Admissible Consensus and Consensualization for Singular Multi-agent Systems (Engineering Applications of Computational Methods, 11) [1st ed. 2023] 9811969892, 9789811969898

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Engineering Applications of Computational Methods 11

Jianxiang Xi · Le Wang · Xiaogang Yang · Jiuan Gao · Ruitao Lu

Admissible Consensus and Consensualization for Singular Multi-agent Systems

Engineering Applications of Computational Methods Volume 11

Series Editors Liang Gao, State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan, Hubei, China Akhil Garg, School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan, Hubei, China

The book series Engineering Applications of Computational Methods addresses the numerous applications of mathematical theory and latest computational or numerical methods in various fields of engineering. It emphasizes the practical application of these methods, with possible aspects in programming. New and developing computational methods using big data, machine learning and AI are discussed in this book series, and could be applied to engineering fields, such as manufacturing, industrial engineering, control engineering, civil engineering, energy engineering and material engineering. The book series Engineering Applications of Computational Methods aims to introduce important computational methods adopted in different engineering projects to researchers and engineers. The individual book volumes in the series are thematic. The goal of each volume is to give readers a comprehensive overview of how the computational methods in a certain engineering area can be used. As a collection, the series provides valuable resources to a wide audience in academia, the engineering research community, industry and anyone else who are looking to expand their knowledge of computational methods. This book series is indexed in both the Scopus and Compendex databases.

Jianxiang Xi · Le Wang · Xiaogang Yang · Jiuan Gao · Ruitao Lu

Admissible Consensus and Consensualization for Singular Multi-agent Systems

Jianxiang Xi Rocket Force University of Engineering Xi’an, Shaanxi, China

Le Wang Rocket Force University of Engineering Xi’an, Shaanxi, China

Xiaogang Yang Rocket Force University of Engineering Xi’an, Shaanxi, China

Jiuan Gao Rocket Force University of Engineering Xi’an, Shaanxi, China

Ruitao Lu Rocket Force University of Engineering Xi’an, Shaanxi, China

ISSN 2662-3366 ISSN 2662-3374 (electronic) Engineering Applications of Computational Methods ISBN 978-981-19-6989-8 ISBN 978-981-19-6990-4 (eBook) https://doi.org/10.1007/978-981-19-6990-4 Jointly published with Science Press, Beijing, China The print edition is not for sale in China (Mainland). Customers from China (Mainland) please order the print book from: Science Press, Beijing, China. © Science Press, Beijing and Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are reserved by the Publishers, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publishers, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publishers nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

This work is dedicated to My family, for all the support and love Jianxiang Xi My wife, Li Zhang, for her beauty and wonderful love Le Wang My family, for all the kindness and love Xiaogang Yang

Preface

There are many biological communities in the magnificent nature, such as swarms of fish, herds, bees, and birds. The individuals of the swarms are fragile, but the groups gathered through interaction and cooperation of them can radiate tenacious vitality, with stronger foraging, migration, benefit seeking, harm avoiding, and other abilities. The animal groups can be regarded as the multi-agent systems, also known as swarm systems, which refer to a networked system composed of multiple autonomous or semi-autonomous agents interacting with each other. The agent is the intelligent unit with abilities such as movement, perception, communication, computation, analysis, and control. Compared with the isolated systems, multi-agent systems own many significant characteristics including the self-organization, extensibility, strong robustness, and low-cost property. According to the type of the algebraic constraints of the agent, multi-agent systems can be divided into the normal ones and the singular ones. Singular multi-agent systems consist of several agents, whose dynamics can be described by singular systems also referred to as descriptor systems, generalized systems, implicit systems, differential algebraic systems, or semi-state systems. It is well known that singular systems can describe more general physical systems than normal ones. Many practical multi-agent systems are singular. For example, multiagent supporting systems, which have potential applications in earthquake damage prevention in buildings, water-floating plants, and large-diameter parabolic antennae or telescopes, are singular when they consist of many independent blocks and each block is supported by several pillars. One of the most interesting research topics of the multi-agent system is the distributed cooperative control, which means to design the distributed control protocols using the local information among neighboring agent transmitted in the communication network such that the collaborative behaviors can be achieved by all agents. During the past two decades, distributed cooperative control of multi-agent systems has aroused many attentions and has been widely investigated by scholars due to its widely applications, such as coordination task of robot swarms, formation flying of multiple unmanned aerial vehicles, and wireless communication of mobile sensor networks, etc. Distributed cooperative control mainly includes the consensus, synchronization, rendezvous, flocking, formation, and containment control, where vii

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consensus is a fundamental research topic to address other cooperative control problems. Consensus control refers to design the distributed consensus control protocols such that a flock of agents can reach the agreement of some relevant states. It should be pointed out that consensus problems of singular multi-agent systems are more complicated and challenging than those of normal multi-agent systems since each agent usually has three types of modes, namely finite-dynamic modes, impulse modes, and non-dynamic modes. In this case, not only consensus, but also regularity and impulse-free property need to be considered simultaneously. This book investigates the admissible consensus and consensualization problems for singular multi-agent systems, where consensualization also refers to consensualizing controller design. This book is broken into eight chapters. Chapter 1 introduces the background and overview of the consensus for the singular multi-agent systems, where the basic knowledges of the consensus and the singular multi-agent systems and the literature review of the consensus control are given. Chapter 2 presents the related fundamental theory of the algebraic graph, the linear algebra, the linear system, and the singular system. Meanwhile, the consensus problems of the highorder linear time-invariant multi-agent systems are discussed. Chapter 3 investigates the admissible consensus and consensualization problems for high-order linear timeinvariant singular multi-agent systems with fixed and switching interaction topologies. The state-space decomposition method is proposed to project the state of a singular multi-agent system onto a consensus subspace and a complement consensus subspace. Then, the explicit expression of the consensus function and the admissible consensus analysis and design criteria are derived on these two subspaces. Chapter 4 discusses the admissible consensus and consensualization problems with time delays in the interaction among agents. The state consensus and the output consensus are tackled with the dynamical output feedback protocols with time delays. Chapter 5 furthers the study on admissible L 2 consensus and consensualization problems of singular multi-agent systems with time delays and external disturbances. A L 2 evaluation approach was proposed to determine the impacts of external disturbances, and LMI criteria for admissible L 2 consensus were presented, which can guarantee the scalability of singular multi-agent systems. Chapter 6 deals with the stable-protocol admissible consensus problems for high-order linear singular multi-agent systems with multiple time-varying delays and switching topologies. The protocol states are asymptotic stable such that the energy expenditure of the dynamical output feedback consensus protocol is finite, which is useful when the resource of the agent is limited. Chapter 7 further considers the energy constraint and the consensus performance of the admissible consensus and consensualization via constructing a cost index. Guaranteed-cost admissible consensus and consensualization criteria are obtained to achieve the trade-off design between the consensus regulation performance and the control energy consumption while achieving admissible consensus. Chapter 8 extends the results of admissible consensus to the admissible formation tracking for singular multi-agent systems with the leader-following structure and the limited energy supply. Sufficient conditions for admissible formation tracking analysis and design with switching transmission topologies containing a spanning tree and a joint spanning tree are given, where the total energy supply is introduced by the Laplacian

Preface

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matrix of a star topology with edge weights being one and the leader being the central node. To summarize, this book presents the research results on the admissible consensus and consensualization of singular multi-agent systems with several constraints, such as the switching and jointly switching interaction topologies, time delays, external disturbances, protocol state conditions, and energy constraints. Moreover, the results on the admissible formation tracking analysis and design are derived based on the admissible consensus theory. This book can serve as a reference to the issues on the admissible consensus and consensualization of singular multi-agent systems. Xi’an, China

Jianxiang Xi Le Wang Xiaogang Yang Jiuan Gao Ruitao Lu

Acknowledgments

We would like to thank our colleagues, collaborators and students for their supports and contributions to this book. In particular, we are appreciated to Profs. Yisheng Zhong, Zongying Shi, Bailong Yang, and Zhiyong Yu, for their professional advice. We thank Yao Yu, Hao Liu, Fanlin Meng, Yanhong Zhang, Jianfei Zheng, Ming He, and Zhong Wang for their useful discussions. We are thankful to Cheng Wang, Junlong Li, Miao Zhao, Donghao Qin, Wanzhen Quan, Hongyao Li, Mingxing Qin, and Chenhao Zhu for their contributions to this book. In addition, we acknowledge Elsevier, John Wiley & Sons, and IEEE for granting us the permission to reuse materials from our publications copyrighted by these publishers in this book. Especially, this book has been supported over the years by the National Science Foundation of China under Grants 62176263, 62103434, 62003363, 61703411, and 61374054, China Postdoctoral Science Foundation under Grant 271004, Science Foundation for Distinguished Youth of Shaanxi Province under Grant 2021JC-35, and Youth Science Foundation of Shaanxi Province under Grant 2021JQ-375.

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Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Background of Consensus for Multi-agent Systems . . . . . . . . . . . . . . 1.1.1 Collective Behavior of Animal Groups in Nature . . . . . . . . . 1.1.2 Cooperative Operation of Multi-agent Systems in Society . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Consensus with Interaction Topology on Graphs . . . . . . . . . . 1.2 Basic Knowledges of Singular Multi-agent Systems . . . . . . . . . . . . . 1.2.1 Models of Singular Multi-agent Systems . . . . . . . . . . . . . . . . 1.2.2 Examples of Singular Multi-agent Systems . . . . . . . . . . . . . . 1.3 Development Direction of Consensus: Literature Reviews . . . . . . . . 1.3.1 Consensus of Multi-agent Systems with Dynamics of Different Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Consensus of Multi-agent Systems with Different Interaction Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Consensus of Multi-agent Systems with Cost Indexes . . . . . 1.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Fundamental Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Algebraic Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Consensus Decomposition of Linear Space . . . . . . . . . . . . . . 2.2.2 Kronecker Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Orthogonal Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Linear Matrix Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Linear System Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Singular System Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 8 13 19 20 22 32 32 41 45 48 49 57 57 57 59 61 61 62 62 62 63 63 65 66

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2.4.2 Constrained System Equivalence and System Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Temporal Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Admissibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.5 Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.6 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.7 Observability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Consensus for High-Order LTI Multi-agent Systems . . . . . . . . . . . . . 2.5.1 Consensus and Consensualization . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Consensus Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Admissible Consensus and Consensualization on Interaction Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Structure and Variance of Interaction Topology . . . . . . . . . . . . . . . . . 3.1.1 Structure of Interaction Topology . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Variance of Interaction Topology . . . . . . . . . . . . . . . . . . . . . . . 3.2 Admissible Consensus and Consensualization with Fixed Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Admissible Consensus Protocol with Fixed Topology . . . . . 3.2.2 Conditions for Admissible Consensus Analysis . . . . . . . . . . . 3.2.3 Admissible Consensus Design Criteria . . . . . . . . . . . . . . . . . . 3.2.4 Consensus Functions on Fixed Topology . . . . . . . . . . . . . . . . 3.2.5 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Admissible Consensus and Consensualization with Switching Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Admissible Consensus Protocol with Switching Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Necessary and Sufficient Conditions for Admissible Consensus Analysis with Switching Topologies . . . . . . . . . . 3.3.3 Admissible Consensus Design Criteria: Switching Topology Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Consensus Functions on Switching Topologies . . . . . . . . . . . 3.3.5 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Admissible Consensus and Consensualization with Time Delays . . . . 4.1 Introduction of Time Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Classification of Time Delays . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 The Influence of Time Delays on Consensus Control . . . . . . 4.2 Delay-Dependent Admissible Consensus and Consensualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Dynamic Output Feedback Consensus Protocol with Time Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.2.2 Necessary and Sufficient Conditions for Delay-Dependent Admissible Consensus Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Delay-Dependent Admissible Consensus Designinse Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Consensus Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.5 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Admissible Output Consensualization with Time Delays . . . . . . . . . 4.3.1 Output Consensus Protocol with Local Delayed Output Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Conditions of Admissible Output Consensus Design with Time Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Output Consensus Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Admissible L2 Consensus and Consensualization with External Disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Properties of External Disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Disturbance Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Disturbance Suppression Methods . . . . . . . . . . . . . . . . . . . . . . 5.2 Problem Description of Admissible L 2 Consensus Control . . . . . . . 5.3 Admissible L 2 Consensus with External Disturbance . . . . . . . . . . . . 5.3.1 Necessary and Sufficient Conditions for Admissible Consensus Without Disturbance . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Admissible L 2 Consensus Function . . . . . . . . . . . . . . . . . . . . . 5.3.3 Admissible L 2 Consensus Analysis and Design Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Admissible Consensus and Consensualization with Protocol State Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Dynamic Output Feedback Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Static Output Feedback Consensus Protocol . . . . . . . . . . . . . . 6.1.2 Dynamic Output Feedback Consensus Protocol . . . . . . . . . . . 6.1.3 Stable Consensus Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Stable-Protocol Admissible Consensus with Time Delays . . . . . . . . 6.2.1 Singular Dynamic Output Feedback Consensus Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Stable-Protocol Admissible Consensus Analysis Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Consensus Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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121 124 129 132 133 134 135 141 144 146 146 149 150 150 151 153 155 155 157 159 166 168 168 173 174 174 175 176 178 178 180 184 189

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6.3 Stable-Protocol Admissible Consensualization with Switching Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Problem Description and Transformation . . . . . . . . . . . . . . . . 6.3.2 Admissible Consensus Criteria for Connected Switching Topology Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Admissible Consensus Criteria for Jointly Connected Switching Topology Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Admissible Consensus and Consensualization with Energy Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Problems of Optimal and Suboptimal Consensus . . . . . . . . . . . . . . . . 7.1.1 Decentralized Optimization Cooperative Control . . . . . . . . . 7.1.2 Global Optimization Cooperative Control . . . . . . . . . . . . . . . 7.2 Problem Description of Guaranteed-Cost Admissible Consensus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Conditions of Guaranteed-Cost Admissible Consensus Analysis and Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Guaranteed-Cost Admissible Consensus Analysis Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Conditions of Guaranteed-Cost Admissible Consensus Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Consensus Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Admissible Formation Tracking with Energy Constraints . . . . . . . . . . 8.1 Problems of Formation Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Time-Invariant Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Time-Varying Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.3 Formation Tracking Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.4 Leaderless Formation Control Problems . . . . . . . . . . . . . . . . . 8.1.5 Formation Control Examples of Multiple UAVs . . . . . . . . . . 8.2 Formation Tracking Control Protocol with Energy Constraint . . . . . 8.3 Energy-Constraint Admissible Formation Tracking Criteria . . . . . . . 8.3.1 Case of Switching Topologies Containing a Spanning Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Case of Switching Topologies Containing a Joint Spanning Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

190 192 200 202 205 210 211 213 214 214 215 218 220 221 226 228 230 236 236 239 240 240 241 242 243 244 246 249 249 256 261 267 270

Notations

C R Cn Cm×n Rm×n 0n 1n j 0m×n In xH xT ex x2 x p A p A2 A>0 A≥0 tr( A) Re(z) Im(z) A−1 AT AH eA λi ( A) λmax (A) λmin (A) det(A) rank(A)

Set of complex numbers Set of real numbers Set of n × 1 complex vectors Set of m × n complex matrices Set of m × n real matrices n × 1 column vector of all zeros n × 1 column vector of all ones Imaginary unit m × n matrices of all zeros n × n identity matrix Hermitian transpose of a vector x Transpose of a vector x Exponential of a real number x 2-norm of a real vector x p-norm of a real vector x Induced 2-norm of a real matrix A Induced p-norm of a real matrix A A positive matrix A A nonnegative matrix A Trace of a matrix A Real part of number z Imaginary part of number z Inverse of a matrix Transpose of a matrix A Hermitian transpose of matrix A Exponential of a real matrix A The ith eigenvalue of matrix A The maximum eigenvalue of a matrix A The minimum eigenvalue of a matrix A Determinant of matrix A Rank of matrix A xvii

xviii

diag(A1 , A2 , . . . , An ) diag(a1 , a2 , . . . , an ) sin cos ≡  ∀ ∃ ⊂ ⊆ ∈ ∈ / ∩ ∪ \ ∅ →  

⊗ max min sup inf ∞

Notations

A block diagonal matrix with diagonal blocks A1 to An A diagonal matrix with diagonal entries a1 to an Sine function Cosine function Identically equal Defined as For all If there exists A strict subset of A subset of Belongs to Does not belong to Intersection Union Excludes Empty set Tends to Left product Summation Kronecker product Maximum Minimum The least upper bound The greatest lower bound Infinity

Acronyms

CCS CLSO CS DARPA DOBC LCE LMI LQG LTI MASS MIMO QMI RBFNN SMC UAV

Complement Consensus Subspace Cascade Leader State Observer Consensus Subspace Defense Advanced Research Projects Agency Disturbance Observer Based Control Limited Control Energy Linear Matrix Inequality Linear Quadratic Gaussian Linear Time-Invariant Multi-Agent Supporting Systems Multiple Input Multiple Output Quadratic Matrix Inequality Radial Basis Function Neural Networks Sliding Mode Control Unmanned Aerial Vehicle

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Chapter 1

Introduction

This chapter presents the background and overview of the consensus for the singular multi-agent systems. The collaborative phenomenon in nature and the cooperative operation in society are introduced to reveal the consensus principle of multi-agent systems, which are formed by the communication and interaction of the different individuals with complex behaviors. Besides, the brief induction of the singular multi-agent system is shown. The model of the singular multi-agent system in the form of the state-space representation is given, and the examples of the singular multi-agent system in various fields are discussed. Moreover, the literature review of the consensus control of the multi-agent systems is given, where the main influence factors of the consensus control are analyzed, including the dynamics of the agent, the interaction topology, and the cost index of the consensus control.

1.1 Background of Consensus for Multi-agent Systems Most systems in nature and human societies can be described by networks with individual members represented by nodes and interactions among nodes denoting as edges, such as biological clusters, animal groups, neural networks, social networks, and electrical power networks. Those systems can be described as multi-agent systems. As a research branch of complex networks, multi-agent systems have inherited many commonalities of complex networks and have become a hot research area closely related to complex networks. Compared with isolated systems, multi-agent systems have the following typical features: (a) self-organization: simple local interactions are used to evolve significant swarm characteristics; (b) expandability: the addition of the individual member is allowed in the operation of the whole system; (c) ruggedness: the failure of the individual member does not affect the execution of the whole task; (d) low-cost: multiple individual members complete complex tasks through complementary functions, which can greatly reduce the performance requirements and design complexity of each agent. © Science Press, Beijing and Springer Nature Singapore Pte Ltd. 2023 J. Xi et al., Admissible Consensus and Consensualization for Singular Multi-agent Systems, Engineering Applications of Computational Methods 11, https://doi.org/10.1007/978-981-19-6990-4_1

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1 Introduction

The cooperative control problems of multi-agent system include swarming [1–3], consensus [4, 5], flocking [6–8], coverage [9–11], and formation [12–14]. Consensus is a basic problem of cooperative control of multi-agent systems, and it is required that all agents can achieve the agreement on some coordination variables under the condition of environmental change. It can be seen that the distributed decision is made by each agent based only on its local information, and this decision can lead to a collective movement of the whole system. The research of multi-agent system usually combines algebraic graph theory, and the information interaction in the multi-agent system can be modeled as an interaction topology of graphs. Based on the mechanisms of the information interaction, each agent requires direct interactions with only a few neighbors to generate the collective decisions for the whole system. In the two past decades, consensus and collective behavior phenomena have been expounded in animal groups, cooperative operations in society, and engineered systems. Through the local interaction between the members of animal groups or the individuals of engineered systems and their neighbors, a consensus state or collective behavior emerges from the group or the system as a whole. In the following, this section describes in detail these three areas: collective behavior of animal groups in nature, cooperative operation of multi-agent systems in society, and consensus with communication topology on graphs, respectively.

1.1.1 Collective Behavior of Animal Groups in Nature The collective behavior of animal groups is a ubiquitous phenomenon in nature, at very different scales and levels of complexity [15]. The collective motions gathered by the aggregation of a large number of animal individuals are the coordinated, orderly, and even shocking scenes in nature. The whole movement makes the group look like a single subject, but every individual in the group has its own state, movement, and reaction pattern. A school of fish move with the ocean current and food orderly and uniformly. When attacked, the fish will quickly gather and disperse, showing a very strict cooperation. In the process of migration, starlings often gather to form a huge group, sometimes even including millions of birds. Microscopically, microorganisms such as bacteria and human melanocytes also engage in collective movements (see Fig. 1.1). In fact, similar large-scale group movement scenes widely exist in nature worlds. These collective phenomena show the characteristics of the distribution, coordination, self-organization, stability, and intelligent emergence. In many cases, united groups are formed as a whole with coordination and adaptability, and spontaneously maintain the stability of the whole group through the interaction between individual members.

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Fig. 1.1 Typical collective behavior of different animal groups in nature. a A school of fish.1 b A pack of wolves.2 c Starling flock.3 d Bacterial colony.4 e Honey-bee colony.5 f Ant colony6

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https://www.nipic.com/show/19764587.html. http://www.16pic.com/pic/pic_6690254.html. 3 https://baijiahao.baidu.com/s?id=1725717924031820727&wfr=spider&for=pc. 4 https://www.sohu.com/a/400715078_610722. 5 https://www.163.com/dy/article/DQQHSR1H0522L8EF.html. 6 https://www.51yuansu.com/sc/rrcxkhaphk.html. 2

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1 Introduction

Typical Collective Behavior Mechanism of Animal Groups

Animal groups in nature can form coordinated and orderly groups through unique mechanisms. For example, bats form groups through the action principle of slowing down the swimming speed when the light disappears; when crowding and collision occur within the ant colony, coordinated swing activities will be carried out every 20 mins; humans form crowd by simply following behavior when there is a lack of normal communication; when there are multiple optional positions in the bee colony, the bee will hit other members that are dancing with its head to assert its own choice, and it will stop dancing until the only choice is determined after being hit many times. In the previous examples, the groups that emerge different collective behavior have no obvious centralized control. Individuals interact with neighbors according to limited local information, which spreads in the whole group and finally produces collective behavior. This mechanism for generating global states from local rules is called self-organization. The analysis of several typical collective behavior mechanisms of animal groups has important implications for the study of multi-agent systems. The collective behavior mechanism of fish schools. As a low-level animal group, fish schools have no fixed leadership in their group structure. With the interaction of all fish individuals, it can spontaneously aggregate from a disordered state to a orderly state. Fish schools generally show the characteristics of consistent movement directions, coordinated movement modes, and orderly group aggregations. On the surface, this collective feature is a disorderly and irregular movement. However, from the perspective of group coordination mechanism, the collective behavior of fish schools mainly depends on the intention of multiple fish individuals to drive the complex emergent behavior of group aggregation. Different individual species and living environments have derived many types of the fish school mechanism. For example, the group area of herring schools presents a variety of shapes, such as circles, elliptical rings, strips, parabolas, or irregular shapes. During the annual migration, herring schools will spontaneously gather into nearly perfect spherical ‘fish balls’ with a diameter of tens of meters to fight against powerful predators along the way (see Fig. 1.2a). This spherical population structure can ensure that the offspring can get enough food on the one hand and can defend against enemies on the other hand. For many fish schools that need the long-distance migration, such as yellow croaker and hairtail, the group area presents a similar queue pattern, forming a neatly arranged fish school structure (see Fig. 1.2b). Since the individual fish in the front row will drive the water flow, the individual fish in the back row can easily swim forward with the water flow without consuming too much energy. The orderly and neat movement of fish schools can be achieved by transmitting information with other fish and perceiving the surrounding environment through vision, hearing, and electrical signals. In the fish school, the visual system of individual fish plays an important role in the interaction mechanism. The individual fish maintains the unity with the movement direction of the fish school through its visual perception system and determines the speed and direction of each fish through the

1.1 Background of Consensus for Multi-agent Systems

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Fig. 1.2 Collective forms of different fish schools. a Collective form of herring school.7 b Collective form of hairtail school8

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Fig. 1.3 Two flight modes of pigeon flocks. a Homing flight of pigeon flocks.9 b Spiral flight of pigeon flocks10

line perception ability, to achieve the purpose of consistent movement direction of the fish school. There are five characteristics in the school of fish: repellency, parallelism, attraction, cohesion, and permutation. Based on these basic characteristics, each fish individual can show strong aggregation by attracting, repelling, and swimming in the same direction, and maintain the high consistency in motion direction and speed. The collective behavior mechanism of pigeon flocks. The pigeon flock is a common bird group, which has special long-distance navigation ability and can form a stable flight group through interaction and cooperation. Recently, the research on the internal mechanism of the pigeon flock behavior is mainly based on the observation and analysis of the pigeon flock flight (see Fig. 1.3). 7

https://baijiahao.baidu.com/s?id=1655137019277571247&wfr=spider&for=pc. https://www.photophoto.cn/tupian/shuiditiantang0173-20243290.html. 9 https://baijiahao.baidu.com/s?id=1709301188624743014. 10 https://dp.pconline.com.cn/dphoto/list_4892401.html. 8

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Many researchers have used three-dimensional simulation imaging: machine vision cooperative positioning, light GPS positioning and other technologies to observe and obtain the individual position, and motion trajectory of the pigeon flock during the long-distance homing flight and the short-distance free flight. The results of related researches show that pigeon flocks can form stable flight clusters through a hierarchical network. The relative position of an individual pigeon in a cluster flight depends on the level of the individual in the hierarchical network. Compared with the collective behavior mechanism of fish schools, the hierarchical network structure of pigeon clusters shows stronger robustness. Vicsek [16] investigated the hierarchical leadership behavior in pigeon flocks. High-precision GPS devices were installed on pigeon flocks with differences in age, flight speed, and navigation experience to collect the flight trajectories of individuals in flock flight and further analyze the direction of flight speed. Based on the flight observation data, Vicsek proposed a correlation function model of the individual flight speed in a pigeon flock, based on which the leader–follower relationship of each pair of pigeons was analyzed and a hierarchical flock leader network was established. The results show that although there is a time delay for different individuals in the flock to follow the leader in flight direction selection, a stable flight flock can be formed within the flock through the hierarchical leadership network. The relative position of an individual in the flock depends on the rank of the individual in the hierarchical network. Moreover, in experiments targeting differences in flight speed and navigation experience, the hierarchical network structure of the pigeon flock demonstrated strong robustness to navigational errors, which suggest that a selforganized hierarchical network system in a pigeon flock might be more effective than a completely equal form of organization as shown in [17]. The collective behavior mechanism of wolf packs. As a cognitive and wellorganized animal groups, wolf packs are good at maintaining their survival advantage in harsh natural environments. Wolf packs rely on learning and understanding of the environment to enhance their environmental adaptability. In addition, wolf packs establish the information exchange among individuals by getting close to their peers or calling them, such that they can expand their perception range and perceive the environment with the strength of the group. The hunting process of the wolf pack perfectly demonstrates the task distribution of the collective behavior (see Fig. 1.4). When a wolf pack hunts, one wolf organizes the attack, and the organizer is not invariable. The organizer can be the executor of the next hunt, and it is not necessarily the head wolf. In the hunting process of wolf packs, there is a strategy of ‘organizingexecuting’. Each wolf individual can receive new task information according to its own perception and release these tasks to other wolves. After being informed of the tasks issued by other individuals, they can participate in the task allocation. The combination of task manager and executor is an important characteristic of wolf packs. Wolf packs have outstanding advantages in cognition, division of labor, and adaptability to the complex environment [18]. Collective behaviors of wolf packs embody the characteristics of active learning, close cooperation, and rapid response, which

1.1 Background of Consensus for Multi-agent Systems

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Fig. 1.4 Collective behavior mechanism of wolf packs. a Task allocation mechanism. b Hunting process of wolf packs11

show the unique nature of the group intelligence. Lampe et al. [19] analyzed the learning and object discrimination ability of wolf packs under the human instruction based on their domesticated behavioral characteristics. It was observed that wolves excelled in the causality discrimination and inferred that wolf packs have strong object memory and association properties. Hiestand [20] studied the environmental cognitive ability of wolf packs through the direction discrimination and tracking tests. It was found that wolf packs showed strong three-dimensional orientation and the agility in a new environment and can eliminate wrong directional guidance by repeated observation.

1.1.1.2

Characteristics of Collective Behavior

Most of the available studies consider that the collective behavior reflects five basic principles [21]: proximity principle, that is, members of a group interact only with neighboring individuals; quality principle, in which the collective behavior can respond to quality factors in the environment; diversity response principle, which requires that the range of group actions should not be too narrow; stability principle, which requires that the group should be able to maintain its own structural stability at each change in the environment; adaptability principle, where the group can change its collective behavior at the right time if the cost is not too high. The collective behavior of animal groups is characterized by the following: Distributed organizational structure. There is no central node in the animal group. Individuals follow simple behavior rules and only have local perception, planning and communication capabilities. Individuals change their behavior patterns in time to adapt to the dynamic environment through information interactions with the environment and neighbor individuals. The collective structure has strong robustness and does not affect the whole system due to the obstacles of one or some individuals. 11

https://baijiahao.baidu.com/s?id=1726792450357173401.

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Simplicity of behavior subject. The abilities of individuals in a group or the rules of followed behaviors are very simple. Every individual performs only one or a limited number of actions and makes a few simple responses to external situations. This behavior makes the group extremely efficient and reflects the emergence of intelligence. However, a collective behavior is not a simple sum of individuals, but a multiplication of capabilities through organization, collaboration, and cooperation among individuals. For example, although individual ants are relatively simple, the whole colony behaves as a highly institutionalized lining organization, which can accomplish complex tasks in many cases that far exceed the capabilities of individual ants. Flexibility of action mode. Flexibility is mainly reflected in the ability of the group to adapt to the environment. When encountering changes in the environment, individuals in a group adapt to the changes by adjusting their behavior. For example, a flock of birds can quickly make a collective escape when encountering a predator, and a flock of fish will change its swirling motion when attacked by a shark to obtain a stronger survival ability. Although the flexibility shown in these groups is contradictory to the stability of the collective group, many groups in nature often combine stability and flexibility. Physicists have proposed the hypothesis that groups in nature work near the critical point of the system’s phase transition, which makes the system to remain stable while being flexible. Intelligence of the whole group. In an animal group, individuals exchange and share information by perceiving the information of their surroundings. According to collective behavioral rules, individuals respond to external stimuli and enhance the survival of the group by adjusting their own state. Individuals in the group adapt their behavior through the state of the environmental feedback to obtain the best adaptation of themselves to the external environment. The evolution of the group includes both temporal and spatial aspects, which are expressed in time as learning of individuals from their own historical experiences and in space as interactive learning with other individuals and the external environment.

1.1.2 Cooperative Operation of Multi-agent Systems in Society In order to obtain adaptive advantages, some individuals in nature gather into a group through a certain relationship. This is also the case in society and engineering. Although the problems and models belong to different disciplines, the internal mechanism is to achieve the common ideal goal in a cooperative method. For groups of different hierarchies and scales in nature or society, the collective behavior and cooperative operation are realized through the local interaction and control. This interaction framework is named as the distributed framework. Compared with the centralized framework, the distributed framework overcomes some limitations, such

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as a single point of failure, high communication requirement and cost, substantial computational burden, limited flexibility and scalability, and lack of privacy. In a distributed framework, some results of modeling, analysis, control, and application of the cooperative operation in society and engineering are described in detail as follows.

1.1.2.1

Cooperative Relation in Social Group

Social group is a group with the positive social ability formed by more than two members relying on the mutual social relation. Sociality is defined as ‘the characteristics that individual members of a social group exhibit in their actions that contribute to the development of the group’. Sociality includes altruism, collaboration, dependency, and consciousness. In social groups, the relationships among people are friendship, collaboration, sharing of information, and so on. In addition, there are also negative effects, including antagonistic, hostile, disagreement, conflict, and so on. For the sociality of a lion herd, a lion herd is generally composed of one or more male lions, multiple female lions, and several young lions. The male lion is responsible for patrolling and guarding the territory. Some female lions are responsible for capturing prey in their territory, while others are responsible for feeding and caring for young lions. With the help of this division of tasks and roles to adapt to survival, a more stable social interrelation is formed among lions and between each lion and the lion herd. This relation makes the comprehensive ability of lions far greater than the superposition of their basic abilities. Therefore, this group of lions is a typical social group and so is human society. In another case, if the lion herd is only composed of one female lion and several young lions, the female lion is a member with the impaired ability. These young lions are members with the beneficial ability, and the comprehensive ability of the lion herd will be less than the superposition of its basic ability. This lion herd is a typical non-social group, because the comprehensive ability of the female lion is severely reduced by the heavy division of tasks for the entire group. In terms of the group stability and the development capacity, the state of non-social groups is characterized by short-term, vulnerability, and instability, and they have little basic nurturing capacity. If it is not maintained by force and deception or by self-sacrifice of a few members with impaired abilities, the non-social group cannot exist in the long term. Benefiting from the social relation formed among members, each member will become a member of the ability to benefit as much as possible. The social ability of the group will be maximized, and the adaptability of the group will continue to increase and develop into a high-quality social group. A special type of the cooperative relation, called an autoeciousness relation, which is degraded when there is only a one-way social relation between the two members [22]. Autoeciousness relation is usually a relatively stable social group over a long period of time. One of them is a one-way parasitic relation in which the member with strong integrative ability exerts coercive predation on other members or groups that are relatively weak, called inactive autoeciousness relation, which is

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1 Introduction

a vicious parasitic relationship. The other is a one-way parasitic relation in which the member with the relatively weak integrative ability passively accepts patronage from other members or groups, called passive autoeciousness relation, which is a kind of good parasitic relationship. In society groups, sociality and social relation are more common survival characteristics. Whether they are at the top or bottom of the ecological chain, the formation of social groups often becomes an efficient and stable way to obtain the greatest evolutionary benefits. Although different types of social groups have their own characteristics, they generally follow the following principles of the social relation: (i) the necessary principle of social relation: the actions of social groups will obtain greater benefits than those of non-social groups. Sociality is necessary for each member of the group. Therefore, the social relation among members and between members and groups is necessary. (ii) The principle of maximum comprehensive income of groups: the comprehensive income reflects the coordinated and balanced relationship between members and groups, local and overall, short-term, and long-term income. The goal of social group action is to maximize the comprehensive income of groups and members. (iii) The inclusive principle of group integrity: the actions of social groups should consider the individual differences of members, and the benefits should be facilitating all members of the group. The group integrity that includes all members is the basic requirement of social groups.

1.1.2.2

Cooperative Control in Engineering

In engineering applications, replacing isolated systems with multi-agent systems can greatly reduce production costs and is more secure and reliable. In addition, some tasks cannot be completed by a single system and require the cooperation of many individuals. To this end, one usually needs to disassemble the task so that different

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Fig. 1.5 Multi-UAV systems. a Civil unmanned aerial vehicle (UAV) formation.12 b Military UAV formation13

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subjects cooperate and collaborate with each other to complete the task. Several examples of engineering applications are listed below (Fig. 1.5). Multi-UAV systems. Distributed formation control of multiple UAV is an important engineering application. Instead of a single, expensive and monolithic UAV, distributed formation control of multiple micro and compact UAVs can accomplish the same maneuver efficiently without costly expenses [23]. By local information interaction, a team of UAVs cooperatively works in a distributed manner such that no member plays a central role while a disabled individual cannot destroy the team function [24]. Formation of multiple UAVs has broad application prospects in large scope searching, cluster operation, formation strike, and multi-target tracking. A typical application project associated with multiple UAVs, called ‘Gremlins’, has been proposed by Defense Advanced Research Projects Agency (DARPA). The project aims to form a swarm of small UAVs with the capabilities of autonomous cooperative, recoverable, and distributed combat. The focus of Gremlins is on exploring key technologies of the air launch and recovery with low costs for small UAV clusters. This design concept provides the combination and synergy of multiple capabilities required for UAVs to perform combat missions. Multi-robot systems. Robots have the advantages of versatility, high efficiency, stability, reliability, and strong repeatability, so multi-robot systems are widely used in industrial production for collaborative robotics. Multi-robot systems have been applied in industrial, military, aerospace, and even medical applications. For example, in industry, multiple robots can carry large objects or move large quantities of cargo. In the military, multiple land robots can be assigned to an exact destination or perform emergency search and rescue missions. In medicine, multiple nanoscale micro-robots can be used to perform more complex surgical procedures [25–27]. Perseverance Mars rover successfully landed on the surface of Mars in 2021 with the Mars ingenuity helicopter. The Mars helicopter sent the map created in real time to the Mars rover, which realized the collaboration between the UAV and the unmanned rover in the sky and the ground, as shown in Fig. 1.6b. Spacecraft formation flying. The study on spacecraft formation flying is a main trend of future space science. By introducing a distributed framework, many inexpensive, simple spacecraft working together can achieve the same objective as a single, expensive, and complicated spacecraft. A typical application of distributed spacecraft system is the distributed remote sensing technology [28–30]. Compared with the traditional technology, distributed remote sensing technology has the advantages of multi-source information access, global coverage, and quick revisit. Micro-/nanosatellites and their distributed implementation have been shown to be well suited for the distributed remote sensing technology because of their unique properties of mass production, low cost, and various forms. For a distributed spacecraft system, the functional implementation usually replies on its formation and attitude, and different 12 13

https://www.sohu.com/a/251704975_190475. https://baijiahao.baidu.com/s?id=1681534309232815225&wfr=spider&for=pc.

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Fig. 1.6 Multi-robot systems. a Multiple industrial manipulator.14 b Ingenuity helicopter and perseverance Mars rover15

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Fig. 1.7 Multi spacecraft systems. a Multiple satellite cooperation.16 b Satellites of Starlink17

formation and attitude guarantee disparate functionality to be realized. The cooperative control technology can be quite useful for multiple spacecrafts to maintain a relative attitude (Fig. 1.7). Smart grid. Smart grid is a small-scale power generation and distributed system that organically integrates distributed power sources, loads, energy storage devices, converters, and monitoring and protection devices [31, 32]. With the key technologies of the cooperative operation control and the energy management, the adverse effects of intermittent distributed power sources on the distributed network can be reduced. The use of distributed power output can be maximized to improve the power supply reliability and power quality. Connecting distributed power sources to the distributed framework in the form of smart grids is generally considered to be one of the effective ways to utilize distributed power sources. To achieve an effective deployment among 14

https://www.sohu.com/a/389954516_100034932. https://baijiahao.baidu.com/s?id=1710610565274180174. 16 https://www.sohu.com/a/225150020_163975. 17 https://baijiahao.baidu.com/s?id=1724717361716140615&wfr=spider&for=pc. 15

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Fig. 1.8 Smart grid systems. a Smart grid structure.18 b City smart grid layout19

distributed energy resources, one needs to properly design the coordination among them. The traditional approach is centralized, where a single control center collects all the necessary information, performs central computation, and provides control signals to the entire system. This prevalent approach has several limitations, such as a single point of failure, limited flexibility and scalability, and high communication requirement and computation burden. Recently, an alternative distributed approach has been proposed to overcome these limitations [33–35] (Fig. 1.8).

1.1.3 Consensus with Interaction Topology on Graphs For animal/society groups, the local information interaction among individuals can determine the movement of the whole group. Similarly, the information flow in the communication network determines the interaction relation among all agents for multi-agent systems. The local information interaction and flow of dynamic systems can be expatiated in detail by a complex network. Complex networks are used to describe the communication mechanism of animal groups [36, 37], social relations [38], smart grids, cellular and metabolic networks [39, 40], neural networks [41, 42] computer systems, and even systems of multiple robots or UAVs. The most remarkable feature of complex network systems is that they can show collective behavior, which cannot be well explained by the individual dynamics of each node. Two important cooperative behaviors are synchronization and consensus [43, 44], both of which imply that all agents agree on a certain status of interests. The interaction topology on the graph is a very effective tool to describe the various types of network models. For example, the topology of social networks affects the spread of information and disease, and the topology of the power grid affects the robustness and stability of power transmission. The interaction topology of multi-agent systems 18 19

https://www.sohu.com/a/125508374_114877. https://www.sohu.com/a/82861436_119733.

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Fig. 1.9 Internet is one of the most complex man-made networks20

affects the consistency and synchronization of the whole system. A review of graph topologies is given in [45]. In the following, we shall discuss the Internet, the power distribution, the regular and random networks, small-world networks, and scale-free networks (Fig. 1.9). The Internet. Many issues arising in control of the Internet or more general communication networks are similar to those ones seen in production systems. In particular, decision making involves choosing and routing of the information from node to node across a network consisting of links. A large number of interconnected computers that are controlled based on limited local information lead to the complex dynamics of the Internet. The key issue related to the Internet is that individual nodes do not have access to global information throughout the network. Then, routing or making decisions must be determined using only available information. In the future, it will be possible to obtain much greater relevant information at each node in the network through some distributed congestion algorithms [46]. The system designer must devise algorithms to make use of this global information regarding varying congestion levels and network topology. The power distribution. A power grid is also called smart electrical/power grid or intelligent grid. A power grid map of Shaanxi Province is shown in Fig. 1.10. The traditional power grids are generally used to transfer power from several central generators to a wide range of users or customers. By utilizing modern transmission technologies, the intelligent power grid can deliver power in a more effective way. Moreover, the intelligent power grid uses two-way flows of electricity and information to create an automated and distributed energy delivery network. Reliability of the power grid is dependent on the reliability of the electric transmission network. 20

https://personalpages.manchester.ac.uk/staff/m.dodge/cybergeography/atlas/topology.html.

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A power grid differs from many other network systems in that capacity must meet demand at every instant of time. If not, the transmission system may become unstable and collapse, with severe economic consequences to follow. A typical transmission network may have hundreds of nodes, so a detailed model is far too complex to provide any insight into planning or design. Therefore, a lot of research about the power network is how to simplify the network in the process of theoretical model study. Then, based on the simplified network model, a reliable distributed power control algorithm is designed. The regular and random networks. In most control problems, the network topologies that include chains, grids, lattices, and fully connected graphs have been completely regular (see Fig. 1.11a). Those simple architectures allowed us to focus on the complexity caused by the nonlinear dynamics of the nodes, without being burdened by any additional complexity in the network structure itself. There is still another kind of graph, random networks. In random networks, there are N agents that are initially disconnected. It is desired to form M edge links among these agents in a random fashion, which is accomplished by selecting two agents at random. Nodes may be chosen more than once, or not at all. The resulting wiring diagram would be a snarl of crisscrossed lines. This is repeated until all the M edges have been disbursed. The result is a physical example of a random graph with N nodes and M edges (see Fig. 1.11b). Now slowly lift a random agent off the whole network. If it is linked with other agents, either directly or indirectly, those are dragged up too. Eventually, an isolated agent, a small group, or a vast network may be pulled up. In the decades since this pioneering work, random networks have been studied deeply within pure mathematics [47]. They have also served as idealized coupling architectures for dynamical models of gene networks, ecosystems, and the spread of infectious diseases and computer viruses. Small-world networks. Random networks represent one extreme of graph organization, namely complete disorganization. Regular networks represent another extreme where the graph is completely organized. An undirected graph is regular if all nodes have the same number of neighbors. Small-world networks provide a method for generating graph topologies that interpolate between these two extremes. In human social groups, we can define a small-world network by friendship links. That is, if two people are friends, there is an advantage between them. In today’s global society, it is possible that any two people chosen at random in the world are connected by a path consisting of friendship links. Moreover, the length of this friendship path is usually equal to six sides. Now, with the advent of communication tools, this shortest friendship path is undoubtedly diminishing. Regular networks and random networks cannot capture the small-world connectivity phenomenon of social networks. In Fig. 1.12, the model starts with a ring lattice of N nodes. Loop through all edges once with probability p to rewire one end of each edge to a randomly selected node from all edges in the graph. Duplicate edges are not allowed. It is seen that the small-world graphs form a continuum between the regular ring lattices and the random graph. However, small-world networks are more highly aggregated than random networks, in the sense that if A links to B and B links to C, then the likelihood that also links to

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Fig. 1.10 Power grid map of Shaanxi Province21

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17

(b)

Fig. 1.11 Schematic illustration of regular and random network architectures. a Regular networks.22 b Random networks23

Fig. 1.12 Solvable model of a small-world network24

C is greatly increased. In addition, Watts [48, 49] speculate that the two properties of short paths and high aggregation apply equally to many natural and social networks. A dynamical system coupled in this manner would show higher signal propagation speed, synchronization, and computational power than a regular network of the same size. The intuition is that short paths can provide high-speed communication channels between distant parts of the system, thus facilitating any dynamic process that requires global coordination and information flow (e.g., synchronization or consensus).

21

https://www.wendangwang.com/doc/3e7facce4890806031765ffa. https://www.photophoto.cn/pic/04888543.html. 23 https://dx.doi.org/10.1038/35065725. 24 https://eldar.cz/cognition/complex/. 22

18

1 Introduction

Fig. 1.13 Scale-free networks25

Scale-free networks. In random networks and small-world networks, the level of each node is the same, because most nodes have the same number of neighbors. However, in the actual network, the connection degree of some nodes may be higher than that of other nodes (see Fig. 1.12). This network has a distinctive feature that cannot be explained by random networks or small-world networks. That is, there are several well-connected nodes directly connected to each other, while most nodes are connected only to these well-connected nodes. With the advent of scaling in random networks, Barabási and Albert [50] studied the properties of graphs with several wellconnected nodes. Two new features that do not exist in random networks or smallworld networks are considered. First, the number of nodes in practical networks increases gradually. Second, new nodes preferentially attach themselves to existing nodes that have a greater degree or social influence. A connection probability growth model that enables new nodes to preferentially connect to larger existing nodes is designed. The model explains the relevant properties in voyage networks and social networks. Scale-free networks grow as new nodes are randomly attached to preexisting nodes. The probability of attachment is proportional to the extent of the target node. Thus, nodes that are richly connected tend to become richer, resulting in the formation of well-connected nodes in voyage networks and social networks. Albert et al. [51] suggested that scale-free networks can resist random failures because some well-connected nodes dominate in their topology. Any node that fails is likely to have a small degree and is consumable. On the other hand, such a network is vulnerable to deliberate attacks on well-connected nodes (Fig. 1.13). Communication networks of multi-agent systems. Most works on the consensus/synchronization of multi-agent systems mainly focus on the analysis of communication network models with the perfect communication. In this case, it is assumed that each agent can receive timely and accurate information from 25

http://ithelp.ithome.com.tw/articles/10187666.

1.2 Basic Knowledges of Singular Multi-agent Systems

2

7

3

1

6

19

4

Information from neighbors

Controller

Executer

5

Fig. 1.14 A communication network model

its neighbors. However, such models cannot reflect the real situation, because the information flow between two neighboring nodes will often be affected by many uncertain factors, including time delays, switching topologies, limited communication capacity, communication noise, and random packet loss. The above constraints should be considered when designing control strategies or algorithms. Therefore, it is necessary to analyze the impact of time delays and switching topologies on the communication network. In many practical complex network systems, due to the remote location or unreliable communication medium of agents, the information exchange between the agent and its neighbors will cause communication delay. Generally speaking, communication delay will have a negative impact on the stability and consistency/synchronization performance. Hence, it is of great significance to study the impact of delay on the coordination performance of complex network systems. In addition, it is also very meaningful to study the collective behavior of complex network systems under the conditions of communication delays and switching topologies. In general, three factors play an important role in determining the evolution of networks, in particular, the ability to reach consensus among agents— that is, (i) the self-dynamics of each agent, (ii) the internal coupling describing how the inputs affect the state evolution, and (iii) the interaction topology among all agents (Fig. 1.14).

1.2 Basic Knowledges of Singular Multi-agent Systems Singular multi-agent systems, also called descriptor multi-agent systems, are a class of systems consist of several agents whose state variables are governed by both differential and algebraic equations, and the dynamics of each agent may be singular and include the normal systems as a special case if the state variables are not constrained by any algebraic equations. Singular multi-agent systems have three types of modes: dynamic ones, static ones, and impulsive ones, while normal multi-agent systems only own dynamic ones. In this case, singular multi-agent systems can describe a wider range of physical systems, such as electrical network systems, power systems, aerospace systems, and chemical processes. To mathematically describe singular multi-agent systems and assure readers about the existence of singular multi-agent

20

1 Introduction

systems in the real world, following two sections are presented to introduce some general models and several practical examples of singular multi-agent systems.

1.2.1 Models of Singular Multi-agent Systems Establishing the mathematical model is the first step to do research on a control system. There are two main methods for control system modeling, one is the transfer function method from classical control theory, which is usually used to deal with the control problems of linear time-invariable system with a single input and single output; the other one is the state-space method from modern control theory, which can deal with multi-input, multi-output problems, time-varying problems, and non-linear problems. Since this book gives deep insight into the structural property of control systems, the state-space method is adopted to describe corresponding models. Without loss of generality, considering a heterogeneous singular multi-agent system that consists of N agents, then for any agent i ∈ {1, 2, . . . , N }, its nonlinear continuous dynamics can be described by the state-space method as follows: 

E i(t)x˙i(t) = f i (xi(t), u i(t), t), yi(t) = gi (x(t), u(t), t),

(1.1)

where t ≥ t0 is the time variable, xi (t) ∈ Rn , u i (t) ∈ Rm and yi (t) ∈ Rl are the state vector, the control input vector and the measured output vector, respectively. f i (·) ˜ and gi (·) are n-dimensional ˜ and l-dimensional vector functions, and E i (t) ∈ Rn×n can be a normal or singular matrix for some t ≥ 0, which means that the rank of E i (t) may be less than n. If f i (·) and gi (·) are linear functions of their arguments, singular multi-agent system (1.1) reduces to the general linear singular multi-agent system as shown below:  E i(t)x˙i(t) = Ai(t)xi(t) + Bi(t)u i(t), (1.2) yi(t) = Ci(t)xi(t) + Di u i(t), ˜ ˜ where Ai (t) ∈ Rn×n , Bi (t) ∈ Rn×m , Ci (t) ∈ Rl×m , and Di (t) ∈ Rl×m are timevarying matrices, also called the system matrices of (1.2). Moreover, if all the system matrices in (1.2) are constant ones, then system (1.2) turns into a time-invariant singular multi-agent system, i.e.,



E i x˙i(t) = Ai xi(t) + Bi u i(t), yi(t) = Ci xi(t) + Di u i(t),

(1.3)

1.2 Basic Knowledges of Singular Multi-agent Systems

21

˜ ˜ where E i , Ai ∈ Rn×n , Bi ∈ Rn×m , Ci ∈ Rl×n , and Di ∈ Rl×m . Considering that the N agents may have the same dynamics, system (1.3) can further reduce to a homogeneous singular multi-agent system by letting

E 1 = E 2 = · · · = E N = E, A1 = A2 = · · · = A N = A, B1 = B2 = · · · = B N = B, and C1 = C2 = · · · = C N = C, D1 = D2 = · · · = D N = D. In this case, one has 

E x˙i(t) = Axi(t) + Bu i(t), y(t) = C xi(t) + Du i(t).

(1.4)

If E and A are square matrix, i.e., n˜ = n, the system is said to be square and to have order n. In this case, when E is a nonsingular matrix, system (1.4) is equivalent to  x˙i(t) = E −1 Axi(t) + E −1 Bu i(t), (1.5) yi(t) = C xi(t) + Du i(t), which becomes a normal system. One can further find that if E equals to an identity matrix, i.e., E = In , system (1.5) degenerates into a standard normal system model. Generally speaking, the research results for singular multi-agent systems hold true for any normal multi-agent systems. Therefore, singular multi-agent systems have more extensive significance than normal ones. It should be noted that (1.4) can be equivalently represented by ⎧        xi(t) B ⎪ ⎪ E 0 x˙i(t) = A 0 ⎪ u i(t), + ⎪ ˙ ⎪ (t) 0 −I (t) I ζ ζ 0 0 i i ⎪ ⎪   ⎨  ε

  ⎪ ⎪

 xi(t) ⎪ ⎪ y = . ⎪ C D ⎪ ⎪  ζi(t) ⎩

A

B

C

That is to say, by introducing an extra descriptor variable to render the D matrix zero, system (1.5) can be transformed to the following form: 

E x˙i(t) = Axi(t) + Bu i(t), y(t) = C xi(t),

(1.6)

22

1 Introduction

and the form (1.6) can be used without loss of generality. In addition, for practical control systems, external disturbances cannot be ignored, and the corresponding models have the following form: 

E x˙i(t) = Axi(t) + Bu i(t) + Bϖ ϖi(t), y(t) = C xi(t),

(1.7)

where Bϖ ∈ Rn×d , and ϖi (t) ∈ Rd is the external disturbance. Since multi-agent systems are usually utilized to perform some cooperative tasks, such as cooperative attacks and joint surveillances, multiple agents may be required to track some objectives. These tracking control problems are typically summarized as the leader-following mathematical model, and the dynamics of singular multi-agent systems with a leader-following structure is given as follows: 

E x˙0(t) = Ax0(t), E x˙i(t) = Axi(t) + Bu i(t),

(1.8)

where agent 0 represents the leader and agent i (i = 1, 2, . . . , N ) is the follower. Singular multi-agent systems (1.6)–(1.8) with E and A being square matrix are the main systems to be investigated in this book.

1.2.2 Examples of Singular Multi-agent Systems Singular multi-agent systems can be found in many fields, such as multiple cartpendulum systems, multi-agent supporting systems (MASS), circuit networks, aerospace engineering, chemical processes, social economic systems, network analysis, biological systems, and so on. In this section, some examples of singular multiagent systems are presented, such that readers can be convinced of the existence of singular multi-agent systems.

1.2.2.1

Singular Mechanical Systems

Example 1.1 (Multiple-cart-pendulum systems) [52]: A simple example of multiagent cart-pendulum systems is shown in Fig. 1.15 that consists of N carts. In Fig. 1.15, m i , m i , and li (i ∈ {1, 2, . . . , N }) are the mass of the cart, the mass of the pendulum, and the length of the pendulum. Let xi1 represent the horizontal position of cart i, xi2 , and xi3 be the horizontal and vertical positions of the pendulum. Then, the dynamics of this system can be described as

1.2 Basic Knowledges of Singular Multi-agent Systems

23

Fig. 1.15 Schematic diagram of multiple-cart-pendulum systems

⎧ m i1 x¨i1(t) = −2λ(xi1(t) − xi2(t)) + u i(t), ⎪ ⎪ ⎪ ⎨ m i x¨i2(t) = −2λ(xi2(t) − xi1(t)), ⎪ m i x¨i3(t) = −2λxi3(t) − m i g, ⎪ ⎪ ⎩ 0 = (x2(t) − x1(t))2 + x32(t) − li2 ,

(1.9)

where λ is a Lagrange multiplier. It can be seen that the last equation in (1.9) describes an algebraic constraint on the variables xi1 (t), xi2 (t), and xi3 (t). If we are interested in the position of the pendulum only, the output equation has the form yi (t) = [xi2 (t), xi3 (t)]T . Let xi4 (t) = x˙i1 (t), xi5 (t) = x˙i2 (t), xi6 (t) = x˙i3 (t) and xi (t) = [xi1 (t), xi2 (t), . . . , xi6 (t)]T , then the singular multi-agent can be modeled as 

E i x˙i(t) = f i (xi(t), u i(t), t), yi(t) = C xi(t),

(1.10)

where ⎡

1 ⎢0 ⎢ ⎢0 ⎢ ⎢ Ei = ⎢ 0 ⎢ ⎢0 ⎢ ⎣0 0

0 1 0 0 0 0 0

0 0 1 0 0 0 0

0 0 0 mi 0 0 0

0 0 0 0 mi 0 0

⎡ ⎤ ⎤ 0 xi4 (t) ⎢ ⎥ 0 ⎥ xi5 (t) ⎢ ⎥ ⎥ ⎢ ⎥ ⎥ 0 ⎥ xi6 (t) ⎢ ⎥ ⎢ ⎥ ⎥ 0 ⎥, f i (xi (t), u i (t), t) = ⎢ −2λ(xi1 (t) − xi2 (t)) + u i (t) ⎥, ⎢ ⎥ ⎥ ⎢ ⎥ 0 ⎥ −2λ(xi2 (t) − xi1 (t)), ⎢ ⎥ ⎥ ⎣ ⎦ mi ⎦ −2λxi3 (t) − m i g, 2 2 2 0 (x2 (t) − x1 (t)) + x3 (t) − li

and  C=

 010000 . 001000

Example 1.2 (MASSs) [53, 54]: MASSs have potential applications in earthquake damage prevention in buildings, water-floating plants, and large-diameter parabolic antennae or telescopes. When a MASS consists of many independent blocks and each block is supported by several pillars, each agent in this MASS becomes a singular system.

24

1 Introduction

Fig. 1.16 Mechanical structure of an agent in the MASS

Consider the case that each agent in a MASS is supported by two pillars called Unit I and Unit II, respectively, as shown in Fig. 1.16, where m is the mass, d is the damping coefficient, and k is the stiffness coefficient. Let xi1 (t), xi2 (t), vi1 (t), and vi2 (t) represent heights and velocities of Unit I and Unit II, respectively. For Unit I, according to [54], one has ¨ + d h(t) ˙ + kh(t) = u i (t), m h(t) where u i (t) is the agent input force. For Unit II, its height and velocity are constrained by Unit I. Let xi1 (t), xi2 (t), vi1 (t), and vi2 (t) represent heights and velocities of Unit I and Unit II, respectively. Then, agent i (i = 1, 2, . . . , N ) can be described by E x˙i (t) = Axi (t) + Bu i (t), where ⎡ ⎤ 10 xiI (t) ⎢0 1 ⎢ viI (t) ⎥ ⎢ ⎥ xi (t) = ⎢ ⎣ xiII (t) ⎦, E = ⎣ 0 0 viII (t) 00 ⎡

0 0 0 0

⎡ ⎤ 0 0 ⎢ ⎥ 0⎥ − mk , A=⎢ ⎣ ⎦ 0 −1 0 0

1 − md 0 −1

0 0 1 0

⎡ ⎤ ⎤ 0 0 ⎢ ⎥ 1⎥ 0⎥ ⎥. , B=⎢ ⎣ ⎦ 0⎦ 0 0 1

Example 1.3 (Rolling ring drive systems) [55]: Rolling ring drives are friction drives which convert the constant rotary movement of a plain shaft into a traversing movement. They operate like nuts on a screw rod but have a pitch, either right-handed or left-handed that can be precisely adjusted and can be equal to zero, and they are widely used in the wire and cable industries for winding and unwinding, traversing, and positioning, also in the food/packaging industries for cutting, separating, distributing, feeding, and cleaning. Figure 1.17 shows the physical model of a rolling drive, where the connections among payloads m 2 , m 3 and the rolling ring drive are simplified to spring-dashpot elements with d and k being the damping coefficient and the stiffness coefficient,

1.2 Basic Knowledges of Singular Multi-agent Systems

25

Fig. 1.17 A rolling ring drive

respectively. Besides, the masses m 1 and m 2 are connected by a rigid massless bar. If φ is regarded as small angles, then the relationship between φ and the translational velocity x˙i3 is linear. Therefore, the dynamics of each rolling ring drive can be modeled as ⎧ xi1 = xi2 , x˙i3 = cφ, θ φ¨ = u i , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ m 1 x¨i1 = λ1 − d1 (x˙i1 − x˙i2 ) − k1 (xi1 − xi2 ), m 2 x¨i2 = −λ1 + d1 (x˙i1 − x˙i2 ) + d2 (x˙i3 − x˙i2 ) ⎪ ⎪ ⎪ + k1 (xi1 − xi2 ) + k2 (xi3 − xi2 ), ⎪ ⎪ ⎪ ⎩ m 3 x¨i3 = λ2 − d2 (x˙i3 − x˙i2 ) − k2 (xi3 − xi2 ), where λ1 and λ2 are the Lagrangian multipliers, and c is transformation coefficient. In general, the translational positions xi2 and xi3 are the state variables that can be measured. Hence, let u i (t) = u i this system can be described as the following state-space equations: 

E x˙i(t) = Axi(t) + Bu i(t), yi(t) = C xi(t),

˙ x˙i1 , x˙i2 , x˙i3 , λ1 , λ2 ]T , yi (t) = yi = where xi (t) = xi = [φ, xi1 , xi2 , xi3 , φ, [xi2 , xi3 ]T and ⎡

⎤ ⎡ ⎡ ⎤ ⎤ ⎡ T ⎤T Cp I 0 0 0 I 0 0 E = ⎣ 0 M 0 ⎦, A = ⎣ −K −D J ⎦, B = ⎣ L ⎦, C = ⎣ CvT ⎦ 0 0 0 H G 0 0 0 with M = diag{1, m 1 , m 2 , m 3 }, I being a fourth-order identity matrix and ⎡

0 ⎢0 D=⎢ ⎣0 0

0 0 d1 −d1 −d1 d1 + d2 0 −d2

⎤ ⎡ 0 0 0 0 ⎥ ⎢ 0 ⎥ 0 k1 −k1 , K =⎢ ⎣ 0 −k1 k1 + k2 −d2 ⎦ d2 0 0 −k2

⎤ 0 0 ⎥ ⎥, −k2 ⎦ k2

26

1 Introduction

⎡ ⎤T ⎡ ⎤ ⎡ ⎤ 00 0 0 θ −1 0 ⎢0 0⎥ ⎢0 1 ⎥ ⎢ 0 ⎥ ⎢ −1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ L=⎢ ⎣ 0 ⎦, J = ⎣ 0 −1 ⎦, G = ⎣ 0 0 ⎦ , H = ⎣ 1 01 1 0 0 0     0010 0000 Cp = , Cv = . 0001 0000 ⎡

1.2.2.2

⎤T −c 0 ⎥ ⎥ , 0 ⎦ 0

Singular Electrical Circuit Systems

Example 1.4 (Circuit network system) [56]: Consider a multiple RLC circuit network system as shown in Fig. 1.18. The resistor, inductor, and capacity are denoted as R, L, and C0 , respectively. Meanwhile, VR (t), VL (t), and VC (t) stand for the corresponding voltages. Vsi(t) in the figure is the voltage source and is regarded as the control input. Based on the Kirchhoff’s law [57] and some fundamental circuit theories, one can obtain the following equations, ⎧ ˙ L I(t) = VL(t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ V (t) = 1 I(t), C C0 ⎪ ⎪ ⎪ R I(t) = VR(t), ⎪ ⎪ ⎩ VL(t) + VC(t) + VR(t) = VS(t).

(1.11)

Choosing the state vector as xi (t) = [I (t), VL (t), VC (t), VR (t)]T , output vector as yi (t) = VC (t), and control input vector as u i (t), then (1.11) can be written in the form of following linear singular multi-agent system: 

E x˙i(t) = Axi(t) + Bu i(t), yi(t) = C xi(t),

where Fig. 1.18 An RLC electrical circuit network system

(1.12)

1.2 Basic Knowledges of Singular Multi-agent Systems

27

Fig. 1.19 A two-loop RLC circuit network

⎡

L ⎢0 E =⎢ ⎣0 0

0 0 0 0

0 1 0 0

⎤ ⎡ 0 0 ⎢ 1 0⎥ ⎥, A = ⎢ C0 ⎣ −R 0⎦ 0 0

1 0 0 1

0 0 0 1

⎤ ⎡ ⎤ ⎡ ⎤T 0 0 0 ⎢ ⎥ ⎢ ⎥ 0⎥ ⎥, B = ⎢ 0 ⎥, C = ⎢ 0 ⎥ . ⎦ ⎣ ⎦ ⎣ 0 0 ⎦ 1 −1 −1 1

Example 1.5 (Two-loop circuit network system) [58]: In this example, there are two current loops in the RLC circuit network as shown in Fig. 1.19, which are denoted as I1 (t) and I2 (t), respectively. VC1 (t) and VC2 (t) are the voltages of C1 and C2 . Choose the state vector as x = [VC1 (t), VC2 (t), I2 (t), I1 (t)]T and the output variable yi (t) as yi (t) = VC2 (t). Then, again according to the Kirchhoff’s second law, we can establish the same state-space representation as (1.11) for the system, where ⎡

C1 ⎢ 0 E =⎢ ⎣ 0 0

1.2.2.3

0 C2 0 0

0 0 −L 0

⎡ ⎤ 0 0 ⎢ ⎥ 0⎥ 0 ,A=⎢ ⎣ −1 0⎦ 0 1

0 0 1 0

0 1 0 R

⎡ ⎤ ⎡ ⎤T ⎤ 1 0 0 ⎢ ⎥ ⎢ ⎥ 0⎥ 0 ⎥ 1⎥ ⎥ . ,B =⎢ ,C = ⎢ ⎣ ⎦ ⎣ ⎦ 1 0 0⎦ R −1 0

Multi-link Manipulator Systems

Example 1.6 (Three-link planar manipulator) [56]: Fig. 1.20 illustrates a cleaning mobile manipulator named ‘KUKA’, whose task is to clean the surfaces of some buildings. This mobile manipulator is part of a class of service robots commonly used in emergency and disaster relief, public services, and environmental protection [59]. The mechanical dynamic characteristic of the cleaning robot named SKYWASH was also studied by some researchers [60].

28

1 Introduction

Fig. 1.20 A cleaning mobile manipulator26

Fig. 1.21 A tree-link planar manipulator

The structure of a three-link planar manipulator can be simplified as shown in Fig. 1.21, and the area between points P1 and P2 is the working region. The manipulator completes the task by repeatedly moving its end-effector from point P1 to point P2 with the specified contact force under the time limit. Suppose that the cleaning surface is a rigid body, there are two constraints which influence the mobility of the manipulator: (i) the limitation on the motion in the x direction, usually required as x ≤ 1; (ii) the orthogonality of the third arm to the cleaning surface, which can be described by α1 + α2 + α3 = 0. Therefore, the dynamics of the three-link planar manipulator in Fig. 1.20 can be described by the following equations in the joint coordinates: 

Mα(α)θ¨ + Cα(α, α) ˙ + G α(α) = u α + FαT μ, ψα(α) = 0,

(1.13)

where α = [α1 , α2 , α3 ]T is the vector of joint displacements, u α ∈ R3 is the vector of control torques applied at joints, Fα = ∂ϕα/∂α, μ ∈ R2 represents the vector of the Lagrangian multipliers, FαT μ is the generalized constraint force. The constraint function ψα (α) is given by  ψα(α) =

26

 l1 cos α1 + l2 cos(α1 + α2 ) + l3 cos(α1 + α2 + α3 ) − l . α1 + α2 + α3

https://www.kuka.com/en-cn.

