Robust Event-Triggered Control of Nonlinear Systems [1st ed.] 9789811550126, 9789811550133

Table of contents :
Front Matter ....Pages i-xx
Front Matter ....Pages 1-1
Introduction (Tengfei Liu, Pengpeng Zhang, Zhong-Ping Jiang)....Pages 3-17
Basic Stability and Small-Gain Tools for System Synthesis (Tengfei Liu, Pengpeng Zhang, Zhong-Ping Jiang)....Pages 19-40
Front Matter ....Pages 41-41
A Small-Gain Paradigm for Event-Triggered Control (Tengfei Liu, Pengpeng Zhang, Zhong-Ping Jiang)....Pages 43-72
Dynamic Event Triggers (Tengfei Liu, Pengpeng Zhang, Zhong-Ping Jiang)....Pages 73-104
Event-Triggered Input-to-State Stabilization (Tengfei Liu, Pengpeng Zhang, Zhong-Ping Jiang)....Pages 105-140
Front Matter ....Pages 141-141
Event-Triggered Control of Nonlinear Uncertain Systems in the Lower-Triangular Form (Tengfei Liu, Pengpeng Zhang, Zhong-Ping Jiang)....Pages 143-174
Event-Triggered Control of Nonholonomic Systems (Tengfei Liu, Pengpeng Zhang, Zhong-Ping Jiang)....Pages 175-206
Back Matter ....Pages 207-253

Citation preview

Research on Intelligent Manufacturing

Tengfei Liu Pengpeng Zhang Zhong-Ping Jiang

Robust Event-Triggered Control of Nonlinear Systems

Research on Intelligent Manufacturing Editors-in-Chief Han Ding, Huazhong University of Science and Technology, Wuhan, Hubei, China Ronglei Sun, Huazhong University of Science and Technology, Wuhan, Hubei, China Series Editors Kok-Meng Lee, Georgia Institute of Technology, Atlanta, GA, USA Cheng’ en Wang, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai, China Yongchun Fang, College of Computer and Control Engineering, Nankai University, Tianjin, China Yusheng Shi, School of Materials Science and Engineering, Huazhong University of Science and Technology, Wuhan, Hubei, China Hong Qiao, Institute of Automation, Chinese Academy of Sciences, Beijing, China Shudong Sun, School of Mechanical Engineering, Northwestern Polytechnical University, Xi’an, Shaanxi, China Zhijiang Du, State Key Laboratory of Robotics and System, Harbin Institute of Technology, Harbin, Heilongjiang, China Dinghua Zhang, School of Mechanical Engineering, Northwestern Polytechnical University, Xi’an, Shaanxi, China Xianming Zhang, School of Mechanical and Automotive Engineering, South China University of Technology, Guangzhou, Guangdong, China Dapeng Fan, College of Mechatronic Engineering and Automation, National University of Defense Technology, Changsha, Hunan, China Xinjian Gu, School of Mechanical Engineering, Zhejiang University, Hangzhou, Zhejiang, China Bo Tao, School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan, Hubei, China Jianda Han, College of Artificial Intelligence, Nankai University, Tianjin, China Yongcheng Lin, College of Mechanical and Electrical Engineering, Central South University, Changsha, Hunan, China Zhenhua Xiong, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai, China

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Tengfei Liu Pengpeng Zhang Zhong-Ping Jiang •

•

Robust Event-Triggered Control of Nonlinear Systems

123

Tengfei Liu Northeastern University Shenyang, China

Pengpeng Zhang Northeastern University Shenyang, China

Zhong-Ping Jiang New York University New York, USA

ISSN 2523-3386 ISSN 2523-3394 (electronic) Research on Intelligent Manufacturing ISBN 978-981-15-5012-6 ISBN 978-981-15-5013-3 (eBook) https://doi.org/10.1007/978-981-15-5013-3 Jointly published with Huazhong University of Science and Technology Press The print edition is not for sale in China (Mainland). Customers from China (Mainland) please order the print book from: Huazhong University of Science and Technology Press. © Huazhong University of Science and Technology Press, Wuhan and Springer Nature Singapore Pte Ltd. 2020 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, express 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

To His Family —Tengfei Liu To Hongliu and Parents —Pengpeng Zhang To the Memory of His Parents —Zhong-Ping Jiang

Foreword

As in numerous other areas of human endeavor, advances in communication and computation technologies have led to new opportunities for implementation of feedback controllers. Control over networks has enabled uses of feedback which had previously not been contemplated and has also given rise to new theoretical challenges in sampled-data control. With the network traffic demands typically outpacing the advances in the network bandwidth, it is important to devise feedback algorithms that do not waste the network’s communication capacity. Eventtriggered control has emerged as the most popular paradigm for transferring feedback information in an economical, “as needed” manner. The exploration of “aperiodic,” “time-varying,” “adaptive,” and “efficient” sampling rates dates back to the late 1950s and the 1960s in the work of authors like (alphabetically listed) Bekey, Bergen, Ciscato, Dorf, Farren, Gupta, Jury, Liff, Martiani, McDaniel, Mitchell, Mullen, Phillips, Tomović, and Wolf. However, it is only after the opening of the opportunities for control over networks over the last two decades, and the advances in feedback design techniques of the preceding two decades, that control with aperiodic sampling emerged as a major theoretical subject in control systems. The work by Astrom and Bernhardsson (1999, 2002) has been particularly influential in that regard. In the literature on event-triggered control, special attention is dedicated to the design of event triggers. Among major concerns is avoiding the Zeno behavior. Among the representative and influential results for linear systems have been those by Gawthrop and Wang (2009); Heemels, Sandee, and Bosch (2009); Lunze and Lehmann (2010); Heemels, Donkers, and Teel (2013); Garcia and Antsaklis (2013); Molin and Hirche (2013); and Selivanov and Fridman (2016). For nonlinear systems, Tabuada’s 2007 work has been among the most impactful. He assumes the existence of a robust controller and uses an ISS-Lyapunov function to characterize the robustness with respect to the sampling error: ð@VÞ=ð@xÞ  f ðx; kðx þ eÞÞ   fiðjxjÞ þ ðjejÞ, where x is the state, e is the sampling error, fi and  are class K1 functions. The sampling event is triggered as soon as ðjejÞ  rfiðjxjÞ with constant 0\r\1. For Zeno-free behavior, Tabuada

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requires that both fi1 and  be Lipschitz at the origin, which is not necessarily satisfied for LTI systems with quadratic Lyapunov functions. Postoyan, Tabuada, Nešić, and Anta (2015) relaxed the requirement by using a Lyapunov-based formulation and the hybrid system model developed by Goebel, Sanfelice, and Teel (2009). An event-triggered Sontag formula was proposed by Marchand, Durand, and Castellanos (2013), whereas Antsaklis, along with Yu, Rahnama, and Xia (2013, 2018) proposed event triggers for systems that can be rendered passive relative to the sampling error as the input. The book by Liu, Zhang, and Jiang makes significant advances in event-triggered control of nonlinear systems. Those advances are in (1) the analysis of event-triggered systems using nonlinear small-gain techniques, (2) the design of dynamic event triggers, (3) event-triggered input-to-state stabilization, (4) the advancement of backstepping technique for event-triggered stabilization of strict-feedback and related classes of systems, and (5) the development of event-triggered controls for nonholonomic systems. After two chapters in which they set the stage for presenting their contributions relative to the results by their peers, in Chap. 3, the authors present nonlinear small-gain techniques for event triggered systems. They assume a robustly stabilizing controller and ISS with respect to the sampling error, jxðtÞj  max flðjxð0Þj; tÞ; ðkekÞg, where fl 2 KL and  2 K1 , and consider the event-triggered control system as a feedback connection of the controlled system and the event trigger, which needs to be designed to satisfy the small-gain condition. Zeno-free asymptotic stabilization is achievable if the ISS gain  is Lipschitz at the origin. The authors’ small-gain design avoids the requirement of availability of a Lyapunov function and, in comparison to the ubiquitous “emulation approach,” goes beyond the existence of an event trigger and leads to constructive designs for strict-feedback systems. In Chap. 4, the authors study dynamic event triggers. Prior to their work, dynamic triggers have been proposed, for example, for linear multi-agent systems, by Seyboth, Dimarogonas, and Johansson (2013), but the exponentially converging threshold signals for bounding the sampling errors, which are common for linear plants, cannot, in general, avoid the Zeno behavior for nonlinear systems. The authors introduce non-exponentially converging threshold signals to the event triggers and, by considering the observation error system as dynamic uncertainty, solve the event-triggered observer-based output-feedback stabilization. In Chap. 5, the authors turn their attention to control of nonlinear systems with external disturbances and dynamic uncertainties. In the presence of disturbances, several modifications have been proposed in the past. Event triggers with threshold offsets are often employed to avoid Zeno behavior, but they sacrifice asymptotic convergence even if the system is disturbance-free. Alternatively, a fixed time interval may be used after each triggering event or the triggering condition may be checked only at predetermined time instants but such modifications may not furnish

Foreword

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global guarantees for nonlinear systems. The authors introduce event triggers with a new term to estimate the influence of the unmeasured disturbance. While they achieve input-to-state stabilization with respect to the external disturbance with their event triggers, when the system is disturbance-free they recover global asymptotic stabilization. In Chap. 6, the authors pursue event-triggered design for specific classes of systems with a strict-feedback (lower-triangular) structure. The prior literature on event-triggered control of nonlinear systems has been limited to assuming the existence of controllers that are robust with respect to sampling errors. Such results give the appearance of generality of the class of systems but restrictions on the structure of the class of systems are implicit in the assumptions of robustness with respect to the sampling errors. Constructing controllers that actually guarantee robustness to the sampling errors, as a part of an overall event-triggered design process, has been rare. The challenge in achieving such robustness is closely related with the problem of stabilization by measurement-feedback control. This problem was studied, among few others, by Freeman and Kokotović (1996), who developed a backstepping design with set-valued maps to cover the measurement errors and constructed flattened Lyapunov functions to solve the measurement feedback control problem of nonlinear strict-feedback systems. However, this and other measurement-feedback designs do not readily fulfill the conditions required by the prior event-triggered control results. This book’s authors employ set-valued maps in a small-gain-based constructive design to deal with the sampling errors and make the gain from the sampling error to the system state satisfy the condition for event-triggered control. In Chap. 7, the authors present developments for uncertain nonholonomic systems in the “perturbed chained form,” a generalization of the chained form introduced by Murray and Sastry (1993). Robustification of state- or output-feedback control of nonholonomic systems to sampling/measurement errors has received no attention except by Morin, Pomet, and Samson (1998) who considered unicycles with perturbed heading angle measurements. This book’s authors extend Astolfi’s 1996 discontinuous state-scaling technique by using set-valued maps to cover the influence of the sampling errors and transform the closed-loop system into an interconnection of input-to-state stable subsystems. The challenge is to guarantee the nonsingularity of the discontinuous state-scaling in the presence of the event-triggered sampling errors, which the authors solve through a coordinated design of the event triggers for the unscaled state and the scaled states. This book represents quite an achievement in both event-triggered and nonlinear control. Besides results I expressly mention here, the book contains a number of other results, including those for systems with state delays. Other classes of infinite-dimensional systems are potentially attractive targets for the small-gain concepts that are being employed to good effect in this book. These targets are various classes of parabolic and hyperbolic PDEs with boundary control for which

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interest has emerged, over the last few years, in designing event-triggered controllers. Event-triggered system identification and adaptive control algorithms are equally relevant emerging topics. San Diego, California March 2020

Miroslav Krstić

Preface

Event-triggered control is aimed at monitoring the performance of dynamical control systems through updating the control action only when the system needs attention. Its basic idea can be traced back to the 1960s, and a new wave of research interest was triggered around 2000 with the development of embedded and networked control systems. Since then, extensive research interest has been developed in event-triggered control within the automatic control community. The event-triggered paradigm has been proved to be quite effective in reducing the utilization of communication and computation resources, while still guaranteeing desirable closed-loop behavior. The recent development of the literature has also evidenced additional advantages of event-triggered control in solving sampled-data control problems for nonlinear plants. This book is inspired by the theoretical interest of the new dynamical behaviors created by the interplay between the controllers and the event triggers. In the context of event-triggered control, the sampling errors due to event-triggering can be considered as the measurement errors. These sampling errors and exogeneous disturbances, which may be small or even exponentially decaying, can cause an otherwise exponentially stable system to chatter, let alone instability. However, how to maintain robustness of event-triggered controllers is known to be one of the fundamentally challenging problems in nonlinear control theory. In this book, the closed-loop event-triggered systems are considered as interconnected systems, and the notion of input-to-state stability (ISS) and the associated gains are employed to characterize the interaction between the controllers and the event triggers. It is shown that the nonlinear small-gain theorem is extremely useful for robust integration of the components, especially for nonlinear uncertain systems. The small-gain idea also gives rise to new dynamic event triggers for robust stabilization of nonlinear systems subject to dynamic uncertainties and external disturbances. In addition to the new event triggers, this book proposes new input-to-state stabilizing controllers for two benchmark examples: lower-triangular nonlinear systems and nonholonomic chained systems. Although we focus on the robustness

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with respect to the event-triggered sampling errors, the designs are of independent interest in solving the measurement feedback control problem for the benchmark systems. The materials in this book are based on the authors’ recent research results on robust event-triggered control of nonlinear systems. To make the exposition self-contained, the stability notions and the nonlinear small-gain tools that are used in the book are introduced in Part I. Part II focuses on the designs of robust event triggers for a class of nonlinear control systems subject to dynamic uncertainties and external disturbances. Part III aims to solve the event-triggered control problem for some benchmark nonlinear systems. It is particularly shown how controllers can be designed for the robustness with respect to the sampling errors. The Appendix gives supplementary materials on the notions of the Lipschitz continuity and graph theory, and the proofs of the technical lemmas which seem too mathematical to be placed in the main text. We believe that nontrivial refinement of previously developed nonlinear control tools including ISS and small-gain theorem would play a crucial role in solving the emerging control problems with networked and distributed controllers. The research in this book is by no means complete. As a matter of fact, most of the results presented in this book can be extended to more general systems. This book can either be treated as a regular monograph on event-triggered control or be used as a comprehensive tutorial on ISS and small-gain tools and methods and their applications to specific nonlinear control problems. The first author wishes to express his sincere gratitude to Professor Zhong-Ping Jiang, who is also a coauthor of this book, for the continuous support and invaluable encouragement during the long-term collaboration. The first author is grateful to Prof. David J. Hill at The University of Hong Kong for introducing him to nonlinear control, and to Prof. Tianyou Chai at Northeastern University, China for the career opportunities. The first author would like to offer his special thanks to his family, Lina, Qingdai and Qingxuan, for lots of happiness and love. The second author would like to thank Prof. Zhong-Ping Jiang and Prof. Tengfei Liu for the guidance, and his labmates at the State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, China for sharing their ideas on nonlinear control. The second author would also like to thank his family for their persistent support and understanding. The third author would like to thank his family Sophie, Jenny and Jack for demonstrating the robustness of their “event-triggered” controllers during the writing of this book. Without their support, he would not be able to devote more time and energy to the completion of this book and the long-term collaboration with Dr. Tengfei Liu. It is also a pleasure to thank Prof. Laurent Praly for being an inspiring mentor and leader whose visions have influenced generations of students in the field of nonlinear control. The authors would like to thank the editorial staff, in particular, Li Shen, Nobuko Hirota, and Arulmurugan Venkatasalam, of Springer Nature, and Daokai Yu of Huazhong University of Science & Technology Press for their efforts in publishing

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the book. They would also like to thank Jingwei Xu and Mengxi Wang for the hand-drawn image on the cover. The research presented in this book was supported partly by National Natural Science Foundation of China, and partly by the U.S. National Science Foundation. Shenyang, China Shenyang, China New York, USA February, 2020

Tengfei Liu Pengpeng Zhang Zhong-Ping Jiang

Contents

Part I

Introduction and Preliminaries

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Event-Triggered Control Challenges for Nonlinear Systems . . . . . 1.2 An Overview of This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Basic Stability and Small-Gain Tools for System Synthesis . . . . . 2.1 Lyapunov Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Input-to-State Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 ISS-Lyapunov Functions . . . . . . . . . . . . . . . . . . . . . . . 2.3 Nonlinear Small-Gain Theorem . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Trajectory-Based Small-Gain Theorem . . . . . . . . . . . . . 2.3.2 Lyapunov-Based Small-Gain Theorem . . . . . . . . . . . . . 2.4 Nonlinear Cyclic-Small-Gain Theorem . . . . . . . . . . . . . . . . . . . 2.4.1 Continuous-Time Systems . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Equivalence Between Cyclic-Small-Gain and Gains Less Than the Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Robust Event Triggers

3 A Small-Gain Paradigm for Event-Triggered Control . . . . 3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Lyapunov-Based Small-Gain Design of Event Triggers . . 3.3 Trajectory-Based Small-Gain Design of Event Triggers . 3.4 Event/Self-triggered Control in the Presence of External Disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Event Triggers with Positive Offsets . . . . . . . . . . 3.4.2 Self-triggered Control . . . . . . . . . . . . . . . . . . . . 3.4.3 Trajectory-Based Formulation . . . . . . . . . . . . . . .

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3.5 Input-to-State Stabilization of Discrete-Time Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . 3.5.2 Event-Triggered Robust Stabilization . . . . . . . . 3.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Dynamic Event Triggers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Dynamic Event Triggers with Partial-State Feedback . . . . . . 4.1.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Design of a Dynamic Event Trigger . . . . . . . . . . . . . 4.1.3 An Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 An Application to Decentralized Event-Triggered Control . . . 4.2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Design of Decentralized Event-Triggers . . . . . . . . . . 4.2.3 An Example: Decentralized Event-Triggered Control of a Class of First-Order Nonlinear Systems . . . . . . . 4.2.4 The Linear Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Event-Triggered Input-to-State Stabilization . . . . . . . . . . . . . . . . . 5.1 Robust Event-Triggered Control of Nonlinear Systems Under a Global Sector-Bound Condition . . . . . . . . . . . . . . . . . 5.2 Robust Event-Triggered Control of Nonlinear Systems without a Global Sector-Bound Condition . . . . . . . . . . . . . . . . 5.3 Robustness Analysis in the Presence of Dynamic Uncertainties . 5.3.1 Event Trigger Design . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Small-Gain Synthesis: ISS of the Closed-Loop EventTriggered System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 A Special Case for Systems Under a Global Sector-Bound Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 An Extension to Systems Subject to Both Dynamic Uncertainties and External Disturbances . . . . . . . . . . . . . . . . . . 5.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part III

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Constructive Designs for Event-Triggered Control

6 Event-Triggered Control of Nonlinear Uncertain Systems in the Lower-Triangular Form . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Event-Triggered State-Feedback Control . . . . . . . . . . . . . . 6.1.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3 Small-Gain Synthesis . . . . . . . . . . . . . . . . . . . . . . . 6.2 Event-Triggered Control of Nonlinear Time-Delay Systems 6.2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Constructive Control Design with Set-Valued Maps .

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6.2.3 Main Result of Event-Triggered Stabilization . . . . . 6.2.4 Extension to Systems with Distributed Time-Delays 6.3 Event-Triggered Output-Feedback Control . . . . . . . . . . . . . 6.3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Observer-Based Output-Feedback Controller . . . . . . 6.3.3 ISS Property of the Subsystems . . . . . . . . . . . . . . . 6.3.4 Event-Triggered Output-Feedback Control . . . . . . . . 6.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Event-Triggered Control of Nonholonomic Systems . . . . . 7.1 Event-Triggered State Feedback Control of Uncertain Nonholonomic Systems . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . 7.1.2 Event-Triggered State-Feedback Controller . . . . 7.1.3 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.4 A Brief Discussion on x0 ðt0 Þ ¼ 0 . . . . . . . . . . . 7.2 Event-Triggered Output Feedback Control of Uncertain Nonholonomic Systems . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . 7.2.2 Event-Triggered Output-Feedback Controller . . . 7.2.3 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Appendix A: Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 Appendix B: Technical Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 Appendix C: Gain Assignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Appendix D: Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

Notations and Abbreviations

Notations R Rþ Rn Z Zþ N xT jxj jAj sgnðxÞ k u kD kuk1 :¼  f g ‚max ‚min t þ (t ) @ rVðxÞ Id

The set of real numbers The set of nonnegative real numbers The n-dimensional Euclidean space The set of integers The set of nonnegative integers The set of natural numbers Transpose of vector x Euclidean norm of vector x Induced Euclidean norm of matrix A The sign of x 2 R: sgnðxÞ ¼ 1 if x [ 0; sgnðxÞ ¼ 0 if x ¼ 0; sgnðxÞ ¼ 1 if x\0 ess supt2D juðtÞj with D  R þ for u : R þ ! Rn kukD with D ¼ ½0; 1Þ Equal by definition Identically equal Composition of functions f and g Largest eigenvalue Smallest eigenvalue Time right after (right before) t Partial derivative Gradient of function V at x The identity function

xix

xx

Notations and Abbreviations

Abbreviations a.e. AG AS GAS GS IOpS IOS ISpS ISS RS WRS

Almost Everywhere Asymptotic Gain Asymptotic Stability Global Asymptotic Stability Global Stability Input-to-Output Practical Stability Input-to-Output Stability Input-to-State Practical Stability Input-to-State Stability Robust Stability Weakly Robust Stability

Part I

Introduction and Preliminaries

Chapter 1

Introduction

1.1 Event-Triggered Control Challenges for Nonlinear Systems Data-sampling is essential for digitalized implementation of control strategies. Periodic data-sampling abstracts continuous-time systems at predetermined arithmetic time instants, and has been playing a central role in the fruitful literature of sampleddata control. Since all the sampling intervals are required to be equal and small enough to guarantee the “worst-case” performance, periodic sampled-data control may not be cost-effective for the systems subject to information constraints. This, to some extent, motivates the latest wave of research interest in event-triggered control (Fig. 1.1). The study of event-triggered control can be traced back to the 1960s. Specific applications include relay feedback control [255], pulse-width-modulation control [213], interrupt-based control [95], satellite attitude control [53], internal combustion engine control [89], and so on. The recent development of the literature has evidenced the the new opportunities and the new challenges created by event-triggered control, especially when the controlled systems involve nonlinear and uncertain dynamics. Regarding the resulting advantage in solving control problems, the following example shows that, for some nonlinear systems for which periodic sampled-data control may only promise semi-global stabilization, event-triggered control achieves global stabilization. Example 1.1 Consider the first-order nonlinear system x(t) ˙ = x 3 (t) + u(t),

© Huazhong University of Science and Technology Press, Wuhan and Springer Nature Singapore Pte Ltd. 2020 T. Liu et al., Robust Event-Triggered Control of Nonlinear Systems, Research on Intelligent Manufacturing, https://doi.org/10.1007/978-981-15-5013-3_1

(1.1)

3

4

1 Introduction Controller

Plant

Controller

Plant

Event Trigger Sampler

Sampler

Fig. 1.1 A comparison between periodic sampled-data control and event-triggered control

where x ∈ R is the state, and u ∈ R is the control input. The system is asymptotically stabilized at the origin by the control law u(t) = −9x 3 (t).

