˆv(t) + ηe−2γ (t−t0 )

V˙ (t) ≤ xˆ T (t)

2

(8.44)

  ≤ −ˆyT (t) Θ˜ ⊗ In yˆ (t) + ηe−2γ (t−t0 ) ,

where ⎛  Θ˜ = ⎝

α 2 aξ (L ) −

2αρ1 +αρ2 +βρ1 2

 − χ1 Ξ

⎞ 

0 

As long as aξ (L ) > max

2αρ1 +αρ2 +βρ1 2α 2

+

χ1 , α α2 β 2

˜ lecting 0 < K ≤ λmin (Θ)/λ max (Ω), we have

0 β 2 aξ (L ) − α −

+

2βρ2 +αρ2 +βρ1 2β 2

2βρ2 +αρ2 +βρ1 2

+

χ2 β2

⎠.  − χ2 Ξ



V˙ (t) ≤ −KV (t) + ηe−2γ (t−t0 ) .

, we have Θ˜ > 0. Se-

(8.45)

If 2γ = K, then we have &

V (t) ≤ V (t0 ) +

' η exp(−ζ (t − t0 )), |2γ − K |

(8.46)

157

158

Second-Order Consensus of Continuous-Time Multi-Agent Systems

where ζ = min{2γ , K } > 0. Choosing 

0 < μ2 < min β, 0 < μ1
0.

(8.48)

Therefore we get μ1 xˆ T (t)(Ξ ⊗ In )ˆx(t) + μ2 vˆ T (t)(Ξ ⊗ In )ˆv(t)   ≤ V (t0 ) + |2γ η−K | exp(−ζ (t − t0 )), t ≥ t0 ,

(8.49)

that is, −1 &  '     η xˆ (t)2 + vˆ (t)2 ≤ min {μ1 , μ2 } V (t0 ) + exp(−ζ (t − t0 )), t ≥ t0 . |2γ − K | λmin (Ξ )

(8.50) Thus the second-order consensus in multi-agent systems (8.3) is achieved asymptotically. Next, we will show that the Zeno behavior is excluded, that is, for agents i = 1, 2, . . . , N , tki +1 − tki > 0 for all k ∈ N. For t ∈ [tki , tki +1 ), letting ui (tki , tkj (t) ) =

N j j i i α N j=1, j =i Gij (xj (tk (t) ) − xi (tk ))+β j=1, j =i Gij (vj (tk (t) ) − vi (tk )) and ρ = max {ρ1 , 1 + ρ2 }, we have the following formula: D+ (exi (t)1 + evi (t)1 ) ≤ ˙exi (t)1 + ˙evi (t)1 ≤ max {ρ1 , 1 + ρ2 } [exi (t)1 + evi (t)1 ]         j +ρ1 xi (tki )1 + (1 + ρ2 )vi (tki )1 + ui (tki , tk (t) ) 1  i    = ρ [exi (t)1 + evi (t)1 ] + ρ1 xi (tk ) 1       j +(1 + ρ2 )vi (tki )1 + ui (tki , tk (t) )

(8.51)

1









with exi (tki ) = 0 and evi (tki ) = 0. Let χ (tki , tkj (t) ) = ρ1 xi (tki )1 + (1 + ρ2 )vi (tki )1 +    i j  ui (tk , tk (t) ) . Integrating from tki to t yields 1

j

exi (t)1 + evi (t)1 ≤

χ (tki , tk (t) ) ρ

j

exp(ρ(t − tki )) −

χ (tki , tk (t) ) ρ

, t ∈ [tki , tki +1 ).

(8.52)

Consensus analysis of multi-agent systems with second-order nonlinear dynamics and general directed topology

Taking t = tki−+1 , we have j

j

χ (tki , tk (t) ) χ (tki , tk (t) )     i i exi (ti− ) + evi (ti− ) ≤ t − t )) − exp(ρ( . k+1 k k+1 1 k+1 1 ρ ρ

(8.53)

Note that for the next sampling time instant tki +1 , we have the equality     exi (ti− ) + evi (ti− ) k+1 1 k+1 1   

 i     i j j = ϕ1 xi (tk ) − xj (tk (t) ) + vi (tki ) − vj (tk (t) ) + ϕ2 e−γ (tk+1 −t0 ) .

(8.54)

j∈Ni

Combining (8.51) and (8.52), we have ϕ1

  

 i i i     i j j xi (tk ) − xj (tk (t) ) + vi (tki ) − vj (tk (t) ) + ϕ2 e−γ (tk −t0 ) + ϕ2 e−γ (tk+1 −tk )

j∈Ni j

≤

χ (tki , tk (t) ) ρ

(8.55)

j

exp(ρ(tki +1

χ (t i , t ) − tki )) − k ρ k (t) .

We will show that there is no accommodation point in the sampling time instant sequence, that is, limk→∞ tki = ∞. If not, suppose that limk→∞ tki = ϑ < ∞. Then j

ϕ2 e−γ (ϑ−t0 ) + ϕ2 ≤

χ (tki , tk (t) ) ρ

j

−

χ (tki , tk (t) ) ρ

= 0,

(8.56)

which implies e−γ (ϑ−t0 ) + 1 ≤ 0, a contradiction. Thus we have limk→∞ tki = ∞ for i = 1, 2, . . . , N. Assume that there exists k ∈ N such that Δtki = tki +1 − tki → 0. Then ϕ1

  

 i     i j j xi (tk ) − xj (tk (t) ) + vi (tki ) − vj (tk (t) ) + ϕ2 e−γ (tk −t0 ) + ϕ2

j∈Ni j

≤

χ (tki , tk (t) ) ρ

(8.57)

j

−

χ (tki , tk (t) ) ρ

= 0.