1.2 Basic Knowledges of Singular Multi-agent Systems

29

Mα (α) ∈ R3×3 is the mass matrix and is given by Mα (α) = [m i j (α)]3×3 with     m 11 (α) = m 1l12 + m 2 l12 + l22 + 2l1l2 cos α1 + m 3 l12 + l22 + l32 + 2l1l2 cos α2 + m 3 (2l2 l3 cos α3 + 2l2 l3 cos(α2 + α3 )),     m 12 (α) = m 2 l22 + l1l2 cos α2 + m 3 l22 + l32 + l1l2 cos α2 + 2l2 l3 cos α3 + m 3 (l1l3 cos(α2 + α3 )),   m 22 (α) = m 2 l22 + m 3 l22 + l32 + 2l2 l3 cos α3 ,   m 23 (α) = m 3 l32 + l2 l3 cos α3 , m 33 (α) = m 3l32 . Cα (α, α) ˙ ∈ R3 is the centrifugal and Coriolis vector, and is given by Cα (α, α) ˙ = CI (α)ϕN + CII (α)ϕS , where T

ϕN = [α˙ 1 α˙ 2 α˙ 1 α˙ 3 α˙ 2 α˙ 3 ]T , ϕS = α˙ 12 α˙ 22 α˙ 32 ,



 and CI (α) = cI,i j (α) 3×3 , CII (α) = cII,i j (α) 3×3 with cI,11(α) = −2m 2 l1l2 sin α2 − 2m 3l1 (l2 sin α2 + l3 sin(α2 + α3 )), cI,12(α) = −2m 3l3 (l2 sin α3 + l1 sin(α2 + α3 )), cI,13(α) = cI,12(α), cI,21(α) = cI,32(α) = cI,33 (α) = 0, cI,22(α) = cI,23(α) = −cI,31 |!(α) = −2m 3l2 l3 sin α3 , cII,11(α) = cII,22(α) = cII,33(α) = 0, cII,21(α) = −cII,12 (α) = (m 2 + m 3 )l1l2 sin α2 + m 3l1l3 sin(α2 + α3 ), cII,31(α) = −cII,13(α) = m 3l3 (l2 sin α3 + l1 sin(α2 + α3 )), cII,32(α) = −cII,23(α) = m 3l2 l3 sin α3 . G α (α) ∈ R3 is the vector of gravity, which is given by G Tα (α) [g1 (α), g2 (α), g3 (α)]T with g1 (α) = gm 1l1 cos α1 + gm 2 (l1 cos α1 + l2 cos(α1 + α2 ))n + gm 3 (l1 cos α1 + l2 cos(α1 + α2 ) + l3 cos(α1 + α2 + α3 )), g2 (α) = gm 2 l2 cos(α1 + α2 )n

=

30

1 Introduction

+ gm 3 (l2 cos(α1 + α2 ) + l3 cos(α1 + α2 + α3 )), g3 (α) = gm 3l3 cos(α1 + α2 + α3 ). Next, we can linearize the system in Cartesian coordinates since the constraint of the environment as well as the statement of the task is often easily described in these coordinates. Let z = [x, y, φ]T be the Cartesian vector representing the position and the orientation of the end-effector. In Cartesian coordinates, the description (1.13) becomes  Mz(α)¨z + C z(α, α) ˙ + G z(α) = u z + FzT μ, (1.14) ψz(α) = 0, where Mz(α) = J −T(α)Mα(α) J −1(α), G z(α) = J −T(α)G α(α),

 C z(α, α) ˙ = J −T(α) Cα(α, α) ˙ − Mα(α)J −1(α) J˙(α)α˙ , and u z = J −T (α)u α with the Jacobian J (α) satisfying z˙ = J (α)α˙ and ⎤ −l1 s1 − l2 s12 − l3 s123 −l2 s12 − l3 s123 −l3 s123 J (α) = ⎣ l1 c1 + l2 c12 + l3 c123 l2 c12 + l3 c123 l3 c123 ⎦ 1 1 1 ⎡

with s1 = sin α1 , s12 = sin(α1 + α2 ), s123 = sin(α1 + α2 + α3 ), c1 = cos α1 , c2 = cos(α1 + α2 ), c123 = (α1 + α2 + α3 ). Noticing the relations x = l1 c1 + l2 c12 + l3 c123 and φ = α1 + α2 + α3 , we can obtain that ψz (z) = F0 z − L 0 with 

 100 F0 = , 001

  l L0 = . 0

Therefore, Fz =

∂ψz (z) = F0 . ∂z

1.2 Basic Knowledges of Singular Multi-agent Systems

31

This means that in Cartesian coordinates the considered robot model has linear constraints. This outcome is favorable in linearizing the model. Let the system parameters be m 1 = 10 kg, m 2 = 5 kg, m 3 = 2.5 kg, l1 = 2 m, l2 = 1.5 m, l3 = 1 m and l = 2 m, Δl = 1 m and g = 9.8 m/s2 . Choose the working point of linearization as z ω = [l, l + Δl/2, 0]T and z˙ ω = [0, y˙ω , 0]T with y˙ω = 0.3 m/s and z¨ ω = [0, 0, 0]T . Furthermore, denote δz = z − z ω , δz˙ = z˙ − z˙ ω and δz¨ = z¨ − z¨ ω . Then, the linearized model of (1.14) can be obtained as follows: M0 δ z¨ + D0 δ z˙ + K 0 δz = S0 δu + F0T δμ,

(1.15)

F0 δz = 0,

(1.16)

with ⎡

⎤ 9.3766 −3.9725 3.9725 M0 = Mz |z=zω = ⎣ −3.9725 15.9091 −13.4091 ⎦, 3.9725 −13.4091 13.4091 ⎡ ⎤ | −0.7607 −0.7758 0.7758 ∂C z || D0 = = ⎣ 1.6103 1.6423 −1.6423 ⎦, ∂ z˙ | z = z ω −1.6103 −1.6423 1.6423 z˙ = z˙ ω ⎡ ⎤ | | 33.7447 34.6197 −69.23920 ∂C z || ∂ G z || K0 = + = ⎣ 34.9062 0.8431 −0.8431 ⎦, ∂z |z=zω ∂z |z=zω z˙ =˙z ω −34.9062 −0.8431 −34.1354 ⎡ ⎤ −0.216598 −0.338060 0.554659 | S0 = J −T |α=αω = ⎣ 0.458506 −0.845153 0.386648 ⎦, −0.458506 0.845153 0.613353 and 

 100 F0 = Fz = , 001 δu = u − u ω , δμ = μ − μω . Note that αω is determined by z ω through inverse kinematics [61]. μω is a twodimensional vector. The first element of μω is chosen to be equal to the desired contact force in the x direction, and the second element is chosen to be zero. Then, μω is suitably determined such that the first equation of (1.15) remains balanced. Let x(t) = x = [δzT , δzT˙ , δμT ]T with δz = [δx , δ y , δφ ]T and δμ = [δμ1 , δμz ]T . Choose δ y , δμ1 , and δμ2 as the tracking outputs, then, the system (1.15)–(1.16) can be rewritten

32

1 Introduction

as the following state-space form: 

E x˙i(t) = Axi (t) + Bu i(t), yi(t) = C xi(t),

(1.17)

where ⎡

⎤ ⎡ ⎤ ⎤ ⎡ I 0 0 0 0 I 0 E = ⎣ 0 M0 0 ⎦, A = ⎣ −K 0 −D0 F0T ⎦, B = ⎣ S0 ⎦, F0 0 0 0 0 0 0 and ⎡

0100000 C = ⎣0 0 0 0 0 0 1 0000000

⎤ 0 0 ⎦, D = 0. 1

1.3 Development Direction of Consensus: Literature Reviews As a fundamental problem of the cooperative control of multi-agent systems, consensus has been widely investigated by researchers. Consensus control refers to designing distributed controllers using local information such that the multi-agent systems realize an agreement of some relative states. In the following, a literature review of the consensus of multi-agent systems with different constraints is presented.

1.3.1 Consensus of Multi-agent Systems with Dynamics of Different Orders From the perspective of the dynamic characteristics of the agents, the interaction mechanism of the agent dynamics on the consensus control is analyzed for several typical continuous-time linear multi-agent systems.

1.3.1.1

Consensus of First-Order Multi-agent Systems

In the early stages of the consensus research, scholars mainly concerned about how to reach the consensus of an isolated coordination variable. For example, a group of agents should achieve an agreement on the target position such that they can

1.3 Development Direction of Consensus: Literature Reviews

33

be gathered at a certain location. In this case, the dynamics of each agent can be described as the first-order integral as follows: x(t) ˙ = u(t) (i = 1, 2, . . . , N ),

(1.18)

where xi (t) ∈ R is the coordination variable, u i (t) is the control input, i.e., the control protocol. If limt→∞ (xi (t)−x j (t)) = 0 (i, j = 1, 2, . . . , N ), then multi-agent system (1.18) is said to achieve the consensus. Olfati-Saber and Murray investigated the χ consensus problems for first-order multi-agent systems with switching topologies and time delays in [62], where χ is called the consensus function; i.e., the common value that is achieved by all agents. If the value of the consensus function is the mean value of the coordination variables of all agents, then it is said to be the average consensus. It should be noted that if the interaction topology is undirected, then∑consensus N xi (0). function is the average of initial states of all agents; that is, χ (t) = 1/N i=1 According to the difference of the consensus control protocol, consensus of firstorder multi-agent systems can be divided into the fixed topology case, the timevarying topology case, the delay-dependent case, etc. For the fixed topology case, a typical consensus protocol was adopted in [62, 63] as: u i (t) =

∑

  wi j x j (t) − xi (t) ,

(1.19)

j∈Ni

where Ni represents the neighboring set of agent i and wi j is the interaction weight from agent j to agent i. Let x(t) = [x1T (t), x2T (t), . . . , x NT (t)]T , then the dynamics of the first-order multi-agent system can be described as the following linear timeinvariant (LTI) system: x(t) ˙ = −L x(t),

(1.20)

where L is the Laplacian matrix of the interaction topology of the multi-agent system. It can be found that the stability of system (1.18) is dependent on the location of the eigenvalues of the Laplacian matrix L on the complex plane. Hence, the spectral property of L is critical in the consensus problems of multi-agent systems. In particular, the eigenvalues of the Laplacian matrix L of the undirected topology are real numbers and satisfy that 0 = λ1 < λ2 ≤ · · · ≤ λ N , where the minimum nonzero eigenvalue λ2 is called the algebraic connectivity. It was proven in [62] that the convergence speed of the multi-agent system is determined by the minimum nonzero eigenvalue λ2 , and the bigger the λ2 is, the faster the consensus achieved. For the directed topology, Lin et al. [63] pointed out that the sufficient and necessary condition of the consensus achievement is that there exists a global access node in the directed topology, which means that the Laplacian matrix only has a zero eigenvalue, and the other eigenvalues have positive real parts. Since the neighboring relationship among agents may be changed in the movement of agents, the interaction topology can be time varying. In this case, the consensus control protocol is described as follows:

34

1 Introduction

u i (t) =

∑

  wi j (t) x j (t) − xi (t) ,

(1.21)

j∈Ni (t)

where Ni (t) represents the time-varying neighboring set of agent i and wi j (t) is the time-varying interaction weight from agent j to agent i. With protocol (1.21), the dynamics of the multi-agent system can be modeled as: x(t) ˙ = −L(t)x(t),

(1.22)

where L(t) denotes the time-varying Laplacian matrix. The time-varying topology can be divided into twofold: (i) switching topology with time-varying neighboring sets and time-invariant interaction weights; (ii) weighting perturbation; that is, the neighboring sets are time-variant while the interaction weights own the boundednorm perturbations. Olfati-Saber and Murray [62] pointed out that if all the switching topologies are strongly connected and balanced, then system (1.22) can reach consensus for any given switching signal and initial states. In [64], Ren and Beard proved that multi-agent system can reach consensus if and only if the interaction topology contains a spanning tree, and proved that if the union of the interaction topologies within serval switching intervals contains a spanning tree, then system (1.22) can reach consensus. By the linear matrix inequality (LMI) tools, the conditions of the consensus reached by system (1.22) were given in [65]. Note that if interaction topology is still connected after removing the edges with the weight wi j (t) < δ, then it is said to be δ-connected. It can be found in [3] that if there exist ( t+T δ > 0 and T > 0 such that L = t L(s)ds is δ-connected for ∀t ∈ [0, +∞), then system (1.22) can reach consensus. In [66], Lin et al. proposed the H∞ consensus control method to deal with the consensus problems with disturbances and weighting perturbations. For multi-agent systems, there exist communication delays in the information transmission among agents, which may affect the system performance and even destroy the stability of system. Notice that the delay may be time-varying due to the movement of agents, congestion of communication channels, and change of physical characteristics of transmission media. For the delay-dependent case, the consensus control protocol can be described in the following form: u i (t) =

∑

  wi j x j (t − τ ) − xi (t − τ ) ,

(1.23)

j∈Ni

where τ is a constant time delay. With protocol (1.23), the dynamics of the multi-agent system can be modeled as x(t) ˙ = −L x(t − τ ).

(1.24)

In [67], Bliman and Ferrari-Trecate proved that system (1.24) with connected undirected topology achieves consensus if and only if τ ∈ [0, π/(2λ N )), where λ N

1.3 Development Direction of Consensus: Literature Reviews

35

is the maximum eigenvalue of L. It can be found that λ N is the measurement index of the delay robustness for first-order multi-agent systems. A smaller λ N indicates a bigger upper bound of the delay. Tian and Liu [68] proposed a consensus control protocol with both input delays and transmission delays as follows: u i (t) =

∑

  wi j x j (t − τi j − Di ) − xi (t − Di ) ,

(1.25)

j∈Ni

where τi j is the transmission delay from agent j to agent i and Di is the input delay. By the frequency domain analysis method, Tian ∑ and Liu concluded that if the interaction topology owns a spanning tree and Di j∈Ni wi j < 1/2, then firstorder multi-agent systems can achieve consensus by protocol (1.25). Moreover, the consensus problems for first-order multi-agent systems with multiple time-varying delays were addressed in [69, 70]. More early works on the consensus of first-order multi-agent systems can be found in [71, 72]. Based on these early works, many researchers have conducted extensive and significant research on the consensus of first-order multi-agent systems (see [73–75] for details), including jointly connected switching topologies, random noises, robust control, event-triggered control, and group consensus.

1.3.1.2

Consensus of Second-Order Multi-agent Systems

The consensus of first-order multi-agent systems is determined by the structure of the communication topology. However, different from the first-order multi-agent system, one of the most significant characteristics of the second-order multi-agent system is that the consensus achievability is related to both communication topologies and the dynamics of agents. Hence, the consensus analysis and design of second-order multi-agent systems are more challenging than those of the first-order ones. In the following, a literature review of consensus of second-order multi-agent systems is presented in three aspects: that is, consensus with fixed topologies, consensus with time delays, and other consensus issues. In the relevant literatures of the consensus of second-order multi-agent systems, researchers mainly describe the dynamics of agents as the second-order integrator, i.e.,  x˙i (t) = vi (t), (1.26) v˙i (t) = u i (t), where i = 1, 2, . . . , N , xi (t) and vi (t) represent the position and velocity of agent i, respectively. u i (t) is the control input of agent i. If limt→+∞ (xi (t) − x j (t)) = 0 and limt→∞ (vi (t) − v j (t)) = 0 (i, j = 1, 2, . . . , N ), then multi-agent system (1.26) is said to achieve consensus.

36

1 Introduction

For the case of consensus with fixed topologies, Ren and Atkins [74] proposed the following consensus control protocol: u i (t) =

∑

    wi j x j (t) − xi (t) + γ v j (t) − vi (t) ,

(1.27)

j∈Ni

where γ > 0 is a bounded scaling factor. Let x(t) = [x1T (t), x2T (t), . . . , x NT (t)]T and v(t) = [v1T (t), v2T (t), . . . , v TN (t)]T , then the dynamics of system (1.26) with protocol (1.27) can be described as: 

    x(t) ˙ 0 I x(t) = . v(t) ˙ −L −γ L v(t)

(1.28)

According to the above analysis, Ren et al. obtained the following conclusions: (i)

The necessary condition of the consensus of system (1.28) is that the communication topology contains a spanning tree. (ii) A sufficient condition of the consensus of system (1.28) is that the communication topology contains a spanning tree and the scaling factor γ satisfies that ⎡ | 2 |  , γ > max √ λi /=0 π i) |λi | cos 2 − tan−1 −Re(λ Im(λi ) where λi is the eigenvalues of −L with Re(λi ) and Im(λi ) being the real part and the imagine part of λi , respectively. (iii) If system (1.28) achieves consensus, then the consensus function satisfies that   lim xi (t) − p T x(0) − t p T v(0) = 0,

t→+∞

  lim vi (t) − p T v(0) = 0,

t→+∞

where p is the left eigenvector associated with the zero eigenvalue of −L. It can be found that the connectivity of the communication topology is a significant condition for the consensus of both first-order and second-order multi-agent systems. Moreover, the consensus function of first-order multi-agent systems is a constant, while it may be time-varying for second-order multi-agent systems. For the consensus of second-order multi-agent systems with time delays, Lin et al. proposed the following consensus control protocol: u i (t) =

∑ j∈Ni

     wi j k1 x j (t − τ ) − xi (t − τ ) + k2 v j (t − τ ) − vi (t − τ ) , (1.29)

1.3 Development Direction of Consensus: Literature Reviews

37

where k1 > 0, k2 > 0, and τ > 0 is the constant time delay. Let ξ(t) = [x1T (t), v1T (t), x2T (t), v2T (t), . . . , x NT (t), v TN (t)]T , then the dynamics of system (1.26) with protocol (1.29) can be modeled as ξ˙ (t) =



01 IN ⊗ 00

! !  0 0 ξ(t − τ ). ξ(t) − L ⊗ k1 k2

(1.30)

By the frequency domain analysis method, Lin et al. [73] proved that system (1.30) with connected undirected topologies can achieve consensus if and only if τ ∈ [0, τ ), where arccos τ=/ k22 λ N 2k1

+

/ 1 k22 λ N 2k1

/

+ k22 λ N 2k1

k22 λ N 2k1

2

!2 +1

,

+ k12 λ2N

λ N is the maximum eigenvalue of the Laplacian matrix L. In [5], Yu et al. proposed the following consensus control protocol: u i (t) =

∑

  wi j −k1 x j (t − τ ) − k2 v j (t − τ ) ,

(1.31)

j∈Ni

where k1 > 0, k2 > 0, and τ > 0 is the constant time delay with the upper bound τ = max2≤i≤N {θi1 /ωi1 } satisfied that 0 ≤ θi1 ≤ 2π , cos θi1 = 2 2 , sin θi1 = Re(λi )ωi1 k2 + Im(λi )k1 /ωi1 , and ωi1 = Re(λi )k1 − Im(λi )ωi1 k2 /ωi1 / / ||λi ||2 k22 + ||λi ||4 k24 + 4||λi ||2 k12 /2. Note that due to the different structures between protocol (1.29) and protocol (1.31), the upper bounds of the time delays are different. Hence, it can be found that the design of the consensus control protocol is an important issue of consensus for second-order multi-agent systems.

1.3.1.3

Consensus of High-Order Multi-agent Systems

In many practical applications, multi-agent systems cannot be modeled as the firstorder or second-order ones. In this case, the dynamics of multi-agent systems are of high order, whose coordinate variables are more than second order. In [76], the dynamics of each agent is described as the following high-order integrator: ⎧ (0) ξ˙i (t) = ξi(1)(t), ⎪ ⎪ ⎪ ⎨ .. . (l−2) ⎪ ˙ ⎪ ξ (t) = ξi(l−1)(t), ⎪ ⎩ i (l−1) ξ˙i (t) = u i(t),

(1.32)

38

1 Introduction

where ξi(k) (t) ∈ Rm (m > 0, i = 1, 2, . . . , N , k = 0, 1, . . . , l − 1) denotes the coordinate variable of agent i and u i (t) ∈ Rm is the control input. Note that multiagent system (1.32) is said to achieve the state consensus if limt→+∞ (ξi(k) (t) − ξ (k) j (t)) = 0, i, j = 1, 2, . . . , N , k = 0, 1, . . . , l − 1. In [76], the following state consensus control protocol is proposed. u i (t) = −

∑ j∈Ni

gi j wi j

l−1 ∑

  (k) γk ξ (k) (t) − ξ (t) , i = 1, 2, . . . N , j i

(1.33)

k=0

where gi j > 0, wi j > 0 and γk > 0. When l = 3, it is proven that system (1.32) achieves state consensus if and only if there are only three zero eigenvalues of the following matrix ⎡

⎤ 0 I 0 ⎡=⎣ 0 0 I ⎦, −γ0 L −γ1 L −γ2 L and other eigenvalues have negative real parts. Besides, if limt→+∞ (ξi(0) (t) − ξ (0) j (t)) = 0, i, j = 1, 2, . . . , N , then system (1.32) is said to achieve the output consensus. To address the output consensus for high-order multi-agent systems, a consensus control protocol was proposed in [77] as follows: u i (t) = −

l−1 ∑ k=1

K 1 ξi(k) (t) + K 2

∑

  (0) wi j ξ (0) j (t) − ξi (t) , i = 1, 2, . . . N ,

j∈Ni

(1.34) where K 1 and K 2 are the control gain matrices with proper dimensions. According to protocol (1.34), the output consensus criterion was given by the LMIs. Note that the high-order integrator can be regarded as the extension of the first-order integrator in high order, which is a special case of high-order multi-agent systems. In general, the dynamics of high-order multi-agent system can be described in the state-space form as follows:  x˙i(t) = Axi(t) + Bu i(t), (1.35) yi(t) = C xi(t), where i = 1, 2, . . . , N , A ∈ Rd×d , B ∈ Rd×m , C ∈ Rq×d , xi (t) is the state, yi (t) is the output, and u i (t) is the control input. If limt→+∞ (xi (t) − x j (t)) = 0 (i, j = 1, 2, . . . , N ), then system (1.35) is said to achieve the state consensus. Both states and outputs of system (1.35) can be utilized to construct the consensus control protocol. In [78], Xiao and Wang considered the following state feedback consensus control protocol:

1.3 Development Direction of Consensus: Literature Reviews

u i (t) = K 1 xi (t) + K 2

∑

39

  wi j x j (t) − xi (t) ,

(1.36)

j∈Ni

where K 1 and K 2 are the control gain matrices with proper dimensions. With protocol (1.36), the dynamics of system (1.35) can be rewritten as: x(t) ˙ = (I N ⊗ (A + B K 1 ) − L ⊗ B K 2 )x(t),

(1.37)

where L is the Laplacian matrix. Xiao and Wang proved that system (1.37) achieves the state consensus if and only if the following conditions hold: (i) the ranks of A + B K 1 and ( A + B K 1 )2 are equal; (ii) the matrix A + B K 1 has no eigenvalue with positive real parts; (iii) A + B K 1 − λi B K 2 (i = 2, 3, . . . , N ) is Hurwitz with λi being the nonzero eigenvalue of L. In [79], Ma and Zhang proposed the following static output feedback consensus control protocol: u i (t) = K

∑

  wi j y j (t) − yi (t) ,

(1.38)

j∈Ni

where K is the control gain matrix with proper dimensions. Ma and Zhang proved that if rank(C) = rank([C T , P T B]T ) with P being the solution of the Riccati equation AT P + P A − P B B T P + I = 0, then system (1.35) can achieve consensus by protocol (1.38) if and only if the interaction topology contains a spanning tree and (A, B) is stabilizable. In [80], Li et al. proposed the following dynamical output feedback consensus control protocol: ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

⎛ v˙i(t) = ( A + B K )vi(t) + F ⎝c

∑ j∈Ni

⎞ ∑    wi j C v j(t) − vi (t) − c wi j y j(t) − yi(t) ⎠ 

j∈Ni

,

(1.39)

u i(t) = K vi(t),

where K and F are the control gain matrices with proper dimensions. By the separation principle, Li et al. proved that system (1.35) can achieve consensus by protocol (1.39) if and only if ( A, B, C) is stabilizable and detectable. Note that the consensus region was introduced to analyze the robustness of the multi-agent systems, and the protocol states are need for protocol (1.39). Xi et al. [81] proposed the partial stability method to tackle the output consensus problems for high-order multi-agent systems and given the output analysis and design criteria. Moreover, the explicit expression of the output consensus function was determined to describe the macroscopic movement of the whole multi-agent system. In [82], an output consensus control protocol was designed with the reduced-order observer based on the estimation errors, which can drive the multi-agent system to the output consensus if the interaction topology owns

40

1 Introduction

a spanning tree. You et al. [83] proposed an event-triggered dynamical output feedback consensus control protocol to address the output consensus tracking problems for high-order multi-agent systems with actuator failures.

1.3.1.4

Consensus of Singular Multi-agent Systems

If the dynamics of agents contains the singular algebra constraints, or the dynamics of the multi-agent systems is composed of the slow and fast sub-dynamics, then it should be modeled as the singular multi-agent systems as shown in [55], whose dynamics can be described as follows 

E x˙i(t) = Axi(t) + Bu i(t), yi(t) = C xi(t),

(1.40)

where i = 1, 2, . . . , N , E ∈ Rd×d , A ∈ Rd×d , B ∈ Rd×n , C ∈ Rh×d . xi (t) ∈ Rd and yi (t) ∈ Rh represent the states and the outputs of agent i, respectively. u i (t) ∈ Rn is the control input. The matrix E satisfies that rank(E) < d, which indicates that the dynamics of the singular multi-agent system contains not only dynamical modes, but the static modes and impulse modes. In recent years, many scholars investigate the consensus problems of singular multi-agent systems and obtained several profound and interesting results. Xi et al. [53] investigated the admissible consensus problems for singular multiagent systems with fixed topologies, where the sufficient conditions of the admissible consensus analysis and design were given. Note that the criteria only contain three LMIs independent of the number of agents, which can ensure the scalability of the multi-agent systems. Yang and Liu [84] addressed the consensus problems of singular multi-agent systems with fixed topologies by proposing the dynamical output feedback consensus control protocols. In [85], Gao et al. designed an observerbased consensus control protocol via the dynamical output feedback and given the admissible consensus criteria in forms of the generalized Riccati equation, which can combine the consensus control protocols with the full-order observers and the reduced-order observers into a unified framework. In [86], the admissible output consensus problems were studied, where a dynamical output feedback consensus protocol with the time delay was constructed and the admissible output consensus design criteria were given. Liu et al. [87] considered the output regulation problems for heterogeneous multi-agent systems and proposed the distributed adaptive control protocol via the state feedback and the output feedback. According to the consensus principle, the formation-containment problems were investigated for singular multiagent systems in [88], where a distributed observer was proposed without the eigenvalues related to the leader to estimate the convex hull of the states of the leader. Meanwhile, the state feedback and the output feedback formation-containment control protocols were given.

1.3 Development Direction of Consensus: Literature Reviews

41

Recently, many new results of the consensus control for singular multi-agent systems have been emerged as shown in [89–91], including the consensus tracking, the admissible consensus, and H∞ tracking design. Compared with the consensus of normal multi-agent system, the research results on the consensus control of singular multi-agent systems indicate that the consensus analysis and design of singular multiagent systems are more challenging since the impulse-free properties should be satisfied. Hence, the relevant conclusions of the consensus of normal multi-agent systems are difficult to apply directly to that of the singular multi-agent systems.

1.3.2 Consensus of Multi-agent Systems with Different Interaction Topologies In the past two decades, the distributed cooperative control based on the algebra graph framework has been widely investigated by the researchers. In this section, a literature review on the consensus control is presented from the point of view of interaction topology structure and the topology variance.

1.3.2.1

Consensus of Multi-agent Systems with Different Interaction Topology Structures

The interaction topology structure of multi-agent systems has significant influence on the consensus control. In the interaction topology, if there exists an isolated node that has no link to all other nodes, then it cannot receive the cooperative control signal from other nodes through the interaction topology. In this case, the consensus cannot be achieved by the isolated node. Hence, it can be found that the interaction topology has a direct effect on the ability of multi-agent systems to achieve the consensus control target. From the topology structure perspective, the interaction topology can be divided into the leaderless ones and the leader-following ones. Correspondingly, the consensus of multi-agent system can be distinguished as the leaderless consensus and the leader-following consensus according to the topology structure, where the leader-following consensus is also called the consensus tracking. For the leaderless consensus, all agents exchange the neighboring information to achieve a collaborative behavior autonomously, which cannot be specified previously. Noting that a virtual leader, also called the consensus function, can describe the expected trajectory of all agents. Besides, there exists a real leader in the leader-following consensus, where the leader can be regarded as an objective to be tracked by all followers. In the following, a review on the leaderless consensus and the leader-following consensus is presented. • The case of the leaderless interaction topology structure

42

1 Introduction

There exists no leader node in the leaderless interaction topology, where all nodes own the ability of sending and receiving information from each other and are equal in the meaning of communication transmissions. When the consensus is achieved by the coordinate variables of each agent, the movement trajectory of the whole multiagent system is determined by the motion mode of all agents. A typical leaderless interaction topology structure is shown in Fig. 1.22. In the fixed topology, the communication links among agents are time-invariant, and the corresponding Laplacian matrix is constant. Since the structure of the fixed topology is clear and the model is simple and easy to analyze, the researchers usually first consider the fixed topology cases when investigating the consensus control. In recent two decades, the research on the consensus control of fixed topologies has been emerging and has achieved a series of important research results. For the leaderless consensus, some research results on the undirected fixed topologies were presented in [92–95]. Considering the influence of time delays, Li et al. [92] gave the consensus criterions for high-order multi-agent systems with undirected connected topologies. In [93], the consensus control protocol was constructed via the dynamical output feedback for the stochastic multi-agent systems, which can ensure the predefined upper bound constraints of the mean-square consensus performance. Utilizing the event-triggered consensus strategy, the consensus control methods for the matching nonlinear uncertainties and the external disturbances were given in [94, 95], respectively. Note that the event-triggered control protocol is updated only when a particular event occurs, which can effectively reduce the communication consumptions of the consensus control. For the consensus-based formation control with undirected fixed topologies, Cheng et al. [96] proposed the adaptive event-triggered formation control protocol by the state feedback and the output feedback, which can achieve the fully distributed control. Because the Laplacian matrix of the undirected topology is symmetric, it can be diagonalized to decompose the system associated with the eigenvalues, which can facilitate the analysis of the consensus convergence. However, when studying the consensus control problems of the directed topology cases, it is necessary to consider the impact of the asymmetric Laplacian matrix, which is more complicated and challenging. Under the condition of directed fixed topologies, an event-triggered Fig. 1.22 Leaderless structure for undirected fixed interaction topology

3 4

2

5

1

... N

1.3 Development Direction of Consensus: Literature Reviews

43

Fig. 1.23 Leader-following structure for undirected fixed interaction topology

0

3

2 1

4 5

...

N

consensus control protocol was designed for second-order multi-agent systems in [97], and the sufficient conditions of the analysis design were given. The output consensus design criteria were given in [98] for heterogeneous multi-agent systems with time delays considering the directed fixed topologies. Liu et al. [99] modeled the interaction topology as the directed graph and investigated the asynchronous sampled-data event-triggered consensus problems for singular multi-agent systems. Zhao et al. [100] gave the consensus-based time-varying formation design criteria for high-order multi-agent systems with directed fixed topologies and achieved the fully distributed time-varying formation control by the scaling adaptive strategy. • The case of the leader-following interaction topology structure Different from the leaderless case, there exists the leader node in the leaderfollowing topology, which sends only information without receiving information from followers. There exists at least a follower that can receive the information from the leader, and the followers can communicate with each other. In this sense, the leader can be regarded as the target that is tracked by the followers in the consensus motions, which means that the consensus trajectory is determined by the motion mode of the leader when the consensus is achieved by the multi-agent systems. A typical leader-following interaction topology structure is shown in Fig. 1.23. For the fixed topology with leader-following structures, Wang and Huang [101] investigated the adaptive consensus tracking problems for the multi-agent systems with the Euler–Lagrange dynamics, where the interaction topology among followers was modeled as the connected undirected graph, and the dynamics of the leader was generated by the exosystems. In [102], a consensus tracking control protocol was designed with the input saturation for general high-order multi-agent systems, and the consensus tracking was achieved on the condition of the fixed topology. Shariati and Zhao [103] studied the robust output consensus tracking control problems for the uncertain multi-agent systems with time-varying delays, where the directed topology was required to contain a spanning tree with the leader locating at the root node of the spanning tree. Under the same topology condition, an adaptive event-triggered consensus tracking control protocol was established in [104] such that the nonlinear multi-agent systems with external disturbances reach the consensus

44

1 Introduction

tracking. Considering the undirected topology among followers, Huang et al. [105] proposed a consensus-based formation control algorithm for the second-order multiagent systems with multiple time delays, and the upper bound of the maximum time delay was determined by the frequency analysis approach.

1.3.2.2

Consensus of Multi-agent Systems with Switching Topologies

In practical applications, due to the disturbances of the transmission or the communication security, there may exist the exchange of the communication link relationship among the agents, which can be modeled as the switching topology. In this case, the Laplacian matrix of the switching topology is no longer time-invariant, but is piecewise continuous, and the neighboring set of each agent is time-varying. It is usually required for the switching topology that the topology in the topology set is connected, or the union of the topology is connected in a period; that is, the switching topologies are jointly connected. In particular, the connectivity of the topology means that a directed topology contains at least a spanning tree, or the undirected topology is connected. Compared with the fixed topology case, the consensus control with the switching topology is more complicated and challenging. In [106], the consensus control problem was investigated with the leaderless switching topology, and the consensus design criterion was derived under the condition that each topology in the switching topology set is connected and undirected. For the first-order multi-agent system with external disturbances, the robust consensus control problem was studied in [107] with the undirected switching topology. For a class of stochastic multi-agent systems with external disturbances and directed switching topologies, Zhou et al. [108] constructed a consensus control protocol by the dynamical output feedback and given the sufficient conditions of the weighted H∞ consensus design. In [109], a consensus control protocol was proposed with the sampled measurement output feedback, which can drive the multi-agent system to the state consensus with jointly connected switching topologies. For the heterogeneous multi-agent systems consisting of the first-order dynamics and second-order dynamics with jointly connected switching topologies, Chen et al. [110] studied the scaled consensus problem. It is required for the scaled consensus that the agents are convergent to different consensus states, which are constraint by certain ratios. For the case of the leader-following switching topology, Jiang et al. [111] proposed the sampled-data H∞ consensus tracking control protocol with the time delay and external disturbances, which ensures the H∞ performance of the disturbance rejection of the consensus. In [112], an event-triggered fault-tolerant consensus tracking control protocol was constructed with the semi-Markov switching topologies, and the sufficient condition of the consensus tracking was given. For the heterogeneous second-order nonlinear multi-agent system with directed switching topologies, the consensus tracking control problems were studied in [113], where the exponential convergence of the consensus tracking was proven by the switching Lyapunov function method. An event-triggered consensus tracking control protocol was designed with the directed switching topologies and external disturbances in [114]. For a

1.3 Development Direction of Consensus: Literature Reviews

45

class of heterogeneous lower triangular nonlinear multi-agent systems, Liu and Huang [115] studied the cooperative robust adaptive output regulation problems, and the theoretical results were applied to the consensus tracking control of the multi-agent systems with the Lorentz chaotic dynamics. Wang [116] designed the adaptive dynamical output feedback formation control protocol, which can achieve the fully distributed time-varying formation control for the second-order multi-agent systems with the leader-following directed switching topologies. From the existing related works on the consensus control of multi-agent systems, it can be found that the main methods of the consensus control can be divided into the state-space decomposition method [117] and the state error method [118]. The state-space decomposition method adopts the nonsingular transformation matrices to decompose the dynamics of the closed-loop systems into the consensus dynamics and the consensus complement dynamics. The consensus dynamics is utilized to determine the explicit expression of the consensus function, which can describe the macroscopic motion mode of the whole multi-agent. The consensus complement dynamics is adopted to describe the relative motion mode among each agent, and the consensus can be achieved if the relative motions of all agents are stable. Hence, the consensus of the multi-agent systems can be transformed to the stability of the consensus complement dynamics by the state-space decomposition method. The state error method utilizes the state errors or the output errors among agents to construct the feedback control. If the errors are stable, then the consensus can be achieved by the multi-agent systems. For the leader-following case, the errors are those of coordinate variables between leaders and followers. For the leaderless case, the errors are those of coordinate variables among all agents. It should be noticed that the state error method can be utilized to both the leaderless and the leader-following consensus control but cannot determine the consensus function. The state-space decomposition method can determine the consensus function, which takes advantage in the leaderless case.

1.3.3 Consensus of Multi-agent Systems with Cost Indexes In many practical applications, it is required for multi-agent systems not only to achieve the consensus, but also to guarantee the consensus regulation performances as well as the control energy consumptions, which can be summarized as the optimized consensus control problems. In the past decade, the optimized consensus problem with the cost index has been widely investigated by the scholars. According to interaction type of the cost index, the optimized consensus control can be divided into the decentralized ones and the global ones.

46

1.3.3.1

1 Introduction

Consensus of Multi-agent Systems with Decentralized Cost Indexes

The decentralized cost index consists of the local cost index of each agent, which can be described as follows: min

N ∑

gi (x),

(1.41)

i=1

where gi (x) is the local cost index of agent i. The decentralized optimized consensus control is to achieve the control goal by optimizing the cost index (1.41), which can be adopted to the performance evaluation and the resource scheduling problems of the multi-agent systems. According to the average consensus, Nedic et al. [119] investigated the optimization problems of multi-agent systems, where the objective function is optimized at each iteration of the algorithm by the gradient optimization method. In [120], the optimization problem was converted to an intersection computation problem by assuming that the intersection of the convex solution set of each agent is nonempty. By the convex and non-smooth analysis, the properties of the distance functions for the global optimal solution set were established. The distributed optimization problems with nonconvex objective functions for multi-agent systems were studied in [121], where the nonconvexity of the objective function was tackled by introducing and updating the dynamic coupling gains. Chen and Li [122] proposed the distributed optimization algorithm to calculate the optimal solution for the multiagent systems in the fixed time, and the fixed-time convergence was proven by the convex optimization and the fixed-time Lyapunov function method. For a class of uncertain multiple input multiple output (MIMO) nonlinear multi-agent systems, the distributed optimization protocol was designed by the pseudo gradient technique, the internal model principle, and the adaptive strategy in [123]. Note that the decentralized cost index is the sum of the local cost indexes of agents. In this sense, the local cost index only reacts on the corresponding agent and has no interaction between each other. Although the cost of each agent can be optimized, it is difficult to optimize the global cost of the whole multi-agent system. From the perspective of the distributed interaction and the overall optimization of the whole system, the optimization of the global cost index should also be considered.

1.3.3.2

Consensus of Multi-agent Systems with Global Cost Indexes

The global cost index is described as follows: (t J= 0



 x T (t)Qx(t) + u T (t)Ru(t) dt,

(1.42)

1.3 Development Direction of Consensus: Literature Reviews

47

where x(t) and u(t) represent the global coordinate variables and control inputs of the multi-agent systems, Q and R are the positive definite symmetric weighting matrices. For the first-order multi-agent system, Cao and Ren [124] constructed the interaction-free cost index and the interaction-related cost index and derived the optimal consensus criteria via the Riccati equation, which can ensure that the cost index is minimized while achieving the consensus. In [125], the optimal output consensus control problem was addressed, where the minimum value of the cost index was obtained. The main results of [124, 125] indicated that the minimum value of the cost index is obtained only when the interaction topology is a complete graph, which means that there exists the edge for any two nodes. By introducing the inverse optimal theory and the partial stability method, Movric and Lewis [126] derived the sufficient condition of the optimal consensus in the form of Riccati equation for the general high-order multi-agent systems with fixed topologies. Zhang et al. [127] gave the sufficient and necessary conditions of the global optimal consensus for the general high-order multi-agent systems with directed topologies, where it is required that the Laplacian matrix of the topology is diagonalized. From the above analysis, it can be found that the constraints of the cost index and the interaction topology are limited for the global optimal consensus. Hence, many researchers have turned to study the suboptimal consensus control in more relaxed constraints. The suboptimal consensus control means to determine the upper bound of the cost index when achieving the consensus, where the cost index is established by the consensus regulation performance and the control energy consumption. In this sense, the suboptimal consensus is called the guaranteed-cost consensus or guaranteed-performance consensus. The guaranteed-performance consensus control considers the constraint of the consensus regulation performance, while the guaranteed-cost consensus control considers both the consensus regulation performance and the control energy consumption. For the guaranteed-performance consensus of second-order multiagent systems, Guan et al. [128] gave the guaranteed-performance consensus control criterion by the impulsive control method, and the upper bound of the cost index was determined, i.e., guaranteed-performance cost. The guaranteed-performance consensus control problem was investigated for second-order multi-agent systems with time delays in [129]. For general high-order multi-agent systems, the fully distributed guaranteed-performance consensus analysis and design criteria were given in [130] by the proposed translation-adaptive method, and the guaranteedperformance cost was determined. For the high-order Lipschitz nonlinear multiagent systems with jointly connected switching topologies, Zheng et al. [131] proposed the sufficient conditions of the guaranteed-performance consensus design for the leaderless and leader-following topology structures. In [132], the nonlinear consensus control protocol was designed for singular multi-agent systems, where the guaranteed-performance consensus control criterion was given by the Riccati equation. The cost index in [128–132] can be summarized in the following form:

48

1 Introduction

(+∞ J= εT (t)Pε(t)dt,

(1.43)

0

where ε(t) denotes the error vector of the global coordinate variables and P is the positive definite symmetric weighting matrix. It can be found from (1.43) that the weighting matrix P determines the interaction relationship of the global consensus regulation performance on the multi-agent system. For the guaranteed-cost consensus control, Wang et al. [133] introduced the guaranteed-cost control of the isolated systems into the multi-agent systems and given the guaranteed-cost consensus analysis and design criteria for the first-order multi-agent systems with fixed topologies. For the second-order multi-agent system with heterogeneous inertias, the sufficient condition of the guaranteed-cost consensus control was given in [134], and the upper bound of the cost index was determined, i.e., the guaranteed cost. For general high-order multi-agent systems, the event-triggered consensus and sampled-data consensus were investigated in [135, 136], where the consensus control problems were transformed to the stability problems of the delayed closed-loop systems by the input delay method, and the guaranteed-cost consensus control criteria were given by the LMI technique. In [135, 136], the dimensions of the variables of the criteria were associated with the number of the agents, so the scalability of the multi-agent systems cannot be ensured. For the general high-order multi-agent systems with time delays, the guaranteed-cost consensus analysis and design criteria were proposed in [137], which can ensure the scalability of the multiagent systems. For the singular multi-agent systems with fast and slow sub-dynamics, the distributed guaranteed-cost consensus control criterion was derived in [138] under the condition of the undirected topologies. Based on the consensus control method, the robust H∞ guaranteed-cost time-varying formation tracking control protocol was proposed in [139] for the general high-order multi-agent systems with the external disturbances and time delays.

1.4 Notes In this chapter, the background of the consensus for the singular multi-agent system was introduced. The ideas of the consensus were shown by the collective behavior of the animal groups and the cooperative operation of multi-agent systems in society. The interactive relationship among agents was described by the interaction topology on graphs. Moreover, some examples and models of the singular multi-agent systems were presented to give the basic knowledge of the singular multi-agent systems. Finally, the literature review was summarized to show the development direction of the consensus.

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128. Guan ZH, Hu B, Chi M, He DX, Cheng XM (2014) Guaranteed performance consensus in second-order multi-agent systems with hybrid impulsive control. Automatica 50(9):2415– 2418 129. Yu SZ, Yu ZY, Jiang HJ, Hu C (2018) Leader-following guaranteed performance consensus for second-order multi-agent systems with and without communication delays. IET Control Theory Appl 12(15):2055–2066 130. Xi JX, Wang C, Liu H, Wang L (2019) Completely distributed guaranteed-performance consensualization for high-order multiagent systems with switching topologies. IEEE Trans Syst Man Cybern Syst 49(7):1338–1348 131. Zheng T, Xi JX, Yuan M (2018) Guaranteed-performance consensus design for Lipschitz nonlinear multi-agent systems with jointly connected topologies. Int J Robust Nonlinear Control 29(11):3627–3649 132. Gao ZY, Zhang HG, Duan J, Cai YL (2020) Guaranteed-performance consensus for descriptor nonlinear multi-agent systems based on distributed nonlinear consensus protocol. Neurocomputing 383:359–367 133. Wang Z, Xi JX, Yao ZC (2015) Guaranteed cost consensus for multi-agent systems with fixed topologies. Asian J Control 17(2):729–735 134. Yu ZY, Jiang HJ, Mei XH, Hu C (2018) Guaranteed cost consensus for second-order multiagent systems with heterogeneous inertias. Appl Math Comput 338:739–757 135. Zhou XJ, Shi P, Lim CC, Yang CH, Gui WH (2015) Event based guaranteed-cost consensus for distributed multi-agent systems. J Franklin Inst 352(9):3546–3563 136. Zhao YD, Zhang WD (2017) Guaranteed cost consensus protocol design for linear multi-agent systems with sampled-data information: an input delay approach. ISA Trans 67:87–97 137. Wang Z, He M, Zheng T, Fan ZL (2018) Guaranteed cost consensus for high-dimensional multi-agent systems with time-varying delays. IEEE-CAA J Autom Sin 5(1):181–189

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Chapter 2

Fundamental Theory

This chapter gives the related fundamental theory for the admissible consensus of the singular multi-agent systems, including the basic knowledge of the algebraic graph theory, the linear algebra, the theories of the linear system, and the singular system. Besides, the consensus problems of the high-order LTI multi-agent systems are discussed to introduce the basic knowledge of the consensus control theory.

2.1 Algebraic Graph Theory In this monograph, we are concerned with the behaviors and interactions of dynamical systems that are interconnected by the links in a communication or sensing network. This network is modeled as a graph with the nodes (or agents) denoting the dynamical systems, where the directed edges in the graph represent the information transmission channels between the systems. The value of the node might represent physical quantities including attitude, position, temperature, voltage, and so on. The fundamental control issues concern how the graph topology interacts with the local feedback control protocols of the agents to produce overall behaviors of the interconnected nodes. We call this the study of multi-agent dynamical systems on graphs.

2.1.1 Definitions Suppose that N agents interact with each other through a communication or sensing network or a combination of both. Thus, we can model the interactions among agents by a directed graph G = (V (G), E(G)), which consists of a node set V (G) = {v1 , v2 , . . . , v N } and an edge set E(G) ⊆ {(vi , v j ), vi , v j ∈ V (G)} (∀i, j ∈ {1, 2, . . . , N }). Each edge ei j is denoted by (vi , v j ), where vi is regarded © Science Press, Beijing and Springer Nature Singapore Pte Ltd. 2023 J. Xi et al., Admissible Consensus and Consensualization for Singular Multi-agent Systems, Engineering Applications of Computational Methods 11, https://doi.org/10.1007/978-981-19-6990-4_2

57

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2 Fundamental Theory

as the parent node (or head), and v j is the child one (or tail). A weighted adjacency matrix of the graph is set as W = [wi j ] ∈ R N ×N , where wi j is the edge weight of (v j , vi ) with wi j > 0 if and only if (v j , vi ) ∈ E(G) and wi j = 0 otherwise. Note the order of the indices i and j in this definition. Moreover, it is assumed that each agent does not contain the self-loop; that is, wii = 0. { } The set of neighbors for vi is denoted by Ni = v j ∈ V (G) : (v j , vi ) ∈ E(G) . The in-degree of vi is defined as degin (vi ) =

Ni ∑

wi j .

j=1

The out-degree of vi is defined as degout (vi ) =

Ni ∑

w ji .

j=1

It can be found that the in(out)-degree of node vi is the sum of the edge weights with vi being the edge head (tail). For node vi of a graph, if the in-degree is equal to the out-degree; that is degin (vi ) = degout (vi ), then the graph is a balance one. Define the diagonal in-degree matrix D = diag{degin (v1 ), degin (v2 ), . . . , degin (v N )} and the Laplacian matrix L = D − W . Note that the row sums of the Laplacian matrix L are all zero. The Laplacian matrix is of extreme importance in the study of dynamical multi-agent systems on graphs. More details on graph theory can refer to [1, 2]. If for any ei j ∈ E(G), e ji ∈ E(G) simultaneously, then G is an undirected graph, otherwise, G is a directed graph. (1) For an undirected graph, if there is not any isolated node, then it is connected. (2) For a directed graph, a directed path from node vi to v j is represented as a sequence of ordered edges with the form as (vi , vi1 ), (vi1 , vi2 ), …, (vil , v j ), where vik ∈ V (G) (k = 1, 2, . . . , l). A directed graph is said to have a spanning tree if there exists at least one node vi (∀i ∈ {1, 2, . . . , N }) having a directed path to every other nodes, where node vi having no parent node and is the root node of the spanning tree. If for any two different nodes vi and v j , there exists a directed path from node vi to node v j , then G is said to be strongly connected. If for any two different nodes vi and v j , there exists a node vk that has directed paths to node vi and v j , then G is said to be weakly connected. But for an undirected graph, it is equivalent whether the graph is weakly or strongly connected, both cases are called as connected. The following two lemmas show some basic properties of the Laplacian matrix. Lemma 2.1 [3] For a directed graph G with N nodes, let L ∈ R N ×N be the Laplacian matrix and 1 = [1, 1, . . . , 1]T ∈ R N , then it holds that (i)

L at least has one 0 eigenvalue, and 1 is the associated eigenvector; that is, L1 = 0;

2.1 Algebraic Graph Theory

59

(ii) If G has a spanning tree, then 0 is a simple eigenvalue of L, and all the other N − 1 eigenvalues have positive real parts; (iii) If G does not have a spanning tree, then L has at least two 0 eigenvalues with the geometric multiplicity being not less than 2. Lemma 2.2 [4] For an undirected graph G with N nodes, let L ∈ R N ×N be the Laplacian matrix and 1 = [1, 1, . . . , 1]T ∈ R N , then it follows that (i)

L has at least one 0 eigenvalue, and 1 is the associated eigenvector satisfying L1 = 0; (ii) If G is connected, then 0 is a simple eigenvalue of L, and all the rest N − 1 eigenvalues are positives, which are expressed in ascending order of numerical size as 0 = λ1 ≤ λ2 ≤ · · · ≤ λ N . The minimum nonzero eigenvalue λ2 is called the algebraic connectivity; (iii) If G is unconnected, then L has at least two 0 eigenvalues with the geometric multiplicity being equal to the algebraic multiplicity.

2.1.2 Example In this section, some examples are given to deepen the understanding of the previous definition. (1) Laplacian matrix Figure 2.1 illustrates a balance graph, where the edge weight is set as 1; that is, wi j = 1 (∀i, j ∈ {1, 2, . . . , 9}). The in-degree matrix D and the weighted adjacency matrix W in Fig. 2.1 are, respectively, shown as follows: ⎡

1 ⎢0 ⎢ ⎢0 ⎢ ⎢0 ⎢ ⎢ D = ⎢0 ⎢ ⎢0 ⎢ ⎢0 ⎢ ⎣0 0

0 1 0 0 0 0 0 0 0

0 0 2 0 0 0 0 0 0

0 0 0 3 0 0 0 0 0

0 0 0 0 1 0 0 0 0

0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 2 0 0

0 0 0 0 0 0 0 1 0

⎡ ⎤ 0100 0 ⎢0 0 1 0 0⎥ ⎢ ⎥ ⎢1 0 0 1 0⎥ ⎢ ⎥ ⎢0 0 1 0 ⎥ 0⎥ ⎢ ⎢ ⎥ 0 ⎥, W = ⎢ 0 0 0 0 ⎢ ⎥ ⎢0 0 0 1 0⎥ ⎢ ⎥ ⎢0 0 0 1 0⎥ ⎢ ⎥ ⎣0 0 0 0 ⎦ 0 0000 1

0 0 0 1 0 0 0 0 0

0 0 0 0 1 0 0 0 0

0 0 0 1 0 0 0 1 0

0 0 0 0 0 0 0 0 1

⎤ 0 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ ⎥ 0 ⎥. ⎥ 0⎥ ⎥ 1⎥ ⎥ 0⎦ 0

Thus, the associated Laplacian matrix L of the balance graph is

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2 Fundamental Theory

Fig. 2.1 Balance graph

⎡

1 ⎢ 0 ⎢ ⎢ −1 ⎢ ⎢ 0 ⎢ ⎢ L = D−W =⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎣ 0 0

2

3

1

5

−1 0 0 1 −1 0 0 2 −1 0 −1 3 0 0 0 0 0 −1 0 0 −1 0 0 0 0 0 0

0 0 0 0 0 0 −1 0 1 −1 0 1 0 0 0 0 0 0

4

7

8

6

9

⎤ 0 0 0 0 0 0 ⎥ ⎥ 0 0 0 ⎥ ⎥ −1 0 0 ⎥ ⎥ ⎥ 0 0 0 ⎥. ⎥ 0 0 0 ⎥ ⎥ 2 0 −1 ⎥ ⎥ −1 1 0 ⎦ 0 −1 1

(2) Undirected graph Comparing the two graphs in Fig. 2.2, each node has at least one connected edge with others in (a), while node 2 is isolated in (b), which implies that the graph in (a) is connected and the one in (b) unconnected. (3) Directed graph Figure 2.3a depicts a directed graph having a spanning tree, which has a directed path from node 1 to all the other nodes, but it is not strongly connected because it is not connected between any two nodes, such as nodes 1 and 4, nodes 1 and 5. In Fig. 2.3b, there does not exist any one node that has a directed path to all the other nodes, which means that the directed graph does not have a spanning tree. As shown in Fig. 2.4, there has at least one directed path between any two nodes, so the directed graph is strongly connected. Fig. 2.2 Undirected graph being connected/ unconnected

1

1

2

3

4

5

(a)

2

3

4

5

(b)

2.2 Linear Algebra

61

Fig. 2.3 Directed graph with/without spanning tree

1

1

2

3

4

2

5

3

4

5

(b)

(a)

Fig. 2.4 Directed graph being strongly connected

1 2

3

4

5

2.2 Linear Algebra 2.2.1 Consensus Decomposition of Linear Space Define λi (i = 1, 2, . . . , N ) as the eigenvalues of L ∈ R N , where the associated eigenvector of λ1 = 0 is u 1 = 1. Define a nonsingular matrix U = [u 1 , u 2 , . . . , u N ] ∈ C N ×N . Let ck ∈ Rv (k = 1, 2, . . . , v) be linearly independent vectors and p j = u i ⊗ ck ( j = (i − 1)ν + k, i = 1, 2, . . . , N , k = 1, 2, . . . , ν). A consensus subspace (CS) is defined as the subspace C(U ) spanned by pk = u 1 ⊗ ck = 1 ⊗ ck (k = 1, 2, . . . , v), and a complement consensus subspace (CCS) is defined as the subspace C(U ) spanned by pv+1 , pv+2 , . . . , p N v . Note that p j ( j = 1, 2, . . . , N v) are linearly independent. The following conclusion can be obtained. Lemma 2.3 C(U ) ⊕ C(U ) = C N v . Remark 2.1 From Lemma 2.3, one sees that any C N v can be uniquely projected onto C(U ) and C(U ). In the following chapters, the value of v will be determined by the dimension of the state or output of each agent in the multi-agent systems. The decomposition of the state space or output space of the multi-agent system is called the state space decomposition or output space decomposition.

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2 Fundamental Theory

2.2.2 Kronecker Product m×d

Definition 2.1 For matrices A = [ai j ] ∈ R product can be defined as

and B = [bi j ] ∈ R

p×q

, the Kronecker

⎡

⎤ a11 B a12 B · · · a1n B ⎢ a21 B a22 B · · · a2n B ⎥ ⎢ ⎥ A ⊗ B = [ai j B] = ⎢ . .. .. .. ⎥. ⎣ .. . . . ⎦ am1 B am2 B · · · amn B Lemma 2.4 [5] For any appropriate dimensions matrices A, B, C, and D, the Kronecker product has the following properties: (i) (ii) (iii) (iv) (v)

k(A ⊗ B) = (k A) ⊗ B = A ⊗ (k B), (A + B) ⊗ C = A ⊗ C + B ⊗ C, (A ⊗ B)(C ⊗ D) = (AC) ⊗ (B D), (A ⊗ B)T = AT ⊗ B T , (A ⊗ B)−1 = A−1 ⊗ B −1 .

2.2.3 Orthogonal Matrix Definition 2.2 If U is also square, then UU T = I ; that is, U T = U −1 , and U is an orthogonal matrix. Lemma 2.5 If A is a real symmetric matrix, then there is an orthogonal matrix U that diagonalizes A; that is, U T AU = D, where D is a diagonal matrix.

2.2.4 Linear Matrix Inequality  Lemma 2.6 (Schur complement [6]) For a given matrix S = S11 ∈ Rr ×r , the following statements are equivalent:

S11 S12 , where ∗ S22

(i) S < 0, T −1 (ii) S11 < 0, S22 − S12 S11 S12 < 0, −1 T (iii) S22 < 0, S11 − S12 S22 S12 < 0. Lemma 2.7 [7] For Q T = Q > 0 and S with compatible dimensions, the following inequality holds S T + S ≤ Q + S T Q −1 S.

2.3 Linear System Theory

63

Lemma 2.8 If A is a real symmetric matrix, then there is an orthogonal matrix U that diagonalizes A; that is, U T AU = D, where D is a diagonal matrix.

2.2.5 Eigenvalues and Eigenvectors Eigenvalues and eigenvectors have their importance in the design of consensus control because the eigenvalues of Laplacian matrix are required. The definition of eigenvalues and eigenvectors is given as: Definition 2.3 Let A be a square matrix A ∈ Rn×n . A scalar λ is called an eigenvalue of A if there exists a nonzero (column) vector v such that Av = λv. Any vector satisfying this relation is called an eigenvector of A belonging to the eigenvalue λ.



 Lemma 2.9 [8] If Φ0 + Re λ˜ i Φ1 + Im λ˜ i Φ2 < 0 (i = 1, 2, 3, 4), then Φ0 + Re(λi )Φ1 + Im(λi )Φ2 < 0 (i = 2, 3, . . . , N ).

2.3 Linear System Theory Consider the following LTI system 

x(t) ˙ = Ax(t) + Bu(t), y(t) = C x(t),

(2.1)

where A ∈ Rn×n , B ∈ Rn×m , and C ∈ Rq×n , and x(t) ∈ Rn , u(t) ∈ Rm and y(t) ∈ Rq is the state, control input and output, respectively. Definition 2.4 For any initial state x(0), if there exists control input u(t) such that the state x(t) of system (2.1) can converge to the origin in a finite time, then system (2.1) is called controllable or (A, B) is controllable. Definition 2.5 If matrix A is Hurwitz, then system (2.1) is asymptotically stable. Definition 2.6 If there exists a matrix K ∈ Rm×n such that A + B K is Hurwitz, then system (2.1) is stabilizable or (A, B) is stabilizable. Lemma 2.10 [9] If rank[B, AB, . . . , An−1 B] = n, then (A, B) is controllable.

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2 Fundamental Theory

Lemma 2.11 (Popov–Belevitch–Hautus (PBH) test for controllability [6]) If rank[s I − A, B] = n (∀s ∈ C), then (A, B) is controllable. Definition 2.7 If any initial state x(0) of system (2.1) can be uniquely determined by the control input u(t) and output y(t) in a finite time, then system (2.1) is called observable or (C, A) is observable. Lemma 2.12 [9] If rank[C T , AT C T , . . . , (An−1 )T C T ]T = n, then (C, A) is observable. Lemma 2.13 (PBH test for observability [9]) If [C T , s I − AT ]T = n (∀s ∈ C), then (C, A) is observable. Lemma 2.14 [9] For system (2.1), the following statements are equivalent: (i) System (2.1) is asymptotically stable, (ii) For any given positive matrix R, the Lyapunov function AT P + P A + R = 0 has a positive definite solution P, (iii) There exists a positive definite matrix R such that the Lyapunov function AT P + P A + R = 0 has a unique positive definite solution P, (iv) There exists a positive definite matrix P such that AT P + P A < 0. Lemma 2.15 [10] System (2.1) is stabilizable if and only if rank[s I − A, B] = n + + (∀s ∈ C ), where C = {s|s ∈ C, Re(s) ≥ 0} represents the closed right complex space. Lemma 2.16 If there exists a gain matrix K ∈ Rn×q such that A + K C is Hurwitz, then system (2.1) is detectable or (C, A) is detectable. Lemma 2.17 [10] System (2.1) is detectable if and only if rank[s I − AT , C T ]T = n + (∀s ∈ C ). Consider the following LTI system 

where y(t) =



yo (t) yo (t)



 , A=

x(t) ˙ = Ax(t), y(t) = C x(t),

(2.2)

A11 A12 and C = [I, 0]. A21 A22

Definition 2.8 If for any given ε > 0, there exists δ = δ(ε) > 0 such that ||y(0)|| < δ ⇒ ||yo (t)|| < ε (∀t ≥ 0), then system (2.2) is said to be stable with respect to yo (t). Definition 2.9 If system (2.2) is stable with respect to yo (t) and limt→∞ yo (t) = 0, then system (2.2) is said to be asymptotically stable with respect to yo (t).

2.4 Singular System Theory

65

Lemma 2.18 [11] If ( A22 , A12 ) is completely observable, then system (2.2) is asymptotically stable with respect to yo (t) if and only if A is Hurwitz. If (A22 , A12 ) is not completely observable, then there always exists a nonsingular matrix T such that      D1 0 F1 , E 1 0 , T −1 A21 = , (T −1 A22 T , A12 T ) = F2 D2 D3 where (D1 , E 1 ) is completely observable. The following results can be obtained. Lemma 2.19 [11] If (A22 , A12 ) is not completely observable, then system (2.2) is asymptotically stable with respect to yo (t) if and only if 

A11 E 1 F1 D1



is Hurwitz.

2.4 Singular System Theory Some basic concepts about LTI singular systems are recalled in this section, which will be used in subsequent chapters. According to Sect. 1.2, singular multi-agent systems present a useful tool for system modeling since they allow to describe a system by both dynamic equations and algebraic constraints, and it is necessary to give a quick reminder of the fundamental definitions and results of singular systems, such as regularity, admissibility, equivalent forms, system decomposition, temporal response, controllability, observability, and duality. Most of the results given in this section can be found in [12–22]. Based on (1.6), a LTI singular system including a control input u(t) and a measurement output y(t) can be written as 

E x(t) ˙ = Ax(t) + Bu(t), y(t) = C x(t),

(2.3)

where x(t) ∈ Rn , y(t) ∈ Rl and u(t) ∈ Rm are the state, measurement, and control input vectors, respectively. The system matrices E, A ∈ Rn×n are square matrices, and E may be singular, i.e., rank(E) =r ≤ n.

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2 Fundamental Theory

2.4.1 Regularity Regularity is one of the most basic properties that distinguish singular systems from normal systems. Any normal system is regular, but not for singular systems. Satisfying regularity is usually the most basic requirement for the design of generalized control systems. Especially, if a singular system is regular, then it has a unique solution for any initial condition and any continuous input function [23]. Definition 2.10 Singular system (2.3), or the matrix pair (E, A) of (2.3) is said to be regular if there exists some s ∈ C such that det(s E − A) /= 0. It can be seen from Definition 2.10 that the regularity of singular system (2.3) is equivalent to the regularity of matrix pair (E, A), which is independent of the choice of matrices B and C of the system. To illustrate the physical mean of regularity, one ˙ = Ax(t) + Bu as follows can examine the Laplace transformed version of E x(t) s E x(s) − E x(0) = Ax(s) + Bu(s), which can be arranged as (s E − A)x(s) = E x(0) + Bu(s), It is observed that, if the system is regular, then there exist some s such that x(s) = (s E − A)−1 (E x(0) + Bu(s)), which guarantees the existence and uniqueness of x(s) for any initial condition and any input function. Otherwise, the matrix s E − A is of rank deficiency, and there exists a nonzero vector θ (s) such that (s E − A)θ (s) = 0. Consequently, one can state that, if the system has a solution denoted x(s), then x(s) + αθ (s) is also a solution for any α, which means that a solution to this system is not unique, and it is also obvious that there may be no solution to this system. More detail knowledge about characterizations of regularity can be found in [17].

2.4.2 Constrained System Equivalence and System Decomposition Take a linear nonsingular transformation on singular system (2.3) by x(t) = Q −1 x(t), one can obtain that

2.4 Singular System Theory



67

˙ = P AQx(t) + P Bu(t), P E Q x(t) y(t) = C Qx(t) + Du(t),

(2.4)

where P and Q are two invertible matrices. The states of the two singular systems before and after the transformation are one-to-one correspondence, and the two singular systems are said to be constrained equivalent. This transformation is usually called constrained equivalent transformation. For simplicity, the singular system (2.3) can be represented by the triple (E, A, C), and the following definition reveals the specific meaning of system equivalence. Definition 2.11 (Constrained System Equivalence) Singular systems (E 1 , A1 , C1 ) and (E 2 , A2 , C2 ) are termed restricted system equivalent if there exist nonsingular matrices P and Q such that 

M 0 0 I



s E 1 − A1 B1 C 0



 N 0 s E 2 − A2 B2 . = C2 0 0 I

In fact, constrained equivalent transformation has the properties of maintaining the structural characteristics of singular systems, such as eigenvalue, regularity, stability, controllability, and observability (these concepts will be introduced in the following sections). Therefore, when analyzing and synthesizing the singular system with complex structure, the constrained equivalent transformation is often used to transform the singular system into a simple and special form. This method will often be seen in the later discussion. Among many equivalent representations, there are three particular form of constrained system equivalence for system analysis and control. They are referred to [24]. The First Equivalent Form can be obtained by the following Lemma. Lemma 2.20 [24] If singular system (2.3) is regular, there exist nonsingular matrices P and Q such that  PEQ =

    Ir 0 B1 A1 0 , PB = , C Q = C1 C2 , , P AQ = B2 0 N 0 In−r

where N ∈ R(n−r )×(n−r ) is a nilpotent matrix. Let Q −1 x(t) = [x1 (t)/x2 (t)] with x1 (t) ∈ Rr and x2 (t) ∈ Rn−r , one can obtain the First Equivalent Form as x˙1 (t) = A1 x1 (t) + B1 u(t), y1 (t) = C1 x1 (t),

(2.5)

N x˙2 (t) = x2 (t) + B2 u(t), y2 (t) = C2 x1 (t),

(2.6)

y(t) = y1 (t) + y2 (t).

(2.7)

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2 Fundamental Theory

The form (2.5)–(2.7) is referred to as the Kronecker–Weierstrass decomposition [24], which can be viewed as an equivalent condition for regularity, and is also referred to by some scholars as slow-fast decomposition [14]. The subsystem (2.5) is called the slow subsystem, and (2.6) is called the fast subsystem. The Second Equivalent Form that does not depend upon the regularity of systems is also called the SVD form. This form can be obtained via a singular value decomposition on E and followed by scaling of the bases. Under the SVD form, singular system (2.3) is decomposed by two nonsingular matrices U and V as     B˜ I 0 A1 A2 , U B = ˜ 1 , C V = C˜ 1 C˜ 2 U EV = , U AV = A3 A4 B2 00 

 T Likewise, let V −1 x(t) = x˜1T (t), x˜1T (t) with x˜1 (t) ∈ Rr and x˜2 (t) ∈ Rn−r , one can obtain the Second Equivalent Form as x˙˜1 (t) = A11 x˜1 (t) + A12 x˜2 (t) + B˜ 1 u(t),

(2.8)

0 = A21 x˜1 (t) + A22 x˜2 (t) + B˜ 2 u(t),

(2.9)

y(t) = C˜ 1 x˜1 (t) + C˜ 2 x˜2 (t).

(2.10)

This form was discussed in [25] for using it to examine general system properties and to derive a linear-quadratic regulator for continuous-time descriptor systems. Similar to the Kronecker–Weierstrass decomposition, U and V for SVD form are in general but not unique. Lemma 2.21 (The Third Equivalent Form) [24]: If singular system (2.3) is regular, then it is equivalent to 

ˆ ˆ Eˆ x(t) ˙ = (I − α E)x(t) + Bu(t), y(t) = C x(t),

where Eˆ = P2 E, Bˆ = P2 B, P2 = (α E + A)−1 , Q 2 = In , and α satisfies det(α E + A) /= 0. Noted that for a fixed α, the Third Equivalent Form is unique in the sense of algebraic equivalence.