(1.2)

To introduce data-sampling to the feedback loop, we consider the sampled-data control law u(t) = −9x 3 (tk ), t ∈ [tk , tk+1 ), k ∈ S ⊆ Z+ ,

(1.3)

where tk with t0 = 0 represents the sampling time instants, and S is the set of the indices of the sampling time instants. We first consider tk = kT with T > 0 being the sampling period. In this case, by substituting (1.3) into (1.1), we have x(t) ˙ = x 3 (t) − 9x 3 (kT ).

(1.4)

For specific k ∈ Z+ and T > 0, suppose that x(t) is defined for t ∈ [kT, (k + 1)T ). If x(kT ) > 0, then it is easily checked that x(t) ≤ x(kT ) for t ∈ [kT, (k + 1)T ), and thus  x((k + 1)T ) = x(kT ) +

(k+1)T

(x 3 (τ ) − 9x 3 (kT )) dτ

kT

≤ x(kT ) − 8x 3 (kT )T.

(1.5)

√ If moreover √x(kT ) > 1/(2 T ), then x((k + 1)T ) < −x(kT ). Similarly, if x(kT ) < −1/(2 T ), then x((k + 1)T ) > −x(kT ). It is observed that, given spea range of x(kT ) such that if x(kT ) is cific k ∈ Z+ and T > 0, there always exists √ outside the range, i.e., |x(kT )| > |1/(2 T )|, then |x((k + 1)T )| > |x(kT )|, as long as x(t) is defined for t ∈ [kT, (k + 1)T ). Figure 1.2 gives a numerical simulation result of periodic sampled-data control with initial state x(0) = −2.5 and sampling period T = 0.05. It is shown that x(t) diverges.

1.1 Event-Triggered Control Challenges for Nonlinear Systems

5

115

system state

10 0 -10 -20

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.04

0.05

0.06

0.07

control input

time 500 0 -500 -1000 -1500

0

0.01

0.02

0.03 time

Fig. 1.2 Periodic sampled-data control of a first-order nonlinear system in Example 1.1

To enlarge the validity range of the sampled-data control law, one may consider reducing the sampling period T . This means semi-global stabilization [146, 206, 207]. Event-triggered sampling makes it possible to achieve global stabilization. We employ the notion of input-to-state stability to characterize the influence of the sampling error. Define w(t) = x(tk ) − x(t) for t ∈ [tk , tk+1 ), k ∈ S as the sampling error. Then, we have x(t) ˙ = x 3 (t) − 9(x(t) + w(t))3 =: g(x(t), w(t)).

(1.6)

It can be checked that V (x) = 0.5x 2 is an ISS-Lyapunov function (see Sect. 2.2) satisfying V (x) ≥ 2w2 ⇒ ∇V (x)g(x, w) ≤ −0.4V 2 (x).

(1.7)

Clearly, asymptotic stabilization is achieved if the sampling mechanism always guarantees |w(t)|2 ≤ 0.5V (x). Based on this idea, one may set tk+1 = inf{t > tk : |x(t) − x(tk )| ≥ 0.5|x(t)|}.

(1.8)

Since the mechanism (1.8) is used to trigger the sampling event based on the system state, it is called an event trigger. The term 0.5|x(t)| is called threshold signal.

1 Introduction

control input

system state

6 1 0 -1 -2 -3

0

0.5

1

1.5

2

2.5 time

3

3.5

4

4.5

5

0

0.5

1

1.5

2

2.5 time

3

3.5

4

4.5

5

40 20 0

Fig. 1.3 The state trajectory and control input of event-triggered control of a first-order nonlinear system in Example 1.1

Different from periodic sampling, the event trigger (1.8) does not explicitly guarantee a positive lower bound of the inter-sampling intervals. This is one of the major issues that should be carefully addressed in the design of event-triggered controllers. Figure 1.3 gives a numerical simulation result of event-triggered control with event trigger (1.8). It is shown that with the same initial state x(0), event-triggered control achieves asymptotic convergence. Moreover, most of the inter-sampling intervals of the event-triggered control strategy are larger than the sampling period T = 0.05 which is used for periodic sampled-data control; see Fig. 1.4.

inter-sampling intervals

10

1

100 10-1 10-2 10

-3

10-4

inter-sampling intervals 0.05 0

0.5

1

1.5

2

2.5 time

3

3.5

4

4.5

5

Fig. 1.4 The inter-sampling intervals of event-triggered control of a first-order nonlinear system in Example 1.1

1.1 Event-Triggered Control Challenges for Nonlinear Systems

7

The theoretical interest of event-triggered control also lies in the nontrivial phenomenon caused by the hybrid nature of the closed-loop event-triggered systems. The following example shows that an exponentially converging disturbance makes the performance of a normal static event trigger to deteriorate, even though the plant is basically an integrator. Example 1.2 Consider system z˙ (t) = −z(t), x(t) ˙ = z(t) + u(t),

(1.9) (1.10)

where z ∈ R and x ∈ R are the states, and u ∈ R is the control input. Suppose that only x is available to feedback control design. Since the z-subsystem is autonomous and exponentially stable at the origin, we may consider z as an exponentially converging disturbance of the x-subsystem. Normally, the system (1.9)–(1.10) is asymptotically stabilized at the origin if the xsubsystem is input-to-state stabilized by a continuous control law with z as the input. Now, we consider the sampled-data control law u(t) = −x(tk ), t ∈ [tk , tk+1 ), k ∈ S.

(1.11)

To analyze the influence of data-sampling, define w(t) = x(tk ) − x(t) for t ∈ [tk , tk+1 ), k ∈ S as the sampling error. Then, we have x(t) ˙ = −x(t) − w(t) + z(t).

(1.12)

The x-subsystem is input-to-state stabilized with z as the input if |w(t)| ≤ 0.2|x(t)|

(1.13)

holds for all t ≥ 0. Thus, one may consider the event trigger tk+1 = inf{t > tk : |x(t) − x(tk )| ≥ 0.2|x(t)|}.

(1.14)

Indeed, in this case, the z-subsystem admits an ISS-Lyapunov function V (x) = 0.5x 2 which satisfies V (x) ≥ 12.5z 2 ⇒ ∇V (x)x˙ ≤ 1.2V (x).

(1.15)

But it is still questionable whether (1.13) is guaranteed for all t ≥ 0, due to the possible accumulation of the sampling times. According to the definition of the event trigger (1.14), it holds that   t    (z(τ ) − x(tk )) dτ  ≤ 0.2|x(t)|   tk

(1.16)

1 Introduction

inter-sampling intervals

states and control input

8

z x u

1 0 -1 0

1

2

3

4 time

5

6

7

8

0

1

2

3

4 time

5

6

7

8

100 10

-2

10

-4

Fig. 1.5 Event-triggered control of a linear system subject to an exponentially converging disturbance in Example 1.2: static event trigger

for all t ∈ [tk , tk+1 ). Note that (1.14) implies that |x(t) − x(tk )| ≤ 0.25|x(tk )|. Then, we have   tk+1    ≤ 0.25|x(tk )|.  (z(τ ) − x(t )) dτ (1.17) k   tk

Recall z(t) ≤ z(tk )e−(t−tk ) , and denote tk+1 − tk = Δk . Direct calculation yields:   z(tk )(1 − e−Δk ) − x(tk )Δk  ≤ 0.25|x(tk )|.

(1.18)

For specific z(tk ) = 0, the maximal solution of Δk to the inequality (1.18) can be arbitrarily close to zero as x(tk ) can be arbitrarily close to zero. This means that the event trigger (1.14) cannot guarantee a positive lower bound for all the inter-sampling intervals. Figure 1.5 gives a simulation result of event triggered control with the event trigger (1.14). The initial state is chosen as z(0) = 1 and x(0) = −1. It is shown that the inter-sampling intervals converge to zero at t = 0.968. This is known as Zeno behavior in the literature of hybrid systems [75]. The threshold signals of the event triggers (1.8) and (1.14) are defined as static functions of the system states. An alternative design is to employ a time-varying threshold signal. In particular, for the system in this example, one may consider tk+1 = inf{t > tk : |x(t) − x(tk )| ≥ ae−bt }, t0 = 0,

(1.19)

where a and b are positive constants. Instead of solely depending on x, the threshold signal ae−bt converges in accordance with the convergence of the closed-loop eventtriggered system.

inter-sampling intervals

states and control input

1.1 Event-Triggered Control Challenges for Nonlinear Systems

9 z x u

1 0 -1 0

1

2

3

4

5 time

6

7

8

9

10

0

1

2

3

4

5 time

6

7

8

9

10

100

Fig. 1.6 Event-triggered control of a linear system subject to an exponentially converging disturbance in Example 1.2: dynamic event trigger with an exponentially converging threshold signal

We choose a = 0.5 and b = 0.4. Figure 1.6 shows the simulation result with initial states z(0) = 1 and x(0) = −1. The effectiveness of dynamic event triggers in dealing with dynamic uncertainties is to be proved in Chap. 4. The problem becomes more complicated if the plant involves nonlinear dynamics. For example, if we replace the z-subsystem (1.9) with z˙ (t) = −z 3 (t),

(1.20)

then z(t) does not converge to zero exponentially, and the dynamic event trigger (1.13) with the exponentially converging threshold signal does not work well any more. Figure 1.7 shows the simulation result with the same event trigger and the same initial states. Infinitely fast sampling occurs. Due to a similar reason as for Example 1.2, non-converging external disturbances may also cause infinitely fast sampling. The following example briefly discusses two intuitive solutions to event-triggered control subject to external disturbances. Example 1.3 Consider the first-order linear system x(t) ˙ = x(t) + u(t) + d(t),

(1.21)

where x ∈ R is the state, u ∈ R is the control input, and d ∈ R represents the external disturbance. It is assumed that d is piece-wise continuous and bounded for all t ∈ [0, ∞). Consider the sampled-data control law u(t) = −2x(tk ), t ∈ [tk , tk+1 ), k ∈ S.

(1.22)

10

1 Introduction

Fig. 1.7 Event-triggered control of a linear system subject to an non-exponentially converging disturbance in Example 1.2: dynamic event trigger with an exponentially converging threshold signal

If d ≡ 0, then one may employ the following event trigger for asymptotic stabilization: tk+1 = inf{t > tk : |x(t) − x(tk )| ≥ 0.14|x(t)|}, t0 = 0.

(1.23)

However, due to a similar reason as for Example 1.2, the nonzero d may cause infinitely fast sampling. One solution is to introduce a positive constant offset  to the threshold signal as tk+1 = inf{t > tk : |x(t) − x(tk )| ≥ max{0.14|x(t)|, }}, t0 = 0.

(1.24)

With such treatment, the threshold signal max{0.14|x(t)|, } never goes to zero, and there is always some “room” between any two triggering events. We employ a numerical simulation to evaluate the event trigger. Figure 1.8 shows the disturbance d(t) that is used for the simulation. Figure 1.9 compares the state trajectories and inter-sampling intervals with event trigger (1.24) with different values of . The corresponding control inputs are given by Fig. 1.10. It can be observed that both the inter-sampling intervals and the “steady-state” control error depend on . Smaller  leads to smaller inter-sampling intervals and smaller control error, and vice versa. The control error does not converge to zero even if the external disturbance decays. Moreover, the inter-sampling intervals also depend on the magnitude of the external disturbance. Intuitively, larger external disturbance results in smaller intersampling intervals. Note that the sampling theorem (see [20]) suggests a disturbanceindependent sampling period for stabilization of the linear system! The discussion above motivates an alternative solution to the event-triggered control problem in the presence of disturbances: a combination of event-triggered sam-

1.1 Event-Triggered Control Challenges for Nonlinear Systems

11

0.8 0.6

disturbance d

0.4 0.2 0 -0.2 -0.4 -0.6 -0.8

0

2

4

6

8

10 time

12

14

16

18

20

system state

Fig. 1.8 The disturbance d used for the simulation of the event trigger (1.24) in Example 1.3 0.6

x with =0.06 x with =0.02

0.4 0.2 0

inter-samping intervals

0

2

4

6

8

10 time

12

14

16

18

20

100 inter-samping intervals with =0.06 inter-samping intervals with =0.02 0

2

4

6

8

10 time

12

14

16

18

20

Fig. 1.9 The state trajectories and inter-sampling intervals with the event trigger (1.24) with different values of  in Example 1.3

pling and periodic sampling. That is, a fixed interval TΔ is enforced after each triggering event: tk+1 = inf{t ≥ tk + TΔ : |x(t) − x(tk )| ≥ 0.14|x(t)|}, t0 = 0.

(1.25)

We employ a numerical simulation to evaluate the time-regularized event trigger. Figure 1.11 shows the disturbances with different magnitudes that are used in the simulation. The initial state x(0) = 0.5 is chosen for the simulation. The state trajectories, control inputs and inter-sampling intervals are given by Fig. 1.12. It is shown that the time-regularized event trigger guarantees a positive lower bound of the inter-sampling intervals, which does not depend on the magnitude of the exter-

12

1 Introduction 0.2

control input

0 -0.2 -0.4 -0.6 u with =0.06 u with =0.02

-0.8 -1

0

2

4

6

8

10 time

12

14

16

18

20

Fig. 1.10 The event-triggered control inputs with the event trigger (1.24) with different values of  in Example 1.3 2

d1

1.5

d2 d3

disturbance

1 0.5 0 -0.5 -1 -1.5 -2

0

1

2

3

4

5

6

7

8

9

10

Fig. 1.11 The disturbances with different magnitudes that are used in the simulation of the timeregularized event trigger (1.25) in Example 1.3

nal disturbance. Also, with the time-regularized event trigger, the convergence of the external disturbance results in the convergence of the control error. However, it is also recognized that when the external disturbance is large, the closed-loop event-triggered system works much like a periodic sampled-data system. This goes against the original motivation of event-triggered control to save communication and computation resources.

1.1 Event-Triggered Control Challenges for Nonlinear Systems d=d 1

d=d

1

1 x u

1

0

5 time

10

-2

0

5 time

inter-samping intervals 0

5 time

-2

10

0

5 time

10

100 inter-samping intervals

-2

10

10

x u

-1

100

-2

3

0 x u

-1

100 10

d=d

2

0

0 -1 -2

13

0

5 time

inter-samping intervals

-2

10

10

0

5 time

10

Fig. 1.12 The state trajectories, control inputs and inter-sampling intervals with the timeregularized event trigger (1.25) with TΔ = 0.05 in Example 1.3

The following example shows that, when the plant involves nonlinear dynamics, one may not even be able to find a TΔ > 0 for the time-regularized event trigger for global stabilization. Example 1.4 Consider the first-order nonlinear system x(t) ˙ = x 3 (t) + u(t) + d(t)

(1.26)

where the variables are with the same meaning as for the system (1.21). The sampled-data control law of the system is designed as u(t) = −8x 3 (tk ) − 4x(tk ), t ∈ [tk , tk+1 ), k ∈ S.

(1.27)

To address the influence of the external disturbance, we consider the timeregularized event trigger: tk+1 = inf{t ≥ tk + TΔ : |x(t) − x(tk )| ≥ 0.49|x(t)|}, t0 = 0

(1.28)

with TΔ > 0 being a constant. We study the motion of x(t) during the interval [tk , tk+1 ). In the case of x(tk ) > 0 and 6x 3 (tk ) + 4x(tk ) > d [0,∞) , it can be directly proved that x(t) ≤ x(tk ) for t ∈ [tk , tk+1 ). Thus, x(t) ˙ = x 3 (t) − 8x 3 (tk ) − 4x(tk ) + d(t) ≤ x 3 (tk ) − 8x 3 (tk ) − 4x(tk ) + d(t) ≤ −x 3 (tk ).

(1.29)

14

1 Introduction 3

disturbance d

2 1 0 -1 -2 -3

0

0.2

0.4

0.6

0.8

1 time

1.2

1.4

1.6

1.8

2

Fig. 1.13 The disturbance d used for the simulation of the event trigger (1.28) in Example 1.4

An estimation of x(tk+1 ) can be made as 

tk+1

x(tk+1 ) = 

x(t)dt ˙ + x(tk )

tk

≤

tk+1

(−x 3 (tk ))dt + x(tk )

tk

= −(tk+1 − tk )x 3 (tk ) + x(tk ) ≤ −TΔ x 3 (tk ) + x(tk ).

(1.30)

It can be observed that for any specific TΔ > 0, one can find a constant x ∗ > 0 such that if x(tk ) > x ∗ and 6x 3 (tk ) + 4x(tk ) > d [0,∞) , then x(tk+1 ) < −x(tk ). Due to symmetry, for any specific TΔ > 0, one can find a constant x ∗ > 0 such that if x(tk ) < −x ∗ and −6x 3 (tk ) − 4x(tk ) > d [0,∞) , then x(tk+1 ) > −x(tk ). This means that one cannot find a sampling period TΔ > 0 to globally asymptotically stabilize the system composed of (1.26) and (1.27). This is verified by a numerical simulation. The disturbance that is used in the simulation is shown in Fig. 1.13. Figure 1.14 shows the simulation results of the event trigger (1.28) for different values of x(0) and TΔ . It is shown that when x(0) or TΔ is large, event trigger (1.28) may lead to the divergence of x. Moreover, larger d may also lead to the divergence of x(t). For this case, we leave to the discussions to the interested reader. The technical challenge of event-triggered control is also due to the fundamental difficulty of designing controllers for nonlinear systems for the robustness with respect to sampling errors.

1.1 Event-Triggered Control Challenges for Nonlinear Systems

20 0 -20 -40 -60 -80

Case (a)

x u 0

0.5 1 time

4 10 Case (b)

10

0

0

0.02 time

-1

0.04

inter-samping intervals

100

10

x u

0

0

1.5

4 10 Case (c)

1

x u

5

15

0

100 inter-samping intervals

inter-samping intervals 0

0.5 1 time

1.5

0.1 time

0

0.5 1 time

1.5

0

0.5 1 time

1.5

Fig. 1.14 The state trajectories, control inputs and inter-sampling intervals with the event trigger (1.28) in Example 1.4. Case a: x(0) = 2 and TΔ = 0.01; Case b: x(0) = 5 and TΔ = 0.01; Case c: x(0) = 2 and TΔ = 0.06

Example 1.5 Consider the first-order nonlinear system x˙ = x 2 + u,

(1.31)

where x ∈ R is the state, and u ∈ R is the control input. If there is no measurement error, we can design a feedback linearizing control law u = −x 2 − 0.1x such that the closed-loop system is x˙ = −0.1x, which is asymptotically, and even exponentially, stable at the origin. If the measurement of the state is subject to an additive measurement error, denoted by w, then the measurement feedback control law is u = −(x + w)2 − 0.1(x + w) and the resulting closed-loop system is x˙ = −(0.1 + 2w)x − w2 − 0.1w,

(1.32)

which clearly does not admit the bounded-input bounded-state (BIBS) stability property when w is considered as the input. Just check w < −0.05. Another example is from [64, Chap. 6], and is used to show that a control law which guarantees global stability under perfect state feedback will not provide global robustness with respect to the state measurement error. Example 1.6 Consider the first-order nonlinear system 2

x˙ = xe x + u, where x ∈ R is the state, and u ∈ R is the control input.

(1.33)

16

1 Introduction k=5 15

x

10

5

0 −0.7

−0.6

−0.5

−0.4

w

−0.3

−0.2

−0.1

0

Fig. 1.15 Example of the stability region of the system (1.36) with k = 5

A globally stabilizing control law with perfect state feedback is 2

u = −kxe x ,

(1.34)

where k is a constant satisfying k > 1. Suppose that the measurement of x is subject to an additive measurement error w. Then, the control law above should be modified as u = −k(x + w)e(x+w) , 2

(1.35)

and the resulting closed-loop system is x˙ = xe x − k(x + w)e(x+w)   2 2 = e x x − k(x + w)e2xw+w . 2

2

(1.36)

Given an arbitrarily small constant measurement error w = 0, there always exists an initial value x(0) such that the state trajectory x(t) diverges to infinity in some finite time; see Fig. 1.15. That is, the measurement error makes the closed-loop system lose global stability.

1.2 An Overview of This Book This book introduces a systematic small-gain approach to robust event-triggered control of nonlinear systems. The rest of the book is organized as follows.

1.2 An Overview of This Book

17

Part I gives the background and motivation of the study in this chapter, and introduces some stability notions and the small-gain tools in Chap. 2. Part II focuses on the synthesis of robust event triggers for the nonlinear control systems subject to dynamic uncertainties and external disturbances. Specifically, Chap. 3 introduces the small-gain approach to event-triggered control of nonlinear systems. In Chap. 4, a class of dynamic event triggers are designed for the nonlinear control systems with dynamic uncertainties, and shown to be useful in solving the event-triggered control problems for nonlinear systems with partial state or output feedback. By introducing an estimation term to the event triggers, Chap. 5 extends the idea of dynamic event triggers to the more general case in which the control systems are subject to external disturbances which may not be converging. Part III aims to solve the event-triggered control problem for some benchmark nonlinear systems. It is particularly shown how controllers can be designed for the robustness with respect to the sampling errors. Nonlinear systems in the lower-triangular forms and nonholonomic systems are considered in Chaps. 6 and 7, respectively.

Chapter 2

Basic Stability and Small-Gain Tools for System Synthesis

This chapter reviews some basic notions of stability, including Lyapunov stability and input-to-state stability, and introduces the small-gain tools that will be used for the synthesis of event-triggered control systems.