This means that ϕ1 = ϕ2 = 0, which contradicts with ϕ1 > 0 and ϕ2 > 0. Consequently, the Zeno behavior is excluded for all agents during the sampling process, that is, tki +1 − tki > 0 for all k ∈ N and i = 1, 2, . . . , N. The proof of Theorem 8.17 is completed. Remark 8.19. (i) Inequality (8.16) holds as long as relatively large coupling strengths α and β are selected. (ii) Suppose that the network is undirected and Assumption 8.1 holds. Then condition (8.16) can be modified as 



2αρ1 + αρ2 + βρ1 χ1 α 2βρ2 + αρ2 + βρ1 χ2 λ2 (L ) > max + 2, 2 + + 2 . 2α 2 α β 2β 2 β

159

160

Second-Order Consensus of Continuous-Time Multi-Agent Systems

(iii) When ϕ1 = 0 and ϕ2 = 0, the threshold of the sampling event is equal to zero, which implies that the sampling will occur for all t ≥ t0 . That is, the multi-agent system (8.5) with distributed event-triggered sampling mechanism reduces to the multi-agent system (8.3) with continuous-time information interaction. Therefore the results obtained in [46] can be seen as a particular case of our results. Remark 8.20. The designed event in this chapter can exclude the Zeno behavior for all agents during the whole running process of the system, even after the consensus has been achieved. This is a great improvement of the previous work, which can only guarantee that there is no Zeno behavior for at least one agent before the consensus is reached [6,8]. In addition, it is also an enhancement of the work [50], which could not reach precise consensus by adding a constant term to avoid the Zeno behavior.

8.4. Second-order consensus in networks containing a directed spanning tree Note that (8.17) is obtained by assuming that the network is strongly connected. It is well known that the assumption is stronger than that the network contains a directed spanning tree. Next, we will consider the consensus problem of multi-agent system (8.3) that has a directed spanning tree. Let the Laplacian matrix L of graph G be written in the Frobenius normal form [4]: ⎛

L¯ 11 ⎜ ⎜ O .

.. .

... ... .. .

O

O

...

P T LP = L¯ = ⎜ ⎜ .. ⎝

L¯ 21 L¯ 22

⎞

L¯ 1p L¯ 2p ⎟ ⎟ L¯ pp

.. .

⎟, ⎟ ⎠

(8.58)

where P is a permutation matrix, L¯ kk ∈ Rmk ×mk are irreducible matrices for all k = 1, 2, . . . , p, which are uniquely determined to within simultaneous permutation of ¯ 1, G ¯ 2, . . . , G ¯ p be the their lines, but their ordering is not necessarily unique. Let G strongly connected components of the underlying graph G of the directed network (8.3) with the connected matrices L¯ 11 , L¯ 22 , . . . , L¯ pp . The matrix L¯ can be interpreted as follows: The nodes and their adjacency edges in L¯ kk constitute an irreducible subgraph of the network, and L¯ kj (j > k) represents the influence of subgraph L¯ jj to L¯ kk . Denote by Si (1 ≤ i ≤ p) the set of nodes associated with the connected matrix L¯ ii .

k k ¯ ¯k +D ¯k ¯ k , where m Let L¯ kk = A j=1 Aij = 0 for i = 1, 2, . . . , mk , and D is a diagonal ¯ k ≥ 0 and D ¯ k = 0 for all k = 1, 2, . . . , p − 1. We can change matrix. We can see that D the order of the node indexes to obtain the Frobenius normal form (8.58). Without loss of generality, we assume that the adjacency matrix G of graph is already in the Frobenius ¯ normal form, that is, L = L.

Consensus analysis of multi-agent systems with second-order nonlinear dynamics and general directed topology

Definition 8.21. ([46]) For a network containing a directed spanning tree and the Laplacian matrix in the form (8.58), the general algebraic connectivity of the ith strongly  connected component 1 ≤ i ≤ p − 1 is defined by . T . −1 . −1 .  Ξ¯ i x Ξ¯ i Lˆ¯ ii Ξ¯ i Ξ¯ i x . T . Ξ¯ i x Ξ¯ i x  . −1 . −1 / −1 / −1  yT Ξ¯ i Lˆ¯ ii Ξ¯ i y , Ξ¯ L¯ˆ Ξ¯ = min =λ

xT Lˆ¯ ii x = min bξ¯i (L¯ ii ) = min T ¯ i x x =0 x =0 x Ξ

y =0

yT y

i

min

where Lˆ¯ ii = (Ξ¯ i L¯ ii + L¯ iiT Ξ¯ i )/2, Ξ¯ i = diag(ξ¯i1 , . . . , ξ¯imi ),

¯ i = 0 and mi ξ¯ij = 1. ξ¯i = (ξ¯i1 , . . . , ξ¯imi )T with ξ¯iT A j=1

ii

i

/ . . Ξ¯ i = diag( ξ¯i1 , . . . , ξ¯imi ), and

Lemma 8.22. ([46]) If the graph   G with Laplacian matrix L has a directed spanning tree, then ¯ ¯ min1≤q≤p−1 aξ¯p (Lpp ), bξ¯q (Lqq ) > 0.

Let Nq = qk=1 mk . Then the multi-agent system (8.3) can be decomposed into q subsystems denoted by the index set Sq = {Nq−1 + 1, Nq−1 + 2, . . . , Nq }, q = 1, 2, . . . , p, with N0 = 0. The dynamics of agents in the set Sq (q = 1, 2, . . . , p − 1) can be described by ⎧ ⎪ x˙ i (t) = vi (t) ⎪ ⎪

⎪ ⎪ v˙ i (t) = f (xi (t), vi (t), t) + α Gij (xj (tkj (t) ) − xi (tki )) ⎪ ⎪ ⎪ j∈Sp ⎪ ⎪

⎨ j G ( v ( t ) − vi (tki )) +β ij j k (t) Sp : j∈Sp ⎪   ⎪ ⎪ Δ j i ⎪ ⎪ t ∈ [ t , tki +1 ), k (t) = arg min t − tl , ⎪ k ⎪ j ⎪ l∈N: t≥tl ⎪ ⎪ ⎩ k ∈ N, i = Np−1 + 1, Np−1 + 2, . . . , Np ,

(8.59)

and the dynamics of agents in the set can be described by ⎧ x˙ i (t) = vi (t), ⎪ ⎪

⎪ ⎪ ⎪ x Gij (xj (tkj (t) ) − xi (tki )) ˙ i (t) = f (xi (t), vi (t), t) + α ⎪ ⎪ ⎪ j∈Sq ⎪ ⎪

j ⎪ ⎪ G ( v ( t ) − vi (tki )) +β ⎪ ij j k (t) ⎪ ⎪ j ∈Sq ⎪ ⎪



⎨ Gij (xj (tkj (t) ) − xi (tki )) +α Sq : q 0, ϕ2 > 0, and β > 0, and t0 = t0i , i = 1, 2, . . . , N, is the initial time. For all 1 ≤ q ≤ p − 1, the general algebraic connectivity degree satisfies the following inequalities: ⎧   q q   η η q q 1 2 1 ⎪ ⎨ bξ¯q L¯ qq > max ρα1 + αρ22+βρ + α12 − ψm , α+βρ + αρ22+βρ + β22 − ψm , α2 β2 β2  p p ⎪ ⎩ aξ¯ (L¯ pp ) > max 2αρ1 +αρ22 +βρ1 + χ12 , α2 + 2βρ2 +αρ22 +βρ1 + χ22 . p 2α α β 2β β