2.4.3 Temporal Response Suppose that h is the degree of the nilpotent matrix N in (2.6), that is, N h−1 /= 0 and N h = 0. The subsystem (2.5) is a normal state-space system, whose temporal response for a given input u(t) and initial condition x1 (0) can be written as

2.4 Singular System Theory

69

⎧ t ⎪ ⎪ ⎪ A1 t ⎪ x1 (t) = e x1 (0) + e A1 (t−τ ) B1 u(τ )dτ , ⎪ ⎪ ⎪ ⎨ 0

⎪ t ⎪ ⎪ ⎪ A t 1 ⎪ y1 (t) = C1 e x1 (0) + C1 e A1 (t−τ ) B1 u(τ )dτ . ⎪ ⎪ ⎩

(2.11)

0

Then we consider the subsystem (2.6). Suppose that u(t) ∈ C h−1 , where C h−1 stands for the set of h − 1 times continuously differentiable functions. Then we have the following relations 

N k x2(k) (t) = N k−1 x2(k−1) (t) + N k−1 B2 u (k−1) (t), k ∈ {1, 2, . . . h − 1}, 0 = N k−1 x2(h−1) (t) + N k−1 B2 u (h−1) (t), k = h,

(2.12)

where x2(k) (t) means the nth-order derivation of x2 (t). Hence, the expression of x2 (t) can be obtained as x2 (t) = N x˙2 − B2 u(t) ˙ = N 2 x˙2(2) (t) − B2 u(t) − N B2 u(t) ··· =−

h−1 ∑

N k B2 u (k) (t),

(2.13)

k=0

which gives y2 (t) = −

h−1 ∑

C2 N k B2 u (k) (t).

(2.14)

k=0

Hence, the temporal response y(t) of the singular system (2.7) is ⎛ y(t) = C1 ⎝e A1 t x1 (0) +

t 0

⎞ e A1 (t−τ ) B1 u(τ )dτ ⎠ −

h−1 ∑

C2 N k B2 u (k) (t).

(2.15)

k=0

It is observed that the response of the subsystem (2.15) depends on the matrix A1 , initial condition x1 (0), as well as the input u(t), while the response of the subsystem (2.15) depends only on the derivative of the input u(t) on time t. That is why we also call these two subsystems slow subsystem and fast subsystem, respectively. If t → 0+ , then we can deduce the following constraint on the initial condition

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x(0+ ) = N1

  ∑ h−1 I 0 N k B2 u (k) (0+ ). x1 (t) − N1 0 I

(2.16)

K =0

Any initial condition satisfying (2.16) is called an admissible condition. From this point of view, only one initial condition is allowed, and hence, only one solution exists for each choice of u(t). In [23], the authors used the theory of distributions and generalized this viewpoint to allow arbitrary initial conditions. Under this theory, for the fast subsystem, we have x2 (t) = −

h−1 ∑ k=1

δ (k−1) N x20 −

h−1 ∑

N k B2 u (k) (t),

(2.17)

k=0

where δ is the Dirac delta. As pointed out in [15], the form of (2.17) suggests that in any conventional sense the dynamics of the overall system are concentrated in the slow subsystem in (2.5). With the theory of distributions, we can represent the systems whose initial conditions are not admissible or those who contain ‘jump’ behaviors. For example, when we switch an electrical circuit on, there will be a jump in the current or voltage at this moment. For these cases, the first term of (2.15) can transform the system into an admissible state.

2.4.4 Admissibility Stability is a fundamental concept for state-space systems, which can be characterized, by one of the definitions, that the system has no poles located in the right-hand plane including the imaginary axis. Under the singular framework, a similar more complicated concept called admissibility plays the same role as stability for statespace systems. Let σ (X ) represent the pole domain of matrix X , then we can give the following definitions. Definition 2.12 (Admissibility) Reference [24]: (i) (ii) (iii) (iv) (v)

The singular system (2.3) is said to be regular if det(s E − A) is not identically null; The singular system (2.3) is said to be impulse-free if deg(det(s E − A)) = rank(E); The singular system (2.3) is said to be stable if all the roots of det(s E − A) = 0 have negative real parts; The singular system (2.3) is said to be stabilizable if there exists a matrix K ∈ Rn×m that σ ([E, A + B K ]) ⊂ C−1 ; The singular system (2.3) is said to be admissible if it is regular, impulse-free, and stable.

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71

From the definition, the admissibility of a singular system concerns stability, as well as regularity and impulsiveness. The latter two are intrinsic properties of conventional state-space systems and are not necessarily considered in the state-space case. Next, some equivalent conditions for admissibility are proposed. Lemma 2.22 [24] Suppose that the singular system (2.3) is regular and there exist nonsingular matrices P and Q such that the First Equivalent Form (2.5)–(2.7) holds. Then (i) This system is said to be impulse-free if and only if N = 0; (ii) This system is said to be stable if and only if σ ( A1 ) ⊂ C−1 ; (iii) This system is said to be admissible if and only if N = 0 and σ (A1 ) ⊂ C−1 . Lemma 2.23 [24] Consider the singular system (2.3) and suppose that there exist nonsingular matrices P and Q such that the Second Equivalent Form (2.8)–(2.10) holds. Then (i) This system is said to be impulse-free if and only if |A4 | /= 0; (ii) This system is said to be admissible if and only if |A4 | /= 0; and σ (A1 − −1 A2 A−1 4 A3 ) ⊂ C . Lemma 2.24 [24] Consider the singular system (2.3), there are two propositions holds: (i) The pair (E, A) is regular and impulse-free if and only if the following equation holds  E 0 rank = n + rank(E); A E (ii) The triplet (E, A, B) is impulse controllable if and only if the following equation holds  E 0 0 rank = n + rank(E). A E B Lemma 2.25 [24] The singular system E ζ˙ (t) = (A − λB)ζ (t) is admissible if and only if there exists a matrix R such that (Ʌ A − ϒλ Ʌ B )T R + R T (Ʌ A − ϒλ Ʌ B ) < 0, ɅTE R = R T Ʌ E ≥ 0. Lemma 2.26 [24] For E, A ∈ Rn×n , B ∈ Rn×m and R = R T > 0, assume that (E, A) is impulse-free and (E, A, B) is stabilizable, then the following Riccati equation

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2 Fundamental Theory



AT Q + Q T A − Q T B B T Q + R = 0 E T Q = QT E ≥ 0

has at least one admissible solution Q, which is unique in the sense of E T Q. Furthermore, if the singular system is regular and the matrices P and Q exist to render it to the First Equivalent Form, then the transfer function of this system can be written as G(s) = C1 (s I − A1 )−1 B1 + C2 (sN − I )−1 B2 .

(2.18)

For an impulse-free system, that is, N = 0, we have G(s) = C1 (s I − A1 )−1 B1 − C2 B2 .

(2.19)

It is noted that the term C2 (sN − I )−1 B2 leads to polynomial terms of s if both B2 and C2 are nonzero. Hence, the impulse-free assumption guarantees the properness of the transfer function. The converse statement is, however, not true. Clearly, if either B2 or C2 vanishes, the transfer function is still proper, even if the system is impulsive. Hence, given a stable transfer function G(s) and its corresponding system (E, A, B, C), the admissibility of this system cannot be concluded. Now we briefly discuss the issue of generalized eigenvalues of a matrix pencil. Consider a matrix pencil λE − A, where E and A are both real n × n matrices, and λ is a scalar. First, we assure that this pencil is regular, that is, |λE − A| /= 0 for all λ. The generalized eigenvalues are defined as those λ for which |λE − A| = 0. Lemma 2.27 Grade 1 infinite generalized eigenvectors of the pencil (s E − A) satisfy Evi1 = 0. Definition 2.13 Grade k (k ≥ 2) infinite generalized eigenvectors of the pencil (s E − A) corresponding to the ith grade 1 infinite generalized eigenvectors satisfy Evik+1 = Avik . Moreover, the finite generalized eigenvalues of s E − A are called the finite dynamic modes. The infinite generalized eigenvalues of s E − A with the grade 1 infinite generalized eigenvectors determine the static modes, while the infinite generalized eigenvalues with the grade k (k ≥ 2) infinite generalized eigenvectors are the impulsive modes. Let q be the degree of the polynomial |λE − A|. One can state that the matrix pencil λE − A has q finite generalized eigenvalues and n − q infinite generalized eigenvalues, where the number of static modes is n − r and the number of impulsive modes is r − q.

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73

2.4.5 Controllability In this section, we introduce controllability for singular systems in a way that reduces to the state-space definition when E = I . We suppose that the singular system is regular, and it is transformed into the First Equivalent Form as follows θs : x˙1 = A1 x1 + B1 u, y1 = C1 x1 , θ f : N x˙2 = x2 + B2 u, y2 = C2 x2 , y = y1 + y2 ,

(2.20)

where x1 ∈ Rn 1 , x2 ∈ Rn 2 , n 1 + n 2 = n and N is a nilpotent matrix with degree h. Define C ip be the i times piecewise continuously differentiable maps on R with range depending on context; (ii) I be the set of admissible initial conditions, that is, (i)

 I=

x1 x2

: x1 ∈ Rn 1 , x2 = −

h−1 ∑

N −k B2 u (k) (0), u ∈ Cmh−1 ;

(2.21)

k=0

(iii) ⟨X, Y ⟩ = β + Xβ + X 2 β + · · · + X n−1 β, where X is a square matrix, n is the order of X, the product XY is well defined and β = Im(Y ). Definition 2.14 (Reachable State) [17]: A state xr is reachable from a state x(0) if there exists u(t) ∈ Cmh−1 such that x(tr ) = xr for some tr > 0. Lemma 2.28 [17] Let R(0) be the set of reachable states from x(0) = 0. Then, R(0) = ⟨A1 , B1 ⟩ ⊕ ⟨N , B2 ⟩.

(2.22)

Lemma 2.29 [17] Let R(x) be the set of reachable states from x ∈ I. Then the complete set of reachable states R is R = ∪ R(x) = Rn 1 ⊕ ⟨N , B2 ⟩.

(2.23)

x∈I

We can adopt the conventional definition of controllability for singular systems. Definition 2.15 (C-controllability) [26]: The singular system (2.20) is said to be completely controllable (C-controllable) if one can reach any state from any initial state. Within the singular framework, we can also define two different types of controllability as follows.

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Definition 2.16 (R-controllability) [26]: The singular system (2.20) is said to be controllable within the set of reachable states (R-controllable) if, from any initial state x0 ∈ I, there exists u(t) ∈ Cmh−1 such that x(t f ) ∈ R for any t f > 0. Note that, for state-space systems, R-controllability and C-controllability are equivalent. This is, however, not the case for singular systems. Definition 2.17 (Imp-controllability) [14]: The singular system (2.20) is said to be impulse controllable (Imp-controllable) if for every w ∈ Rn 2 there exists u(t) ∈ Cmh−1 such that the fast subsystem θ f satisfies h−1 ! " ∑ x2 t f = δt(k−1) N k w, ∀t f > 0. f k=1

Lemma 2.30 (Regarding C-controllability) [17]: (a) The following propositions are equivalent: (a.1) (a.2) (a.3) (a.4) (a.5) (a.6)

The singular system (2.3) is C-controllable. θs and θ f are both controllable. ⟨A1 , B1 ⟩ ⊕ ⟨N , B2 ⟩ = Rn 1 +n 2 . rank([s E − A, B]) = n, for a finite s ∈ R and rank([E,B]) = n. Im(λE − A) ⊕ Im(B) = Rn and Im(E) ⊕ Im(B) = Rn . The matrix ⎡ is full row rank, ⎤ −A B ⎥ ⎢ E −A B ⎥ ⎢ ⎥ ⎢ . .. ⎥. ⎡=⎢ E B ⎥ ⎢ ⎥ ⎢ . . . . −A .. ⎦ ⎣ E B ⎡

(b) The following statements are equivalent: (b.1) (b.2) (b.3) (b.4) (b.5)

θs is controllable. The singular system (2.20) is R-controllable. ⟨A1 , B1 ⟩ = Rn 1 . rank([s E − AB]) = n, for a finite s ∈ R. Im(λE − A) ⊕ Im(B) = Rn .

(c) The following statements are equivalent: (c.1) (c.2) (c.3) (c.4) (c.5)

θs is controllable. The singular system (2.3) is R-controllable. ⟨A1 , B1 ⟩ = Rn 1 . rank([s E − A, B]) = n, for a finite s ∈ R. Im(λE − A) ⊕ Im(B) = Rn .

(2.24)

2.4 Singular System Theory

75

(d) The following statements are equivalent: θ f is controllable. ⟨N , B2 ⟩ = Rn 2 . rank([E, B]) = n. Im(E) ⊕ Im(B) = Rn . Im(N ) ⊕ Im(B2 ) = Rn 2 . The rows of B2 corresponding to the bottom rows of all Jordan blocks of N are linearly independent. (d.7) v T (sN − I )−1 B2 = 0 for constant vector v implies that v = 0. (d.1) (d.2) (d.3) (d.4) (d.5) (d.6)

Lemma 2.31 (Regarding R-controllability) [17]: The following statements are equivalent. (i) (ii) (iii) (iv) (v)

The singular system (2.20) is R-controllable. θs is controllable. ⟨A1 , B1 ⟩ = Rn 1 . rank([s E − A, B]) = n, for a finite s ∈ R. Im(λE − A) ⊕ Im(B) = Rn .

Lemma 2.32 (Regarding Imp-controllability) [17]: The following statements are equivalent. (i) (ii) (iii) (iv) (v)

The singular system (2.3) is Imp-controllable. θ f is Imp-controllable. Ker(N ) ⊕ ⟨N , B2 ⟩ = Rn 2 . Im(N ) = ⟨N , B2 ⟩. Im(N ) ⊗ Ker(N )  ⊗ Im(B2 ) = Rn 2 .  A E B (vi) rank = n + r. E 0 0 (vii) The rows of B2 corresponding to the bottom rows of the nontrivial Jordan blocks of N are linearly independent. (viii) v T N (sN − I )−1 B2 = 0 for constant vector v implies that v = 0. It is observed that the conditions characterizing R-controllability are only concerned with the slow subsystem θs . The response of the fast subsystem depends and its derivatives. Any reachable state of θ f w ∈ ⟨A1 , B1 ⟩ can be written only on u(t) ∑ h−1 ηk N k B2 . Then it is easy to find an input u(t) satisfying u (k) (t f ) = ηk , as w = k=0 ∑ h−1 for k = 0, 1, . . . , h−1 (e.g., u(t) = k=0 ηk (t −t f )k /k!) which leads to x2 (t f ) = w. Hence, the fast subsystem has no impact on R-controllability. Imp-controllability guarantees the ability to generate a maximal set of impulses, at each instant, in the following sense: suppose E and A are given but B and u are allowed to vary over all values.

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2.4.6 Duality As known, there is a strong sense of symmetry between controllability and observability for the state-space setting. We now extend this idea to singular systems. Corresponding to (2.4), we define the dual system θ 

E T x(t) ˙ = AT x(t) + C T u(t), y(t) = B T x(t).

(2.25)

Then we have the following statements. Lemma 2.33 (Duality) [26]: The singular system (2.3) is C-controllable (C-observable) if and only if the system (2.25) is C-observable (C-controllable). (ii) The singular system (2.3) is R-controllable (R-observable) if and only if the system (2.25) is R-observable (R-controllable). (iii) The singular system (2.3) is Imp-controllable (Imp-observable) if and only if the system (2.25) is Imp-observable (Imp-controllable). (i)

2.4.7 Observability In this section, we introduce observability for singular systems in a way that allows for a set of results analogous to the last section. Similarly, the concepts, that is, C-observability, R-observability, and Imp-observability, are defined. Definition 2.18 (C-observability) [26]: The singular system (2.20) is said to be completely observable (C-observability) if knowledge of u(t) and y(t) for t ∈ [0, ∞] is sufficient to determine the initial condition x0 . Definition 2.19 (R-controllability) [26]: The singular system (2.20) is said to be observable within the set of reachable states (R-observable) if, for t ≥ 0, x(t) ∈ I can be computed from u(τ ) and y(τ ) for any τ ∈ [0, t]. Definition 2.20 (Imp-observability) [14]: The singular system (2.20) is said to be impulse observable (Imp-observable) if, for every w ∈ Rn 2 , knowledge of y(t) for t ∈ [0, ∞] to determine x2 (t). x2 (t) =

h−1 ∑

δt(k−1) N k w. f

k=1

Lemma 2.34 (Regarding C-observability) [17]: (a) The following statements are equivalent.

(2.26)

2.4 Singular System Theory

(a.1) (a.2) (a.3) (a.4) (a.5) (a.6)

77

The singular system (2.3) is C-observable. θ⟨ s and θ⟩f are ⟨ both observable. ⟩ AT1 ,! = Rn 1 +n 2 . C1T ⊗ N T , C2T " " ! T T T = n, for a finite s ∈ R and rank E T , C T = n. rank s E − A , C Ker(λE − A) ∩ Ker(C) = {0} and Ker(E) ∩ Ker(C) = {0}. The matrix D is full row rank. ⎤ −AT CT ⎥ ⎢ E T −AT CT ⎥ ⎢ ⎥ ⎢ . . T T ⎥. ⎢ . D=⎢ C E ⎥ ⎥ ⎢ .. .. ⎦ ⎣ . −AT . T T E C ⎡

(b) The following statements are equivalent. (b.1) (b.2) (b.3) (b.4) (b.5)

θs is observable. The singular system (2.20) is R-observable. ⟨AT1 ,! C1T ⟩ = Rn1 . " rank s E T − AT C T = n, for a finite s ∈ R. Ker(λE − A) ∩ Ker(C) = {0}.

(c) The following statements are equivalent. θ f is observable. ⟨N T! , C2T ⟩ = R"n2 . rank E T , C T = n. Ker(E) ∩ Ker(C) = {0}. Ker(N ) ∩ Ker(C2 ) = {0}. The rows of C2T corresponding to the bottom rows of all Jordan blocks of N T are linearly independent. (c.7) C2 (sN − I )−1 v = 0 for constant vector v implies that v = 0. (c.1) (c.2) (c.3) (c.4) (c.5) (c.6)

Lemma 2.35 (Regarding R-observability) [17]: The following statements are equivalent. (i) (ii) (iii) (iv) (v)

The singular system (2.20) is R-observable. θs is observable. ⟨AT1 ,! C1T ⟩ = Rn1 . " rank s E T − AT C T = n, for a finite. Ker(λE − A) ∩ Ker(C) = {0}.

Lemma 2.36 (Regarding Imp-observability) [14]: The following statements are equivalent. (i) (ii)

The singular system (2.20) is Imp-observable. θ f is Imp-observable.

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2 Fundamental Theory

! " Im(N T ) ∩ Ker ⟨N!T , C2T ⟩ = "{0}. Ker(N T ) = N Ker ⟨N T , C2T ⟩ . Ker(N ) ∩ Ker(C  2 ) = {0}. ) ∩TIm(N A E T BT = n + r. (vi) rank ET 0 0 (vii) The rows of C2T corresponding to the bottom rows of the nontrivial Jordan blocks (i) N T are linearly independent. (viii) C2 (sN − I )−1 N v = 0 for constant vector v implies that v = 0. (iii) (iv) (v)

Similar to R-controllability, the characterizations for evaluating R-observability are only concerned with the slow subsystem θs .

2.5 Consensus for High-Order LTI Multi-agent Systems Consider a high-order LTI multi-agent system with N homogeneous agents which only interact with each other locally. Assume that all the agents share a common state-space Rd , and let xi (t) ∈ Rd (i ∈ {1, 2, . . . , N }) denote the state of agent i which needs to be coordinated, then the dynamics of agent i can be described by x˙i (t) = Axi (t) + Bu i (t), where A ∈ Rd×d , B ∈ Rd×m , and u i (t) is the consensus protocol, which can be designed according to the state information available to agent i. Consider a consensus protocol of the form: u i (t) = K 1 xi (t) + K 2

∑

! " wi j x j (t) − xi (t) ,

j∈Ni

where i, j ∈ {1, 2, . . . , N }, K 1 , K 2 ∈ Rm×d , and Ni denotes the set of neighbors of agent i. Under this consensus protocol, the dynamics of a multi-agent system with N agents can be described as the following high-order continuous-time LTI system: x(t) ˙ = (I N ⊗ (A + B K 1 ) − L ⊗ B K 2 )x(t),

(2.27)

 T where x(t) = x1T (t), x2T (t), . . . , x NT (t) and I N is an N × N identity matrix. Remark 2.2 In (2.27), A denotes the intrinsic dynamics of each agent. The selffeedback matrix K 1 is used to assign the consensus modes of system (2.27) by changing the dynamics of each agent. The neighboring feedback matrix K 2 indicates the interaction between any two neighboring agents. For examples, in a selfpropelled particle multi-agent system, wi j (i, j ∈ {1, 2, . . . , N }) denotes the interaction strength between agent i and agent j, and K 2 describes the attraction interaction among agents, which propels the states of all agents to be identical.

2.5 Consensus for High-Order LTI Multi-agent Systems

79

Now the definitions of consensus and consensualization are presented. Definition 2.21 (Consensus) For given matrices K 1 and K 2 , system (2.27) is said to achieve consensus if for any given bounded initial state, there exists a vector function c(t) ∈ Rd dependent of the initial state such that limt→∞ (x(t)−1⊗c(t)) = 0, where c(t) is called a consensus function. Definition 2.22 (Consensualization) System (2.27) is said to be consensualizable if there exist control gains K 1 and K 2 such that system (2.27) achieves consensus for any bounded initial conditions.

2.5.1 Consensus and Consensualization For the Laplacian matrix L ∈ R N ×N of the interaction topology G, there exists a nonsingular matrix U ∈ C N ×N such that U −1 LU = JL where JL is a Jordan canonical form of L. For simplicity of notation, let H = I N ⊗(A+ B K 1 )− L ⊗ B K 2 , and 0 is applied to denote zero matrices of any size with zero vectors with zero number as special cases and to denote subspaces consisting of zero matrices. Let x(t) ˜ = [x˜1H (t), x˜2H (t), . . . , x˜ NH (t)]H = (U −1 ⊗ Id )x(t), where Id is a d × d identity matrix, then by Lemma 2.1 system (2.27) can be transformed into ˙˜ = (I N ⊗ (A + B K 1 ) − JL ⊗ B K 2 )x(t) ˜ x(t) ⎤ ⎡ A + B K1 0 0 ⎥ ⎢ A + B K 1 − λ L ,2 B K 2 × ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ . . ˜ =⎢ .. .. ⎥x(t), ⎥ ⎢ ⎥ ⎢ ⎦ ⎣ × A + B K 1 − λ L ,N −1 B K 2 0 A + B K 1 − λ L ,N B K 2

(2.28) where the symbol × represents the term δ B K 2 , where δ = 0 or δ = −1 which is related to the structure of JL . In that follows, two subspaces of the N × d-dimensional complex space C N d will be introduced, which are the foundation of the state space decomposition method. For H ∈ R N d×N d , there exists a nonsingular matrix P = [ p1 , p2 , . . . , p N d ] ∈ C N d×N d such that P −1 H P = J H , where J H is the Jordan canonical form of H . P is comprised of the eigenvectors and generalized eigenvectors of H . Let Jλ be a Jordan block of order k corresponding to the eigenvalue λ of H , then k column vectors of P can be obtained by the following equations 

(λI − H ) pλ,1 = 0, (λI − H ) pλ,i = − pλ,i−1 (i = 2, 3, . . . , k),

(2.29)

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where pλ,1 is an eigenvector of H associated with λ, and pλ,i (i = 2, 3, . . . , k) are generalized eigenvectors of H . By (2.28), H is similar to I N ⊗(A+ B K 1 )− JL ⊗ B K 2 , so eigenvalues of H consist of eigenvalues of A + B K 1 and A + B K 1 − λ L ,i B K 2 (i = 2, 3, . . . , N ). Based on this fact, we introduce the following two subspaces. Definition 2.23 Let H ∈ R N d×N d and p j ∈ C N d ( j = 1, 2, . . . , N d) be linear independent eigenvectors and generalized eigenvectors of H . p j ( j = 1, 2, . . . , d) and p j ( j = d + 1, d + 2, . . . , N d) are eigenvectors and generalized eigenvectors of H corresponding to eigenvalues of A + B K 1 and A + B K 1 − λ L ,i B K 2 (i = 2, 3, . . . , N ), respectively. A CS is defined as the subspace C(H ) spanned by p j ( j = 1, 2, . . . , d) and a CCS as the subspace C(H ) spanned by p j ( j = d + 1, d + 2, . . . , N d). Base on the fact that P is nonsingular and (2.29), the following lemma can be easily obtained. Lemma 2.37 C(H )⊕C(H ) = C N d , where C(H ) and C(H ) are invariant subspaces of H . The following lemma presents a kind of structure of p j ( j = 1, 2, . . . , d) used in this book. Lemma 2.38 Let c1 , c2 , . . . , cd ∈ Cd be linear independent eigenvectors and generalized eigenvectors of A+ B K 1 , then p j = 1⊗c j ( j = 1, 2, . . . , d) are eigenvectors and generalized eigenvectors of H corresponding to eigenvalues of A + B K 1 . Proof Consider a k-order Jordan block of J A+B K 1 which is the Jordan canonical form of A + B K 1 , and let c1 ∈ Cd be an eigenvector associated with the eigenvalue λ of A + B K 1 and c2 , c3 , . . . , ck ∈ Cd are generalized eigenvectors induced by c1 . Since L1 = 0, one obtains H p1 = (I N ⊗ ( A + B K 1 ) − L ⊗ B K 2 )(1 ⊗ c1 ) = λp1 . Hence, p1 = 1 ⊗ c1 is an eigenvector of H . By (λI − (A + B K 1 ))c j = −c j−1 , one has ( A + B K 1 )c j = λc j + c j−1 ( j = 2, 3, . . . , k). It follows that H p j = (I N ⊗ (A + B K 1 ) − L ⊗ B K 2 )(1 ⊗ c j ) = λp j + p j−1 , hence p j = 1 ⊗ c j ( j = 2, 3, . . . , k) are generalized eigenvectors of H . By considering all Jordan blocks of J A+B K 1 , one can obtain the conclusion of Lemma 2.38. ⟁ With the above preparation, now we present a necessary and sufficient condition for system (2.27) to achieve consensus.

2.5 Consensus for High-Order LTI Multi-agent Systems

81

Theorem 2.1 Assume that the interaction topology G has a spanning tree, then system (2.27) achieves consensus if and only if A+B K 1 −λ L ,i B K 2 (i = 2, 3, . . . , N ) are Hurwitz. Proof Since the interaction topology G has a spanning tree, by Lemma 2.1, λ L ,i (i = 2, 3, . . . , N ) are nonzero. By (2.28), eigenvalues of H consist of eigenvalues of A + B K 1 and A + B K 1 − λ L ,i B K 2 (i = 2, 3, . . . , N ). Let P be a nonsingular matrix such that P −1 H P = J H = diag{JC , JC }, where JC and JC denote the Jordan blocks of H corresponding to eigenvalues of A + B K 1 and A + B K 1 − λ L ,i B K 2 (i = 2, 3, . . . , N ), respectively. ∑N d Necessity: Consider an initial state x(0) = j=d+1 α j (0) p j ∈ C(H ), where α j (0) /= 0 ( j = d + 1, d + 2, . . . , N d). The response of system (2.27) due to x(0) is x(t) = Pe JH t P −1 x(0) = P



H e JC t 0  H 0, . . . , 0, αd+1 (0), . . . , α HN d (0) . JC t 0 e (2.30)

We prove the conclusion by contradiction. If A + B K 1 − λ L ,i B K 2 (i = H H (0), αd+2 (0), . . . , α HN d (0)]H 2, 3, . . . , N ) are not Hurwitz, then the limit of e JC t [αd+1 as t → ∞ does not exist or is nonzero. By (2.30), the limit of x(t) as t → ∞ does not exist or is nonzero. By Lemma 2.37, C(H ) is an invariant subspace of H , hence x(t) ∈ C(H ). Since system (2.27) achieves consensus, there exists a vector function c(t) ∈ Rd such that limt→∞ (x(t) − 1 ⊗ c(t)) = 0. Since c1 , c2 , . . .∑ , cd are linear d independent, there exist γ j (t) ( j = 1, 2, . . . , d) such that c(t) = j=1 γ j (t)c j . According to the structure of p j ( j = 1, 2, . . . , d) given in Lemma 2.38, one has ∑ limt→∞ (x(t) − dj=1 γ j (t) p j ) = 0, which means x(t) ∈ C(H ) as t → ∞. Since C(H ) ∩ C(H ) = 0, one has limt→∞ x(t) = 0. A contradiction is obtained. Hence, it is necessary that A + B K 1 − λ L ,i B K 2 (i = 2, 3, . . . , N ) are Hurwitz. Sufficiency: Consider any initial state x(0) ∈ R N d . By Lemma 2.37, x(0) can be uniquely decomposed as follows x(0) = xC (0) + xC (0),

(2.31)

∑ ∑ d where xC (0) = dj=1 α j (0) p j and xC (0) = Nj=d+1 α j (0) p j . By Lemma 2.37, the response of system (2.27) due to xC (0) is xC (t) = Pe

JH t

P

−1

H e JC t 0  H 0, . . . , 0, αd+1 xC (0) = P (0), . . . , α HN d (0) . 0 e JC t (2.32) 

Since A + B K 1 − λ L ,i B K 2 (i = 2, 3, . . . , N ) are Hurwitz, one can see that

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 H H H lim e JC t αd+1 (0), αd+2 (0), . . . , α HN d (0) = 0.

t→∞

(2.33)

From (2.32) and (2.33), lim x (t) t→∞ C

= 0.

(2.34)

By Lemma 2.37, the response of system (2.27) due to xC (0) is xC (t) = Pe JH t P −1 xC (0) = P



H e JC t 0  H α1 (0), . . . , αdH (0), 0, . . . , 0 . JC t 0 e (2.35)

Let  H  H H β1 (t), β2H (t), . . . , βdH (t) = e JC t α1H (0), α2H (0), . . . , αdH (0) , then  H xC (t) = P β1H (t), . . . , βdH (t), 0, . . . , 0 .

(2.36)

Since the first d column vectors of P are p j = 1 ⊗ c j ( j = 1, 2, . . . , d) by Lemma 2.38, one has xC (t) = 1 ⊗

d ∑

β j (t)c j .

(2.37)

j=1

From (2.34) and (2.37), it holds that x(t) → xC (t), as t → ∞. Hence, there exists a vector function c(t) ∈ Rd such that lim (x(t) − 1 ⊗ c(t)) = 0.

t→∞

The proof of Theorem 2.1 is completed.

⟁

Remark 2.3 From the proof of Theorem 2.1, xC (0) and xC (0) determine the disagreement dynamics and consensus dynamics of system (2.27) respectively; that is, system (2.27) achieves consensus if and only if its response due to xC (0) is asymptotically stable, and if system (2.27) achieves consensus, then the consensus function, which describes the agreement state of each agent, is determined by its response due to xC (0).

2.5 Consensus for High-Order LTI Multi-agent Systems

83

Corollary 2.1 Assume that the interaction topology G does not have a spanning tree. Then system (2.27) achieves consensus if and only if A + B K 1 and A + B K 1 − λ L ,i B K 2 (λ L ,i /= 0, i ∈ {2, 3, . . . , N }) are Hurwitz. Proof Since the interaction topology G does not have a spanning tree, L at least has two zero eigenvalues. Thus at least two diagonal blocks of I N ⊗(A+B K 1 )− JL ⊗B K 2 shown in (2.28) are A + B K 1 . By a similar analysis as Theorem 2.1, it is necessary that A+ B K 1 and A+ B K 1 −λ L ,i B K 2 (λ L ,i /= 0, i ∈ {2, 3, . . . , N }) are Hurwitz. If A + B K 1 and A + B K 1 − λ L ,i B K 2 (λ L ,i /= 0, i ∈ {2, 3, . . . , N }) are Hurwitz, then system (2.27) is asymptotically stable. Obviously, system (2.27) achieves consensus and the consensus function is 0. Therefore, system (2.27) achieves consensus if and only if A + B K 1 and A + B K 1 − λ L ,i B K 2 (λ L ,i /= 0, i ∈ {2, 3, . . . , N }) are Hurwitz. ⟁ Remark 2.4 If the interaction topology G does not have a spanning tree, by Corollary 2.1, consensus problems of a multi-agent system can be converted into asymptotic stability problems in R N d . Since there exist interactions among agents, system (2.27) may not achieve consensus even if each agent has been stabilized. The following lemma is useful to obtain the condition of consensualization. Lemma 2.39 [27]: For any matrices R T = R > 0 and Q = D T D ≥ 0, if ( A, B) is stabilizable and (A, D) is detectable, then the algebraic Riccati equation P A + AT P − P B R −1 B T P + Q = 0 has a unique solution P T = P > 0, and A − ρ B R −1 B T P is Hurwitz for all ρ with Re(ρ) > 0.5. Theorem 2.2 System (2.27) is consensualizable if and only if (A, B) is stabilizable. Proof Necessity is obvious by Theorem 2.1 and Corollary 2.1. Sufficiency: If ( A, B) is stabilizable, then (A + B K 1 , B) is stabilizable. If G has a spanning tree, then Re(λ L ,i ) > 0 (i = 2, 3, . . . , N ) by Lemma 2.1. Let λmin = −1 T B P, where P and R can be min{Re(λ L ,i ) (i = 2, 3, . . . , N )} and K 2 = λ−1 min R obtained by Lemma 2.39, then A + B K 1 − λ L ,i B K 2 (i = 2, 3, . . . , N ) are Hurwitz. By Theorem 2.1, the conclusion can be obtained. If G does not have a spanning tree, then A + B K 1 should be Hurwitz by Corollary 2.1. In this case, let K 2 = 0, one can obtain the conclusion. ⟁

2.5.2 Consensus Functions The following theorem provides a general method to determine the consensus function. Theorem 2.3 Let any x(0) ∈ R N d be the initial state of system (2.27). If system (2.27) achieves consensus, then the consensus function c(t) satisfies

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! " lim c(t) − e( A+B K 1 )t [Id , 0, . . . , 0]PC(H ),C(H ) x(0) = 0,

t→∞

where PC(H ),C(H ) = [ p1 , . . . , pd , 0, . . . , 0]P −1 is an oblique projector onto C(H ) along C(H ). Proof By Lemma 2.37, x(0) can be uniquely decomposed as x(0) = xC (0) + xC (0), where xC (0) ∈ C(H ) and xC (0) ∈ C(H ). Since system (2.27) achieves consensus, the response of system (2.27) due to xC (0) satisfies limt→∞ xC (t) = 0 by Theorem 2.1. Then xC (0) determines the consensus function c(t). As C(H ) ⊕ C(H ) = C N d , there exists an oblique projector PC(H ),C(H ) = [ p1 , . . . , pd , 0, . . . , 0]P −1 such that xC (0) = PC(H ),C(H ) x(0). Since basis vectors of C(H ) satisfy p j = 1 ⊗ c j ( j = 1, 2, . . . , d) by Lemma 2.38, there exists a vector c0 ∈ Rd such that xC (0) = 1 ⊗ c0 . Then one has c0 = [Id , 0, . . . , 0]PC(H ),C(H ) x(0). Since L1 = 0 by Lemma 2.1, one can obtain " ! xC (t) = e H t (1 ⊗ c0 ) = 1 ⊗ e( A+B K 1 )t c0 . Therefore, one has ! " lim c(t) − e( A+B K 1 )t [Id , 0, . . . , 0]PC(H ),C(H ) x(0) = 0.

t→∞

The proof of Theorem 2.3 is completed.

⟁

Remark 2.5 Intuitionally, the consensus function of a multi-agent system with a given initial state describes its macroscopic property as a whole. The χ -consensus problem, which determines the consensus function, was proposed by Olfati-Saber and Murray [28]. For any first-order multi-agent system, if the interaction topology is balanced and strongly connected, then the consensus function is constant, and its value is the average value of the initial state of each agent. But when interaction topologies are directed and the dynamics of each agent is of high order, to the best of our knowledge, there is no general method to determine the consensus functions in the literature. Theorem 2.3 shows that the consensus function can be represented in a simple transformation of x(0) by the oblique projector PC(H ),C(H ) . For a high-order multi-agent system with a time-invariant consensus function, the conclusion also holds. In the practical applications, the consensus function of system (2.27) may be different according to real situations. The following corollary can be obtained directly by Theorems 2.2 and 2.3. Corollary 2.2 The consensus modes of system (2.27) with a nonzero consensus function can be designed if and only if the following two conditions are satisfied simultaneously: (i) The interaction topology G has a spanning tree; (ii) (A, B) is stabilized.

References

85

2.6 Notes In this chapter, the basic knowledge and fundamental theory for the admissible consensus of the singular multi-agent systems were given. Firstly, some basic knowledges of the algebraic graph theory were presented to describe the interaction topology of the multi-agent systems. Secondly, the linear algebra related to the consensus problems of the multi-agent systems were shown. Thirdly, the fundamental concepts and lemmas of the linear system theory and the singular system theory were given. Moreover, the consensus and consensualization for the highorder LTI multi-agent systems were addressed to give the first sight of the consensus problems.

References 1. Godsil C, Royal G (2001) Algebraic graph theory. Springer-Verlag, New York, NY 2. Lewis FL, Zhang H, Hengster-Movric K, Das A (2013) Cooperative control of multi-agent systems: optimal and adaptive design approaches. Springer Science & Business Media, Berlin, Germany 3. Ren W, Beard RW (2005) Consensus seeking in multiagent systems under dynamically changing interaction topologies. IEEE Trans Autom Control 50(5):655–661 4. Dong XW (2015) Formation and containment control for high order linear swarm systems. Springer-Verlag, New York, NY 5. Horn RA, Johnson CR (1989) Topics in matrix analysis. Cambridge University Press, Cambridge 6. Boyd S, Ghaoui LE, Feron E, Balakrishnan V (1994) Linear matrix inequalities in system and control theory. SIAM, Philadelphia 7. Xie L, Souza CE (1992) Robust control for linear systems with norm-bounded time-varying uncertainty. IEEE Trans Autom Control 37(8):1188–1191 8. Xi JX, Shi ZY, Zhong YS (2011) Consensus analysis and design for high-order linear swarm systems with time-varying delays. Phys A 390(23):4114–4123 9. Williams RL, Lawrence DA (2007) Linear state-space control systems. Wiley, Hoboken 10. Chen CT (1999) Linear system theory and design. Oxford University Press, New York 11. Xi JX, Shi ZY, Zhong YS (2012) Output consensus analysis and design for high-order linear swarm systems: partial stability method. Automatica 48:2335–2343 12. Roth WE (1952) The equations AX − YB = C and AX − XB = C in matrices. Proc Am Math Soc 3(3):392–396 13. Rosenbrock HH (1974) Structural properties of linear dynamic systems. Int J Control 20(2):191–202 14. Cobb D (1984) Controllability, observability, and duality in singular systems. IEEE Trans Autom Control 29(12):1076–1082 15. Cobb D (1981) Feedback and pole placement in descriptor variable systems. Int J Control 33(6):1135–1146 16. Verghese GC, Lévy BC, Kailath T (1981) A generalized state-space for singular systems. IEEE Trans Autom Control 26(4):811–831 17. Yip EL, Sincovec RF (1981) Solvability, controllability, and observability of continuous descriptor systems. IEEE Trans Autom Control 26(3):702–707 18. Ailon A (1987) Controllability of generalized linear time-invariant systems. IEEE Trans Autom Control 32(5):429–432

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19. Christodoulou MA, Paraskevopoulos PN (1985) Solvability, controllability and observability of singular systems. J Optim Theory Appl 45(1):53–72 20. Hou M (2004) Controllability and elimination of impulsive modes in descriptor systems. IEEE Trans Autom Control 49(10):1723–1727 21. Lewis FL (1986) Further remarks on the Cayley-Hamilton theorem and Leverrier’s method for the matrix pencil. IEEE Trans Autom Control 31(9):869–870 22. Ishihara JY, Terra MH (2002) On the Lyapunov theorem for singular systems. IEEE Trans Autom Control 47(11):1926–1930 23. Cobb D (1983) Descriptor variable system and optimal state regulation. IEEE Trans Autom Control 28(5):601–611 24. Dai LY (1989) Singular control systems. Springer, Berlin 25. Bender DJ, Laub AJ (1987) The linear-quadratic optimal regulator for descriptor systems. IEEE Trans Autom Control 32(8):672–688 26. Duan GR (2010) Analysis and design of descriptor linear systems. Springer, New York 27. Anderson BDO, Moore JB (1989) Optimal control: linear quadratic methods. Prentice-Hall 28. Olfati-Saber R, Murray RM (2004) Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans Autom Control 49(9):1520–1533

Chapter 3

Admissible Consensus and Consensualization on Interaction Topology

This chapter investigates the admissible consensus analysis and consensualizing controller design problems for high-order LTI singular multi-agent systems. Singular systems are a more nature representation of dynamic systems and can describe a larger class of systems than normal ones. Moreover, when coordinated variables include fast-variation and slow-variation components or there exist algebraic constraints among coordinated variables, each agent in multi-agent systems can only be modeled as a singular system rather than a normal one as shown in [1, 2], where the models of three typical singular multi-agent systems, i.e., three-link manipulator networks, a specific type of circuit networks and MASSs with each block supported by several pillars, were presented. Consensus problems of singular multi-agent systems are more complicated and challenging than those of normal multi-agent systems since each agent usually has three types of modes, namely, finite-dynamic modes, impulse modes, and non-dynamic modes. We analyze admissible consensus under, respectively, fixed and switching interaction graphs. For the fixed interaction graphs, firstly, by projecting the state of a singular multi-agent system onto a CS and a CCS, a necessary and sufficient condition for admissible consensus is presented in terms of LMIs. Then, an approach to decrease the calculation complexity is proposed, by which only three LMIs independent of the number of agents need to be checked. Finally, by using the changing variable method, sufficient conditions for admissible consensualization are shown. An explicit expression of the consensus function is given, and it is shown that the modes of the consensus function can be arbitrarily placed if each agent is R-controllable and impulse controllable, and the interaction topology has a spanning tree. For the switching interaction graphs, Firstly, by the state projection on the CS and the CCS, admissible consensus problems are converted into admissible problems of a reduced order switching subsystem, and an explicit expression of the consensus function is given on the basis of the First Equivalent Form. Secondly, a necessary and sufficient condition for admissible consensualization is proposed, which can guarantee

© Science Press, Beijing and Springer Nature Singapore Pte Ltd. 2023 J. Xi et al., Admissible Consensus and Consensualization for Singular Multi-agent Systems, Engineering Applications of Computational Methods 11, https://doi.org/10.1007/978-981-19-6990-4_3

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the scalability of singular multi-agent systems since it is independent of the number of agents. Finally, numerical simulations are presented to demonstrate theoretical results.

3.1 Structure and Variance of Interaction Topology Suppose that the agent of a multi-agent system interacts with each other via a communication network, which can be modeled by undirected or directed graphs. The basic definitions of the graphs have been shown in Sect. 2.1. For a given graph, it is naturally concerned of two important features; that is, the structure and the variance of the graph. The structure of the graph refers to the class and the relationship of the nodes, while the variance of the graph indicates the dynamically changing of the graph with respect to time. In the following, the structure and the variance of the interaction topology will be discussed.

3.1.1 Structure of Interaction Topology The structure of the interaction topology can be distinguished as the leaderless topology and the leader-following topology. For simplicity of the expression, the graph G in Sect. 2.1 is utilized to denote a leaderless topology. Different from the leaderless topology, there exists a true leader, labeled as agent 0, besides the followers 1 to N in the leader-following topology. In this sense, a leader-following topology can be described as the graph G l = (Vl (G), El (G)), where Vl (G) = V (G) ∪ {0} with {0} being the node set of the leader and El (G) ⊆ Vl (G) × Vl (G). Let the graph G in Sect. 2.1 represent the follower subgraph of the leader-following graph. In the leader-following topology, the leader can transmit the information to the followers, but it cannot receive information from the followers, which means that the edge from the leader node to the follower node is directed. For the leader-following topology, the leader is assumed to locate at the root node of the spanning tree of the graph. The edge weights from the leader to the followers is defined as wi0 for i = 1, 2, . . . , N , where wi0 > 0 if (0, i) ∈ El (G) and wi0 = 0 otherwise. In this case, the adjacency matrix of the leader-following graph can be defined as Wl (G) = [wi j ] ∈ R(N +1)×(N +1) and the Laplacian can be described as:  Ll =

 0 0 , −ll L + Ʌl

where Ʌl = diag{w10 , w20 , . . . , w N 0 }, ll = [w10 , w20 , . . . , w N 0 ]T , and L is the Laplacian matrix of G.

3.1 Structure and Variance of Interaction Topology

89

In the following, the properties of the interaction mechanism matrices for the leaderless and the leader-following topologies are discussed. For the leaderless topology, the interaction mechanism matrix is the Laplacian matrix of the complete graph denoted by I N − N −1 1 N 1TN ; i.e., ⎡

⎤ N − 1 −1 · · · −1 ⎢ −1 N − 1 · · · −1 ⎥ ⎢ ⎥ N ×N , ⎢ . .. .. ⎥ ∈ R .. ⎣ .. . . . ⎦ −1 −1 · · · N − 1 which indicates that all agents denoted by the nodes play equal roles to determine the interaction mechanism in the consensus achieving process for multi-agent systems. However, the interaction mechanism of the leader-following topology is determined by the leader, and the interaction mechanism matrix is the Laplacian matrix of the star graph represented by ⎡

−1 · · · 1 ··· .. . . . . −1 0 · · ·

N ⎢ −1 ⎢ ⎢ . ⎣ ..

⎤ −1 0 ⎥ ⎥ (N +1)×(N +1) . .. ⎥ ∈ R . ⎦ 1

The illustrations of the complete graph and the star graph are shown in Fig. 3.1.

1

1

2

6

3

5

6

4

(a) Complete graph. Fig. 3.1 Examples for complete graphs and star graphs

2

0

3

4

5 (b) Star graph.

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3 Admissible Consensus and Consensualization on Interaction Topology

3.1.2 Variance of Interaction Topology In the above analysis, the clear definitions of the structure of the graph are given with the fixed interaction topology, which means that the interaction topology is time-invariant. However, some common issues of the multi-agent systems on the interaction topologies are the possible communication link failures, the changing of the weights of the communication links, and the packet losses of the information transmissions, to name just a few. Therefore, it is necessary to consider the influence of the variance of the interaction topology on the consensus control of multi-agent systems. For a interaction topology described by a graph, if the edge weights of the graph are time-varying, then the directed graph is called a time-varying graph representing by G(t) = (V , E(t), W (t)). To discuss the variance of a time-varying graph, the following concepts are given. Definition 3.1 (Union of Graph [3]) The union of the time-varying graph G(t) within the time interval [T0 , T1 ) is a graph set with the same node set V , and the adjacency matrix Wˆ = [wˆ i j ] satisfies that

T1 wˆ i j =

wi j (τ )dτ, T0

and the edge set Eˆ is induced from Wˆ . Definition 3.2 (Joint Connectivity) A time-varying graph G(t) is said to be jointly connected if the following conditions hold: (i) A time-varying directed graph may not be strongly connected or contain a spanning tree, but its union can be. (ii) The union of the undirected time-varying graph is connected, but each graph in the union may not be connected. Definition 3.3 (Joint (δ, Δ)-connectivity [4]) The time-varying graph G(t) is said to be jointly (δ, Δ)-connected (strongly connected or containing a spanning tree for directed graph and connected for undirected graph) if there exist positive numbers δ and Δ such that the edge weights satisfy that T 0 +Δ

wi j (τ )dτ ≥ δ, T0

where i, j ∈ V for a connected graph (strongly connected or containing a spanning tree for directed graph and connected for undirected graph). Next, a special case of the time-varying graph is considered, which is piecewise fixed within some time intervals [5]. Suppose that there exists an infinite sequence

3.2 Admissible Consensus and Consensualization with Fixed Topology

91

of bounded and contiguous time intervals [T p , T p+1 ) ( p ∈ N) with T0 = 0 and 0 < T p+1 − T p ≤ Tmax . For the time interval [T p , T p+1 ), consider a sequence of uniformly bounded and non-overlapping time subintervals such that [T p0 , T p1 ), [T p1 , T p2 ), . . . , [T ps p −1 , T ps p ), with T p0 = T p , T ps p = T p+1 and T pm+1 − T pm ≥ Tmin (0 ≤ m ≤ s p − 1) for some integer s p ≥ 1 and a positive constant Tmin . In this case, the time-varying graph G(t) remains fixed over time intervals [T pm , T pm+1 ) for 0 ≤ m ≤ s p − 1 with p ∈ N. Let {G p | p ∈ 1, 2, . . . , ϒ } be the set of all possible graphs for the interaction topologies with the integer ϒ > 1, then define a right continuous signal σ (t) : [0, +∞) → {1, 2, . . . , ϒ}, such that G σ (t) = (V , E σ (t) , Wσ (t) ) with Wσ (t) = [wiσj(t) ] switches at T pm and remains fixed during time subintervals [T pm , T pm+1 ). Note that the piecewise fixed time-varying topology is also called the switching topology, which has been widely investigated for the consensus of the multi-agent systems. Without loss of generality, the properties of the jointly connected switching topology can be concluded with respect to the time intervals [T p , T p+1 ) ( p ∈ N) as follows: The interaction topologies switch at T p0 , T p1 , . . . , T ps p , and are fixed during [T pm , T pm+1 ) for 0 ≤ m ≤ s p − 1 with p ∈ N, where the interaction topology may not be connected, but the union of the interaction topologies across each time interval [T p , T p+1 ) ( p ∈ N) is connected.

3.2 Admissible Consensus and Consensualization with Fixed Topology In this section, admissible consensus analysis and design problems of high-order LTI singular multi-agent systems are addressed. A CS and a CCS are introduced, the direct sum of which is the state space of a singular multi-agent system with N agents. By state projection onto the two subspaces, a necessary and sufficient condition with N − 1 LMI constrains is proposed for the system to achieve admissible consensus. One of important features of multi-agent systems is large scale, so the feasibility of the N − 1 LMI constrains may be difficult to check when N is large. To improve the calculation efficiency, by encapsulating the complex eigenvalues of the Laplacian matrix into a convex set, an approach is given to reduce the N − 1 LMI constrains into three ones. Consensualizing controller design problems are investigated and sufficient conditions for admissible consensualization are given in terms of LMIs. Analysis and design problems of the consensus function are discussed, and an explicit expression of the consensus function is shown, which is determined by finite-dynamic modes and is independent of impulse modes and non-dynamic modes.

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3.2.1 Admissible Consensus Protocol with Fixed Topology Consider a singular multi-agent system with N agents which only interact with each other locally. A directed graph G can be used to describe the interaction topology of the system. For i, j ∈ {1, 2, . . . , N }, the node vi in G represents agent i, the edge (vi , v j ) ∈ ε(G) corresponds to the interaction channel from agent i to agent j, and w ji denotes the interaction strength of the channel (vi , v j ). It is assumed that all the agents share a common state space Rd . Let xi (t) ∈ Rd denote the state of agent i (i ∈ {1, 2, . . . , N }), which needs to be coordinated. If there exist some algebraic constraints on the coordinated variable xi (t) (i ∈ {1, 2, . . . , N }), then the dynamics of agent i can be described by E x˙i (t) = Axi (t) + Bu i (t)

(3.1)

where A ∈ Rd×d , B ∈ Rd×m , E ∈ Rd×d with rank(E) = r ≤ d, and u i (t) is the consensus protocol which is designed according to the information available to agent i. The following consensus protocol is applied: u i (t) = K 1 xi (t) + K 2





wi j x j (t) − xi (t) ,

(3.2)

j∈Ni

where (i ∈ {1, 2, . . . , N }), K 1 and K 2 are constant matrices with appropriate dimensions. Let x(t) = [x1T (t), x2T (t), . . . , x NT (t)]T , then the dynamics of a singular multi-agent system with N agents can be described by (I N ⊗ E)x(t) ˙ = (I N ⊗ (A + B K 1 ) − L ⊗ B K 2 )x(t),

(3.3)

where L is the Laplacian matrix of G. Remark 3.1 In (3.3), (E, A) denotes the intrinsic dynamics of each agent, which usually includes three types of modes, namely, finite-dynamic modes, impulse modes, and non-dynamic modes. The state feedback matrix K 1 is used to eliminate impulse modes and to place finite-dynamic modes of (E, A). The feedback matrix K 2 indicates the attractive interaction among neighboring agents, which is designed to propel the states of all agents to be identical. In the following, the definitions of admissible consensus and consensualization are presented, respectively. Definition 3.4 System (3.3) is said to achieve admissible consensus if it is regular and impulse-free, and there exists a vector-valued function c(t) dependent of the admissible bounded x(0) such that limt→∞ (x(t) − 1 ⊗ c(t)) = 0, where c(t) is called a consensus function. Definition 3.5 System (3.3) is said to be admissibly consensualizable if there exist K 1 and K 2 such that it achieves admissible consensus.

3.2 Admissible Consensus and Consensualization with Fixed Topology

93

The following three problems are investigated: (i) under what conditions system (3.3) achieves admissible consensus; (ii) how to determine K 1 and K 2 such that system (3.3) achieves admissible consensus; (iii) how to determine the consensus function c(t) if system (3.3) achieves admissible consensus.

3.2.2 Conditions for Admissible Consensus Analysis In this section, firstly, two subspaces of CNd , namely, a CS and a CCS, are constructed, and a necessary and sufficient condition with 2(N −1) LMI constraints for admissible consensus is presented by state projection onto the two subspaces. Then an approach to decrease the calculation complexity is shown, which includes only three LMI constraints independent of the number of agents. H    Let U −1 LU = JL where U = 1, U , U −1 = υ H , U˜ H and JL is the Jordan canonical form of L. λi (i = 1, 2, . . . , N ) denote the eigenvalues of L where λ1 = 0 with the associated eigenvector 1. Let Id denote a d × d identity matrix, c j ∈ Rd ( j = 1, 2, . . . , d) be linearly independent vectors, and ei ∈ R N (i = 1, 2, . . . , N ) with a 1 as its ith component and 0 elsewhere. The following two subspaces of CNd are introduced. Definition 3.6 Let p j = (U ⊗ Id )(ei ⊗ ck ) ( j = (i − 1)d + k; i = 1, 2, . . . , N ; k = 1, 2, . . . , d). A CS is defined as the subspace C(U ) spanned by p1 , p2 , . . . , pd and a CCS as the subspace C(U ) spanned by pd+1 , pd+2 , . . . , pNd . Let x(t) ˜ = (U −1 ⊗ Id )x(t) = [x˜1H (t), x˜2H (t), . . . , x˜ NH (t)]H , then system (3.3) can be transformed into ˙˜ = (I N ⊗ ( A + B K 1 ) − JL ⊗ B K 2 )x(t). (I N ⊗ E)x(t) ˜

(3.4)

If the interaction topology G associated with L has a spanning tree, by Lemma 2.1, one has υ LU = 0 and U˜ LU = J˜L where J˜L consists of Jordan blocks associated with λ2 , . . . , λ N . If G does not have a spanning tree, by Lemma 2.1, one can set that λ2 = 0 in J˜L and υ LU = 0 without loss of generality. Hence, JL can be written as JL = diag{0, J˜L }. Let ς (t) = [x˜2H (t), . . . , x˜ NH (t)]H , then system (3.4) can be rewritten as follows E x˙˜1 (t) = (A + B K 1 )x˜1 (t),

(3.5)

  (I N −1 ⊗ E)ς˙ (t) = I N −1 ⊗ (A + B K 1 ) − J˜L ⊗ B K 2 ς (t).

(3.6)

By using LMI techniques, the following theorem presents a necessary and sufficient condition for admissible consensus.

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3 Admissible Consensus and Consensualization on Interaction Topology

Theorem 3.1 System (3.3) achieves admissible consensus if and only if the following conditions hold simultaneously   E 0 = d + rank(E); (i) rank A + B K1 E (ii) There exist Ri (i = 2, 3, . . . , N ) such that ɅTE Ri = RiT Ʌ E ≥ 0,

T

Ʌ A+B K 1 − ϒλi Ʌ B K 2 Ri + RiT Ʌ A+B K 1 − ϒλi Ʌ B K 2 < 0; (iii) The interaction topology G has a spanning tree. Proof Necessity: Because nonsingular transformations cannot change the regularity and impulse-free property and U −1 ⊗ Id is nonsingular, system (3.3) is regular and impulse-free if and only if system (3.4) is regular and impulse-free. From Lemma 2.24, subsystem(3.5) is  regular and impulse-free if and only if condition (i) holds. Due to JL = diag 0, J˜L , subsystems (3.5) and (3.6) are independent of each other. System (3.4) can be equivalently described by subsystems (3.5) and (3.6), so condition (i) is necessary. In the following, we prove that condition (ii) is necessary by contradiction. Assume that system (3.3) achieves admissible consensus, then it is necessary that subsystems E ζ˙i (t) = (A + B K 1 − λi B K 2 )ζi (t) (i = 2, 3, . . . , N ) are regular and impulse-free according to the structures of J˜L in (3.6). If condition (ii) does not hold for a λi (i ∈ {2, 3, . . . , N }), then the subsystem E ζ˙i (t) = (A + B K 1 − λi B K 2 )ζi (t) is not asymptotically stable by Lemma 2.25; that is, the limit of ζi (t) as t → ∞ is nonzero or does not exist. In this case, by (3.6), the limit of ς (t) as t → ∞ is nonzero or does not exist. Define  H xC (t) = (U ⊗ Id ) x˜1H (t), 0 . Due to 

x˜1H (t), 0

H

= e1 ⊗ x˜1 (t),

one has xC (t) = 1 ⊗ x˜1 (t).

(3.7)

3.2 Admissible Consensus and Consensualization with Fixed Topology

95

Because c j ( j = 1, 2, . . . , d) are linearly independent, there exist α j (t) ( j =  1, 2, . . . , d) such that x˜1 (t) = dj=1 α j (t)c j . From (3.7), it can be obtained that xC (t) =

d 

α j (t) p j .

(3.8)

j=1

Define  H xC (t) = (U ⊗ Id ) 0, ς H (t) , then there exist α j (t) (j = d + 1, d + 2, . . . , Nd) such that xC (t) = (U ⊗ Id )

d N  

α(i−1)d+k (t)(ei ⊗ ck ) =

i=2 k=1

Nd 

αj (t)pj .

(3.9)

j=d+1

Since x(t) ˜ = (U −1 ⊗ Id )x(t), one has x(t) = xC (t) + xC (t).

(3.10)

Since system (3.3) achieves admissible consensus, by (3.7) and (3.10), from Lemma 2.3, one can obtain that lim x (t) t→∞ C

= 0,

which means that limt→∞ ς (t) = 0. A contradiction is obtained. Hence, condition (ii) is necessary for system (3.1) to achieve admissible consensus. Sufficiency: If conditions (i) and (ii) hold, then by Lemmas 2.23 and 2.24, system (3.3) is regular and impulse-free, and subsystem (3.6) is admissible; that is, limt→∞ ς (t) = 0. In this case, one has limt→∞ xC (t) = 0. By (3.7) and (3.10), system (3.3) achieves ⟁ admissible consensus. The proof of Theorem 3.1 is completed. Remark 3.2 If E = I , then system (3.3) becomes a high-order LTI normal multiagent system. In this case, the conclusions of Theorem 3.1 are equivalent to the ones of Theorem 2.1 in Sect. 2.5.1. Remark 3.3 By the above analysis, the consensus property of system (3.3) is jointly determined by the following three factors: the consensus protocol, the interaction topology, and the dynamics of each agent. Especially, If the interaction topology G does not have a spanning tree, then L at least has two zero eigenvalues by Lemma 2.1. In this case, it can be shown that system (3.3) achieves admissible consensus if and only if it is admissible. This case is trivial for singular multi-agent systems. Note that condition (ii) in Theorem 3.1 includes N − 1 LMI constraints, so it may be difficult to check the feasibility of these constraints when a singular multi-agent

96

3 Admissible Consensus and Consensualization on Interaction Topology

system consists of numerous agents. The following theorem presents a method to improve the calculation efficiency by encapsulating all complex eigenvalues of the Laplacian matrix L into a convex set. Let λ˜ 1 = ζmin + ςmax j, λ˜ 2 = ζmin + ςmin j, λ˜ 3 = ζmax + ςmax j and λ˜ 4 = ζmax + ςmin j, where j 2 = −1 and ζmin = min{Re(λi ), i = 2, 3, . . . , N }, ζmax = max{Re(λi ), i = 2, 3, . . . , N }, ςmin = min{Im(λi ), i = 2, 3, . . . , N }, ςmax = max{Im(λi ), i = 2, 3, . . . , N }. Theorem 3.2 System (3.3) achieves admissible consensus if the following conditions hold simultaneously   E 0 = d + rank(E); (i) rank A + B K1 E (ii) There exists a matrix R such that E T R = R T E ≥ 0,

T

Ʌ A+B K 1 − ϒλ˜ i Ʌ B K 2 Ʌ R + ɅTR Ʌ A+B K 1 − ϒλ˜ i Ʌ B K 2 < 0 (i = 1, 3).

(iii) The interaction topology G has a spanning tree.

T

Proof Let Ξi = Ʌ A+B K 1 − ϒλi Ʌ B K 2 Ʌ R + ɅTR Ʌ A+B K 1 − ϒλi Ʌ B K 2 (i = 2, 3, . . . , N ), then one has Ξi = Θ0 + Re(λi )Θ1 + Im(λi )Θ2 , where  Θ0 =

 0 (A + B K 1 )T R + R T (A + B K 1 ) , 0 (A + B K 1 )T R + R T (A + B K 1 )   0 −(B K 2 )T R − R T (B K 2 ) Θ1 = , 0 −(B K 2 )T R − R T (B K 2 )   0 −(B K 2 )T R + R T (B K 2 ) Θ2 = . 0 (B K 2 )T R − R T (B K 2 )



T

Similarly, let Φi = Ʌ A+B K 1 − ϒλ˜ i Ʌ B K 2 Ʌ R + ɅTR Ʌ A+B K 1 − ϒλ˜ i Ʌ B K 2 (i = 1, 2, 3, 4), then one can obtain Φi = Θ0 + Re(λ˜ i )Θ1 + Im(λ˜ i )Θ2 .

3.2 Admissible Consensus and Consensualization with Fixed Topology

97

Let  T =

 0I , I 0

then due to ςmax = −ςmin , it can be shown that T −1 Φ1 T = Φ2 , T −1 Φ3 T = Φ4 , which mean that Φ1 < 0 ⇔ Φ2 < 0 and Φ3 < 0 ⇔ Φ4 < 0, respectively. Now, we show that if Φ1 < 0 and Φ3 < 0, then Ξi < 0 (i = 2, 3, . . . , N ). Consider any λi = Re(λi ) + Im(λi ) j (i = 2, 3, . . . , N ). There exist β Ri ∈ [0, 1] and β I i ∈ [0, 1] such that Re(λi ) = β Ri ζmin + (1 − β Ri )ζmax , Im(λi ) = β I i ςmin + (1 − β I i )ςmax . Let κi1 κi2 κi3 κi4

= β Ri (1 − β I i ), = β Ri β I i , = (1 − β Ri )(1 − β I i ), = β I i (1 − β Ri ),

then one can see that κik ∈ [0, 1] (k = 1, 2, 3, 4) and that

4

Ξi = Θ0 + Re(λi )Θ1 + Im(λi )Θ2 =

k=1

4 

κik = 1. It can be shown

κik Φk .

k=1

Therefore, the condition that Φ1 < 0 and Φ3 < 0 (hence, Φ2 < 0 and Φ4 < 0) guarantees that Ξi < 0 (i = 2, 3, . . . , N ). By the above analysis, if conditions (i) and (ii) hold simultaneously, then system ⟁ (3.3) achieves admissible consensus from Theorem 3.1. Remark 3.4 One of properties of multi-agent systems is large scale, so the solvability of presented methods is critically important. The LMI tools were used to deal with consensus problems of first-order normal multi-agent systems with N agents in [6], where the dimension of each variable of LMI conditions is N − 1. It is time-cost and memory-cost to check those LMI conditions when N is large. By the conclusions in Theorem 3.2, only three LMI constraints independent of the number of agents are required to be checked, so the calculation complexity is decreased greatly if N is very large. However, it should be pointed out that conclusions of Theorem 3.2 are

98

3 Admissible Consensus and Consensualization on Interaction Topology

conservative compared with that of Theorem 3.1 since there can exist solutions Ri (i = 2, 3, . . . , N ) for condition (ii) in Theorem 3.1 even if there is no solution R for condition (ii) in Theorem 3.2. Moreover, the calculation complexity of the approach in Theorem 3.2 is less than that of Lemma 5 in [7]. The conditions in Theorem 3.2 are dependent on the eigenvalues of L, which may be difficult to obtain when N is huge. In the following, an approach is given to make LMI conditions for admissible consensus independent of the eigenvalues of L. By the Gersgorin in [8], all the eigenvalues of  disk theorem   L lie in the   ˜ ˜ ˜ ˜ ˜ ˜ disk s ∈ C : s − d M  ≤ d M , where d M = max d1 , d2 , . . . , d N with d˜i (i = 1, 2, . . . , N ) denoting the in-degree of node i of the interaction topology G, so it can be obtained that ζmax ≤ 2d˜M and ςmax ≤ d˜M . Let λmin ∈ (0, ζmin ]. The problem how to determine ζmin has been studied in [9, 10]. The following corollary can be directly obtained from Theorem 3.2. Corollary 3.1 System (3.3) achieves admissible consensus if conditions (i), (ii), and (iii) with λ˜ 1 = λmin + d˜M j and λ˜ 3 = 2d˜M + d˜M j in Theorem 3.2 hold simultaneously.