2.1 Lyapunov Stability The stabilization problem is one of the most fundamental problems in control theory. Many control problems can be transformed into stabilization problems. This section reviews some basic concepts of Lyapunov stability [81, 138, 182] for dynamical systems without external inputs. Consider the system x˙ = f (x),

(2.1)

where f : Rn → Rn is a locally Lipschitz function. Assume that the origin is an equilibrium of the system, i.e., f (0) = 0. Note that if an equilibrium other than the origin, say x e , is of interest, one may use a coordinate transformation x  = x − x e to move the equilibrium to the origin. Therefore, the assumption of the equilibrium at the origin is without loss of generality. Denote x(t, x0 ) or simply x(t) as the solution of the system (2.1) with initial condition x(0) = x0 , and let [0, Tmax ) with 0 < Tmax ≤ ∞ be the right maximal interval for the definition of x(t, x0 ). The standard definition of Lyapunov stability is usually given by using “-δ” terms, which can be found in the standard textbooks of nonlinear systems; see, e.g., [81, 138]. Definition 2.1 employs the comparison functions α ∈ K and β ∈ KL for convenience of the comparison between Lyapunov stability and ISS. © Huazhong University of Science and Technology Press, Wuhan and Springer Nature Singapore Pte Ltd. 2020 T. Liu et al., Robust Event-Triggered Control of Nonlinear Systems, Research on Intelligent Manufacturing, https://doi.org/10.1007/978-981-15-5013-3_2

19

20

2 Basic Stability and Small-Gain Tools for System Synthesis

Definition 2.1 The system (2.1) is • stable at the origin if there exist an α ∈ K and a constant c > 0 such that for any |x0 | ≤ c, |x(t)| ≤ α(|x0 |)

(2.2)

for all t ≥ 0; • globally stable (GS) at the origin if property (2.2) holds for all initial states x0 ∈ Rn ; • asymptotically stable (AS) at the origin if there exist a β ∈ KL and a constant c > 0 such that for any |x0 | ≤ c, |x(t)| ≤ β(|x0 |, t)

(2.3)

for all t ≥ 0; • globally asymptotically stable (GAS) at the origin if condition (2.3) holds for any initial state x0 ∈ Rn . A proof of the equivalence between the standard definition and Definition 2.1 can be found in [138, Appendix C.6]. See also the discussions in [81, Definitions 2.9 and 24.2]. With the standard definition, GAS at the origin can be defined based on GS by adding the global convergence property at the origin: limt→∞ x(t) = 0 for all x0 ∈ Rn ; see [138, Definition 4.1]. It can be observed that GAS is more than global convergence. Theorem 2.1, which is known as Lyapunov’s Second Theorem (or the Lyapunov Direct Method), gives sufficient conditions for stability and AS. Theorem 2.1 Let the origin be an equilibrium of the system (2.1) and Ω ⊂ Rn be a domain containing the origin. Let V : Ω → R+ be a continuously differentiable function such that V (0) = 0, V (x) > 0, ∀x ∈ Ω\{0},

(2.4) (2.5)

∇V (x) f (x) ≤ 0, ∀x ∈ Ω.

(2.6)

Then, the system (2.1) is stable at the origin. Moreover, if ∇V (x) f (x) < 0, ∀x ∈ Ω\{0},

(2.7)

then the system (2.1) is AS at the origin. A function V that satisfies (2.4)–(2.6) is called a Lyapunov function. If moreover, V satisfies (2.7), then it is called a strict Lyapunov function [23]. However, the condition for AS in Theorem 2.1 does not guarantee GAS even if the Ω is replaced with Rn . Theorem 2.2 gives the additional conditions for GAS.

2.1 Lyapunov Stability

21

Theorem 2.2 Suppose that the origin is an equilibrium of the system (2.1). Let V : Rn → R+ be a continuously differentiable function such that V (0) = 0,

(2.8)

V (x) > 0, ∀x ∈ R \{0}, |x| → ∞ ⇒ V (x) → ∞,

(2.9) (2.10)

∇V (x) f (x) < 0, ∀x ∈ Rn \{0}.

(2.11)

n

Then, the system (2.1) is globally asymptotically stable at the origin. Condition (2.8)–(2.10) is equivalent to the statement that V is positive definite and radially unbounded, which can be represented with comparison functions α, α ∈ K∞ as α(|x|) ≤ V (x) ≤ α(|x|)

(2.12)

for all x ∈ Rn . Moreover, condition (2.11) is equivalent to the existence of a continuous and positive definite function α such that ∇V (x) f (x) ≤ −α(V (x))

(2.13)

holds for all x ∈ Rn . See [138, Lemma 4.3] for the details. Theorems 2.1 and 2.2 give sufficient conditions for stability, AS and GAS. A proof of the converse Lyapunov theorem for the necessity of the conditions can be found in [138].

2.2 Input-to-State Stability The notion of input-to-state stability (ISS), invented by E. D. Sontag, has been proved to be powerful in characterizing both the internal and external stability of nonlinear systems with external inputs.

2.2.1 Definition Consider the system x˙ = f (x, u),

(2.14)

where x ∈ Rn is the state, u ∈ Rm represents the input, and f : Rn × Rm → Rn is a locally Lipschitz function and satisfies f (0, 0) = 0. By considering the input u

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2 Basic Stability and Small-Gain Tools for System Synthesis

as a function of time, assume that u is measurable and locally essentially bounded. Recall that u is locally essentially bounded if for any t ≥ 0, u [0,t] exists. Denote x(t, x0 , u), or simply x(t), as the solution of the system (2.1) with initial condition x(0) = x0 and input u. In [235], the original definition of ISS is given in the “plus” form; see (2.17). For convenience of discussions, we mainly use the definition in the equivalent “max” form. The equivalence is discussed later. Definition 2.2 The system (2.14) is said to be input-to-state stable (ISS) if there exist β ∈ KL and γ ∈ K such that for any initial state x(0) = x0 and any measurable and locally essentially bounded input u, the solution x(t) satisfies |x(t)| ≤ max{β(|x0 |, t), γ( u ∞ )}

(2.15)

for all t ≥ 0. Here, γ is called the ISS gain of the system. Notice that, if u ≡ 0, then Definition 2.2 is reduced to the definition of GAS at the origin given by Definition 2.1. Due to causality, x(t) depends on x0 and the past inputs {u(τ ) : 0 ≤ τ ≤ t}, and thus, the u ∞ in (2.15) can be replaced with u [0,t] . Since max{a, b} ≤ a + b ≤ max{(1 + 1/c)a, (1 + c)b}

(2.16)

holds for any a, b ≥ 0 and any c > 0, property (2.15) in the “max” form is equivalent to |x(t)| ≤ β  (|x0 |, t) + γ  ( u ∞ ),

(2.17)

where β  ∈ KL and γ  ∈ K. With property (2.17), x(t) asymptotically converges to within the region defined by |x| ≤ γ  ( u ∞ ), i.e., lim |x(t)| ≤ γ  ( u ∞ ).

t→∞

(2.18)

As shown in Fig. 2.1, γ  describes the “steady-state” performance of the system, and is usually called the asymptotic gain (AG), while the “transient performance” is described by β  . Intuitively, since only large values of t determine the value limt→∞ |x(t)|, one may replace the γ  ( u ∞ ) in (2.18) with γ  (limt→∞ |u(t)|) or limt→∞ γ  (|u(t)|). See [239, 241] for more detailed discussions. When the system (2.14) is reduced to a linear time-invariant system, a necessary and sufficient condition for the ISS property can be derived. Theorem 2.3 The linear time-invariant system x˙ = Ax + Bu is ISS if and only if A is Hurwitz.

(2.19)

2.2 Input-to-State Stability

23

x(t)

x0

β  (|x0 |, t) + γ  (u∞ )

γ  (u∞ ) t

0 Fig. 2.1 Asymptotic gain property of ISS

Proof. With initial condition x(0) = x0 and input u, the solution of the system (2.19) is  t e A(t−τ ) Bu(τ )dτ , (2.20) x(t) = e At x0 + 0

which implies 

∞

|x(t)| ≤ |e At ||x0 | +

 |e Aτ |dτ |B| u ∞ .

(2.21)

0

∞ If A is Hurwitz, i.e., every eigenvalue of Ahas negativereal part, then 0 |e As |ds < ∞ ∞. Define β  (s, t) = |e At |s and γ  (s) = 0 |e Aτ |dτ |B|s for s, t ∈ R+ . Clearly, β  ∈ KL and γ  ∈ K∞ . Then, the system (2.19) is ISS in the sense of (2.17). The sufficiency part is proved. For the necessity, one may consider the case of u ≡ 0. In this case, the ISS of the system (2.19) implies GAS of system x˙ = Ax

(2.22)

at the origin. According to linear systems theory [33], the system (2.22) is GAS at the origin if and only if A is Hurwitz.  Based on the proof of Theorem 2.3, one may consider the ISS property (2.17) as a nonlinear modification of property (2.21) of linear systems. Lemma 2.1 shows that any KL function β(s, t) can be considered as a nonlinear modification of se−t .

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2 Basic Stability and Small-Gain Tools for System Synthesis

Lemma 2.1 For any β ∈ KL, there exist α1 , α2 ∈ K∞ such that β(s, t) ≤ α2 (α1 (s)e−t )

(2.23)

for all s, t ≥ 0. See [236, Proposition 7] and its proof therein. According to Lemma 2.1, if property (2.17) holds, then there exist α1 , α2 ∈ K∞ such that |x(t)| ≤ α2 (α1 (|x0 |)e−t ) + γ  ( u ∞ ),

(2.24)

which shows a close analogy of ISS to the solution property (2.21) of the linear system (2.19) with A being Hurwitz. Also, with Lemma 2.1, property (2.15) implies |x(t)| ≤ max{α2 (α1 (|x0 |)e−t ), γ( u ∞ )},

(2.25)

where α1 , α2 are appropriate class K∞ functions. This means, for any x0 and any u ∞ satisfying α2 ◦ α1 (|x0 |) > γ( u ∞ ), there exists a finite time t ∗ = log(α1 (|x0 |)) − log(α2−1 ◦ γ( u ∞ )), after which the solution x(t) is within the range defined by |x| ≤ γ( u ∞ ). This shows the difference between the ISS gain γ defined in (2.15) and the asymptotic gain γ  defined in (2.17). From Definition 2.2, an ISS system is always forward complete, i.e., for any initial state x(0) = x0 and any measurable and locally essentially bound input u, the solution x(t) is defined for all t ≥ 0. Moreover, it has the uniformly bounded-input bounded-state (UBIBS) property. Definition 2.3 The system (2.14) is said to have the UBIBS property if there exist σ1 , σ2 ∈ K such that for any initial state x(0) = x0 and any measurable and locally essentially bounded input u, the solution x(t) of (2.14) satisfies |x(t)| ≤ max{σ1 (|x0 |), σ2 ( u ∞ )}

(2.26)

for all t ≥ 0. Recall Definition A.6 for class KL functions. If the system (2.14) is ISS satisfying (2.15), then it admits property (2.26) with σ1 (s) = β(s, 0) and σ2 (s) = γ(s) for s ∈ R+ . More importantly, ISS is equivalent to the conjunction of UBIBS and AG [241]. This result can be used for the proof of the ISS small-gain theorem for interconnected nonlinear systems; see detailed discussions in Sects. 2.3 and 2.4. Theorem 2.4 The system (2.14) is ISS if and only if it has the properties of UBIBS and AG in the sense of (2.26) and (2.18), respectively.

2.2 Input-to-State Stability

25

2.2.2 ISS-Lyapunov Functions For the system (2.14), the equivalence between ISS and the existence of ISSLyapunov functions was originally presented in [240]. Theorem 2.5 The system (2.14) is ISS if and only if it admits a continuously differentiable function V : Rn → R+ , for which 1. there exist α, α ∈ K∞ such that α(|x|) ≤ V (x) ≤ α(|x|), ∀x,

(2.27)

2. there exist a γ ∈ K and a continuous, positive definite α such that V (x) ≥ γ(|u|) ⇒ ∇V (x) f (x, u) ≤ −α(V (x)), ∀x, u.

(2.28)

A function V satisfying (2.27) and (2.28) is called an ISS-Lyapunov function and γ is called an Lyapunov-based ISS gain. ISS-Lyapunov functions defined with (2.28) are said to be in the gain margin form. It can be observed that, under condition (2.28), the state x ultimately converges to within the region such that V (x) ≤ γ( u ∞ ). If input u ≡ 0, then the sufficiency part of Theorem 2.5 is reduced to Theorem 2.2 for GAS. An equivalent formulation to (2.28) is in the dissipation form: ∇V (x) f (x, u) ≤ −α (V (x)) + γ  (|u|),

(2.29)

where α ∈ K∞ and γ  ∈ K. The proof of the sufficiency part of Theorem 2.5 can be found in the original ISS paper [235], while the necessity part was proved for the first time in [240]. Here, according to [235, 240], we give a sketch of the proof, which could be helpful in understanding ISS-Lyapunov functions. With property (2.28), it can be proved that there exists a β ∈ KL satisfying β(s, 0) = s for all s ∈ R+ such that V (x(t)) ≤ β(V (x(0)), t),

(2.30)

as long as V (x(t)) ≥ γ( u ∞ ). This means V (x(t)) ≤ max{β(V (x(0)), t), γ( u ∞ )}

(2.31)

¯ t) = α−1 (β(α(s), t)) and γ(s) ¯ = α−1 ◦ γ(s) for s, t ∈ R+ . for all t ≥ 0. Define β(s, ¯ Then, β ∈ KL, γ¯ ∈ K, and ¯ |x(t)| ≤ max{β(|x(0)|, t), γ( u ¯ ∞ )}

(2.32)

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2 Basic Stability and Small-Gain Tools for System Synthesis

ISS-Lyapunov function

ISS

WRS

Fig. 2.2 Equivalence between ISS and the existence of an ISS-Lyapunov function

holds for all t ≥ 0. The sufficiency part of Theorem 2.5 is proved. It should be noted that an ISS-Lyapunov function is not necessarily continuously differentiable. Sometimes, it is more convenient to construct locally Lipschitz ISSLyapunov functions, which are still sufficient for ISS. According to Rademacher’s theorem [60, p. 216], a locally Lipschitz function is continuously differentiable almost everywhere. For a locally Lipschitz ISS-Lyapunov function V, condition (2.28) holds for almost all x. In this case, the arguments used in the original ISS paper [235] are still valid to show that the existence of such a V implies ISS. The necessity part of Theorem 2.5 can be proved by constructing ISS-Lyapunov functions. The proof given in [240] employs the notion of weakly robust stability (WRS), and the basic idea is shown in Fig. 2.2. The WRS property describes the capability of a system to handle state-dependent perturbations. The system (2.14) is said to be WRS if it admits a stability margin ρ ∈ K∞ such that the system x˙ = f (x, d(t)ρ(|x|))

(2.33)

is GAS at the origin uniformly with respect to time, for all possible d : R+ → B m . Recall that B m represents the unit ball with center at the origin in Rm . It is proved in [240] that ISS implies WRS for the system (2.14), and the Lyapunov function of the system (2.33) can be used as the ISS-Lyapunov function for the system (2.14). The proof of the existence of a Lyapunov function for a WRS system is related to the converse Lyapunov theorem. Reference [159] presents a result on the construction of smooth Lyapunov functions for weakly robustly stable systems. A property stronger than WRS is the robust stability (RS) property [240], which considers systems with state-dependent perturbations in the more general form: x˙ = f (x, δ(t, x)),

(2.34)

where the perturbation term δ(t, x) might be caused by uncertainty of the system dynamics. The system (2.14) is said to be RS with a gain margin ρ ∈ K if the perturbed system (2.34) is uniformly GAS at the origin as long as |δ(t, x)| ≤ ρ(|x|). The equivalence between RS and ISS has also been proved in [240].

2.3 Nonlinear Small-Gain Theorem

27

2.3 Nonlinear Small-Gain Theorem The small-gain theorem is a useful tool for the analysis and control design of interconnected systems. With the ISS small-gain theorem, the ISS property of an interconnected system composed of two ISS subsystems can be tested by checking the composition of the ISS gains.

2.3.1 Trajectory-Based Small-Gain Theorem Consider an interconnected nonlinear system composed of two subsystems x˙1 = f 1 (x, u 1 ) x˙2 = f 2 (x, u 2 ),

(2.35) (2.36)

where x = [x1T , x2T ]T with x1 ∈ Rn 1 and x2 ∈ Rn 2 is the state, u 1 ∈ Rm 1 and u 2 ∈ Rm 2 are the external inputs, and f 1 : Rn 1 +n 2 × Rm 1 → Rn 1 and f 2 : Rn 1 +n 2 × Rm 2 → Rn 2 are locally Lipschitz functions satisfying f 1 (0, 0) = 0 and f 2 (0, 0) = 0. For convenience of notations, define u = [u 1T , u 2T ]T . By considering u as a function of time, we assume that it is measurable and locally essentially bounded (Fig. 2.3). For i = 1, 2, assume that each xi -subsystem is ISS with x3−i and u i as the inputs. Specifically, for each i = 1, 2, there exist βi ∈ KL and γi(3−i) , γiu ∈ K such that for any initial state xi (0) = xi0 and any measurable and locally essentially bounded inputs x3−i and u i , |xi (t)| ≤ max{βi (|xi0 |, t),γi(3−i) ( x3−i ∞ ), γiu ( u i ∞ )}

(2.37)

holds for all t ≥ 0. Here, the ISS property of each individual subsystem is expressed using the “max” form, which is mathematically equivalent to the “plus” form used in [122], possibly with different pairs (β, γ).

u1

x1 -subsystem

x2 -subsystem Fig. 2.3 An interconnected system with external inputs

u2

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2 Basic Stability and Small-Gain Tools for System Synthesis

With the discussions below, we show that the interconnected system is ISS with u as the input if γ12 ◦ γ21 < Id.

(2.38)

By γ12 ◦ γ21 < Id, we mean γ12 (γ21 (s)) < s for all s > 0. It should be noted that for any γ12 , γ21 ∈ K, γ12 ◦ γ21 < Id ⇔ γ21 ◦ γ12 < Id.

(2.39)

Indeed, for the implication “⇒”, assume that γ21 ◦ γ12 < Id does not hold. That is, there exists a positive s such that γ21 (γ12 (s)) ≥ s. Then, γ12 ◦ γ21 (γ12 (s)) ≥ γ12 (s), which leads to a contradiction with γ12 ◦ γ21 < Id. By symmetry, the other implication “⇐” holds. Theorem 2.6 presents a trajectory-based ISS small-gain result. Theorem 2.6 Consider the interconnected system composed of two subsystems in the form of (2.35)–(2.36) satisfying (2.37). The interconnected system is ISS with u as the input if the small-gain condition (2.38) is satisfied. The proof of Theorem 2.6 is given in Appendix D.1. A special, yet interesting, case of the interconnected system is a cascade system for which one of the gains γ12 and γ21 is zero. In this case, the small-gain condition is satisfied automatically. If, moreover, (u 1 , u 2 ) = (0, 0), then Theorem 2.6 recovers [138, Lemma 4.7] as a special case for GAS. The small-gain result developed in [122] also takes into account the more general case in which the subsystems are interconnected with each other by outputs instead of states. Consider the following interconnected system: x˙i = f i (xi , y3−i , u i )

(2.40)

yi = h i (xi )

(2.41)

where, for i = 1, 2, xi ∈ Rni is the state, u i ∈ Rm i is the input, yi ∈ Rli is the output, and f i and h i are locally Lipschitz functions satisfying fi (0, 0, 0) = 0 and h i (0) = 0. Assume that each ith subsystem is unboundedness observable (UO) with zero offset and input-to-output stable (IOS) with y3−i , u i as the inputs and yi as the output. Specifically, there exist αiO ∈ K∞ , βi ∈ KL, γi(3−i) ∈ K, and γiu ∈ K such that   |xi (t)| ≤ αiO |xi (0)| + y3−i [0,t] + u i [0,t] |yi (t)| ≤ max{βi (|xi (0)|, t), γi(3−i) ( y3−i [0,t] ), γiu ( u i [0,t] )}

(2.42) (2.43)

for all t ∈ [0, Tmax ), where [0, Tmax ) with 0 < Tmax ≤ ∞ is the right maximal interval for the definition of (x1 (t), x2 (t)). Theorem 2.7 gives a small-gain result for the interconnected IOS system.

2.3 Nonlinear Small-Gain Theorem

29

Theorem 2.7 Consider the interconnected system (2.40)–(2.41) satisfying (2.42) and (2.43) for i = 1, 2. Then the interconnected system is UO and IOS if γ12 ◦ γ21 < Id.

(2.44)

Theorem 2.7 does not assume the forward completeness of the subsystems. Following the discussions in [122], IOS and UO of the subsystems imply the forward completeness of the subsystems. If the small-gain condition is satisfied, then the forward completeness of the interconnected system is guaranteed by the IOS and UO properties of the subsystems. In [125], Theorem 2.7 is generalized for large-scale nonlinear systems composed of more than two subsystems. Reference [122] also takes into account the issue of practical stability by introducing the notion of input-to-output practical stability (IOpS) property, and the smallgain theorem therein is more general than Theorem 2.7. Further extensions of [122] can be found in [124]. References [128, 155, 157] as well as the book [130] have given the extensions of the small-gain theorem to more general complex systems such as hybrid systems and systems modeled by retarded functional differential equations.

2.3.2 Lyapunov-Based Small-Gain Theorem Lyapunov functions play an irreplaceable role in the analysis and synthesis of nonlinear control systems. With the Lyapunov-based formulation of ISS, the ISS property of nonlinear systems is often tested by constructing ISS-Lyapunov functions. This subsection reviews the Lyapunov-based ISS small-gain theorem developed in [118] for interconnected ISS systems. In particular, it is shown that if an interconnected system satisfies the Lyapunov-based ISS small-gain condition, then ISS-Lyapunov functions can be constructed for the system by using the ISS-Lyapunov functions of the subsystems. For the interconnected system (2.35)–(2.36), assume that each xi -subsystem for i = 1, 2 admits a continuously differentiable ISS-Lyapunov function Vi : Rni → R+ satisfying 1. there exist αi , αi ∈ K∞ such that αi (|xi |) ≤ Vi (xi ) ≤ αi (|xi |), ∀xi ;

(2.45)

2. there exist γi(3−i) , γiu ∈ K and a continuous, positive definite αi such that Vi (xi ) ≥ max{γi(3−i) (V3−i (x3−i )), γiu (|u i |)} ⇒∇Vi (xi ) f i (x, u i ) ≤ −αi (Vi (xi )), ∀x, u i .

(2.46)

Theorem 2.8 gives a Lyapunov formulation of the ISS small-gain theorem.

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2 Basic Stability and Small-Gain Tools for System Synthesis

Theorem 2.8 The interconnected system (2.35)–(2.36) with each xi -subsystem admitting an ISS-Lyapunov function Vi satisfying (2.45)–(2.46) is ISS if γ12 ◦ γ21 < Id.