(8.65)

The second-order is achievable globally, where the chosen positive vector in aξ¯p (L¯ pp ) sat m isfies ξ¯pT L¯ pp = 0, i=p1 ξ¯pi = 1, ξ¯p = (ξ¯p1 , ξ˙p2 , . . . , ξ¯pmp )T > 0, and the vector ξ¯q in bξ¯q (L¯ qq )

¯ q = 0, mq ξ¯qi = 1, ξ¯q = (ξ¯q1 , ξ˙q2 , . . . , ξ¯qmq )T > 0, 1 ≤ q ≤ p − 1; satisfies ξ¯qT A i=1 ⎡ ⎣

   p

p

M

2

p

χ1 = α max{α, β} ⎡ ⎣ p

χ2 = β max{α, β}

   p

p

λmin (Ξ¯ p )   

3Πij  φ1 Sp n M

2

+

+

⎤

2 /  p    p Πij  φ1 Sp n M ⎦ 2κ





3Πij  φ1 Sp n

+ β max{α, β} ⎤

2 / p    p Πij  φ2 Sp n M ⎦ 2κ

   p  p   Πij  φ1 Sp n M

2λmin (Ξ¯ p )    p  p   Πij  φ1 Sp n

,

M + α max{α, β} , ¯ λmin (Ξ¯ p ) 2 λ min (Ξp )          ϕ1 SpM  Sp     p p ¯ ¯     , SpM  = max {|Ni |} , γ > 0, Πij  = max  (Ξp Lpp )ij , φ1 =   i∈Sp M 1≤i, j≤Sp  1 − ϕ1 SpM  Sp  ⎫ ⎧   ⎨ ϕ2 Sp  1 1 ⎬ p     , κ > 0, 0 ≤ ϕ1 < min     ,  2 , ϕ2 ≥ 0, φ2 =   ⎩ SM  S  Sq  ⎭ 1 − ϕ1 SpM  Sp  p p

Ξ¯ p = diag(ξ¯p1 , ξ¯p2 , . . . , ξ˙¯pmp ), Ξ¯ q = diag(ξ¯q1 , ξ¯q2 , . . . , ξ˙¯qmq ), ψmq = min{ψi }, i∈Sq

163

164

Second-Order Consensus of Continuous-Time Multi-Agent Systems

 2   ϕ1 Sq  ϕ2 Sq  1 q ψi = =  2 , χ2 =  2 , 0 < ϕ1 <  2 ,   Sq  1 − ϕ1 Sq  j=Nq +1 ⎧ 1 − ϕ1 Sq 2 ⎫ /    ⎨   q   ⎬   Sq nχ q Ξ¯ L¯  Ξ¯ L¯   nSq χ q  3nSq χ1 q 2 1 + β max{α, β} λ q qqΞ¯ 1 η1 = α max{α, β} λ q qqΞ¯ 1 , + 2 2 κ 2 q ⎩ q min min ⎭ ⎧  /  2 ⎫   ⎨   q   ⎬   Sq nχ q   Ξ¯ q L¯ qq   nSq χ q    ¯ ¯ Ξ L 3n Sq χ1 q 2 1  1 + α max{α, β} η2 = β max{α, β} λ q qqΞ¯ 1 + . 2 2κ 2 λmin Ξ¯ q q ⎩ min ⎭ Np

Gij , χ1q

Furthermore, the Zeno behavior can be excluded. Proof. From Theorem 8.17 we know that under condition (8.65), the second-order consensus can be achieved in the pth strongly connected component. By the inductive method it suffices to show that (8.64) is globally stable at the origin. Thus consider the following Lyapunov functional candidate: V (˜xq (t), v˜ q (t), t)

  T q ¯ ¯ ¯ ¯ + 2αβψm Ξ¯ q L ) + ( Ξ L ) αβ ( Ξ q qq q qq = 12 x˜ T ˜ qT (t) q (t ), v α Ξ¯ q    T   x˜ q (t) , = 12 x˜ T ˜ qT (t) Ω¯ q ⊗ In q (t ), v v˜ q (t) 

T



α Ξ¯ q β Ξ¯ q



 ⊗ In

x˜ q (t) v˜ q (t)



(8.66) T T (t), v˜ N (t), . . . , where x˜ q (t) = [xˆ TNq−1 +1 (t), xˆ TNq−1 +2 (t), . . . , xˆ TNq (t)]T and v˜ q (t) = [˜vN q−1 +1 q−1 +2 T T v˜ Nq (t)] . Taking the time derivative of V (˜xq (t), v˜ q (t), t) along (8.63) and using Assumption 8.1 yield

V˙ (˜xq (t), v˜ q (t), t)

     T = x˜ T Ξ¯ q L¯ qq + (Ξ¯ q L¯ qq ) + 2αβψmq Ξ¯ q ⊗ In v˜ q (t) + α v˜ qT (t)(Ξ¯ q ⊗ In )˜vq (t) q (t ) αβ +

Nq i=Nq−1 +1

   ξqi α xˆ T ˆ iT (t) f (xi (t), vi (t), t) − f (x∗ (t), v∗ (t), t) i (t ) + β v ⎡

Nq

Lij (exj (t) + xˆ j (t)) − αψi xˆ i (t) −α Nq ⎢ ⎢  T j=Nq−1 +1 T + ξqi α xˆ i (t) + β vˆ i (t) ⎢ Nq

⎣ i=Nq−1 +1 Lij (evj (t) + vˆ j (t)) − βψi vˆ i (t) −β

⎤ ⎥ ⎥ ⎥ ⎦

j=Nq−1 +1

' &     αρ2 + βρ1 2 2 q ¯ x˜ Tq (t) Ξ¯ q ⊗ In x˜ q (t)+ ≤ −α bξ¯q Lqq − α ψm + αρ1 +

2

(8.67)

Consensus analysis of multi-agent systems with second-order nonlinear dynamics and general directed topology