3.2.3 Admissible Consensus Design Criteria In this section, sufficient conditions for admissible consensualization are presented in terms of LMIs. By the proof of Theorem 3.1, subsystems (3.5) and (3.6) describe the consensus dynamics and disagreement dynamics of system (3.3), respectively. Based on this fact, K 1 and K 2 are determined via the following two steps: (i) By choosing K 1 , place poles of (E, A) to eliminate impulsive modes and to design the consensus function; (ii) Design K 2 such that subsystem (3.6) is admissible. First of all, an approach is shown to determine K 1 such that (E, A + B K 1 ) is impulse-free and its finite-dynamic modes can be arbitrarily placed, which is needed to state the main conclusions about admissible consensualization and the consensus function. Let T0 and Z 0 be nonsingular matrices satisfying T0 E Z 0 = diag{Ir , 0} and denote     A11 A12 B1 , T0 B = . T0 AZ 0 = A21 A22 B2 If (E, A, B) is impulse controllable, then there exists K 11 such that A22 + B2 K 11 is invertible, which means that (E, A + B K 1 ) with K 1 = [0, K 11 ]Z 0−1 is impulse-free. It is not difficult to find that the following two matrices are nonsingular,  T1 =

 Ir −(A12 + B1 K 11 )(A22 + B2 K 11 )−1 , 0 Id−r

3.2 Admissible Consensus and Consensualization with Fixed Topology

99



 Ir 0 Z1 = . −(A22 + B2 K 11 )−1 A21 (A22 + B2 K 11 )−1 Direct computation shows that  T1

       A 0 B1 A11 A12 + B1 K 11 B1 Z1 = , T1 Z1 = , A21 A22 + B2 K 11 B2 B2 0 Id−r

where A = A11 − (A12 + B1 K 11 )(A22 + B2 K 11 )−1 A21 , B 1 = B1 − (A12 + B1 K 11 )(A22 + B2 K 11 )−1 B2 . If the triplet (E, A, B) is R-controllable, then ( A, B 1 ) is controllable. Hence, there exists a matrix K 12 such that the poles of A + B 1 K 12 can be placed into any given symmetric set on the complex plane, which means that (E, A + B K 1 ) is impulse-free and its finite-dynamic modes can be arbitrarily placed by K 1 = [0, K 11 ]Z 0−1 + [K 12 , 0]Z 1−1 Z 0−1 . In this case, it can be shown that  T1 T0 E Z 0 Z 1 =  T1 T0 (A + B K 1 )Z 0 Z 1 =

 Ir 0 , 0 0

 A + B 1 K 12 0 . B2 K 12 Id−r

(3.11)

(3.12)

From (3.11) and (3.12), by Definition 2.10, one can see that (E, A + B K 1 ) is regular. In the sequel, based on the above analysis, an approach to determine the gain matrices of protocol (3.2) is presented by using LMI techniques. Theorem 3.3 Suppose that the interaction topology G has a spanning tree and the triplet (E, A, B) is impulse controllable. Let K 1 be a gain such that the pair (E, A + B K 1 ) is impulse-free. Assume that for the gain K 1 , there exist K˜ 2 and an invertible matrix R˜ such that R˜ T E T = E R˜ ≥ 0, ɅTR˜ ɅTA+B K 1 + Ʌ A+B K 1 Ʌ R˜ − ɅTB K˜ ϒλ˜T − ϒλ˜ i Ʌ B K˜ 2 < 0 (i = 1, 3). 2

i

Then, the system (3.3) is admissibly consensualizable by K 1 and K 2 = K˜ 2 R˜ −1 . Proof Because the triplet (E, A, B) is impulse controllable, there exists K 1 such that (E, A + B K 1 ) is impulse-free. By the conditions in Theorem 3.3, one obtains that

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3 Admissible Consensus and Consensualization on Interaction Topology

E T R = R T E ≥ 0,

(3.13)

ɅTA+B K 1 Ʌ R + ɅTR Ʌ A+B K 1 − ɅTB K 2 ϒλ˜T Ʌ R − ɅTR ϒλ˜ i Ʌ B K 2 < 0 (i = 1, 3), (3.14) i

˜ then the result follows from Theorem 3.2. ⟁ where R := R˜ −1 . Define K˜ 2 = K 2 R,

3.2.4 Consensus Functions on Fixed Topology An approach to determine the consensus function is shown on the basis of the mode decomposition, and an explicit expression of the consensus function is given, which is completely determined by finite-dynamic modes and is independent of non-dynamic modes. Theorem 3.4 If the triplet (E, A, B) is R-controllable and impulse controllable, the interaction topology G has a spanning tree, and system (3.3) achieves admissible consensus, then the modes of the consensus function c(t) can be arbitrarily placed by K 1 = [0, K 11 ]Z 0−1 + [K 12 , 0]Z 1−1 Z 0−1 and satisfies    T lim c(t) − Z 0 Z 1 Ir , −(B2 K 12 )T e( A+B 1 K 12 )t [Ir , 0]Z 1−1 Z 0−1 (υ ⊗ Id )x(0) = 0.

t→∞

Proof Since G has a spanning tree, by Theorem 3.1 and Remark 3.3, the consensus function can be nonzero. Since (E, A, B) is R-controllable and impulse controllable, the impulsive modes of (E, A) can be eliminated and its finite-dynamic modes can be arbitrarily placed by K 1 = [0, K 11 ]Z 0−1 + [K 12 , 0]Z 1−1 Z 0−1 . If system (3.3) achieves admissible consensus, then subsystem (3.5) determines the consensus function. By Lemma 2.3, one has xC (0) = PC(U ),C(U ) x(0),

(3.15)

  where PC(U ),C(U ) = [ p1 , p2 , . . . , pd , 0, . . . , 0]P −1 with P = p1 , p2 , . . . , pNd is an oblique projector onto C(U ) along C(U ). Let C = [c1 , c2 , . . . , cd ], then it can be shown by Definition 3.6 that P = U ⊗ C. Due to [ p1 , p2 , . . . , pd , 0, . . . , 0] = [1, 0] ⊗ C, one has

3.2 Admissible Consensus and Consensualization with Fixed Topology

PC(U ),C(U ) = ([1, 0]U −1 ) ⊗ Id

101

(3.16)

From (3.7), one can obtain that x˜1 (0) = [Id , 0, . . . , 0]xC (0).

(3.17)

Since [1, 0]U −1 = (1 ⊗ υ), from (3.15) to (3.17), it can be shown that x˜1 (0) = (υ ⊗ Id )x(0).

(3.18)

By (3.11)–(3.12) and (3.18), the conclusion of Theorem 3.4 can be obtained. ⟁ For first-order multi-agent systems, the average consensus problem was extensively addressed in Refs. [6, 11], which mainly concerns that under what conditions the consensus function of a multi-agent system is the average of states of all agents. Obviously, if multi-agent systems with the same initial states but different topologies achieve average consensus, consensus functions of these multi-agent systems are identical. For high-order multi-agent systems, average consensus is difficult to be achieved since each agent has more complex dynamics. However, there exists a similar problem; that is, under what conditions consensus functions for multi-agent systems with the same initial states but different topologies are same. The following corollary gives a topology condition for this problem. Corollary 3.2 If the triplet (E, A, B) is R-controllable and impulse controllable, the interaction topology G is balanced and has a spanning tree, and system (3.3) achieves admissible consensus, then the consensus function c(t) satisfies  N    1 −1 −1 T T (A+B 1 K 12 )t lim c(t) − Z 0 Z 1 Ir , −(B2 K 12 ) e xi (0) = 0. [Ir , 0]Z 1 Z 0 t→∞ N i=1 

Proof If G is balanced, then 1T L = 0. Since U consists of eigenvectors and generalized eigenvectors of L, it can be shown that 1T U = 0. In this case, one has υ = 1T /N . ⟁ By Theorem 3.4, the conclusion of Corollary 3.2 can be obtained. Remark 3.5 In [12], Cai et al. pointed out that the motions of a multi-agent system consist of the absolute motion as a whole and the relative motions among agents, but they did not give an approach to determine the absolute motion. The consensus function describes the absolute motion, which is important to analyze and design a multi-agent system. In Sect. 2.5.2, we presented an explicit expression of the consensus function for a high-order LTI normal multi-agent system based on the initial state decomposition. Since E may be singular, the method in Sect. 2.5.2 cannot be applied to determine the consensus function of system (3.3). Based on the mode decomposition, Theorem 3.4 gives a general approach to determine the consensus functions of high-order LTI singular multi-agent systems. Moreover, from Corollary 3.2, the average consensus problem for first-order multi-agent systems proposed in [11] is a special case of the above works.

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3 Admissible Consensus and Consensualization on Interaction Topology

3.2.5 Numerical Simulation MASSs [13] have potential applications in earthquake damage-preventing buildings, water-floating plants and large-diameter parabolic antennae or telescopes. We investigated cooperative control problems of MASS by using consensus techniques in [7, 14], where it was assumed that each agent is described by a normal second-order system. However, when a MASS consists of many independent blocks and each block is supported by several pillars, each agent in this MASS becomes a singular system. In this section, theoretical results shown in previous sections are applied to deal with cooperative control problems of singular MASS. Consider the case that each agent in a MASS is supported by two pillars called Unit I and Unit II, respectively, as shown in Fig. 3.2, where m is the mass, d is the damping coefficient, and k is the stiffness coefficient. Let xiI (t), xiII (t), viI (t) and viII (t) represent heights and velocities of Unit I and Unit II, respectively, then agent i (i ∈ {1, 2, . . . , N }) can be described by (3.3), where ⎡

⎤ ⎡ xiI (t) 10 ⎢ viI (t) ⎥ ⎢0 1 ⎥ ⎢ xi (t) = ⎢ ⎣ xiII (t) ⎦, E = ⎣ 0 0 viII (t) 00

0 0 0 0

⎤ ⎡ 0 0 ⎥ ⎢ 0⎥ − mk , A=⎢ ⎦ ⎣ 0 −1 0 0

1 − md 0 −1

0 0 1 0

⎤ ⎡ ⎤ 0 0 ⎥ ⎢ 1⎥ 0⎥ ⎥. , B=⎢ ⎦ ⎣ 0⎦ 0 0 1

Example 3.1 (Scalability) Consider a MASS consisting of N agents, where the parameters of each agent are chosen as m = 1.6, d = 50.5 and k = 12. Let K 1 = [−0.25, 0, 0, 0] and K 2 = [0, 0, 0.0148, 0]. Figure 3.3a shows its interaction topology whose edges are labeled from 1 to N + 1, and the weight of edge k is 0.01k (k ∈ {1, 2, . . . N + 1}). Since the topology has a link structure, it is not difficult to determine the eigenvalues of its Laplacian matrix. Under the same experiment conditions, for different N , the calculation time to check the feasibility of condition (ii) in Theorems 3.1 and 3.2 is shown in Table 3.1. It can be found that the method in Theorem 3.2 significantly improves the calculation efficiency, especially for very large N .

Fig. 3.2 Model of each agent in a MASS

m

m

d

k

Unit I

d

k

Unit II

3.2 Admissible Consensus and Consensualization with Fixed Topology Table 3.1 Calculation time to check feasibility

103

N = 100 (s)

N = 1000 (s)

N = 10,000 (s)

Theorem 3.1

0.6978

7.8796

83.8873

Theorem 3.2

0.0125

0.0128

0.0123

Example 3.2 (Water-floating plant) All agents in a water-floating plant try to keep the plant horizontal. It is not difficult to see that the plant will keep horizontal if all agents achieves admissible consensus on the height; that is, there exists a function c(t) such that limt→∞ (xiI (t) − c(t)) = 0 and xiI (t) = xiII (t) (i = 1, 2, . . . , N ). The consensus function c(t) would continuously oscillate since the water disturbs the plant all the time. The parameters of each agent are chosen as m = 1.6, d = 50.5 and k = 1200. Since the triplet (E, A, B) is R-controllable and impulse controllable, the modes of the consensus function are placed into at ±0.5j with j2 = −1 by K 1 = [749.7500, 31.5625, 0, 0]. By Theorem 3.3, one can obtain that K 2 = [0.7801, 0.3628, −0.2338, 0.1170]. The interaction topology of the plant is shown in Fig. 3.3b and it is assumed that its adjacent matrix is a 0–1 matrix without loss of generality. The initial state is set as x(0) = [−7, 7, −7, 7, 0, −2, 0, −2, 7, −6, 7, −6]T . The position and velocity trajectories of the water-floating plant are shown in Fig. 3.4, where the trajectories marked by circles denote the curves of the consensus function. In the simulation, the trajectories of the positions and velocities of Unit I and Unit II overlap all the time, respectively, so one can see that the plant keeps horizontal. Moreover, state trajectories of the plant converge to ones marked by circles, which demonstrates the conclusion of Theorem 3.4. Remark 3.6 By choosing appropriate parameters, (3.1) can also be used to model self-propelled particle multi-agent systems, where each agent has complex dynamics.

1

3

6

4 5

2

4

7

1

8

5 6

2

2

3

(a) Fig. 3.3 Interaction topologies of MASS

N+1 N

1

3

(b)

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3 Admissible Consensus and Consensualization on Interaction Topology

Fig. 3.4 State trajectories of the water-floating plant

By a similar analysis in [14], many very interesting collective phenomena for selfpropelled particles, such as aggregation of particles with a zero velocity and formation of particles with a nonzero velocity, can be well explained. By numerical simulations, heading consensus of self-propelled particles was illustrated in [15, 16]. Based on graph theory, Jadbabaie et al. gave a theoretical explanation of the observed collective phenomena in [15, 16]. It should be pointed out that each agent is modeled as a first-order integrator in [15–17], which means that the state of each particle is timeinvariant if this particle does not interact with others. In many practical self-propelled particle multi-agent systems, the states of each particle are time-varying even if it does not interact with other particles. Collective phenomena of these particle multi-agent systems can be well explained by the approaches in the current section.

3.3 Admissible Consensus and Consensualization with Switching Topologies In this section, we address admissible consensus and consensualization problems for high-order LTI singular multi-agent systems with switching topologies. By state projection, a necessary and sufficient condition for admissible consensus is presented, and admissible consensus problems are transformed into admissible problems of a switching singular subsystem with a lower dimension. An approach is given to determine the consensus function, and it is shown that switching movements do not impact the consensus function. An admissible consensualization criterion is presented, which shows that it is necessary and sufficient for admissible consensualization that each agent is stabilizable.

3.3 Admissible Consensus and Consensualization with Switching Topologies

105

3.3.1 Admissible Consensus Protocol with Switching Topologies Consider the singular multi-agent system (3.1). The interaction topology of multiagent system (3.1) is described by an undirected graph, where each vertex represents an agent, the edge between any two vertices stands for the interaction channel between them, and the weight of the edge denotes the interaction strength. The following consensus protocol is applied: u i (t) = K





wi j x j (t) − xi (t) ,

(3.19)

j∈Ni (t)

where K ∈ Rm×d is a gain matrix and Ni (t) denotes the time-varying neighbor set of agent i. The current section mainly discusses the case with switching interaction topologies. Let the finite set S with an index set I ⊂ N denote all possible interaction topologies, where N is the set of natural numbers. σ (t) : [0, ∞) → I is a switching signal, whose value at time t is the index of the interaction topology at time t. It is assumed that Assumption 3.1 The switching movements 0 < t1 < · · · < tk < · · · satisfy inf k (tk+1 − tk ) = Td > 0. Assumption 3.2 All interaction topologies in S are connected.  T Let x(t) = x1T (t), x2T (t), . . . , x NT (t) , then the dynamics of multi-agent system (3.1) with protocol (3.19) can be rewritten as a vector form as follows

˙ = I N ⊗ A − L σ (t) ⊗ B K x(t), (I N ⊗ E)x(t)

(3.20)

where L σ (t) is the Laplacian matrix of the associated interaction topology G σ (t) . Definition 3.7 Multi-agent system (3.20) is said to achieve admissible consensus if for any given admissible bounded x(0), there exists a vector-valued function c(t) dependent on x(0) such that limt→∞ (x(t) − 1 ⊗ c(t)) = 0, where c(t) is called a consensus function. Definition 3.8 Multi-agent system (3.20) is said to be admissibly consensualizable by protocol (3.20) if there exists a gain matrix K such that multi-agent system (3.20) achieves admissible consensus. The current section focuses on the following three problems: (i) under what conditions multi-agent system (3.20) achieves admissible consensus; (ii) how to determine the consensus function c(t) if multi-agent system (3.20) achieves admissible consensus; and (iii) how to determine the gain matrix K such that multi-agent system (3.20) achieves admissible consensus.

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3 Admissible Consensus and Consensualization on Interaction Topology

3.3.2 Necessary and Sufficient Conditions for Admissible Consensus Analysis with Switching Topologies In this section, consensus problems of multi-agent system (3.20) are transformed into admissible problems of an (N − 1)d dimensional subsystem. Let Id denote a d × d identity matrix, c j ∈ Rd ( j = 1, 2, . . . , d) be linearly N independent column vectors,  ei√∈ R (i = 1, 2, . . . , N ) with a 1 as its ith component and 0 elsewhere, and U = 1/ N , U˜ be an orthogonal matrix. It can be shown by Lemma 2.1 that   0 0 T . U L σ (t) U = 0 U˜ T L σ (t) U˜  T

Let U T ⊗ Id x(t) = xCT (t), xCT (t) , multi-agent system (3.20) can be converted into two subsystems as follows E x˙C (t) = AxC (t),   (I N −1 ⊗ E)x˙C (t) = I N −1 ⊗ A − U˜ T L σ (t) U˜ ⊗ B K xC (t).

(3.21) (3.22)

The following theorem gives a necessary and sufficient condition for multi-agent system (3.20) to achieve admissible consensus. Theorem 3.5 Multi-agent system (3.20) achieves admissible consensus if and only if subsystem (3.22) is admissible and  rank

E 0 A E

 = d + rank(E).

Proof Since nonsingular transformations cannot change the regularity and impulsefree property and U ⊗ Id is nonsingular, multi-agent system (3.20) is regular and impulse-free if and only if both subsystems (3.21) and (3.22) are regular and impulsefree. By Lemma 2.24, the regularity and impulse-free property of subsystem (3.22) and   E 0 rank = d + rank(E) A E are necessary and sufficient for the regularity and impulse-free property of Multiagent system (3.20). Because c j ( j = 1, 2, . . . , d) are linearly independent, there exist α j (t) ( j = 1, 2, . . . , d) such that

3.3 Admissible Consensus and Consensualization with Switching Topologies

xC (t) =

d 

α j (t)c j .

107

(3.23)

j=1

Let 1 x˜C (t) ≙ √ 1 ⊗ xC (t), N

(3.24)

then one can obtain by (3.23) that x˜C (t) =

d 

α j (t) U e1 ⊗ c j ∈ C(U ).

(3.25)

j=1

Furthermore, it can be shown that  T x˜C (t) = (U ⊗ Id ) xCT (t), 0 .

(3.26)

Moreover, there exist α j (t) (j = d + 1, d + 2, . . . , Nd) such that N  d   xC (t) = 0, I(N −1)d α(i−1)d+k (t)(ei ⊗ ck ).

(3.27)

i=2 k=1

Let x˜C (t) ≙

N  d 

α(i−1)d+k (t)(U ei ⊗ ck ) ∈ C(U ).

(3.28)

i=2 k=1

By (3.27) and (3.28), one can see that  T x˜C (t) = (U ⊗ Id ) 0, xCT (t) .

(3.29)

 T

From (3.26) and (3.29), due to U T ⊗ Id x(t) = xCT (t), xCT (t) , one has x(t) = x˜C (t) + x˜C (t).

(3.30)

From Lemma 2.3, by (3.27), (3.28) and (3.30), it is necessary and sufficient for multi-agent system (3.20) to achieve consensus that limt→∞ xC (t); that is, subsystem (3.22) is asymptotically stable. By the above analysis, the conclusion of Theorem 3.5 can be obtained. ⟁

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3 Admissible Consensus and Consensualization on Interaction Topology

3.3.3 Admissible Consensus Design Criteria: Switching Topology Case A typical feather of multi-agent systems is of large scale, so consensualization criterions should be independent of the number of agents to guarantee scalability. This section presents a necessary and sufficient condition for admissible consensualization, which is independent of the number of agents. Let λ1σ (t) , λ2σ (t) , . . . , λσN(t) denote the eigenvalues of L σ (t) , where λ1σ (t) = 0 with √   the associated eigenvector 1/ N . Let λmin = min λik (∀k ∈ I, i = 2, 3, . . . , N ) , then it can be shown that by Lemma 2.1 and Assumption 3.2. The following theorem gives an approach to determine the gain matrix K . Theorem 3.6 Assume that (E, A) is regular and impulse-free. Then multi-agent system (3.1) is admissibly consensualizable by protocol (3.19) if and only if T (E, A, B) is stabilizable. In this case, K = λ−1 min B Q/2, where Q is a solution of the Riccati equation in Lemma 2.26.   Proof Because U T L σ (t) U = diag 0, U˜ T L σ (t) U˜ and all interaction topologies in S are undirected, there exists an orthogonal matrix U σ (t) such that   T ˜ σ (t) = diag λ2σ (t) , λ3σ (t) , . . . , λσN(t) . U σ (t) U˜ T L σ (t) UU   T 

T

T

T T Let ησ (t) (t) = U σ (t) ⊗ Id xC (t) = η2σ (t) (t) , η3σ (t) (t) , . . . , ησN(t) (t) , then by Assumption 3.1, subsystem (3.22) can be transformed into

i E η˙ σ (t) (t) = A − λiσ (t) B K ηiσ (t) (t) (i = 2, 3, . . . , N ). Thus, necessity is obvious. In the following, we proceed to prove sufficiency. Consider the following Lyapunov function candidate

V (t) = xCT (t) I N −1 ⊗ Q T E xC (t), where Q is a solution of Riccati equation of Lemma 2.25. By taking the derivative of V (t) with respect to time t along the solution of subsystem (3.22), one can obtain that 

 V˙ (t) = xCT (t) I N −1 ⊗ AT Q + Q T A − U˜ T L σ (t) U˜ ⊗ Q T B K + K T B T Q xC (t) =

N  i

T

ησ (t) (t) AT Q + Q T A − λiσ (t) Q T B K + K T B T Q ηiσ (t) (t). i=2

T Let K = λ−1 min B Q/2, then it can be shown by (3.1) that

3.3 Admissible Consensus and Consensualization with Switching Topologies

V˙ (t) =

109

N  i

T

T

i T ησ (t) (t) −R + 1 − λiσ (t) λ−1 min Q B B Q ησ (t) (t). i=2

T ˙ Due to 1 − λiσ (t) λ−1 min ≤ 0 and R = R > 0, then one can see that V (t) ≡ 0 if i and only if ησ (t) (t) ≡ 0 (i = 2, 3, . . . , N ), which means that ησ (t) (t) ≡ 0. Due to

xC (t) = U σ (t) ⊗ Id ησ (t) (t), subsystem (3.22) is asymptotically stable. In this case, by the property of Kronecker products, one can show that

(I N −1 ⊗ Q T )(I N −1 ⊗ E) = (I N −1 ⊗ E T )(I N −1 ⊗ Q) ≥ 0,   (I N −1 ⊗ Q T ) I N −1 ⊗ A − U˜ T L σ (t) U˜ ⊗ B K T  + I N −1 ⊗ A − U˜ T L σ (t) U˜ ⊗ B K (I N −1 ⊗ Q) < 0, which can guarantee that subsystem (3.22) is regular and impulse-free by Lemma 2.2 ⟁ in [18]. By Theorem 3.5, the conclusion of Theorem 3.6 can be obtained. Remark 3.7 The LMI tool was extensively applied to deal with consensus problems for normal multi-agent systems with time-varying topologies (e.g., [19] for the cases with switching topologies and [2] for the cases with topology uncertainties). However, it should be pointed out that the dimensions of all variables of LMI consensus conditions in [2, 19] are dependent of the number of agents, so it is timecost and memory-cost to check them, which means that those conditions cannot guarantee the scalability of multi-agent systems. Since the consensualization criterion in Theorem 3 is independent of the number of agents, it can guarantee scalability. Remark 3.8 The algebraic Riccati equation was used to investigate consensualization problems in [14, 20–22]. Necessary and sufficient conditions for consensualization were given in [14, 22], where it was assumed that interaction topologies are fixed. High-order linear multi-agent systems with switching topologies were dealt with in [20, 21], where the dynamics of each agent was modeled as a normal system. Since singular multi-agent systems have more general structures than normal ones, the methods in [20, 21] cannot be directly used to investigate consensualization problems for high-order LTI singular multi-agent systems with switching topologies. Theorem 3.6 gives a general method to deal with consensus design problems for linear singular multi-agent systems with switching topologies.

3.3.4 Consensus Functions on Switching Topologies From the proof of Theorem 3.5, x˜C (t) and x˜C (t) can be regarded as the state projection of multi-agent system (3.20) on the CS C(U ) and the CCS C(U ), respectively. Based on this fact, we give an approach to determine the consensus function.

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3 Admissible Consensus and Consensualization on Interaction Topology

Let T and Z be nonsingular matrices such that  T EZ =

   Ir 0 A˜ 0 , , T AZ = 0 0 0 Id−r

which are usually called the First Equivalent Form as shown in [18]. Theorem 3.7 If multi-agent system (3.20) achieves admissible consensus, then the consensus function satisfies 

 ˜ T At

lim c(t) − Z [Ir , 0] e [Ir , 0]Z

−1

t→∞

N 1  xi (0) N i=1

 = 0.

  Proof Let PC(U ),C(U ) = [ p1 , p2 , . . . , pd , 0, . . . , 0]P −1 with P = p1 , p2 , . . . , pNd is an oblique projector onto C(U ) along C(U ). Let C = [c1 , c2 , . . . , cd ], then by the property of Kronecker products, it can be shown that



P = U ⊗ C,

(3.31)

 1 √ 1, 0 ⊗ C = [ p1 , p2 , . . . , pd , 0, . . . , 0]. N

(3.32)

Hence, one can see by (3.31) and (3.32) that  PC(U ),C(U ) =

 

1 T 1 11 ⊗ Id . √ 1, 0 ⊗ C U T ⊗ C −1 = N N

(3.33)

Due to x˜C (t) ∈ C(U ), from Lemma 2.3 it can be obtained by (3.33) that  x˜c (0) = PC(U ),C(U ) x(0) = 1 ⊗

 N 1  xi (0) . N i=1

(3.34)

By (3.24) and (3.34), one has N 1  xC (0) = √ xi (0). N i=1

(3.35)

From the proof of Theorem 3.5, by (3.24) and (3.30), one can obtain that 

 1 lim x(t) − √ 1 ⊗ xC (t) = 0, t→∞ N

(3.36)

3.3 Admissible Consensus and Consensualization with Switching Topologies

111

which means that subsystem (3.31) determines the consensus function. By (3.35) ⟁ and (3.36), the conclusion of Theorem 3.7 can be obtained. Remark 3.9 When multi-agent systems achieve consensus, the states of all agents asymptotically tend to be identical. Consensus functions are usually used to determine the identical state. Average consensus problems for first-order normal multiagent systems with undirected interaction topologies were investigated in [23], where consensus functions are the average of states of all agents. Let E = I and A = 0, then multi-agent system (3.20) becomes a typical first-order multi-agent system. Thus, by Theorem 3.7, it is not difficult to see that the conclusions in [23] are special cases of the above works. Xiao and Wang gave an explicit expression of the consensus function for high-order normal multi-agent systems in [2], where it was supposed that the consensus function is time-invariant. The initial state projection approach in Sect. 2.5.2 can determine time-varying consensus functions, but the approach cannot reveal the influences of topology structures and topology variances on consensus functions. Theorem 3.7 shows that switching movements do not impact consensus functions of singular multi-agent systems.

3.3.5 Numerical Simulation Consider a singular multi-agent system with six agents, where the dynamics of each agent is described by (3.1) with ⎡

1 ⎢0 E =⎢ ⎣0 0

0 1 0 0

0 0 0 0

⎤ ⎡ 0 3 ⎢9 0⎥ ⎥, A = ⎢ ⎣1 0⎦ 0 2

5 2 3 −5

3 0 1 −2

⎤ ⎡ ⎤ 0 02 ⎢ ⎥ 2⎥ ⎥, B = ⎢ 3 0 ⎥. ⎦ ⎣ 0 0 1⎦ 1 10

Figure 3.5 shows the switching topology set S which consists of four undirected interaction topologies, and it is supposed that the adjacency matrix of each topology is a 0–1 matrix. The interaction topologies of the singular multi-agent system are randomly chosen from S with Td = 0.5 s. Let ⎡

16 ⎢0 R=⎢ ⎣0 0

0 5 4 0

0 4 4 0

⎤ 0 0⎥ ⎥. 0⎦ 9

Since (E, A, B) is stabilizable and all interaction topologies in S are connected, it can be obtained by Theorem 3.6 that

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3 Admissible Consensus and Consensualization on Interaction Topology

2

2

1

3

1

3

6

4

6

4

5

5

G1

G2

2

2

1

3

1

3

6

4

6

4

5

5

G3

G4

Fig. 3.5 Switching topology set S

⎡

1 ⎢ 0 Q=⎢ ⎣ −1 −4

0 1 −3 −1

0 0 1 0

⎤ 0 0⎥ ⎥, 0⎦ 1



 −1.1761 4.8219 0.2319 −0.6556 K = . 0.7752 −9.7153 −0.3558 −1.0792 Figure 3.7 shows the state trajectories of the singular multi-agent system with the switching signal σ (t) given in Fig. 3.6, where the trajectories marked by circles denote the curves of the consensus function given by Theorem 3.7. It is clear that the singular multi-agent system achieves admissible consensus, and it demonstrates the conclusion of Theorem 3.7 that the state trajectories converge to the ones marked by circles. Moreover, by repeating this simulation with different switching signals, it is found that state trajectories always converge to the same curves although they may be different, which means that switching movements do not influence the consensus function although they are critical for singular multi-agent systems to achieve admissible consensus.

3.4 Notes

113

Fig. 3.6 Switching signal σ (t)

Fig. 3.7 State trajectories with switching signal σ (t)

3.4 Notes In this chapter, admissible consensus and consensualization problems for high-order singular multi-agent systems with fixed and switching topologies were investigated. Based on state projection decomposition, a necessary and sufficient condition with

114

3 Admissible Consensus and Consensualization on Interaction Topology

2(N − 1) LMI constraints was given for the system to achieve admissible consensus with fixed topology. An approach was proposed to improve the calculation efficiency, by which 2(N − 1) LMI constraints are reduced into three ones. An approach to consensualize the system was presented in term of LMIs and the design problem of the consensus function was investigated. Especially, it was shown that the consensus function is only determined by finite-dynamic modes and is independent of impulse modes and non-dynamic modes of each agent. Furthermore, similar to method of the fixed topology, admissible consensus problems with switching topologies were transformed into admissible ones by the state projection, and an explicit expression of the consensus function was presented, which is associated with the average of initial states of all agents but is independent of switching movements. Based on the Riccati equation methods, an approach was given to determine the gain matrices of consensus protocols, which is independent of the number of agents of multi-agent systems. Numerical simulations are presented to demonstrate the above theoretical results.

References 1. Xi JX, Meng FL, Shi ZY, Zhong YS (2012) Time-dependent admissible consensualization for singular time-delayed swarm systems. Syst Control Lett 61(11):1089–1096 2. Xiao F, Wang L (2007) Consensus problems for high-dimensional multi-agent systems. IET Control Theory Appl 1(3):830–837 3. Papachristodoulou A, Jadbabaie A, Münz U (2010) Effects of delay in multi-agent consensus and oscillator synchronization. IEEE Trans Autom Control 55(6):1471–1477 4. Anderson BD, Shi G, Trumpf J (2016) Convergence and state reconstruction of timevarying multi-agent systems from complete observability theory. IEEE Trans Autom Control 62(5):2519–2523 5. Qin JH, Ma QC, Gao HJ, Zheng WX, Kang Y (2022) Consensus over switching network topology: characterizing system parameters and joint connectivity. Springer-Verlag Press, Switzerland 6. Lin P, Jia YM (2008) Average consensus in networks of multi-agents with both switching topology and coupling time-delay. Phys A 387(1):303–313 7. Xi JX, Shi ZY, Zhong YS (2011) Consensus analysis and design for high-order linear swarm systems with time-varying delays. Phys A 390(23):4114–4123 8. Horn RA, Johnson CA (1990) Matrix analysis. Cambridge University 9. Berman A, Zhang XD (2000) Lower bounds for the eigenvalues of Laplacian matrices. Linear Algebra Appl 316(1–3):13–20 10. Yoonsoo K, Mesbah MI (2006) On maximizing the second smallest eigenvalue of a statedependent graph Laplacian. IEEE Trans Autom Control 51(1):116–120 11. Olfati-Saber R, Murray RM (2004) Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans Autom Control 49(9):1520–1533 12. Cai N, Xi JX, Zhong YS (2015) Swarm stability of high-order linear time-invariant swarm systems. IET Control Theory Appl 46(8):1458–1471 13. Ma S, Hackwood S, Beni G (1994) Multi-agent supporting systems (MASS): control with centralized estimator of disturbance. In: IEEE/RSJ international conference on intelligent robots and systems 14. Xi JX, Cai N, Zhong YS (2010) Consensus problems for high-order linear time-invariant swarm systems. Phys A 389(24):5619–5627

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15. Vicsek T, Czirók A, Ben-Jacob E, Cohen I, Shochet O (1995) Novel type of phase transition in a system of self-driven particles. Phys Rev Lett 75(6):1226–1229 16. Czirók A, Vicsek T (2000) Collective behavior of interacting self-propelled particles. Phys A 281(1–4):17–29 17. Jadbabaie A, Lin J, Morse AS (2003) Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Trans Autom Control 48(6):988–1001 18. Masubuchi I, Kamitane Y, Ohara A, Suda N (1997) H ∞ control for descriptor systems: a matrix inequalities approach. Automatica 33(4):669–673 19. Cepeda-Gomez R, Olgac N (2013) Exact stability analysis of second-order leaderless and leader-follower consensus protocols with rationally-independent multiple time delays. Syst Control Lett 62(6):482–495 20. Ni W, Cheng D (2010) Leader-following consensus of multi-agent systems under fixed and switching topologies. Syst Control Lett 59(3):209–217 21. Su Y, Huang J (2012) Solvability of two consensus problems for a class of linear multi-agent systems. IEEE Trans Autom Control 57(6):1420–1430 22. Yang XR, Liu GP (2012) Necessary and sufficient consensus conditions of descriptor multiagent systems. IEEE Trans Circuits Syst I Regul Pap 59(11):2669–2677 23. Sun YG, Wang L, Xie G (2008) Average consensus in networks of dynamic agents with switching topologies and multiple time-varying delays. Syst Control Lett 57(2):175–183

Chapter 4

Admissible Consensus and Consensualization with Time Delays

In this chapter, we investigate admissible consensus and dynamic output consensus problems of singular swarm systems with time delays. By state decomposition and a nonsingular transformation similar to the separation principle, the consensus problem is converted into the stability problems of multiple singular subsystems associated with eigenvalues of the Laplacian matrix of the interaction topology, and the gain matrices are assigned to different subsystems, so they can be designed, respectively. Furthermore, sufficient conditions are given, which are expressed by strict LMIs and can ensure scalability since they only include eight LMI constraints independent of the number of agents. Moreover, an explicit expression of consensus functions is shown and the impacts of protocol states and interaction topologies on consensus functions are determined. Especially, it is shown that time delays do not influence consensus functions and the singular swarm systems with the same initial states but different interaction topologies have the same consensus function if interaction topologies are balanced.

4.1 Introduction of Time Delays In practical applications, the process of information measurement, calculation and execution often takes a certain amount of time. In this case, the process of information processing by each agent will cause time delays, and there are generally time delays in the process of information exchange among agents. The time delays will influence the overall performance of the multi-agent system, even destroy the consistency regulation performance or stability of the multi-agent systems, or lead to undesirable oscillations in the practical engineering systems. It can be seen that the admissible consensus and output consensus problem under time delays is a very important task.

© Science Press, Beijing and Springer Nature Singapore Pte Ltd. 2023 J. Xi et al., Admissible Consensus and Consensualization for Singular Multi-agent Systems, Engineering Applications of Computational Methods 11, https://doi.org/10.1007/978-981-19-6990-4_4

117

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4 Admissible Consensus and Consensualization with Time Delays

4.1.1 Classification of Time Delays The chapter focuses on the consensus design and analysis of multi-agent systems with two different types of time delays, which are distinguished by the length of time delays. One is that the length of time delays is constant, which is called constant delay, and the other is time-dependent, which is called time-varying delay. The multi-agent systems with time-varying delay can be described as ⎧

x˙i (t) = Axi (t) + Bxi (t − τ (t)), xi (t) = φi (t),

where xi (t) stands for the state of agent i, φi (t) is the initial condition, τ (t) denotes the continuous time-varying delay, which satisfies the condition τ1 ≤ τ (t) ≤ τ2 . τ1 and τ2 are positive constants, which represent the time delay upper bound and lower bound of the system, respectively. If the time-varying delay satisfies the condition τ (t) ≡ τ , then multi-agent systems will be described with constant delay as follows ⎧

x˙i (t) = Axi (t) + Bxi (t − τ ), xi (t) = φi (t).

4.1.2 The Influence of Time Delays on Consensus Control For the existing consensus conditions, the consensus criterion can be divided according to whether it depends on the size of time delays. Here, we take a constant delay for example. One is a delay-dependent criterion; that is, the system can achieve consensus for some values of delay greater than zero satisfying the criterion, while the system cannot reach consensus for other values greater than zero. The other is the time-delay-independent criterion; that is, for all values with time-delay greater than zero, the system can achieve consensus. The characteristics are as follows: (i) The delay-dependent criterion depends on the system delay, and it needs to know the accurate information of the system delay time. On the contrary, the time-delayindependent criterion does not need any time-delay information of the system. For the delay-independent criterion, the system is uniform under any delay. (ii) For delaydependent criteria, only when the delay is less than the prescribed value, the system can reach consensus. Comprehensive analysis shows that the designed controller is conservative because there is no delay-related information in the delay-independent criterion. On the contrary, the delay-dependent criterion contains the information of time delays, so that the consistency of the system depends on the time delay.

4.2 Delay-Dependent Admissible Consensus and Consensualization

119

For the delay-independent criterion, the Lyapunov–Krasovskii function is usually selected in the following form ∫t V (t, x(t)) = x (t)P x(t) +

x T (s)Qx(s)ds,

T

t−τ (t)

where P and Q are positive definite symmetric matrices. According to Lyapunov stability theorem, only if V (t, x(t)) is greater than zero, and the derivative of V (t, x(t)) is less than zero; that is, the multi-agent systems can achieve consensus control, as shown in the following V˙ (t, x(t)) =

[

x(t) x(t − τ )

]T [

AT P + P A + Q P B ∗ −Q

][

] x(t) < 0. x(t − τ )

Obviously, the inequality does not contain the relevant information of time delays, so for any time delay τ , as long as the above linear matrix inequalities with respect to variables P and Q have feasible solutions, the system can achieve consensus. For the delay-dependent criterion, the double integral term is added as follows ∫0 ∫ t V1 (t) =

x˙ T (s)R x(s)dsdθ. ˙ −τ t+θ

The partial derivative of V1 (t) for time t, one can obtain that ˙ − V˙1 (t, x(t)) = τ x˙ (t)R x(t)

∫t

T

x˙ T (s)R x(s)ds. ˙

t−τ (t)

∫ t It is Tworth noting that the treatment of the above indefinite integral term ˙ brings certain conservatism, and there is no fixed solution at t−τ (t) x˙ (s)R x(s)ds present. Therefore, many scholars are committed to constructing Lyapunov functions of the system.

4.2 Delay-Dependent Admissible Consensus and Consensualization Admissible consensus analysis and design problems for high-order LTI singular multi-agent systems with time delays are investigated. Firstly, by state decomposition, the admissible consensus problem is transformed into admissible problems of

120

4 Admissible Consensus and Consensualization with Time Delays

multiple singular subsystems with lower dimensions. Then, LMI criteria for admissible consensualization are presented, which only involve eight LMI constraints independent of the number of agents. Moreover, an explicit expression of the consensus function which is independent of time delays is presented, and the impacts of protocol states and interaction topologies on the consensus function are revealed. Finally, a numerical example is given to illustrate the effectiveness of theoretical results.

4.2.1 Dynamic Output Feedback Consensus Protocol with Time Delays Consider a singular multi-agent system as follows ⎧

E x˙i (t) = Axi (t) + Bu i (t), yi (t) = C xi (t),

(4.1)

where i = 1, 2, . . . , N , E ∈ Rd×d with rank(E) = r ≤ d, A ∈ Rd×d , B ∈ Rd×m , C ∈ Rq×d , xi (t), yi (t) and u i (t) are the state, the output and the control input, respectively. Construct the following consensus protocol ⎧ ) ( ∑ E z˙ i (t) = (A + B K 0 C + B K 1 )z i (t) − K 2 C wi j z j (t − τ ) − z i (t − τ ) ⎪ ⎪ ⎪ j∈Ni ⎨ ( ) ∑ + K2 wi j y j (t − τ ) − yi (t − τ ) , ⎪ j∈Ni ⎪ ⎪ ⎩ u i (t) = K 0 yi (t) + K 1 z i (t), (4.2) where z i (t) ∈ Rd is the state of the protocol, K 0 , K 1 , and K 2 are gain matrices with appropriate dimensions and 0 ≤ τ ≤ τ is a constant delay with τ as its upper bound. [ T ]T Let κi (t) = xi (t), z iT (t) = (i = 1, 2, . . . , N ) and κ(t) ]T [ T T T κ1 (t), κ2 (t), . . . , κ N (t) , then the dynamics of multi-agent system (4.1) with protocol (4.2) can be described by ⎧

) ( ) ( ˙ = I N ⊗ A κ(t) − L ⊗ B κ(t − τ ), t ∈ [0, ∞), (I N ⊗ Ʌ E )κ(t) κ(t) = φ(t), t ∈ [−τ, 0],

(4.3)

where φ(t) is a compatible bounded vector-valued function, and [ A=

] ] [ B K1 A + B K0C 0 0 , B= . 0 A + B K0C + B K1 K 2 C −K 2 C

Definition 4.1 Multi-agent system (4.3) is said to achieve admissible consensus if it is regular and impulse-free, and for any given admissible bounded φ(t)

4.2 Delay-Dependent Admissible Consensus and Consensualization

121

(t ∈ [−τ, 0]), there exists a 2d-vector-valued function c(t) dependent of κ(0) such that limt→∞ (κ(t) − 1 ⊗ c(t)) = 0, where c(t) is called a consensus function. Definition 4.2 Multi-agent system (4.1) is said to be admissibly consensualizable by protocol (4.2) if there exist K 0 , K 1 , and K 2 such that multi-agent system (4.3) achieves admissible consensus. It should be pointed out that it is necessary for admissible consensus that the initial condition φ(t) (t ∈ [−τ, 0]) is admissible (see [1, 2] for more details about the admissible property of the initial condition), but this is not required for normal multi-agent systems to achieve consensus. The chapter addresses the following three problems: (i) under what conditions multi-agent system (4.3) achieves admissible consensus; (ii) how to determine K 0 , K 1 , and K 2 such that multi-agent system (4.3) achieves admissible consensus; (iii) how to determine the consensus function if multi-agent system (4.3) achieves admissible consensus.

4.2.2 Necessary and Sufficient Conditions for Delay-Dependent Admissible Consensus Analysis In this section, based on state decomposition, the admissible consensus problem for multi-agent system (4.3) is converted into the stability problems of multiple subsystems with [ lower ] dimensions. Let U = u 1 , U with U = [u 2 , u 3 , . . . , u N ] be a nonsingular matrix such that U −1 LU = JL , where JL is the Jordan canonical form of L. Let λ1 , λ2 , . . . √ , λN denote the eigenvalues of L, where λ1 = 0 with associated eigenvector u 1 = 1/ N and 0 ≤ Re(λ2 ) ≤ · · · ≤ Re(λ N ). Let ) [ ( ]H H H κ(t) = U −1 ⊗ I2d κ(t) = κ H 1 (t), κ 2 (t), . . . , κ N (t) ,

(4.4)

then by Lemmas 2.1 and 2.2, multi-agent system (4.3) can be converted into the following two parts Ʌ E κ˙ 1 (t) = Aκ 1 (t),

(4.5)

) ( ) ( ˙ = I N −1 ⊗ A η(t) − J L ⊗ B η(t − τ ), (I N −1 ⊗ Ʌ E )η(t)

(4.6)

[ ]H H H where η(t) = κ H and J L consists of the Jordan blocks of L 2 (t), κ 3 (t), . . . , κ N (t) associated with λ2 , λ3 , . . . , λ N . The following theorem shows a necessary and sufficient condition for multi-agent system (4.3) to achieve admissible consensus.

122

4 Admissible Consensus and Consensualization with Time Delays

Theorem 4.1 Multi-agent system (4.3) achieves admissible consensus if and only if the following conditions hold simultaneously (i) (E, A + B K 0 C + B K 1 ) is admissible; (ii) The systems E ς˙i (t) = (A + B K 0 C)ςi (t) + λi K 2 Cςi (t − τ ) (i = 2, 3, . . . , N ) are admissible. Proof Let c1 , c2 , . . . , c2d ∈ R2d be linearly independent vectors and p j = u i ⊗ ck , then one has p j = (U ⊗ I2d )(ei ⊗ ck ), where j = 2(i −1)d +k, i = 1, 2, . . . , N , k = 1, 2, . . . , 2d, and ei (i = 1, 2, . . . , N ) are N-dimensional vectors with a 1 as its ith entry and 0 elsewhere. Since U ⊗ I2d is nonsingular, p j ( j = 1, 2, . . . , 2d) are linearly independent. Hence, there exist α j (t) ( j = 1, 2, . . . , 2N d) such that κ(t) = κC (t) + κC (t),

(4.7)

where κC (t) ≙

2d ∑

⎛ α j (t) p j = (U ⊗ I2d )⎝e1 ⊗

j=1

κC (t) ≙

2N d ∑

2d ∑

⎞ α j (t)c j ⎠,

j=1

α j (t) p j = (U ⊗ I2d )

j=2d+1

2d N ∑ ∑

α2(i−1)d+k (t)(ei ⊗ ck ).

i=2 k=1

) ( Due to κ(t) = U −1 ⊗ I2d κ(t), one can obtain from (4.7) that ]H [ κC (t) = (U ⊗ I2d ) κ H 1 (t), 0 ,

(4.8)

]H [ κC (t) = (U ⊗ I2d ) 0, ηH (t) .

(4.9)

Since [

κH 1 (t), 0

]H

= e1 ⊗ κ 1 (t),

it can be shown by (4.8) that 1 κC (t) = √ 1 ⊗ κ 1 (t). N

(4.10)

Because κC (t) and κC (t) are linearly independent, by (4.9) and (4.10), multiagent system (4.3) achieves admissible consensus if and only if it is regular and

4.2 Delay-Dependent Admissible Consensus and Consensualization

123

impulse-free and limt→∞ κC (t) = 0 for any initial condition; that is, subsystem (4.6) is asymptotically stable. Since U ⊗ I2d is nonsingular, multi-agent system (4.3) is regular and impulse-free if and only if both subsystems (4.5) and (4.6) are regular and impulse-free. Let [ P=

] I −I , I 0

[ ]H and P −1 κ i (t) = ζiH (t), ςiH (t) (i = 2, 3, . . . , N ), then according to the structure of J L , one sees that subsystem (4.6) is admissible if and only if the following systems [

E 0 0 E

][

] [ ][ ] 0 A + B K0C + B K1 ζi (t) ζ˙i (t) = ς˙i (t) 0 A + B K 0 C ςi (t) ][ ] [ ζi (t − τ ) 0 λi K 2 C + 0 λi K 2 C ςi (t − τ )

(4.11)

are admissible. From (4.11), it can be shown that subsystem (4.5) is regular and impulse-free and subsystem (4.6) is admissible if and only if conditions (i) and (ii) □ hold. Thus, the conclusion of Theorem 4.1 can be obtained. Remark 4.1 The motions of a multi-agent system consist of the absolute motion as a whole and the relative motions among agents which describe its macroscopic and microcosmic behaviors, respectively. By the proof of Theorem 4.1, subsystems (4.5) and (4.6) determine the absolute motion and relative motions of multi-agent system (4.3), respectively. Conditions (i) and (ii) in Theorem 4.1 can ensure that the absolute motion is regular and impulse-free and the relative motions are admissible simultaneously. If the interaction topology G does not have a spanning tree, then the Laplacian matrix L at least has two zero eigenvalues by Lemma 2.1. From Theorem 4.1, the following corollary can be obtained directly. Corollary 4.1 If the interaction topology G does not have a spanning tree, then multi-agent system (4.3) achieves admissible consensus if and only if it is admissible. From Theorem 4.1, multi-agent system (4.3) with τ = 0 achieves admissible consensus if and only if (E, A + B K 0 C + B K 1 ) and (E, A + B K 0 C + λi K 2 C) (i = 2, 3, . . . , N ) are admissible. Let T and Z be nonsingular matrices such that T E Z = diag{Ir , 0} and ] A˜ 11 A˜ 12 , T (A + B K 0 C + B K 1 )Z = ˜ ˜ A21 A22 ] [ Ai11 Ai12 T (A + B K 0 C + λi K 2 C)Z = (i = 2, 3, . . . , N ). Ai21 Ai22 [

124

4 Admissible Consensus and Consensualization with Time Delays

The following corollary gives a necessary and sufficient condition of admissible consensus for singular multi-agent systems without time delays. Corollary 4.2 Multi-agent system (4.3) with τ = 0 achieves admissible consensus ˜ if and only if A˜ 22 and Ai22 (i = 2, 3, . . . , N ) are invertible, and A˜ 11 + A˜ 12 A˜ −1 22 A21 −1 and Ai11 + Ai12 Ai22 Ai21 (i = 2, 3, . . . , N ) are Hurwitz. For the cases where E is nonsingular; that is, rank(E) = r = d. From Theorem 4.1, the following corollary can be obtained. Corollary 4.3 If E is nonsingular, then multi-agent system (4.3) with τ = 0 achieves admissible consensus if and only if E −1 (A + B K 0 C + B K 1 ) and E −1 (A + B K 0 C + λi K 2 C) (i = 2, 3, . . . , N ) are Hurwitz. Remark 4.2 By the above analysis, the admissible consensus of multi-agent system (4.3) is jointly determined by the consensus protocol, the interaction topology and the dynamics of each agent. The consensus protocol represents some attractive interaction among agents, which makes the states of all agents converge to the same variables. The interaction topology among agents should be sufficiently connected. Otherwise, by Corollary 4.1, admissible consensus problems become asymptotic stability problems if the interaction topology does not have a spanning tree. This case is trivial for multi-agent systems. Moreover, the dynamics of each agent is a very important factor for high-order singular multi-agent systems to achieve consensus. In contrast, consensus for first-order multi-agent systems (e.g., [3–5]) is completely determined by the consensus protocol and the interaction topology.

4.2.3 Delay-Dependent Admissible Consensus Designinse Criteria From Theorem 4.1, it is necessary for multi-agent system (4.3) to achieve admissible consensus that (E, A, B) is impulse controllable and (E, A, C) is impulse observable. In this case, there must exist a K 0 such that (E, A + B K 0 C) is impulse-free by Theorem 3.5.1 in [6], where an approach to determine K 0 is given. This section focuses on determining K 1 and K 2 such that multi-agent system (4.3) achieves admissible consensus. It is well-known that the LMI tool is powerful to deal with time-delayed systems, but if an LMI criterion is of high dimension, then it is time-cost and memory-cost to check it. Since an important feature of multi-agent systems is large scale, the presented criterions for admissible consensualization should be independent of the number of agents to ensure scalability. The following lemma can be used to improve the calculation efficiency. Let λ˜ 1,2 = Re(λ2 ) ± jμ M , λ˜ 3,4 = Re(λ N ) ± j μ M with μ M = max{Im(λi ), i = 2, 3, . . . , N } and Φ0 , Φ1 and Φ2 be real symmetric matrices independent of λi (i = 2, 3, . . . , N ) and λ˜ i (i = 1, 2, 3, 4).

4.2 Delay-Dependent Admissible Consensus and Consensualization

125

The following theorem presents LMI criteria for admissible consensualization of multi-agent system (4.3), which can guarantee scalability. Theorem 4.2 For any τ ∈ [0, τ ], multi-agent system (4.1) can be admissibly consensualized by protocol (4.2) if there exist 0 < Q T = Q ∈ R2d×2d , 0 < S T = S ∈ T R2d×2d , W, G˜ ∈ Rd×d , 2d × 2d real matrices R, X , Y , M12 , M11 = M11 and T m×d d×q M22 = M22 , K 1 ∈ R and K 2 ∈ R such that ɅTE R = R T Ʌ E ≥ 0,

(4.12)

⎡

⎤ Ξ11 Ξ12 Ξi13 Ξi = ⎣ ∗ Ξ22 Ξi23 ⎦ < 0 (i = 1, 2, 3, 4), ∗ ∗ −Q ⎡ ⎤ M11 M12 X Θ = ⎣ ∗ M22 Y ⎦ > 0, ∗ ∗ S G˜ T E T = E G˜ ≥ 0, )T ( G˜ T (A + B K 0 C)T + (A + B K 0 C)G˜ + B K 1 + B K 1 < 0,

(4.13)

(4.14)

(4.15) (4.16)

where Ξ11 = ɅTW Ʌ A+B K 0 C + ɅTA+B K 0 C ɅW + Q + X Ʌ E + ɅTE X T + τ M11 , Ξ12 = R T − ɅTW + ɅTA+B K 0 C ɅW + ɅTE Y T + τ M12 , Ξ22 = −ɅTW − ɅW + τ S + τ M22 , Ξi13 = Ʌ K 2 ϒλ˜ i ɅC − X Ʌ E , Ξi23 = Ʌ K 2 ϒλ˜ i ɅC − Y Ʌ E . In this case, multi-agent system (4.3) achieves admissible consensus by protocol (4.2) with K 1 = K 1 G˜ −1 and K 2 = W −T K 2 . Proof First of all, consider the asymptotic stability of the following systems E ς˙i (t) = (A + B K 0 C)ςi (t) + λ˜ i K 2 Cςi (t − τ ) (i = 1, 2, 3, 4).

(4.17)

According to the decomposition of real and imaginary parts, systems (4.17) are admissible if and only if

126

4 Admissible Consensus and Consensualization with Time Delays

Ʌ E ς˙ˆi (t) = Ʌ A+B K 0 C ςˆi (t) + Ʌ K 2 ϒλ˜ i ɅC ςˆi (t − τ ) (i = 1, 2, 3, 4)

(4.18)

[ ]T are admissible, where ςˆi (t) = Re(ςi (t))T , Im(ςi (t))T . Construct the following Lyapunov–Krasovskii functional candidate Vi (t) = Vi1 (t) + Vi2 (t) + Vi3 (t),

(4.19)

where Vi1 (t) = ςˆiT (t)R T Ʌ E ςˆi (t), ∫t Vi2 (t) =

ςˆiT (s)Q ςˆi (s)ds, t−τ

∫0

∫t

Vi3 (t) =

ςˆ˙iT (s)ɅTE SɅ E ς˙ˆi (s)dsdθ.

−τ t+θ

By taking the derivative of these functions with respect to time t, along the solution of systems (4.18), one has ) ] ] [ T T ]([ [ 0 0 I R ɅW T , ς ˆ ϕ (t) (t) + V˙i1 (t) = 2ϕi1 − τ (t ) i1 i Ʌ K 2 ϒλ˜ i ɅC 0 ɅTW Ʌ A+B K 0 C −I (4.20) V˙i2 (t) = ςˆiT (t)Q ςˆi (t) − ςˆiT (t − τ )Q ςˆi (t − τ ), V˙i3 (t) ≤ τ ς˙ˆiT (t)ɅTE SɅ E ς˙ˆi (t) −

∫t

ς˙ˆiT (s)ɅTE SɅ E ς˙ˆi (s)ds,

(4.21)

(4.22)

t−τ

[ ]T where ϕi1 (t) = ςˆiT (t), ɅTE ς˙ˆiT (t) . Due to ∫t

ς˙ˆi (s)ds = ςˆi (t) − ςˆi (t − τ ),

t−τ

for any real matrices X and Y with appropriate dimensions, it can be shown that ⎞ ⎛ ] ∫t X ɅE ⎝ T Φi1 ≙ 2ϕi1 (t) ς˙ˆi (s)ds ⎠ = 0. ςˆi (t) − ςˆi (t − τ ) − Y ɅE [

t−τ

(4.23)

4.2 Delay-Dependent Admissible Consensus and Consensualization

[ In addition, for any real symmetric matrix M =

M11 M12 ∗ M22

127

] > 0, one has

∫t Φi2 ≙

T τ ϕi1 (t)Mϕi1 (t)

−

T ϕi1 (t)Mϕi1 (t)ds ≥ 0.

(4.24)

t−τ

Let K 2 = W T K 2 , then from (4.19) to (4.24), one has V˙i (t) ≤ V˙i1 (t) + V˙i2 (t) + V˙i3 (t) + Φi1 (t) + Φi2 (t) ∫t T T ϕi3 (t, s)Θϕi3 (t, s)ds, ≤ ϕi2 (t)Ξi ϕi2 (t) − t−τ

[ ]T [ T ]T T where ϕi2 (t) = ϕi1 (t), ςˆiT (t − τ ) and ϕi3 (t, s) = ϕi1 (t), ɅTE ς˙ˆiT (s) . It can be shown that if LMIs (4.12)–(4.14) are feasible, then systems (4.18) are asymptotically stable. In the sequel, we prove that LMIs (4.12)–(4.14) can guarantee that systems (4.18) are regular and impulse-free. Let E˜ = diag{Ʌ E , 0}, Q˜ = diag{Q, τ S}, and R˜ =

[

] [ ] ] [ [ ] 0 0 ˜ X 0 0 I R 0 ˜ ˜ ,X = , A1 = , A2 = , Ʌ K 2 ϒλ˜ i ɅC 0 Y 0 Ʌ A+B K 0 C −I ɅW ɅW

then by Schur complement in [7], it can be shown that LMIs (4.12)–(4.14) are feasible if and only if the following inequalities hold E˜ T R˜ = R˜ T E˜ ≥ 0,

(4.25)

R˜ T A˜ 1 + A˜ T1 R˜ + Q˜ + τ M + X˜ E˜ + E˜ T X˜ T ( ( ) )T + R˜ T A˜ 2 − X˜ E˜ Q˜ −1 R˜ T A˜ 2 − X˜ E˜ < 0.

(4.26)

By Assumption 2.1 in [8], one has (

( ) ( )T ( ) )T R˜ T A˜ 2 − X˜ E˜ + R˜ T A˜ 2 − X˜ E˜ − Q˜ ≤ R˜ T A˜ 2 − X˜ E˜ Q˜ −1 R˜ T A˜ 2 − X˜ E˜ . (4.27) By (4.26) and (4.27), one can obtain that ( ) ( )T R˜ T A˜ 1 + A˜ 2 + A˜ 1 + A˜ 2 R˜ < 0.

(4.28)

128

4 Admissible Consensus and Consensualization with Time Delays

( ) ˜ A˜ 1 + A˜ 2 is regular and impulse-free. It can be shown By (4.25) and (4.28), E, that ) ( ) ( det sɅ E − Ʌ A+B K 0 C+λ˜ i K 2 C = det s E˜ − A˜ 1 − A˜ 2 , )) ( ( )) ( ( deg det sɅ E − Ʌ A+B K 0 C+λ˜ i K 2 C = deg det s E˜ − A˜ 1 − A˜ 2 = rank(Ʌ E ). Hence, systems (4.18) are regular and impulse-free. Moreover, (E, A + B K 0 C + B K 1 ) is admissible if and only if there exists a matrix G ∈ Rd×d such that T

E T G = G E ≥ 0, T

(A + B K 0 C + B K 1 )T G + G (A + B K 0 C + B K 1 ) < 0.

(4.29) (4.30)

−1 ˜ then LMIs (4.15) and From (4.30), G is invertible. Let G˜ = G and K 1 = K 1 G, (4.16) can be obtained. By Theorem 4.1 and Lemma 2.9, multi-agent system (4.1) can be admissibly consensualized by protocol (4.2) with K 1 = K 1 G˜ −1 and K 2 = W −T K 2 if LMIs (4.12)–(4.16) are feasible. The proof Theorem 4.2 is completed. □

Remark 4.3 Although the approach in Lemma 2.9 can guarantee the scalability of multi-agent systems, it may result in some conservation if some eigenvalues of Laplacian matrices of interaction topologies are complex. However, no conservation was brought in if all the eigenvalues of Laplacian matrices of interaction topologies are real. As a special case, all the eigenvalues of the Laplacian matrix of an undirected topology, which was extensively used to model multi-agent systems in Refs. [9–12], are real. Remark 4.4 The LMI is a valid tool to analyze the influence of time delays. By using LMI techniques, stabilization criteria for isolated time-delayed systems with dynamic output feedback were given in [13–15] and consensualization criteria for time-delayed multi-agent systems with dynamic output feedback consensus protocols were shown in [16]. It should be pointed out that the in [13–16] include both LMI and matrix equality constraints, which can be checked via the cone complementarity linearization algorithm proposed by Ghaoui et al. in [17], but the convergence of the algorithm was not proven and some conservation may be brought in. Because multi-agent system (4.1) with protocol (4.2) satisfies some separation principle, the admissible consensualization criterion in Theorem 4.2 can be expressed as strict LMIs, which can be easily checked via FEASP solver in the Matlab’s LMI Toolbox [18]. Hence, our approach has less conservation than ones in [13–16]. Moreover, it is well-known that delay-dependent LMI criteria usually have less conservation than delay-independent ones. By a number of numerical simulations, it is revealed that the LMI criterion in Theorem 4.2 usually is feasible when τ is small, but it may be unfeasible when τ is very large. This coincides with the above analysis.

4.2 Delay-Dependent Admissible Consensus and Consensualization

129

4.2.4 Consensus Functions If multi-agent system (4.3) achieves admissible consensus, then the states of all agents tend to be identical as time tends to infinity. A challenging problem is how to determine the identical state; that is, the absolute motion as a whole. This section gives an explicit expression of the consensus function and determines the influence of the agreement state of protocols on the agreement state of agents. Let T1 , T2 , Z 1 , and Z 2 be nonsingular matrices such that T1 E Z 1 = diag{Ir , 0}, T2 E Z 2 = diag{Ir , 0} and ] ] [ A1 0 A2 0 , T2 (A + B K 0 C + B K 1 )Z 2 = . 0 Id−r 0 Id−r

[ T1 (A + B K 0 C)Z 1 =

Theorem 4.3 If multi-agent system (4.3) achieves admissible consensus, then the consensus function c(t) satisfies ( [ ]T ) = 0, lim c(t) − cxT (t) + czT (t), 0

t→∞

where ( ) ) 1 ( cx (t) = Z 1 [Ir , 0]T e A1 t [Ir , 0]Z 1−1 √ eT1 U −1 ⊗ Id x(0) , N ∫t cz (t) = Z 1 [Ir , 0]

T

e A1 (t−s) [Ir , 0]Z 1−1 B K 1 cz (s)ds,

0

cz (t) = Z 2 [Ir , 0] e

T A2 t

[Ir , 0]Z 2−1

(

) ) 1 ( T −1 √ e1 U ⊗ Id z(0) . N

Proof By (4.10), one can obtain that (

) 1 lim c(t) − √ κ 1 (t) = 0. t→∞ N

(4.31)

Due to [I, 0, . . . , 0] = eT1 ⊗ I, ) ( κ(t) = U −1 ⊗ I2d κ(t), it can be shown that ( ) κ 1 (0) = eT1 U −1 ⊗ I2d κ(0).

(4.32)

130

4 Admissible Consensus and Consensualization with Time Delays

[ ]T [ ]T Let κ 1 (0) = κ T1x (0), κ T1z (0) , x(0) = x1T (0), x2T (0), . . . , x NT (0) , and z(0) = ]T [ ]T [ T z 1 (0), z 2T (0), . . . , z TN (0) , then due to κi (0) = xiT (0), z iT (0) , one can obtain by (4.32) that ( ) κ 1x (0) = eT1 U −1 ⊗ Id x(0),

(4.33)

( ) κ 1z (0) = eT1 U −1 ⊗ Id z(0).

(4.34)

[ ]T Let c(t) = cxT (t), czT (t) , where cx (t) and cz (t) denote the agreement states of agents and protocols respectively. From (4.5) and (4.31), one has 1 cz (t) = √ Z 2 [Ir , 0]T e A2 t [Ir , 0]Z 2−1 κ 1z (0). N

(4.35)

Since (E, A + B K 0 C + B K 1 ) is admissible, one has lim cz (t) = 0.

t→∞

(4.36)

Let Ʌ E ς˙ˆi (t) = Ʌ A+B K 0 C ςˆi (t) + Ʌ K 2 ϒλ˜ i ɅC ςˆi (t − τ ) (i = 1, 2, 3, 4),

(4.37)

where cx (t) and cz (t) represent the impacts of x(0) and cz (t) (i.e., z(0)) on the agreement state of agents, respectively. It can be shown that 1 cx (t) = √ Z 1 [Ir , 0]T e A1 t [Ir , 0]Z 1−1 κ 1x (0), N ∫t cz (t) = Z 1 [Ir , 0]

T

e A1 (t−s) [Ir , 0]Z 1−1 B K 1 cz (s)ds.

(4.38)

(4.39)

0

From (4.3) to (4.39), the conclusion of Theorem 4.3 can be obtained.

□

Theorem 4.3 shows that protocol states of all agents tend to zero as time tends to infinity if admissible consensus is achieved. Especially, if z(0) = 0, then cz (t) ≡ 0 and cz (t) ≡ 0. In this case, the agreement state of a multi-agent system is completely determined by its intrinsic properties. Moreover, the constant delay does not influence the consensus function although it is an important factor for multi-agent system (4.3) to achieve admissible consensus. In Theorem 4.3, the consensus function is associated with the interaction topology G. An interesting problem is under what conditions consensus functions for multiagent systems with the same initial states, but different interaction topologies are

4.2 Delay-Dependent Admissible Consensus and Consensualization

131

identical. The following corollary shows that the consensus function is independent of the interaction topology if it is balanced. Corollary 4.4 If multi-agent system (4.3) achieves admissible consensus and the interaction topology G is balanced, then the consensus function c(t) satisfies ( [ ]T ) = 0, lim c(t) − cxT (t) + czT (t), 0

t→∞

where ( cx (t) = Z 1 [Ir , 0]T e A1 t [Ir , 0]Z 1−1 ∫t cz (t) = Z 1 [Ir , 0]

T

) N 1 ∑ xi (0) , N i=1

e A1 (t−s) [Ir , 0]Z 1−1 B K 1 cz (s)ds,

0

cz (t) = Z 2 [Ir , 0]T e A2 t [Ir , 0]Z 2−1

(

) N 1 ∑ z i (0) . N i=1

Proof Let Jλi be a Jordan block of order k corresponding to the eigenvalue λi (i ∈ {2, 3, . . . , N }) of L, then k column vectors of U can be obtained by the following equations ⎧

(λI − L)u λi ,1 = 0, (λI − L)u λi , j = −u λi , j−1 ( j = 2, 3, . . . , k),

where u λi ,1 is an eigenvector of L associated with λi , and u λi , j ( j = 2, 3, . . . , k) are generalized eigenvectors of L. If G is balanced, then√1T L = 0. One can see that 1T u λi , j = 0 ( j = 1, √ 2, . . . , k). Hence, one has 1T U / N = 0. Thus, it can be T −1 T shown that e1 U = 1 / N . By Theorem 4.3, the conclusion of Corollary 4.4 can be obtained. □ Remark 4.5 We proposed an initial state projection approach to determine timevarying consensus functions in Sect. 2.5.2, but the approach cannot reveal the influence of interaction topologies on consensus functions. Note that in [3, 19], the consensus protocols were constructed based on state information and their methods cannot determine the impacts of protocol states on the agreement state of agents for dynamic output feedback consensus protocols. Theorem 4.3 presents a general approach to determine the consensus function, which can reveal the impacts of the interaction topology and protocol states on the agreement state of agents.