(2.47)

The proof of Theorem 2.8 is given in Appendix D.2. The following example is from [116], and is used to show the effectiveness of the nonlinear small-gain theorem in tackling nonlinear stabilization problems. Example 2.1 Consider the system introduced in [116, 234]: ξ˙1 = ξ12 ξ2 , ξ˙2 = u,

(2.48)

where [ξ1 , ξ2 ]T ∈ R2 is the state, and u ∈ R is the control input. Note that this system is not feedback linearizable. Apply the following change of coordinates: z = ξ1 , x = ξ1 + ξ2

(2.49)

to arrive at the system z˙ = −z 3 + z 2 x,

(2.50)

x˙ = u − z + z x, y = x.

(2.51) (2.52)

3

2

Obviously, this system is not feedback linearizable. Let the output be defined by y = x. The controller u for the system (2.50)–(2.52) is designed as u = −(1 + 72x 2 )x.

(2.53)

Then, the z-subsystem and the x-subsystem are ISS with ISS-Lyapunov functions Vz (z) = 0.5z 2 and Vx (x) = 0.5x 2 satisfying Vz (z) ≥ 4Vx (x) ⇒ ∇Vz (z)˙z ≤ −2Vz2 (z), Vx (x) ≥ 1/9Vz (z) ⇒ ∇Vx (x)x˙ ≤ −2Vx (x) −

(2.54) 144Vx2 (x).

(2.55)

By using the ISS small-gain theorem given by Theorem 2.8, it can be verified that the controller defined by (2.53) globally asymptotically stabilizes the system (2.48). Following the integrator backstepping [127, Lemma IB], we get a globally asymptotically stable controller for the system (2.50)–(2.52) u = −x − z 2 x.

(2.56)

2.3 Nonlinear Small-Gain Theorem

31

Also, by means of the ISS property of the z-subsystem in (2.50), Ref. [234] gives a global asymptotic stabilizer for the system (2.50)–(2.52) u = −x + z 3 − z 2 x.

(2.57)

In sharp contrast to the controllers (2.56) and (2.57), the controller (2.53) only makes use of the information of the scalar output y of the system (2.50)–(2.52). Another appealing advantage of the small-gain approach is that the control law (2.53) also globally asymptotically stabilizes the following system with input and state driven unmodeled dynamics z˙ = −z 3 + z 2 (x + φ(t, z, x, u))

(2.58)

x˙ = u − z + z u

(2.59)

3

2

where φ is any continuously differentiable function satisfying |φ(t, z, x, u)| ≤ 0.05|u|1/3 for all (t, z, x, u) in R+ × R3 . However, the other two methods fail to apply to the system (2.58)–(2.59). This section mainly reviews the results for continuous-time interconnected systems described by differential equations, while the counterparts for discrete-time systems [123, 145] and hybrid systems [155, 157, 204, 205] have also been developed based on the corresponding extensions of ISS. The interconnected hybrid systems studied in [157] may involve both stable and unstable dynamics. If a system can be transformed into an interconnection of ISS subsystems through control designs, then one may employ the ISS small-gain theorem to analyze the stability property of the closed-loop system. The gain assignment technique has been developed such that an appropriate ISS gain can be assigned to a system by means of feedback, and has been recognized to be a key step in applying the ISS small-gain theorem to nonlinear control designs; see, e.g., [116, 117, 122, 223], for small-gain control designs for nonlinear uncertain systems based on the gain assignment technique.

2.4 Nonlinear Cyclic-Small-Gain Theorem The small-gain theorem introduced in Sect. 2.3 has found wide applications in stability analysis, stabilization, robust adaptive control, observer design, output regulation and other bio-system problems [115] for interconnected nonlinear systems. Although one may recursively apply the small-gain theorem to the interconnected systems involving more than one cycles, refined small-gain criteria are highly desirable to handle large-scale interconnected nonlinear systems more efficiently.

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2 Basic Stability and Small-Gain Tools for System Synthesis

2.4.1 Continuous-Time Systems Consider the following large-scale interconnected system containing N subsystems: x˙i = f i (x, u i ) , i = 1, . . . , N ,

(2.60)

 T where x = x1T , . . . , x NT with xi ∈ Rni is the state, u i ∈ Rm i represents the external

inputs, and each f i : Rn+m i → Rni with n = Nj=1 n j is a locally Lipschitz function  T satisfying f i (0, 0) = 0. The external input u = u 1T , . . . , u TN is a measurable and

N locally essentially bounded function from R+ to Rm with m = i=1 m i . Denote f (x, u) = [ f 1T (x, u 1 ), . . . , f NT (x, u N )]T . Assume that for i = 1, . . . , N , each xi -subsystem admits a continuously differentiable ISS-Lyapunov function Vi : Rni → R+ satisfying 1. there exist αi , αi ∈ K∞ such that αi (|xi |) ≤ Vi (xi ) ≤ αi (|xi |), ∀xi ;

(2.61)

2. there exist γi j ∈ K ∪ {0} ( j = 1, . . . , N , j = i) and γui ∈ K ∪ {0} such that  Vi (xi ) ≥ max γi j (V j (x j )), γui (|u i |) j=i

⇒∇Vi (xi ) f i (x, u i ) ≤ −αi (Vi (xi )), ∀x, u i ,

(2.62)

where αi is a continuous and positive definite function. For systems that are formulated in the dissipation form, property 2 above should be replaced by  2 . there exist αi ∈ K∞ , σi j ∈ K ∪ {0} ( j = 1, . . . , N , j = i) and σui ∈ K ∪ {0} such that

  (|u i |) . ∇Vi (xi ) f i (x, u i ) ≤ −αi (Vi (xi )) + max σi j (V j (x j )), σui

(2.63)

Due to the equivalence of the two kinds of ISS-Lyapunov functions, we only consider the gain margin form in the following discussions. By considering the subsystems as vertices and the nonzero gain interconnections as directed links, the gain interconnection structure of the interconnected system can be represented by a digraph, called the gain digraph. Then, concepts from graph theory, such as path, reachability, and simple cycle, can be used to describe the gain interconnections in the large-scale interconnected system. Since the gains are defined with Lyapunov functions, for i = 1, . . . , N , each xi -subsystem is represented by its Lyapunov function Vi . Theorem 2.9 gives the cyclic-small-gain criterion for continuous-time large-scale interconnected systems with subsystems admitting ISS-Lyapunov functions.

2.4 Nonlinear Cyclic-Small-Gain Theorem

33

Theorem 2.9 Consider the continuous-time large-scale interconnected system (2.60) with each xi -subsystem admitting a continuously differentiable ISS-Lyapunov function Vi satisfying (2.61)–(2.62). Then, it is ISS with x as the state and u as the input if for every simple cycle (Vi1 , Vi2 , . . . , Vir , Vi1 ) in the gain digraph, γi1 i2 ◦ γi2 i3 ◦ · · · ◦ γir i1 < Id,

(2.64)

where r = 2, . . . , N and 1 ≤ i j ≤ N , i j = i j  if j = j  . Basic Idea of Constructing ISS-Lyapunov Functions The small-gain theorem introduced in Sect. 2.3 considers the case in which the interconnected system (2.60) contains two subsystems, i.e., N = 2. In this case, if γ12 ◦ γ21 < Id, then the system is ISS and an ISS-Lyapunov function can be constructed as: V (x) = max{V1 (x1 ), σ(V2 (x2 ))},

(2.65)

where σ ∈ K∞ is continuously differentiable on (0, ∞) and satisfies σ > γ12 , σ −1 > γ21 .

(2.66)

Recall the fact that γ12 ◦ γ21 < Id ⇔ γ21 ◦ γ12 < Id. By using Lemma B.2 twice, there exist γˆ 12 , γˆ 21 ∈ K∞ which are continuously differentiable on (0, ∞) and satisfy γˆ 12 > γ12 , γˆ 21 > γ21 and γˆ 12 ◦ γˆ 21 < Id. Thus, with γ12 , γ21 replaced by γˆ 12 , γˆ 21 (as shown Fig. 2.4), the small-gain condition is still satisfied. If we choose σ = γˆ 12 , then condition (2.66) is satisfied and the resulting ISSLyapunov function is  V (x) = max V1 (x1 ), γˆ 12 (V2 (x2 )) .

(2.67)

Since γˆ 12 is a modification of the ISS gain γ12 , the term γˆ 12 (V2 (x2 )) can be considered as the “potential influence” of V2 acting on V1 with modified gain γˆ 12 .

γ21 V1

γˆ21 V2

γ12 Fig. 2.4 The replacement of the ISS gains

⇒

V1

V2 γˆ12

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2 Basic Stability and Small-Gain Tools for System Synthesis

A Class of ISS-Lyapunov Functions for Large-Scale Interconnected Nonlinear Systems Based on the idea of potential influence, a class of ISS-Lyapunov functions are constructed for large-scale interconnected systems satisfying the cyclic-small-gain condition. Consider an interconnected system in the form of (2.60) with the cyclic-small-gain condition (2.64) satisfied. For each i ∗ = 1, . . . , N , it holds that γi ∗ i2 ◦ γi2 i3 ◦ · · · ◦ γir i ∗ < Id

(2.68)

for r = 2, . . . , N , 1 ≤ i j ≤ N , i j = i ∗ , i j = i j  if j = j  . By using Lemma B.2, if γi ∗ i2 = 0, then one can find a γˆ i ∗ i2 ∈ K∞ which is continuously differentiable on (0, ∞) and satisfies γˆ i ∗ i2 > γi ∗ i2 such that (2.68) still holds with γi ∗ i2 replaced by γˆ i ∗ i2 . By repeating this procedure for all the γi ∗ i2 with i ∗ = 1, . . . , N , i 2 = i ∗ , there exist γˆ (·) ’s such that 1. γˆ (·) ∈ K∞ and γˆ (·) > γ(·) if γ(·) ∈ K; γˆ (·) = 0 if γ(·) = 0. 2. γˆ (·) ’s are continuously differentiable on (0, ∞). 3. for each r = 2, . . . , N , γˆ i1 i2 ◦ · · · ◦ γˆ ir i1 < Id

(2.69)

holds for all 1 ≤ i j ≤ N and i j = i j  if j = j  . With the treatment above, all the nonzero gains in the interconnected system are replaced by the γˆ (·) ’s, which are of class K∞ and continuously differentiable on (0, ∞) such that the cyclic-small-gain condition is still satisfied. Note that the replacement of the nonzero gains does not influence the gain digraph. In the large-scale interconnected system, the potential influence acting on the pth subsystem from all the subsystems can be described as V[ p] =

[ p]

V j (x)

(2.70)

j=1,...,N

with     [ p] V j (x) = γˆ i [ p] i [ p] ◦ · · · ◦ γˆ i [ p] i [ p] Vi [ p] xi [ p] , 1

[ p]

[ p]

2

j−1 j

j

[ p]

j

[ p]

where i 1 = p, i k ∈ {1, . . . , N }, k ∈ {1, . . . , j}, i k = i k  if k = k  , for j = [ p] 1, . . . , N . Clearly, each element in V j (x) corresponds to a simple path ending at V p in the gain digraph.

2.4 Nonlinear Cyclic-Small-Gain Theorem

35

Note that γˆ (·) ∈ K∞ ∪ {0}. It is easy to verify that max V[ p] is positive definite and radially unbounded with respect to the Lyapunov functions of the subsystems with indices belonging to RS( p). T  Correspondingly, the potential influence of the external input u = u 1T , . . . , u TN acting on the p-th subsystem can be described as U[ p] =

[ p]

Uj

(2.71)

j=1,...,N

with   [ p] U j = γˆ i [ p] i [ p] ◦ · · · ◦ γˆ i [ p] i [ p] ◦ γui [ p] (|u i [ p] |) 1

2

j−1 j

j

j−1

(2.72)

for j = 1, . . . , N . Define ⎛ VΠ (x) = max VΠ (x) = max ⎝

⎞ V[ p] (x)⎠

(2.73)

p∈Π

where set Π ⊆ {1, . . . , N } satisfies

(RS( p)) = {1, . . . , N }.

(2.74)

p∈Π

  [ p] It can be directly verified that max is positive definite and radially V p∈Π unbounded with respect to max {V1 , . . . , VN } and thus with respect to x, i.e., there exist α, α ∈ K∞ such that α(|x|) ≤ VΠ (x) ≤ α(|x|) for all x. It can also be observed that VΠ is locally Lipschitz on Rn \ {0}. Thanks to Rademacher’s theorem (see, e.g., [60, p. 216]), VΠ is differentiable almost everywhere. Accordingly, define ⎛ u Π = max UΠ = max ⎝

⎞ U[ p] ⎠ .

(2.75)

p∈Π

It can be verified that there exists a γ u ∈ K∞ such that u Π ≤ γ u (|u|) for all u. Reference [162] shows that VΠ (x) is an ISS-Lyapunov function (not necessarily continuously differentiable) of the interconnected system with u Π as the new input. One can find a continuous and positive definite function αΠ such that VΠ (x) ≥ u Π ⇒ ∇VΠ (x) f (x, u) ≤ −αΠ (VΠ (x)).

(2.76)

36

2 Basic Stability and Small-Gain Tools for System Synthesis

V2

Fig. 2.5 The gain digraph of the interconnected system (2.77)

V1

V3

Example 2.2 Consider an interconnected nonlinear system composed of three subsystems: x˙i = f i (x), i = 1, 2, 3,

(2.77)

where xi ∈ Rni is the state of the i-th subsystem, x = [x1T , x2T , x3T ]T , and f i : Rn 1 +n 2 +n 3 → Rni is a locally Lipschitz function satisfying f i (0) = 0. Suppose that each xi -subsystem has an ISS-Lyapunov function Vi , which is positive definite and radially unbounded, and satisfies  Vi (xi ) ≥ max γi j (V j (x j )) ⇒ ∇Vi (xi ) f i (x) ≤ −αi (Vi (xi )), ∀x, j=i

(2.78)

where γi j ∈ K ∪ {0} represents the ISS gains and αi is a continuous and positive definite function. The gain digraph of the interconnected system defined above is shown in Fig. 2.5. Since there is no external input, the interconnected system is asymptotically stable at the origin if it satisfies the cyclic-small-gain condition given by Theorem 2.9. For the gain digraph shown in Fig. 2.5, RS(i) = {1, 2, 3} for i = 1, 2, 3. Different ISS-Lyapunov functions VΠ ’s can be constructed by choosing different Π ’s. For example,  V{1} (x) = max V1 (x1 ), γˆ 12 (V2 (x2 )), γˆ 13 ◦ γˆ 32 (V2 (x2 )), γˆ 13 (V3 (x3 )) ,  V{2} (x) = max V2 (x2 ), γˆ 21 (V1 (x1 )), γˆ 21 ◦ γˆ 13 (V3 (x3 )) .

(2.79) (2.80)

There are two terms depending on V2 (x2 ) in the definition of V{1} (x), because there are two simple paths leading from V2 to V1 in the gain digraph. If the gain digraph of an interconnected system is disconnected, then it is impossible to find one single subsystem which is reachable from all the other subsystems, and the Π defined by (2.74) should contain more than one element to construct a positive definite and radially unbounded VΠ .

2.4 Nonlinear Cyclic-Small-Gain Theorem

37

V2

Fig. 2.6 A disconnected gain digraph

V4

V5 V1

V3

Example 2.3 Consider an interconnected system with the gain digraph shown in Fig. 2.6. Since RS(1) ∪ RS(5) = {1, 2, 3, 4, 5}, we choose Π = {1, 5}, with which an ISS-Lyapunov function is constructed as  VΠ (x) = max V1 (x1 ), γˆ 13 (V3 (x3 )),γˆ 13 ◦ γˆ 32 (V2 (x2 )), V5 (x5 ), γˆ 54 (V4 (x4 ))



(2.81)

where γˆ (·) ’s are the appropriately modified ISS gains. It can be observed that max{V1 (x1 ), γˆ 13 (V3 (x3 )), γˆ 13 ◦ γˆ 32 (V2 (x2 ))} and max{V5 (x5 ), γˆ 54 (V4 (x4 ))} are the Lyapunov functions of the (x1 , x2 , x3 )-subsystem (the part on the left-hand side in Fig. 2.6) and the (x4 , x5 )-subsystem (the part on the right-hand side), respectively. In fact, the Lyapunov function for a system that is composed of disconnected subsystems can be directly defined as the maximum of the Lyapunov functions of all the disconnected subsystems. With Π = {1, 5}, define  u Π = max u 1 , γˆ 13 ◦ γu3 (u 3 ), γˆ 13 ◦ γˆ 32 ◦ γu2 (u 2 ), u 5 , γˆ 54 ◦ γu4 (u 4 ) .

(2.82)

Then, there exists a continuous and positive definite function αΠ such that VΠ (x) ≥ u Π ⇒ ∇VΠ (x) f (x, u) ≤ −αΠ (VΠ (x)), a.e.

(2.83)

To analyze the influence of the external inputs to each subsystem, one may first transform the Lyapunov-based ISS property into the trajectory-based ISS property: for any initial state x(0) = x0 ,  VΠ (x(t)) ≤ max β(VΠ (x0 ), t), u Π [0,t]

(2.84)

where β ∈ KL. Consider the x2 -subsystem for example. From the definition of VΠ , −1 −1 ◦ γˆ 13 (VΠ (x)) for all x, and the influence of the external it holds that V2 (x2 ) ≤ γˆ 32 inputs on the x2 -subsystem can be estimated through the following IOS property by

38

2 Basic Stability and Small-Gain Tools for System Synthesis

considering V2 (x2 ) as the output:    −1 −1 −1 −1 u Π [0,t] ◦ γˆ 13 ◦ γˆ 13 V2 (x2 (t)) ≤ max γˆ 32 (β(VΠ (x0 ), t)) , γˆ 32

(2.85)

for any initial state x(0) = x0 . According to the definition of u Π in (2.82), property (2.85) implies that the x2 -subsystem is influenced by u 4 and u 5 . However, due to the disconnected system structure, u 4 and u 5 do not influence the x2 -subsystem. For a more accurate estimation, one may just use the Lyapunov function of the (x1 , x2 , x3 )subsystem to estimate the influence of u 1 , u 2 , and u 3 on the x2 -subsystem.

2.4.2 Equivalence Between Cyclic-Small-Gain and Gains Less Than the Identity This subsection proposes a result on the equivalence between cyclic-small-gain and gains less than Id. The proof of the equivalence is based on the fact that, for a continuous-time or discrete-time system, if V is an ISS-Lyapunov function, then for any σ ∈ K∞ being locally Lipschitz on (0, ∞), σ(V ) is also an ISS-Lyapunov function. Consider a continuous-time system for example. Assume that the system x˙ = f (x, u) with state x ∈ Rn and external input u ∈ Rm is ISS with V : Rn → R+ as an ISS-Lyapunov function satisfying V (x) ≥ γu (|u|) ⇒ ∇V (x) f (x, u) ≤ −α(V (x)), a.e.,

(2.86)

where γ ∈ K and α is a continuous and positive definite function. Then, for any σ ∈ K∞ being locally Lipschitz on (0, ∞), V¯ := σ(V ) is also continuously differentiable almost everywhere, and there exists a continuous and positive definite function α¯ such that V¯ (x) ≥ σ ◦ γ(|u|) ⇒ ∇ V¯ (x) f (x, u) ≤ −α( ¯ V¯ (x)), a.e.

(2.87)

Theorem 2.10 gives the equivalence between cyclic-small-gain and gains less than the identity. A detailed proof is given by [170]. Theorem 2.10 With (2.61) and (2.62) satisfied, suppose that the large-scale system composed of (2.60) satisfies the cyclic-small-gain condition (2.64). For each i = 1, . . . , N , there exists a σi ∈ K∞ being locally Lipschitz on (0, ∞) such that V¯i (xi ) = σi (Vi (xi )) is an ISS-Lyapunov function of the xi -subsystem, that satisfies

(2.88)

2.4 Nonlinear Cyclic-Small-Gain Theorem

39

 V¯i (xi ) ≥ max γ¯ i j (V¯ j (x j )), γ¯ ui (|u i |) j=i

⇒∇ V¯i (xi ) f i (x, u i ) ≤ −αi (V¯i (xi )) a.e.

(2.89)

where γ¯ i j ∈ K ∪ {0} satisfies γ¯ i j < Id, γ¯ ui = σi ◦ γui , and αi is continuous and positive definite. With the satisfaction of the cyclic-small-gain condition (2.64), property (2.89) can be proved by directly using the main result of [170].

2.5 Notes Small-gain condition, i.e., a loop-gain less than unity, is one way to ensure stability of interconnected systems. In the past twenty years, tremendous efforts have been made for stability analysis and control design of interconnected nonlinear systems. The idea of small-gain theorem has originally been studied with the gain property taking a linear or affine form; see, e.g., [49, 281] for input-output feedback systems, as well as the recent works [29, 80]. The small-gain theorem for nonlinear feedback systems with non-affine gains was presented in [94, 185] within the input-output context. Taking an explicit advantage of Sontag’s seminal work on ISS [235, 239–241], the first generalized, nonlinear ISS small-gain theorem is proposed in [122]. The IOS counterpart of the small-gain theorem is also available in [122]. As a fundamental difference with respect to the earlier small-gain theorems, in the ISS or IOS framework, the role of the initial conditions is made explicit to ensure asymptotic stability in the Lyapunov sense as well as bounded-input bounded-output stability. A new small-gain design tool is presented for the first time in [116, 122] for robust global stabilization of nonlinear systems with dynamic uncertainties. In parallel, Teel presents a small-gain tool for the analysis and synthesis of control systems with saturation in [249]. A Lyapunov reformulation of the ISS small-gain theorem can be found in [118]. Some recent extensions of the ISS small-gain theorem can be found in [45, 113, 114, 125, 130, 221, 226]. To the best of the authors’ knowledge, Teel [248] stated an extension of the nonlinear small-gain theorem for the first time, for networks of discrete-time ISS systems. Shortly, the authors of [45, 46, 226] developed a matrix-small-gain criterion for networks with plus-type interconnections. In [113, 125], a more general cyclic-small-gain theorem for networks of IOS systems was developed. The corresponding Lyapunov formulations have been developed in [162, 163]. The small-gain methods have also been introduced for hybrid systems, which involve both continuous-time and discrete-time dynamics; see e.g., [30, 43, 93, 128, 131, 155, 156, 205]. A cyclic-small-gain theorem for hybrid dynamical networks with the impulses of the subsystems triggered asynchronously is developed in [170]. Reference [24] presents a small-gain result for large-scale interconnected systems

40

2 Basic Stability and Small-Gain Tools for System Synthesis

composed of hybrid subsystems. A time-delay version of the cyclic-small-gain theorem can be found in [253]. As a powerful tool, the ISS small-gain theorem has been included by standard textbooks on nonlinear systems; see, e.g., [106, 138]. See also the book [130] and the references cited therein for other more recent developments along the line of ISS small-gain theorem. There have also been numerous successful applications of the small-gain theorem to nonlinear control designs. The applications of the small-gain theorem to output regulation and global stabilization of nonlinear feedforward systems can be found in [35, 37, 38, 98, 99]. References [32, 157, 204] employ the small-gain theorem for networked and quantized control designs. In [219], the authors employ a modified small-gain theorem to solve the stability problem arising from observer-based control designs. Another interesting application of the small-gain theorem lies in robust adaptive dynamic programming; see e.g., [108, 112]. This chapter mainly focuses on continuous-time interconnected systems described by differential equations, while the counterparts of the results for discrete-time systems [123, 145] and hybrid systems [155, 157, 204, 205] have also been developed based on the corresponding extensions of ISS. The interconnected hybrid systems studied in [157] may involve both stable and unstable dynamics. In the discontinuous case, i.e., systems involving discontinuous dynamics, the cyclic-small-gain condition is still valid as long as the subsystems are ISS. See [88] for the extension of the original ISS small-gain theorem for discontinuous systems. This chapter has also presented cyclic-small-gain results for continuous-time dynamical networks modeled by differential equations with the ISS property of the subsystems formulated by ISS-Lyapunov functions based on recent results in [113, 125, 162], while the counterparts of the results for discrete-time and hybrid dynamical networks [163, 170] have been developed based on the corresponding extensions of ISS.