& ' αρ2 + βρ1 T 2 2 q ¯ −β bξ¯q (Lqq ) − β ψm + α + βρ2 + v˜ q (t)(Ξ¯ q ⊗ In )˜vq (t)− 2  q  q   ¯ ¯ ¯ ¯ α 2 x˜ T ˜T q (t ) (Ξq Lqq ) ⊗ In e˜x (t ) − αβ x q (t ) (Ξq Lqq ) ⊗ In e˜v (t )−     αβ v˜ qT (t) (Ξ¯ q L¯ qq ) ⊗ In e˜xq (t) − β 2 v˜ qT (t) (Ξ¯ q L¯ qq ) ⊗ In e˜vq (t),

where e˜qx (t) = [exT(Nq−1 +1) (t), exT(Nq−1 +2) (t), . . . , exT(Nq ) (t)]T and ψmq = mini∈Sq {ψi }. The distributed event-triggered sampling rule (8.64) yields the following inequality: exi (t)1 + evi (t)1    

     i j j ≤ ϕ1 xi (tk ) − xj (tk (t) ) + vi (tki ) − vj (tk (t) ) 1

(8.68)

exi (t)1 + evi (t)1                ≤ ϕ1 Sq  x˜ q (t)1 + v˜ q (t)1 + e˜qx (t) + e˜qv (t)

(8.69)

j∈Sq ∩Ni

1

+ϕ2 exp(−γ (t − t0 )), t ≥ t0 .

By a similar computation process to get (8.36) we have

1

1

+ϕ2 exp(−γ (t − t0 )).

It follows from (8.69) that       x  e˜q (t) + e˜qv (t) 1 1  2        ϕ1 Sq   v˜ q (t) + ϕ2 S q  2 exp(−γ (t − t0 ))   2 ≤ x ( t ) + ˜ q 1 1 1−ϕ1 Sq  1−ϕ1 Sq      Δ q  q = χ1 x˜ q (t)1 + v˜ q (t)1 + χ2 exp(−γ (t − t0 )).

(8.70)

Therefore, substituting (8.70) into the last two terms in (8.67), we get the following two inequalities:    q  q ¯ ¯ ¯ ¯ −αα x˜ T ˜T q (t ) (Ξq Lqq ) ⊗ In e˜x (t ) − αβ x q (t ) (Ξq Lqq ) ⊗ In e˜v (t )   q   q ⎧       Sq  χ q x˜ q (t)2 + nSq χ1 x˜ q (t)2 + nSq χ1 v˜ q (t)2 ⎪ n ⎪ 1 2 2 ⎨    /  2 ≤ α max{α, β}Ξ¯ q L¯ qq 1 Sq nχ q   2 ⎪ x˜ q (t)2 + κ 2 exp(−2γ (t − t0 )) ⎪ ⎩ + 2κ

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

,

(8.71) and

⎧    2 nSq χ q  2 nSq χ q  2 q 1 1   +   ⎪ n S v ( t ) + x ( t ) v˜ q (t) χ ˜ ˜ q 1 q q ⎪ 2 2 ⎨    /  2 ≤ β max{α, β}Ξ¯ q L¯ qq 1 Sq nχ q   2 ⎪ v˜ q (t)2 + κ 2 exp(−2γ (t − t0 )) ⎪ + ⎩ 2κ

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

. (8.72)

165

166

Second-Order Consensus of Continuous-Time Multi-Agent Systems

Combining (8.71) and (8.72), from (8.67) we have V˙ (˜xq (t), v˜ q (t), t)

      q 1 x˜ Tq (t) Ξ¯ q ⊗ In x˜ q (t) ≤ −α 2 bξ¯q L¯ qq − α 2 ψm + αρ1 + αρ2 +βρ 2   q 1 v˜ qT (t)(Ξ¯ q ⊗ In )˜vq (t) + −β 2 bξ¯q (L¯ qq ) − β 2 ψm + α + βρ2 + αρ2 +βρ 2 ⎧  /  2 ⎫   ⎨   ⎬   q Sq nχ q   Ξ¯ q L¯ qq    nS χ q 2 +α max{α, β} λ Ξ¯ 1 n Sq  χ1 + 2q 1 + x˜ Tq (t) Ξ¯ q ⊗ In x˜ q (t) 2κ q ⎩ min ⎭      q   Ξ¯ L¯  nSq χ1 x˜ Tq (t) Ξ¯ q ⊗ In x˜ q (t) +β max{α, β} λ q qqΞ¯ 1 2 q min     Ξ¯ L¯   nSq χ q  1 v˜ qT (t)(Ξ¯ q ⊗ In )˜vq (t) +α max{α, β} λ q qqΞ¯ 1 2 q min ⎧  /  2 ⎫   ⎨   ⎬   q Sq nχ q Ξ¯ q L¯ qq    nS χ q 2 v˜ T (t)(Ξ¯ q ⊗ In )˜vq (t) +β max{α, β} λ Ξ¯ 1 n Sq  χ1 + 2q 1 + 2κ q ⎩ min ⎭ q     + (α + β) max{α, β}Ξ¯ q L¯ qq 1 κ 2 exp(−2γ (t − t0 ))       Δ q 1 = −α 2 bξ¯q L¯ qq − α 2 ψm + αρ1 + αρ2 +βρ x˜ Tq (t) Ξ¯ q ⊗ In x˜ q (t)+ 2   q 1 −β 2 bξ¯q (L¯ qq ) − β 2 ψm + α + βρ2 + αρ2 +βρ v˜ qT (t)(Ξ¯ q ⊗ In )˜vq (t)+ 2   q q q ¯ ˜ q (t) + η2 v˜ qT (t)(Ξ¯ q ⊗ In )˜vq (t) + η3 exp(−2γ (t − t0 )) η1 x˜ T q (t ) Ξq ⊗ In x       q q 1 ¯ ˜ q (t)+ ˜T = −α 2 bξ¯q L¯ qq − α 2 ψm + αρ1 + αρ2 +βρ + η q (t ) Ξq ⊗ In x 1 x 2   q q 1 −β 2 bξ¯q (L¯ qq ) − β 2 ψm + α + βρ2 + αρ2 +βρ + η2 v˜ qT (t)(Ξ¯ q ⊗ In )˜vq (t)+ 2 q

η3 exp(−2γ (t − t0 ))  T  Ω˜ q ⊗ In = x˜ T ˜ qT (t) q (t ), v 



x˜ q (t) v˜ q (t)

 q

+ η3 exp(−2γ (t − t0 )).