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4 Admissible Consensus and Consensualization with Time Delays

4.2.5 Numerical Simulation Consider a multi-agent system with nine agents and the dynamics of each agent is described by (4.1) with ⎡

1 ⎢0 E =⎢ ⎣0 0

0 1 0 0

0 0 0 0

⎡ ⎤ 0 0 ⎢ 0 0⎥ ⎥, A = ⎢ ⎣ −1 0⎦ 0 0

1 0 1 −2

0 0 −1 0

⎤ ⎡ ⎤ ⎡ ⎤T 1 0 1 ⎢1⎥ ⎢0⎥ 0 ⎥ ⎥, B = ⎢ ⎥, C = ⎢ ⎥ . ⎣0⎦ ⎣0⎦ 0 ⎦ −1 0 0

Figure 4.1 illustrates an interaction topology of the multi-agent system, and it is assumed that its adjacent matrix is a 0–1 matrix without loss of generality. The initial states of agents are x1 (0) = [9, 8, −1, −16]T , x3 (0) = [3, 2, −1, −4]T , x5 (0) = [−9, −8, 1, 16]T , x7 (0) = [−1, 5, 6, −10]T , x9 (0) = [−1 , −3, −2, 6]T .

x2 (0) = [6, 4, −2, −8]T , x4 (0) = [−6, 15, 21, −30]T , x6 (0) = [2, 4, 2, −8]T , x8 (0) = [1, 4, 3, −8]T ,

Let K 0 = 0 and τ = 0.05, then one can obtain by Theorem 4.2 that K 1 = = [10.1103, −14.5558, −2.3115, 9.4432] and K 2 [−0.8843, −0.3610, −0.5450, −0.0881]. In Fig. 4.2, the state trajectories of the multi-agent system are shown, and the trajectories marked by circles are the curves of the consensus function given in Theorem 4.3. Figure 4.3 illustrates the trajectories of z iT (t)z i (t) (i = 1, 2, . . . , 9). One sees that the singular multi-agent system achieves admissible consensus. Moreover, the state trajectories converge to the ones marked by circles and limt→∞ z iT (t)z i (t) = 0 (i = 1, 2, . . . , 9); that is, limt→∞ cz (t) = 0, which coincides with the conclusion of Theorem 4.3.

2

1

3

4

5

Fig. 4.1 Directed interaction topology

7

6

8

9

4.3 Admissible Output Consensualization with Time Delays

133

Fig. 4.2 State trajectories

Fig. 4.3 Trajectories of z iT (t)z i (t) (i = 1, 2, . . . , 9)

4.3 Admissible Output Consensualization with Time Delays In this section, admissible output consensualization problems for high-order LTI singular multi-agent systems with constant time delays are dealt with. Firstly, based on the observability decomposition, a dynamic output feedback consensus protocol is proposed, which makes singular multi-agent systems satisfy some separation principle and can simplify admissible output consensus design problems. Then, LMI criteria for admissible output consensualization are presented, which can guarantee the regular and impulse-free properties of singular multi-agent systems directly. Moreover, an approach to determine the output consensus function is presented based

134

4 Admissible Consensus and Consensualization with Time Delays

on the First Equivalent Form and the impacts of initial states of consensus protocols and dynamic agents are determined, respectively.

4.3.1 Output Consensus Protocol with Local Delayed Output Information Consider a group of N homogenous agents with singular linear dynamics. The dynamics of the ith agent is described by ⎧

E x˙i (t) = Axi (t) + Bu i (t), yi (t) = C xi (t),

(4.40)

where i ∈ {1, 2, . . . , N }, A ∈ Rd×d , B ∈ Rd× p , C ∈ Rq×d , E ∈ Rd×d with rank(E) ≤ d, xi (t), u i (t) and yi (t) are the state, the control input and the measured output, respectively. Let T be a nonsingular matrix with a compatible dimension such that ] ] [ [ [ ] Eo 0 Ao 0 Bo , T −1 AT = , T −1 B = , C T = [Co , 0], T −1 E T = Bo E o˜ E o Ao˜ Ao where E o ∈ Rm×m , Ao ∈ Rm×m , Bo ∈ Rm× p , Co ∈ Rq×m and the triple (E o , Ao , Co ) is observable. It is assumed that agent i collects the delayed output information of neighboring agents by the following protocol ⎧ ( ∑ E o ω˙ i (t) = (Ao + Bo K A )ωi (t) + K B wi j y j (t − τ ) ⎪ ⎪ ⎪ j∈Ni ⎪ ⎨ ) − yi (t − τ ) + υi (t − τ ) − υ j (t − τ ) , ⎪ ⎪ υi (t) = Co ωi (t), ⎪ ⎪ ⎩ u i (t) = K A ωi (t),

(4.41)

where yi (t) = φ yi (t) and υi (t) = φωi (t) for t ∈ [−τ, 0) are admissible bounded vector-valued functions. Now, the definitions of the admissible output consensus and consensualization are introduced, respectively. Definition 4.3 Multi-agent (4.40) with protocol (4.41) is said to achieve admissible output consensus if it is regular and ( impulse-free ) and there exists a vector-valued function c y (t) such that limt→∞ yi (t) − c y (t) = 0 (i = 1, 2, . . . , N ) for any admissible bounded φ yi (t) and φωi (t) (i = 1, 2, . . . , N ; t ∈ [−τ, 0)), where c y (t) is called the output consensus function.

4.3 Admissible Output Consensualization with Time Delays

135

Definition 4.4 Multi-agent (4.40) is said to be admissibly output consensualizable by protocol (4.41) if there exist K A and K B such that it achieves admissible output consensus. This chapter investigates the following two admissible output consensus problems: (i) How to determine gain matrices K A and K B in protocol (4.41) such that multi-agent system (4.40) achieves admissible output consensus; and (ii) How to determine the output consensus function c y (t) if multi-agent system (4.40) achieves admissible output consensus.

4.3.2 Conditions of Admissible Output Consensus Design with Time Delays In this section, firstly, admissible output consensus problems are transformed into admissible ones of multiple subsystems with lower dimensions and LMI consensualization criteria are proposed. Then, an explicit expression of the output consensus function is given, which is independent of the constant delay. [ T ]T T Let T −1 xi (t) = xio (t), xio (t) , then the dynamics of each agent can be rewritten as ][ ][ ] [ ] [ ] ⎧[ x˙io (t) xio (t) Ao 0 Bo ⎨ Eo 0 u i (t), = + (4.42) Bo E o˜ E o x˙io (t) Ao˜ Ao xio (t) ⎩ yi (t) = Co xio (t), where i = 1, 2, . . . , N . Since the unobservable part does not impact the measured output yi (t) (i = 1, 2, . . . , N ), multi-agent system (4.40) achieves admissible output consensus if the observable parts of all agents achieve admissible consensus. Let [ T ]T T w (t) = x1o (t), ω1T (t), x2o (t), ω2T (t), . . . , x NT o (t), ωTN (t) , then the observable part of multi-agent system (4.40) with protocol (4.41) can be rewritten in a global form as ( ]) ( ]) [ [ Bo K A Eo 0 Ao IN ⊗ w ˙ (t) = I N ⊗ w (t) 0 Ao + Bo K A 0 Eo ( [ ]) 0 0 − L⊗ w (t − τ ). (4.43) K B Co −K B Co Let λ1 , λ2 , . . . λ N denote the eigenvalues [ √ ]of the Laplacian matrix L, then there exists an invertible matrix U = 1/ N , U ∈ R N ×N with U T = U −1 such that U T LU = diag{λ1 , λ2 , . . . λ N }. Since we model the interaction topology among agents as a connected undirected graph, one can set that 0 ( = λ1 )< λ2 ≤ · · · ≤ λ N by Lemma 2.2 without loss of generality. Let U T ⊗ I2m w (t) = ]T [ T ˆ 2T (t), . . . , w ˆ NT (t) , then system (4.43) can be transformed into w ˆ 1 (t), w

136

4 Admissible Consensus and Consensualization with Time Delays

[

] ] [ Bo K A Eo 0 ˙ Ao w ˆ 1 (t), (4.44) w ˆ 1 (t) = 0 Ao + Bo K A 0 Eo [ ] ] ] [ [ Bo K A Eo 0 ˙ Ao 0 0 w ˆ i (t) − w ˆ i (t − τ ), w ˆ i (t) = 0 Ao + Bo K A 0 Eo λi K B Co −λi K B Co (4.45) where i = 2, 3, . . . , N . Actually, systems (4.44) and (4.45) describe consensus and disagreement dynamics of multi-agent system (4.40), respectively. Let [ Q=

] I −I . I 0

[ T ]T and Q −1 w ˆ i (t) = w ˜ i (t), w iT (t) (i = 2, 3, . . . , N ), then systems (4.45) can be converted into ˙˜ i (t) = (Ao + Bo K A )w Eo w ˜ i (t) + λi K B Co w i (t − τ ),

(4.46)

˙ i (t) = Ao w i (t) + λi K B Co w i (t − τ ), Eo w

(4.47)

where i = 2, 3, . . . , N . By the LMI tool, the following theorem gives an approach to determine gain matrices K A and K B in protocol (4.41) such that multi-agent system (4.40) achieves admissible output consensus. Theorem 4.4 Multi-agent system (4.40) is admissibly output consensualizable T T by protocol (4.41) if there exist M = M > 0, S = S > 0, R, Q 1 , Q 2 , X 11 , X 12 , X 13 , X 22 , X 23 , X 33 , Y 1 , Y 2 and Y 3 such that T

R E oT = E o R ≥ 0, T

T

R ATo + Ao R + K A BoT + Bo K A < 0, T

E oT Q 1 = Q 1 E o ≥ 0, ⎤ T Ξ11 Ξ12 λi K B Co + τ X 13 − Y 1 E o + E oT Y 3 ⎥ ⎢ Ξi = ⎣ ∗ Ξ22 λi K B Co + τ X 23 − Y 2 E o ⎦ < 0 (i = 2, N ), T T ∗ ∗ −M + τ X 33 − Y 3 E o − E o Y 3

(4.48) (4.49) (4.50)

⎡

(4.51)

4.3 Admissible Output Consensualization with Time Delays

⎡

X 11 ⎢ ∗ Φ=⎢ ⎣ ∗ ∗

X 12 X 22 ∗ ∗

X 13 X 23 X 33 ∗

137

⎤ Y1 Y2 ⎥ ⎥ ≥ 0, Y3 ⎦ S

(4.52)

where T

T

Ξ11 = Q 2 Ao + ATo Q 2 + M + τ X 11 + Y 1 E o + E oT Y 1 , T T T Ξ12 = Q 1 − Q 2 + ATo Q 2 + τ X 12 + E oT Y 2 , T Ξ22 = −Q 2 − Q 2 + τ S + τ X 22 . In this case, K A = K A R

−1

−T

and K B = Q 2 K B .

Proof Firstly, we transform admissible output consensus problems into admissible ( ) [ T ]T ˆ 1 (t), w ˆ 2T (t), . . . , w ˆ NT (t) , one can obtain that ones. Due to U T ⊗ I2m w (t) = w w (t) = wc (t) + wc (t),

(4.53)

]T [ T wc (t) = (U ⊗ I2m ) w ˆ 1 (t), 0 ,

(4.54)

]T [ wc (t) = (U ⊗ I2m ) 0, w ˆ NT (t) . ˆ 2T (t), . . . , w

(4.55)

where

[ T ]T ˆ 1 (t), 0 = e1 ⊗ w Since w ˆ 1 (t), where e1 is an N -dimensional column vector with first element 1 and 0 elsewhere, one can obtain by (4.54) that 1 wc (t) = U e1 ⊗ w ˆ 1 (t) = √ 1 ⊗ w ˆ 1 (t). N

(4.56)

By (4.52), (4.53), and (4.56), if systems (4.45) are admissible, then ( ) 1 lim w (t) − √ 1 ⊗ w ˆ 1 (t) = 0, t→∞ N

(4.57)

which means that the observable part of multi-agent system (4.40) with protocol (4.41) achieves admissible consensus; that is, multi-agent system (4.40) achieves admissible output consensus, and system (4.44) can be used to determine the output consensus function. Since Q is invertible, multi-agent system (4.40) achieves admissible output consensus if systems (4.46) and (4.47) are admissible. Then, we give an approach to determine K A and K B such that systems (4.46) and (4.48) are admissible in terms of LMIs. According to the structures of system (4.46), if system (4.47) is admissible, then by Lemma 2.1, system (4.46) is admissible if and only if there exists a matrix R such that

138

4 Admissible Consensus and Consensualization with Time Delays

E oT R = R T E o ≥ 0,

(4.58)

(Ao + Bo K A )T R + R T (Ao + Bo K A ) < 0.

(4.59)

From (4.59), R is invertible. Let R = R −1 and K A = K A R, then LMIs (4.48) and (4.49) can be obtained. In the sequel, we deal with the admissible property of systems (4.47). To the end, consider the following Lyapunov–Krasovskii functional candidate Vi (t) = Vi1 (t) + Vi2 (t) + Vi3 (t) + Vi4 (t),

(4.60)

where Vi1 (t) = w iT (t)E oT Q 1 w i (t), ∫t Vi2 (t) =

w iT (s)Mw i (s)ds, t−τ

∫0 ∫ t Vi3 (t) =

˙ iT (s)E oT S E o w ˙ i (s)dsdθ, w

−τ t+θ

∫ t ∫θ Vi4 (t) =

χ T (s, θ )Φχ (s, θ )dsdθ, 0 θ −τ

[ ]T ˙ iT (θ ), w iT (θ − τ ), E o w ˙ iT (s) . Taking the derivative and χ (s, θ ) = w iT (θ ), E o w of Vi (t) with respect to time t along the solutions of systems (4.47), one can obtain that [

]T [

][

] ˙ i (t) Eo w ˙ i (t) + Ao w i (t) + λi K B Co w i (t − τ ) −E o w ) ]T [ ][ ] [ ]([ ] [ T T 0 0 I w i (t) Q1 Q2 w i (t) w i (t − τ ) , + =2 T ˙ i (t) ˙ i (t) λi K B Co Ao −I Eo w Eo w 0 Q2

V˙i1 (t) = 2

w i (t) ˙ i (t) Eo w

T

T

Q1 Q2 T 0 Q2

V˙i2 (t) = w iT (t)Mw i (t) − w iT (t − τ )Mw i (t − τ ), V˙i3 (t) =

˙ iT (s)E oT S E o w ˙ i (s) τw

∫t − t−τ

˙ iT (s)E oT S E o w ˙ i (s)ds, w

(4.61) (4.62)

(4.63)

4.3 Admissible Output Consensualization with Time Delays

139

⎡

⎤⎡ ⎡ ⎤T ⎡ ⎤ ⎤T w i (t) X 11 X 12 X 13 w i (t) w i (t) ˙ i (t) ⎦ ⎣ ∗ X 22 X 23 ⎦⎣ E o w ˙ i (t) ⎦ + 2⎣ E o w ˙ i (t) ⎦ V˙i4 (t) = τ ⎣ E o w w i (t − τ ) ∗ ∗ X 33 w i (t − τ ) w i (t − τ ) ⎡ ⎤ t ∫ Y 1 Eo ˙ iT (s)E oT S E o w ˙ i (s)ds. (4.64) × ⎣ Y 2 E o ⎦(w i (t) − w i (t − τ )) + w Y 3 Eo t−τ [ ]T ˙ iT (t), w iT (t − τ ) and K B = Q T2 K B , then one can Let μi (t) = w iT (t), E o w obtain by (4.61)–(4.64) that V˙i (t) ≤ μiT (t)Ξi μi (t) < 0, which means that systems (4.47) are asymptotically stable. In the following, we show that LMIs (4.50)–(4.52) can guarantee the regular and impulse-free properties of systems (4.47). By (4.52), one can see that ⎡

⎤ X 11 X 12 X 13 ⎣ ∗ X 22 X 23 ⎦ ≥ 0. ∗ ∗ X 33 −T

Hence, let K B = Q 2 K B , then one can show that T

T

E Q = Q E ≥ 0,

(4.65)

T Z + Y˜ E + E Y˜ T > 0,

(4.66)

( T ) T T T T T Q Ao + Ao Q + Z + Y E + E Y + Q A K i + E Y˜ T − Y E )−1 ( T )T ( T T Q A K i + E Y˜ T − Y E < 0, × Z + Y˜ E + E Y˜ T

(4.67)

where ] ] ] [ [ [ ] Q1 0 Eo 0 0 I 0 0 , Ao = , AK i = , ,Q = Q2 Q2 0 0 Ao −I λi K B Co 0 ] ] [ [ [ ] M 0 Y1 0 ˜ Y3 0 ,Y = ,Y = Z= . Y2 0 0 τS 0 0 [

E=

By Lemma 2.9, one has (

T T Q A K i + E Y˜ T − Y E

)(

T Z + Y˜ E + E Y˜ T

)−1 (

T T Q A K i + E Y˜ T − Y E

)T

140

4 Admissible Consensus and Consensualization with Time Delays

( T )T T T T T ≥ Q A K i + E Y˜ T − Y E + Q A K i + E Y˜ T − Y E − Z − Y˜ E − E Y˜ T . (4.68) By (4.67) and (4.68), it can be shown that Q

T(

) ( )T Ao + A K i + Ao + A K i Q < 0.

(4.69)

( ) From (4.65) and (4.68), by Lemma 2.1, one can deduce that the pair E, Ao + A K i is regular and impulse-free. By (4.6) and (4.67), one can obtain that T

T

T

T

Q Ao + Ao Q + Y E + E Y < 0.

(4.70)

Let T and Z be nonsingular matrices such that ] [ ] A11 A12 Ir 0 , , T Ao Z = A21 A22 0 0 ] ] [ [ −T T −1 Q 11 Q 12 Y 11 Y 12 , Z YT = . T QZ = Q 21 Q 22 Y 21 Y 22 [

T EZ =

By (4.6), one has Q 12 = 0. Thus, by (4.70), one can obtain that T

T

A22 Q 22 + Q 22 A22 < 0, ( ) which means that A22 is nonsingular; that is, the pair E, Ao is regular and impulsefree. It can be shown that | | |κ E o − Ao | = |κ E − Ao |, |) ( ) (| deg(|κ E o − Ao |) = deg |κ E − Ao | = rank E = rank(E o ) = r, | | |κ E o − Ao − λi K B Co | = |κ E − Ao − A K i |, |) (| deg(|κ E o − Ao − λi K B Co |) = deg |κ E − Ao − A K i | ( ) = rank E = rank(E o ) = r, which means that the pairs (E o , Ao ) and (E o , Ao + λi K B Co ) are regular and impulsefree. By Definition 4.4, systems (4.47) are regular and impulse-free. The proof of Theorem 4.4 is completed. Remark 4.6 Because only the observable part of multi-agent system (4.40) impacts the admissible output consensus, protocol (4.41) is constructed by the observable

4.3 Admissible Output Consensualization with Time Delays

141

part (E o , Ao , Bo , Co ). Moreover, protocol (4.41) has a specific structure, which makes multi-agent system (4.40) satisfies some separation principle; that is, the disagreement dynamics of multi-agent system (4.40) can be decomposed into two parts as shown in (4.46) and (4.47). In this case, the admissible output consensualization criterion in Theorem 4.4 can be expressed in a strict LMI form, which can be easily checked via the FEASP solver in the Matlab’s LMI Toolbox [18]. However, state consensualization criteria for multi-agent systems with time delays and dynamic output feedback consensus protocols contain both LMI and matrix equality constraints, which cannot be directly checked by the FEASP solver but can be checked by an iterative algorithm namely the cone complementarity linearization algorithm in [17]. It should be pointed out that the convergence of this algorithm was not proven and some conservatism may be brought in. Remark 4.7 To deal with both stabilization problems for isolated systems (e.g., [15]) and consensualization problems for multi-agent systems (e.g., [16]), the changing variable method was usually used, which restricts the structures of some variables and may bring in some conservatism. In Theorem 4.4, since the gain matrix K B lies on the left side of Co , which is introduced by the specific structure of protocol (4.41), the gain matrix can be determined by the LMI tool without using the changing variable method. Moreover, the regular property ensures the existence and uniqueness of the solutions of singular systems, and the impulse-free property is used to avoid the impacts of impulse modes. Admissible output consensualization criteria in Theorem 4.4 can directly guarantee both regular and impulse-free properties of multi-agent system (4.40). Remark 4.8 If the LMI criteria for admissible output consensualization are associated with the Laplacian matrices of interaction topologies of multi-agent systems as shown in [11], then it is time-cost to check those criteria when multi-agent systems contain a large number of agents. Since the LMI criterion in Theorem 4.4 is only associated with the second small and maximum eigenvalues of the Laplacian matrix, the calculation complexity is lower. Furthermore, if multi-agent systems consist of a large number of agents, the eigenvalues of the Laplacian matrix are difficult to be obtained. Actually, the eigenvalue estimation approach with the lower calculation complexity was proposed to estimate the second small and maximum eigenvalues of the Laplacian matrix in [20, 21]. Hence, it is not required to obtain the precise values of the second small and maximum eigenvalues.

4.3.3 Output Consensus Functions Let T˜ and Z˜ be nonsingular matrices such that T˜ E o Z˜ =

[

[ ] Ir 0 ˜ A˜ , T Ao Z˜ = ˜ 11 A21 0 0

] [ ] A˜ 12 ˜ B˜ 1 , T B . = o ˜ A22 B˜ 2

(4.71)

142

4 Admissible Consensus and Consensualization with Time Delays

If LMIs (4.50)–(4.52) are feasible, then the pair (E o , Ao ) is impulse-free; that is, A˜ 22 is invertible. Let Tˆ =

[

] [ ] 0 Ir Ir −A12 A−1 22 , Z ˆ = −1 , 0 Im−r −A−1 22 A21 A22

then one can show that [ A˜ Tˆ ˜ 11 A21

(4.72)

] ] [ ˜ A˜ 12 ˆ A˜ 11 − A˜ 12 A˜ −1 22 A21 0 . Z = A˜ 22 0 Im−r

] [ Let K A Z˜ = K˜ A1 , K˜ A2 with K˜ A1 ∈ R p×r and K˜ A2 ∈ R p×(m−r ) , then it can be obtained that ] [ A˜ 11 + B˜ 1 K˜ A1 A˜ 12 + B˜ 1 K˜ A2 ˜ ˜ . T (Ao + Bo K A ) Z = ˜ A21 + B˜ 2 K˜ A1 A˜ 22 + B˜ 2 K˜ A2 If LMIs (4.40) and (4.48) are feasible, then the pair (E o , Ao + Bo K A ) is impulsefree, which means that A˜ 22 + B˜ 2 K˜ A2 is invertible. One can set that [ ◠

T =

)( )−1 ] ( Ir − A˜ 12 + B˜ 1 K˜ A2 A˜ 22 + B˜ 2 K˜ A2 0

[ ◠

Z=

( − A˜ 22 + B˜ 2 K˜ A2

Im−r Ir )−1 (

,

] 0 )( )−1 , A˜ 21 + B˜ 2 K˜ A1 A˜ 22 + B˜ 2 K˜ A2

then it can be shown that ◠

◠

T T˜ ( Ao + Bo K A ) Z˜ Z [ ] )( )−1 ( ) ( A˜ 11 + B˜ 1 K˜ A1 − A˜ 12 + B˜ 1 K˜ A2 A˜ 22 + B˜ 2 K˜ A2 A˜ 21 + B˜ 2 K˜ A1 0 = 0 Im−r (4.73) From the proof of Theorem 4.4, system (4.44) can determine the output consensus function. The following theorem gives an explicit expression of the out consensus function. Theorem 4.5 If multi-agent system (4.40) achieves admissible output consensus, then the output consensus function c y (t) satisfies ( ) lim c y (t) − c y0 (t) − c yω (t) = 0,

t→∞

4.3 Admissible Output Consensualization with Time Delays

143

where ( ˜ ˜ ˜ c y0 (t) = Co Z˜ Zˆ [Ir , 0]T e( A11 − A12 A22

−1

c yω (t) = Co Z˜ Zˆ [Ir , 0]T

∫t

˜

A˜ 21 )t

˜

[Ir , 0] Zˆ −1 Z˜ −1 [Im , 0]T −1

˜ −1 A˜ 21 )(t−s)

e( A11 − A12 A22

0 ( T A˜ 11 + B˜ 1 K˜ A1 −( A˜ 12 + B˜ 1 K˜ A2

◠

cω (t) = Z˜ Z [Ir , 0] e ◠ −1

[Ir , 0] Z

(

˜ −1

Z

) N 1 ∑ xi (0) , N i=1

[Ir , 0] Zˆ −1 Z˜ −1 Bo K A cω (s)ds, −1

)( A˜ 22 + B˜ 2 K˜ A2 ) ( A˜ 21 + B˜ 2 K˜ A1 )

) t

) N 1 ∑ ωi (0) . N i=1

[ ]T T T Proof Let w ˆ 1 (t) = w ˆ 1y (t), w ˆ 1ω (t) with w ˆ 1y (t) ∈ Rm and w ˆ 1ω (t) ∈ Rm , then ]T ( ) [ T ˆ 2T (t), . . . , w ˆ NT (t) , one has due to U T ⊗ I2m w (t) = w ˆ 1 (t), w ] N [ )( ) ( 1 ∑ xio (0) . w ˆ 1 (0) = eT1 ⊗ I2m U T ⊗ I2m w (0) = √ N i=1 ωi (0)

(4.74)

From (4.57), one can see that ( ) 1 lim c y (t) − √ Co w ˆ 1y (t) = 0. t→∞ N

(4.75)

By (4.44), (4.71), (4.73) and (4.74), one can show that (

◠

w ˆ 1ω (t) = Z˜ Z [Ir , 0]T e ◠ −1

× [Ir , 0] Z

A˜ 11 + B˜ 1 K˜ A1 −( A˜ 12 + B˜ 1 K˜ A2

−1

)( A˜ 22 + B˜ 2 K˜ A2 ) ( A˜ 21 + B˜ 2 K˜ A1 )

) ( N 1 ∑ −1 ˜ ωi (0) . Z √ N i=1

) t

(4.76)

From (4.44), (4.71), (4.72) and (4.74), one can obtain that ( ˜ T ( A˜ 11 − A˜ 12 A˜ −1 22 A21 )t

w ˆ y0 (t) = Z˜ Zˆ [Ir , 0] e

w ˆ yω (t) = Z˜ Zˆ [Ir , 0]T

∫t

˜

˜

ˆ −1

[Ir , 0] Z

˜ −1 A˜ 21 )(t−s)

e( A11 − A12 A22

˜ −1

Z

) N 1 ∑ xio (0) , (4.77) √ N i=1

ˆ 1ω (s)ds. [Ir , 0] Zˆ −1 Z˜ −1 Bo K A w

0

(4.78)

144

4 Admissible Consensus and Consensualization with Time Delays

[ T ]T T Due to w ˆ 1y (t) = w ˆ y0 (t) + w ˆ yω (t) and T −1 xi (0) = xio (0), xio (0) , from (4.75) to (4.78), the conclusion of Theorem 4.5 can be obtained. Remark 4.9 By Theorem 4.4, it is necessary for multi-agent systems to achieve admissible output consensus that the pair (E o , Ao + Bo K A ) is admissible, so cω (t) tends to zero as time tends to infinity if multi-agent system (4.40) achieves admissible output consensus. However, c yω (t) may not tend to zero as time tends to infinity. Furthermore, if the initial states of protocol (4.41) are zero, that is; ωi (0) = 0 (i = 1, 2, . . . , N ), then cω (t) ≡ 0 and c yω (t) ≡ 0. In this case, the output consensus function is completely determined by the dynamics of each agent and the average of initial states of all agents. It should be also pointed out that the time delay does not influence the output consensus function although it is a critically important factor for admissible output consensualization.

4.3.4 Numerical Simulation This section gives a numerical example to demonstrate the effectiveness of theoretical results shown in the above section. Consider a singular multi-agent system with six agents, where the interaction topology is shown in Fig. 4.4 and the dynamics of each agent modeled by (4.40) with ⎡ ⎤ 0.2857 0 0.1429 0 −0.1429 0.1429 ⎢ −0.2857 −1 −0.1429 2 1.1429 −0.1429 ⎥ ⎢ ⎥ ⎢ ⎥ 1 2 0 −2 −1 0 ⎢ ⎥ E =⎢ ⎥, ⎢ ⎥ 0 −1 0 2 1 0 ⎢ ⎥ ⎣ −0.2857 0 −0.1429 0 0.1429 −0.1429 ⎦ 0.1429 −2 −0.4286 2 0.4286 0.5714 ⎡ ⎤ 0.7857 1.4286 0.6429 −0.7143 0.0714 −0.5714 ⎢ −2.2857 −1.4286 −3.1429 −2.2857 −1.5714 0.5714 ⎥ ⎢ ⎥ ⎢ ⎥ −2 2.5000 5 1.5000 1 ⎢ 2.5000 ⎥ A=⎢ ⎥, ⎢ −2.5000 ⎥ −1 −2.5000 −2 −1.5000 0 ⎢ ⎥ ⎣ 1.2143 2.5714 1.3571 −1.2857 0.9286 −0.4286 ⎦ −4.3571 −6.2857 −5.9286 0.1429 −0.2143 1.7143 ⎡ ⎤ 1.1429 0.7143 ⎢ −3.1429 −0.7143 ⎥ ⎥ ⎢ ⎥ ⎢ 5 0 ⎥ ⎢ B=⎢ ⎥, ⎥ ⎢ −3 0 ⎥ ⎢ ⎣ −0.1429 0.2857 ⎦ −4.4286 −0.1429

4.3 Admissible Output Consensualization with Time Delays

145

[

] 726 6 21 C= . 3 3 2 −1 0 0 The transform matrix for observability decomposition is chosen as ⎡

2 ⎢1 ⎢ ⎢ ⎢1 T =⎢ ⎢1 ⎢ ⎣1 0

−1 1 0 1 0 0

1 1 1 0 0 1

2 0 2 −1 0 0

0 0 1 0 1 4

⎤ 1 0⎥ ⎥ ⎥ 0⎥ ⎥. 0⎥ ⎥ 0⎦ 1

Let τ = 0.01. By checking LMI criteria in Theorem 4.4, one can obtain that [ KA =

] 0.9616 0.0546 −0.2574 0.0119 , −0.6200 0.6111 0.1001 0.0219 ⎡ ⎤ −19.0692 0 ⎢ −10.6318 0 ⎥ ⎥. KB = ⎢ ⎣ −2.2707 0 ⎦ 4.1471 0.0130

Figure 4.5 depicts the output trajectories of the singular multi-agent system, where the curves marked by circles describe the trajectories of the output consensus function in Theorem 4.5. One can find that the output trajectories converge to the ones marked by circles; that is, the singular multi-agent achieves admissible output consensus and the expression in Theorem 4.5 is a valid candidate of the output consensus function. Fig. 4.4 Interaction topology

2 3

1 0.01 0.01

0.01

0.01

4

6 5

146

4 Admissible Consensus and Consensualization with Time Delays

Fig. 4.5 Output trajectories

4.4 Notes In this chapter, admissible consensus and admissible output consensus problems for singular multi-agent systems with time delays were investigated. By using LMI techniques, admissible output consensualization conditions were proposed, which are independent of the number of agents and have less conservatism since the changing variable method was not used to determine gain matrices of consensus protocols. Moreover, an approach to determine consensus functions was presented, which can determine the impacts of time delays, initial conditions, and interaction topologies on consensus functions. Especially, it was shown that the single constant delay does impact the output consensus function, but it is critically important for singular multiagent systems to achieve admissible output consensus.

References 1. Dai LY (1989) Singular control systems. Springer, Berlin 2. Xu S, Dooren VP, Stefan R, Lam J (2002) Robust stability and stabilization for singular systems with state delay and parameter uncertainty. IEEE Trans Autom Control 47(7):1122–1128 3. Olfati-Saber R, Murray RM (2004) Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans Autom Control 49(9):1520–1533 4. Ren W, Beard RW (2005) Consensus seeking in multiagent systems under dynamically changing interaction topologies. IEEE Trans Autom Control 50(5):655–661 5. Jadbabaie A, Lin J, Morse AS (2003) Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Trans Autom Control 48(6):988–1001 6. Xi JX, Wang C, Liu H, Wang L (2019) Completely distributed guaranteed-performance consensualization for high-order multiagent systems with switching topologies. IEEE Trans Syst Man Cybern Syst 49(7):1338–1348 7. Boyd S, Ghaoui LE, Feron E, Balakrishnan V (1994) Linear matrix inequalities in system and control theory. SIAM, Philadelphia 8. Xie L, Souza CE (1992) Robust H ∞ control for linear systems with norm-bounded time-varying uncertainty. IEEE Trans Autom Control 37(8):1188–1191 9. Lin P, Jia YM (2008) Average consensus in networks of multi-agents with both switching topology and coupling time-delay. Phys A 387(1):303–313

References

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10. Bliman PA, Ferrari-Trecate G (2008) Average consensus problems in networks of agents with delayed communications. Automatica 44(8):1985–1995 11. Sun YG, Wang L, Xie G (2008) Average consensus in networks of dynamic agents with switching topologies and multiple time-varying delays. Syst Control Lett 57(2):175–183 12. Sun YG, Wang L (2009) Consensus of multi-agent systems in directed networks with nonuniform time-varying delays. IEEE Trans Autom Control 54(7):1607–1613 13. Gao H, Lam J, Wang C, Wang Y (2004) Delay-dependent output-feedback stabilisation of discrete-time systems with time-varying state delay. IET Control Theory Appl 151(6):691–698 14. Chen WH, Guan ZH, Lu X (2004) Delay-dependent output feedback guaranteed cost control for uncertain time-delay systems. Automatica 40(7):1263–1268 15. He Y, Wu M, Liu GP, She JH (2008) Output feedback stabilization for a discrete-time system with a time-varying delay. IEEE Trans Autom Control 53(10):2372–2377 16. Xi JX, Shi ZY, Zhong YS (2012) Consensus and consensualization of high-order swarm systems with time delays and external disturbances. J Dyn Syst Meas Control 134(4):1–7 17. Ghaoui LE, Oustry F, Aitrami M (1997) A cone complementarity linearization algorithm for static output-feedback and related problems. IEEE Trans Autom Control 42(8):1171–1176 18. Gahinet P, Nemirovskii A, Laub AJ, Chilali M (1995) LMI control toolbox user’s guide. The Math Works, Natick, MA 19. Xiao F, Wang L (2007) Consensus problems for high-dimensional multi-agent systems. IET Control Theory Appl 1(3):830–837 20. Berman A, Zhang XD (2000) Lower bounds for the eigenvalues of Laplacian matrices. Linear Algebra Appl 316(1–3):13–20 21. Yoonsoo K, Mesbah MI (2006) On maximizing the second smallest eigenvalue of a statedependent graph Laplacian. IEEE Trans Autom Control 51(1):116–120

Chapter 5

Admissible L2 Consensus and Consensualization with External Disturbances

In this chapter, admissible consensus analysis and design problems for high-order LTI singular multi-agent systems with external disturbances are discussed. External disturbances generally exist in the actual physical systems under complex environments, and it is usually difficult to model their dynamics accurately. For instance, UAVs suffer wind disturbances more or less when flying outdoor, mechanical systems are disturbed by reciprocating collisions between components or between components and boundaries, and the power system is incurred with the random variation of electrical load and the interference of generator set regulation. These external disturbances can destroy the stability of the singular multi-agent system and should not be ignored in some important practical control processes. However, many literatures (e.g., Refs. [1–4]) investigated the consensus analysis and design problems under the assumption that multi-agent systems are free of disturbances. Although [5–7] considered external disturbances for normal multi-agent systems and proposed some useful robust consensus control methods to enhance the ability of resisting disturbances, these anti-disturbance methods for normal multi-agent systems are not applicable to singular multi-agent systems, for the later ones have more complex dynamic structure. Hence, anti-disturbance cooperative control researches for singular multi-agent systems are worth researching seriously. The current chapter discusses admissible L 2 consensus and consensualization problems of singular multi-agent systems with time delays and external disturbances. One can find that Sect. 4.2 actually becomes a special case of the current chapter if disturbances are removed from the systems. Firstly, by the state-space decomposition, a necessary and sufficient condition for admissible consensus is given, which shows that singular multi-agent systems achieve admissible consensus if and only if the relative motions among agents are admissible. Secondly, according to different influences of initial states of agents and external disturbances, explicit expressions of consensus functions, which describe the absolute motions of singular multi-agent systems as a whole, are presented based on the First Equivalent Form, and it is shown that time delays do not impact consensus functions. Moreover, the L 2 evaluation approach is proposed to determine the impacts of external disturbances and LMI © Science Press, Beijing and Springer Nature Singapore Pte Ltd. 2023 J. Xi et al., Admissible Consensus and Consensualization for Singular Multi-agent Systems, Engineering Applications of Computational Methods 11, https://doi.org/10.1007/978-981-19-6990-4_5

149

150

5 Admissible L 2 Consensus and Consensualization with External …

criteria for admissible L 2 consensus and consensualization are given, respectively, which only depend on the maximum and second small eigenvalues of the Laplacian matrix of the interaction topology. Finally, numerical simulations are presented to demonstrate theoretical results.

5.1 Properties of External Disturbances Disturbance attenuation, compensation, and rejection are hot topics in the control field, due to the increasing complexity of controlled plants and higher demand for system accuracy, reliability, and real-time performance. In most of the practical control systems, disturbances can be categorized into Gaussian noise, norm-bounded random disturbances, disturbances described by exogenous systems, as well as equivalent disturbances representing the unmodeled dynamics and system uncertainties. As a result, disturbance attenuation and rejection are technically challenging. In addition, similar challenges also exist for the filtering problems, where various types of disturbances and the existence of uncertainties and nonlinearities limit the use of conventional methods such as Kalman filter and the minimum variance filter. In order to better understand the characteristics of disturbances in control system, main disturbance types and disturbance rejection methods are reviewed in the following sections.

5.1.1 Disturbance Types Disturbances can be divided into deterministic disturbances with explicit dynamic models and random disturbances with unknown dynamic models. Deterministic disturbance refers to the disturbance with known dynamic characteristics but unknown initial value. For example, for a specific model, the disturbance frequency is usually fixed or periodic and can be obtained, but the amplitude, phase angle, and initial value are unavailable. On the other side, random disturbances mean that the disturbances of the system are random signals, and we only know that the disturbance signals are energy bounded or amplitude bounded. For systems with known dynamic disturbances, a number of researches focus on periodic perturbed systems containing sinusoidal disturbances. This type of systems has a wide application background. For instance, in the aircraft flight vibration control system, the resonant component of the wind shear stress borne by the wing is a kind of sinusoidal disturbing force [8, 9], and the similar disturbing force also exists in the real-time vibration control system of offshore platform with the resonant component of wind or wave force borne by offshore structures [10–12]. Under the condition of known frequency, the commonly used control methods include the methods: internal model principal method, adaptive method, and repetitive control method [13]. These

5.1 Properties of External Disturbances

151

methods are essentially closely related. Ref. [14] points out that a standard adaptive feedforward control algorithm is equivalent to internal model control law. For disturbances with unknown dynamics, a great deal of researches concentrates on stochastic bounded perturbation systems. Since the influence of unknown disturbances on system performance is random, the optimal control results are often conservative. H∞ control can only give an upper bound of the relationships between the system output norm and the disturbance norm [15]. For the system with predictable disturbance, rolling optimization is often applied to predictive control to compensate for mismatched models and disturbances [16]. However, due to the limitation of prediction accuracy, it is difficult to obtain the optimal control parameters. For sinusoidal disturbances with unknown frequency, Bodson and Douglas [17, 18] proposed two adaptive algorithms for disturbance suppression. One is called indirect method, which gives different frequency estimation methods and combines the repetitive control method. The other is called direct method, which estimates the frequency, phase, and amplitude of disturbance as a whole, and combines phase-locked loop algorithm and sinusoidal disturbance suppression algorithm. Compared with deterministic disturbances, it is more difficult and meaningful to resist random disturbances, which is also the main topic discussed of current chapter.

5.1.2 Disturbance Suppression Methods Since the 1980s, the research on disturbance suppression has developed rapidly. For different types of disturbances, a series of effective disturbance suppression and compensation methods have been proposed for various kinds of control systems. Some disturbance compensation methods are combined with robust control theory to obtain more effective control effect and have been widely used. Here are some typical disturbance suppression methods shown below.

5.1.2.1

Internal Model Control

For systems with known disturbance dynamic characteristics, the internal model principle is a useful control method. The internal model principle of linear system was developed in the 1970s and proposed by Francis and Wonham [19]. It has been a mature theory at present and was well developed in [20–24]. The essence of the internal model principle is to accurately copy the modes of external disturbances in the closed-loop system and rely on the accurate cancelation of the internal model and unstable vibration modes—that is, to construct a servo compensator to achieve the purpose of non-static error tracking or disturbance suppression. The important advantage of disturbance suppression based on internal model principle is that it is highly insensitive to the changes of controlled system and compensator parameters except internal model. When the parameters of the controlled system and the compensator are perturbed, even if the perturbation range is quite large, the control

152

5 Admissible L 2 Consensus and Consensualization with External …

system can still achieve non-static error as long as the closed-loop system remains asymptotically stable. The excellent robustness of this method makes it widely used in the field of process control.

5.1.2.2

Sliding Mode Control

Sliding mode control (SMC) method [25–27] does not require high accuracy of system model. In this method, a sliding surface is applied to eliminate the influence of parameter uncertainty and external disturbance on the system. As long as the upper bound of modeling error and external disturbance is known, the asymptotically stable controller can be subsequently designed [28]. For example, Ref. [29] proposed a robust SMC method for time-delayed uncertain systems and presented a sufficient condition for the existence of sliding surface. Ref. [30] designed a SMC method based on disturbance observer, which can maintain the original control performance even if affected by disturbances, and its switching gain only needs to be greater than the peak of disturbance estimation error. For uncertain systems or systems with external disturbances, SMC needs to obtain the bound of the uncertainties or external disturbances to satisfy the matching conditions. If the control systems that cannot meet these conditions, the designed controller can be conservative. Therefore, this method needs further research for the disturbances that do not meet the matching conditions. Adaptive control can avoid the use of these conditions, and some literatures adopt a combinatorial method that combines SMC and adaptive control [31–33].

5.1.2.3

Disturbance Observer Based Control

The DOBC method was originated in the late 1980s [34, 35]. It has been widely used in the field of robot control since it appeared and is now one of the mainstream antidisturbance methods [36–38]. It utilizes the inverse model of the control object and a low-pass filter to construct a disturbance observer, and the disturbance observer is used to estimate the external disturbance in real time, such that a compensation term can be introduced into the control to eliminate the influence of external disturbance based on the disturbance observer. In the past two decades, this method is also extended to nonlinear systems [39, 40]. The advantages of simple structure, good robustness, and low cost of disturbance observer method make it widely used in the fields of optical disk drive system [41] magnetic bearing control [42] and robot teleoperation [43]. However, since the use of the inverse model of the control object, the physical realization of the DOBC method requires additional Q-filters, which brings difficulty to achieve the expected disturbance suppression effect. In addition, when the system has unstable zero pole or time-delayed factors, the inverse model must introduce the same instability factors, and additional technical means need to be applied to eliminate the influence of these adverse factors, which results in the increase of the order of the system and the difficulty of debugging [44, 45].

5.2 Problem Description of Admissible L 2 Consensus Control

5.1.2.4

153

H∞ Robust Control and L 2 Disturbance Suppression Method

Robust control theory solves two difficult control problems: that is, the conventional frequency domain theory is not suitable for MIMO system control design, and LQG theory is not suitable for model perturbation [46]. Besides, the rapid development of computer technology and the emergence of standard software development toolbox make it easier to implement. There are many uncertain factors in the operation of most systems, such as the perturbation of internal parameters and external load of asynchronous motor. H∞ robust control theory is introduced to address this kind of uncertain model systems. As an effective and important robust control method, L 2 disturbance suppression technology has been captured great attention and is often used to suppress the disturbances caused by the change of system parameters or by the complex environment [47–49]. This technology has been successfully applied in many industrial fields, such as induction motor speed regulation system, underactuated surface vessel system, and multimachine power systems [50–52]. L 2 gain disturbance suppression method is related to passivity theory. The operation of passive system is always accompanied by energy dissipation. When the system has external input, the growth of system energy is less than the total external energy injected. If the system has no external input, then the system will gradually stabilize to the equilibrium point. The main idea of L 2 disturbance suppression method is to make the closed-loop control system become a passive system by reasonably designing the L 2 gain controller of the system [53]. Usually, one can firstly set a suppression parameter γ , a penalty signal z, and the expected equilibrium point x0 , and then design the positive definite energy storage function v(x) to make the γ -dissipation hold, such that the following objectives can be achieved: The L 2 gain from the disturbance w to the penalty input z is less than the desired level; (ii) When w disappears, the closed-loop system satisfies Lyapunov stability and has asymptotic stability near the equilibrium point; (iii) If the disturbance w is bounded, then the state variable x is bounded. (i)

In this chapter, the L 2 disturbance suppression idea is developed to suppress the disturbance of singular multi-agent systems.

5.2 Problem Description of Admissible L 2 Consensus Control Consider a singular multi-agent system as follows: E x˙i (t) = Axi (t) + Bu i (t) + Bw wi (t),

(5.1)

where i = 1, 2, . . . , N , E, A ∈ Rd×d with rank(E) = r ≤ d, B ∈ Rd×m , Bw ∈ Rd×n , and xi (t), u i (t) and wi (t) ∈ L 2e are the state, the control input, and the external

154

5 Admissible L 2 Consensus and Consensualization with External …

disturbance, respectively. The interaction topology of this multi-agent system can be described by an undirected graph, where each node denotes an agent, the edge between any two nodes stands for the interaction channel between them, and the weight of the edge corresponds to the interaction strength. Construct the following consensus protocol u i (t) = K 1 xi (t) + K 2

∑

( ) wi j x j (t − τ ) − xi (t − τ ) ,

(5.2)

j∈Ni

where K 1 and K 2 are gain matrices with compatible dimensions and 0 ≤ τ ≤ τ [ ]T is a constant delay with τ its upper bound. Let x(t) = x1T (t), x2T (t), . . . , x NT (t) [ ]T and w (t) = w1T (t), w2T (t), . . . , w NT (t) , then the dynamics of multi-agent system (5.1) with protocol (5.2) can be described by ⎧ ˙ = (I N ⊗ (A + B K 1 ))x(t) − (L ⊗ B K 2 )x(t − τ ) ⎪ ⎨ (I N ⊗ E)x(t) + (I N ⊗ Bw )w (t), t ∈ [0, +∞), ⎪ ⎩ x(t) = φ(t), t ∈ [−τ, 0],

(5.3)

where φ(t) (t ∈ [−τ, 0]) is an admissible bounded vector-valued function. Let 0 = λ1 ≤ λ2 ≤ · · · ≤ λ N denote eigenvalues of the Laplacian matrix U such matrix L, then by Lemma 2.1, there exists a [nonsingular √ that ] U T LU = diag{0, λ2 , . . . , λ N }, where U = u 1 , U with u 1 = 1/ N and U = [u 2 , u 3 , . . . , u N ]. Let ( T ) [ ]T U ⊗ Id x(t) = xaT (t), xrT (t) ,

(5.4)

( T ) [ ]T U ⊗ In w (t) = waT (t), wrT (t) ,

(5.5)

where xa (t) ∈ Rd and wa (t) ∈ Rn . For any given T ≥ 0, ||xr ||22 ≤ γ 2 ||wr ||22 with ∫T ∫T γ > 0 represents 0 xrT (t)xr (t)dt ≤ γ 2 0 wrT (t)wr (t)dt. Actually, xa (t) and xr (t) describe the absolute motion and relative motions of multi-agent system (5.3). In this case, ||xr ||2 ≤ γ ||wr ||2 determines the impacts of external disturbances on relative motions. In the following, the definitions of admissible L 2 consensus and admissible L 2 consensualization are presented respectively. Definition 5.1 Multi-agent system (5.3) is said to achieve admissible L 2 consensus if it is regular and impulse-free, and for any given admissible bounded φ(t) (t ∈ [−τ, 0]), there exists a d-dimensional vector-valued function c(t) dependent on x(0) and w (t) such that limt→∞ (x(t) − 1 ⊗ c(t)) = 0 and ||xr ||2 ≤ γ ||wr ||2 when x(0) = 0, where c(t) is called a consensus function. Definition 5.2 Multi-agent system (5.1) is said to be admissibly L 2 consensualizable by protocol (5.2) if there exist K 1 and K 2 such that multi-agent system (5.3) achieves admissible L 2 consensus.

5.3 Admissible L 2 Consensus with External Disturbance

155

The current paper addresses the following three problems: (i) under what conditions multi-agent system (5.3) achieves admissible consensus and admissible L 2 consensus, respectively; (ii) how to determine the consensus function if multi-agent system (5.3) achieves admissible L 2 consensus; (iii) how to determine K 1 and K 2 such that multi-agent system (5.3) achieves admissible L 2 consensus. Remark 5.1 For many practical multi-agent systems, the information delay usually consists of two parts: the process delay and the transmission delay. Since all agents are homogeneous, their process delays are approximately identical. However, the transmission delays may be different since the distances among agents may be different. In the case, an upper bound of these transmission delays can be chosen as a common constant delay. A typical example is network control systems where multiple servers are placed in different cities. If the distances among cities are not the same, then the transmission delays among servers are different and the maximum transmission delay can usually be determined. Thus, a latency strategy can be used to match the maximum transmission delay if the practical transmission delay between two servers is less than the maximum one. Moreover, multi-agent systems with multiple time delays can be dealt with by the LMI tool as shown in Refs. [54, 55], but consensus criteria cannot guarantee the scalability of multi-agent systems since the dimensions of all variables in those criteria are dependent on the number of agents. However, LMI criteria of admissible L 2 consensus for multi-agent systems with a single delay given in this chapter can guarantee the scalability.

5.3 Admissible L 2 Consensus with External Disturbance In this section, we first propose a necessary and sufficient condition for admissible consensus and an approach to determine consensus functions, and then present the sufficient conditions for admissible L 2 consensus and consensualization in terms of LMIs.

5.3.1 Necessary and Sufficient Conditions for Admissible Consensus Without Disturbance In this section, the admissible consensus problem for multi-agent system (5.3) with w (t) ≡ 0 is converted into the stability problems of multiple singular time-delayed systems with lower dimensions, and an approach to determine consensus functions is given. [ T ]T Let xr (t) = xr 2 (t), xrT3 (t), . . . , xrTN (t) and wr (t) = ]T [ T T T wr 2 (t), wr 3 (t), . . . , wr N (t) , then by (5.4) and (5.5), multi-agent system (5.3) can be transformed into

156

5 Admissible L 2 Consensus and Consensualization with External …

E x˙a (t) = (A + B K 1 )xa (t) + Bw wa (t),

(5.6)

E x˙ri (t) = (A + B K 1 )xri (t) − λi B K 2 xri (t − τ ) + Bw wri (t).

(5.7)

The following theorem gives a necessary and sufficient condition for multi-agent system (5.3) with w (t) ≡ 0 to achieve admissible consensus. Theorem 5.1 Multi-agent system (5.3) with w (t) ≡ 0 achieves admissible consensus if and only if systems E x˙ri (t) = (A + B K 1 )xri (t) − λi B K 2 xri (t − τ ) (i = 2, 3, . . . , N ) are admissible. Proof Because nonsingular transformations cannot change the regularity and impulse-free property and U ⊗ Id and U ⊗ In are nonsingular, multi-agent system (5.3) is regular and impulse-free if and only if subsystems (5.6) and (5.7) are regular and impulse-free. Similar to the proof of Lemma 2.2 in [4], it can be shown that subsystems (5.7) are regular and impulse-free if and only if (E, A + B K 1 ) and (E, A + B K 1 − λi B K 2 ) are regular and impulse-free. Moreover, one can see that if subsystems (5.7) are regular and impulse-free, then subsystem (5.6) is also regular and impulse-free. Let ]T [ x˜a (t) = (U ⊗ Id ) xaT (t), 0 ,

(5.8)

]T [ x˜r (t) = (U ⊗ Id ) 0, xrT (t) ,

(5.9)

then one can see by (5.4) that x(t) = x˜a (t) + x˜r (t).

(5.10)

Due to [

xaT (t), 0

]T

= e1 ⊗ xa (t),

where e1 denotes an N -dimensional vector with first entry 1 and 0 elsewhere, one can obtain by (5.8) that 1 x˜a (t) = U e1 ⊗ xa (t) = √ 1 ⊗ xa (t). N

(5.11)

From (5.8) and (5.9), x˜a (t) and x˜r (t) are linearly independent. Hence, by (5.10) and (5.11), multi-agent system (5.3) with w (t) ≡ 0 achieves consensus if and only if subsystems (5.7) with wri (t) ≡ 0 (i = 2, 3, . . . , N ) are asymptotically stable. By the above analysis, multi-agent system (5.3) with w (t) ≡ 0 achieves admissible consensus if and only if systems E x˙ri (t) = (A + B K 1 )xri (t) −

5.3 Admissible L 2 Consensus with External Disturbance

157

λi B K 2 xri (t − τ ) (i = 2, 3, . . . , N ) are admissible. The proof of Theorem 5.1 is □ completed. Remark 5.2 The motions of a multi-agent system consist of the absolute motion as a whole and the relative motions among agents which describe its macroscopic and microcosmic behaviors, respectively. By the proof of Theorem 5.1, subsystems (5.6) and (5.7) determine the absolute motion and relative motions of multi-agent system (5.3), respectively; that is, xa (t) and xr (t) describe the macroscopic and microcosmic behaviors of multi-agent system (5.3), respectively. Obviously, multi-agent system (5.3) achieves admissible consensus if and only if the relative motions are admissible. Furthermore, the regularity and impulse-free property of the relative motions can guarantee that the absolute motion is regular and impulse-free. Moreover, it should be pointed out that the absolute motion may not be asymptotically stable.

5.3.2 Admissible L 2 Consensus Function In the sequel, we address the problem that how to determine consensus functions which describe the behaviors of absolute motions. Let T and Z be nonsingular matrices such that T E Z = diag{Ir , 0} and denote [ T AZ =

] [ ] A11 A12 B1 , TB = . A21 A22 B2

If (E, A, B) is impulse controllable, then there exists K 11 such that A22 + B2 K 11 is nonsingular; that is, (E, A + B K 1 ) with K 1 = [0, K 11 ]Z −1 is impulse-free. Let [

] Ir −(A12 + B1 K 11 )( A22 + B2 K 11 )−1 , 0 Id−r [ ] Ir 0 ˜ , Z= −(A22 + B2 K 11 )−1 A21 (A22 + B2 K 11 )−1 T˜ =

then one can obtain that [ [ ] ] A11 A12 + B1 K 11 ˜ A˜ 0 Z= T˜ , 0 Id−r A21 A22 + B2 K 11 where A˜ = A11 − (A12 + B1 K 11 )( A22 + B2 K 11 )−1 A21 . The following theorem gives an explicit expression of the consensus function and determines the influence of the external disturbances and initial states of agents on the consensus function, respectively.

158

5 Admissible L 2 Consensus and Consensualization with External …

Theorem 5.2 If multi-agent system (5.3) achieves admissible L 2 consensus, then the consensus function c(t) satisfies lim (c(t) − cx (t) − cw (t)) = 0,

t→∞

where ( ˜ T At

cx (t) = Z Z˜ [Ir , 0] e [Ir , 0] Z˜ −1 Z −1

cw (t) = Z Z˜ [Ir , 0]

(

∫t T

) N 1 ∑ xi (0) , N i=1

e

˜ A(t−s)

˜ −1

[Ir , 0] Z

Z

−1

Bw

0

) N 1 ∑ wi (s) ds. N i=1

Proof From (5.4) and (5.5), one has ( ) xa (t) = [Id , 0] U T ⊗ Id x(t), ) ( wa (t) = [In , 0] U T ⊗ In w (t). Due to [I, 0] = eT1 ⊗ I , one can obtain that N ) ( 1 ∑ xa (0) = eT1 U T ⊗ Id x(0) = √ xi (0), N i=1

(5.12)

N ) ( 1 ∑ wa (t) = eT1 U T ⊗ In w (t) = √ wi (t). N i=1

(5.13)

By (5.6), one has ˜ cx (t) = Z Z˜ [Ir , 0]T e At [Ir , 0] Z˜ −1 Z −1 xa (0),

cw (t) = Z Z˜ [Ir , 0]T

∫t

˜ e A(t−s) [Ir , 0] Z˜ −1 Z −1 Bw wa (s)ds.

(5.14)

(5.15)

0

By (5.11), one can obtain that ( ) 1 lim c(t) − √ xa (t) = 0. t→∞ N From (5.12) to (5.16), the conclusion of Theorem 5.2 can be obtained.

(5.16) □

5.3 Admissible L 2 Consensus with External Disturbance

159

Theorem 5.2 shows that the constant delay does not influence the consensus function although it is an important factor for multi-agent system (5.3) to achieve admissible L 2 consensus. However, the external disturbances impact the consensus function. Furthermore, multi-agent systems with the same initial states but different undirected interaction topologies have the same c(t), which means that the consensus function is independent of interaction topologies. It should be pointed out that the consensus function may be dependent on interaction topologies if interaction topologies are directed. Moreover, since the time delay is a constant, the delayed term can be written as −(L ⊗ B K 2 )x(t − τ ) in (5.3). In this case, by L1 = 0, multi-agent system (5.3) can be decomposed into subsystems (5.6) and (5.7), where subsystem (5.6) determines the consensus function and is independent of the time delay. Hence, the intrinsic reason why the consensus function is independent of the time delay is that the delay is single. Remark 5.3 We proposed a state projection method to determine time-varying consensus functions in Sect. 2.5.2, where time delays and external disturbances were not considered. Theorem 5.2 presents a general approach to determine consensus functions of singular multi-agent systems, which can determine the impacts of external disturbances, time delays, and interaction topologies on consensus functions.

5.3.3 Admissible L 2 Consensus Analysis and Design Criteria In this section, by using LMI techniques, sufficient conditions for admissible L 2 consensus and consensualization are presented, respectively, which are scalable since they are only dependent on the maximum and second small eigenvalue of Laplacian matrix L. Theorem 5.3 For any τ ∈ [0, τ ] and a given γ > 0, multi-agent system (5.3) achieves admissible L 2 consensus if there exist 0 < Q T = Q ∈ Rd , 0 < S T = S ∈ T T and M22 = M22 such Rd , d × d real matrices R, R1 , R2 , X , Y , M12 , M11 = M11 that E T R = R T E ≥ 0, ⎡

⎤ Ξ12 Ξi13 R1T Bw Ξ22 Ξi23 R2T Bw ⎥ ⎥ < 0 (i = 2, N ), ∗ −Q 0 ⎦ ∗ ∗ −γ 2 I ⎡ ⎤ M11 M12 X Θ = ⎣ ∗ M22 Y ⎦ > 0, ∗ ∗ S

Ξ11 ⎢ ∗ Ξi = ⎢ ⎣ ∗ ∗

(5.17)

(5.18)

(5.19)

160

5 Admissible L 2 Consensus and Consensualization with External …

where Ξ11 = R1T (A + B K 1 ) + (A + B K 1 )T R1 + Q + X E + E T X T + τ M11 + I, Ξ12 = R T − R1T + (A + B K 1 )T R2 + E T Y T + τ M12 , Ξi13 = −λi R1T B K 2 − X E, Ξ22 = −R2T − R2T + τ S + τ M22 , Ξi23 = −λi R2T B K 2 − Y E. Proof First of all, consider the asymptotic stability of the following system E x˙ri (t) = ( A + B K 1 )xri (t) − λi B K 2 xri (t − τ ) (i ∈ {2, 3, . . . , N }).

(5.20)

Construct the following Lyapunov–Krasovskii functional candidate Vi (t) = Vi1 (t) + Vi2 (t) + Vi3 (t),

(5.21)

where Vi1 (t) = xriT (t)R T E xri (t), ∫t Vi2 (t) =

xriT (s)Qxri (s)ds, t−τ

∫0 ∫ t Vi3 (t) =

x˙riT (s)E T S E x˙ri (s)dsdθ. −τ t+θ

Let Rˆ =

[

] R 0 , R1 R2

then by taking the derivative of these functions with respect to t along the solution of system (5.20), one has V˙i1 (t) = 2

[

xri (t) E x˙ri (t)

]T

Rˆ T

([

0 I A + B K 1 −I

][

] ] [ ) xri (t) 0 xri (t − τ ) , − E x˙ri (t) λi B K 2 (5.22)

V˙i2 (t) = xriT (t)Qxri (t) − xriT (t − τ )Qxri (t − τ ),

(5.23)

5.3 Admissible L 2 Consensus with External Disturbance

V˙i3 (t) ≤ τ x˙riT (t)E T S E x˙ri (t) −

161

∫t x˙riT (s)E T S E x˙ri (s)ds.

(5.24)

t−τ

Due to ∫t x˙ri (s)ds = xri (t) − xri (t − τ ), t−τ

for any real matrices X and Y with appropriate dimensions, it can be shown that [ T Φi1 Δ 2ϕi1 (t)

⎛

]

XE ⎝ xri (t) − xri (t − τ ) − YE

∫t

⎞ x˙ri (s)ds ⎠ = 0

(5.25)

t−τ

[ ]T where ϕi1 (t) = xriT (t), E T x˙riT (t) . In addition, for any real symmetric matrix M = [ ] M11 M12 > 0, one has ∗ M22 ∫t Φi2 Δ

T τ ϕi1 (t)Mϕi1 (t)

−

T ϕi1 (t)Mϕi1 (t)ds ≥ 0.

(5.26)

t−τ

From (5.21) to (5.26), by Schur complement in [56], one has V˙i (t) ≤ V˙i1 (t) + V˙i2 (t) + V˙i3 (t) + Φi1 (t) + Φi2 (t) ∫t T T ˆ i ϕi2 (t) − ≤ ϕi2 (t)Ξ ϕi3 (t, s)Θϕi3 (t, s)ds,

(5.27)

t−τ

[ T ]T where ϕi2 (t) = xri (t), E T x˙riT (t), xriT (t − τ ) , [ T ]T xri (t), E T x˙riT (t), E T x˙riT (s) and

ϕi3 (t, s)

=

⎡

⎤ Ξ11 − I Ξ12 Ξi13 ˆi = ⎣ Ξ ∗ Ξ22 Ξi23 ⎦. ∗ ∗ −Q ˆ i < 0 and (5.17) and (5.19) hold, then system (5.20) is asymptotically stable. If Ξ ˆ i < 0, Θ < 0 and (5.17) can guarantee that system In the sequel, we show that Ξ (5.20) is regular and impulse-free. Let Eˆ = diag{E, 0}, Qˆ = diag{Q, τ S}, and

162

5 Admissible L 2 Consensus and Consensualization with External …

Aˆ 1 =

[

] ] [ [ ] X 0 0 I 0 0 ˆ ˆ , A2 = ,X = , Y 0 A + B K 1 −I −λi B K 2 0

ˆ i < 0 are feasible if and then it can be shown by Schur complement that (5.17) and Ξ only if the following inequalities hold Eˆ T Rˆ = Rˆ T Eˆ ≥ 0,

(5.28)

Rˆ T Aˆ 1 + Aˆ T1 Rˆ + Qˆ + τ M + Xˆ Eˆ + Eˆ T Xˆ T ( ( ) )T + Rˆ T Aˆ 2 − Xˆ Eˆ Qˆ −1 Rˆ T Aˆ 2 − Xˆ Eˆ < 0.

(5.29)

It can be shown that (

( ) ( )T ( ) )T Rˆ T Aˆ 2 − Xˆ Eˆ + Rˆ T Aˆ 2 − Xˆ Eˆ − Qˆ ≤ Rˆ T Aˆ 2 − Xˆ Eˆ Qˆ −1 Rˆ T Aˆ 2 − Xˆ Eˆ . (5.30) By (5.29) and (5.30), one can obtain that ( ) ( )T Rˆ T Aˆ 1 + Aˆ 2 + Aˆ 1 + Aˆ 2 Rˆ + τ M < 0.

(5.31)

Due to M > 0, Rˆ is invertible from (5.31). Let Tˆ and Zˆ be nonsingular matrices such that Tˆ Eˆ Zˆ = diag{Ir , 0} and Tˆ Aˆ 1 Zˆ =

[

[ [ ] ] ] Xˆ 11 Xˆ 12 Aˆ 11 Aˆ 12 ˆ −T ˆ ˆ Rˆ 11 Rˆ 12 ˆ T ˆ ˆ −1 = R Z = X T , T , Z . Aˆ 21 Aˆ 22 Rˆ 21 Rˆ 22 Xˆ 21 Xˆ 22

By (5.28), one can show that Rˆ 12 = 0.

(5.32)

Due to Qˆ > 0 and M > 0, it can be obtained by (5.29) that Rˆ T Aˆ 1 + Aˆ T1 Rˆ + Xˆ Eˆ + Eˆ T Xˆ T < 0.

(5.33)

Pre- and post-multiplying (5.33) by Zˆ T and Zˆ , respectively, one can obtain that T ˆ Aˆ T22 Rˆ 22 + Rˆ 22 A22 < 0.

(5.34)

( ) ˆ Aˆ 1 is regular and impulse-free. Due to From (5.34), Aˆ 22 is invertible, so E,

5.3 Admissible L 2 Consensus with External Disturbance

163

) ( det(s E − A − B K 1 ) = det s Eˆ − Aˆ 1 , )) ( ( deg(det(s E − A − B K 1 )) = deg det s Eˆ − Aˆ 1 = rank(E), one can see that (E, A +(B K 1 ) is regular ) and impulse-free. By (5.28) and (5.31), ˆ Aˆ 1 + Aˆ 2 is regular and impulse-free. It can be shown from Lemma 2.2 in [57], E, that ) ( det(s E − A − B K 1 + λi B K 2 ) = det s Eˆ − Aˆ 1 − Aˆ 2 , )) ( ( deg(det(s E − A − B K 1 + λi B K 2 )) = deg det s Eˆ − Aˆ 1 − Aˆ 2 = rank(E), which means that (E, A + B K 1 − λi B K 2 ) is regular and impulse-free. Hence, system (5.20) is regular and impulse-free. Now discuss the performance of multi-agent system (5.3) with the disturbance w (t). Consider the case that system (5.20) includes the external disturbance; that is, E x˙ri (t) = ( A + B K 1 )xri (t) − λi B K 2 xri (t − τ ) + Bw wri (t) (i ∈ {2, 3, . . . , N }). Consider the following cost performance, for any T ≥ 0, ∫T JT =

(

) xriT (t)xri (t) − γ 2 wriT (t)wri (t) dt.

0

For a zero-valued initial condition, i.e., φ(t) ≡ 0 (t ∈ [−τ, 0]), by a similar analysis as the stability of system (5.20), it can be shown that ∫T JT =

(

) xriT (t)xri (t) − γ 2 wriT (t)wri (t) + V˙ (t) dt − V (T ) + V (0)

0

∫T T ϕi4 (t)Ξi ϕi4 (t)dt − V (T ),

≤ 0

[ T ]T where ϕi4 (t) = ϕi2 (t), wriT (t) . If Ξi < 0, then JT ≤ 0; that is, ∫T

∫T xriT (t)xri (t)dt

0

Hence, one has

≤γ

wriT (t)wri (t)dt.

2 0

164

5 Admissible L 2 Consensus and Consensualization with External … N ∫ ∑

T

N ∫ ∑

T

xriT (t)xri (t)dt

≤γ

2

i=2 0

wriT (t)wri (t)dt,

i=2 0

which means that ∫T

∫T xrT (t)xr (t)dt

0

≤γ

wrT (t)wr (t)dt.