Part II

Robust Event Triggers

Chapter 3

A Small-Gain Paradigm for Event-Triggered Control

This chapter proposes a small-gain approach to event-triggered control of nonlinear systems. In particular, we consider an event-triggered control system as an interconnection of two parts, the controlled system and the event trigger, and use gains to characterize the interaction between the two parts. Then, we discuss the forward completeness and stability issues by using small-gain arguments, and develop a class of static event triggers to solve the event-triggered stabilization problem for a class of nonlinear uncertain systems.

3.1 Problem Formulation Consider an event-triggered control system x(t) ˙ = f (x(t), u(t))

(3.1)

u(t) = κ(x(tk )), t ∈ [tk , tk+1 ), k ∈ S ⊆ Z+ ,

(3.2)

where x ∈ Rn is the state, u ∈ Rm is the control input, f : Rn × Rm → Rn is a locally Lipschitz function representing system dynamics, κ : Rn → Rm is a locally Lipschitz function representing the control law. It is assumed that f (0, κ(0)) = 0. In event-triggered control, the time sequence {tk }k∈S with t0 = 0 is determined online by an appropriately designed event trigger, with S being the set of the indices of all the sampling times. Because of possible finite-time accumulation of tk and finite escape time, the state trajectory x(t) may only be defined on some finite interval. Suppose that x(·) is right maximally defined on [0, Tmax ) with 0 < Tmax ≤ ∞. With respect to the possible finite-time accumulation of tk and finite escape time, there are three cases: © Huazhong University of Science and Technology Press, Wuhan and Springer Nature Singapore Pte Ltd. 2020 T. Liu et al., Robust Event-Triggered Control of Nonlinear Systems, Research on Intelligent Manufacturing, https://doi.org/10.1007/978-981-15-5013-3_3

43

44

3 A Small-Gain Paradigm for Event-Triggered Control

(1) S = Z+ and limk→∞ tk < ∞, which means Zeno behavior [74, 75]. (2) S = Z+ and limk→∞ tk = ∞. In this case, x(t) is defined on [0, ∞). (3) S is a finite set {0, . . . , k ∗ } with k ∗ ∈ Z+ , i.e., there is a finite number of sampling time instants. In this case, tk ∗ < Tmax and we denote tk ∗ +1 = Tmax to simplify the discussions.  In any case above, x(t) is defined for all t ∈ k∈S [tk , tk+1 ). By appropriately designing an event trigger, we will show that inf k∈S {tk+1 − tk } > 0, which means that Case (1) is impossible. Also, by means of small-gain arguments, we will prove that Tmax = ∞ for Case (3). To explicitly represent the influence of data-sampling, define w(t) = x(tk ) − x(t), t ∈ [tk , tk+1 ), k ∈ S

(3.3)

as the sampling error, and rewrite the control law (3.2) as u(t) = κ(x(t) + w(t)).

(3.4)

Then, we have the closed-loop system x(t) ˙ = f (x(t), κ(x(t) + w(t))).

(3.5)

Figure 3.1 gives the block diagram of the closed-loop event-triggered system, from which one may recognize the relationship between event-triggered control and robust control. It is an essential requirement that the closed-loop system is robust with respect to the sampling error w. Moreover, in the problem setting of event-triggered control, the sampling error w is adjustable by triggering the data-sampling events online, and it is desired that the system state x is asymptotically steered to the origin by appropriately adjusting w. For physical realization of event-triggered sampling, a positive lower bound of the inter-sampling intervals should be guaranteed throughout the process of event-triggered control, i.e.,

w(t)

x(t) ˙ = f (x(t), κ(x(t) + w(t)))

− + x(tk )

event trigger

Fig. 3.1 Event-triggered control problem considered as a robust control problem

x(t)

3.1 Problem Formulation

45

inf {tk+1 − tk } > 0, k∈S

(3.6)

to avoid infinitely fast sampling. A natural approach to event-triggered control contains two steps: 1. Designing a continuous-time controller which guarantees the robustness of the closed-loop system with respect to the sampling error; 2. Designing an event trigger to restrict the sampling error to be within the robustness margin. As a standard notion of robust stability for nonlinear systems, ISS has been used for event-triggered control of nonlinear systems. This chapter uses gains to characterize the interaction between the controlled system and the event trigger, and gives a smallgain condition for event-triggered stabilization of nonlinear uncertain systems. Both Lyapunov-based and trajectory-based formulations are discussed.

3.2 Lyapunov-Based Small-Gain Design of Event Triggers In this section, we focus on the design of the event triggers, and directly assume the existence of controllers for the plants such that the closed-loop systems are robust with respect to the sampling errors. The robustness is characterized by ISS-Lyapunov functions. Assumption 3.1 The system (3.5) is ISS with w as the input, and admits a continuously differentiable ISS-Lyapunov function V : Rn → R+ satisfying the following conditions: 1. there exist α, α ∈ K∞ such that α(|x|) ≤ V (x) ≤ α(|x|), ∀x,

(3.7)

2. there exist a γ ∈ K and a continuous, positive definite α such that V (x) ≥ γ(|w|) ⇒ ∇V (x) f (x, κ(x + w)) ≤ −α(V (x)), ∀x, w.

(3.8)

Under Assumption 3.1, if the event trigger is designed such that |w(t)| ≤ ρ(|x(t)|)

(3.9)

for all t ≥ 0 with ρ ∈ K satisfying α−1 ◦ γ ◦ ρ < Id,

(3.10)

then with the robust stability property of ISS, the state x asymptotically converges to the origin.

46

3 A Small-Gain Paradigm for Event-Triggered Control

Based on this idea, the event trigger is designed as: if x(tk ) = 0, then tk+1 = inf {t > tk : |x(t) − x(tk )| ≥ ρ(|x(t)|)} , t0 = 0.

(3.11)

If x(tk ) = 0 or {t > tk : |x(t) − x(tk )| ≥ ρ(|x(t)|)} = ∅, then the data sampling event is not triggered and in this case, we set tk+1 = ∞. Note that, under the assumption of f (0, κ(0)) = 0, if x(tk ) = 0, then u(t) = κ(x(tk )) = 0 keeps the state at the origin for all t ∈ [tk , ∞). With the event trigger proposed above, given tk and x(tk ) = 0, tk+1 is the first time instant after tk such that ρ(|x(tk+1 )|) − |x(tk+1 ) − x(tk )| = 0.

(3.12)

Since ρ(|x(tk )|) − |x(tk ) − x(tk )| > 0 for any x(tk ) = 0 and x(t) is continuous on the time-line, the proposed event trigger guarantees that ρ(|x(t)|) − |x(t) − x(tk )| ≥ 0

(3.13)

for t ∈ [tk , tk+1 ). Recall the definition of w in (3.3). Property (3.13) implies that |w(t)| ≤ ρ(|x(t)|)

(3.14)

holds for t ∈ [tk , tk+1 ). At  this stage, it cannot be readily guaranteed that (3.14) holdsfor all t ≥ 0, as k∈S [tk , tk+1 ) may not cover the whole time-line, i.e., R+ \ k∈S [tk , tk+1 ) = ∅. As mentioned above, for physical implementation of (3.14) with event-triggered sampling, a positive lower bound of the inter-sampling intervals should be guaranteed throughout the event-triggered control procedure, i.e., inf k∈S {tk+1 − tk } > 0, to avoid infinitely fast sampling. Theorem 3.1 presents a condition on the ISS gain γ in (3.8), which guarantees the existence of a ρ for the event trigger (3.11) such that inf k∈S {tk+1 − tk } > 0, and the closed-loop event-triggered system is asymptotically stable at the origin. Theorem 3.1 Consider the event-triggered controlled system (3.5) with locally Lipschitz f and κ satisfying f (0, κ(0)) = 0 and w defined in (3.3). If Assumption 3.1 is satisfied with α−1 ◦ γ being Lipschitz on compact sets, then one can find a ρ such that (a) (b) (c)

ρ ∈ K∞ is Lipschitz on compact sets; ρ−1 is Lipschitz on compact sets; ρ satisfies (3.10).

Moreover, with the sampling time instants triggered by (3.11), for any specific initial state x(0), the system state x satisfies ˘ |x(t)| ≤ β(|x(0)|, t)

(3.15)

3.2 Lyapunov-Based Small-Gain Design of Event Triggers Fig. 3.2 An illustration of Θ1 (x(tk )) ⊆ Θ2 (x(tk ))

x2

47

Θ2 Θ1 x(tk ) ρ

1 0

x1

for all t ≥ 0, with β˘ ∈ KL, and the inter-sampling intervals are lower bounded by a positive constant. Proof With α−1 ◦ γ ∈ K being Lipschitz on compact sets, one can always find a γ¯ ∈ K∞ being Lipschitz on compact sets such that γ¯ > α−1 ◦ γ. By choosing ρ = γ¯ −1 , we have ρ ◦ α−1 ◦ γ = γ¯ −1 ◦ α−1 ◦ γ < Id, and ρ−1 = γ¯ is Lipschitz on compact sets. Along each trajectory of the closed-loop system, for each k ∈ S with state x(tk ) at time instant tk , define   Θ1 (x(tk )) = x ∈ Rn : |x − x(tk )| ≤ ρ ◦ (Id + ρ)−1 (|x(tk )|) ,   Θ2 (x(tk )) = x ∈ Rn : |x − x(tk )| ≤ ρ(|x|) .

(3.16) (3.17)

By directly using Lemma B.3, it can be proved that Θ1 (x(tk )) ⊆ Θ2 (x(tk )). An illustration with x = [x1 , x2 ]T ∈ R2 is given in Fig. 3.2. Given a ρ ∈ K∞ such that ρ−1 is Lipschitz on compact sets, it can be proved that (ρ ◦ (Id + ρ)−1 )−1 = (Id + ρ) ◦ ρ−1 = ρ−1 + Id is Lipschitz on compact sets, and there exists a continuous, positive function ρ˘ : R+ → R+ such that (ρ−1 + of ρ, ˆ one has s = ≤ ρ(s)s ˘  =: ρ(s) ˆ for s ∈ R+ . By using  the definition  Id)(s)−1 ¯ Here, it can be ρ˘ ◦ ρˆ (s) ρˆ−1 (s), and thus ρˆ−1 (s) = s/ ρ˘ ◦ ρˆ−1 (s) =: ρ(s)s. checked that ρ¯ : R+ → R+ is continuous and positive. Thus, ¯ ρ ◦ (Id + ρ)−1 (s) = (ρ−1 + Id)−1 (s) ≥ ρˆ−1 (s) = ρ(s)s.

(3.18)

Property (3.18) implies that, if ¯ |x − x(tk )| ≤ ρ(|x(t k )|)|x(tk )|,

(3.19)

48

3 A Small-Gain Paradigm for Event-Triggered Control

then x ∈ Θ1 (x(tk )). Also, for any x ∈ Θ1 (x(tk )), by using the locally Lipschitz property of f and κ, it holds that | f (x, κ(x(tk )))| = | f (x, κ(x(tk )))| = | f (x − x(tk ) + x(tk ), κ(x(tk )))|   ≤ L f¯ |[x T − x T (tk ), x T (tk )]T | |[x T − x T (tk ), x T (tk )]T | ¯ ≤ L(|x(t (3.20) k )|)|x(tk )|, where L f¯ and L¯ are continuous, positive functions defined on R+ . Property (3.19) is used for the last inequality. Then, the minimum time Tkmin needed for the state of the closed-loop system starting at x(tk ) to go outside the region Θ1 (x(tk )) can be estimated by Tkmin ≥

ρ(|x(t ¯ ρ(|x(t ¯ k )|)|x(tk )| k )|) = , ¯L(|x(tk )|)|x(tk )| ¯L(|x(tk )|)

(3.21)

which is well defined and strictly larger than zero for any x(tk ). Since Θ1 (x(tk )) ⊆ Θ2 (x(tk )) and x(t) is continuous on the time-line, the minimum time needed for the state starting at x(tk ) to go outside Θ2 (x(tk )) is not less than Tkmin . By directly using (3.21), one has T0min ≥

ρ(|x(0)|) ¯ . ¯ L(|x(0)|)

(3.22)

If S = {0}, then w(t) is continuous and (3.13) holds for t ∈ [0, ∞). We now consider the case of S = {0}. Suppose that for a specific k ∈ Z+ \{0}, the event trigger (3.11) guarantees that for t ∈ [0, tk ), w(t) is piece-wise continuous and (3.14) holds. Under conditions (3.7) and (3.8), along the trajectories of the system (3.5), it holds that V˙ (x(t)) ≤ −α(V (x(t)))

(3.23)

˘ |x(t)| ≤ β(|x(0)|, t)

(3.24)

for all t ∈ [0, tk ), and thus,

for all t ∈ [0, tk ), with β˘ ∈ KL. Due to the continuity of x(t) with respect to ˘ 0). This, together with property t, x(tk ) = limt→tk− x(t). Thus, |x(tk )| ≤ β(|x(0)|, (3.21), implies that  Tkmin ≥ min

 ρ(|x|) ¯ ˘ : |x| ≤ β(|x(0)|, 0) . ¯ L(|x|)

(3.25)

3.2 Lyapunov-Based Small-Gain Design of Event Triggers

49

This means that for t ∈ [0, tk+1 ), w(t) is piece-wise continuous and (3.14) holds. By induction, w(t) is piece-wise continuous and (3.13) holds for t ∈ [0, tk+1 ) for any k ∈ S. If S is an infinite set, then limk→∞ tk+1 = ∞ by using (3.22); if S is a finite set, say {0, . . . , k ∗ }, then tk ∗ +1 = ∞. In both cases, w(t) is piece-wise continuous and (3.13) holds for t ∈ [0, ∞). With the robust stability property of ISS, property (3.24) holds for t ≥ 0. This ends the proof of Theorem 3.1. The proof of Theorem 3.1 naturally leads to a self-triggered sampling strategy [86, 187, 189], which computes tk+1 by using tk and x(tk ), and thus does not continuously monitor the trajectory of x(t). Suppose that Assumption 3.1 is satisfied for the closedloop system composed of (3.1) and (3.4) with locally Lipschitz f and κ. With property (3.21), given tk and x(tk ), tk+1 can be computed as tk+1 =

ρ(|x(t ¯ k )|) + tk ¯L(|x(tk )|)

(3.26)

for k ∈ Z+ . Based on the proof of Theorem 3.1, it can be verified that ρ(|x(t)|) − |x(t) − x(tk )| ≥ 0 holds for all t ∈ [tk , tk+1 ), k ∈ Z+ , and all the inter-sampling intervals are lower bounded by a positive constant. With Assumption 3.1 satisfied, the closed-loop event-triggered system with x as the state is globally asymptotically stable at the origin. The following example shows the validity of Theorem 3.1 for event-triggered control of linear time-invariant systems. Example 3.1 Consider the linear system x˙ = Ax + Bu

(3.27)

where x ∈ Rn is the state, and u ∈ Rm is the control input. According to Theorem 3.1, the condition for event-triggered control is fulfilled, if the system (3.27) is controllable. One finds a K such that A − B K is Hurwitz and designs a control law u = −K (x + w) with w being the sampling error. Then, x˙ = Ax − B K (x + w) = (A − B K )x − B K w.

(3.28)

According to the Lyapunov theorem for linear systems [33, Theorem 5.5], for any given symmetric matrix Q > 0, there is an unique symmetric matrix P > 0 satisfying the Lyapunov equation (A − B K )T P + P(A − B K ) = −Q. Define V (x) = x T P x. Then, V (x) satisfies (3.7) and (3.8) with

(3.29)

50

3 A Small-Gain Paradigm for Event-Triggered Control

α(s) = λmin (P)s 2 , α(s) = λmax (P)s 2 , γ(s) =

4λmax (P)|P B K |2 2 λmin (Q) s s , α(s) = −c 2 2 (1 − c) (λmin (Q)) λmax (P)

for s ∈ R+ , where c is a positive constant satisfying 0 < c < 1. Since both α and γ are quadratic functions, α−1 ◦ γ is linear. The following example shows the effectiveness of Theorem 3.1 for event-triggered control of a nonlinear uncertain system. Example 3.2 Consider the nonlinear system x(t) ˙ = ax 2 (t) + b sin(x(t)) + u(t) u(t) = −(8|x(tk )| + 4)x(tk ), t ∈ [tk , tk+1 ), k ∈ S,

(3.30) (3.31)

where x ∈ R is the state, u ∈ R is the control input, and a and b are unknown constants and satisfy 1 < a < 2 and 0.5 < b < 1, respectively. Define V (x) = |x|. Then, Assumption 3.1 is satisfied with α(s) = s, α(s) = s, ρd (s) = 2s and α(s) = s for s ∈ R+ . With Theorem 3.1, the event trigger is designed as tk+1 = inf{t > tk : |x(t) − x(tk )| ≥ ρw (|x(t)|)}

(3.32)

where ρw (s) = 0.5s for s ∈ R+ . Figures 3.3, 3.4 and 3.5 show that the objective of event-triggered stabilization is achieved for the nonlinear uncertain system (3.30). Specifically, the inter-sampling intervals are lower bound by a positive constant, and the system state x converges to the origin.

x x(t )

1

k

system state

0.8 0.6 0.4 0.2 0 −0.2 0

0.5

time

1

1.5

Fig. 3.3 The state trajectory and its sampled data with the event trigger (3.32) in Example 3.2

3.3 Trajectory-Based Small-Gain Design of Event Triggers

51

2 0

control input

-2 -4 -6 -8 -10 -12 -14

0

0.5

1

1.5

time

threshold signal and sanpling error

Fig. 3.4 The control input with the event trigger (3.32) in Example 3.2

threshold signal sampling error

0.5

0

0

0.5

1

1.5

1

1.5

Inter-samping intervals

time 100 10

-2

10

-4

0

0.5 time

Fig. 3.5 The trajectories of the threshold signal ρw (|x|), the sampling error w, and the intersampling intervals with the event trigger (3.32) in Example 3.2

3.3 Trajectory-Based Small-Gain Design of Event Triggers An ISS-Lyapunov function readily determines the ISS properties of a system, but even if a nonlinear system has been designed to be ISS, constructing an ISS-Lyapunov function may not be straightforward. By using the relationship between ISS and robust stability, the study in this section shows how an event trigger can be designed without the Lyapunov formulation. In this section, Assumption 3.2 is made to replace Assumption 3.1.

52

3 A Small-Gain Paradigm for Event-Triggered Control

Assumption 3.2 The system (3.5) is ISS with w as the input, that is, there exist β ∈ KL and γ ∈ K such that for any initial state x(0) and any piece-wise continuous and bounded w, it holds that |x(t)| ≤ max {β(|x(0)|, t), γ(w∞ )}

(3.33)

for all t ≥ 0. Under Assumption 3.2, with the robust stability property of ISS, if the event trigger is designed such that |w(t)| ≤ ρ(|x(t)|) for all t ≥ 0 with ρ ∈ K satisfying ρ ◦ γ < Id,

(3.34)

then x(t) asymptotically converges to the origin. Based on this idea, the event trigger considered in this section is still defined as (3.11), but the function ρ is chosen to satisfy (3.34). Theorem 3.2 gives the condition on the ISS gain γ defined by (3.33), and the additional conditions on ρ for the event trigger (3.11) to avoid infinitely fast sampling, and to achieve asymptotic stabilization at the origin. Theorem 3.2 Consider the event-triggered controlled system (3.5) with locally Lipschitz f and κ satisfying f (0, κ(0)) = 0 and w defined in (3.3). If Assumption 3.2 is satisfied with γ being Lipschitz on compact sets, then one can find a ρ such that (a) (b) (c)

ρ ∈ K∞ is Lipschitz on compact sets; ρ−1 is Lipschitz on compact sets; ρ satisfies (3.34).

Moreover, with the sampling time instants triggered by (3.11), for any specific initial state x(0), the system state x satisfies ˆ |x(t)| ≤ β(|x(0)|, t)

(3.35)

for all t ≥ 0, with βˆ ∈ KL, and the inter-sampling intervals are lower bounded by a positive constant. Proof The basic idea of the proof of Theorem 3.2 is similar to that of Theorem 3.1. The major difference lies in that Theorem 3.2 is based on the trajectory-based small-gain theorem (see Theorem 2.6). A sketch of the proof is given here. With γ ∈ K being Lipschitz on compact sets, one can always find a γ¯ ∈ K∞ being Lipschitz on compact sets such that γ¯ > γ. By choosing ρ = γ¯ −1 , we have ρ ◦ γ = γ¯ −1 ◦ γ < Id, and ρ−1 = γ¯ is Lipschitz on compact sets. Then, one can find a positive lower bound of the inter-sampling intervals such that inf k∈S {tk+1 − tk } > 0. Then, it is proved that (3.35) holds for all t ≥ 0 by using the trajectory-based smallgain theorem given in Sect. 2.3.1.  The following example shows the validity of Theorem 3.2 for event-triggered control of linear time-invariant systems.

3.3 Trajectory-Based Small-Gain Design of Event Triggers

53

Example 3.3 Consider the linear system (3.27) where x ∈ Rn is the state, and u ∈ Rm is the control input. Suppose that it is controllable. Then, one can find a K such that A − B K is Hurwitz and design a control law u = −K (x + w) with w representing the measurement error caused by data sampling. For any initial state x(0) and any piece-wise continuous and bounded w, the solution of the closed-loop system is x(t) = e

(A−B K )t

x(0) −

t

e(A−B K )(t−τ ) B K w(τ )dτ .