(8.73) As long as bξ¯q

  q q  η1 αρ2 + βρ1 η2 ρ1 αρ2 + βρ1 q α + βρ2 q ¯ Lqq > max + 2 − ψm , + + 2 − ψm , + α 2α 2 α β2 2β 2 β



(8.74) 



we obtain Ω˜ q > 0. Selecting 0 < Kp ≤ λmin Ω˜ q /λmax (Ω¯ q ), we have V˙ (˜xq (t), v˜ q (t), t) ≤ −Kq V (˜xq (t), v˜ q (t), t) + η3q e−2γ (t−t0 ) .

(8.75)

Consensus analysis of multi-agent systems with second-order nonlinear dynamics and general directed topology

Thus we have



 q η3  exp(−ζq (t − t0 )), V (˜xq (t), v˜ q (t), t) ≤ V (˜xq (t0 ), v˜ q (t0 ), t0 ) +  2γ − Kq  

(8.76)



where ζq = min Kq , 2γ > 0 and 2γ = Kq . Choose μq1 and μq2 to satisfy 0 < μq2 < β − 2αβ b

α2

¯ ξ¯q (Lqq )+2αβψm q

, 

0 < μq1 < 2αβ bξ¯q (L¯ qq ) + 2αβψmq − α 2 β − μq2

−1

(8.77) .

Then by Schur complement (Lemma 8.7) we have 

0
max + 2 − ψm , + + 2 − ψm . + α 2α 2 α β2 2β 2 β



167

168

Second-Order Consensus of Continuous-Time Multi-Agent Systems

Remark 8.25. In Theorems 8.17 and 8.23 the parameter κ is a relaxed factor. As long as

  2 √  κ is selected as a relatively large positive real number, the terms Πij M φ2 Nn /2κ /   2 Sq  nχ q /2κ in Theorem 8.23 can be neglected. Therefore in Theorem 8.17 and 2

inequality (8.16) in Theorem 8.17 can be modified as 



2αρ1 + αρ2 + βρ1 χ¯ 1 α 2βρ2 + αρ2 + βρ1 χ¯ 2 + 2, 2 + + 2 , aξ (L ) > max 2α 2 α β 2β 2 β where 



3Πij M φ1 Nn 2

χ¯ 1 = α max{α, β}

λmin (Ξ )

+ β max{α, β}

  Πij  φ1 Nn M

2λmin (Ξ )

and 

χ2 = β max{α, β}



3Πij M φ1 Nn 2

λmin (Ξ )

+ α max{α, β}

  Πij  φ1 Nn M

2λmin (Ξ )

.

Similarly, inequality (8.65) in Theorem 8.23 can be modified as follows: for 1 ≤ q ≤ p − 1, ⎧   q q   η¯ η¯ q α+βρ q αρ +βρ αρ +βρ ⎪ ⎨ bξ¯q L¯ qq > max ρα1 + 22α2 1 + α12 − ψm , β 2 2 + 22β 2 1 + β22 − ψm ,  p p ⎪ ⎩ a ¯ (L¯ pp ) > max 2αρ1 +αρ22 +βρ1 + χ¯12 , α2 + 2βρ2 +αρ22 +βρ1 + χ¯22 , ξp 2α α β 2β β

where

p χ¯ 1

= α max{α, β}

   p 3Πij 

p  φ Sp n M 1 2 λmin (Ξ¯ p )

+ β max{α, β}

   p

p

  p   p Πij  φ1 Sp n M

2λmin (Ξ¯ p )



3Πij  φ1 Sp n p

χ¯ 2 = β max{α, β}

M

2

λmin (Ξ¯ p )

q

    Ξ¯ q L¯ qq  3nSq χ q 1  1 ¯ 2 min Ξq

q

    Ξ¯ L¯  3nS χ q β} λ q qqΞ¯ 1 2q 1 q min

η¯ 1 = α max{α, β} λ η¯ 2 = β max{α,

+ α max{α, β}

,

   p  p   Πij  φ1 Sp n M

2λmin (Ξ¯ p )   Ξ¯ q L¯ qq   1 ¯ min Ξq

+ β max{α, β} λ + α max{α,

  Ξ¯ L¯  β} λ q qqΞ¯ 1 q min





,

 q

nSq χ1 2

   q  n Sq χ 1

2

, .

8.5. Illustrative examples To verify the effectiveness of the proposed theorems and corollaries, we give an example. Consider the multi-agent network (8.5) consisting of N identical Chua’s circuits [46]

Consensus analysis of multi-agent systems with second-order nonlinear dynamics and general directed topology

described by x˙ i (t) = vi (t), v˙ i (t) = f (xi (t), vi (t), t) + α +β

N

j=1, j =i

N

j=1, j =i

Gij (xj (tkj (t) ) − xi (tki ))

Gij (vj (tkj (t) ) − vi (tki )), 

Δ

(8.81)

i = 1, 2, . . . , N , 

t ∈ [tki , tki +1 ), k (t) = arg min t − tlj , k ∈ N, j

l∈N: t≥tl

where xi (t), vi (t) ∈ R3 and ⎛

⎞

10(−vi1 (t) + vi2 (t) − l(vi1 (t))) ⎜ ⎟ f (xi (t), vi (t), t) = ⎝ vi1 (t) − vi2 (t) + vi3 (t) ⎠ −18vi2 (t) 

(8.82)



with l(vi1 (t)) = − 34 vi1 (t) + 12 − 43 + 34 (|vi1 (t) + 1| − |vi1 (t) − 1|). By computation we have ρ1 = 0 [27]. Choose N = 12, α = 5, and β = 8. The network structure is shown in Fig. 8.1 with the weight near the edge. The vertices of the network are ordered from 1 to 12, and the network is composed of 4 strongly connected components as shown in Fig. 8.1. From Fig. 8.1 we have the Laplacian matrix ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ L=⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

3 −2 0 0 0 0 0 0 0 0 0 0

0 8 −1 0 0 0 0 0 0 0 0 0

−3

0 1 0 0 0 0 0 0 0 0 0

0 0 −6 0 0 0 2 −2 0 2 −3 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 11 −3 0 0 0 0 0

0 0 0 0 −2 0 3 0 0 0 0 0

0 0 0 0 0 −8 0 2 −1 0 0 0

0 0 0 0 0 0 0 −2 4 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −3 0 2 −2 0 4 −3 0

0 0 0 0 0 0 0 0 0 0 −4 3

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

It is easy to see that the Laplacian matrix L is already in the Frobenius normal form with ⎛

3 ⎜ L¯ 11 = ⎝ −2 0

0 8 −1

−3

⎞

⎛

3 ⎟ ⎜ 0 ⎠ = ⎝ −2 0 1

0 2 −1

−3

⎞

⎛

0 ⎟ ⎜ 0 ⎠+⎝ 0 1 0

0 6 0

⎞

0 ⎟ ¯1 ¯1 +D , 0 ⎠=A 0

169

170

Second-Order Consensus of Continuous-Time Multi-Agent Systems

Figure 8.1 Structure of a small network (the vertices in the dotted boxes are in the same strongly connected components).