2 0

ˆ i < 0. By the above analysis, from Theorem 5.1, It is obvious that if Ξi < 0, then Ξ multi-agent system (5.3) achieves admissible L 2 consensus if LMIs (5.17)–(5.19) are feasible for both λ2 and λ N since LMI has the convex property. The proof of □ Theorem 5.3 is completed. Remark 5.4 An H∞ control approach was proposed to deal with normal multiagent systems with external disturbances in Refs. [7, 58], where a controlled output function was defined on the basis of the average of states of all agents, and the H∞ norm of the closed-loop transfer function matrix from external disturbances to the controlled output was applied to evaluate the impact of external disturbances. However, the H∞ control approach in Refs. [7, 58] cannot be used to investigate the cases that consensus functions are not the average of states of all agents. In Ref. [59], we gave an evaluation criterion of the influence of external disturbances, where it was not required that consensus functions are the average of states of all agents, but it was assumed that the dynamics of each agent is described by a normal system. The evaluation approach of the influence of external disturbances in Theorem 5.3 requires fewer limitations about the consensus function and the dynamics of each agent. Since it is necessary for admissible L 2 consensualization of multi-agent system (5.3) that (E, A, B) is impulse controllable, one can choose K 1 = [0, K 11 ]Z −1 such that (E, A + B K 1 ) is impulse-free. By the changing variable method, the following theorem gives an approach to determine K 2 such that multi-agent system (5.3) achieves admissible L 2 consensus. Theorem 5.4 For any τ ∈ [0, τ ] and a given γ > 0, multi-agent system (5.3) can be T admissibly L 2 consensualized by protocol (5.2) if there exist 0 < Q = Q ∈ Rd×d , T T 0 < S = S ∈ Rd×d , d × d real matrices R, R 1 , R 2 , M 12 , M 11 = M 11 and T M 22 = M 22 , and K 2 ∈ Rm×d such that T

R E T = E R ≥ 0,

(5.35)

5.3 Admissible L 2 Consensus with External Disturbance

⎡

Ξ11 ⎢ ⎢ ∗ ⎢ ⎢ Ξi = ⎢ ∗ ⎢ ∗ ⎢ ⎣ ∗ ∗

T

Ξ12 Ξ22 ∗ ∗ ∗ ∗

0 τ R1 T Ξi23 τ R 2 −Q 0 ∗ −τ S ∗ ∗ ∗ ∗ ⎡ M 11 Θ=⎣ ∗ ∗

0 Bw 0 0 −γ 2 I ∗

T⎤ R ⎥ 0 ⎥ ⎥ 0 ⎥ < 0 (i = 2, N ), ⎥ 0 ⎥ ⎥ 0 ⎦ −I ⎤

M 12 0 M 22 S ⎦ ≥ 0, ∗ S

165

(5.36)

(5.37)

where T

Ξ11 = R 1 + R 1 + Q + τ M 11 , ( )T T T Ξ12 = R 2 − R 1 + A R + B K 1 R + R E T + τ M 12 , T

Ξ22 = −R 2 − R 2 + τ M 22 , Ξi23 = −λi B K 2 − E R. −1

In this case, K 2 = K 2 R . Proof By (5.31), Rˆ is invertible. It can be shown that Rˆ −1 =

[

] R 0 , R1 R2

where R = R −1 , R 1 = −R2−1 R1 R −1 and R 2 = R2−1 . By Schur complement, Ξi < 0 is equivalent to ⎡

ψ11 ⎢ ∗ ⎢ ⎢ ψi = ⎢ ∗ ⎢ ⎣ ∗ ∗

⎤ ψi12 [0, τ I ]T ψ13 [I, 0]T −Q 0 0 0 ⎥ ⎥ ⎥ 0 ⎥ < 0, ∗ −τ S −1 0 ⎥ ∗ ∗ −γ 2 I 0 ⎦ ∗ ∗ ∗ −I

where ψ11 = Rˆ T Aˆ 1 + Aˆ T1 Rˆ + Xˆ Eˆ + Eˆ T Xˆ T + diag{Q, 0} + τ M,

166

5 Admissible L 2 Consensus and Consensualization with External …

] [ T T T ψi12 = Ξi13 , Ξi23 , ]T [ T T ψ13 = Bw R1 , Bw R2 . Let X = R1T and Y = R2T , then one has [ ] [ ] 0 −T X ˆ R = . Y I T Let S = S −1 , Q = R Q R and M = Rˆ −T M Rˆ −1 =

[

] M 11 M 12 . Pre- and ∗ M 22

T

post-multiplying } (5.17) by { { R and }R, and pre- and post-multiplying (5.19) by −T −1 ˆ diag R , S and diag Rˆ −1 , S −1 , respectively, LMIs (5.35) and (5.37) can be { T T } obtained, respectively. Pre- and post-multiplying ψi by diag Rˆ − , R , I, I, I and { } diag Rˆ −1 , R, I, I, I , LMI (5.36) can be obtained. The proof of Theorem 5.4 is completed. □ Remark 5.5 If multi-agent system (5.3) achieves admissible L 2 consensus and w (t) ∈ L 2 ∩ L ∞ , then xr (t) ∈ L 2 . It can be shown that x˙r (t) ∈ L ∞ by (5.7). Thus, one has limt→∞ xr (t) = 0 by Barbalat’s Lemma in Ref. [60], which means that limt→∞ (x(t) − 1 ⊗ c(t)) = 0 and the relative motions are admissible.

5.4 Numerical Simulation Consider a singular multi-agent system where each agent is described by (5.1) with ⎡

1 ⎢0 E =⎢ ⎣0 0

0 1 0 0

0 0 0 0

⎤ ⎡ 0 0 ⎢ 0 0⎥ ⎥, A = ⎢ ⎣ −1 0⎦ 0 0

1 0 1 −1

0 0 −1 0

⎡ ⎤ ⎤ ⎡ ⎤ 1 1 0 ⎢1⎥ ⎢1⎥ 0 ⎥ ⎥, B = ⎢ ⎥, Bw = ⎢ ⎥. ⎣0⎦ ⎣0⎦ 0 ⎦ 0 −2 0

An undirected interaction topology of the singular multi-agent system is illustrated in Fig. 5.1, and it is assumed that its adjacent matrix is a 0–1 matrix without loss of generality. The initial states are x1 (0) = [9, 8, −1, −4]T , x4 (0) = [−6, 15, 21, −7.5]T , x7 (0) = [−1, 5, 6, −2.5]T ,

x2 (0) = [6, 4, −2, −2]T , x5 (0) = [−9, −8, 1, 4]T , x8 (0) = [1, 4, 3, −2]T ,

x3 (0) = [3, 2, −1, −1]T , x6 (0) = [2, 4, 2, −2]T , x9 (0) = [−1, −3, −2, −1.5]T .

5.4 Numerical Simulation Fig. 5.1 Undirected interaction topology

167

2

3

1

5

4

7

8

6

9

Let K 1 = 0, τ = 0.05 and γ = 3, then one can obtain by Theorem 5.4 that K 2 = [1.2946, 4.2559, 0.0002, 2.0000]. In Fig. 5.2, the state trajectories of the singular multi-agent system with ]T [ w (t) = 0, e−0.5t cos(t), 0, e−2t sin(t), e−t , 0, 0.11 sin(t), 0, e−t are given, and the trajectories marked by circles are function given ∫ t the curves of the∫ consensus t in Theorem 5.2. The trajectories of 0 xrT (t)xr (t)dt and 0 wrT (t)wr (t)dt under the zero initial condition are depicted in Fig. 5.3. One sees that the singular multi-agent system achieves admissible L 2 consensus with ||xr ||2 ≤ 3||wr ||2 . Moreover, the state trajectories converge to the ones marked by circles, which illustrates the conclusion of Theorem 5.2.

Fig. 5.2 State trajectories

168

Fig. 5.3 Trajectories of

5 Admissible L 2 Consensus and Consensualization with External …

∫t 0

xrT (t)xr (t)dt and

∫t 0

wrT (t)wr (t)dt

5.5 Notes Admissible L 2 consensus problems for high-order linear time-delayed singular multiagent systems were dealt with. Admissible consensus problems were transformed into admissible problems for multiple lower-dimensional time-delayed singular systems, and an explicit expression of the consensus function was given based on the First Equivalent Form. An L 2 evaluation approach was proposed to determine the impacts of external disturbances, and LMI criterions for admissible L 2 consensus were presented, which are only associated with the second small and maximum eigenvalues and can guarantee the scalability of singular multi-agent systems since the dimensions of all variables in those criteria are independent of the number of agents. Moreover, by using changing variable method, sufficient conditions for admissible L 2 consensualization were shown.

References 1. Ma CQ, Zhang JF (2010) Necessary and sufficient conditions for consensusability of linear multi-agent systems. IEEE Trans Autom Control 55(5):1263–1268 2. Xi JX, Shi ZY, Zhong YS (2012) Output consensus analysis and design for high-order linear swarm systems: partial stability method. Automatica 48:2335–2343 3. Yang XR, Liu GP (2012) Necessary and sufficient consensus conditions of descriptor multiagent systems. IEEE Trans Circuits Syst I Regul Pap 59(11):2669–2677 4. Long C, Hou ZG, Lin Y, Min T, Zhang W (2011) Solving a modified consensus problem of linear multi-agent systems. Automatica 47(10):2218–2223

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Chapter 6

Admissible Consensus and Consensualization with Protocol State Constraints

This chapter investigates the stable-protocol admissible consensus analysis and design problems for high-order LTI singular multi-agent systems with multiple time delays and switching topologies. It is well known that singular systems can better describe physical systems than normal ones. Consensus problems for singular multi-agent systems are much more complicated because both consistency and regularity and impulse-free property need to be considered. Singular multi-agent systems with state feedback consensus protocols were dealt with in Sect. 3.2. Yang and Liu [1] constructed static output feedback consensus protocols and presented necessary and sufficient conditions for consensus. Singular multi-agent systems with dynamic output feedback consensus protocols were addressed in [2], where it was assumed that the time delay is single and constant and interaction topologies are fixed. However, in most practical cases, multiple time delays are more likely to occur. Especially, due to electronic interference, equipment fault, and active communication silence, the impact of switching topologies on the admissible consensus of singular multi-agent systems cannot be ignored. We investigate admissible consensus analysis problems of singular multi-agent systems by using dynamic output feedback consensus protocols with multiple timevarying delays, where both connected switching topology cases and jointly connected switching topology cases are discussed. Firstly, admissible consensus problems for high-order LTI singular multi-agent systems with multiple time delays are investigated. Based on the state-space decomposition, necessary and sufficient conditions for admissible consensus are presented and admissible consensus problems are transformed into admissible problems of multiple lower dimensional singular systems. And based on different influences of initial states of agents, initial states of protocols, time delays and topology variances, an explicit expression of the consensus function is given, and a simpler expression is presented when the interaction topology is fixed and the time delay is single. Secondly, stable-protocol admissible consensus analysis and design problems for singular multi-agent systems with both connected switching topologies and jointly connected switching topologies are investigated.

© Science Press, Beijing and Springer Nature Singapore Pte Ltd. 2023 J. Xi et al., Admissible Consensus and Consensualization for Singular Multi-agent Systems, Engineering Applications of Computational Methods 11, https://doi.org/10.1007/978-981-19-6990-4_6

173

174

6 Admissible Consensus and Consensualization with Protocol State …

Based on outputs and protocol states among neighboring agents, a singular observertype consensus protocol is applied. Based on the orthonormal projection between the CS and the CCS, a necessary and sufficient condition for stable-protocol admissible consensus is proposed and an explicit expression of the consensus function which determines the consensus behavior of all agents is presented. Moreover, stable-protocol admissible consensus conditions for connected switching topology cases and jointly connected switching topology cases are given, respectively. Finally, numerical simulations are presented to demonstrate theoretical results.

6.1 Dynamic Output Feedback Protocol In the previous section, the consensus protocol of multi-agent systems is designed based on the relative states of neighboring agents, and it is assumed that all states of neighboring agents are known. However, in many circumstances, the state information might not be fully available. Instead, each agent can have access to the local output information of its neighboring agents, such as part of the state information. By using the output information of neighboring agents, two basic types of consensus protocols have been proposed to realize state consensus in some literatures. One is the static output feedback consensus protocol, as shown in [3], and the other is the dynamic output feedback consensus protocol, as given in [4–6]. For multi-agent systems with static output feedback consensus protocols, the input of each agent tends to zero when consensus is achieved, which means that the control energy is limited. However, for multi-agent systems with dynamic output feedback consensus protocols, the input of each agent may not tend to zero even if consensus is achieved, which means that a continuous expenditure of energy may be required. In addition, in many practical applications, output consensus instead of state consensus needs to be achieved. For example, for network control systems, since velocity agreement can reduce the queuing delay and package loss rate, all routers are usually needed to achieve consensus on the data-processing rate, but the account of data of each router can be different. In this section, we will design consensus protocols using local output information.

6.1.1 Static Output Feedback Consensus Protocol Here we consider a system consisting of N agents and the dynamics of ith agent is described as follows: ⎧ x˙i (t) = Axi (t) + Bu i (t), (6.1) yi (t) = C xi (t),

6.1 Dynamic Output Feedback Protocol

175

where i = 1, 2, . . . , N , A ∈ Rn×n , B ∈ Rn×m and C ∈ Rq×n , and xi (t) ∈ Rn , u i (t) ∈ Rm and yi (t) ∈ Rq is the state, control input and output, respectively. Nowadays, the following state-protocol is often used as shown in [7, 8]: u i (t) = K

∑

( ) wi j x j (t) − xi (t) ,

(6.2)

j∈Ni

where i = 1, 2, . . . , N , Ni is the set of neighbors for ith agent, and K is the control gain matrix. Consensus protocol (6.2) not only is distributed but also depends on the errors of states between ith agent and its neighbors. A feature of consensus protocol (6.2) is to use all relative state information between agents and their neighbors. However, in practice, it is sometimes difficult to directly measure the relative state information of all agents, and only the relative output information of agents can be used. Therefore, it is more practical to consider outputbased consensus protocols. Especially, when all states are measurable, consensus protocols based on outputs and states are equivalent. Thus, we will consider the output feedback case directly. It is supposed that system (6.1) satisfies that ( A, B) is stabilizable and C is of full row rank. Consider the following static output feedback consensus protocol: u i (t) = K

∑

( ) wi j y j (t) − yi (t) .

(6.3)

j∈Ni

[ ]T [ ]T Let x(t) = x1T (t), x2T (t), . . . , x NT (t) and y(t) = y1T (t), y2T (t), . . . , y NT (t) , then the dynamics of systems (6.1) can be described as ⎧

x(t) ˙ = (I N ⊗ A − L ⊗ B K C)x(t), y(t) = (I N ⊗ C)x(t).

(6.4)

Definition 6.1 For a given K, system (6.4) is said to achieve output consensus if for any given bounded x(0), there exists c(t) such that limt→∞ (y(t) − 1 ⊗ c(t)) = 0, where c(t) is called an output consensus function. Definition 6.2 System (6.4) is said to be output consensualizable by static output feedback consensus protocol (6.3) if there exists a gain matrix K such that it achieves output consensus.

6.1.2 Dynamic Output Feedback Consensus Protocol When applying static output feedback control protocols to deal with consensus problems, numerical algorithms are needed to determine the gain matrix, but the algorithms do not guarantee the convergence. Consider a multi-agent system (6.1), and

176

6 Admissible Consensus and Consensualization with Protocol State …

the dynamic output feedback consensus protocol is designed as follows: ⎧ ( ( ) ) ∑ ∑ ⎨ z˙ i (t) = K A z i (t) − K B wi j z j (t) − z i (t) + K C wi j y j (t) − yi (t) , ⎩ u (t) = K z (t), i D i

j∈Ni

j∈Ni

(6.5) where z i (t) ∈ Rn (i ∈ {1, 2, . . . , N }) is the state of the protocol, and K A , K B , K C and K D are constant gain matrices with appropriate dimensions. Definition 6.3 System (6.1) is said to be output consensualizable by dynamic output feedback consensus protocol (6.5) if there exist gain matrices K A , K B , K C , and K D such that it achieves output consensus. According to the dynamic output feedback consensus protocol (6.5), it can be found that the input of each agent may not tend to zero even if output consensus is achieved. For the cases where the resource of each agent is limited, it is expected that the control input for each agent tends to 0 when output consensus is achieved. Limited control energy (LCE) output consensus is defined as follows. Definition 6.4 System (6.1) is said to achieve LCE output consensus if it achieves output consensus and limt→∞ z i (t) = 0 (i = 1, 2, . . . , N ).

6.1.3 Stable Consensus Protocol For multi-agent systems with dynamic output feedback consensus protocols, the input to each agent may not converge to zero even if consensus or synchronization is reached. In this case, continuous energy consumption is required. Since the resources of each agent are usually limited, it is expected that the control inputs of each agent converge to zero if consensus is reached, which can be guaranteed by a stable control protocol. Consider a homogeneous multi-agent system with N agents as (6.1), and it is assumed that the output matrix C is of full row rank. When C is of full row rank, ] [ T T is nonsingular. there exists C ∈ Rα×n with α + q = n such that T = C T , C Denote ] [ [ ] −1 A11 A12 B1 , TB = , T AT = A21 A22 B2 where A11 ∈ Rq×q and B 1 ∈ Rq×m . Let T˜ be a nonsingular matrix such that (T˜ −1 A22 T˜ , A12 T˜ ) =

([

) ] ] [ Do 0 , Eo 0 , D o D˜ o

6.1 Dynamic Output Feedback Protocol

T˜ −1 A21 =

[

177

] ] [ Fo FK −1 ˜ , T B2 = , Fo FK

( ) where D o ∈ Rβ×β and D o , E o is completely observable. Denote [ Ao =

] ] [ [ ] A11 E o B1 , Bo = , C o = Iq , 0 . F o Do FK

Based on Lemmas 2.18 and 2.19, the stable consensus protocol is considered as follows: ∑ ⎧ ( ) wi j z j (t) − z i (t) ⎪ ⎪ z˙ i (t) = ( Ao + Bo K 1 )z i (t) − K 2 Co ⎪ ⎪ j∈Ni (t) ⎨ ∑ ( ) (6.6) y w (t) − y (t) , + K 2 ij j i ⎪ ⎪ ⎪ j∈N (t) i ⎪ ⎩ u i (t) = K 1 z i (t), where z i (t) ∈ Rq+β (i ∈ {1, 2, . . . , N }) is the state of the protocol, K 1 and K 2 are gain matrices with appropriate dimensions. [ T ]T Let vi (t) = z i (t), xiT (t) (i ∈ {1, 2, . . . , N }), v(t) = ] [ T ]T [ T T y1 (t), y2T (t), . . . , y NT (t) and z(t) = v1 (t), v2T (t), . . . , v TN (t) , y(t) = ]T [ T z 1 (t), z 2T (t), . . . , z TN (t) , then, the dynamics of system (6.1) with stable consensus protocol (6.6) can be described by ⎧ ( ) ⎨ v(t) ˙ = I N ⊗ A˜ − L σ (t) ⊗ B˜ v(t), ( ) ⎩ y(t) = I N ⊗ C˜ v(t),

(6.7)

where A˜ =

[

] [ ] −K 2 Co K 2 C Ao + Bo K 1 0 ˜ , B= , C˜ = C[0, Id ]. A B K1 0 0

Definition 6.5 Multi-agent system (6.7) is said to achieve stable-protocol output consensus if for any given bounded v(0), there exists a vector-valued function c(t) dependent on v(0) such that limt→∞ z(t) = 0 and limt→∞ (y(t) − 1 ⊗ c(t)) = 0, where c(t) is called an output consensus function. Definition 6.6 Multi-agent system (6.7) is said to be stable-protocol output consensualizable by stable consensus protocol (6.6) if there exist K 1 and K 2 such that multi-agent system (6.7) achieves stable-protocol output consensus.

178

6 Admissible Consensus and Consensualization with Protocol State …

6.2 Stable-Protocol Admissible Consensus with Time Delays In this section, stable-protocol admissible consensus analysis and design problems of high-order LTI singular multi-agent systems with multiple time delays and topology variances are addressed. The impacts of multiple time-varying delays and topology variances on consensus are investigated, and the impacts of initial states of agents and protocols, time delays, and topology variances on consensus functions are determined, respectively. Firstly, basic concepts about singular systems are given and the problem description is presented. Secondly, necessary and sufficient conditions for admissible consensus are given and explicit expressions of consensus functions are shown. Finally, a numerical simulation is given to demonstrate theoretical results.

6.2.1 Singular Dynamic Output Feedback Consensus Protocol Consider a multi-agent system with N homogeneous agents as follows ⎧

E 0 x˙i (t) = Axi (t) + Bu i (t), yi (t) = C xi (t),

(6.8)

where i = 1, 2, . . . , N , A ∈ Rd×d , B ∈ Rd×m , C ∈ Rq×d , E 0 ∈ Rd×d with rank(E 0 ) = r0 ≤ d, xi (t), yi (t), and u i (t) are the state, the output and the control input, respectively. The interaction topology of this multi-agent system can be described by a directed graph, where each node stands for an agent, the edge between any two nodes denotes the interaction channel between them, and the weight of the edge corresponds to the interaction strength. Consider the following singular dynamic output feedback consensus protocol ( ( ( ) )) ∑ ⎧ E 1 z˙ i (t) = A z z i (t) + B y wi j (t) y j t − τi j (t) − yi t − τi j (t) ⎪ ⎪ ⎪ v j ∈Ni (t) ⎪ ( ( ( ) )) ⎨ ∑ + Bz wi j (t) z j t − τi j (t) − z i t − τi j (t) , v j ∈Ni (t) ⎪ ⎪ ( ( ( ) )) ∑ ⎪ ⎪ (t) = K y (t) + K z z i (t) + K y wi j (t) y j t − τi j (t) − yi t − τi j (t) , u ⎩ i 0 i v j ∈Ni (t)

(6.9) where E 1 ∈ Rh×h with rank(E 1 ) = r1 ≤ h, z i (t) ∈ Rh (i ∈ {1, 2, . . . , N }) is the protocol state, K 0 , K z , K y , A z , B y , and Bz are gain matrices with compatible dimensions, Ni (t) and wi j (t) are the neighbor set of agent i and the interaction strength of the edge from agent j to agent i respectively, and τi j (t) is a time-varying delay. It is supposed that

6.2 Stable-Protocol Admissible Consensus with Time Delays

179

Assumption 6.1 There }exist r different time delays τk (t) ∈ { τi j (t) : i, j ∈ {1, 2, . . . , N } (k = 1, 2, . . . , r ) with 0 ≤ τk (t) ≤ τ k < ∞ for t ≥ 0, where τ k (k = 1, 2, [ . . . , r] ) are known constants. Define matrices L k (t) = lki j (t) ∈ R N ×N as follows

lki j (t) =

⎧ −wi j (t), ⎪ ⎪ ⎨ 0, ⎪ ⎪ ⎩−

N ∑

m=1,m/=i

j /= i, τi j (t) = τk (t), j /= i, τi j (t) /= τk (t), lkim (t), j = i,

where i, j = 1, 2, . . . , N and k = 1, 2, . . . , r . It can be shown that L k (t)1 = 0 (k = [ ]T 1, 2, . . . , r ), where 1 = [1, 1, . . . , 1]T ∈ R N . Let x(t) = x1T (t), x2T (t), . . . , x NT (t) [ T ] T and z(t) = z 1 (t), z 2T (t), . . . , z TN (t) , then the dynamics of multi-agent system (6.1) with protocol (6.2) can be described by ⎧ r ( ) ∑ ⎪ ⎪ L k (t) ⊗ B y C x(t − τk (t)) ⊗ E ⊗ A z (t) = − (I )˙ (I )z(t) N 1 N z ⎪ ⎪ ⎪ k=1 ⎪ ⎪ r ∑ ⎪ ⎨ − (L (t) ⊗ B )z(t − τ (t)), k

z

k

k=1 ⎪ r ( ) ∑ ⎪ ⎪ ⎪ L k (t) ⊗ B K y C x(t − τk (t)) ⊗ E ⊗ + B K C))x(t) − x(t) ˙ = (I (A ) (I N 0 N 0 ⎪ ⎪ ⎪ k=1 ⎪ ⎩ + (I N ⊗ B K z )z(t), (6.10)

[ ]T where x T (t), z T (t) = φ(t) (t ∈ [−τ , 0]) with τ = max{τ 1 , τ 2 , . . . , τ r } is a bounded admissible vector-valued function. Definition 6.7 Multi-agent system (6.10) is said to achieve stable-protocol admissible consensus if for any given φ(t) (t ∈ [−τ , 0]), it is regular and impulse-free, and there exists a vector-valued function c(t) such that lim t→∞ (x(t) − 1 ⊗ c(t)) = 0 and limt→∞ z(t) = 0, where c(t) is called a consensus function. It should be pointed out that if the states of consensus protocols do not tend to zero as time tends to infinity, then the control input of each agent may not tend to zero when consensus is achieved. In this case, a continuous energy expenditure is required. Since the resource of each agent is usually limited, it is expected that limt→∞ z(t) = 0. We give criteria for multi-agent system (6.10) to achieve stable-protocol and admissible consensus and will present explicit expressions of the consensus function c(t) of multi-agent system (6.10). Moreover, the problem will be discussed that under what conditions the consensus function of multi-agent system (6.10) is uniquely determined by x(0) and z(0) and is independent of topology variances and timevarying delays.

180

6 Admissible Consensus and Consensualization with Protocol State …

Remark 6.1 The model of multi-agent system (6.10) possesses generality since each agent of multi-agent system (6.10) is described by a high-order LTI singular system, the consensus protocol is constructed by using the output information and multiple time-varying delays and topology variances are considered. Especially, the models of singular multi-agent systems with fixed topologies and/or a single constant delay discussed in [1, 2] are special cases of multi-agent system (6.10). Moreover, when E 0 = Id and E 1 = Ih , multi-agent system (6.10) becomes a normal one. It is not difficult to find that normal LTI multi-agent systems with linear consensus protocols are also special cases of multi-agent system (6.10).

6.2.2 Stable-Protocol Admissible Consensus Analysis Conditions This section presents necessary and sufficient conditions for stable-protocol admissible consensus, respectively, and the consensus problem is transformed into asymptotic stability problems. ] [ √ ] [ H H Let U = 1/ N , U˜ ∈ R N ×N be a nonsingular matrix with U −1 = υ H , U . Due to L k (t)1 = 0, it can be shown that U

−1

[

0 υ L k (t)U˜ L k (t)U = 0 U L k (t)U˜

] (i = 1, 2, . . . , N ).

(6.11)

Let ( −1 ) [ ]H U ⊗ Id x(t) = xwH (t), xrH (t) ,

(6.12)

( −1 ) [ H ]H U ⊗ Ih z(t) = z w (t), zrH (t) ,

(6.13)

then multi-agent system (6.10) can be converted into ⎧ ) r ( ∑ ⎪ ˜ ⎪ E υ L z ˙ (t) = A z (t) − (t) U ⊗ B C xr (t − τk (t)) z w k z ⎪ 1 w ⎪ ⎪ k=1 ⎪ ( ) ⎪ r ∑ ⎪ ⎨ υ L k (t)U˜ ⊗ B y zr (t − τk (t)), − k=1 ) ⎪ r ( ∑ ⎪ ⎪ ˜ ⎪ υ L E x ˙ (t) = + B K C)x (t) − (t) U ⊗ B K C xr (t − τk (t)) (A 0 w k y ⎪ 0 w ⎪ ⎪ k=1 ⎪ ⎩ + B K z z w (t), (6.14)

6.2 Stable-Protocol Admissible Consensus with Time Delays

181

⎧ ) r ( ∑ ⎪ ˜ ⎪ ⊗ E (t) = ⊗ A (t) − U L (t) U ⊗ B C xr (t − τk (t)) z (I (I )˙ )z 1 r N −1 z r k y ⎪ N −1 ⎪ ⎪ k=1 ⎪ ( ) ⎪ r ∑ ⎪ ⎨ − U L (t)U˜ ⊗ B z (t − τ (t)), k

z

r

k

k=1 ) ⎪ r ( ∑ ⎪ ⎪ ˜ ⎪ ⊗ E (t) = ⊗ + B K C))x (t) − U L (t) U ⊗ B K C x ˙ (I (I (A ) 0 r N −1 0 r k y ⎪ N −1 ⎪ ⎪ k=1 ⎪ ⎩ · xr (t − τk (t)) + (I N −1 ⊗ B K z )zr (t). (6.15)

The following theorem gives a necessary and sufficient condition for multi-agent system (6.10) to achieve stable-protocol admissible consensus. Theorem 6.1 Multi-agent system (6.10) achieves stable-protocol admissible consensus if and only if the system (E 1 , A z ) and system (6.15) are admissible. Proof Because nonsingular transformations cannot change the regularity and impulse-free property, multi-agent system (6.10) is regular and impulse-free if and only if systems (6.14) and (6.15) are regular and impulse-free. Similar to the proof of Lemma 2 in [9], it can be shown that system (6.14) is regular and impulse-free if and only if both (E 0 , A + B K 0 C) and (E 1 , A z ) are regular and impulse-free. According to the property of Kronecker product, it is necessary for the regularity and impulsefree property of system (6.15) that (E 0 , A + B K 0 C) and (E 1 , A z ) are regular and impulse-free. Hence, multi-agent system (6.10) is regular and impulse-free if and only if the system (E 1 , A z ) and system (6.15) are regular and impulse-free. In the sequel, we prove that it is necessary and sufficient for multi-agent system (6.10) to achieve stable-protocol consensus that the system (E 1 , A z ) and system (6.15) are asymptotically stable. Let ]H [ x˜w (t) = (U ⊗ Id ) xwH (t), 0 ,

(6.16)

]H [ H z˜ w (t) = (U ⊗ Ih ) z w (t), 0 .

(6.17)

Due to [

xwH (t), 0

]H

= e1 ⊗ xw (t),

where e1 is an N -dimensional column vector with 1 as its first element and 0 elsewhere, one has 1 x˜w (t) = U e1 ⊗ xw (t) = √ 1 ⊗ xw (t). N

(6.18)

]H [ x˜r (t) = (U ⊗ Id ) 0, xrH (t) ,

(6.19)

Let

182

6 Admissible Consensus and Consensualization with Protocol State …

]H [ z˜r (t) = (U ⊗ Ih ) 0, zrH (t) ,

(6.20)

then one can obtain by (6.12) and (6.13) that x(t) = x˜w (t) + x˜r (t),

(6.21)

z(t) = z˜ w (t) + z˜r (t).

(6.22)

Because U ⊗ Id is nonsingular, from (6.16) and (6.19), x˜w (t) and x˜r (t) are linearly independent. Similarly, z˜ w (t) and z˜r (t) are linearly independent by (6.17) and (6.20). Necessity: Assume that multi-agent system (6.10) achieves stable-protocol admissible consensus but the system (E 1 , A z ) is not asymptotically stable, then the limit of z w (t) as t → ∞ is nonzero or does not exist for some z w (0). Since z˜ w (t) and z˜r (t) are linearly independent, the limit of z(t) as t → ∞ is nonzero or does not exist by (6.17) and (6.22), which contradicts with limt→∞ z(t) = 0. If system (6.15) is not asymptotically stable, then the limits of zr (t) and/or xr (t) as t → ∞ are nonzero or [ ]H do not exist for some xrH (0), zrH (0) . In this case, if the limit of zr (t) as t → ∞ is nonzero or does not exist, then the limit of z(t) as t → ∞ is nonzero or does not exist since z˜ w (t) and z˜r (t) are linearly independent. A contradiction is obtained. If the limit of xr (t) as t → ∞ is nonzero or does not exist, then since x˜w (t) and x˜r (t) are linearly independent, there does not exist a vector-valued function c(t) such that limt→∞ (x(t) − 1 ⊗ c(t)) = 0 by (6.16), (6.18) and (6.21). Thus, necessity is proven. Sufficiency: If system (6.15) is asymptotically stable, then one has lim xr (t) = 0,

t→∞

lim zr (t) = 0.

t→∞

In this case, if (E 1 , A z ) is asymptotically stable, then one can obtain that lim z w (t) = 0.

t→∞

(6.23)

From (6.19) and (6.20), it can be shown that lim x˜r (t) = 0,

(6.24)

lim z˜r (t) = 0.

(6.25)

t→∞

t→∞

By (6.17), (6.22), (6.23), and (6.25), one has

6.2 Stable-Protocol Admissible Consensus with Time Delays

lim z(t) = 0.

t→∞

183

(6.26)

From (6.18), (6.21), and (6.24), it can be shown that ( ) 1 lim x(t) − √ 1 ⊗ xw (t) = 0. t→∞ N

(6.27)

From (6.26) and (6.27), multi-agent system (6.10) achieves stable-protocol consensus. By the above analysis, it is necessary and sufficient for multi-agent system (6.10) to achieve stable-protocol admissible consensus that the system (E 1 , A z ) and system □ (6.15) are admissible. Remark 6.2 As shown in Sect. 3.2, the motions of multi-agent systems consist of the relative motions among agents and the whole motion, which describe the microcosmic and macroscopic behaviors of multi-agent systems, respectively. From the proof of Theorem 6.1, xr (t) and zr (t) describe the relative motions among agents of multi-agent system (6.10), and xw (t) and z w (t) determine the whole motion of multiagent system (6.10). It is clear that multi-agent system (6.10) achieves admissible consensus if and only if the relative motions among agents are admissible. When K z ≡ 0, protocol (6.9) becomes a static output feedback protocol. In this case, (E 1 , A z ) does not impact consensus properties of the multi-agent system, so one can set that E 1 = A z = Bz = 0 and B y = 0. Thus, the following corollary can be obtained directly, which gives a necessary and sufficient condition for admissible consensus. Corollary 6.1 Multi-agent system (6.10) with K z ≡ 0 achieves admissible consensus if and only if the system (I N −1 ⊗ E 0 )x˙r (t) = (I N −1 ⊗ (A + B K 0 C))xr (t) r ( ) ∑ U L k (t)U˜ ⊗ B K y C xr (t − τk (t)) − k=1

is admissible. For the case that the interaction topology is fixed and the time delay does not exist; that is, τ1 (t) = τ2 (t) = · · · = τr (t) = 0. Let λ1 = 0, λ2 , . . . , λ N denote the eigenvalues of the Laplacian matrix L. By Lemmas in Sect. 3.2, there exists an invertible matrix U such that U −1 LU = diag{0, Jλ }, where Jλ consists of Jordan blocks of L associated with λ2 , λ3 , . . . , λ N . Thus, the following three corollaries can be obtained. Corollary 6.2 Multi-agent system (6.10) with a fixed topology but without time delays achieves stable-protocol admissible consensus if and only if (E 1 , A z ) and

184

6 Admissible Consensus and Consensualization with Protocol State …

([

] [ ]) −λi B y C E1 0 A z − λi Bz , (i = 2, 3, . . . , N ) A + B K 0 C − λi B y C 0 E0 B Kz

are admissible. Corollary 6.3 Multi-agent system (6.10) with a fixed topology, K z ≡ ) achieves admissible consensus if and only if (0 but without time delays E 0 , A + B K 0 C − λi B K y C (i = 2, 3, . . . , N ) are admissible. Corollary 6.4 Multi-agent system (6.10) with a fixed topology, K z ≡ 0, C ( = I but without time ) delays achieves admissible consensus if and only if E 0 , A + B K 0 − λi B K y (i = 2, 3, . . . , N ) are admissible. Remark 6.3 Theorem 6.1 and Corollaries 6.1–6.4 transform the consensus problem of multi-agent system (6.10) into the admissible problem of multiple lower dimensional singular systems, which can be dealt with by many existing methods. For the cases with fixed topologies but without time delays, eigenvalue analysis methods and general Riccati equation methods given in [10] can be used to investigate admissible analysis and design problems, respectively. Admissible problems for singular systems with time delays can be dealt with by LMI tools. Common Lyapunov function methods were usually used to study switched singular systems, as shown in [11–13], where admissible criterions include multiple LMI constraints. Moreover, some researchers discussed consensus problems of singular multi-agent systems. One can see that the consensus results for singular multi-agent systems with fixed topologies and a single constant delay in [2] are special cases of Theorem 6.1. By Corollaries 6.2 and 6.4, consensus conclusions for state feedback cases in [1] and for static output feedback cases in Sect. 3.2 are also their special cases.

6.2.3 Consensus Functions This section gives explicit expressions of consensus functions of multi-agent system (6.10) on the basis of the First Equivalent Form, and presents sufficient conditions under which the consensus function is independent of time-varying interaction topologies and time delays. There exist nonsingular matrices T0 , T1 , Z 0 , and Z 1 such that ] [ ] Ir0 0 A˜ 0 0 , T0 E 0 Z 0 = , T0 (A + B K 0 C)Z 0 = 0 0 0 Id−r0 ] [ [ ] I 0 A˜ z 0 , T1 E 1 Z 1 = r1 , T1 A z Z 1 = 0 0 0 Ih−r1 [

which are the First Equivalent Forms of (E 0 , A + B K 0 C) and (E 1 , A z ), respectively. Dai [10] introduced the concept of the First Equivalent Form and showed that there

6.2 Stable-Protocol Admissible Consensus with Time Delays

185

must exist two nonsingular matrices such that a singular system can be transformed into the First Equivalent Form. Theorem 6.2 If multi-agent system (6.10) achieves stable-protocol admissible consensus, then the consensus function c(t) satisfies ) ( 1 lim c(t) − √ (cx0 (t) + cz0 (t) + cxr (t) + czx (t) + czz (t)) = 0, t→∞ N where [ ]T ˜ [ ] cx0 (t) = Z 0 Ir0 , 0 e A0 t Ir0 , 0 Z 0−1 (υ ⊗ Id )x(0), [

cz0 (t) = Z 0 Ir0 , 0 e

[ A˜ z s

]T

∫t

]

[ [ ] ]T ˜ e A0 (t−s) Ir0 , 0 Z 0−1 B K z Z 1 Ir1 , 0

0

Ir1 , 0 Z 1−1 (υ ⊗ Id )z(0)ds, [

cxr (t) = −Z 0 Ir0 , 0

r ]T ∑

∫t

[ ] ˜ e A0 (t−s) Ir0 , 0

k=1 0

Z 0−1 [

czx (t) = −Z 0 Ir0 , 0 [

× Ir1 , 0

]

[

czz (t) = −Z 0 Ir0 , 0 [

× Ir1 , 0

]

( ) υ L k (s)U˜ ⊗ B K y C xr (s − τk (s))ds,

r ]T ∑

∫ t ∫s

k=1 0

Z 1−1

0

( ) υ L k (θ )U˜ ⊗ B y C xr (θ − τk (θ ))dθ ds,

r ]T ∑

∫ t ∫s

k=1 0

Z 1−1

[ [ ] ]T ˜ ˜ e A0 (t−s) Ir0 , 0 Z 0−1 B K z Z 1 Ir1 , 0 e Az (s−θ)

(

[ [ ] ]T ˜ ˜ e A0 (t−s) Ir0 , 0 Z 0−1 B K z Z 1 Ir1 , 0 e Az (s−θ)

0

) υ L k (θ )U˜ ⊗ Bz zr (θ − τk (θ ))dθ ds.

Proof From (6.12) and (6.13), one can see that ( ) xw (0) = [Id , 0, . . . , 0] U −1 ⊗ Id x(0), ) ( z w (0) = [Ih , 0, . . . , 0] U −1 ⊗ Ih z(0). Due to [I, 0, . . . , 0] = e1T ⊗ I , by the property of Kronecker product, it can be shown that

186

6 Admissible Consensus and Consensualization with Protocol State …

) ( xw (0) = e1T U −1 ⊗ Id x(0),

(6.28)

( ) z w (0) = e1T U −1 ⊗ Ih z(0).

(6.29)

z w (t) = z w0 (t) + z wx (t) + z wz (t),

(6.30)

From (6.14), one has

where [ ]T ˜ [ ] z w0 (t) = Z 1 Ir1 , 0 e Az t Ir1 , 0 Z 1−1 z w (0), r ∫t ( ) [ [ ]T ∑ ] ˜ e Az (t−s) Ir1 , 0 Z 1−1 υ L k (s)U˜ ⊗ B y C xr (s − τk (s))ds, z wx (t) = −Z 1 Ir1 , 0 k=1 0

[

z wz (t) = −Z 1 Ir1 , 0

r ]T ∑

∫t

( ) [ ] ˜ e Az (t−s) Ir1 , 0 Z 1−1 υ L k (s)U˜ ⊗ Bz zr (s − τk (s))ds.

k=1 0

Moreover, one can see by (6.14) that xw (t) = xw0 (t) + xwz (t) + xwx (t),

(6.31)

where ]T ˜ [ ] [ xw0 (t) = Z 0 Ir0 , 0 e A0 t Ir0 , 0 Z 0−1 xw (0), ∫t [ [ ]T ] ˜ xwz (t) = Z 0 Ir0 , 0 e A0 (t−s) Ir0 , 0 Z 0−1 B K z z w (s)ds, 0

[

xwx (t) = −Z 0 Ir0 , 0

r ]T ∑

∫t

[ ] ˜ e A0 (t−s) Ir0 , 0

k=1 0

( ) Z 0−1 υ L k (s)U˜ ⊗ B K y C xr (s − τk (s))ds. If multi-agent system (6.10) achieves stable-protocol admissible consensus, then one can obtain by (6.27) that ( ) 1 lim c(t) − √ xw (t) = 0. t→∞ N

(6.32)

From (6.28) to (6.32), by eT1 U −1 = υ, the conclusion of Theorem 6.2 can be obtained. □

6.2 Stable-Protocol Admissible Consensus with Time Delays

187

From Theorem 6.2, the consensus function c(t) is not unique, but as the time tends to infinity, √c(t) converges to the function (cx0 (t) + cz0 (t) + cxr (t) + czx (t) + czz (t))/ N , where cx0 (t) and cz0 (t) determine the influences of the initial states of agents and protocols on the consensus function c(t), respectively, and cxr (t), czx (t) and czz (t) describe the impacts of time delays and topology variances on the consensus function c(t), respectively. Furthermore, cxr (t), czx (t), and czz (t) can also be regarded as the impacts of relative motions among agents on the whole motion of multi-agent system (6.10). Since topology variances and multiple time delays are considered, the expression of the consensus function given in Theorem 6.2 seems complex. However, when the interaction topology is fixed and the time delay is single, the expression of the consensus function can be simplified. Due to U −1 LU = diag{0, Jλ }, one has υ L U˜ = 0. In this case, the following corollary can be obtained directly, which shows that the single delay does not influence the consensus function. Corollary 6.5 If multi-agent system (6.10) with a fixed topology and a single timevarying delay achieves stable-protocol admissible consensus, then the consensus function c(t) satisfies ) ( 1 lim c(t) − √ (cx0 (t) + cz0 (t)) = 0, t→∞ N where [ ]T ˜ [ ] cx0 (t) = Z 0 Ir0 , 0 e A0 t Ir0 , 0 Z 0−1 (υ ⊗ Id )x(0), ∫t ]T ] ]T [ [ [ ˜ e A0 (t−s) Ir0 , 0 Z 0−1 B K z Z 1 Ir1 , 0 cz0 (t) = Z 0 Ir0 , 0 e

[ A˜ z s

]

0

Ir1 , 0 Z 1−1 (υ ⊗ Id )z(0)ds.

The following theorem gives sufficient conditions under which multi-agent systems with the same initial states but different interaction topologies and different time delays have an identical consensus function. Theorem 6.3 If multi-agent system (6.10) achieves stable-protocol admissible consensus and interaction topologies associated with L k (t) (k = 1, 2, . . . , r ) are balanced, then the consensus function c(t) satisfies lim (c(t) − (cx0 (t) + cz0 (t))) = 0,

t→∞

where [

]T

cx0 (t) = Z 0 Ir0 , 0 e

[ A˜ 0 t

Ir0 , 0

]

( Z 0−1

) N 1 ∑ xi (0) , N i=1

188

6 Admissible Consensus and Consensualization with Protocol State …

[

cz0 (t) = Z 0 Ir0 , 0

]T

∫t

[ [ ] ]T ˜ e A0 (t−s) Ir0 , 0 Z 0−1 B K z Z 1 Ir1 , 0

0

e

[ A˜ z s

Ir1 , 0

]

Z 1−1 ds

(

) N 1 ∑ z i (0) . N i=1

Proof Because interaction topologies associated with L k (t) are balanced, it can be shown that 1T L k (t) = 0 (k = 1, 2, . . . , r ). √ According to the structure of U , one has υ = 1T / N , which means that υ L k (t) = 0 (k = 1, 2, . . . , r ). Hence, one can obtain that cxr (t) = czx (t) = czz (t) = 0. Moreover, it can be shown that N 1 ∑ xi (0), (υ ⊗ Id )x(0) = √ N i=1

(6.33)

N 1 ∑ z i (0). (υ ⊗ Ih )z(0) = √ N i=1

(6.34)

By (6.33) and (6.34), the conclusion of Theorem 6.3 can be obtained by □ Theorem 6.2. Remark 6.4 Olfati-Saber and Murray [7] first proposed the χ -consensus problem to determine the whole motions of first-order multi-agent systems; that is, consensus functions. Xiao et al. [14] presented an approach to give explicit expressions of consensus functions of high-order multi-agent systems, where it was supposed that consensus functions are time-invariant. In [7, 14], each agent was modeled by a normal system, and their methods are no longer valid for singular cases. In [2], it was shown that the single constant delay does not influence consensus functions of singular multi-agent systems with fixed topologies, but its approach cannot be applied to deal with the cases where topology variances and multiple time-varying delays are considered. Theorems 6.2 and 6.3 determine the impacts of topology variances and time delays on consensus functions for singular multi-agent systems with time-varying topologies and multiple time-varying delays. Remark 6.5 The consensus functions in Theorem 6.3 are associated with the average of the initial states of all agents but are independent of topology changes and timevarying delays. It should be pointed out that it is a general assumption that the gain matrices of consensus protocols are time-invariant as shown in [15–17] and the

6.2 Stable-Protocol Admissible Consensus with Time Delays

189

conclusion in Theorem 6.3 does not hold if the gain matrices of consensus protocols are time-varying.

6.2.4 Numerical Simulation Consider a singular multi-agent system with nine agents, where the dynamics of each agent is described by (6.8) with ⎡

1 ⎢0 E =⎢ ⎣0 0

0 1 0 0

0 0 0 0

⎡ ⎤ 0 0 1 0 ⎢ 0 0 0 0⎥ ⎥, A = ⎢ ⎣ −1 1 −1 0⎦ 0 0 −2 0

⎡ ⎤ ⎤ ⎡ ⎤T 1 0 1 ⎢1⎥ ⎢0⎥ 0 ⎥ ⎥, B = ⎢ ⎥, C = ⎢ ⎥ . ⎣0⎦ ⎣0⎦ 0 ⎦ −1 0 0

The interaction topology set S consists of two balanced directed topologies G 1 with τ1 = 0.05 and G 2 with τ2 = 0.1 as shown in Fig. 6.1, where it is assumed that the adjacency matrix of each graph is a 0–1 matrix. The interaction topologies of the singular multi-agent system are randomly chosen from S with the dwell time 1. The initial states are given as follows x1 (0) = [9, 8, −1, −16]T , x2 (0) = [6, 4, −2, −8]T , x3 (0) = [3, 2, −1, −4]T , x4 (0) = [−6, 15, 21, −30]T , x5 (0) = [−9, −8, 1, 16]T , x6 (0) = [2, 4, 2, −8]T , x7 (0) = [−1, 5, 6, −10]T , x8 (0) = [1, 4, 3, −8]T , x9 (0) = [−1 , −3, −2, 6]T .

Fig. 6.1 Interaction topology set S

2

1

3

5

2

1

3

5

4

9

8

6

7

9

8

6

7

G1

4

G2

190

6 Admissible Consensus and Consensualization with Protocol State …

Let E 1 = E 0 , K 0 = 0, and K y = 0, then by common Lyapunov function methods and LMI techniques, one can obtain that K z = [10.1103, −14.5558, −2.3115, 9.4432], B y = [−0.8843, 0.3610, 0.5450, −0.0881]T , ⎡

0 ⎢ 10.1103 Az = ⎢ ⎣ −1.0000 0

⎤ ⎡ 1.0000 0 1.0000 0.8843 0 ⎢ −0.3610 0 −14.5558 −2.3115 9.4432 ⎥ ⎥, Bz = ⎢ ⎦ ⎣ −0.5450 0 1.0000 −1.0000 0 −2.0000 0 −1.0000 0.0881 0

0 0 0 0

⎤ 0 0⎥ ⎥, 0⎦ 0

which can guarantee that consensus conditions in Theorem 6.1 hold. The state trajectories of the singular multi-agent system with the switching signal given in Fig. 6.3 are shown in Fig. 6.2, and the trajectories marked by circles denote the curves of the consensus function obtained by Theorem 6.3. In Fig. 6.4, the trajectories of z iT (t)z i (t) (i = 1, 2, . . . , 9) are given. One can find that lim t→∞ z iT (t)z i (t) = 0 (i = 1, 2, . . . , 9); that is, limt→∞ z(t) = 0. It is clear that the singular multiagent system achieves stable-protocol admissible consensus, and it demonstrates the conclusion of Theorem 6.3 that the state trajectories converge to the ones marked by circles. Moreover, by repeating this simulation with different switching signals, it is found that state trajectories may be different, but they always converge to the same curves. This means that switching topologies and time delays do not impact the consensus function if interaction topologies are balanced.

6.3 Stable-Protocol Admissible Consensualization with Switching Topologies In this section, stable-protocol admissible consensus analysis and design problems of high-order LTI singular multi-agent systems with dynamic output feedback consensus protocols, where both connected switching topology cases and jointly connected switching topology cases are discussed. Based on outputs and protocol states among neighboring agents, a singular observer-type consensus protocol is applied, whose structures can guarantee the stability of the consensus protocol and can simplify consensus problems. Furthermore, based on the orthonormal projection between the CS and the CCS, a necessary and sufficient condition for stableprotocol admissible consensualization is proposed and an explicit expression of the consensus function which determines the consensus behavior of all agents in a complex network is presented. Moreover, based on the generalized Riccati equation and Barbalat’s lemma, stable-protocol admissible consensus conditions for

6.3 Stable-Protocol Admissible Consensualization with Switching Topologies agent 1 agent 3 agent 5 agent 7 agent 9

191

agent 2 agent 4 agent 6 agent 8

agent 1 agent 3 agent 5 agent 7 agent 9

agent 2 agent 4 agent 6 agent 8

agent 1 agent 3 agent 5 agent 7 agent 9

agent 2 agent 4 agent 6 agent 8

t/s

t/s

agent 1 agent 3 agent 5 agent 7 agent 9

agent 2 agent 4 agent 6 agent 8

t/s

t/s

Fig. 6.2 State trajectories Fig. 6.3 Switching signal

t/s

192

6 Admissible Consensus and Consensualization with Protocol State …

Fig. 6.4 Trajectories of z iT (t)z i (t)

t/s

connected switching topology cases and jointly connected switching topology cases are given, respectively.

6.3.1 Problem Description and Transformation Consider the following homogeneous singular multi-agent system ⎧

E x˙i (t) = Axi (t) + Bu i (t), yi (t) = C xi (t),

(6.35)

where i = 1, 2, . . . , N , A ∈ Rd×d , B ∈ Rd× p , C ∈ Rq×d , E ∈ Rd×d with rank(E) = r ≤ d, xi (t), u i (t) and yi (t) are the state, control input, and measured output of agent i, respectively. It is assumed that Assumption 6.2 The pair (E, A) is regular and impulse-free. Note that u i (t) (i = 1, 2, . . . , N ) can only use information from neighboring agents of agent i. An undirected graph G = (V (G), E(G)) can be used to illustrate the neighboring relationship among agents, i.e., the interaction topology, where V (G) = {1, 2, . . . , N } is composed of all agents, ( j, i ) ∈ E(G) means that agent j is one of the neighbors of agent i, and wi j denotes the communication strength of the edge ( j, i ). The following distributed consensus protocol is applied:

6.3 Stable-Protocol Admissible Consensualization with Switching Topologies

( ) ⎧ ∑ E υ˙ i (t) = (A + B K 1 )υi (t) − K 2 C wi j υ j (t) − υi (t) ⎪ ⎪ ⎪ j∈Ni (t) ⎨ ( ) ∑ + K2 wi j y j (t) − yi (t) , ⎪ j∈Ni (t) ⎪ ⎪ ⎩ u i (t) = K 1 υi (t),

193

(6.36)

where υi (t) is the protocol state, K 1 and K 2 are gain matrices with compatible dimensions and Ni (t) represents the time-varying neighbor set of agent i (i ∈ {1, 2, . . . , N }). We focus on switching interaction topology cases; that is, the neighbor set of agent i (i ∈ {1, 2, . . . , N }) may be time-varying, the communication strength between agents i and j is wi j when agent i can receive the information of agent j and the communication strength between agents i and j is 0 when there is no communication between them. Let the finite set l with an index set ψ ⊂ N contain all possible interaction topologies for switching, where N denotes the natural number set and σ (t) : [ 0, ∞) → ψ denote the switching signal, whose value at time t is the index of the interaction topology at time t. It is clear that if the switching signal σ (t) is identically equal to a constant, then the neighbor set is time-invariant, which means that the interaction topology is fixed. Here, G σ (t) denotes the switching interaction topology. G σ (t) is said to be connected if each topology in l is connected and jointly connected if some topologies in l may not be connected but the union of interaction topologies across a series of time intervals is connected. Moreover, it is supposed that Assumption 6.3 The switching sequences 0 < t1 < · · · < tk < · · · satisfy inf k (tk+1 − tk ) = Td > 0. [ ]T Let ηi (t) = xiT (t), υiT (t) (i = 1, 2, . . . , N ), then it can be obtained by (6.35) and (6.36) that [

] ] ∑ [ [ ] ) ( E 0 A B K1 0 0 ηi (t) − wi j η j (t) − ηi (t) . η˙ i (t) = 0 A + B K1 0 E −K 2 C K 2 C j∈Ni (t)

(6.37) [ ]T Let η(t) = η1T (t), η2T (t), . . . , ηTN (t) and L σ (t) denote the Laplacian matrix of the interaction topology G σ (t) , then from (6.37), the dynamics of multi-agent system (6.35) with protocol (6.36) can be written in a compact form as (

[

E 0 IN ⊗ 0 E

])

[

( η(t) ˙ =

IN ⊗

] ]) [ A B K1 0 0 − L σ (t) ⊗ η(t). 0 A + B K1 K 2 C −K 2 C (6.38)

Now, the definitions of the stable-protocol admissible consensus and consensualization are given, respectively, as follows.

194

6 Admissible Consensus and Consensualization with Protocol State …

Definition 6.8 Multi-agent system (6.38) is said to achieve stable-protocol admissible consensus if for any admissible bounded initial state η(0), it is regular and impulse-free and there exists a vector-valued function s(t) dependent on η(0) such that limt→∞ (xi (t) − s(t)) = 0 and limt→∞ υi (t) = 0 (i = 1, 2, . . . , N ), where s(t) is called a consensus function. Definition 6.9 Multi-agent system (6.35) is said to be stable-protocol admissibly consensus by protocol (6.36) if there exist K 1 and K 2 such that it achieves stableprotocol admissible consensus. Remark 6.6 When outputs instead of states are available to neighboring agents, two types of consensus protocols were constructed in the literature; that is, static output feedback ones and dynamic output feedback ones. In [18], static output feedback consensus protocols were applied and quadratic matrix inequality (QMI) criteria for consensus were proposed, where an iterative algorithm was given to determine the gain matrices of consensus protocols. In [4], dynamic output feedback consensus protocols were proposed and LMI criteria for consensus were presented, where changing variable methods which may bring in some conservative were used. It should be pointed out that both QMI and LMI approaches are not analytic; that is, the existence of the solutions of consensus criteria cannot be guaranteed. To obtain criteria, we introduce the protocol state error term ( ) ∑ analytic consensus −K 2 C j∈N (t) wi j υ j (t) − υi (t) in protocol (6.36). i

The following consensus problems are investigated: (i) under what conditions multi-agent system (6.35) achieves stable-protocol admissible consensus; (ii) how to determine the influences of the agent initial states, the protocol initial states, and switching topologies on the consensus function when multi-agent system (6.38) achieves stable-protocol admissible consensus; (iii) under what conditions multiagent system (6.35) can be stable-protocol admissibly consensus by protocol (2) for connected switching topologies and jointly connected switching topologies, respectively. Firstly, two subspaces of the Nd-dimensional real space are introduced and multi-agent system (6.38) is decomposed into two independent parts, which can be used to describe the consensus and non-consensus dynamics. Then, the structures of the orthonormal projection operators between the two subspaces are given and consensus problems are converted into admissible ones. Since the interaction topology G σ (t) is undirected, L σ (t) is symmetric. According continuous orthonormal to Assumption 6.3 and Lemma 2.2, there exists a piecewise √ matrix Uσ (t) = [u 1 , u 2 (t), . . . , u N (t)] with u 1 = 1/ N such that } { UσT(t) L σ (t) Uσ (t) = Ʌσ (t) = diag λσ (t),1 , λσ (t),2 , . . . , λσ (t),N where 0 = λσ (t),1 ≤ λσ (t),2 ≤ · · · ≤ λσ (t),N denote the eigenvalues of L σ (t) . Let sl ∈ Rd (l = 1, 2, . . . , d) be orthonormal column vectors. According to Lemma 2.3, the two subspaces of R N d are defined as follows.

6.3 Stable-Protocol Admissible Consensualization with Switching Topologies

195

Definition 6.10 Let h j = u 1 ⊗ s j ( j = 1, 2, . . . , d) and h j (t) = u i (t) ⊗ sl ( j = (i − 1)d + l; i = 2, 3, . . . , N ; l = 1, 2, . . . , d). A CS is defined as the subspace C spanned by h 1 , h 2 , . . . , h d and a CCS as the subspace C spanned by h d+1 (t), h d+2 (t), . . . , h N d (t). Note that since u 1 and si (i = 1, 2, . . . , d) are time-invariant, h i (i = 1, 2, . . . , d) are time-invariant. Hence, the subspace C is time-invariant. Moreover, because sl (l = 1, 2, . . . , d) are orthonormal and u 1 , u 2 (t), . . . , u N (t) are orthonormal, it can be shown that h 1 , h 2 , . . . , h N , h N +1 (t), . . . , h N d (t) are orthonormal. Therefore, C and C are orthonormal. Since the orthonormal subspace of the subspace C is unique and time-invariant, C is unique and time-invariant although h d+1 (t), h d+2 (t), . . . , h N d (t) may be time-varying. √ Furthermore, due to u 1 = 1/ N , any vector in C has the form 1 ⊗ s˜ , where s˜ [ ]T is a d-dimensional column vector. Let x(t) = x1T (t), x2T (t), . . . , x NT (t) , then it is not difficult to find that if x(t) converges to the subspace C, then all agents achieve consensus. ( ) [ ]T T T T T Let UσT(t) ⊗ I2d η(t) = xsT (t), υsT (t), xs,2 (t), υs,2 (t), . . . , xs,N (t), υs,N (t) with xs (t), υs (t), xs,i (t), υs,i (t) ∈ Rd (i = 2, 3, . . . , N ). Since UσT(t) ⊗ I2d is piecewise continuous, multi-agent system (6.38) can be transformed into ⎧

E x˙s (t) = Axs (t) + B K 1 υs (t), (6.39) E υ˙ s (t) = ( A + B K 1 )υs (t), [ ] [ ] ([ ])[ ] ][ E 0 0 0 A B K1 xs,i (t) x˙s,i (t) − λσ (t),i = , 0 A + B K1 υs,i (t) 0 E υ˙ s,i (t) K 2 C −K 2 C (6.40) where i = 2, 3, . . . , N . Let [

] [ ] ][ x˜s,i (t) I 0 xs,i (t) = , υ˜ s,i (t) υs,i (t) I −I

then system (6.39) can be converted into ⎧

E x˙˜s,i (t) = (A ( + B K 1 )x˜s,i (t)) − B K 1 υ˜ s,i (t), E υ˙˜ s,i (t) = A + λσ (t),i K 2 C υ˜ s,i (t),

(6.41)

where i = 2, 3, . . . , N . Actually, systems (6.39) and (6.41) can be applied to describe the consensus and non-consensus dynamics of multi-agent system (6.38), respec[ ]−1  [h 1 , h 2 , . . . , h d , 0] h 1 , . . . , h d , h d+1 (t), . . . , h N d (t) tively. Let PC,C denote an orthonormal projection operator onto C along C and PC,C  [ ] 0, h d+1 (t), h d+2 (t), . . . , h Nd (t) [h 1 , . . . , h d , h d+1 (t), . . . , h N d (t)]−1 represent an

196

6 Admissible Consensus and Consensualization with Protocol State …

orthonormal projection operator onto C along C. The following lemma shows that PC,C and PC,C are time-invariant and gives their structures. Lemma 6.1 The orthonormal projection operators PC,C and PC,C satisfy that PC,C = PC,C

1 1 N ×N ⊗ Id , N

] [ 1 N −1 −1TN −1 ⊗ Id . = N −1 N −1 N I − 1 N −1 1TN −1

Proof Let S = [s1 , s2 , . . . , s N ], then one can obtain by Definition 6.10 that 1 [h 1 , h 2 , . . . , h d , 0] = √ [1, 0] ⊗ S, N ]−1 ( )−1 [ = Uσ (t) ⊗ S = UσT(t) ⊗ S T . h 1 , . . . , h d , h d+1 (t), . . . , h N d (t)

(6.42) (6.43)

By (6.42) and (6.43), one can obtain that 1 1 PC,C = √ [1, 0]UσT(t) ⊗ Id = 1 N ×N ⊗ Id . N N

(6.44)

]T [ T Furthermore, let U˜ σ (t) = [u 2 (t), u 3 (t), . . . , u N (t)] = κσT(t) , U σ (t) with κσ (t) ∈ R N −1 and U σ (t) ∈ R(N −1)×(N −1) , then it can be shown that [

] h d+1 (t), h d+2 (t), . . . , h N d (t) = U˜ σ (t) ⊗ S.

(6.45)

Thus, one can obtain by (6.43) and (6.45) that ( )( ) PC,C = [0, U˜ σ (t) ] ⊗ S UσT(t) ⊗ S T = [0, U˜ σ (t) ]UσT(t) ⊗ Id .

(6.46)

Due to Uσ (t) UσT(t)

1 = N

[

1 + N κσ (t) κσT(t)

T

1TN −1 + N κσ (t) U σ (t)

T

1 N −1 + NU σ (t) κσT(t) 1 N −1 1TN −1 + NU σ (t) U σ (t)

one has 1 + N κσ (t) κσT(t) = N , T T 1 N −1 + N κσ (t) U σ (t) = 0, T 1 N −1 1TN −1 + NU σ (t) U σ (t) = N I.

] = I,

(6.47)

6.3 Stable-Protocol Admissible Consensualization with Switching Topologies

197

Thus, one can obtain that [ ] ] [ T [ ] T 1 N −1 κ κ κ U −1TN −1 σ (t) σ (t) T σ (t) σ (t) . 0, U˜ σ (t) Uσ (t) = = T N −1 N −1 N I − 1 N −1 1TN −1 U σ (t) κσT(t) U σ (t) U σ (t) (6.48) By (6.45) and (6.48), it can be shown that PC,C =

] [ 1 N −1 −1TN −1 ⊗ Id . N −1 N −1 N I − 1 N −1 1TN −1

The proof of Lemma 6.1 is completed.

□

In the sequel, a necessary and sufficient condition for stable-protocol admissible consensus of multi-agent system (6.38) is shown. Theorem 6.4 Multi-agent system (6.38) achieves stable-protocol admissible) ( consensus if and only if the pairs (E, A + B K 1 ) and E, A + λσ (t),i K 2 C (i = 2, 3, . . . , N ) are admissible. Proof Due to x(t) = (I N ⊗ [Id , 0])η(t), one can show by Lemma 6.1 that 1 (1 N ×N ⊗ Id )(I N ⊗ [Id , 0])η(t) N 1 = √ ([1, 0] ⊗ [Id , 0])η(t) ˜ N 1 = √ 1 ⊗ xs (t). N

PC,C x(t) =

(6.49)

[ ]T Let υ(t) = υ1T (t), υ2T (t), . . . , υ NT (t) . Since υ(t) = (I N ⊗ [0, Id ])η(t), by a similar analysis, one can obtain that 1 PC,C υ(t) = √ 1 ⊗ υs (t). N

(6.50)

Moreover, it can be shown from Lemma 6.1 that ([ ] ) 1 N −1 −1TN −1 ⊗ Id (I N ⊗ [Id , 0])η(t) PC,C x(t) = −1 N −1 N I − 1 N −1 1TN −1 N ] ([ ) = 0, U˜ σ (t) ⊗ [Id , 0] η(t) ˜ ( )[ ]T T T T = U˜ σ (t) ⊗ Id xs,2 (t), xs,3 (t), . . . , xs,N (t) . (6.51) Due to UσT(t) Uσ (t) = I , then one can obtain that

198

6 Admissible Consensus and Consensualization with Protocol State … T

T

κσT(t) κσ (t) + U σ (t) U σ (t) = I, which means that U˜ σT(t) U˜ σ (t) = I.

(6.52)

( )T ∑N T From (6.51) and (6.52), one has PC,C x(t) PC,C x(t) = i=2 x s,i (t)x s,i (t). Thus, one can obtain that limt→∞ PC,C x(t) = 0 if and only if limt→∞ xs,i (t) = 0 (i = 2, 3, . . . , N ). By a similar analysis, one has )[ ( ]T T T T PC,C υ(t) = U˜ σ (t) ⊗ Id υs,2 (t), υs,3 (t), . . . , υs,N (t) .