(3.36)

0

for t ≥ 0. It can be verified that x(t) property  satisfies  (3.33) with β(s, t) = (1 + ∞ 1/c)|e(A−B K )t |s and γ(s) = (1 + c) 0 |e(A−B K )τ |dτ s, where c can be chosen as any positive constant. Clearly, γ is Lipschitz on compact sets.

3.4 Event/Self-triggered Control in the Presence of External Disturbances Theorems 3.1 and 3.2 do not take into account the influence of external disturbances. To study the influence of the disturbances, we consider the following system x(t) ˙ = f (x(t), u(t), d(t)),

(3.37)

where d ∈ Rn d represents the external disturbances, and the other variables are defined as for (3.1). It is assumed that d is piece-wise continuous and bounded. The control law is still in the form of (3.2). With w defined in (3.3) as the sampling error, the control law (3.2) can be rewritten as (3.4). By substituting (3.4) into (3.37), we have x(t) ˙ = f (x(t), κ(x(t) + w(t)), d(t)) =: g(x(t), w(t), d(t)).

(3.38)

In accordance with Assumption 3.1 for the disturbance-free case, we make the following assumption on the system (3.38). Assumption 3.3 The system (3.38) is ISS with w and d as the inputs, and admits a continuously differentiable ISS-Lyapunov function V : Rn → R+ satisfying the following conditions: 1. there exist α, α ∈ K∞ such that α(|x|) ≤ V (x) ≤ α(|x|), ∀x, 2. there exist γ, γ d ∈ K and a continuous, positive definite α such that

(3.39)

54

3 A Small-Gain Paradigm for Event-Triggered Control

V (x) ≥ max{γ(|w|), γ d (|d|)} ⇒∇V (x)g(x, w, d) ≤ −α(V (x)), ∀x, w, d.

(3.40)

Under Assumption 3.3, if the event trigger is still capable of guaranteeing (3.13) with ρ ∈ K such that α−1 ◦ γ ◦ ρ < Id, then by using the robust stability property of ISS, we can prove that V (x) ≥ γ d (|d|) ⇒ ∇V (x)g(x, w, d) ≤ −α(V (x)).

(3.41)

According to (3.13), if x converges to the origin, then the upper bound of |w(t)| = |x(tk ) − x(t)| converges to zero. However, in the presence of the external disturbance d, the system dynamics f (x(t), κ(x(t) + w(t)), d(t)) may not converge to zero as x converges to the origin. This means that the inter-sampling interval tk+1 − tk could be arbitrarily small.

3.4.1 Event Triggers with Positive Offsets An intuitive solution to dealing with the external disturbance is to introduce a positive offset to the threshold signal of the event trigger. We modify the event trigger (3.11) as tk+1 = inf{t > tk : |x(t) − x(tk )| ≥ max{ρ(|x(t)|), }},

(3.42)

where ρ is a class K∞ function satisfying α−1 ◦ γ ◦ ρ < Id and constant > 0. In this case, corresponding to (3.13), we have |x(t) − x(tk )| < max{ρ(|x(t)|), }

(3.43)

for t ∈ [tk , tk+1 ), k ∈ Z+ . Accordingly, property (3.41) should be modified as V (x) ≥ max{γ( ), γ d (|d|)} ⇒ ∇V (x)g(x, w, d) ≤ −α(V (x)).

(3.44)

It should be noted that, with > 0, tk+1 − tk > 0 is guaranteed for all k ∈ S and the function ρ−1 is no longer required to be Lipschitz on compact sets. This result is given by Theorem 3.3 without proof. Theorem 3.3 Consider the event-triggered controlled system (3.38) with g being locally Lipschitz and satisfying g(0, 0, 0) = 0, and w defined in (3.3), and the event trigger (3.42). Suppose that Assumption 3.3 is satisfied. Then, for any specific initial state x(0), the system state x(t) satisfies (3.44) for all t ≥ 0, and the inter-sampling intervals are lower bounded by a positive constant. For such an event-triggered control system, even if d ≡ 0, only practical convergence can be guaranteed. Indeed, x can only be guaranteed to converge to within a

3.4 Event/Self-triggered Control in the Presence of External Disturbances

55

neighborhood of the origin defined by |x| ≤ α−1 ◦ γ( ). In the next subsection, we employ a self trigger to address this issue, under the assumption of an a priori known upper bound of d∞ .

3.4.2 Self-triggered Control We show that if an upper bound of d∞ is known a priori, then we can design a self trigger such that x is steered to within a neighborhood of the origin with its size depending solely on d∞ . Moreover, if d(t) converges to zero, then x(t) asymptotically converges to the origin as t → ∞. Assumption 3.4 There is a known constant B d ≥ 0 such that d∞ ≤ B d .

(3.45)

Assume that g in (3.38) is locally Lipschitz and g(0, 0, 0) = 0. Then, with Lemma  −1 being Lipschitz on compact sets, B.4, for any specific χ, χd ∈ K∞ with χ−1 , χd one can find a continuous, positive and nondecreasing L g such that   |g(x, w, d)| ≤ L g (max {|x|, |w|, |d|}) max χ(|x|), |w|, χd (|d|)

(3.46)

for all x, w, d.  −1 being locally Lipchitz, the self trigger By choosing χ, χd ∈ K∞ with χ−1 , χd is designed as tk+1 = tk +

Lg



1  , max χ(|x(t ¯ ¯ d (B d ) k )|), χ 

(3.47)

where χ(s) ¯ = max{χ(s), s} and χ¯ d (s) = max{χd (s), s} for s ∈ R+ . Theorem 3.4 provides the main result of this subsection. Theorem 3.4 Consider the system (3.38) with locally Lipschitz g satisfying g(0, 0, 0) = 0 and w defined in (3.3). If Assumption 3.3 is satisfied with α−1 ◦ γ being Lipschitz on compact sets, then one can find a ρ such that • ρ ∈ K∞ satisfies α−1 ◦ γ ◦ ρ < Id,

(3.48)

and • ρ−1 is Lipschitz on compact sets. Moreover, under Assumption 3.4, by choosing χ = ρ ◦ (Id + ρ)−1 and χd ∈ K∞ with  d −1 χ being Lipschitz on compact sets for the self trigger (3.47), for any specific initial state x(0), the system state x satisfies

56

3 A Small-Gain Paradigm for Event-Triggered Control

˘ |x(t)| ≤ max{β(|x(0)|, t), γ˘ d (d∞ )}

(3.49)

for all t ≥ 0, with β˘ ∈ KL and γ˘ d ∈ K, and the inter-sampling intervals are lower bounded by a positive constant. Proof Note that χ = ρ ◦ (Id + ρ)−1 implies χ−1 = Id + ρ−1 . If ρ−1 is Lipschitz on  −1 is chosen compact sets, then χ−1 is Lipschitz on compact sets. Also note that χd to be Lipschitz on compact sets. For the locally Lipschitz g satisfying g(0, 0, 0) = 0, by using Lemma B.4, one can find a continuous, positive, and nondecreasing L g such that (3.46) holds. We first prove that the self trigger achieves that |x(t) − x(tk )| ≤ max{χ(|x(tk )|), χd (d∞ )}

(3.50)

for t ∈ [tk , tk+1 ). By taking the integration of both the sides of (3.38), one has

t

x(t) − x(tk ) =

g(x(tk ), w(τ ), d(τ ))dτ ,

(3.51)

|g(x(tk ), w(τ ), d(τ ))|dτ .

(3.52)

tk

and thus,

t

|x(t) − x(tk )| ≤ tk

Denote   (x(tk ), d∞ ) = x : |x − x(tk )| ≤ max{χ(|x(tk )|), χd (d∞ )}

(3.53)

Then, the minimum time needed for x(t) starting at x(tk ) to go outside the region (x(tk ), d∞ ) can be estimated by max{χ(|x(tk )|), χd (d∞ )} C(x(tk ), d∞ ) max{χ(|x(tk )|), χd (d∞ )}   ≥ ¯ L g max{χ(|x(t ¯ d (d∞ )} max{χ(|x(tk )|), χd (d∞ )} k )|), χ 1   = ¯ L g max{χ(|x(t ¯ d (d∞ )} k )|), χ 1  , ≥ ¯ L g max{χ(|x(t ¯ d (B d )} k )|), χ where χ(s) ¯ = max{χ(s), s} and χ¯ d (s) = max{χd (s), s} for s ∈ R+ , and

(3.54)

3.4 Event/Self-triggered Control in the Presence of External Disturbances

57

 C(x(tk ), d∞ ) = max |g(x(tk ), w, d)| : |w| ≤ max{χ(|x(tk )|), χd (d∞ )},  |d| ≤ d∞ . (3.55) Thus, the proposed self trigger (3.47) guarantees (3.50). With Lemma B.3, (3.50) implies |w(t)| = |x(t) − x(tk )| ≤ max{ρ(|x(t)|), χd (d∞ )}

(3.56)

for t ∈ [tk , tk+1 ). Recall that α−1 ◦ γ ◦ ρ < Id. Using the robust stability property of ISS and employing a similar induction procedure as for the proof of Theorem 3.1, one can prove that (3.49) holds for all t ≥ 0.  With the asymptotic gain property of ISS, if d(t) converges to zero, then x(t) asymptotically converges to the origin as t → ∞.

3.4.3 Trajectory-Based Formulation Corresponding to Assumption 3.2 in Sect. 3.3, we make the following assumption on the system (3.38). Assumption 3.5 The system (3.38) is ISS with w and d as the inputs, that is, there exist β ∈ KL and γ, γ d ∈ K such that for any initial state x(0) and any piece-wise continuous, bounded w and d, it holds that   |x(t)| ≤ max β(|x(0)|, t), γ(w∞ ), γ d (d∞ )

(3.57)

for all t ≥ 0. Under Assumption 3.5, if the event trigger is still capable of guaranteeing (3.43) with ρ ∈ K such that ρ ◦ γ < Id, then by using the robust stability property of ISS, we can prove that  ˘ |x(t)| ≤ max β(|x(0)|, t), γ( ), ˘ γ˘ d (d∞ )

(3.58)

with β˘ ∈ KL and γ, ˘ γ˘ d ∈ K. This result is given by Theorem 3.5 without proof. Theorem 3.5 Consider the event-triggered controlled system (3.38) with locally Lipschitz g satisfying g(0, 0, 0) = 0 and w defined in (3.3). If Assumption 3.5 is satisfied, with the sampling time instants triggered by (3.42) and ρ ∈ K∞ satisfying ρ ◦ γ < Id, for any specific initial state x(0), the system state x(t) satisfies (3.58) for all t ≥ 0, and the inter-sampling intervals are lower bounded by a positive constant.

58

3 A Small-Gain Paradigm for Event-Triggered Control

The self trigger (3.47) is still valid under Assumption 3.4. This result is given by Theorem 3.6 without proof. Theorem 3.6 Consider the event-triggered controlled system (3.38) with locally Lipschitz g and w defined in (3.3). If Assumption 3.5 holds with a γ being Lipschitz on compact sets, then one can find a ρ ∈ K∞ such that • ρ satisfies ρ ◦ γ < Id,

(3.59)

and • ρ−1 is Lipschitz on compact sets. Moreover, under Assumption 3.4, by choosing χ = ρ ◦ (Id + ρ)−1 and χd ∈ K∞ with  d −1 χ being Lipschitz on compact sets for the self trigger (3.47), for any specific initial state x(0), the system state x satisfies ˘ |x(t)| ≤ max{β(|x(0)|, t), γ(d ˘ ∞ )}

(3.60)

for all t ≥ 0, with β˘ ∈ KL and γ˘ ∈ K, and the inter-sampling intervals are lower bounded by a positive constant.

3.5 Input-to-State Stabilization of Discrete-Time Nonlinear Systems The notion of gains is generic for not only continuous-time systems but also the systems involving discrete-time dynamics. This section extends the small-gain approach to event-triggered control of a class of nonlinear discrete-time systems in the presence of external disturbances. In particular, we focus on the practically interesting situation where the controller update at step k is determined by the feedback information x(k − 1) which is measured at step k − 1. The presence of external disturbance makes it hard to predict the magnitude of x(k) based on x(k − 1). In this section, refined tools of ISS and the nonlinear small-gain theorem are developed to estimate the influence of external disturbances before an ISS-induced design is proposed to solve the problem.

3.5.1 Problem Formulation Consider a class of discrete-time nonlinear systems subject to external disturbances: x(k + 1) − x(k) = f (x(k), u(k), d(k)), k ∈ Z+

(3.61)

3.5 Input-to-State Stabilization of Discrete-Time Nonlinear Systems

59

where x ∈ Rn is the state, u ∈ Rm is the control input, d ∈ Rn d represents external disturbances, and f : Rn × Rm × Rn d → Rn is a continuous function representing the rate of change of state x. We consider the state-feedback control law in the form of u(k) = κ(x(ki )), ki ≤ k < ki+1 , i ∈ S

(3.62)

where κ : Rn → Rm is a continuous function, and {ki }i∈S is the sequence of steps when the controller is updated with S ⊆ Z+ being the set of the indices of all the sampling steps and k0 = 0. In practice, some response time is often required for data transmission after triggering an event. In this section, we consider the case in which whether the control law is updated at step k depends on x(k − 1). In particular, the event trigger proposed in this section is defined in the form of ki+1 = min {k ≥ ki : ϕ(x(k), x(ki )) > 0} + 1

(3.63)

where ϕ : Rn × Rn → R is the triggering function, to be designed later. Define w(k) = x(ki ) − x(k), ki ≤ k < ki+1 , i ∈ S

(3.64)

as the event-triggered sampling error. By substituting (3.62) and (3.64) into (3.61), we obtain the closed-loop system x(k + 1) − x(k) = f (x(k), κ(x(k) + w(k)), d(k)) := f˜(x(k), w(k), d(k)),

(3.65)

for which, the sampling error w and the external disturbance d are considered as the inputs. Without loss of generality, assume f˜(0, 0, 0) = 0. To focus on the design of the event trigger, we assume that the system (3.61) already has a stabilizing control law (3.62), and that the closed-loop system (3.65) is ISS with respect to w and d. It should be mentioned that the design of a stabilizing control law is closely related to the literature of robust control of discrete-time nonlinear systems subject to external disturbances, see, e.g., [102, 103, 147, 202]. In this section, the ISS property of discrete-time systems is described by ISS-Lyapunov functions. See [123] for the original development of ISS-Lyapunov functions for discrete-time systems. Assumption 3.6 The system (3.65) is ISS with w and d as the inputs, and admits an ISS-Lyapunov function V : Rn → R+ satisfying the following conditions: 1. there exist α, α ∈ K∞ such that α(|x|) ≤ V (x) ≤ α(|x|), ∀x;

(3.66)

60

3 A Small-Gain Paradigm for Event-Triggered Control

d(k)

x(k + 1) = f¯(x(k), w(k), d(k))

x(k)

w(k) x(ki ) −

+

S

Fig. 3.6 The block diagram of the closed-loop system (3.65), where S represents the event trigger defined by (3.63)

2. there exist functions α ∈ K∞ and γ, γ d ∈ K, such that V ( f¯(x, w, d)) − V (x) ≤ −α(|x|) + max{γ(|w|), γ d (|d|)}, ∀x, w, d (3.67) where f¯(x, w, d) = f˜(x, w, d) + x. Without loss of generality, the following assumption is made on f˜. Assumption 3.7 There exist functions ψ xf˜ , ψ wf˜ , ψ df˜ ∈ K∞ such that | f˜(x, w, d)| ≤ ψ xf˜ (|x|) + ψ wf˜ (|w|) + ψ df˜ (|d|)

(3.68)

for all x ∈ Rn , w ∈ Rn and d ∈ Rn d . Assumption 3.7 can be globally satisfied as long as f˜ is continuous and f˜(0, 0, 0) = 0. Specifically, the functions ψ xf˜ , ψ wf˜ , ψ df˜ can be chosen as follows: ψ xf˜ (s) = s + max{| f˜(x, w, d)| : |w| ≤ |x|, |d| ≤ |x|, |x| ≤ s}, ψ w (s) = s + max{| f˜(x, w, d)| : f˜

|x| ≤ |w|, |d| ≤ |w|, |w| ≤ s} and ψ df˜ (s) = s + max{| f˜(x, w, d)| : |x| ≤ |d|, |w| ≤ |d|, |d| ≤ s}. According to the definition of w(k) in (3.64), the closed-loop event-triggered system can be represented in the feedback form shown in Fig. 3.6.

3.5.2 Event-Triggered Robust Stabilization Following the standard line of robustness analysis, this subsection studies the eventtriggered robust stabilization problem for a class of discrete-time nonlinear systems. Both the cases of state-dependent disturbance and state-independent disturbance are considered. An ISS-induced design is proposed for the triggering function ϕ in (3.63).

3.5 Input-to-State Stabilization of Discrete-Time Nonlinear Systems

x(k + 1) = f¯(x(k), w(k), d(k))

d(k)

61

x(k)

|w(k)| ≤ ρw (|x(k)|)

w(k)

|d(k)| ≤ ρd (|x(k)|) Fig. 3.7 The block diagram of the closed-loop event-triggered system (3.65) with state-dependent disturbance

State-Dependent Disturbance We suppose that the external disturbance d is upper bounded by a function of the system state x. In particular, there exists a ρd ∈ K∞ such that |d(k)| ≤ ρd (|x(k)|)

(3.69)

for all k ∈ Z+ . Motivated by the results in Sects. 3.2 and 3.3, we propose an event trigger such that the sampling error w satisfies |w(k)| ≤ ρw (|x(k)|)

(3.70)

for all k ∈ Z+ , where ρw is an appropriately chosen class K∞ function. Then, the closed-loop system (3.65) with interconnections satisfying (3.69) and (3.70) can be described as a network of three blocks as shown in Fig. 3.7. Considering the network structure and the gain interconnection, we employ the cyclic-small-gain theorem to analyze the robust stability of the closed-loop system. Loosely speaking, according to the discrete-time cyclic-small-gain theorem [125, 162], the closed-loop system is stable at the origin if the following cyclic-small-gain conditions are satisfied: α−1 ◦ (Id − )−1 ◦ γ d ◦ ρd < Id, α

−1

◦ (Id − )

−1

w

◦ γ ◦ ρ < Id,

(3.71) (3.72)

where α is defined in Assumption 3.6, and is a continuous and positive definite function satisfying (Id − ) ∈ K∞ . Since w is determined by the event trigger, we find an appropriate event trigger to satisfy both conditions (3.70) and (3.72). Specifically, the function ϕ of the event trigger (3.63) is defined as follows:

62

3 A Small-Gain Paradigm for Event-Triggered Control

ϕ(r1 , r2 ) = −χx (|r1 |) + χw (|r1 − r2 |)

(3.73)

for all r1 , r2 ∈ Rn , with functions χx and χw to be chosen later. From (3.73), it holds that ϕ(x(k), x(ki )) = −χx (|x(k)|) + χw (|x(k) − x(ki )|) = −χx (|x(k)|) + χw (|w(k)|),

(3.74)

which, together with (3.63), determines the sequence of sampling steps ki . Compared with the triggering functions used in Sects. 3.2 and 3.3, the triggering function ϕ is in a more general form. The main result on event-triggered stabilization for the case of state-dependent disturbance is given by Theorem 3.7. Theorem 3.7 Consider the system (3.65). Under Assumptions 3.6 and 3.7, if the external disturbance d satisfies (3.69) and (3.71), then global asymptotic stabilization can be achieved with the event trigger (3.63) and (3.73), where χx = ρw ◦ (Id + ρw )−1 − (ψ xf˜ + ψ df˜ ◦ ρd ) χw =

ψ wf˜

+ Id

(3.75) (3.76)

with ρw ∈ K∞ satisfying (3.72). Proof The event trigger (3.63) uses x(k) and w(k) to determine whether x(k + 1) should be sampled for controller update. Note that |w(0)| = 0 ≤ ρw (|x(0)|). With ϕ defined in (3.73), we prove that for any given k ∈ Z+ and any x(k) and w(k), |w(k + 1)| ≤ ρw (|x(k + 1)|).

(3.77)

We consider two cases: (a) ϕ(x(k), x(ki )) > 0; (b) ϕ(x(k), x(ki )) ≤ 0. Case (a): By directly using (3.63), we have ki+1 = k + 1, and thus |w(k + 1)| = 0,

(3.78)

which guarantees (3.77). Case (b): In this case, x(k + 1) is not sampled, which means ki+1 > k + 1. Thus, we have |w(k + 1)| = |x(k + 1) − x(ki )|.

(3.79)

Recall that x(k + 1) is determined by x(k), x(ki ) and d(k), and also d(k) satisfies (3.69). We estimate an upper bound of |w(k + 1)| by using x(k) and x(ki ). From (3.79), we have

3.5 Input-to-State Stabilization of Discrete-Time Nonlinear Systems

63

|w(k + 1)| = |x(k + 1) − x(k) + x(k) − x(ki )| ≤ |x(k + 1) − x(k)| + |x(k) − x(ki )|.

(3.80)

By substituting (3.68) and (3.69) into the right-hand side of (3.80), we get an upper bound of w(k + 1) as |w(k + 1)| ≤ |x(k + 1) − x(k)| + |x(k) − x(ki )| ≤ ψ xf˜ (|x(k)|) + ψ wf˜ (|x(k) − x(ki )|) + ψ df˜ (|d(k)|) + |x(k) − x(ki )| ≤ ψ xf˜ (|x(k)|) + ψ df˜ ◦ ρd (|x(k)|) + (ψ wf˜ + Id)(|x(k) − x(ki )|) ≤ (ψ xf˜ + ψ df˜ ◦ ρd )(|x(k)|) + (ψ wf˜ + Id)(|x(ki ) − x(k)|).

(3.81)

Using ϕ(x(k), x(ki )) ≤ 0, we have (ψ xf˜ + ψ df˜ ◦ ρd )(|x(k)|) − ρw ◦ (Id + ρw )−1 (|x(k)|) + (ψ wf˜ + Id)(|x(ki ) − x(k)|) ≤ 0, and thus (ψ xf˜ + ψ df˜ ◦ ρd )(|x(k)|) + (ψ wf˜ + Id)(|x(ki ) − x(k)|) ≤ ρw ◦ (Id + ρw )−1 (|x(k)|).

(3.82)

From the first and the last inequalities of (3.81), we also have |x(k + 1) − x(k)| ≤ ρw ◦ (Id + ρw )−1 (|x(k)|).