⎛ ⎜ ⎜ ⎝

L¯ 22 = ⎜

−2

2 0 −3 0

0 0 11 −3

2 0 0

¯2+D ¯ 2, =A



L¯ 33 =

−2

2 −1

2 ⎜ L¯ 44 = ⎝ 0 −3 0

⎜ L¯ 13 = ⎝ 0

0

⎟ ⎜ ⎟ ⎜ ⎟=⎜ ⎠ ⎝

 =

⎛ ⎜ ⎜ L¯ 24 = ⎜ ⎝

−2

2 0 −3 0

2 0 0

2 −2 −1 1 ⎞

−2

4 0

⎛

0 0 3 −3



 +

0 −2 0 3

0 0

0 ⎜ L¯ 12 = ⎝ −6 0

0 0 0 0

0 0 0 0

0 0 0

⎞

⎛

⎟ ⎜ ⎟ ⎜ ⎟+⎜ ⎠ ⎝

⎞



⎟ ⎟ ⎟, ⎠

L¯ 34 =

0 0

0 0 8 0

0 0 0 0

⎞ ⎟ ⎟ ⎟ ⎠

⎞

0 0 0

⎛

L¯ 23 = ⎜

0 −3

0 0 0 0

¯3+D ¯ 3, =A

⎜ ⎜ ⎝

0 0 0 0 ⎟ 0 0 0 ⎠, 0 0 0 0

0 0 0 0



⎞

⎜ L¯ 14 = ⎝ 0

0 0 0 0

0 3

⎛

0 ⎟ −4 ⎠ , 3

⎞

0 ⎟ 0 ⎠, 0

⎛



4

⎛

⎛

⎞

0 −2 0 3

0 0

0 ⎟ 0 ⎠, 0 0 0 −8 0

0 0 0 0

⎞ ⎟ ⎟ ⎟, ⎠

 .

By computation we have that the nonnegative left eigenvector associated with zero

eigenvalue satisfying N i=1 ξi = 1 is ξ = (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.4615  , 0.2308, M  0.3077). In addition, p = 4, |S1 | = 3, |S2 | = 4, |S3 | = 2, |S4 | = 3, S4  = 1, ξ¯1 = (0.1818, 0.2727, 0.5455)T , ξ¯2 = (0.3, 0.3, 0.3, 0.3)T , ξ¯3 = (0.3333, 0.6667)T , ξ¯4 = (0.4615, 0.2308, 0.3077)T , ⎛ ⎜ Ξ¯ 1 = ⎝

0.1818 0 0

0 0.2727 0

⎞

0 ⎟ 0 ⎠, 0.5455

⎛ ⎜ ⎜ Ξ¯ 2 = ⎜ ⎝

0.3 0 0 0

0 0.3 0 0

0 0 0.3 0

0 0 0 0.3

⎞ ⎟ ⎟ ⎟, ⎠

Consensus analysis of multi-agent systems with second-order nonlinear dynamics and general directed topology

 Ξ¯ 3 =

0.3333 0

0 0.6667

⎛



and

,

0.4615 ⎜ Ξ¯ 4 = ⎝ 0 0

0 0.2308 0

⎞

0 ⎟ 0 ⎠. 0.3077

Therefore we have the general algebraic connectivity degree as bξ¯1 (L¯ 11 ) = 0.5120, bξ¯2 (L¯ 22 ) = 0.4772, bξ¯3 (L¯ 33 ) = 1.2679, aξ¯4 (L¯ 44 ) = 3.6330. Selecting [ϕ1 , ϕ2 , γ , κ] = [0.0014, 2, 1, 500], we have φ14 = 0.0042, φ24 = 6.0253, 

 χ11 , χ12 , χ13 , χ14 = [0.0128, 0.0229, 0.0056, 13.9691] ,  1 2 3 4 χ2 , χ2 , χ2 , χ2 = [6.0766, 8.1833, 4.0225, 17.6132] ,  1 2 3 η1 , η1 , η1 = [158.6884, 354.5973, 31.1249] ,  1 2 3 η2 , η2 , η2 = [200.1529, 447.2536, 39.2576] ,  1  ψm , ψm2 , ψm3 = [33, 15, 9] .

We further check the conditions in Theorem 8.23. By computation we further have  max

ρ1 α

+

αρ2 +βρ1 2α 2

+

η11 α2

− ψm1 ,

α+βρ2 β2

+

αρ2 +βρ1 2β 2

+

η21 β2

− ψm1



= max {−26.2138, −29.0747} = −26.2138 < 0.5120 = bξ¯1 (L¯ 11 ),   2 η12 2 , α+βρ2 + αρ2 +βρ1 + η2 − ψ 2 1 + − ψ max ρα1 + αρ22+βρ 2 2 2 2 2 m m α α β 2β β = max {−0.3774, −7.2138} = −0.3774 < 0.4772 = bξ¯2 (L¯ 22 ),   3 η13 3 , α+βρ2 + αρ2 +βρ1 + η2 − ψ 3 1 max ρα1 + αρ22+βρ + − ψ m m α2 α2 β2 2β 2 β2 = max {−7.3163, −7.5887} = −7.3163 < 1.2679 = bξ¯3 (L¯ 33 ),   χ4 χ4 2 +βρ1 2 +βρ1 max 2αρ1 +αρ + α12 , βα2 + 2βρ2 +αρ + β22 2α 2 2β 2 = max {0.9975, 1.0731} = 1.0731 < 3.6330 = aξ¯4 (L¯ 44 ).