(6.53)

Hence, it can be shown that limt→∞ PC,C υ(t) = 0 if and only if limt→∞ υs,i (t) = 0 (i = 2, 3, . . . , N ). From (6.40) and (6.41), limt→∞ PC,C υ(t) = 0 and limt→∞ PC,C x(t) = 0 if and only if the pairs (E, A + B K 1 ) ) ( and E, A + λσ (t),i K 2 C (i = 2, 3, . . . , N ) are admissible. Moreover, by (6.39), limt→∞ υs (t) = 0 if and only if the pair (E, A + B K 1 ) is admissible. From (6.50) and (6.53), limt→∞ υ(t) = 0 if and only if limt→∞ υs (t) = 0 and limt→∞ υs,i (t) = 0 (i = 2, 3, . . . , N ). Furthermore, the nonsingular transformation cannot change the regularity and impulse-free property, and it is necessary and sufficient for consensus that the state projection onto C tends to zero as time tends to infinity by Lemma 2.3, admissible consensus if and so multi-agent system (6.38) achieves stable-protocol ( ) only if the pairs (E, A + B K 1 ) and E, A + λσ (t),i K 2 C (i = 2, 3, . . . , N ) are □ admissible. The proof of Theorem 6.4 is completed. Remark 6.7 From Theorem 6.4, the stable-protocol admissible consensus property of multi-agent system (6.38) is jointly determined by the dynamics of each agent (E, A), the gain matrices K 1 and K 2 , and the nonzero eigenvalues of the Laplacian matrices of the switching interaction topologies. Moreover, the projections of the agent state x(t) and the protocol state υ(t) onto the CS C are determined by xs (t) and υs (t), respectively; which means that system (6.39) describes the consensus dynamics of multi-agent system (6.38). Especially, if multi-agent system (6.38) achieves stableprotocol admissible consensus, then limt→∞ υs (t) = 0 since the pair (E, A + B K 1 ) is admissible, but the limit of xs (t) may not tend to zero as time tends to infinity since the pair (E, A) may not be admissible. Furthermore, the state projections of x(t) and υ(t) onto the CCS C are associated with xs,i (t) and υs,i (t) (i = 2, 3, . . . , N ), respectively; that is, system (6.40) describes the non-consensus dynamics of multiagent system (6.38). As shown in [10], there exist nonsingular matrices T and Z such that [ T EZ =

] [ ] ] [ B1 Ir 0 A11 A12 , TB = , , T AZ = A21 A22 B2 0 0

6.3 Stable-Protocol Admissible Consensualization with Switching Topologies

199

which is called the Second Equivalent Form. By Assumption 6.2, A22 is nonsingular. Moreover, one can see that T˜ =

[

] [ ] 0 Ir Ir −A12 A−1 22 , Z ˜ = −1 , −A−1 0 Id−r 22 A21 A22

are nonsingular. Direct computation shows that ] ] [ ] [ ] [ [ B1 B˜ 1 A11 A12 ˜ A˜ 0 ˜ ˜ , T = , T Z= A21 A22 B2 B2 0 Id−r −1 ˜ ˜ where A˜ = A11 − A12 A−1 22 A21 and B1 = B1 − A12 A22 B2 . If there exists K 1 such that A˜ + B˜ 1 K˜ 1 is Hurwitz, then one has

T˜ T E Z Z˜ = T˜ T (A + B K 1 )Z Z˜ =

[

[

] Ir 0 , 0 0

] A˜ + B˜ 1 K˜ 1 0 , B2 K˜ 1 Id−r

(6.54)

(6.55)

] K˜ 1 , 0 Z˜ −1 Z −1 . In this case, it is not difficult to find that the pair √ (E, A + B K 1 ) is admissible. Moreover, since the first row of UσT(t) is 1T / N , it can be shown that where K 1 =

[

N ) ( 1 ∑ xs (0) = [Id , 0] UσT(t) ⊗ I2d η(0) = √ xi (0), N i=1

(6.56)

N ) ( 1 ∑ υs (0) = [0, Id , 0] UσT(t) ⊗ I2d η(0) = √ υi (0). N i=1

(6.57)

From( the proof of Theorem 6.4, it can be obtained that √ ) limt→∞ s(t) − xs (t)/ N = 0; that is, system (6.39) determines the consensus function. By the above analysis, the following theorem can be obtained, which presents an approach to determine the consensus function. Theorem 6.5 If multi-agent system (6.38) achieves stable-protocol admissible consensus, then the consensus function satisfies lim (s(t) − sx0 (t) − sxυ (t)) = 0,

t→∞

where

200

6 Admissible Consensus and Consensualization with Protocol State …

( ˜ T At

sx0 (t) = Z Z˜ [Ir , 0] e [Ir , 0] Z sxυ (t) = Z Z˜ [Ir , 0]T

∫t

˜ −1

Z

−1

) N 1 ∑ xi (0) , N i=1

˜ e A(t−τ ) [Ir , 0] Z˜ −1 Z −1 B K 1 υs (τ )dτ,

0

) ( ] N ∑ 1 I ˜ B˜ 1 K˜ 1 )t r ( A+ −1 −1 e υs (t) = Z Z˜ υi (0) . [Ir , 0] Z˜ Z −B2 K˜ 1 N i=1 [

Remark 6.8 Theorem 6.5 shows that switching topologies do not influence consensus functions although they are critically important for multi-agent systems to achieve stable-protocol admissible consensus. Moreover, it can be found that if υi (0) = 0 (i = 1, 2, . . . , N ), then υs (t) ≡ 0 and sxυ (t) ≡ 0. In this case, the consensus function is independent of the consensus protocols and is associated with the initial states of all agents. Moreover, based on the Second Equivalent Form, an approach was proposed to determine consensus functions for multi-agent systems with switching topologies in [19], where it was assumed that all state information of neighboring agents is available. It should be pointed out that when dynamic output feedback consensus protocols are applied, the approach in [19] cannot determine the impacts of the protocol states υi (0) = 0 (i = 1, 2, . . . , N ) on consensus functions. Combining the First and Second Equivalent Forms, Theorem 6.5 shows a general approach to determine the consensus functions of multi-agent systems with dynamic output feedback consensus protocols.

6.3.2 Admissible Consensus Criteria for Connected Switching Topology Cases [ ] If (E, A, B) is stabilizable, then there exists K 1 = K˜ 1 , 0 Z˜ −1 Z −1 such that (E, A + B K 1 ) is admissible, where K˜ 1 is chosen such that A˜ + B˜ 1 K˜ 1 is Hurwitz. By Theorem 6.4, consensus problems of multi-agent system (6.35) with protocol (6.36) are transformed into simultaneous stabilization ones; that is, how to determine K 2 ( ) such that E, A + λσ (t),i K 2 C (i = 2, 3, . . . , N ) can be stabilized simultaneously. In this section, based on the generalized Riccati equation, stable-protocol admissible consensus criteria for multi-agent systems with connected switching topologies are proposed as follows. Theorem 6.6 Multi-agent system (6.35) with connected switching topologies is stable-protocol admissibly consensualization by protocol (6.36) if and only if the triple (E, A, [B) is ]stabilizable and the triple (E, A, C) is detectable. In this T T T case, K 1 = K˜ 1 , 0 Z˜ −1 Z −1 and K 2 = −λ−1 = R > 0, min Q C /2, where R { } λmin = min λl,2 (∀l ∈ ψ) and Q is the solution of the generalized Riccati equation

6.3 Stable-Protocol Admissible Consensualization with Switching Topologies

⎧

201

Q T AT + AQ − Q T C T C Q + R = 0, Q T E T = E Q ≥ 0.

Proof By Theorem 6.4, the necessity is obvious. We here focus on the proof of sufficiency. the pair (E, A) is regular and impulse-free, it can be shown that the pair ( T TSince ) E ,A is regular ( ) and impulse-free. Because the triple (E, A, C) is detectable, the triple E T , AT , C T is stabilizable by the duality principle. Hence, from Lemma 2.26, the following generalized Riccati equation ⎧

Q T AT + AQ − Q T C T C Q + R = 0, Q T E T = E Q ≥ 0,

(6.58)

has at least one admissible solution Q, which is unique in the sense of E Q. For the following systems ( ) E T ζ˙i (t) = AT + λσ (t),i C T K 2T ζi (t) (i = 2, 3, . . . , N ),

(6.59)

consider the following Lyapunov function candidates Vi (t) = ζiT (t)Q T E T ζi (t) (i = 2, 3, . . . , N ), where Q T E T = E Q ≥ 0 and Q is a solution of generalized Riccati Eq. (6.58). Taking the derivative of Vi (t) with respect to t, one can obtain that ( ) V˙i (t) = ζiT (t) Q T AT + AQ + λσ (t),i Q T C T K 2T + λσ (t),i K 2 C Q ζi (t).

(6.60)

For the cases that switching topologies are connected, due to λmin > 0, one can T T set that K 2 = −λ−1 min Q C /2. Thus, one can obtain by (6.58) that ) T T ( ( ) V˙i (t) = ζiT (t) −R + 1 − λσ (t),i λ−1 min Q C C Q ζi (t). T ˙ Since 1 − λiσ (t) λ−1 min ≤ 0 and R = R > 0, it can be shown that Vi (t) ≡ 0 if and only ζi (t) ≡ 0 (i = 2, 3, . . . , N ); that is, systems (6.55) are asymptotically stable. T T In this case, for K 2 = −λ−1 min Q C /2, there exists a matrix Θ such that

ΘT E T)= EΘ ( ≥ 0, ) Θ A + λσ (t),i C T K 2T + A + λσ (t),i K 2 C Θ < 0, T

(

T

which can guarantee that systems (6.59) are regular and ( impulse-free by) Lemma 2.26 in [20]. It is not difficult to see that the pairs 2C ) ( T E,T A + λσ (t),i K T T E if the pairs , A + λ C K (i = 2, 3, . . . , N ) are admissible if and only σ (t),i 2 ( ) (i = 2, 3, . . . , N ) are admissible. Hence, E, A + λσ (t),i K 2 C (i = 2, 3, . . . , N ) T T with K 2 = −λ−1 min Q C /2 are admissible. From Theorem 6.4, the conclusions of □ Theorem 6.6 can be obtained.

202

6 Admissible Consensus and Consensualization with Protocol State …

For the case that the interaction topology is fixed and connected, let 0 = λ1 ≤ λ2 ≤ · · · ≤ λ N denote the eigenvalues of the associated Laplacian matrix, then the following theorem can be obtained directly. Theorem 6.7 Multi-agent system (6.35) with a connected fixed topology is stableprotocol admissibly consensus by protocol (6.36) if and only if the triple [ (E,] A, B) is stabilizable and the triple (E, A, C) is detectable. In this case, K 1 = K˜ 1 , 0 Z˜ −1 Z −1 T T T and K 2 = −λ−1 = R > 0 and Q is the solution of the 2 Q C /2, where R generalized Riccati equation

⎧

Q T AT + AQ − Q T C T C Q + R = 0, Q T E T = E Q ≥ 0.

Remark 6.9 In Sect. 3.2, LMI consensualization criteria for complex networks with fixed topologies were presented, where all states of neighbors were applied to construct consensus protocols. Yang and Liu [1] proposed static output feedback consensus protocols, where it was supposed that the interaction topology is fixed. By the LMI tool, consensualization problems for complex networks with switching topologies were dealt with in [19], where state feedback consensus protocols were used. It should be pointed that LMI consensualization criteria are not analytic. Theorem 6.6 presents analytic consensualization criteria for complex networks with dynamic output feedback consensus protocols including switching topologies.

6.3.3 Admissible Consensus Criteria for Jointly Connected Switching Topology Cases This section focuses on jointly connected switching topology cases and gives sufficient conditions for stable-protocol admissible consensus. [ It is supposed that in each time interval tm , tm+1 ), there is a series of nonoverlapping contiguous subintervals [ j+1

) [ ) [ ) tm0 , tm1 , . . . , tmj−1 , tmj , . . . , tmn m −1 , tmn m , tm0 = tm , tmn m = tm+1 j

with tm − tm ≥ Td ( j = 0, 1, . . . , n m − 1) for n m > 0 and Td > 0. Without loss of generality, it is assumed that the interaction topologies[switch at ) j j+1 tm0 , tm1 , . . . , tmn m −1 ; that is, the interaction topology G σ (t) is fixed during tm , tm [ ) j j+1 ( j = 0, 1, . . . , n m − 1). Furthermore, the interaction topology in tm , tm ( j ∈ {0, 1, . . . , n[m − 1}) may not be connected, but the union of interaction topologies across each tm , tm+1 ) (m = 0, 1, . . .) is connected. By the labeling rule in [21]

6.3 Stable-Protocol Admissible Consensualization with Switching Topologies

203

and searching components among agents labeled 2 to N , the eigenvalues of L m associated with { the interaction topology G m} ∈ l are denoted as λm,2 , λm,3 , . . . , λm,N . Let η(m) = k|λm,k /= 0, k = 2, 3, . . . , N , then the following lemma can be obtained. [ Lemma 6.2 [21]: U The interaction topologies across tm , tm+1 ) are jointly connected if and only if t∈[tm ,tm+1 ) η(t) = {2, 3, . . . , N }. ˜ Let { λm,min denote} the minimum nonzero eigenvalue of L m and λmin = min λm,min (∀m ∈ ψ) , then one has λ˜ min > 0. The following theorem gives stableprotocol admissible consensus criteria for multi-agent systems with jointly connected switching topologies. Theorem 6.8 Multi-agent system (6.35) with jointly connected switching topologies is stable-protocol admissibly consensualization by protocol (6.36) if the triple (E, A, B) is stabilizable, the triple (E, A, C) is detectable and Q T AT + AQ ≤ 0, where R T = R > 0 and Q is the solution of the generalized Riccati equation ⎧

Q T AT + AQ − Q T C T C Q + R = 0, Q T E T = E Q ≥ 0.

(6.61)

[ ] T T In this case, K 1 = K˜ 1 , 0 Z˜ −1 Z −1 and K 2 = −λ˜ −1 min Q C /2. Proof Constructing the Lyapunov function candidate as follows V (t) =

N ∑

ζiT (t)Q T E T ζi (t),

i=2

where Q T E T = E Q ≥ 0 and Q is a solution of the generalized Riccati Eq. (6.61). Taking the derivative of V (t), one can show by (6.59) that V˙ (t) =

N ∑

( ) ζiT (t) Q T AT + AQ + λσ (t),i Q T C T K 2T + λσ (t),i K 2 C Q ζi (t). (6.62)

i=2 T T T T Due to λ˜ min > 0, it can be set that K 2 = −λ˜ −1 min Q C /2. Since Q A + AQ ≤ 0 and λσ (t),i ≥ λ˜ min , one can obtain by (6.62) that

V˙ (t) =

N ∑

( ) T T ζiT (t) Q T AT + AQ − λσ (t),i λ˜ −1 min Q C C Q ζi (t)

i=2

≤

∑

( ) ζiT (t) Q T AT + AQ − Q T C T C Q ζi (t).

(6.63)

i∈η(σ (t))

˜ i (t), then it can be obtained by Let R = R˜ T R˜ with R˜ nonsingular and ζ˜i (t) = Rζ (6.61) and (6.63) that

204

6 Admissible Consensus and Consensualization with Protocol State …

∑

V˙ (t) ≤ −

ζiT (t)Rζi (t) = −

i∈η(σ (t))

∑

ζ˜iT (t)ζ˜i (t) ≤ 0,

(6.64)

i∈η(σ (t))

which means that the limit of V (t) as t → ∞ exists. In the sequel, it is shown that the limit of ζ (t) as t → ∞ tends to zero. Considering the infinite sequence V (tm ) (m = 0, 1, . . .), by the Cauchy convergence criterion, for any δ > 0, there is an integer M > 0 such that, for ∀m > M, one has ∫tm+1 −

∫tm 1

V˙ (t)dt = −

tm

∫tm 2

V˙ (t)dt −

tm0

∫tmm n

V˙ (t)dt − · · · −

V˙ (t)dt < δ.

(6.65)

tmn m −1

tm1

For any l ∈ {0, 1, . . . , n m − 1}, one can show by (6.64) that ∫tm

∫tm

l+1

l+1

V˙ (t)dt ≤ −

tml

∑

ζ˜iT (t)ζ˜i (t)dt ≤ −

i∈η(σ (tml ))

tml

l tm∫ +Td

tml

∑

ζ˜iT (t)ζ˜i (t)dt.

(6.66)

i∈η(σ (tml ))

By (6.65) and (6.66), one can see that n∑ m −1 l=0

l +Td tm∫

∑

ζ˜iT (t)ζ˜i (t)dt < δ,

i∈η(σ (tml ))

tml

which means that l tm∫ +Td

∑

ζ˜iT (t)ζ˜i (t)dt < δ (l = 0, 1, . . . , n m − 1);

i∈η(σ (tml ))

tml

that is, t+T ∫ d

lim

t→∞ t

∑

ζ˜iT (t)ζ˜i (t)dt = 0 (l = 0, 1, . . . , n m − 1).

i∈η(σ (t l ))

Therefore, one can show that

lim

t→∞

t+Td n∑ m −1 ∫ l=0

t

∑

i∈η(σ (t l ))

ζ˜iT (t)ζ˜i (t)dt = 0.

(6.67)

6.3 Stable-Protocol Admissible Consensualization with Switching Topologies

205

U From Lemma 6.2, one can obtain that t∈[tm ,tm+1 ) η(t) = {2, 3, . . . , N }. By (6.67), there exist positive numbers γ2 , γ3 , . . . , γ N such that

lim

t+T ∫ d(∑ N

t→∞ t

) γi ζ˜iT (t)ζ˜i (t)

dt = 0.

(6.68)

i=2

Since V˙ (t) ≤ 0, ζi (t) is bounded. From (6.59), ζ˙i (t) is bounded. Due to ζ˜i (t) = ˜ i (t), ζ˜i (t) and ζ˙˜i (t) are bounded. Hence, γi ζ˜iT (t)ζ˜i (t) is uniformly continuous. By Rζ Barbalat’s lemma in [20], it can be shown that lim γi ζ˜iT (t)ζ˜i (t) = 0 (i = 2, 3, . . . , N ),

t→∞

which means that lim ζi (t) = 0 (i = 2, 3, . . . , N );

t→∞

( ) that is, the pairs E, A + λσ (t),i K 2 C (i = 2, 3, . . . , N ) are asymptotically stable. The proof of the regularity is similar to Theorem 6.6. From Theorem 6.4, the □ conclusions of Theorem 6.8 can be obtained. Remark 6.10 By Theorem 6.7, if Q T E T = E Q ≥ 0 and Q T AT + AQ < 0, then (E, A) is asymptotically stable. In this case, A˜ is Hurwitz. From Theorem 6.5, the consensus function tends to zero as time tends to infinity. If Q T E T = E Q ≥ 0 and Q T AT + AQ = 0, then (E, A) is Lyapunov stable; that is, A˜ is Lyapunov stable. In this case, the consensus function may continuously oscillate or tend to some constant as time tends to infinity but cannot diverge away. If switching topologies are connected, then (E, A) can be unstable by Theorem 6.6. In this case, the consensus function can diverge away.

6.3.4 Numerical Simulation This section presents two examples to demonstrate the effectiveness of theoretical results given in the previous sections. Example 6.1 (Connected switching topology cases) Consider a singular multi-agent system with six agents with the dynamics of each agent modeled by (6.35), where ⎡

1 ⎢0 E =⎢ ⎣0 0

0 1 0 0

0 0 0 0

⎤ ⎡ 0 0 ⎢1 0⎥ ⎥, A = ⎢ ⎣0 0⎦ 0 0

9 2 0 2

1 3 1 0

⎤ ⎡ ⎤ 2 0 [ ] ⎢1⎥ −5 ⎥ ⎥, B = ⎢ ⎥, C = 0 3 0 1 . ⎣0⎦ −2 ⎦ 2010 1 0

206

6 Admissible Consensus and Consensualization with Protocol State …

Figure 6.5 gives the switching topology set l with four undirected interaction topologies, whose adjacency matrices are 0–1. The interaction topologies of the singular multi-agent system are randomly chosen from l with Td = 0.1 s. Let ⎡

16 0 ⎢0 5 R=⎢ ⎣0 4 0 0

0 4 4 0

⎤ 0 0⎥ ⎥. 0⎦ 9

From Theorem 6.6, one can obtain that ⎤ 5.9359 0 0 ⎥ 5.7637 0 0 ⎥, −24.0501 −0.7116 −2.1584 ⎦ −7.2822 0.4638 −1.3111 [ ] K 1 = −126 −20 0 0 ,

⎡

11.7575 ⎢ 5.9359 Q=⎢ ⎣ −25.3947 −10.7888

2

2

1

3

1

3

6

4

6

4

5

5

G1

G2

2

2

1

3

1

3

6

4

6

4

5

5

G3

G4

Fig. 6.5 Switching topology set for connected switching topology cases

6.3 Stable-Protocol Admissible Consensualization with Switching Topologies

207

Fig. 6.6 Switching signal for connected switching topology cases

t/s

⎡

−3.5095 ⎢ −5.0045 K2 = ⎢ ⎣ −0.2319 0.6556

⎤ 0.9398 6.0891 ⎥ ⎥. 0.3558 ⎦ 1.0792

Figure 6.8 depicts the state trajectories of the singular multi-agent system with the switching signal σ (t) shown in Fig. 6.6. The curves of the consensus function given by Theorem 6.5 are marked by circles in Fig. 6.8. Since (E, A) is Lyapunov unstable, the curves of the consensus function diverge away. The trajectories of z iT (t)z i (t) (i = 1, 2, . . . , 6) are depicted in Fig. 6.7. One can see that the state trajectories converge to the ones marked by circles and limt→∞ z iT (t)z i (t) = 0 (i = 1, 2, . . . , 6); that is, the singular multi-agent system achieves stable-protocol admissible consensus. Example 6.2 (Jointly connected switching topology cases) The dynamics of each agent of a multi-agent system with six agents is modeled by (6.35) with ⎡

1 ⎢0 E =⎢ ⎣0 0

0 1 0 0

0 0 0 0

⎤ ⎡ 0 1 ⎥ ⎢ 0⎥ 1 , A=⎢ ⎣ 1 0⎦ 0 −1

−1 1 0 1

1 0 1 −1

⎡ ⎤ ⎤ ⎡ ⎤ 0 0 1000 ⎢ ⎥ ⎥ 1⎥ 1⎥ ⎣ ⎦ , B=⎢ ⎣ 0 ⎦, C = 0 2 0 0 . 0⎦ 0010 1 0

Figure 6.9 shows the switching topology set l with four undirected interaction topologies, each of which is not connected but the union of which is connected. It is assumed that the associated adjacency matrices are 0–1. The interaction topologies are switched as G 1 → G 2 → G 3 → G 4 → · · · with Td = 0.2 s as shown in Fig. 6.10. Let

208

6 Admissible Consensus and Consensualization with Protocol State …

t/s Fig. 6.7 Trajectories of z iT (t)z i (t) for connected switching topology cases

agent 1 agent 2 agent 3 agent 4 agent 5 agent 6

agent 1 agent 2 agent 3 agent 4 agent 5 agent 6

t/s

t/s agent 1 agent 2 agent 3 agent 4 agent 5 agent 6

i3

i4

agent 1 agent 2 agent 3 agent 4 agent 5 agent 6

t/s

Fig. 6.8 State trajectories for connected switching topology cases

t/s

6.3 Stable-Protocol Admissible Consensualization with Switching Topologies

⎡

8 ⎢ −6 R=⎢ ⎣ 0 −6

−6 25 0 9

0 0 1 0

209

⎤ −6 9 ⎥ ⎥. 0 ⎦ 9

By Theorem 6.8, it can be obtained that 2

2

1

3

1

3

6

4

6

4

2 5

5

G1

G2

2 1

2 3

1

1

3

4

6 3

5 G1 G2

4

6

4

6

5

5

G3

G4

Fig. 6.9 Switching topology set for jointly connected switching topology cases

Fig. 6.10 Switching signal for jointly connected switching topology cases

G3

G4

210

6 Admissible Consensus and Consensualization with Protocol State …

t/s Fig. 6.11 Trajectories of z iT (t)z i (t) for jointly connected switching topology cases

⎡

⎤ 2 0 0 0 ⎢ 0 2 0 0⎥ ⎥ Q=⎢ ⎣ −2 3 0 3 ⎦, −3 −2 −3 3 [ ] K 1 = 49 −10 0 0 , ⎡

⎤ −1.0 0 1.0 ⎢ 0 −2.0 −15 ⎥ ⎥. K2 = ⎢ ⎣ 0 0 0 ⎦ 0 0 −1.5 The state trajectories of the singular multi-agent system are shown in Fig. 6.11, where circle markers describe the curves of the consensus function in Theorem 6.5. Because (E, A) is Lyapunov stable, the curves of the consensus function continuously oscillate. The trajectories of z iT (t)z i (t) (i = 1, 2, . . . , 6) are depicted in Fig. 6.12. One can see that limt→∞ z iT (t)z i (t) = 0 (i = 1, 2, . . . , 6). It can be found that the multi-agent system achieves stable-protocol admissible consensus.

6.4 Notes Stable-protocol admissible consensus problems for high-order linear singular multiagent systems with multiple time-varying delays and switching topologies were

References

211 agent 1 agent 2 agent 3 agent 4 agent 5 agent 6

agent 1 agent 2 agent 3 agent 4 agent 5 agent 6

t/s

i3

agent 1 agent 2 agent 3 agent 4 agent 5 agent 6

agent 1 agent 2 agent 3 agent 4 agent 5 agent 6

i4

t/s

t/s

t/s

Fig. 6.12 State trajectories for jointly connected switching topology cases

dealt with. A CS and a CCS were introduced, the direct sum of which are the state spaces of multi-agent systems. Based on the state projection between the two subspaces, a necessary and sufficient condition for stable-protocol admissible consensus was proposed and an explicit expression of the consensus function was given by combining the First and Second Equivalent Forms. Moreover, by the generalized Riccati equation, stable-protocol admissible consensus criteria for multi-agent systems with multiple time-varying delays and connected switching topologies were presented, respectively. Furthermore, stable-protocol admissible consensus criteria for jointly connected switching topology cases were given.

References 1. Yang XR, Liu GP (2012) Necessary and sufficient consensus conditions of descriptor multiagent systems. IEEE Trans Circuits Syst I Regul Pap 59(11):2669–2677 2. Xi JX, Meng FL, Shi ZY, Zhong YS (2012) Time-dependent admissible consensualization for singular time-delayed swarm systems. Syst Control Lett 61(11):1089–1096 3. Ma CQ, Zhang JF (2010) Necessary and sufficient conditions for consensusability of linear multi-agent systems. IEEE Trans Autom Control 55(5):1263–1268

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4. Xi JX, Shi ZY, Zhong YS (2012) Consensus and consensualization of high-order swarm systems with time delays and external disturbances. J Dyn Syst Meas Control 134(4):1–7 5. Seo JH, Shim H, Back J (2009) Consensus of high-order linear systems using dynamic output feedback compensator: low gain approach. Automatica 45(11):2659–2664 6. You K, Xie L (2011) Coordination of discrete-time multi-agent systems via relative output feedback. Int J Robust Nonlinear Control 21(13):1587–1605 7. Olfati-Saber R, Murray RM (2004) Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans Autom Control 49(9):1520–1533 8. Ren W, Beard RW (2005) Consensus seeking in multiagent systems under dynamically changing interaction topologies. IEEE Trans Autom Control 50(5):655–661 9. Xu S, Dooren VP, Stefan R, Lam J (2002) Robust stability and stabilization for singular systems with state delay and parameter uncertainty. IEEE Trans Autom Control 47(7):1122–1128 10. Dai LY (1989) Singular control systems. Springer, Berlin 11. Meng B, Zhang JF (2006) Output feedback based admissible control of switched linear singular systems. Acta Autom Sin 32(2):179–185 12. Ma SP, Zhang CH, Wu Z (2008) Delay-dependent stability and H ∞ control for uncertain discrete switched singular systems with time-delay. Appl Math Comput 206(1):413–424 13. Xia YQ, Boukas EK, Shi P, Zhang JH (2009) Stability and stabilization of continuous-time singular hybrid systems. Automatica 45(6):1504–1509 14. Xiao F, Wang L, Chen J, Gao YP (2009) Finite-time formation control for multi-agent systems. Automatica 45(11):2605–2611 15. Dong XW, Xi JX, Shi XY, Zhong YS (2012) Practical consensus for high-order swarm systems with uncertainties, time delays and external disturbances. Int J Syst Sci 44(10):1–14 16. Xi JX, Shi ZY, Zhong YS (2012) Stable-protocol output consensualization for high-order swarm systems with switching topologies. Int J Robust Nonlinear Control 23(18):2044–2059 17. Su YF, Huang J (2012) Stability of a class of linear switching systems with applications to two consensus problems. IEEE Trans Autom Control 57(6):1420–1430 18. Xi JX, Shi ZY, Zhong YS (2012) Output consensus for high-order linear time-invariant swarm systems. Int J Control 85(4):350–360 19. Xi JX, Yu Y, Zhong YS (2014) Guaranteed-cost consensus for singular multi-agent systems with switching topologies. IEEE Trans Circuits Syst I Regul Pap 61(5):1531–1542 20. Masubuchi I, Kamitane Y, Ohara A, Suda N (1997) H ∞ control for descriptor systems: a matrix inequalities approach. Automatica 33(4):669–673 21. Ni W, Cheng D (2010) Leader-following consensus of multi-agent systems under fixed and switching topologies. Syst Control Lett 59(3):209–217

Chapter 7

Admissible Consensus and Consensualization with Energy Constraints

In this chapter, admissible consensus analysis and design problems for high-order singular multi-agent systems with switching topologies and energy constraints are investigated. Singular multi-agent systems in many application scenarios are usually not only required to realize consensus, but also need to save the energy, which is an optimization consensus control problem with the consensus regulation performance and the energy consumption being considered simultaneously. This optimization cooperative control behavior can be seen in many biological multi-agent systems, such as the flocks of geese and pelicans, which can spontaneously organize themselves into V-shaped formations, and each individual in the formation can use the airflow of other companions to glide continuously for a long time to reduce the energy consumption [1]. As practical engineering examples, the energy supply of mobile robots, miniature UAVs, and deep-sea submarines is also limited and valuable. If the total energy is consumed too fast, it will be difficult for these multiagent systems to complete their assigned tasks. This kind of control problem is also called as the guaranteed-cost control problem by scholars. In [2, 3], guaranteed-cost consensualization for high-order normal multi-agent systems were studied, where a linear quadratic cost function was constructed to evaluate the team performance of all agents as a whole. However, guaranteed-cost consensus for singular multiagent systems were not considered in most literatures, which is more meaningful than normal ones. Besides, due to the motions of agents and the fault of communication devices or circuits, interaction topologies among agents may be switching. In [4–6], the assumption that interaction topologies are fixed was required, and their approaches cannot be used to deal with the cases with switching topologies. Hence, it is of great practical significance to study the admissible guaranteed-cost consensus and consensualization problems for high-order singular multi-agent systems with switching topologies. The current chapter investigates the impacts of switching topologies on consensus for singular multi-agent systems and proposes a distributed guaranteed-cost consensus approach to realize a trade-off design between consensus regulation

© Science Press, Beijing and Springer Nature Singapore Pte Ltd. 2023 J. Xi et al., Admissible Consensus and Consensualization for Singular Multi-agent Systems, Engineering Applications of Computational Methods 11, https://doi.org/10.1007/978-981-19-6990-4_7

213

214

7 Admissible Consensus and Consensualization with Energy Constraints

performances and control energy consumptions. Firstly, a quadratic cost function is constructed by state errors among agents and control inputs of all agents and guaranteed-cost consensus problems are introduced. Then, based on linear matrix inequality techniques, sufficient conditions for guaranteed-cost consensus and consensualization are presented, respectively, which can guarantee the scalability of singular multi-agent systems since the dimensions of all the variables in these conditions are independent of the number of agents. Moreover, an upper bound of the cost function is determined, explicit expressions of consensus functions are given based on the Second Equivalent Form, and it is shown that consensus functions are dependent on the average of initial states of all agents but are independent of switching topologies. Finally, the applications of theoretical results in multi-agent supporting systems are shown.

7.1 Problems of Optimal and Suboptimal Consensus When performing cooperative tasks like consensus, formation, flocking, and containment, etc., practical multi-agent systems usually suffer from the constraints of resource and energy supply for the limitation of fuel tank capacity, the battery power, or the limited payload, and so on. Furthermore, each agent is only allowed to use the information from itself and its neighbors in distributed control mode, which is the critical property of the local feedback control protocol of multi-agent systems. This kind of problem, where the multi-agent system is required to implement distributed cooperative control with realizing expected performance index concurrently, is identified as the optimization cooperative control problem. In the last few decades, many scholars had paid a lot of attention on the optimization cooperative control problems, and according to different expressions of the performance indexes, the optimization cooperative control can be categorized into decentralized and global ones as shown in following sections.

7.1.1 Decentralized Optimization Cooperative Control The decentralized performance index of the multi-agent system is usually composed of the performance index of each agent itself, which is usually described as Jc = min

N 

gi (x),

i=1

where gi (x) : R N → R is the objective function of agent i and denotes a certain performance index of agent i. The decentralized optimization cooperative control is intended to achieve the optimization goal by the performance indexes Jc , which can be

7.1 Problems of Optimal and Suboptimal Consensus

215

used for the performance evaluation and resource scheduling problems of multi-agent systems. Reference [7] takes the performance index of formula Jc as the optimization objective function, and achieved the optimization cooperative control objective by the idea of average consensus, where the gradient optimization method is used for algorithm iteration. Reference [8] firstly assumed that the intersection of convex solution sets of all agents is not empty, then transformed the decentralized optimization cooperative control into an intersection calculation problem. By using convex analysis and nonsmoothed analysis, the critical properties of the global optimal solution set are given. In addition, the distributed optimization control problem of multiagent systems with nonconvex objective function is investigated in Ref. [9], where the nonconvex objective function problem is solved by introducing adaptive dynamic coupling gain. Chen and Li [10] proposed a distributed optimization algorithm for the multi-agent optimization problem with convex objective function Jc , and obtained the optimal solution in a fixed time. They also proved the fixed-time convergence by using convex optimization and fixed-time Lyapunov function method. Moreover, based on the pseudo-gradient technique, the internal model principle, and the adaptive strategy, Ref. [11] designed a distributed optimal control protocol to solve the decentralized optimal cooperative control problems for uncertain nonlinear MIMO multi-agent systems. It can be found that the decentralized performance index function is the sum of the performance function of each single agent. Besides, each performance function only acts on its corresponding agent, and there is no interaction relationship among them, which is the essential reason why it is decentralized. Although the single performance function of each agent can be optimized, the global performance of the whole multi-agent system cannot be reflected. From the perspective of the distributed interaction among agents and the optimization of the overall multi-agent system, the global performance of the multi-agent system should be considered.

7.1.2 Global Optimization Cooperative Control The global performance index of a multi-agent system is usually expressed as t J=

 T  e (t)Qe(t) + u T (t)Ru(t) dt,

0

where e(t) and u(t) denote the global cooperative error variables and control inputs of multi-agent systems, respectively, and Q and R are the corresponding symmetric positive-definite weight matrix. The above equation can be regarded as a tradeoff performance index combining cooperative regulation performances with control energy consumptions. Take the consensus control problem as an example, e(t) denotes the vector of state errors among agents. If time t is infinite, then the index J is an infinite time trade-off performance index corresponding to the asymptotic

216

7 Admissible Consensus and Consensualization with Energy Constraints

consensus. If time t is finite, then the index J is a trade-off performance index generally related to the finite-time consensus. Some scholars have done a lot of researches on the global optimization cooperative control. In [12, 13], it was demonstrated that only when the communication topology of the multi-agent system is a complete graph that the optimal control performance index function can obtain the minimum value. However, a complete graph implies that there exist communication channels between any two agents in the multi-agent system, which is actually a very strict condition in practical applications. Based on inverse optimization and partial stability methods, Movric and Lewis [14] utilized Riccati equation to expound sufficient conditions for general high-order multi-agent systems to achieve optimization cooperative control under fixed topology, and discussed the influence of topology directivity on the optimization results. Zhang et al. [15] extended the results of [14], and proposed sufficient and necessary conditions for achieving global optimal cooperative control under the constraints of a special kind of directed topology, where the corresponding Laplacian matrix was required to be diagonalizable. It can be seen from Refs. [14, 15] that the performance indexes were required to satisfy special forms; that is, the weight matrices cannot be given arbitrarily. According to the analyses of above literatures, the realization of global optimal cooperative control has harsh requirements on the communication topology and the performance index. Hence, many scholars turned to investigate suboptimal cooperative control problems with milder conditions, that is, the performance index is constructed based on the cooperative regulation performance and the control energy consumption. Correspondingly, these kinds of suboptimal control problems can be categorized into the guaranteed-performance control one and the guaranteed-cost cooperative control one, which are introduced as follows. (I) For the guaranteed-performance control, the performance index is only determined by cooperative regulation performances, and the pattern of the performance index function can be summarized as ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

Jp =

∞  N  N 0

wi j εiTj (t)Pεi j (t)dt,

i=1 j=1

εi j (t) = x j (t) − xi (t),

where εi j (t) represents the state error vectors between interacted agents i and j, and P is the symmetric positive-definite weight matrix. For the second-order multi-agent systems, Guan et al. [16] proposed the guaranteed-performance consensus control criterion by the impulsive control method, and determined the upper bound of the performance index, i.e., the guaranteed-performance value. In addition, the problem of leader-following guaranteed-performance consensus control for second-order multi-agent systems with communication time delay is studied in Ref. [17], where the Jensen’s inequality and Newton– Leibniz formula are introduced to solve the consensus problems. Based on the linear matrix inequality, state-space decomposition approach and Lyapunov

7.1 Problems of Optimal and Suboptimal Consensus

217

stability theory, sufficient conditions were derived to achieve the guaranteedperformance consensus. For high-order multi-agent systems, a translation adaptive strategy is proposed in Ref. [18] to deal with the problem of adaptive guaranteed-performance consensus control, where the fully distributed guaranteed-performance consensus design and analysis criteria are given, and the guaranteed-performance value is solved, which is related to the initial states of each agent in the multi-agent system. Under the constraint of joint connected switching topologies, Zheng et al. [19] gave a sufficient condition to address the problem of the guaranteed-performance consensus design of high-order Lipschitz nonlinear multi-agent systems. This reference also analyzed the differences and relationships between the guaranteed-performance values of leaderless case and leader-following case. Reference [20] designed a nonlinear consensus control protocol for singular nonlinear multi-agent systems, and proposed a guaranteed-performance consensus control criterion by the Riccati equation. (II) For the guaranteed-cost control, the performance index is usually the combination of the cooperative regulation performance and the control energy consumption, which can be described as ⎧ Jc = Jx + Ju , ⎪ ⎪ ⎪ ⎪ ⎪ ∞  N N  ⎪ ⎪ ⎪ ⎪ = wi j ηiTj (t)Pηi j (t)dt, J ⎪ x ⎪ ⎪ ⎪ ⎪ 0 i=1 j=1 ⎨ ∞  N ⎪ ⎪ ⎪ Ju = u iT (t)Qu i (t)dt, ⎪ ⎪ ⎪ ⎪ ⎪ 0 i=1 ⎪ ⎪  ⎪ ⎪ ⎪ u (t) = K wi j ηi j (t) i ⎪ ⎩ j∈Ni (t)

where ηi j (t) is the cooperative errors between agents i and j, u i (t) is the control input of agent i, and P and Q are the symmetric positive-definite weight matrices, respectively. It can be found from the above equation that Jc is a distributed cost function for multi-agent system since Ju and Jx are both related to the communication weight wi j . Based on the idea of guaranteed-cost control for isolated systems, Ref. [21] studied the problems of guaranteed-cost consensus for first-order multi-agent systems, and proposed the guaranteedcost consensus analysis and design criteria. Yu et al. [22] considered the influence of heterogeneous inertial forces on second-order multi-agent systems, and presented a sufficient condition for the guaranteed-cost consensus control. The upper bound of the performance index was also calculated, which is the guaranteed-cost value. References [23, 24], respectively, studied the event-based and sampled-data guaranteed-cost consensus control of general high-order multi-agent systems. These

218

7 Admissible Consensus and Consensualization with Energy Constraints

two kinds of problems are transformed into the stability problem of time-delayed closed-loop systems by using the input-delay method, and the guaranteed-cost consensus control criteria are given based on the LMI method. Based on the idea of guaranteed-cost consensus control, Ref. [48] studied the problem of guaranteed-cost time-varying formation tracking control for general high-order multi-agent systems, which considered the effects of external disturbance, time delay, and unknown leader input. The control criteria of the guaranteed-cost robust formation tracking were also given. However, since the guaranteed-cost value is related to the disturbance parameters and dynamics of the time delay, its specific mathematical expression was not given.Please check for the missing reference citation in the sentence “Based on the idea of guaranteed-cost consensus control … external disturbance, time delay, and unknown leader input”.Yu J L, Dong X W, Li Q D, Ren Z. (2020 ) Robust H∞ guaranteed cost time-varying formation tracking for high-order multiagent systems with time-varying delays. IEEE Transactions on Systems, Man and Cybernetics: Systems, 50(4): 1465-1475DOI:10.1109/TSMC.2018.2883516 It can be seen from the researches mentioned above that there are really a lot of guaranteed-cost cooperative control problems that worth studying. As a research branch, the current chapter mainly focuses on guaranteed-cost consensus analysis and design problems for high-order linear singular multi-agent systems with switching topologies, and sufficient conditions to achieve guaranteed-cost consensus and consensualization are presented. Especially, we give a general approach to determine the consensus function and show that switching movements of the interaction topology do not impact the consensus function. However, the state projection approach in [4] is no longer valid when switching topologies are considered and no general approach was given to determine the consensus function in [6]. The results of the current chapter were first published in [25].

7.2 Problem Description of Guaranteed-Cost Admissible Consensus Firstly, some basic lemmas about singular systems are presented and the problem description is given as follow. Lemma 7.1 [26] The pair (E, A) is admissible if and only if there exists a matrix W such that W T E = E T W ≥ 0, AT W + W T A < 0. Next, the problem description is presented as follows. Consider the following singular multi-agent system

7.2 Problem Description of Guaranteed-Cost Admissible Consensus

E x˙i (t) = Axi (t) + Bu i (t) (i = 1, 2, . . . , N ),

219

(7.1)

where A ∈ Rn×n , B ∈ Rn×m , and E ∈ Rn×n with rank(E) = r ≤ n, xi (t), andu i (t) are the state and the control input, respectively. The interaction topology of a singular multi-agent system is described by an undirected graph G, where each vertex represents an agent, the edge between two vertices stands for the interaction channel between them, and the weight of the edge denotes the interaction strength. Consider a distributed consensus protocol as follows u i (t) = K



  wi j x j (t) − xi (t) (i = 1, 2, . . . , N ),

(7.2)

j∈Ni (t)

where K ∈ Rm×n is a gain matrix andNi (t) denotes the time-varying neighbor set of agent i. This chapter mainly focuses on the case with switching interaction topologies. Let the finite set S with an index set  ⊂ N stand for all possible interaction topologies, where N is the set of natural numbers. σ (t) : [ 0, ∞) →  denotes a switching signal, whose value at time t is the index of the interaction topology at time t. It is supposed that Assumption 7.1 All interaction topologies in S are connected. Assumption 7.2 The switching movements 0 < t1 < · · · < tk < · · · satisfy inf k (tk+1 − tk ) = Td > 0.

T Let x(t) = x1T (t), x2T (t), . . . , x NT (t) , then under consensus protocol (7.2), the dynamics of the singular multi-agent system can be rewritten as   ˙ = I N ⊗ A − L σ (t) ⊗ B K x(t), (I N ⊗ E)x(t)

(7.3)

where L σ (t) is the Laplacian matrix of the interaction topology G σ (t) . For given Q T = Q > 0 and R T = R > 0, consider a linear quadratic cost function as follows N  

∞

JC =

i=1 0

⎛

⎞    T   ⎝ wi j x j (t) − xi (t) Q x j (t) − xi (t) + u iT (t)Ru i (t)⎠dt. j∈Ni (t)

(7.4) Associated with the cost function JC , the definitions of the admissible guaranteedcost consensus and consensualization are, respectively, presented as follows. Definition 7.1 System (7.3) is said to achieve admissible guaranteed-cost consensus if for any given admissible bounded initial state x(0), it is regular and impulse-free, and there exist a vector-valued function c(t) dependent on x(0) and a positive scalar χ such that limt→∞ (x(t) − 1 ⊗ c(t)) = 0 and JC ≤ χ , where c(t) is called a consensus function and χ is said to be a guaranteed cost.

220

7 Admissible Consensus and Consensualization with Energy Constraints

Definition 7.2 System (7.1) is said to be admissibly guaranteed cost consensualizable by protocol (7.2) if there exists a gain matrix K such that it achieves admissible guaranteed-cost consensus. Remark 7.1 In (7.1), the pair (E, A) represents the intrinsic dynamics of each agent. Because E may be singular, each agent may include not only dynamic modes but also static modes and impulsive modes. Consensus protocol (7.2) describes the attraction interaction among agents, which can propel the states of all agents to be identical. Moreover, when E is singular, there exist some algebraic constraints among states. The admissible initial state means that x(0) satisfies those algebraic constraints. ∞ T Remark 7.2 In (7.4), the terms  and 0 u i (t)Ru i (t)dt  ∞  T   j∈Ni (t) wi j x j (t) − x i (t) Q x j (t) − x i (t) dt can be regarded as the 0 evaluation indexes of the control energy consumptions and dynamic performances of agent i (i = 1, 2, . . . , N ), respectively. In this sense, the cost function JC can be applied to achieve a trade-off design object between control energy consumptions and consensus regulation performances by choosing Q and R. Moreover, for guaranteed-cost control of linear isolated systems, linear quadratic cost functions constructed by states and control inputs can be applied to achieve a trade-off design object between state regulation performances and control energy consumptions in the literature (e.g., [27, 28]). However, the approaches in [27, 28] cannot be used to investigate guaranteed-cost consensus control for multi-agent systems since states of multi-agent systems may diverge even if consensus is achieved. This chapter will investigate admissible guaranteed-cost consensus analysis and design problems; that is, under what conditions system (7.3) achieves admissible guaranteed-cost consensus and how to determine the gain matrix K such that system (7.3) achieves admissible guaranteed-cost consensus. Moreover, the approaches are presented to determine the consensus function c(t) and the guaranteed cost χ , respectively.

7.3 Conditions of Guaranteed-Cost Admissible Consensus Analysis and Design This section gives LMI conditions for guaranteed-cost consensus and consensualization, respectively, determines an upper bound of the cost function JC , and presents an explicit expression of the consensus function c(t) on the basis of the Second Equivalent Form.

7.3 Conditions of Guaranteed-Cost Admissible Consensus Analysis …

221

7.3.1 Guaranteed-Cost Admissible Consensus Analysis Criteria From Lemma 2.2√and Assumption 7.1, there exists an orthogonal matrix U with the first column 1 N / N such that   U T L σ (t) U = diag 0, Ʌσ (t) ,

(7.5)

where Ʌσ (t) is a symmetric matrix. Let 

 T x(t) ˜ = U T ⊗ In x(t) = xCT (t), xCT (t) ,

(7.6)

where xC (t) ∈ Rn and xC (t) ∈ R(N −1)n , then by (7.5) and (7.6), system (7.3) can be transformed into E x˙C (t) = AxC (t),

(7.7)

  (I N −1 ⊗ E)x˙C (t) = I N −1 ⊗ A − Ʌσ (t) ⊗ B K xC (t).

(7.8)

Actually, subsystem (7.7) and (7.8) describe the consensus dynamics and disagreement dynamics of system (7.3), respectively. Let 0 = λ1k < λ2k ≤ · · · ≤ λkN (∀k ∈ ) denote the eigenvalues of the Lapla  cian matrix of the kth interaction topology in S. Let λ˜ 2 = min λ2k (∀k ∈ ) and   λ˜ N = max λkN (∀k ∈ ) , then one has 0 < λ˜ 2 ≤ λ˜ N from Lemma 2.2 and Assumption 7.1. The following theorem presents LMI criterions for admissible guaranteedcost consensus and shows an approach to determine an upper bound of the cost function JC . Theorem 7.1 System (7.3) achieves admissible guaranteed-cost consensus if  rank

E 0 A E

 = n + rank(E)

and there exists W such that W T E = E T W ≥ 0, ⎡ ⎢ i = ⎣

A − λ˜ i B K

T

  ⎤ W + W T A − λ˜ i B K λ˜ i K T R 2λ˜ i Q ⎥ λ˜ i R K −R 0 ⎦ < 0, 0 −2λ˜ i Q 2λ˜ i Q

where i = 2, N . In this case, the guaranteed cost satisfies

222

7 Admissible Consensus and Consensualization with Energy Constraints

 1 T  x (0) ⊗ W T E x(0), N

χ= where ⎡

⎤ N − 1 −1 · · · −1 ⎢ −1 N − 1 · · · −1 ⎥ ⎢ ⎥

=⎢ . .. .. ⎥. .. ⎣ .. . . . ⎦ −1 −1 · · · N − 1 Proof Let c j ( j = 1, 2, . . . , n) be linearly independent n-dimensional column vectors, then there exist γ j (t) ( j = 1, 2, . . . , N n) such that xC (t) =

n 

γ j (t)c j ,

(7.9)

j=1 N  n 

xC (t) = 0, I(N −1)n γ(i−1)n+k (t)(ei ⊗ ck ).

(7.10)

i=2 k=1

Let 1 x˜C (t)  √ 1 ⊗ xC (t), N x˜C (t) 

n N  

γ(i−1)n+k (t)(U ei ⊗ ck ),

(7.11)

(7.12)

i=2 k=1

then it can be shown by (7.9) and (7.10) that x˜C (t) =

n 

 T 

γ j (t) U e1 ⊗ c j = (U ⊗ In ) xCT (t), 0 ,

(7.13)

T

x˜C (t) = (U ⊗ In ) 0, xCT (t) .

(7.14)

j=1

 T   Due to U T ⊗ In x(t) = xCT (t), xCT (t) , one can show by (7.13) and (7.14) that x(t) = x˜C (t) + x˜C (t).

(7.15)

Since U ⊗ In is nonsingular, from (7.13) and (7.14), x˜C (t) and x˜C (t) are linearly independent. Hence, by (7.11) and (7.15), system (7.13) achieves consensus if and only if subsystem (7.8) is asymptotically stable; that is, limt→∞ xC (t) = 0. Consider the following Lyapunov function candidate

7.3 Conditions of Guaranteed-Cost Admissible Consensus Analysis …

  V (t) = xCT (t) I N −1 ⊗ W T E xC (t).

223

(7.16)

Due to W T E = E T W , by taking the derivative of V (t) with respect to time t along the solution of subsystem (7.8), it can be obtained by Assumption 7.2 that      V˙ (t) = xCT (t) I N −1 ⊗ W T A + AT W − Ʌσ (t) ⊗ W T B K + K T B T W xC (t). (7.17) Let 0 = λ1σ (t) < λ2σ (t) ≤ · · · ≤ λσN(t) denote the eigenvalues of L σ (t) , then one can see by (7.5) that the eigenvalues of Ʌσ (t) are λ2σ (t) , λ3σ (t) , . . . , λσN(t) . Since Ʌσ (t) is symmetric, there exists an orthogonal matrix Uσ (t) such that   UσT(t) Ʌσ (t) Uσ (t) = diag λ2σ (t) , λ3σ (t) , . . . , λσN(t) .

(7.18)

 

T κ(t) = UσT(t) ⊗ In xC (t) = κ2T (t), κ3T (t), . . . , κ NT (t) ,

(7.19)

Let

then it can be shown by (7.17) and (7.18) that V˙ (t) =

N 

    T  κiT (t) W T A − λiσ (t) B K + A − λiσ (t) B K W κi (t).

i=2

The LMI has the convex property and 0 < λ˜ 2 ≤ λ2σ (t) ≤ λ3σ (t) ≤ · · · ≤ λσN(t) ≤ λ˜ N , so the inequalities 

A − λ˜ i B K

T

  W + W T A − λ˜ i B K < 0 (i = 2, N )

(7.20)

can guarantee that    T W T A − λiσ (t) B K + A − λiσ (t) B K W < 0 (i = 2, 3, . . . , N ), which means that V˙ (t) ≤ 0 and V˙ (t) ≡ 0 if and only if κi (t) ≡ 0 (i = 2, 3, . . . , N ). In this case,one has κ(t) ≡ 0 by (7.19). Since Uσ (t) ⊗ In is nonsingular and xC (t) = Uσ (t) ⊗ In κ(t), it can be shown that limt→∞ xC (t) = 0 if inequalities (7.20) hold. In the following, we discuss the problem of determining the guaranteed cost. One can show that N     T     wi j x j (t) − xi (t) Q x j (t) − xi (t) = x T (t) 2L σ (t) ⊗ Q x(t), i=1 j∈Ni (t)

(7.21)

224

7 Admissible Consensus and Consensualization with Energy Constraints N 

  u iT (t)Ru i (t) = x T (t) L 2σ (t) ⊗ K T R K x(t).

(7.22)

i=1

Moreover, it can be obtained by (7.18) and (7.19) that N    xCT (t) 2Ʌσ (t) ⊗ Q xC (t) = 2λiσ (t) κiT (t)Qκi (t),

(7.23)

i=2 N   i 2 T   λσ (t) κi (t)K T R K κi (t). xCT (t) Ʌ2σ (t) ⊗ K T R K xC (t) =

(7.24)

i=2

Due to λ1k = 0 (∀k ∈ ), from (7.21) to (7.24), one has N  

T

JT 

⎞    T   ⎝ wi j x j (t) − xi (t) Q x j (t) − xi (t) + u iT (t)Ru i (t)⎠dt ⎛

j∈Ni (t)

i=1 0

=

N T 

  2  κiT (t) 2λiσ (t) Q + λiσ (t) K T R K κi (t)dt,

(7.25)

i=2 0

where T ≥ 0. For the case that i < 0 (i = 2, N ), by Schur complement in [29] and the convex property of LMIs, one can show from (7.25) that

JT =

N T  

   2  κiT (t) 2λiσ (t) Q + λiσ (t) K T R K κi (t) + V˙ (t) dt

i=2 0

− V (T ) + V (0) < V (0).

(7.26)

Let U=

1TN −1 √1 √ N N 1 N −1 √ U N

! ,

then due to UU T = I , one can see that T

1TN −1 1T −1 U = 0, + N√ N N

(7.27)

1 N −1 1TN −1 T + UU = I N −1 . N

(7.28)

7.3 Conditions of Guaranteed-Cost Admissible Consensus Analysis …

225

 

Since xC (t) = 0, I(N −1)n U T ⊗ In x(t), by (7.16), (7.27) and (7.28), one has V (0) =

1 T x (0) N

"

 # N −1 −1TN −1 T ⊗ W E x(0). −1 N −1 N I N −1 − 1 N −1 1TN −1

(7.29)

From (7.26) and (7.29), let T → ∞, then one has χ=

 1 T  x (0) ⊗ W T E x(0). N

The inequalities W T E = E T W ≥ 0 and i < 0 (i = 2, N ) can guarantee that subsystem (7.8) is regular and impulse-free by Lemma 7.1, and the equation  rank

E 0 A E

 = n + rank(E)

can ensure that subsystem (7.7) is regular and impulse-free by Lemma 7.1. Hence, one can see that system (7.3) is regular and impulse-free since nonsingular transformations do not change the regularity and impulsive properties and U ⊗ In is nonsingular. □ By the above analysis, the conclusion of Theorem 7.1 can be obtained. Remark 7.3 The condition  rank

E 0 A E

 = n + rank(E)

is applied to guarantee that system (7.3) is regular and impulse-free. By Schur complement, i < 0 (i = 2, N ) are equivalent to   T  i = A − λ˜ i B K W + W T A − λ˜ i B K + λ˜ i2 K T R K + 2λ˜ i Q < 0 (i = 2, N ). It is obvious that i =



A − λ˜ i B K

T

  W + W T A − λ˜ i B K < 0 (i = 2, N )

are necessary for i < 0 (i = 2, N ). Actually, W T E = E T W ≥ 0 and i < 0 (i = 2, N ) can ensure that system (7.3) achieves consensus and the terms λ˜ i2 K T R K and 2λ˜ i Q are used to guarantee the required performances shown in (7.4). Moreover, W T E = E T W ≥ 0 and i < 0 (i = 2, N ) are feasible if and only if (E, A, B) is stabilizable, which means that it is necessary for system (7.3) to achieve consensus that (E, A, B) is stabilizable and there always exists W such that W T E = E T W ≥ 0 and i < 0 (i = 2, N ) are feasible. Furthermore, for a required trade-off design object between consensus regulation performances and energy consumptions, which can be determined by choosing Q and R, by a number of simulations, it is shown that LMI conditions in Theorem 7.1 are usually feasible. From the proof of Theorem 7.1, the following corollary can be obtained directly.

226

7 Admissible Consensus and Consensualization with Energy Constraints

Corollary 7.1 System (7.3) achieves admissible consensus if 

E 0 rank A E

 = n + rank(E)

and there exists W such that W T E = E T W ≥ 0, AT W + W T A − λ˜ i K T B T W − λ˜ i W T B K < 0 (i = 2, N ).

7.3.2 Conditions of Guaranteed-Cost Admissible Consensus Design By the changing variable method, the following theorem presents sufficient conditions for admissible guaranteed-cost consensualization. Theorem 7.2 System (7.1) is admissibly guaranteed cost consensualizable by protocol (7.2) if  rank

E 0 A E

 = n + rank(E)

and there exist W˜ and K˜ such that E W˜ = W˜ T E T ≥ 0, ⎡ ˜i = ⎢ ⎣

  ⎤ + A W˜ − λ˜ i B K˜ λ˜ i K˜ T R 2λ˜ i W˜ T Q ⎥ ⎦ < 0 (i = 2, N ). λ˜ i R K˜ −R 0 0 −2λ˜ i Q 2λ˜ i Q W˜

A W˜ − λ˜ i B K˜

T

  In this case, K = K˜ W˜ −1 and χ = x T (0) ⊗ W˜ −T E x(0)/N . Proof If i < 0 (i = 2, N ), then one has 

A − λ˜ i B K

T

  W + W T A − λ˜ i B K < 0

It can be found that W is invertible. Let W˜ = W −1 , then pre- and post-multiplying W T E = E T W ≥ 0 by W˜ T and W˜ , respectively, one can obtain that

7.3 Conditions of Guaranteed-Cost Admissible Consensus Analysis …

227

E W˜ = W˜ T E T ≥ 0. % $ % $ Pre- and post-multiplying i < 0 (i = 2, N ) by diag W˜ T , I, I and diag W˜ , I, I , respectively, and setting K˜ = K W˜ , it can be obtained that ˜ i < 0 (i = 2, N ). □

The proof of Theorem 7.2 is completed.

From Corollary 7.1, by a similar analysis to Theorem 7.2, the following corollary can be obtained, which presents sufficient conditions for admissible consensualization. Corollary 7.2 System (7.1) is admissibly consensualizable by protocol (7.2) if  rank

E 0 A E

 = n + rank(E)

and there exist W˜ and K˜ such that E W˜ = W˜ T E T ≥ 0, A W˜ + W˜ T AT − λ˜ i K˜ T B T − λ˜ i B K˜ < 0 (i = 2, N ). In this case, K = K˜ W˜ −1 . The following algorithm shows a procedure to determine the gain matrix Algorithm Step (1)

Choose positive and symmetric matrices Q and R to obtain the required trade-off objective between consensus regulation performances and energy consumptions

Step (2)

Check the condition ! E 0 rank = n + rank(E). A E

Step (3)

If the condition does not hold, then system (7.1) is not guaranteed cost consensualizable by protocol (7.2). Stop ˜ i < 0 (i = 2, N ) to obtain W˜ and K˜ Check E W˜ = W˜ T E T ≥ 0 and

Step (4)

Set K = K˜ W˜ −1

The LMI conditions in the above conclusions are dependent on the second small and maximum eigenvalues of Laplacian matrices of interaction topologies in the switching set S. When N is huge, the eigenvalues of the Laplacian matrix of each interaction topology may be difficult to obtain. Actually, many approaches were

228

7 Admissible Consensus and Consensualization with Energy Constraints

proposed to estimate the second small and maximum eigenvalues in the literature. For examples, the approaches to estimate the second small eigenvalue were presented in [30, 31] and the Gershgorin disc theorem in [32] can be used to estimate the maximum eigenvalue. Hence, it is not necessary to determine the precise values of λ˜ 2 and λ˜ N . Remark 7.4 By the above analysis, the consensus properties of system (7.3) are jointly determined by consensus protocols, the dynamics of each agent, and interaction topologies. Furthermore, system (7.3) can be regarded as a high-dimensional switching system with a specific structure; that is, interaction topologies are switching but the dynamics of each agent is time invariant, and consensus problems of system (7.3) are converted into admissible problems of reduced-order switching singular subsystems. It should be pointed out that the approaches in the literature about switching isolated systems (e.g., [33–35]) cannot be directly used to deal with consensus problems of singular multi-agent systems with switching topologies due to this specific structure. By the structure property of Laplacian matrices of interaction topologies, we give LMI conditions for admissible consensus and consensualization of system (7.3). Remark 7.5 Consensus problems for multi-agent systems with switching topologies were extensively investigated by LMI tools in the literature (e.g., [36, 37]), where the dimensions of all the variables of LMI consensus criterions are dependent on the number of agents. In this case, it is time cost and memory cost to check those criterions if multi-agent systems consist of numerous agents. It is not difficult to find that LMI conditions in Theorems 7.1 and 7.2 have lower calculation complexity. Moreover, the Riccati equation was applied to deal with consensualization problems of high-order linear normal multi-agent systems with switching topologies in [38, 39], where consensualization criterions have lower calculation complexity, but their approaches are no longer valid when the guaranteed cost is considered.

7.3.3 Consensus Functions When system (7.3) achieves admissible guaranteed-cost consensus, the states of all agents will tend to be identical. The consensus function is often used to describe the identical state in the literature. Let T and Z be invertible matrices such that     Ir 0 A11 A12 , (7.30) T EZ = , T AZ = A21 A22 0 0 which is called the Second Equivalent Form of the pair (E, A) as shown in [40]. The following theorem gives an approach to determine the consensus function of system (7.3).

7.3 Conditions of Guaranteed-Cost Admissible Consensus Analysis …

229

Theorem 7.3 If system (7.3) achieves admissible guaranteed-cost consensus, then consensus function satisfies & lim c(t) − Z

t→∞



T ( A11 −A12 A−1 A21 )t 22 e Ir , −AT21 A−T [Ir , 0]Z −1 22

&

N 1  xi (0) N i=1

'' = 0.

Proof By (7.6), one can see that   xC (0) = [In , 0] U T ⊗ In x(0).

(7.31)

√ Since the first column of U is 1 N / N , it can be shown that 1 e1T U T = √ 1TN . N

(7.32)

Due to [In , 0] = e1T ⊗ In , one can obtain by (7.31) and (7.32) that N   1  xC (0) = e1T U T ⊗ In x(0) = √ xi (0). N i=1

(7.33)

From the proof of Theorem 7.1, by (7.11) and (7.15), one has "

# 1 lim c(t) − √ xC (t) = 0, t→∞ N

(7.34)

which means that subsystem (7.7) determines the consensus function. Let

T T T Z −1 xC (t) = xC1 (t), xC2 (t) , then it can be shown by (7.7) and (7.30) that 

Ir 0 0 0



    A11 A12 xC1 (t) x˙C1 (t) = . A21 A22 xC2 (t) x˙C2 (t)

(7.35)

Due to 

E 0 rank A E

 = n + rank(E),

it can be shown that A22 is nonsingular. Hence, it can be obtained by (7.35) that

230

7 Admissible Consensus and Consensualization with Energy Constraints

(

x˙C1 (t) = A11 xC1 (t) + A12 xC2 (t), xC2 (t) = −A−1 22 A21 x C1 (t).

(7.36)

From (7.33), (7.34), and (7.36), the conclusion of Theorem 7.3 can be obtained. □ By Theorem 7.3, since consensus functions are dependent on the dynamics of each agent and the initial state of each agent but are independent of the variances of interaction topologies, multi-agent systems with the same initial states but different switching movements have the same consensus function. Moreover, the Second Equivalent Form always exists and can be found by some nominal procedures as shown in [40], so the expression in Theorem 7.3 is easily obtained. Remark 7.6 Intuitionally speaking, the consensus function of a multi-agent system describes its macroscopic property as a whole. Based on the Second Equivalent Form, Theorem 7.3 gave a general approach to determine consensus functions of singular multi-agent systems with switching topologies. Besides, in [41], it was shown that some circuits can only be modeled as singular systems rather than normal ones. Synchronization problems of networks consisting of those circuits can be investigated by our approaches. Moreover, multi-agent systems with many three-link manipulators can be applied to clean the facade of a large building as shown in [6]. Since there are specific constraints on the motion of each manipulator, the dynamics of each agent is singular. In this case, the theoretical results in the current paper can also be used.

7.4 Numerical Simulation In [42], the model for the MASS with each agent supporting by one pillar was introduced and it was shown that the MASS has potential applications in earthquake damage-preventing buildings, water-floating plants, and large-diameter parabolic antennae or telescopes. In Sect. 3.2, it was shown that each agent in a MASS can only be modeled by a singular system when this MASS consists of many independent blocks and each block is supported by several pillars. For the case that each agent in a MASS is supported by two pillars called Unit I and Unit II, respectively, as shown in Fig. 7.1, where m the mass, d the damping coefficient, and k the stiffness coefficient, let xiI (t), xiII (t), viI (t), and viII (t) denote the heights and velocities of Unit I and Unit II, respectively, then agent i (i ∈ {1, 2, . . . , N }) can be described by E x˙i (t) = Axi (t) + Bu i (t), where

7.4 Numerical Simulation

231

Fig. 7.1 Model of each agent in a MASS [4]

⎡ ⎤ ⎤ 1000 xiI (t) ⎢0 1 0 0⎥ ⎢ viI (t) ⎥ ⎢ ⎥ ⎥ xi (t) = ⎢ ⎣ xiII (t) ⎦, E = ⎣ 0 0 0 0 ⎦, 0000 viII (t) ⎡ ⎤ ⎡ ⎤ 0 1 00 0 ⎢− k −d 0 0⎥ ⎢1⎥ ⎥ ⎢ ⎥ m m A=⎢ ⎣ −1 0 1 0 ⎦, B = ⎣ 0 ⎦. 0 0 −1 0 1 ⎡

Example 7.1 (Scalability) Consider a MASS consisting of N agents, where the parameters of each agent are chosen as m = 16, d = 25 and k = 12. Let K = [−0.0002, 0.2156, 0.0017, 0.0011], R = 1 and ⎡

1 ⎢0 Q=⎢ ⎣0 0

0 5 2 0

0 2 1 0

⎤ 0 0⎥ ⎥. 0⎦ 9

Figure 7.2 shows an undirected interaction topology with the edges labeled from 1 to N , where the weight of edge k is 0.01k k = {1, 2, . . . N }). Since the topology has a link structure, it is not difficult to determine the eigenvalues of its Laplacian matrix. Consider the following two cases: Case 1: LMIs in Theorem 7.1 with λ˜ i (i = 2, N ); Case 2: LMIs in Theorem 7.1 with λ˜ i = λi (i = 2, 3, . . . , N ). Under the same experiment conditions, for different N , the calculation time to check the feasibility of LMIs is shown in Table 7.1. It can be found that the method in Theorem 7.1 significantly improves the calculation efficiency for a large N .

232

7 Admissible Consensus and Consensualization with Energy Constraints

Fig. 7.2 Undirected interaction topology

3

3

1

N = 10

5 6

4

2

Table 7.1 Calculation time to check feasibility

5

...