(3.83)

By using Lemma B.3, it follows that |x(k + 1) − x(k)| ≤ ρw ◦ (Id + ρw )−1 (|x(k)|) ≤ ρw (|x(k + 1)|).

(3.84)

Hence, (3.77) is proved for Case (b) by combining (3.81), (3.82), and (3.84). From the discussions above, we have |w(k)| ≤ ρw (|x(k)|)

(3.85)

for all k ∈ Z+ . Then, with the satisfaction of (3.71) and (3.72), property (3.67) implies V ( f¯(x, w, d)) − V (x) ≤ −α(|x|) + max{γ ◦ ρw (|x|), γ d ◦ ρd (|x|)} ≤ −α(|x|) + max{(Id − ) ◦ α(|x|), (Id − ) ◦ α(|x|)} ≤ − ◦ α(|x|) =: −α(|x|) ˆ for all x, w, d. This ends the proof of Theorem 3.7.

(3.86) 

64

3 A Small-Gain Paradigm for Event-Triggered Control

It can be observed that |w(k + 1)| ≤ ρw (|x(k + 1)|) cannot be directly guaranteed w by |w(k)| ≤ ρw 0 (|x(k)|) for some ρ0 ∈ K∞ , since the amplitude of x(k + 1) depends on the uncertain f˜(x(k), w(k), d(k)). The problem is solved by introducing a new triggering function ϕ, for which χx is not necessarily positive definite, and χw is defined to be of class K∞ . In the case of χx (|x(k)|) < 0, it can be concluded that ki+1 = k + 1.

(3.87)

If, moreover, χx is negative definite, then (3.87) holds for all k ∈ Z+ , i ∈ Z+ . In this case, each x(k) for k ∈ Z+ should be sampled. If χx is positive definite, then for each i ∈ S, we have ϕ(x(ki ), x(ki )) = −χx (|x(ki )|) ≤ 0, and min{k ≥ ki : ϕ(x(k), x(ki )) > 0} ≥ ki + 1, which implies ki+1 ≥ ki + 2.

(3.88)

As a result, if x(k) is sampled, then x(k + 1) will not be sampled. We employ a simple example to show that the function χx defined in the triggering function (3.73) may not be positive definite. Example 3.4 Consider the discrete-time nonlinear system x1 (k + 1) = x1 (k) + 0.05x2 (k) + u(k) x2 (k + 1) = 0.9x2 (k) +

0.001x12 (k)

(3.89)

+ 0.01d(k)

(3.90)

where x = [x1 , x2 ]T ∈ R is the state, u ∈ R is the control input, and d ∈ R represents the external disturbance. We employ the feedback control law u(k) = −0.05(x1 (ki ) + x2 (ki ))

(3.91)

| f˜(x, w, d)| ≤ 0.12|x| + 0.001|x|2 + 0.071|w| + 0.01|d|

(3.92)

for ki ≤ k < ki+1 , i ∈ S. Then, it holds that

for all x, w, d. To verify the satisfaction of Assumption 3.6, define 1

V (x) = max{V1 (x1 ), V22 (x2 )}

(3.93)

with V1 (x1 ) = |x1 | and V2 (x2 ) = |x2 |. Then, we have V (x(k + 1)) − V (x(k)) ≤ − 0.04V (x(k)) 1

+ max{0.078|w(k)|, 1.1|d(k)| 2 }

(3.94)

3.5 Input-to-State Stabilization of Discrete-Time Nonlinear Systems

65

for all x, w, d, k. For the satisfaction of the conditions (3.71) and (3.72), we choose functions ρw and ρd as ρw (s) = min{0.25s, 0.25s 2 } 1

ρd (s) = min{0.00022s 2 , 0.00031s} for all s ∈ R+ . By using (3.75), we have 1

χx (s) = min{0.2s, 0.032(64s + 1) 2 − 0.032} − 0.11s − 0.001s 2 − 0.01 min{0.00022s 2 , 0.00031s}

(3.95)

for all s ∈ R+ . It can be checked that the function χx defined in (3.95) satisfies χx (s) ≥ 0 and χx (s) < 0 for s ∈ [0, 4.24] and s ∈ (4.24, ∞), respectively. Suppose that the system (3.61) is an Euler approximation of a continuous-time system x(t) ˙ = g(x(t), u(t), d(t)).

(3.96)

f (x(k), u(k), d(k)) = T g(x(k), u(k), d(k))

(3.97)

Then, we have

where constant T > 0 is the sampling period for the continuous time system (3.96). Clearly, by choosing T small enough, one can make ψ xf˜ , ψ wf˜ , ψ df˜ small enough such that χx is positive definite. Intuitively, this means that if T is small, then some x(k) is not necessarily sampled; otherwise, each x(k) should be sampled for controller update. There is a trade-off between the sampling period for continuous-time system and the number of data sampling events for the discrete-time system. State-Independent Disturbance Now, we show that the event trigger proposed above is also effective for input-tostate stabilization of the closed-loop event-triggered system. The analysis in the case of state-dependent disturbance is not valid for this case. Specifically, if |d(k)| > ρd (|x(k)|) for some k, then the event trigger (3.63) with function ϕ defined in (3.73) cannot directly guarantee (3.77), and thus the stability of the closed-loop eventtriggered system cannot be directly proved. This also leads to one major difference between the event-triggered control system and conventional discrete-time nonlinear systems. We show that the closed-loop event-triggered system is ISS, though the resulting ISS gain does not have the standard relation with the ρd satisfying (3.71) or the γ d defined in Assumption 3.6. For convenience of discussions, denote d(−1) = 0.

66

3 A Small-Gain Paradigm for Event-Triggered Control

Theorem 3.8 Consider the system (3.65). Under Assumptions 3.6 and 3.7, if the event trigger (3.63) with function ϕ defined in (3.73) satisfies (3.75) and (3.76), then the closed-loop event-triggered system is ISS with V satisfying V (x(k + 1)) − V (x(k)) ≤ − α(|x(k)|) ˘ + max{γ˘ d (|d(k − 1)|), γ d (|d(k)|)}

(3.98)

for all k ∈ Z+ , where α(s) ˘ = min{α(s), ˆ α(s)}

(3.99)

for all s ∈ R+ with αˆ being defined as in (3.86), and γ˘ d = γ ◦ λd with λd = (ψ xf˜ + ρw ◦ (ψ wf˜ + Id)) ◦ ρd

−1

(3.100)

+ ψ df˜ .

Proof For any specific k ∈ Z+ , we consider the cases of |w(k)| ≤ ρw (|x(k)|) and |w(k)| > ρw (|x(k)|), respectively. Case (a): |w(k)| ≤ ρw (|x(k)|). Under Assumption 3.6, condition (3.72) implies V (x(k + 1)) − V (x(k)) ≤ −α(|x(k)|) + max{(Id − ) ◦ α(|x(k)|), γ d (|d(k)|)}.

(3.101)

If, moreover, (3.69) is satisfied, then property (3.101) implies V (x(k + 1)) − V (x(k)) ≤ −α(|x(k)|) ˆ

(3.102)

where αˆ is defined as in (3.86). If, on the other hand, condition (3.69) is not satisfied, then property (3.101) leads to V (x(k + 1)) − V (x(k)) ≤ −α(|x(k)|) + γ d (|d(k)|).

(3.103)

By combining (3.102) and (3.103), we have V (x(k + 1)) − V (x(k)) ≤ −α(|x(k)|) ˆ + γ d (|d(k)|).

(3.104)

Case (b): |w(k)| > ρw (|x(k)|). In this case, we first prove |w(k − 1)| ≤ ρw (|x(k − 1)|),

(3.105)

|d(k − 1)| > ρ (|x(k − 1)|).

(3.106)

d

3.5 Input-to-State Stabilization of Discrete-Time Nonlinear Systems

67

In order to obtain a contradiction, assume that (3.105) does not hold, that is, |w(k − 1)| > ρw (|x(k − 1)|). Then, ϕ(x(k − 1), x(ki−1 )) > 0, and thus x(k) is sampled, i.e., x(ki ) = x(k). From the definition of w in (3.64), we have |w(k)| = 0 ≤ ρw (|x(k)|). This contradicts with |w(k)| > ρw (|x(k)|), and thus (3.105) is proved. With (3.105) proved, if |d(k − 1)| ≤ ρd (|x(k − 1)|), then following the proof of Theorem 3.7, we have |w(k)| ≤ ρw (|x(k)|). This contradicts with |w(k)| > ρw (|x(k)|), and thus (3.106) is proved. Then, the following relation between |w(k)| and |d(k − 1)| can be observed: |w(k)| = |x(k) − x(ki )| ≤ |x(k) − x(k − 1)| + |w(k − 1)| ≤ ψ xf˜ (|x(k − 1)|) + ψ wf˜ (|w(k − 1)|) + ψ df˜ (|d(k − 1)|) + |w(k − 1)| ≤ (ψ xf˜ + ρw ◦ (ψ wf˜ + Id))(|x(k − 1)|) + ψ df˜ (|d(k − 1)|) ≤ λd (|d(k − 1)|)

(3.107)

where λd (s) = (ψ xf˜ + ρw ◦ (ψ wf˜ + Id)) ◦ ρd From (3.67), we have

−1

+ ψ df˜ (s) for s ∈ R+ .

V (x(k + 1))−V (x(k)) ≤ −α(|x(k)|) + max{γ ◦ λd (|d(k − 1)|), γ d (|d(k)|)} = −α(|x(k)|) + max{γ˘ d (|d(k − 1)|), γ d (|d(k)|)}

(3.108)

where γ˘ d (s) = γ ◦ λd (s) for s ∈ R+ . Then, (3.104) and (3.108) together imply V (x(k + 1)) − V (x(k)) ≤ − α(|x(k)|) ˘ + max{γ˘ d (|d(k − 1)|), γ d (|d(k)|)}

(3.109)

where α(s) ˘ = min{α(s), ˆ α(s)} for s ∈ R+ , which means ISS of the closed-loop event-triggered system. This ends the proof of Theorem 3.8.  It can be observed that the right-hand side of (3.107) depends on d(k − 1). Intuitively, this is caused by the “+1” term in (3.63). More specifically, for the case of |w(k)| > ρw (|x(k)|) in the proof of Theorem 3.8, some upper bound of |w(k)| is needed to guarantee the ISS-like relation between |x| and |d| as given by (3.98). This is achieved by using the event trigger (3.63). Theorems 3.7 and 3.8 are based on Assumptions 3.6 and 3.7. Practically, the properties of the system dynamics given by Assumptions 3.6 and 3.7 are known a priori, and thus the conditions for Theorems 3.7 and 3.8 can be checked before the implementation of the control strategy. No real-time condition checking is needed. We employ a simple example to show the relation between gain margin and ISS gain.

68

3 A Small-Gain Paradigm for Event-Triggered Control

Example 3.5 Consider the discrete-time scalar system x(k + 1) = x(k) + 0.05|x(k)| + 0.01d(k) + 0.1u(k) u(k) = −x(ki ), ki ≤ k < ki+1 , i ∈ S

(3.110)

where x is the state, u is the control input, and d is the external disturbance. With w defined in (3.64), we obtain the closed-loop system x(k + 1) = 0.9x(k) + 0.05|x(k)| + 0.01d(k) − 0.1w(k).

(3.111)

By transforming the system (3.111) into the form of (3.65), it can be proved that | f˜(x, w, d)| ≤ 0.15|x| + 0.1|w| + 0.01|d|

(3.112)

holds for all x, w, d. We define an ISS-Lyapunov function V (x) = |x| for the system (3.111). Then, it holds that V ( f¯(x, w, d)) − V (x) ≤ −0.05V (x) + max{0.2|w|, 0.02|d|}

(3.113)

for all x, w, d. In this example, we consider d(k) = 0.01 sin(0.03k) + 0.01 cos(0.09k) + 0.03.

(3.114)

By choosing = 0.1, ρw = 0.22, ρd = 0.2 satisfying conditions (3.71) and (3.72), we calculate λd = 1.97, and thus γ˘ d = 0.394. Then, the triggering function is chosen as follows ϕ(x(k), x(ki )) = −0.028|x(k)| + 1.1|x(k) − x(ki )|.

(3.115)

With the event trigger (3.63), the simulation result for the gain of the system (3.111) with initial states x(0) = −1 is shown in Fig. 3.8. Clearly, in the case of state-independent disturbance, the γ d in (3.67) is not the correct ISS gain for the event-triggered closed-loop system. The following example gives a numerical simulation to verify the effectiveness of the obtained results for event-triggered stabilization of discrete-time nonlinear systems. Example 3.6 Consider the discrete-time nonlinear system 1

x1 (k + 1) = x1 (k) + 0.01|x2 | 2 (k) + u(k) + 0.01d1 (k) x2 (k + 1) = 0.9x2 (k) +

0.001x12 (k)

+ 0.01d2 (k)

(3.116) (3.117)

3.5 Input-to-State Stabilization of Discrete-Time Nonlinear Systems

69

1

|x| d (k) 1 d (k) 2

|x| and | |

0.8 0.6 0.4 0.2 0

0

5

10

15

20

25 steps

30

35

40

45

50

Fig. 3.8 The trajectories of the system state |x(k)| and related signals γ1d (k) = max{α˜ ◦ γ˘ d (|d(k − 1)|), α˜ ◦ γ d (|d(k)|)} and γ2d (k) = α˜ ◦ γ d (|d(k)|) with α(s) ˜ = α−1 ◦ (α ◦ α−1 )−1 (s) for s ∈ R+ in Example 3.5

where x = [x1 , x2 ]T is the state, u is the control input, and d = [d1 , d2 ]T represents the external disturbance. The control law is chosen as u(k) = −0.05x1 (ki )

(3.118)

for ki ≤ k < ki+1 , i ∈ S. Then, it can be proved that 1 | f˜(x, w, d)| ≤ 0.12|x| + 0.01(|x| 2 + 0.1|x|2 ) + 0.05|w| + 0.015|d|

(3.119)

holds for all x, w, d. To verify the satisfaction of Assumption 3.6, define 1

V (x) = max{V1 (x1 ), V22 (x2 )}

(3.120)

with V1 (x1 ) = |x1 | and V2 (x2 ) = |x2 |. Then, it holds that V (x(k + 1)) − V (x(k)) ≤ − 0.04V (x(k)) 1

+ max{0.06|w(k)|2 , 0.6|d(k)| 2 }

(3.121)

for all x, w, d, k. To satisfy conditions (3.71) and (3.72), we choose functions ρw and ρd as

3 A Small-Gain Paradigm for Event-Triggered Control 0.5 0 -0.5 -1 -1.5

x (k) and x (k ) 2 2 i

x1(k) and x1(ki)

70

x1(k) x1(ki) 0

5

10

15 steps

20

25

2

30 x (k) 2

1

x (k ) 2

i

0 0

5

10

15 steps

20

25

30

Fig. 3.9 The state trajectories with the event trigger (3.63) in Example 3.6

ρw (s) = min{0.32s, 0.32s 2 } 1

ρd (s) = min{0.0008s, 0.0007s 2 } for all s ∈ R+ . Then, we have 1

χx (s) = min{0.25s, 0.1(40s − 1) 2 } − 0.12s − 0.01s 2 1

− 0.001s 2 − 0.01 min{0.0007s 2 , 0.0008s} χw (s) = 1.05s for all s ∈ R+ . In the numerical simulation, the disturbances d1 and d2 are chosen as d1 (k) = (sin(3k) + cos(k/9) − sin(sin(7k))), d2 (k) = (cos(k/3) + sin(5k) + cos(7k) + sin(cos(k/9))). The simulation result with initial states x1 (0) = −1.5 and x2 (0) = 2 is shown in Figs. 3.9, 3.10 and 3.11.

3.6 Notes Early applications that introduce the idea of event-triggered control include [40, 78, 158, 194, 254]. Due to the increasing popularity of networked control systems, recent years have seen a renewed interest in event-triggered control of linear and nonlinear systems.

3.6 Notes

71

0.07 0.06 control input

0.05 0.04 0.03 0.02 0.01 0 -0.01

0

5

10

15 steps

20

25

30

25

30

Fig. 3.10 The control input with the event trigger (3.63) in Example 3.6 7

inter-samping intervals

6 5 4 3 2 1 0

0

5

10

15 steps

20

Fig. 3.11 The inter-sampling intervals with the event trigger (3.63) in Example 3.6

For practical implementation of event-triggered control, infinitely fast sampling should be avoided, that is, the intervals between the sampling time instants should be lower bounded by some positive constant. In the context of event-based control, due to the hybrid nature, the forward completeness analysis is not a trivial task. This problem is examined in this chapter. Significant contributions have been made to the literature of event-triggered control; see, e.g., [14, 17, 19, 86, 87, 91, 148, 245, 276] and the references therein. Specifically, in [19, 91], impulsive control methods are developed to keep the states of first-order stochastic systems inside certain thresholds. In [69, 181], prediction of the real-time system state between the sampling time instants was employed to generate the control signal, and the prediction is corrected by data-sampling when

72

3 A Small-Gain Paradigm for Event-Triggered Control

the difference between the true state and the predicted state is too large. Reference [245] considered the systems which admit controllers to guarantee the robustness with respect to the sampling errors. Then, the event trigger is designed such that the sampling error is bounded by a specific threshold (depending on the real-time system state) for convergence of the system state. In [47], the ISS small-gain theorem is applied to guarantee the stability of the overall system, and a parsimonious event trigger is developed to avoid the infinitely fast sampling. Reference [184] proposes a universal formula for event-based stabilization of general nonlinear systems affine in the control by extending Sontag’s result for continuous-time stabilization [237]. Reference [246] proposes a Lyapunov condition for tracking control of nonlinear systems. The designs have been extended to distributed networked control [47, 262], output-feedback control and decentralized control [55] and systems with quantized measurements [68], to name a few. In the process of event-triggered control, the real-time system state should be continuously monitored, which may not be easy in specific practical scenarios. An alternative approach is the self-triggered control, with which the controller computes the control signal as well as the next sampling time instant such that continuous monitoring of system state is not needed [256]. Recent results on self-triggered control can be found in [8, 10, 50, 187, 210, 227, 261, 262, 274]. When systems are subject to external disturbances, finite-gain L p stability [261, 262] can be realized. Reference [10] presents a self-triggered control technique for homogeneous systems and polynomial systems. Polynomial approximations are employed for self-triggered control of nonlinear systems in [50]. In [210], self-triggered sampling has been successfully applied to robotic networks. The reader may consult the nice tutorials [86, 148] for the recent developments of event-triggered control and self-triggered control. The event-triggered control problem for discrete-time nonlinear systems has also been studied in quite a few recent works. In [58], event-triggered condition and self-triggered formulation are proposed for discrete-time systems without external disturbances by using the notion of ISS. Reference [84] proposes sensor-controller and controller-actuator event triggers, and also model-based periodic event-triggered control strategies to achieve L2 gain property with respect to external disturbances. The event-triggered H∞ control problem for discrete-time nonlinear networked control systems with unreliable communication links is studied in [151]. In [36, 153], the event-triggered consensus problem of discrete-time multi-agent systems is investigated, and the triggering condition is designed based on the measurement error to ensure multi-agent consensus.

Chapter 4

Dynamic Event Triggers

The objective of this chapter is to extend the small-gain-based event-trigger design to nonlinear uncertain systems with partial-state or output feedback. For nonlinear systems with only partial state available to feedback, it is nontrivial to design static event triggers for asymptotic stabilization even if the system is disturbance-free. By introducing dynamics to the event triggers, this chapter proposes a new class of dynamic event triggers to address this issue. An application of the dynamic event triggers to decentralized event-triggered control is also presented. This chapter focuses on the systems without external disturbances, and a more general design which is capable of handling both dynamic uncertainties and external disturbances is introduced in Chap. 5.

4.1 Dynamic Event Triggers with Partial-State Feedback This section studies the event-triggered control problem for nonlinear systems with partial state feedback. We consider a control system that can be transformed into an interconnection of two ISS subsystems with the sampling error as the external input. It is shown that infinitely fast sampling can be avoided and asymptotic stabilization can be achieved by introducing a dynamic threshold signal to the event trigger.

4.1.1 Problem Formulation Suppose that a well-designed control system takes the following interconnected form: z˙ (t) = h(z(t), x(t), w(t)) x(t) ˙ = f (x(t), z(t), w(t)) © Huazhong University of Science and Technology Press, Wuhan and Springer Nature Singapore Pte Ltd. 2020 T. Liu et al., Robust Event-Triggered Control of Nonlinear Systems, Research on Intelligent Manufacturing, https://doi.org/10.1007/978-981-15-5013-3_4

(4.1) (4.2) 73

74

4 Dynamic Event Triggers

where [z T , x T ]T with z ∈ Rm and x ∈ Rn is the state, w ∈ Rn represents the sampling error of x, and h : Rm × Rn × Rn → Rm and f : Rn × Rm × Rn → Rn represent the system dynamics with h(0, 0, 0) = 0 and f (0, 0, 0) = 0. Here, z is supposed to be unavailable to the event trigger. For convenience of notations, define x¯ = [z T , x T ]T . The sampling error w is defined as w(t) = x(tk ) − x(t), t ∈ [tk , tk+1 ), k ∈ S.

(4.3)

It is the basic idea of event-triggered control that the sampling time instants are triggered by comparing the sampling error |x(tk ) − x(t)| with a continuous and positive threshold signal. Define μ : R+ → R+ as the threshold signal. Then, an event trigger often takes the following general form tk+1 = inf {t > tk : |x(tk ) − x(t)| ≥ μ(t)} , k ∈ S.

(4.4)

Recall the definition of w(t) in (4.3). We have |w(t)| ≤ μ(t)

(4.5)

 for all t ∈ k∈S [tk , tk+1 ). In this chapter, the event trigger design problem is solved for the system (4.1)– (4.2) by finding an appropriate threshold signal μ(t) for the event trigger (4.4) such that the following two objectives are achieved at the same time: Objective 1: z(t) and x(t) are defined for all t ≥ 0 and the closed-loop eventtriggered system with [z T , x T , μ]T as the state is globally asymptotically stable at the origin. Objective 2: For any given (z(0), x(0)) and any given μ(0) > 0, the intervals between the sampling time instants are lower bounded by a positive constant. Objective 2 aims to avoid infinitely fast sampling. Here, we do not require the existence of some constant TΔ > 0 such that for any (z(0), x(0)) and any μ(0) > 0, tk+1 − tk ≥ TΔ for all k ∈ S. We assume that both the z-subsystem and the x-subsystem are ISS. More precisely, we make the following assumption on the Lyapunov-based ISS properties of the subsystems. Assumption 4.1 Both the z-subsystem and the x-subsystem are ISS with ISSLyapunov functions Vz : Rm → R+ and Vx : Rn → R+ which are locally Lipschitz on Rm \{0} and Rn \{0}, respectively, and satisfy • there exist αz , αz ∈ K∞ and γzx , γzw ∈ K ∪ {0} such that

4.1 Dynamic Event Triggers with Partial-State Feedback

αz (|z|) ≤ Vz (z) ≤ αz (|z|),

75

(4.6)

max{γzx (Vx (x)), γzw (|w|)}

Vz (z) ≥ ⇒∇Vz (z)h(z, x, w) ≤ −αz (Vz (z)), a.e.