We have that condition (8.65) is satisfied by selecting the above parameters and that the second-order consensus in multi-agent system (8.81) is globally achievable. Let the initial position and velocity conditions be −3 × rand(3, 1) and −2 × rand(3, 1), respectively, and let the total simulation time be five units with step size h = 0.0025. The states of position and velocity of all agents are depicted in Fig. 8.2, from which we see that the second-order consensus is achieved. The distributed event-triggered control inputs are shown in Fig. 8.3. According to the statistics, the average triggered times for all agents are 79, and the sampling  =3.98%. Let exvi(t)= exi (t)1 + evi (t)1 and  rate is 79/2000

    i j Thi (t) = ϕ1 j∈Sq ∩Ni xi (tk ) − xj (tk (t) ) + vi (tki ) − vj (tkj (t) ) + ϕ2 exp(−γ (t − t0 )). The 1 1 time evolutions of exvi (t) and Thi (t) for i = 1 are shown in Fig. 8.4, which expresses the

171

172

Second-Order Consensus of Continuous-Time Multi-Agent Systems

Figure 8.2 The states of position and velocity of all agents.

Figure 8.3 Control inputs.

Figure 8.4 exv1 (t) and its threshold Th1 (t).

Consensus analysis of multi-agent systems with second-order nonlinear dynamics and general directed topology

Figure 8.5 The sampling time instants of all agents (ϕ2 = 0).

Figure 8.6 The sampling time instants of all agents (ϕ2 = 0).

event-triggered time instants. In addition, the event-triggered sampling time instants of all agents are shown in Fig. 8.5, from which we can see that even if the states of neighborhood agents are close, the intervals of successive sampling time instants are strictly positive. Next, we present a simulation result to show the function of ϕ2 exp(−γ (t − t0 )) excludes the Zeno behavior. We take ϕ2 = 0 and keep all other parameters unchanged. The event-triggered sampling time instants of all agents are shown in Fig. 8.6, from which we can see that even if the states of neighborhood agents are close, the interval of successive sampling time instants decay to zero. According to the statistics, the average triggered times for all agents is 792, and the sampling rate is 792/2000 = 39.6%. Therefore the exponential term in the designed event plays a key role in excluding the Zeno behavior.

173

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Second-Order Consensus of Continuous-Time Multi-Agent Systems

8.6. Conclusion Based on the algebraic graph theory, matrix theory, and stability theory, we studied the event-triggered consensus of second-order multi-agent system with nonlinear dynamics and directed topologies, including strongly connected networks, networks with a directed spanning tree. We designed an effective piecewise constant feedback control protocol. We performed detailed analysis of strongly connected networks and networks containing directed spanning tree. Under the given triggered function and triggered condition, we also presented some sufficient conditions. The first one is suitable for the case where the corresponding topology is strongly connected, whereas the second one is suitable for the case that contains a directed spanning tree. In addition, we also showed that not only the consensus can be achieved, but also a continuous communication between neighboring agents is avoided. Moreover, the Zeno behavior is excluded for the triggering time sequence. The obtained results should be of interest to practical control applications involving agents with limited computing capability, limited capabilities of communication and actuation, and limited onboard energy source. A topic of future research will be directed at investigating the event-based consensus problem of multi-agent systems with switching topologies and/or time delays. Moreover, it could be interesting to see how the proposed algorithm scales up as the number of vertices increases. The event-triggered sampling control strategy for consensus on general complex dynamical networks is a very interesting topic with wide potential applications and deserves further investigation in the future.

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Index

A Acceleration term, 39, 106, 117 Adjacency matrix, 3, 108 for agents, 22 leader, 22, 23, 126, 134 Adjacency weighted matrix, 66, 73 Agents consensus, 103 control input, 127, 132 dynamics, 66, 161 in networks, 66, 125 neighbors, 126, 142 networks, 106 position state, 4 states, 21 unmodeled uncertainties, 20, 40, 66 updates, 60 velocity states, 23, 29 velocity variable vectors, 88 Algebraic connectivity, 40, 66, 106, 149, 161 connectivity degree, 163, 171 graph theory, 21, 126 inequality, 41, 92 multiplicity, 10, 93 properties, 40 theory, 95 Asymptotical stability, 48, 51, 130, 133, 136 Autonomous mobile agents, 105

C Chaotic attractor, 44, 68, 79, 100 dynamics, 119 Chattering, 21, 28, 31, 34 Chattering phenomenon, 28 Cluster consensus, 103 Combinational measurements, 125, 141 Communication channels, 60, 140 cost, 139, 145

delays, 141 links, 42, 61, 82, 105 local, 39 networks, 19, 65, 72 noises, 40 time delays, 39 topology, 125, 128, 132 Consensus agents, 103 algorithms, 2, 20, 24, 42, 60, 86, 140 algorithms for agents, 39, 106, 117 algorithms for networks, 142 cluster, 103 criteria, 142 error, 119 error dynamics, 132 information, 105 manifold, 41, 44, 45, 66–70, 72, 78, 94, 107, 114, 119 nonlinear, 69, 87, 88, 91, 93, 95, 99, 102 position, 12, 14, 59, 60, 80, 162 problem, 1, 2, 19, 20, 39, 40, 65, 85, 105, 106, 125, 139, 140, 160 problem for agents, 85, 139 problem in networked dynamical, 65 protocols, 1, 2, 4, 12, 20 state, 46, 70 state vector, 80 velocity, 85 Control input, 1, 2, 4, 19, 24, 25, 27, 28, 30, 31, 33, 34, 39, 65, 106, 117, 125, 128, 142 agents, 127, 132 for agent, 23 Control protocols, 30, 33 Cooperative control, 1, 19, 42, 65, 85, 105 Cooperative control strategies, 20 Coordination control, 39, 85

D Decomposition matrix, 55, 59, 72, 80 Deteriorated communication channels, 66 Deteriorated communication channels existence, 66, 73, 82

177

178

Index

Digraph, 1, 3, 4, 21, 22, 108, 109 Digraph interaction, 3, 21, 29 Divergent consensus, 15 Double integrator, 2, 125, 127 Double integrator dynamics, 106 Dwell time, 66, 79, 100, 132, 134 Dwell time topology, 73 Dynamics agents, 66, 161 network, 102 nonlinear, 28, 40, 44, 58, 66, 68, 79, 82, 85, 86, 91, 94, 103, 106, 107, 110, 140, 142, 145, 167, 174

E Eigenvalue zero, 144 zero algebraic multiplicity, 93 Euclidean norm, 21, 41, 67, 101, 107, 126, 143 Exponential stability, 51, 56, 86, 95, 99, 101

F Flocking, 19, 39, 65, 105, 139, 144 attitude alignment, 105 behavior, 19 Follower agents, 20, 23, 28 Formation control, 1, 39, 65, 105 Frobenius normal form, 160, 169