2

4

N

1

N = 50

N

N = 100

Case 1

0.0359s

0.0475s

0.0571s

Case 2

0.5813s

10.0128s

30.0123s

Example 7.2 (Water-floating plant) All agents in a water-floating plant try to keep the plant horizontal. The plant will keep horizontal if all agents achieve consensus on the heights and velocities. Since the water disturbs the plant all the time, the consensus function oscillates continuously. Let u i (t) = K 0 xi (t) + K



  wi j x j (t) − xi (t) (i = 1, 2, . . . , N ),

j∈Ni (t)

where K 0 is a self-feedback matrix to adjust the dynamics of each agent. The parameters of each agent are chosen as m = 1.6, d = 50.5 and k = 1200. Let K 0 = [749.7500, 31.5625, 0, 0], then the dynamic modes of each agents are placed into at ±0.5j with j2 = −1. Figure 7.3 gives four undirected interaction topologies in the switching set S, and it is assumed that the weights of edges of all interaction topologies are 1 without loss of generality. The interaction topologies of the water-floating plant are randomly chosen from S with Td = 0.1 s. Let ⎡

0 ⎢0 L=⎢ ⎣1 9

0 0 1 8

0 0 0 6

⎤ 0 0⎥ ⎥, 1⎦ 5

then one can see that E L = 0. One can choose that W˜ = G 1 E + LG 2 , where G 1 ∈ R4×4 is symmetric and positive and G 2 ∈ R4×4 . In this case, it can be found that E W˜ = W˜ T E T ≥ 0. Let R = 1 and ⎡

16 ⎢0 Q=⎢ ⎣0 0

0 5 2 0

0 2 4 0

⎤ 0 0⎥ ⎥. 0⎦ 9

7.4 Numerical Simulation

233

2

2 1

6

3

1

3

4

6

4

5

5

G1

G2

2

2

1

3

1

3

6

4

6

4

5

5

G3

G4

Fig. 7.3 Switching topology set S

From Theorem 7.2, by replacing A by A + B K 0 , it can be obtained that K˜ = [−0.0004, 0.2213, 0.0004, 0.0012], ⎤ 0.0026 0.0060 0 0 ⎥ ⎢ 0.0060 0.0335 0 0 ⎥, G1 = ⎢ 4 ⎦ ⎣ 0 0 0 2.6120 × 10 4 0 0 0 2.6120 × 10 ⎡ ⎤ 4.0799 −0.2007 5.0001 −7.8656 ⎢ −2.9640 1.2111 −0.9315 3.4410 ⎥ ⎥ G 2 = 104 × ⎢ ⎣ −1.2379 −0.4718 −2.8677 3.5233 ⎦. −1.1159 −1.0104 −4.0687 4.4247 ⎡

Thus, one can obtain that ⎡

0.0026 ⎢ 0.0060 W˜ = ⎢ ⎣ −0.0034 −0.0003

⎤ 0.0060 0 0 ⎥ 0.0335 0 0 ⎥, −0.0174 −0.0262 0.0002 ⎦ 0.0056 0.0003 −0.0112

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7 Admissible Consensus and Consensualization with Energy Constraints

K = [−26.4128, 11.3324, −0.0152, −0.1042]. The state trajectories of the water-floating plant with the switching signal σ (t) shown in Fig. 7.5 are given in Fig. 7.4, where the trajectories marked by circles describe the curves of the consensus function in Theorem 7.3, and the trajectory of the cost function JT is depicted in Fig. 7.6. One can see that the water-floating plant keeps horizontal and state trajectories converge to the curves of the consensus function. Furthermore, by repeating this simulation with different switching signals, it is found that state trajectories always converge to the same curves. This shows that switching movements do not impact the consensus function. Moreover, for the case that T = 4, due to V (4) = 0, by [36, 43–45], it can be obtained that 4 χ − JT =

  T 

Ʌσ (t) ⊗ W˜ −T B K + K T B T W˜ −1 − 2Q x T (t) 0, I(N −1)n

0



  − Ʌ2σ (t) ⊗ K T R K − I N −1 ⊗ W˜ −T A + AT W˜ −1 0, I(N −1)n x(t)dt = 1.0806 × 105 , which is coincident with Fig. 7.6.

Fig. 7.4 State trajectories

7.4 Numerical Simulation

Fig. 7.5 Switching signal

Fig. 7.6 Trajectory of the cost function JT

235

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7 Admissible Consensus and Consensualization with Energy Constraints

7.5 Notes Guaranteed-cost consensus analysis and design problems for high-order linear singular multi-agent systems with switching topologies were dealt with. LMI conditions for guaranteed-cost consensus and consensualization were presented, respectively, and sufficient conditions for consensus and consensualization were also given, respectively. These conditions are independent of the number of agents, so they can guarantee the scalability of singular multi-agent systems. Moreover, it was shown that an upper bound of the cost function is dependent on initial states of all agents and consensus functions are dependent on the average of initial states of all agents but are independent of switching movements. Since all the eigenvalues of the Laplacian matrices of undirected topologies are real, the convex property can be used to decrease calculation complexity. However, for multi-agent systems with switching directed topologies and nonlinear consensus protocols as discussed in [46, 47], the convex condition does not hold. Further work should be done on guaranteed-cost consensus analysis and design problems for singular multi-agent systems with nonlinear consensus protocols and directed topologies. Moreover, optimal control approaches may be used to deal with the cases that the bound on the guaranteed cost is specified a priori by system designers.

References 1. Weimerskirch H, Martin J, Clerquin Y, Alexandre P, Jiraskova S (2001) Energy saving in flight formation. Nature 413(18):697–698 2. Wang Z, He M, Zheng T, Fan Z (2018) Guaranteed cost consensus for high-dimensional multi-agent systems with time-varying delays. IEEE-CAA J Autom Sin 5(1):181–189 3. Xi JX, Yang XG, Yu ZY (2015) Leader–follower guaranteed-cost consensualization for highorder linear swarm systems with switching topologies. J Franklin Inst 352(4):1343–1363 4. Xi JX, Shi ZY, Zhong YS (2012) Admissible consensus and consensualization of high-order linear time-invariant singular swarm systems. Phys A 391(23):5839–5849 5. Xi JX, Meng FL, Shi ZY, Zhong YS (2012) Time-dependent admissible consensualization for singular time-delayed swarm systems. Syst Control Lett 61(11):1089–1096 6. Yang XR, Liu GP (2012) Necessary and sufficient consensus conditions of descriptor multiagent systems. IEEE Trans Circuits Syst I Regul Pap 59(11):2669–2677 7. Nedic A, Ozdaglar A, Parrilo PA (2010) Constrained consensus and optimization in multiagent networks. IEEE Trans Autom Control 55(4):922–938 8. Shi GD, Johansson KH, Hong YG (2013) Reaching an optimal consensus: dynamical systems that compute intersections of convex sets. IEEE Trans Autom Control 58(3):610–622 9. Li ZH, Ding ZT, Sun JY, Li ZK (2018) Distributed adaptive convex optimization on directed graphs via continuous-time algorithms. IEEE Trans Autom Control 63(5):1434–1441 10. Chen G, Li ZY (2018) A fixed-time convergent algorithm for distributed convex optimization in multi-agent systems. Automatica 95:539–543 11. Li RR, Yang GH (2020) Distributed optimization for a class of uncertain MIMO nonlinear multi-agent systems with arbitrary relative degree. Inf Sci 506:58–77 12. Cao YC, Ren W (2010) Optimal linear-consensus algorithms: an LQR perspective. IEEE Trans Syst Man Cybern Part B Cybern 40(3):819–830

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37. Sun YG, Wang L, Xie G (2008) Average consensus in networks of dynamic agents with switching topologies and multiple time-varying delays. Syst Control Lett 57(2):175–183 38. Ni W, Cheng D (2010) Leader-following consensus of multi-agent systems under fixed and switching topologies. Syst Control Lett 59(3):209–217 39. Xi JX, Shi ZY, Zhong YS (2012) Stable-protocol output consensualization for high-order swarm systems with switching topologies. Int J Robust Nonlinear Control 23(18):2044–2059 40. Dai LY (1989) Singular control systems. Springer, Berlin 41. Newcomb RW (1981) The semistate description of nonlinear time-variable circuits. IEEE Trans Circuits Syst I Regul Pap 28(1):62–71 42. Ma S, Hackwood S, Beni G (1994) Multi-agent supporting systems (MASS): control with centralized estimator of disturbance. In: IEEE/RSJ international conference on intelligent robots and systems 43. Xi JX, Cai N, Zhong YS (2010) Consensus problems for high-order linear time-invariant swarm systems. Phys A 389(24):5619–5627 44. Ma CQ, Zhang JF (2010) Necessary and sufficient conditions for consensusability of linear multi-agent systems. IEEE Trans Autom Control 55(5):1263–1268 45. Li Z, Duan Z, Chen G, Huang L (2010) Consensus of multiagent systems and synchronization of complex networks: a unified viewpoint. IEEE Trans Circuits Syst I Regul Pap 57(1):213– 224 46. Wu CW (2003) Synchronization in coupled arrays of chaotic oscillators with nonreciprocal coupling. IEEE Trans Circuits Syst I Regul Pap 50(2):294–297 47. Chen Y, Lü J, Lin Z (2013) Consensus of discrete-time multi-agent systems with transmission nonlinearly. Automatica 49(6):1768–1775 48. Yu J L, Dong X W, Li Q D, Ren Z. (2020 ) Robust H∞ guaranteed cost time-varying formation tracking for high-order multiagent systems with time-varying delays. IEEE Transactions on Systems, Man and Cybernetics: Systems, 50(4): 1465-1475

Chapter 8

Admissible Formation Tracking with Energy Constraints

This chapter investigates the problems of admissible formation tracking with energy constraints. Different from leaderless formation control problems as noted in [1–4], there exist one or more leader agents in the multi-agent systems for formation tracking problems. Formation tracking problems come from practical task backgrounds such as cooperative attacks and joint surveillances, where multiple agents are required to track some objectives with the specific formation structure. It should be pointed out that two critically important and practically restrict factors for formation tracking were not addressed in most previous work [5–8], that is, algebraic constraints among the cooperative states and the total energy limitation. If there are algebraic constraints, the dynamics of each agent is singular, and it has more special properties than a normal system as mentioned in Sect. 1.2. Furthermore, it is well known that the resource or energy of agents is usually constrained, and the energy constraint can be depicted as optimal or suboptimal analysis and design problems with specific optimization indexes. The optimization indexes can be local or global. Multi-agent systems can realize the local optimization control by optimizing the local objective of each agent as shown in Refs. [9, 10]. In most literatures related to optimization control [11–14], the energy consumption is not considered or cannot be given previously. However, the total energy supply for practical multi-agent systems should be limited previously, which can be limited by some energy index functions. Hence, time-varying formation tracking problems for singular multi-agent systems with the total energy limitation are necessary to be deeply explored. A new formation protocol is put forward to achieve admissible formation tracking in current chapter, that is, followers with specific formation structures track the cooperative state of the leader, where the total energy supply is limited and two types switching transmission topologies are involved. Furthermore, two nonsingular matrices with specific structures are constructed and the whole dynamics of a singular multi-agent system with the formation protocol is divided into two parts to design, which are independent with each other namely the dynamics of the leader and the disagreement dynamics between followers and the leader. Admissible formation tracking design and analysis criteria for singular multi-agent systems with switching © Science Press, Beijing and Springer Nature Singapore Pte Ltd. 2023 J. Xi et al., Admissible Consensus and Consensualization for Singular Multi-agent Systems, Engineering Applications of Computational Methods 11, https://doi.org/10.1007/978-981-19-6990-4_8

239

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8 Admissible Formation Tracking with Energy Constraints

transmission topologies containing a spanning tree are given, where the coupling relationship between the total energy supply and the matrix variable can be described by the Laplacian matrix of a star topology with edge weights being one and the leader being the central node. Moreover, sufficient conditions for admissible formation tracking of singular multi-agent systems with switching transmission topologies containing a joint spanning tree are shown, which are independent of the eigenvalues of Laplacian matrices associated with switching transmission topologies.

8.1 Problems of Formation Control The problem of formation control originates from the study of the group behavior of biological swarms in nature. We often see the swarm activities of fish, birds, and other animals. They generally have intelligent swarm behavior to survive in the harsh living environment. These animal swarms have strict cooperative formation rules, form formations according to a certain geometrical shape, and maintain formation during movement, which can make each individual play its special role on the corresponding position in the group. Consequently, the whole swarm can jointly fight against natural enemies and better complete a series of survival problems such as migration and predation, which has extraordinary biological significance and great enlightenment to other disciplines. The law of survival of the fittest in nature and the life science formed for millions of years enlighten us that when multiple agents work with a formation matching the task, they can exert great time efficiency and energy efficiency that a single agent does not have. In this case, formation research is a very meaningful work and has attracted extensive attention of scholars. It can be predicted that formation control can make great achievements in military, aerospace, industry, and other fields. Formation control studies how to make multiple agents in the system work together to achieve a global macroscopic goal with a static or dynamic geometry structure. According to whether the desired formation changes with time, the formation can be divided into time-invariant formation and time-varying formation, and whether there exist one or more leaders to guide the macroscopic motion of the multi-agent systems classifies the formation control into formation tracking control problems and leaderless formation problems. In the following, detail instructions and some practical application cases of formation control are proposed to further explain its characteristics.

8.1.1 Time-Invariant Formation The geometric configuration of agents in time-invariant formation is fixed. This kind of formation is mainly found in early literatures, especially in the research on robot formation control [15–17]. Traditional control strategies of time-invariant formation contain the following three method: leader-following-based method [18],

8.1 Problems of Formation Control

241

behavior-based method [19], and virtual structure-based method [20]. The basic idea of leader-following-based method is to designate one or more agents in the multiagent system as leaders and the other agents as followers. The leader moves along a global trajectory which is known previously, and followers maintain a specific relative position and angle relationship with the leader or adjacent followers. The basic idea of behavior-based formation control is that each agent in the multi-agent system has several predetermined behavior patterns, such as formation maintenance, collision avoidance, obstacle avoidance, moving to specific targets, etc. These behavior patterns construct a behavior set. Each behavior can produce a corresponding control effect, and the final controller of the subject is obtained by the weighted sum of the control effects of all behaviors. The multi-agent system can dynamically adjust the weight of different behavior control functions to achieve the required behavior patterns. The idea of formation control based on virtual structure method is to regard the desired formation as a rigid virtual structure. Each agent in the multi-agent system corresponds to a specific point on the virtual structure. When the formation moves, each agent can track its corresponding point on the virtual structure, then the whole formation movement can be realized. These three classical control strategies have their own advantages and disadvantages. Leader-following-based method is simple and easy to implement, but its robustness is poor. If the leader is broken down, the whole formation cannot be maintained. In addition, if the leader is particularly flexible or there are many nodes between a follower and the leader, it is prone to large formation errors. The advantage of behavior-based formation control strategy is that it can take into account the behavior modes of formation maintenance, collision avoidance, obstacle avoidance, and moving to specific targets simultaneously, and the method also has a high degree of intelligence. The disadvantage is that the model is too complex to analyze theoretically. The robustness of virtual structure-based method is satisfying, and this method has high formation accuracy. However, the construction of virtual structure will introduce large amount of communication and calculation. Besides, it is not easy for all agents to track the corresponding points on the virtual structure in real time.

8.1.2 Time-Varying Formation Time-varying formations are more practical and general than the time-invariant ones and are also more challenging to achieve. With the development and improvement of cooperative control of multi-agent systems in past decades, more and more researchers begin to use consensus theory to deal with time-varying formation control problems. The main idea of consensus-based formation control is that the states or outputs of each agent in the multi-agent system maintain a specific deviation relative to a common formation reference. At the beginning of formation control, the formation reference function can be unknown to each agent, but through distributed cooperation, all agents can reach an agreement on the formation reference, such that the desired formation can be achieved. Thus, this method usually transforms

242

8 Admissible Formation Tracking with Energy Constraints

the formation control problem into consensus problem by constructing formation error state vectors, and then use the relevant theory of consensus for subsequent analysis and design. In Ref. [1], Ren extended the consensus control protocol to the formation control of second-order multi-agent systems, and proved that the traditional formation control strategies of leader-following, behavior-based method and virtual structure-based method can be considered as special cases of consensus-based formation control method, and the shortcomings of these three control strategies can be overcome to a certain extent. Lafferriere et al. studied the time-varying formation control problem of a special class of high-order LTI systems in Ref. [21]. It should be noted that this kind of special high-order system can be considered as the series of multiple second-order systems in structure, so the analysis difficulty of this kind of system is relatively small. In Ref. [22], Fax and Murray discussed the formation stability of general higher-order LTI multi-agent systems. By introducing a detectable vector field into the multi-agent systems, Porfiri et al. expanded the research results of document [22] in document [7], and investigated the formation and tracking stability of general high-order LTI multi-agent systems. Although the system models in Ref. [7] are all general high-order systems, they only studied the problem of formation stability, but do not address how to achieve the desired formation. In order to tackling this problem, Dong et al. proposed a general time-varying formation control protocol in [23, 24], and gave the necessary and sufficient conditions for multi-agent system to realize a designed time-varying formation based on the consensus method, where the explicit expression of the time-varying formation reference function was also derived. Zhao et al. [25] designed a fully distributed time-varying formation controller based on adaptive method. It is worth noting that this controller does not need to know the global information of the interaction topology when designing the gain matrices. In addition, for a multi-agent system, whether an expected formation is feasible or what conditions should be satisfied to realize the time-varying formation are also important problems that should be concerned. Lin et al. [26] studied the formation feasibility problems of multiple under-actuated wheeled robots. Dong et al. [23, 27], proposed necessary and sufficient conditions for formation feasibility of general high-order multi-agent systems, they also give some useful methods to expand the feasible formation set as well as a protocol design algorithm for multi-agent system to realize time-varying formation.

8.1.3 Formation Tracking Problems Formation tracking problems also called leader-following formation control problems, where multiple followers are needed to track an objective with the specific geometrical structure, where the objective is usually regarded as the leader. Formation tracking has potential applications in joint attacks and cooperative surveillances. In [8], a finite-time leader-following formation protocol was shown and some novel formation criteria for first-order multi-agent systems were given, where each agent

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243

was modeled as a first-order integrator. In this case, if an agent does not receive any information from other agents, then its coordinated state is not changed. Dong et al. [28] molded each agent as a second-order integrator and time-varying leaderfollowing formation criteria were proposed. Especially, their theoretical results were successfully applied to deal with multiple quadrotor formation flying with a leaderfollowing geometric structure. Wang [29] proposed a leader-following adaptive formation control protocol for second-order multi-agent systems under switching directed topologies, which are more general and have stronger robustness. In [30], the dynamics of each agent was described by a general high-order LTI system and formation criteria for multi-agent systems containing multiple leaders were proposed on the basis of Riccati equations. The formation tracking control problem of multiagent system with fixed time convergence is studied in [31], where followers can form a specified formation in a fixed time, and their geometric center moves synchronously with the leader. The formation control protocol is designed based on the formation error system and terminal SMC method. By using Lyapunov stability theorem and fixed time stability theorem, it is proved that the control task is possible to converge in fixed time, and the convergence time can be calculated by several parameters. In [32], Xiong et al. investigated the time-varying formation tracking control of multiagent systems with model uncertainty and without leader speed measurement. For each follower, a new fixed time cascade leader state observer (CLSO) is designed to reconstruct the leader state, and radial basis function neural networks (RBFNN) is used to deal with the model uncertainty online. By taking the square of neural network weight vector norm as a new adaptive parameter, the RBFNN adaptive control scheme based on minimum learning parameter method and fixed time CLSO is constructed to solve the problem of time-varying formation tracking. Moreover, in order to satisfy the requirements of limited communication frequency among multiple agents, event-triggered mechanism is introduced into formation tracking control recently [33, 34].

8.1.4 Leaderless Formation Control Problems For leaderless formation, all agents in multi-agent systems own equal status and make decision in a cooperative manner as shown in [1, 3, 11, 12, 35], and the trajectory of the formation center usually depends on the initial value of agent and formation function as well as the dynamic characteristics of each agent. In Refs. [11, 12], the leaderless formation control problems were investigated by the consensus-based formation control strategy, which has inbuilt advantage in the distributed realization and the theoretical rigorism compared with the behavior-based formation control strategy and the virtual structure formation control strategy, as shown in [1]. Liu et al. [36] proposed some new robust formation criteria for networked multi-agent systems with disturbances and applied the theoretical conclusions to a set of quadcopters. In [37], a new approach was proposed to investigate the robust formation design problem, where an ESO is constructed to deal with the impacts of external disturbances. In [38],

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8 Admissible Formation Tracking with Energy Constraints

the active disturbance rejection control method is introduced into the time-varying formation control problem of UAV multi-agent systems with external disturbances. The external disturbance is also estimated by the ESO, and a distributed formation control protocol is subsequently designed with a compensation part that equal to the estimated disturbances, under which the predetermined time-varying formation can be realized. Distributed time-varying formation control with nonuniform time delays and jointly connected topologies was investigated in [39], where the union of topologies in certain time interval is connected, but each topology in the switching set can be unconnected, this study is more challenging than the one for switchingconnected interaction topology cases. Obstacle avoidance is another hot topic that attracts special attention in leaderless formation control, where each agent in the multi-agent system should not only avoid collision with its neighbors, but also avoid collision with obstacles in an unfamiliar and complex environment. In [40], a new method to realize formation control and obstacle avoidance of multiple-agent systems in limited communication range is proposed. The distributed formation control algorithm is designed by using the improved artificial potential function, which can ensure the fast formation performance and no collision occurring between any two agents. In addition, by proposing the potential exclusion function and using it as the obstacle avoidance function, the agent can perfectly avoid obstacles of different shapes and sizes. Consequently, multiagent can move together and form predefined formation quickly, such as straight line, circle, and pyramid, while avoiding obstacles simultaneously.

8.1.5 Formation Control Examples of Multiple UAVs In recent years, with the development of sensor technology, microelectronics technology, and computer technology, some formation control experiments are gradually appeared. Especially in the formation control of UAVs, many important formation control theories are verified by flight experiments. Ref. [41] introduced the formation flight of two unmanned helicopters realized by the National University of Singapore based on the leader-following method. The two UAVs achieves communication by WiFi modules, and the follower receives the leader’s state information at a frequency of 10 Hz. The results shows that the leader-following system obtained a good tracking performance. Figure 8.1 is a flight snapshot. Professor Dong and his team from Beijing University of Aeronautics and Astronautics in China have conducted a lot of significant formation flight experiments [5, 27, 42]. They mainly utilize the consensus-based method to address timevarying formation problems, and have obtained many remarkable results in formation tracking, formation containment, target enclosing, group formation control, and so on [23, 30, 43, 44]. Figure 8.2 shows a leaderless formation flight experiment in outdoor environment with switching interaction topologies by Dong’s work in [43], where the position and velocity states of each UAV are measured by the GPS module

8.1 Problems of Formation Control

245

Fig. 8.1 Leader-following formation flight experiment [41]

with an accuracy of 1.2 m CEP at a rate of 10 Hz, and the communication among each agent with its neighbors is realized by the ZigBee network. Figure 8.3 shows a captured image of four UAVs in target enclosing, where three follower UAVs enclose the target UAV with the desired time-varying formation [5]. Ref. [45] proposed a distributed formation tracking control protocol by distancebased formation method, in which the leader moves with an unknown velocity. Each follower can only control its distance among its neighbors and itself by measuring the relative positions, and the control law is mathematically verified by using graph theory and nonlinear control theory. As shown in Fig. 8.4, the effectiveness of the proposed algorithm is verified by the flight experiments of five UAVs. Figure 8.5 realized a hand gesture-based distributed formation of multiple UAVs by integrating the retraction balancing control into an onboard model predictive

Fig. 8.2 Leaderless formation flight experiment with four UAVs [43]

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8 Admissible Formation Tracking with Energy Constraints

Fig. 8.3 A captured image of the target enclosing experiment [5]

Fig. 8.4 Distance-based formation control experiment with five quadcopters [45]

control framework [46]. By using virtual reality equipment to generate finger trajectories, this method enables the multiple UAVs to achieve the corresponding formation that is configured by the five fingers. The control method is suitable for twodimensional and three-dimensional space formation, and the formation function can be time varying (e.g., expanding, shrinking, or moving).

8.2 Formation Tracking Control Protocol with Energy Constraint Consider a singular multi-agent system with l + 1 agents, where agent 0 is set as the leader and the other l agents as homogenous followers, then the dynamics of singular multi-agent systems with a leader-following structure is given as follows:

8.2 Formation Tracking Control Protocol with Energy Constraint

247

Fig. 8.5 Screenshot of the progress of generating a formation shape via human–robot interaction [46]



E x˙0 (t) = Ax0 (t), E x˙m (t) = Axm (t) + Bu m (t),

(8.1)

where m = 1, 2, . . . , l, E ∈ Rh×h , A ∈ Rh×h , B ∈ Rh×k , and xm (t) and u m (t) are the cooperative state and the control input, respectively. The rank of E can be less than h; that is, rank(E) ≤ h, so multi-agent system (8.1) is said to be singular. Moreover, let f m (t) (m ∈ {1, 2, . . . , l}) be the formation function of agent m, which is piecewise continuous differentiable. The formation functions f m (t) (m = 1, 2, . . . , l) are used to determine the required formation structure among followers and they are not the trajectories for the followers to track. It is noteworthy that the arbitrary piecewise continuous differentiable formation structure among followers can be designed by f m (t) (m = 1, 2, . . . , l), but it may be not achievable due to structure constraints of (E, A) the system matrix A. Moreover, the following assumption is necessary: Assumption 8.1 The triple (E, A, B) is stabilizable. Due to rank(E) ≤ h, multi-agent system (8.1) has three types of modes, that is, dynamic ones, static ones, and impulsive ones. Impulsive modes bring in devastating impacts on the operation  ofthe whole system, so they must be eliminated. E 0 and the condition rank rank = h + rank(E) is necessary and sufficient for A E guaranteeing (E, A) to be regular and impulse-free, as shown in [47]. The cooperative states corresponding to both dynamic modes and static modes should realize formation tracking. To achieve this objective, a new formation tracking protocol with switching transmission topologies and energy constraints is proposed as follows: ⎧ fl ff u m (t) = u m (t) + u m (t), ⎪ ⎪ ⎪ f l m0 ⎪ K u (x0 (t) − xm (t) + f m (t)), u (t) = wξ(t) ⎪ ⎪ ⎨ mf f mi wξ(t) u m (t) = K u (xi (t) − f i (t) − xm (t) + f m (t)), m i∈Nξ(t) ,i/=0 ⎪ ⎪ ⎪ ⎪ l ∞ ⎪ ⎪ J = u T (t)Qu (t)dt, ⎩ e

m=1 0

m

m

(8.2)

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8 Admissible Formation Tracking with Energy Constraints

where m = 1, 2, . . . , l, K u ∈ Rk×h , Q T = Q > 0 is the symmetric and positive definite weighted matrix, and Je is the practical energy consumption of multi-agent system (8.1) as a whole. Let Je∗ be the total energy supply of the whole singular multi-agent system, which is limited previously and is required to be larger than the practical energy consumption. ξ(t) : [ 0, +∞) → ϕ is the switching signal, where the index ϕ ∈ N of the switching set γs with N denoting the natural number set includes labels of all possible transmission topologies for switching. Topology switching movements satisfy that ti − ti−1 ≥ ζd , where ζd > 0 is the minimum dwell time between switching moments ti−1 and  ti . Thus, it can be found that ξ(t) is piecei = m : (n m , n i ) ∈ E(G ξ(t) ) denotes the set of wise continuous. The symbol Nξ(t) the agents which transmit their information to agent i at time t, which is called the im represents the transmission weight from neighbor set of agent i. The notation wξ(t) im agent m to agent iat time t, where wξ(t) = 0if agent m does not transmit its informaim tion to agent i and wξ(t) > 0 otherwise. The Laplacian matrix of the switching trans  mi mm mi ∈ R(l+1)×(l+1) with lξ(t) = i∈Nξ(t) mission topology G ξ(t) is L ξ(t) = lξ(t) m w ξ(t) mi mi and lξ(t) = −wξ(t) (m /= i ). We present the definition of the admissible formation tracking achievability of singular multi-agent systems with energy constraints.

Definition 8.1 For any given Je∗ > 0, multi-agent system (8.1) is said to be admissible formation tracking achievable by formation protocol (8.2) if it is regular and impulse-free and there exists K u such that limt→∞ (xm (t) − f m (t) − x0 (t)) = 0 (m = 1, 2, . . . , l) and Je ≤ Je∗ for any bounded disagreement admissible initial conditions xm (0) − f m (0) (m = 1, 2, . . . , l). For two kinds of switching transmission topologies, that is, switching transmission topologies containing a spanning tree and switching transmission topologies containing a joint spanning tree, the main purpose of this article is to design the gain matrix such that multi-agent system (8.1) with formation control protocol (8.2) achieves admissible formation tracking with the condition that the total energy supply is predetermined. fl

Remark 8.1 In formation protocol (8.1), u m (t) = 0 if agent m does not obtain any ff information from the leader at time t and u m (t) = 0 if agent m does not receive fl ff the information from any followers at time t; that is, u m (t) and u m (t) describe the control impacts of the leader and following agents on the cooperative state of agent m, respectively. Furthermore, the critical challenge is to establish the coupling relationship between the total energy supply and the gain matrix; that is, the gain matrix K u can make multi-agent system (8.1) achieve admissible formation tracking and can meanwhile restraint the practical energy consumption less than the total energy supply Je∗ . Guaranteed-cost formation tracking criteria for normal multi-agent systems in Ref. [48] do not determined this coupling relationship and the guaranteed cost cannot be limited previously. Besides, essential conclusions and concepts on graph theory and singular systems are available in [47, 49], respectively, and they are neglected due to the limitation of the length.

8.3 Energy-Constraint Admissible Formation Tracking Criteria

249

8.3 Energy-Constraint Admissible Formation Tracking Criteria Two kinds of formation energy-constraint admissible tracking criteria were proposed in this section. Firstly, by a dual nonsingular transformation, sufficient conditions are provided for formation tracking of singular multi-agent systems with energy constraints and switching transmission topologies containing a spanning tree. Then, the main conclusions of singular multi-agent systems with switching transmission topologies containing a spanning tree are extended into those cases with a joint spanning tree.

8.3.1 Case of Switching Topologies Containing a Spanning Tree By a dual nonsingular transformation, the whole dynamics of multi-agent system (8.1) with formation protocol (8.2) is decomposed into the dynamics of the leader and the disagreement dynamics between followers and the leader, and the influence of the asymmetric structure of the leader-following transmission topology is well dealt with. Then, sufficient conditions for admissible formation tracking design and analysis are proposed, respectively, where the total energy supply is previously limited. Let ηm (t) = xm (t) − f m (t), then the dynamics of each follower can be rewritten as E η˙ m (t) = A(ηm (t) + f m (t)) + Bu m (t) − E f˙m (t),

(8.3)

where m = 1, 2, . . . , l. We introduce an auxiliary variable f 0 (t) ≡ 0 and

T set that η0 (t) = x0 (t) − f 0 (t). Let η(t) = η0T (t), ηTf f (t) with η f f (t) =  T T η1 (t), η2T (t), . . . , ηlT (t) and em (m ∈ {1, 2, . . . , l}) be an h-dimensional column vector with the mth component 1 and 0 elsewhere, then it can be found that    fl fl m0 u mf l (t) = wξ(t) K u (η0 (t) − ηm (t)) = emT lξ(t) , −Ʌξ(t) ⊗ K u η(t), where  10  fl 20 l0 T lξ(t) = wξ(t) , wξ(t) , . . . , wξ(t) ,  10 fl 20 l0 . Ʌξ(t) = diag wξ(t) , wξ(t) , . . . , wξ(t) Furthermore, one can see that

(8.4)

250

8 Admissible Formation Tracking with Energy Constraints

u mf f (t) = K u

  ff mi wξ(t) (ηi (t) − ηm (t)) = − emT L ξ(t) ⊗ K u η f f (t),



(8.5)

m i∈Nξ(t) ,i/=0

ff

where L ξ(t) is the Laplacian matrix among followers at time t without considering  T the impacts of the leader. Let f (t) = f 0T (t), f 1T (t), . . . , flT (t) , it can be obtained from (8.3) to (8.5) that   ˙ = Il+1 ⊗ A − L ξ(t) ⊗ B K u η(t) + (Il+1 ⊗ A) f (t) (Il+1 ⊗ E)η(t) − (Il+1 ⊗ E) f˙(t),

(8.6)

where the Laplacian matrix L ξ(t) has a specific structure as follows:  L ξ(t) =

 0 0 . fl ff fl −lξ(t) L ξ(t) + Ʌξ(t)

Now, we show a dual nonsingular transformation to decompose the whole dynamics and convert formation tracking problems into asymptotic stability ones. Introduce the following nonsingular matrix  U=

 1 0 , 1l Il

then the inverse matrix of U is U ff

−1

 1 0 . = −1l Il 

fl

It can be deduced that Ʌξ(t) 1l = lξ(t) , so one has  −1

L ξ(t) U =



0

.

(8.7)

T  −1   U ⊗ Ih η(t) = x0T (t), η˜ T (t) ,

(8.8)

U

ff

fl

L ξ(t) + Ʌξ(t)

One can further show that

 T T where η(t) ˜ = η˜ 1 (t), η˜ 2T (t), . . . , η˜lT (t) with η˜ m (t) = ηm (t) − x0 (t) (m = 1, 2, . . . , l). Since each transmission topology in the switching set contains a spanning tree, zero is the single eigenvalue of the Laplacian matrix L ξ(t) . ff fl Due to the symmetric information transmission among followers, L ξ(t) + Ʌξ(t) is symmetric and positive definite by (8.7). Hence, there exists an orthonormal matrix U ξ(t) satisfying that

8.3 Energy-Constraint Admissible Formation Tracking Criteria

   T ff fl U ξ(t) L ξ(t) + Ʌξ(t) U ξ(t) = diag λ1ξ(t) , λ2ξ(t) , . . . , λlξ(t) ,

251

(8.9)

where 0 < λ1ξ(t) ≤ λ2ξ(t) ≤ · · · ≤ λlξ(t) are nonzero eigenvalues of L ξ(t) . Let  T T U ξ(t) η(t) ˜ = ηT1 (t), ηT2 (t), . . . , ηlT (t) . Due to f 0 (t) ≡ 0 and f˙0 (t) ≡ 0, by (8.8) and (8.9), we can transform subsystem (8.6) into E x˙0 (t) = Ax0 (t),

(8.10)

  .   T T −1 E ηm (t) = A − λm B K η (t) + e U ⊗ A f (t) I [0, ]U u l m ξ(t) m ξ(t)   T − emT U ξ(t) [0, Il ]U −1 ⊗ E f˙(t),

(8.11)

where m = 1, 2, . . . , l. Because ηm (t) = xm (t) − f m (t), η˜ m (t) = ηm (t) − x0 (t) (m = 1, 2, . . . , l) and U ξ(t) is orthonormal, multi-agent system (8.1) with formation protocol (8.1) achieves admissible formation tracking if and only if it is regular and impulse-free and limt→∞ ηm (t) = 0 (m = 1, 2, . . . , N ). It can be found that the whole dynamics of multi-agent system (8.1) with formation protocol (8.2) can be divided into two independent parts as shown in (8.10) and (8.11) by a dual nonsingular transformation, where subsystem (8.10) is identical to the dynamics of the leader and subsystem (8.11) describes the disagreement dynamics of formation tracking. m Let λm min and λmax denote the minimum and maximum eigenvalues of the Laplacianmatrix of the mth switching transmission topology, respectively. Let λmin =  min λimin , ∀i ∈ γs and λmax = max λimax , ∀i ∈ γs , then the following theorem shows sufficient conditions for admissible formation tracking design of singular multi-agent systems with switching transmission topologies containing a spanning tree. Theorem 8.1 For any given Je∗ > 0, multi-agent system (8.1) with switching transmission topologies containing a spanning tree is admissible formation tracking achievable by formation protocol (8.2) if A f m (t) − E f˙m (t) = 0 (m = 1, 2, . . . , l) and there exists an invertible matrix W such that T

W E T = E W ≥ 0,   T T AW + W AT − 2λmin B K u λmin K u Q Ωmin = < 0, ∗ −Q (i)   T T AW + W AT − 2λmax B K u λmax K u Q Ωmax = < 0, ∗ −Q    l −1lT T ⊗ Ih η(0)E T ≤ Je∗ W , (ii) η (0) −1l Il  T where η(0) = x0T (0), x1T (0) − f 1T (0), . . . , xlT (0) − flT (0) . In this case, Ku = K u W

−1

.

252

8 Admissible Formation Tracking with Energy Constraints

Proof Construct the following common Lyapunov function Vm (t) = ηTm (t)E T W ηm (t), where m = 1, 2, . . . , l and E T W = W T E ≥ 0. Taking the derivative of Vm (t) relative to time t along the trajectory of subsystem (8.11), one can deduce that 

T   T A − λm A − λm ξ(t) B K u W + W ξ(t) B K u ηm (t)   T + 2ηTm (t)W T emT U ξ(t) [0, Il ]U −1 ⊗ A f (t)   T − 2ηTm (t)W T emT U ξ(t) [0, Il ]U −1 ⊗ E f˙(t).

V˙m (t) = ηTm (t)

One can further obtain that  T   T A − λm ηTm (t) A − λm ξ(t) B K u W + W ξ(t) B K u ηm (t)   = ηTm (t) AT W + W T A − 2λm ξ(t) B K u ηm (t).

(8.12)

(8.13)

Since the LMI has the convex property, it can be deduced that AT W + W T A − 2λ W T B K u < 0 and AT W + W T A − 2λmax W T B K u < 0 can guarantee that  min T   m T A − λm A − λ < 0 for any switching signal ξ(t). In B K W + W B K u u ξ(t) ξ(t) this case, the matrix W is invertible. It can be found that     T T W T emT U ξ(t) [0, Il ]U −1 ⊗ A f (t) = W T emT U ξ(t) [0, Il ]U −1 ⊗ Ih (I N ⊗ A) f (t)     T T W T emT U ξ(t) [0, Il ]U −1 ⊗ E f˙(t) = W T emT U ξ(t) [0, Il ]U −1 ⊗ Ih (I N ⊗ E) f˙(t).

  T  T T T Let Δ(t) = A f 1 (t) − E f˙1 (t) , A f 2 (t) − E f˙2 (t) , . . . , A fl (t) − E f˙l (t) , then one can obtain that     T T W T emT U ξ(t) [0, Il ]U −1 ⊗ A f (t) + W T emT U ξ(t) [0, Il ]U −1 ⊗ E f˙(t)   T = W T emT U ξ(t) [0, Il ]U −1 ⊗ E Δ(t). (8.14) When A f m (t) − E f˙m (t) = 0 (m = 1, 2, . . . , l), then Δ(t) = 0. From (8.12) to (8.14), if AT W + W T A−2λmin W T B K u < 0 and AT W + W T A−2λmax W T B K u < 0, ˙ then V˙m (t) ≤  0 and Vm (t) = 0 if and only if ηm (t) ≡ 0; that is, limt→∞ ηm (t) = 0. E 0 Due to rank = h + rank(E), subsystem (8.10) is regular and impulse-free. A E E T W = W T E ≥ 0, AT W + W T A − 2λmin W T B K u < 0 and AT W + W T A −

8.3 Energy-Constraint Admissible Formation Tracking Criteria

253

2λmax W T B K u < 0 can also ensure that subsystem (8.11) is regular and impulsefree. In this case, multi-agent system (8.1) with formation protocol (8.2) is regular and impulse-free since the nonsingular transformation does not alter the regular and impulse-free properties. Hence, when the total energy supply is not considered, multiagent system (8.1) with switching transmission topologies containing a spanning tree is admissible formation tracking achievable by formation protocol (8.2). Furthermore, the impacts of the total energy supply are dealt with. One can find by (8.4) and (8.5) that   

  fl fl ff u mf l (t) + u mf f (t) = emT lξ(t) ⊗ K u η0 (t) − emT Ʌξ(t) + L ξ(t) ⊗ K u η f f (t).

  fl fl ff Due to emT lξ(t) , −emT Ʌξ(t) + L ξ(t) 1l+1 = 0, one can deduce that     fl ff ˜ u mf l (t) + u mf f (t) = − emT Ʌξ(t) + L ξ(t) ⊗ K u η(t). By (8.2), one can see that   +∞ 2 fl ff T T ˜ Je = η˜ (t) Ʌξ(t) + L ξ(t) ⊗ K u Q K u η(t)dt.

(8.15)

0

T  T Due to ηT1 (t), ηT2 (t), . . . , ηlT (t) = U ξ(t) η(t), ˜ it can be obtain that η˜ T (t)

  l 2   m 2 T fl ff ˜ = λξ(t) ηm (t)K uT Q K u ηm (t). Ʌξ(t) + L ξ(t) ⊗ K uT Q K u η(t) m=1

(8.16) Let ζe ≥ 0, then one can see by (8.15) and (8.16) that Jeζe

ζe Δ

η˜ T (t)

  2 fl ff ˜ Ʌξ(t) + L ξ(t) ⊗ K uT Q K u η(t)dt

0 l    m 2 T = λξ(t) ηm (t)K uT Q K u ηm (t)dt. ζe

m=1 0

Since

Jeζe

ζe 0

V˙m (t)dt = Vm (ζe ) − Vm (0), it can be shown that

⎞ ⎛ ζe   l    m 2 T ⎝ V˙m (t) + λξ(t) ηm (t)K uT Q K u ηm (t) dt − Vm (ζe ) + Vm (0)⎠ = m=1

0

254

8 Admissible Formation Tracking with Energy Constraints l  e     T   −1 = ηTm (t)W T W −T A − λm + A − λm ξ(t) B K u ξ(t) B K u W ζ

m=1 0 l     2 −T T  + λm K u Q K u W −1 W ηm (t) dt − Vm (ζe ) + Vm (0). (8.17) ξ(t) W m=1

Let W −1 = W and K u = K u W . By Schur Complement lemma in [50] and the T convex property of the LMI, if W E T = E W ≥ 0, Ωmin < 0 and Ωmax < 0, then limt→∞ ηm (t) = 0 (m = 1, 2, . . . , l) and limζe →∞ Vm (ζe ) = 0. Thus, it can be shown by (8.15) and (8.17) that Je ≤

l 

ηTm (0)E T W ηm (0).

(8.18)

m=1

  T ˜ (m = 1, 2, . . . , l), one has Due to ηm (0) = emT U ξ(t) ⊗ Ih η(0) l 

  ηTm (0)E T W ηm (0) = η˜ T (0) Il ⊗ E T W η(0). ˜

(8.19)

m=1

  Since η(0) ˜ = [0, Il ]U −1 ⊗ Ih η(0), one has 





η˜ (0) Il ⊗ E W η(0) ˜ = η (0) T

T

T

  l −1lT T ⊗ E W η(0). −1l Il

(8.20)

From (8.18) to (8.20), it can be found that  Je ≤ η (0) T

  l −1lT T ⊗ E W η(0). −1l Il

(8.21)

Because xm (0) − f m (0) (m = 1, 2, . . . , l) are disagreement, there exists some agents cannot track the cooperative state of the leader; that is, there exists some ηm (0) /= 0 (m ∈ {1, 2, . . . , l}). Hence, one can derive that  η (0) T

  l  l −1lT ⊗ Ih η(0) = ηTm (0)ηm (0) > 0. −1l Il m=1

In this case, one can find a positive scalar α such that Je∗

 = η (0) T

  l −1lT ⊗ α Ih η(0). −1l Il

(8.22)

8.3 Energy-Constraint Admissible Formation Tracking Criteria

255



 l −1lT By (8.21) and (8.22), E W ≤ α Ih can ensure that Je ≤ since is −1l Il positive semi-definite. Thus, one can obtain the conclusion of Theorem 8.1. □ Theorem 8.1 gives an admissible formation tracking design criterion; that is, a new method is proposed to design the gain matrix K u such that multi-agent system (8.1) with formation protocol (8.2) achieves admissible formation tracking. When the gain matrix K u is given, the following corollary shows sufficient conditions for admissible formation tracking analysis with energy constraints and switching transmission topologies containing a spanning tree. T

Je∗

Corollary 8.1 For any given Je∗ > 0 and K u , multi-agent system (8.1) with formation protocol (8.2) and switching transmission topologies containing a spanning tree achieves admissible formation tracking if A f m (t) − E f˙m (t) = 0 (m = 1, 2, . . . , l) and there exists an invertible matrix W such that   E 0 (i) rank = h + rank(E), A E E T W = W T E ≥ 0,  T  W A + AT W − 2λmin W T B K u λmin K uT Q Ωmin = < 0, (ii) ∗ −Q  T  W A + AT W − 2λmax W T B K u λmax K uT Q Ωmax = < 0, ∗ −Q    l −1lT ⊗ Ih η(0)E T W ≤ Je∗ Ih , (iii) ηT (0) −1l Il  T where η(0) = x0T (0), x1T (0) − f 1T (0), . . . , xlT (0) − flT (0) . Remark 8.2 By selecting different formation functions f m (t) (m = 1, 2, . . . , l), followers can form specific geometric structures to track objective, that is, the cooperative state of the leader. However, it should be pointed out that some formation structures may be not feasible due to structure features of the dynamics of each agent. It was revealed that the formation feasible condition is A f m = 0 (m = 1, 2, . . . , l) for timeinvariant formation in [1]. In Theorem 8.1, A f m (t) − E f˙m (t) = 0 (m = 1, 2, . . . , l) can be regarded as the formation feasible condition for time-varying formation, which can be used to restrict the formation time-varying rate, since the dynamics of each agent is singular, the formation feasible condition is different with the normal one as shown in [7, 51]. Moreover, the formation feasible condition essentially reflects the constraint of the dynamics of each agent, which should be satisfied for multi-agent systems to realize formation tracking, and the detailed analysis can be found in [23]. Besides, for leaderless multi-agent systems, switching transmission topologies are symmetric and the orthonormal transformation can be directly used to deal with the impacts of energy constraints. However, information transmissions from the leader to followers are unidirectional, which means that switching transmission topologies are asymmetric. We put forward a dual nonsingular transformation approach to obtain the disagreement dynamics between the leader and followers and to eliminate the

256

8 Admissible Formation Tracking with Energy Constraints

impacts of the asymmetric structure on admissible formation tracking with energy constraints. Remark 8.3 For many practical multi-agent systems, the total energy supply is limited previously and is not always available, so it is necessary to consider the impacts of the energy limitation on admissible formation tracking. Essentially speaking, it is required to design a proper formation control protocol, which can make multi-agent systems realize objective tracking with some specific formation structure and can restrict the practical energy consumption less than the total energy supply. The key challenge is to introduce the total energy supply into admissible formation tracking criteria. By dividing the whole interaction into the interactions among followers and the interactions from the leader to followers and constructing a constant transformation matrix with a specific structure, Theorem 8.1 determines the coupling relationship matrix between the total energy supply and the matrix variable, which is essentially a Laplacian matrix that describes a time-invariant star topology with edge weights being 1 and the leader being the central node. It can also be found that switching movements do not impact this relationship matrix although they are critically important for singular multi-agent systems to realize admissible formation tracking.

8.3.2 Case of Switching Topologies Containing a Joint Spanning Tree This section deals with admissible formation tracking design and analysis problems for singular multi-agent systems with switching transmission topologies containing a joint spanning tree and energy constraints and presents the corresponding admissible formation tracking criteria via the Cauchy convergence criterion and Barbalat’s lemma. 

     The time interval tk , tk+1 ) is divided into tk0 , tk1 , tk1 , tk2 , . . . , tkik −1 , tkik with j+1

j

tk0 = tk and tkik = tk+1 , where tk − tk ≥ ζd ( j = 0, 1, . . . , i k − 1), k is a nonnegative integer and these subintervals are not overlapping and continuous. The transmission topologies switch at tk0 , tk1 , . . . , tkik −1  and the topology structure does not j

j+1

change during the time subinterval tk , tk ( j = 0, 1, . . . , i k − 1). It is supposed that the union of the transmission topologies G tk0 , G tk1 , . . . , G t ik −1 contains a spank ning tree, but each transmission topology may not contain; that is, switching transmission topologies contain a joint spanning tree. Let L ξ (tk0 ) , L ξ (tk1 ) , . . . , L ξ t ik −1  k

denote the Laplacian matrices of G tk0 , G tk1 , . . . , G t ik −1 , respectively. One can set that k ik −1   k+1 L ξ (tkk+1 ) = L , then L is the Laplacian matrix of a transmission j ξ (tk ) j=0 ξ t k topology containing  a spanning tree when switching transmission topologies during the time interval tk , tk+1 ) contain a joint spanning tree.

8.3 Energy-Constraint Admissible Formation Tracking Criteria

257

By the LMI tool, sufficient conditions for admissible formation tracking design of multi-agent system (8.1) with formation protocol (8.2) are presented, where switching transmission topologies contain a joint spanning tree and the total energy supply is limited previously. Theorem 8.2 For any given Je∗ > 0, multi-agent system (8.1) with switching transmission topologies containing a joint spanning tree is admissible formation tracking ˙ achievable  by formation protocol (8.2) if A f m (t) − E f m (t) = 0 (m = 1, 2, . . . , l), E 0 rank = h + rank(E), there exist an invertible matrix W and a scalar β such A E that



W T E T = E W ≥ 0,



A W + W T AT ≤ 0,   T T T A W + W A − β B B 0.5β B Q Ω= < 0, ∗ −Q    l −1lT ⊗ Ih η(0)E T ≤ Je∗ W , (ii) ηT (0) −1l Il  T where η(0) = x0T (0), x1T (0) − f 1T (0), . . . , xlT (0) − flT (0) . In this case, (i)



K u = 0.5β B T W −1 . Proof According to the analysis of Theorem 8.1, when A f m (t) − E f˙m (t) = 0 (m = 1, 2, . . . , l), the disagreement dynamics of multi-agent system (8.1) with formation protocol (8.2) can be transformed by the nonsingular matrix U into     . ff fl ˜ ˜ = Il ⊗ A − L ξ(t) + L ξ(t) ⊗ B K u η(t). (Il ⊗ E) η(t)

(8.23)

Consider a common Lyapunov function as follows   V˜ (t) = η˜ T (t) Il ⊗ E T W˜ η(t), ˜

(8.24)

where E T W˜ = W˜ T E ≥ 0. Taking the derivative of V˜ (t) relative to time t along the trajectory of subsystem (8.23), it can be found that       ff fl ˜ (8.25) V˙˜ (t) = η˜ T (t) Il ⊗ W˜ T A + AT W˜ − 2 L ξ(t) + L ξ(t) ⊗ W˜ T B K u η(t). ff

fl

Since the transmission topology among followers is undirected, L ξ(t) + L ξ(t) is symmetric and positive semi-definite and its eigenvalues are the ones of the whole ff fl Laplacian matrix L ξ(t) except a zero eigenvalue. Actually, L ξ(t) + L ξ(t) may have multiple zero eigenvalues for switching transmission topologies containing a joint

258

8 Admissible Formation Tracking with Energy Constraints

spanning tree. Let K u = 0.5β B T W˜ with β > 0. If W˜ T A + AT W˜ ≤ 0, then one can obtain by (8.25) that V˙˜ (t) ≤ −β η˜ T (t)



  ff fl L ξ(t) + L ξ(t) ⊗ W˜ T B B T W˜ η(t) ˜ ≤ 0.

(8.26)

From (8.23), one can find that η(t) ˜ is uniformly continuous. Due to Il ⊗ E T W ≥ 0, ˜ it can be found that V (t) is not negative and is uniformly continuous. In (8.26), it is indicated that the value of V˜ (t) does not increase as t increases, so V˜ (t) is bounded and the limit of V˜ (t) as t tends to infinity is a constant, which can be nonzero. Next, we show that the limit of V˜ (t) as t tends to infinity is zero instead of a nonzero constant. By the Cauchy convergence criterion, for any constant σ > 0 and the infinite sequence V˜ (tk ) (k = 0, 1, . . .), there exists a positive constant N ∈ N such that ∀k > N : V˜ (tk ) − V˜ (tk+1 ) < σ , which means that tk+1 · − V˜ (t)dt < σ. tk

 Since the time interval tk , tk+1 ) can be divided into multiple overlapping and continuous subintervals, one can find that tk

1

−

V˜ (t)dt −

tk0 j+1

tk

tk ik

2

·

·

V˜ (t)dt − · · · −

·

V˜ (t)dt < σ.

(8.27)

i −1 tkk

tk1

j

Due to tk − tk ≥ ζd > onecan obtain  0 ( j = 0, 1, . . . , i k − 1),   from  (8.26)  j j+1 j j ˜ ˜ ˜ ˜ < , which is equivalent to − V tk + ςd − V tk that V tk + ςd ≥ V tk      j+1 j − V˜ tk − V˜ tk ; that is, j

t k +ζd

−

j+1

·

V˜ (t)dt ≤ −

j

tk

·

V˜ (t)dt.

j

tk

tk

By (8.27) and (8.28), one has j

i k −1 j=0

t k +ζd

β η˜ T (t) j

tk

which means that

 L

ff j ξ(tk )

+L

fl j ξ(tk )



 ⊗ W˜ T B B T W˜ η(t)dt ˜ < σ,

(8.28)

8.3 Energy-Constraint Admissible Formation Tracking Criteria t+ζd i k −1 

lim

t→∞

β η˜ T (t)

 L

j=0 t

ff j ξ(tk )

+L

fl j ξ(tk )



259

 ⊗ W˜ T B B T W˜ η(t)dt ˜ = 0.

·

(8.29)

·

˜ Since ˜ is bounded. From (8.23), η(t) ˜ is bounded. In this case,  V (t) ≤ 0, η(t) ff fl T T T ˜ ˜ η˜ (t) L j + L j ⊗ W B B W η(t) ˜ is uniformly continuous. By Barbalat’s ξ(tk )

ξ(tk )

lemma in [52], it can be found that lim

t→∞

i k −1

β η˜ T (t)

 L

j=0

Due to L ξ (tkk+1 ) =

ik −1 j=0

ff j ξ(tk )

+L

fl j ξ(tk )



 ⊗ W˜ T B B T W˜ η(t) ˜ = 0.

(8.30)

L ξ t j  , one can show that k

⎡ ⎢ U −1 L ξ (tkk+1 ) U = ⎣

0

⎤ i k −1  j=0

L

ff j ξ(tk )

+L

fl j ξ(tk )

 ⎥. ⎦

For switching transmission topologies containing a joint spanning tree, L ξ (tkk+1 ) is the Laplacian matrix of a transmission topology containing a spanning tree, so  ik −1  f f fl the matrix j=0 L j + L j is symmetric and positive definite; that is, there ξ(tk ) ξ(tk ) exists an orthonormal matrix Uξ(tkk+1 ) such that ⎛

i k −1  T ff L j U ξ(tkk+1 ) ⎝ ξ(tk ) j=0

+

fl L j ξ(tk )

⎞  & % ⎠U ξ(t k+1 ) = diag λ1 k+1 , λ2 k+1 , . . . , λl k+1 > 0. ξ(t ) ξ(t ) ξ(t ) k k

k

k

(8.31)  T T Due to U ξ(tkk+1 ) η(t) ˜ = ηT1 (t), ηT2 (t), . . . , ηlT (t) , from (8.30) to (8.31), one can see that   T  W˜ ηm (t) B B T W˜ ηm (t) = 0 (m = 1, 2, . . . , l). lim βλm ξ(t k+1 )

t→∞

k

Since it is required that W˜ T A + AT W˜ − βλm W˜ T B B T W˜ < 0 (m = ξ(tkk+1 ) 1, 2, . . . , l), W˜ is nonsingular.  Because itis necessary that the triple (E, A, B) is stabilizable, its dual system E T , AT , B T is detectable. Hence, limt→∞ ηm (t) =









0 (m = 1, 2, . . . , l) if W T E T = E W ≥ 0, A W + W T AT ≤ 0 and A W + W T AT − β B B T < 0 with W = W˜ −1 , which can guarantee that W˜ T A+AT W˜ − W˜ T B B T W < 0 by (8.31); that is, the impacts of topology information can βλm ξ(t k+1 ) k

260

8 Admissible Formation Tracking with Energy Constraints

be neglected under the condition that switching topologies contain a joint spanning tree and each agent is Lyapunov stable. Furthermore, the proofs of regular and impulse-free properties as well as the impacts of energy constrains are resemble with □ Theorem 8.1. Thus, we can obtain the conclusion of Theorem 8.2. On the basis of the analysis mentioned above, the following corollary can be obtained, which shows an admissible formation tracking analysis criterion for switching transmission topologies containing a joint spanning tree and energy constraints; that is, an approach is given to check whether or not multi-agent system (8.1) with formation protocol (8.2) can achieve admissible formation tracking for the given gain matrix K u . Corollary 8.2 For any given Je∗ > 0 and K u , multi-agent system (8.1) with formation protocol (8.2) and switching transmission topologies containing a joint spanning tree achieves admissible formation tracking if A f m (t) − E f˙m (t) = 0 (m = 1, 2, . . . , l) and there exists a matrix W˜ such that   E 0 (i) rank = h + rank(E), A E E T W˜ = W˜ T E ≥ 0, W˜ T A + AT W˜ = 0 (ii)   T T ˜ T ˜T ˜ = −W B K u − K u B W K u Q < 0, Ω ∗ −Q    T l −1 l ⊗ Ih η(0)E T W˜ ≤ Je∗ Ih , (iii) ηT (0) −1l Il  T where η(0) = x0T (0), x1T (0) − f 1T (0), . . . , xlT (0) − flT (0) . Remark 8.4 Compared Theorem 8.2 with Theorem 8.1, there are two critically important distinctions. The first one is that each agent for singular multi-agent systems with switching transmission topologies containing a spanning tree may tend to be unstable, in other words, the dynamic modes of each agent can be located on the right-half plane of the complex plane. In this case, the cooperative state of the leader for all followers to track can be divergent. However, for multi-agent systems with switching transmission topologies containing a joint spanning tree, it is needed



that the dynamics of each agent is Lyapunov stable, ensured by W T E T = E W ≥ 0



and A W + W T AT ≤ 0 in Theorem 8.2. Under this circumstance, the cooperative state of the leader converges to zero or continuously oscillates since the dynamic modes of each agent may be located on the left-half closed plane of the complex plane. The second one is that admissible formation tracking criteria in Theorem 8.1 are related to the minimum and maximum eigenvalues of Laplacian matrices of switching transmission topologies. However, the ones in Theorem 8.2 do not depend on eigenvalues of Laplacian matrices although it is needed that the dynamics of each agent is Lyapunov stable.

8.4 Numerical Simulation

261

8.4 Numerical Simulation In order to further illustrate the validity of the theoretical results derived in the sections above, two corresponding simulation examples are presented in this section, where the dynamics of each agent of the singular multi-agent system is set to be a T  three-order system with the states of each agent xm (t) = xm1 (t), xm2 (t), xm3 (t) ,   1 T and formation functions f m (t) = f m (t), f m2 (t), f m3 (t) (m = 0, 1, 2, . . . , l). The transmission weights of each topology are assumed to be 0–1 for the convenience of analysis. Example 8.1 (Spanning tree case) Consider a singular multi-agent system with one leader and five followers, whose switching transmission topologies contain a spanning tree which can be seen in Fig. 8.6. The system matrix triple (E, A, B) of each agent described in multi-agent system (8.1) is set as ⎡

⎤ ⎡ ⎡ ⎤ ⎤ 100 0 10 6 E = ⎣ 0 3 0 ⎦, A = ⎣ −1 0 0 ⎦, B = ⎣ −9 ⎦. 000 0 11 0 The formation function of followers can be designed as ⎡

⎤ √ √ −6 3 cos 33 t + 2π(m−1) 5 ⎢ √  ⎥ √ ⎢ ⎥ f m (t) = ⎢ 6 3 sin 33 t + 2π(m−1) ⎥ (m = 1, 2, . . . , 5), 5 ⎣ ⎦ √ √ 2π(m−1) −6 3 sin 33 t + 5 it can be found that f m satisfies that A f m (t) − E f˙m (t) = 0 in Theorem 8.1. Let Q = 1 and Je∗ = 25. The transmission topologies are switched randomly among G 1 , G 2 , G 3 , and G 4 with ζd = 1s as shown in Fig. 8.7. Meanwhile, the initial states of each agent are chosen as x0 (0) = [1, 0, 0]T , x1 (0) = [1, −2, 2]T , 1

1 3

2

5

G1

2

0

0

4

1 3

2

G2

0

5

4

3

2

0

5

4

1 3

G3

Fig. 8.6 Switching transmission topology set containing a spanning tree

5

4

G4

262

8 Admissible Formation Tracking with Energy Constraints

Fig. 8.7 Switching signal ξ(t) for the spanning tree case

x2 (0) = [1, 1, −1]T , x3 (0) = [−1.5, −1.3, 1.3]T , x4 (0) = [−0.3, 1.1, −1.1]T , x5 (0) = [1.3, −0.5, 0.5]T . From Theorem 8.1, it can be obtained by the FEASP solver of the Matlab’s LMI Toolbox that K u = [2.7311, −1.9082, −1.5000]. ⎡

⎤ 72.0821 0.0690 0 ⎦. W = ⎣ 0.0230 69.9266 0 −0.0230 −69.9266 −34.7579 Furthermore, the gain matrix K u in the formation tracking protocol (8.2) can be designed as Ku = K u W

−1

  = 0.0379, 0.0158, 0.0432 .

Figure 8.8 depicts the state trajectories of the leader and five followers for the singular multi-agent system at t = 0 s, t = 10 s, t = 15 s, and t = 20 s. The positions of these colored circles represent the states of six agents, respectively. It can be seen from Fig. 8.8a, b that the followers form a regular pentagon centering on the leader. Meanwhile, it can also be found from Fig. 8.8b–d that the achieved

8.4 Numerical Simulation

263

Fig. 8.8 State trajectories of followers and leader for the spanning tree case. a t = 0 s. b t = 10 s. c t = 15 s. d t = 20 s

formation continues to rotate around the leader with the development of simulation time, which means that the formation is time varying. In Fig. 8.9, the trajectories of xm (t) − f m (t) (m = 1, 2, . . . , 5) are depicted, and the red circle markers indicate the trajectory of the leader x0 (t). One can obtain from Fig. 8.9 that xm (t) − f m (t) converges to the states of x0 (t) asymptotically, that is, lim (xm (t) − f m (t) − x0 (t)) = 0. Figure 8.10 shows that the energy cost function t→+∞

Je (t) converges to a finite value less than the total energy supply Je∗ . According to the above analysis of simulation results, it can be obtained that the singular multiagent system (8.1) with energy constraints and switching transmission topologies containing a spanning tree is admissible formation tracking achievable by formation protocol (8.2).

Example 8.2 (Joint spanning tree case) In this example, another third-order singular multi-agent system composed of one leader and six followers is considered, where the switching transmission topologies containing a joint spanning tree are shown in Fig. 8.11. The system matrix triple (E, A, B) of each agent is set as

264

8 Admissible Formation Tracking with Energy Constraints

Fig. 8.9 Trajectories of xm (t) − f m (t) and x0 (t) for the spanning tree case

(a)

(b)

(c)

8.4 Numerical Simulation

265

Fig. 8.10 Curves of Je (t) and Je∗ for the spanning tree case

⎡

⎤ ⎡ ⎤ ⎡ ⎤ 100 0 20 3 0 E = ⎣ 0 1 0 ⎦, A = ⎣ −2 0 0 ⎦, B = ⎣ 3 −8 ⎦. 000 0 02 4 4

3

2

2

3

5

6

G1

G2

1

1 3

2

4

5 6

5

4

6

6

G3

G4

G2

G1 1 3

2

5

4

3

0

0 4

3 0

5

4

6

2

2

0

0 4

1

1

1

0 5 6

G3

Fig. 8.11 Switching transmission topologies contain a joint spanning tree

G4

266

8 Admissible Formation Tracking with Energy Constraints

The formation function is given as      T π (m − 1) π (m − 1) f m (t) = 3 cos 2t + , 3 sin 2t + , 0 (m = 1, 2, . . . , 6). 3 3 It should be pointed out that the desired formation of this singular multi-agent system is a time-varying hexagon according to the function f m (t), which satisfies that A f m (t) − E f˙m (t) = 0 in Theorem 8.2. Figure 8.12 indicates that the switching signal cycles every four times with ζd = 1 s; that is, the switching order of transmission topologies is G 1 → G 2 → G 3 → G 4 → G 1 → G 2 · · · . It should be noticed that the switching order can be set in other ways, but the union of the transmission topologies is required to contain a joint spanning tree during the time interval. Let Q = diag{1, 1}, Je∗ = 40, and the initial state value of leader is set as x0 (0) = [2, −2, 0]T , and the ones for followers are set as x1 (0) = [−8.5, 6.5, 0]T , x2 (0) = [4, 1.2, 0]T , x3 (0) = [6.3, −3.1, 0]T , x4 (0) = [−3, −8, 19]T , x5 (0) = [−1.3, 3.5, 0]T , x6 (0) = [−7, 7.9, 0]T . According to Theorem 8.2, it can be obtained based on the LMI toolbox that

Fig. 8.12 Switching signal ξ(t) for the joint spanning tree case

8.5 Notes

267

⎡

⎤ 18.72 0 0 W = ⎣ 0 18.72 0 ⎦, β = 1.5. 0 0 −22.15

Then, the gain matrix K u in the formation tracking protocol can be designed as

K u = 0.5β B W T

−1



 0.6011 −0.2004 0.3047 = . 0.4808 −0.1202 −0.1693

Figure 8.13 shows the state trajectories of the five followers and the leader for the singular multi-agent system at t = 1 s, t = 12 s, t = 14 s, and t = 16 s. It can be seen from Fig. 8.13a, b that the followers form a regular hexagon, and the leader locates at the formation center. Figure 8.13b–d indicates that the hexagon keeps rotating around the leader with the development of simulation time; that is, the formation is time varying. Figure 8.14 describes the trajectories of xm (t) − f m (t) (m = 1, 2, . . . , 6), and the red circle markers indicate the trajectory of the leader x0 (t). It can be found from Fig. 8.14 that xm (t)− f m (t) converges to x0 (t) asymptotically, which means that lim (xm (t) − f m (t) − x0 (t)) = 0. Figure 8.15 shows that the energy cost function t→+∞

Je (t) converges to a finite value, which is less than the total energy supply Je∗ . Hence, it can be found from Figs. 8.13, 8.14 and 8.15 that the singular multi-agent system (8.1) with energy constraints and switching transmission topologies containing a joint spanning tree is admissible formation tracking achievable by formation protocol (8.2).

8.5 Notes For leader-following singular multi-agent systems with the limited energy supply and switching transmission topologies, a new formation protocol was proposed to realize formation tracking. By the LMI techniques, a dual nonsingular transformation approach was shown to determine the disagreement dynamics between followers and the leader and to convert the admissible formation tracking problem into asymptotic stability ones. Especially, the impacts of asymmetric structures of transmission topologies on energy constraints were well dealt with. Furthermore, sufficient conditions for admissible formation tracking of singular multi-agent systems with switching transmission topologies containing a spanning tree were given, where the total energy supply is introduced by the Laplacian matrix of a star topology with edge weights being one and the leader being the central node. Moreover, admissible formation tracking design as well as analysis criteria was presented for singular multi-agent systems with switching transmission topologies containing a joint spanning tree, in which it was required to ensure that each agent is Lyapunov stable, but they are not related to the minimum and maximum eigenvalues of Laplacian matrices of switching transmission topologies.

268

8 Admissible Formation Tracking with Energy Constraints

Fig. 8.13 State trajectories of followers and leader for joint spanning tree case. a t = 5 s. b t = 12 s. c t = 14 s. d t = 16 s

8.5 Notes

269

Fig. 8.14 Trajectories of the leader and followers for the joint spanning tree case

(a)

(b)

(c)

270

8 Admissible Formation Tracking with Energy Constraints

Fig. 8.15 Curves of curves of Je (t) and Je∗ for joint spanning tree case

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