G Globally exponential stability, 95

H Harmonic oscillators, 2, 20, 40, 65, 140

I Illustrative examples, 11, 29, 58, 79, 99, 117, 134, 168 Independent Bernoulli random variables, 42, 87 Inequality algebraic, 41, 92 Infinite subsequence, 87, 94, 102 Information channel, 73 consensus, 105 exchange, 3, 21, 42, 108, 125 flow, 66 graph, 106 interaction, 43 link, 42, 87

local, 65, 85, 128 network, 41, 44 relative, 139 state, 94, 127 velocity, 28, 73, 88 Inherent nonlinear dynamics, 88, 102, 106, 134, 140, 142 function, 102 Interaction digraph, 3, 21, 29 graphs, 139 information, 43 local, 2 network, 40, 66, 126 protocols, 4, 16, 65, 87, 106 random graphs, 42 topology, 2, 8, 11, 29, 65, 86, 139 Interactive network, 106 topology, 121 Interconnection rules, 1, 19, 39 Isolated agents, 66, 79, 86, 88, 94, 101, 102

K Kronecker product, 4, 21, 41, 67, 107, 146

L Laplacian matrix, 2–5, 12, 22, 23, 42, 43, 49, 54, 68, 69, 72, 73, 85, 87–90, 93, 94, 101, 106, 108, 109, 116, 125, 126, 144, 145, 147, 149, 150, 160–162, 169 Leader adjacency matrix, 22, 23, 126, 134 agent, 20, 23–25, 28–30, 33, 86 node, 28 Leaderless consensus, 85 Linear matrix inequality (LMI), 109, 146 Linear protocol, 2, 4, 8, 10, 11 Local communication, 39 information, 65, 85, 128 information exchange, 4, 88 interaction, 2 interaction mechanism, 39 states, 125 Logical path, 73, 74 Lyapunov function, 25, 50, 52, 72, 74, 86, 87, 91, 94, 95, 102, 111, 130 Lyapunov theory, 20, 67, 72

Index

M

O

Matrix factorizations, 113 form, 4, 24 measure, 57 norms, 55, 72 theory, 8, 20, 174 Measurement errors, 128, 129, 131, 133, 155 errors velocity, 147, 155 state value, 128 Mobile agents, 1, 139 Model formulation, 3, 142 Multiple agents networks, 102 stable flocking, 106

Opinion dynamics, 42

N Negative weights, 61, 73, 82 Neighborhood states agents, 173 Neighboring agents, 4, 28, 42, 66, 85, 105, 125, 141, 174 Neighboring information, 24 Neighbors agents, 126, 142 Network bandwidth, 140 consensus protocols, 106 dynamics, 102 information, 41, 44 interaction, 40, 66, 126 interactive topologies, 107 load, 142 random, 41, 43, 52, 55, 82 random switching, 45, 58–60, 86 structure, 169 topology, 2, 20, 22, 23, 27, 40, 61, 65, 66, 101, 103 Networked control systems, 141 dynamic systems, 106 systems, 105 Nonlinear consensus, 69, 87, 88, 91, 93, 95, 99, 102 dynamics, 28, 40, 44, 58, 66, 68, 79, 82, 85, 86, 91, 94, 103, 106, 107, 110, 140, 142, 145, 167, 174 function, 29, 72, 90 system, 45, 94

P Permutation matrix, 143, 160 Piecewise continuous functions, 48, 51, 55 Pinning control schemes, 20, 28 technique, 40, 66, 106 Position consensus, 12, 14, 59, 60, 80, 162 consensus manifold, 114 relative, 66 solution space, 114 states, 59, 80, 119, 129, 171 vector, 127 Potential topologies, 79 Protocols consensus, 1, 2, 4, 12, 20 interaction, 4, 16, 65, 87, 106

Q Quantized average consensus problem, 105

R Random dynamical network, 94 graph, 41–43, 73 Laplacian matrix, 43 theory, 67 Laplacian matrix, 51, 55 network, 41, 43, 52, 55, 82 network topology, 66 switching, 40, 41, 44, 61, 86, 94, 95, 99, 102, 103 directed topology, 86 moments, 88 network, 45, 57–60, 86 nonlinear system, 86, 87, 90, 94, 95 process, 94 sequence, 86, 94, 100, 101 system, 102 variables, 42, 43, 87, 88 Randomly interconnected pairwise, 88 jumps, 66, 73 switching, 45

179

180

Index

Relative information, 139 position, 66 state deviations, 28 Resultant error, 66, 86, 94, 101 Resultant error system, 86

Transition matrix, 48, 52 Transversal components, 56, 66, 107 Triggered times, 171, 173 Trust consensus, 106

S

Undirected networks, 3, 22, 144 random network, 42 Unmanned air vehicle (UAV), 105 Unmanned autonomous vehicle (UAV), 1, 39 Unmodeled uncertainties, 23 uncertainties agents, 20, 40, 66

Sampling condition, 155 event, 135, 145, 151, 155, 160 instant, 131 period, 140, 141 process, 159, 167 rate, 171, 173 scheme, 127 time instant, 142, 155, 159 Saturation function, 21, 28, 30, 31, 33, 34 Schur complement, 91, 109, 112, 118, 146, 152, 158, 167 Simulation results, 11, 12, 16, 82, 102, 119 Sliding mode, 25, 27, 33 Sliding mode neighborhood, 31, 34 State information, 94, 127 transition, 9 variables, 46, 70, 117 vector, 41, 44, 45, 55, 66, 69, 70, 80, 107, 114 Swarming, 39, 139, 144 Symmetric matrix, 67, 108 positive definite matrix, 108, 111, 112, 114, 116

T Terminal sliding mode, 23 Terrestrial planet finder (TPF), 40 Topology communication, 125, 128, 132 dwell time, 73 interaction, 2, 8, 11, 29, 65, 86, 139 network, 2, 20, 22, 23, 27, 40, 61, 65, 66, 101, 103 sequence, 73 switchings, 88

U

V Vector form, 45, 69, 89, 111 Velocity consensus, 85 consensus manifold, 114 errors, 59, 60, 80 information, 28, 73, 88 initial values, 119 measurement errors, 147, 155 state, 4, 12, 14, 59, 60, 66, 79, 80, 85, 99, 100, 102, 119, 144, 151, 162 state information, 68 state vectors, 68, 110 term, 39 vectors, 43 Virtual leader, 40, 106

W Weight matrix, 42 Wireless communication, 125 communication networks, 125 networks, 125

Z Zeno behavior, 128, 132, 141, 142, 147, 151, 158–160, 164, 167, 173, 174