(4.7)

• there exist αx , αx ∈ K∞ and γxz , γxw ∈ K ∪ {0} such that αx (|x|) ≤ Vx (x) ≤ αx (|x|), Vx (x) ≥ max{γxz (Vz (z)), γxw (|w|)} ⇒∇Vx (x) f (x, z, w) ≤ −αx (Vx (x)), a.e.

(4.8) (4.9)

Example 4.1 shows how an event-triggered control system can be transformed into the form of (4.1)–(4.2) satisfying Assumption 4.1. Example 4.1 Consider system z˙ (t) = −z 3 (t)

(4.10)

x(t) ˙ = u(t) + z(t)

(4.11)

where z ∈ R and x ∈ R are the state variables, u ∈ R is the control input. We consider the case where only x is available to feedback control design, and employ the feedback control law u(t) = −x(tk ), t ∈ [tk , tk+1 ), k ∈ S.

(4.12)

By using (4.3) and (4.12), we have x(t) ˙ = −x(t) − w(t) + z(t).

(4.13)

Thus, the closed-loop system composed of (4.10) and (4.13) is in the form of (4.1)–(4.2) with h(z, x, w) = −z 3 and f (x, z, w) = −x − w + z. To verify the satisfaction of Assumption 4.1, we define Vz (z) = |z| and Vx (x) = |x|. Clearly, Vz and Vx are locally Lipschitz. It can be directly checked that Vz and Vx satisfy (4.6) and (4.8), respectively, with αz , αz , αx , αx = Id. Also, direct calculation yields: ∇Vz (z)h(z, x, w) = −|z|3 = −Vz3 (z) a.e. ∇Vx (x) f (x, z, w) ≤ −|x| + |w| + |z| = −Vx (x) + Vz (z) + |w| ≤ −Vx (x) + 2 max {Vz (z), |w|} a.e.

(4.14)

(4.15)

Then, property (4.15) implies Vx (x) ≥ 4 max {Vz (z), |w|} ⇒ ∇Vx (x) f (x, z, w) ≤ −0.5Vx (x) a.e.

(4.16)

76

4 Dynamic Event Triggers

Thus, properties (4.7) and (4.9) are satisfied with γzx (s) = 0, γzw (s) = 0, αz (s) = s 3 , γxz (s) = 4s, γxw (s) = 4s and αx (s) = 0.5s for s ∈ R+ . In [77, 230], the event-triggered control problem was studied and exponentially converging threshold signals were used in the context of distributed control. The dynamic threshold signal has been shown to be powerful in dealing with the unmeasured states. Based on this idea, we first try the threshold signal μ(t) defined by μ(t) = μ(0)e−ct

(4.17)

for all t ≥ 0, with initial state μ(0) > 0 and constant c > 0. Equivalently, μ(t) is the solution of the initial value problem μ(t) ˙ = −cμ(t)

(4.18)

for all t ≥ 0. Example 4.2 shows that an exponentially converging μ(t) may lead to infinitely fast sampling, if the closed-loop system involves nonlinear dynamics. Example 4.2 Consider the system composed of (4.10) and (4.13), which is in the form of (4.1)–(4.2) with h(z, x, w) = −z 3 and f (x, z, w) = −x − w + z. It is shown in Example 4.1 that the system satisfies Assumption 4.1. Moreover, the system is a cascade connection of the z-subsystem and the x-subsystem, and thus the nonlinear small-gain condition between γxz and γzx , i.e., γxz ◦ γzx < Id

(4.19)

holds trivially. As a consequence, the cascade system is ISS with w as the input. See [118, 122] for the details. However, we show that for some initial states, there does not exist an exponentially converging threshold signal in the form of (4.17) to avoid infinitely fast sampling. It can be proved that, for any given z(0), x(0), μ(0) and positive constant c, there exist positive constants m 1 , m 2 , m 3 , c∗ such that ∗

| f (x(t), z(t), w(t))| ≥ m 1 e−t + m 2 e−c t − m 3 e−ct for all t ∈

for all t ∈



k∈S [tk , tk+1 ).



(4.20)

Moreover, there exist z(0), x(0), μ(0) such that

m 1 ≥ 0, m 2 > 0, m 2 ≥ 2m 3 , 2c∗ ≤ c,

(4.21)

x(t) > z(t) > 0, f (x(t), z(t), w(t)) < 0

(4.22) (4.23)

k∈S [tk , tk+1 ).

See Appendix D.3 for a detailed proof.

4.1 Dynamic Event Triggers with Partial-State Feedback

77

Properties (4.20) and (4.21) together imply | f (x(t), z(t), w(t))| ≥

m 2 −c∗ t e 2

(4.24)

 for all t ∈ k∈S [tk , tk+1 ). Given tk , we estimate the upper bound of Δtk = tk+1 − tk . With property (4.23), by using the triggering condition (4.4), we have μ(tk+1 ) = |x(tk ) − x(tk+1 )|   tk+1   f (x(τ ), z(τ ), w(τ ))dτ  =  tk  tk+1 = | f (x(τ ), z(τ ), w(τ ))|dτ .

(4.25)

tk

Then, by using (4.17) and (4.24), we have μ(0)e−c(tk +Δtk ) ≥

m 2 −c∗ (tk +Δtk ) e Δtk , 2

(4.26)

which implies Δtk ec

∗

(tk +Δtk )

≤

2μ(0) m2

(4.27)

by using 2c∗ ≤ c. Now, we show S = Z+ . Suppose S = {0, 1, . . . , k ∗ } with k ∗ being a positive integer. In this case, by using the standard small-gain theorem in [122], we can guarantee the boundedness of z(t) and x(t) for all t ∈ [0, Tmax ). Due to continuation, z(t) and x(t) are defined for all t ∈ [0, ∞), i.e., Tmax = ∞. Recall that x(tk ∗ ) > 0 (see (4.22)). Since the z-subsystem is globally asymptotically stable at the origin, one can find a finite time t ∗ ≥ tk ∗ such that z(t) ≤

1 x(tk ∗ ) 2

(4.28)

for all t ∈ [t ∗ , ∞), and thus 1 x(t) ˙ = −x(tk ∗ ) + z(t) ≤ − x(tk ∗ ) 2

(4.29)

for all t ∈ [t ∗ , ∞). This contradicts with the boundedness of x(t). Thus, S = Z+ . Suppose that infinitely fast sampling does not occur. Then, limk→∞ tk = ∞ and inf k∈Z+ Δtk > 0. However, if limk→∞ tk = ∞, then property (4.27) implies limk→∞ Δtk = 0. Thus, infinitely fast sampling occurs. The simulation result shown in Figs. 4.1, 4.2 and 4.3 verifies the theoretical analysis.

78

4 Dynamic Event Triggers 1

z x

system state

0.5

0

-0.5

-1

0

5

10

15

20

25

time

Fig. 4.1 The state trajectories with the threshold signal μ(t) = e−t in Example 4.2

control input

0.5

0

-0.5

-1

0

5

10

15

20

25

time

Fig. 4.2 The control input with the threshold signal μ(t) = e−t in Example 4.2

Note that the system dynamics are assumed to be known in Example 4.2. The problem would be more complicated for nonlinear uncertain systems. In particular, in the system setup of event-triggered control, because the sampling error w is updated at discrete time instants (which depend on x and μ), the forward completeness of the system may not be trivially guaranteed. From the discussions in Example 4.2, it can be observed that the problem is caused by the nonlinearity z 3 in the dynamics of the z-subsystem. Intuitively, the signal z(t) does not converge to the origin exponentially, and the exponential convergence of μ(t) is too fast compared with the convergence rate of | f (x(t), z(t), w(t))|.

threshold signal and |x|

4.1 Dynamic Event Triggers with Partial-State Feedback

79

1 |x|

0.5 0

0

5

10

15

20

25

15

20

25

inter-samping intervals

time 10

0

0

5

10 time

Fig. 4.3 The trajectories of the threshold signal μ and inter-sampling intervals with the threshold signal μ(t) = e−t in Example 4.2

To overcome the limitation of the exponentially decreasing threshold signal, we consider threshold signals generated by more general dynamic systems in the form of μ(t) ˙ = −Ω(μ(t))

(4.30)

with initial condition μ(0) > 0, where Ω : R+ → R+ is Lipschitz on compact sets and positive definite. Clearly, (4.18) is a special case of (4.30). In the following discussions, the initial condition μ(0) is only required to be strictly larger than zero, and our event trigger is proved to be valid as long as this condition is satisfied. Under Assumption 4.1, we develop a condition on the ISS gains of the subsystems under which event-triggered control can be realized without infinitely fast sampling. We consider the interconnected system composed of the z-subsystem (4.1), the xsubsystem (4.2) and the μ-subsystem (4.30) subject to (4.5). Under Assumption 4.1, if w(t) is well defined for all t ≥ 0 and the small-gain condition (4.19) is satisfied, then the interconnected system is asymptotically stable at the origin. Moreover, according to the Lyapunov-based ISS cyclic-small-gain theorem in Sect. 2.4, we can construct a Lyapunov function for the interconnected system:   V0 (z, x, μ) = max γˆ xz (Vz (z)), Vx (x), γˆ xw (μ), γˆ xz ◦ γˆ zw (μ) .

(4.31)

(·) (·) (·) If γ(·) is nonzero, then the corresponding γˆ (·) in (4.31) is chosen such that γˆ (·) ∈ K∞ and it is continuously differentiable on (0, ∞) and slightly larger than its correspond(·) (·) (·) ; if γ(·) = 0, then γˆ (·) = 0. Moreover, γˆ xz satisfies γˆ xz ◦ γzx < Id. ing γ(·) w w Define γ˘ x (s) = max{γˆ x (s), γˆ xz ◦ γˆ zw (s)} for s ∈ R+ . Clearly, γ˘ xw is a class K∞ function being locally Lipschitz on (0, ∞). It is a standard result that

80

4 Dynamic Event Triggers

 −1 V (z, x, μ) = γ˘ xw (V0 (z, x, μ))  

 −1 −1 = max γ˘ xw ◦ γˆ xz (Vz (z)), γ˘ xw (Vx (x)), μ =: max {σz (Vz (z)), σx (Vx (x)), μ}

(4.32)

is also a Lyapunov function of the interconnected system. Note that σz and σx are locally Lipschitz on (0, ∞) and thus continuously differentiable almost everywhere on (0, ∞). Moreover, Lemma 4.1 shows that the interconnection ISS gains are less than the identity function by considering V¯z (z) = σz (Vz (z)) and V¯x (x) = σx (Vx (x)) as the ISS-Lyapunov functions of the z-subsystem and the x-subsystem, respectively. This result is used in the following discussions on the convergence rate of the closed-loop event-triggered system; see the proof of Lemma 4.2. Lemma 4.1 Under Assumption 4.1, if (4.19) is satisfied, then there exist class K∞ functions γ¯ zx , γ¯ zw , γ¯ xz , γ¯ xw , all less than Id, such that V¯z (z) ≥ max{γ¯ zx (V¯x (x)), γ¯ zw (|w|)} ⇒∇ V¯z (z)h(z, x, w) ≤ −α¯ z (V¯z (z)) a.e.

(4.33)

V¯x (x) ≥ max{γ¯ xz (V¯z (z)), γ¯ xw (|w|)} ⇒∇ V¯x (x) f (x, z, w) ≤ −α¯ x (V¯x (x)) a.e.

(4.34)

where α¯ z can be any continuous and positive definite function satisfying α¯ z (s) ≤ ∂σz (σz−1 (s))αz (σz−1 (s)) for almost all s > 0 and α¯ x can be any continuous and positive definite function satisfying α¯ x (s) ≤ ∂σx (σx−1 (s))αx (σx−1 (s)) for almost all s > 0. The proof of Lemma 4.1 is given in Appendix D.4. It is shown in the following discussions  that, to avoid infinitely fast sampling and at the same time, to guarantee k∈S [tk , tk+1 ) = [0, ∞), the decreasing rate of μ(t) should be chosen in accordance with the decreasing rate of V (z(t), x(t), μ(t)). According to the definition of V in (4.32), the decreasing rate of V (z(t), x(t), μ(t)) depends on the decreasing rates of Vz (z(t)), Vx (x(t)) and μ(t). Lemma 4.2 gives a condition on Ω under which the lower bound of the decreasing rate of V (z(t), x(t), μ(t)) can be estimated by Ω. Lemma 4.2 Consider the interconnected system composed of (4.1), (4.2), (4.3) and (4.30). It is assumed that w satisfies (4.5). Under Assumption 4.1, if (4.19) is satisfied, and if Ω is Lipschitz on compact sets and positive definite, then there exists a continuous, locally Lipschitz and positive definite αV such that for any V (z(0), x(0), μ(0)), V (z(t), x(t), μ(t)) ≤ η(t) holds for all t ∈



(4.35)

k∈S [tk , tk+1 ), where η(t) is the solution of the initial value problem

4.1 Dynamic Event Triggers with Partial-State Feedback

η(t) ˙ = −αV (η(t))

81

(4.36)

with initial condition η(0) = V (z(0), x(0), μ(0)). Moreover, if there exists a constant Δ > 0 such that (A)

Ω satisfies   Ω(s) ≤ min ∂σz (σz−1 (s))αz (σz−1 (s)), ∂σx (σx−1 (s))αx (σx−1 (s))

(4.37)

for almost all s ∈ (0, Δ)  with σz and σx defined in (4.32), and there exists a T O ∈ k∈S [tk , tk+1 ) such that V (z(t), x(t), μ(t)) ≤ Δ for all T O ≤ t < sup k∈S [tk , tk+1 ),  then (4.35) holds for all T O ≤ t < sup k∈S [tk , tk+1 ) with η(t) being the solution of the initial value problem defined by (4.36) with αV = Ω for T O ≤ t < sup k∈S [tk , tk+1 ) with initial condition η(T O ) = V (z(T O ), x(T O ), μ(T O )). (B)

The proof of Lemma 4.2 is given in Appendix D.5. With respect to condition (A) in Lemma 4.2, given specific ∂σz (σz−1 (s)) αz (σz−1 (s)) and ∂σx (σx−1 (s))αx (σx−1 (s)), one can always find an Ω satisfying the condition. In the following subsection, we will show how Condition (B) can be satisfied by appropriately choosing Ω.

4.1.2 Design of a Dynamic Event Trigger The main result of this section is given in Theorem 4.1. Theorem 4.1 Consider the interconnected system composed of (4.1), (4.2), (4.5) and (4.30) with Assumption 4.1 and (4.19) satisfied. Then, the objectives 1 and 2 given in Sect. 4.1.1 are achievable if • Ω is chosen to be positive definite and Lipschitz on compact sets, • there exists a constant Δ > 0 such that Ω(s)/s is nondecreasing for s ∈ (0, Δ] and Ω satisfies (4.37) for almost all s ∈ (0, Δ), and  −1 −1  • σz ◦ αz and σx ◦ αx are Lipschitz on compact sets. Proof Due to the positive definiteness of Ω, the threshold signal μ(t) generated by (4.30) satisfies 0 ≤ μ(t) ≤ μ(0)

(4.38)

for all t ≥ 0. Moreover, since Ω is chosen to be Lipschitz on compact sets, there exists a constant c¯ > 0 such that Ω(s) ≤ cs ¯

(4.39)

82

4 Dynamic Event Triggers

for 0 ≤ s ≤ μ(0), and thus μ(t) ˙ = −Ω(μ(t)) ≥ −cμ(t) ¯

(4.40)

along the trajectory of μ with initial state μ(0). A direct application of the comparison principle (see, e.g., [138, Lemma 3.4]) implies ¯ −t) μ(τ ) ≥ μ(t)e−c(τ

(4.41)

for all τ ≥ t ≥ 0. Also, by using the event trigger (4.4), we have − ) = |x(tk+1 ) − x(tk )| μ(tk+1   tk+1    f (x(τ ), z(τ ), w(τ ))dτ  = tk  tk+1 | f (x(τ ), z(τ ), w(τ ))| dτ . ≤

(4.42)

tk

If the conditions of Theorem 4.1 are satisfied, thenwith Lemma 4.2, the function V defined in (4.32) has property (4.35) for all t ∈ k∈S [tk , tk+1 ) with η(t) being generated by (4.36). Due to the positive definiteness of αV , for any initial condition V (z(0), x(0), μ(0)), V (z(t), x(t), μ(t)) ≤ V (z(0), x(0), μ(0))

(4.43)

 for all t ∈ k∈S [tk , tk+1 ). If V (z(0), x(0), μ(0)) ≤ Δ, then define T ∗ = 0; otherwise, define T ∗ as the first time instant such that η(T ∗ ) = Δ, where η(t) is the solution of the initial value problem (4.36) with η(0) = V (z(0), x(0), μ(0)). Recall that, according to Lemma 4.2, property (4.35) holds for all t ∈ k∈S [tk , tk+1 ). Let us estimate the lower bound of Δtk = tk+1 − tk by considering the cases of tk ≤ T ∗ and tk > T ∗ separately. Case 1: tk ≤ T ∗ . In this case, we prove that given specific z(0), x(0), μ(0) and specific T ∗ , there exists a Δ0 > 0 such that Δtk ≥ Δ0 . In this way, we can also guarantee that z(t), x(t) and w(t) are defined for all t ∈ [0, T ∗ ] and thus T ∗ ∈  k∈S [tk , tk+1 ). Property (4.43) means that there exists a finite Δs > 0 depending on the initial state such that   T [z (t), x T (t), μ(t)]T  ≤ Δs for all t ∈



k∈S [tk , tk+1 ).

Thus, there exists a Δ f such that

(4.44)

4.1 Dynamic Event Triggers with Partial-State Feedback

| f (z(t), x(t), w(t))| ≤ Δ f for all t ∈



k∈S [tk , tk+1 ).

μ(0)e

83

(4.45)

Then, by also using properties (4.41) and (4.42), we have

−c(t ¯ k +Δtk )

 ≤

tk+1

| f (x(τ ), z(τ ), w(τ ))| dτ

tk

≤ (tk+1 − tk )Δ f = Δtk Δ f ,

(4.46)

i.e., ¯ k +Δtk ) ≥ Δtk ec(t

μ(0) . Δf

(4.47)

μ(0) . Δf

(4.48)

If tk ≤ T ∗ , it is concluded that ¯ Δtk ec(T

∗

+Δtk )

≥

∗

¯ +Δ0 ) = μ(0)/Δ f . Then, Δ0 can be chosen to satisfy Δ0 ec(T ∗ Case 2: tk > T . In this case, we prove that given specific z(T ∗ ), x(T ∗ ) and μ(T ∗ ), there exists a Δ1 > 0 such that Δtk ≥ Δ1 . By using (4.35) and the definition of T ∗ , we have

V (z(t), x(t), μ(t)) ≤ Δ

(4.49)

 for all T ∗ ≤ t < sup k∈S [tk , tk+1 ). Consider an η1 (t) defined by η˙1 (t) = −Ω(η1 (t))

(4.50)

for all t > T ∗ with η1 (T ∗ ) = V (z(T ∗ ), x(T ∗ ), μ(T ∗ )). Then,  by using Lemma 4.2, we have V (z(t), x(t), μ(t)) ≤ η1 (t) for all T ∗ < t < sup k∈S [tk , tk+1 ). Also, by using the definition of V in (4.32), we have V (z(t), x(t), μ(t)) ≥ μ(t) for all t ∈  k∈S [tk , tk+1 ). Thus, μ(t) ≤ V (z(t), x(t), μ(t)) ≤ η1 (t)

(4.51)

 for all T ∗ < t < sup k∈S [tk , tk+1 ). With a similar reasoning as for (4.41), it can be proved that the η1 (t) defined by (4.50) is strictly positive for all T ∗ < t < sup k∈S [tk , tk+1 ). Define kμ =

η1 (T ∗ ) . μ(T ∗ )

(4.52)

84

4 Dynamic Event Triggers

Then, according to (4.51), kμ ≥ 1. We prove that η1 (t) ≤ kμ μ(t)

(4.53)

 for all T ∗ < t < sup k∈S [tk , tk+1 ). Since Ω(s)/s is non-decreasing for all s ∈ (0, Δ], we have   Ω η1 /kμ Ω(η1 ) ≥ , η1 η1 /kμ

(4.54)

which implies Ω(η1 )/kμ ≥ Ω(η1 /kμ ) for η1 ∈ (0, Δ]. Then, by using (4.50), we have  1 1 1 η˙1 (t) = − Ω(η1 (t)) ≤ −Ω η1 (t) (4.55) kμ kμ kμ  for all T ∗ < t < sup k∈S [tk , tk+1 ). Property (4.53) can then be proved by using the comparison principle (see, e.g., [138, Lemma 3.4]) for the system (4.55) with η1 /kμ as the state and the system (4.30) with μ as the state. −1 −1   and σx ◦ αx are Lipschitz on compact sets, then one can find If σz ◦ αz constants k z , k x > 0 such that η1 (t) ≥ V (z(t), x(t), μ(t)) ≥ max {k z (|z(t)|), k x (|x(t)|)} . for all T ∗ < t < sup



k∈S [tk , tk+1 ).

μ(t) ≥

(4.56)

Then, property (4.53) implies

1 max {k z (|z(t)|), k x (|x(t)|)} kμ

(4.57)

 for all T ∗ < t < sup k∈S [tk , tk+1 ). By using the locally Lipschitz property of f , there exists a constant k f > 0 such that | f (z, x, w)| ≤ k f max {|z|, |x|, μ}

(4.58)

for all (z, x, μ) satisfying V (z, x, μ) ≤ V (z(T ∗ ), x(T ∗ ), μ(T ∗ )). Then, properties (4.42), (4.57) and (4.58) together imply max {|z(τ )|, |x(τ )|, μ(τ )}   ≤ (tk+1 − tk )k f max kμ μ(τ )/k z , kμ μ(τ )/k x , μ(τ ) tk ≤τ ≤tk+1   ≤ Δtk k f max kμ /k z , kμ /k x , 1 μ(tk ).

μ(tk+1 ) ≤ (tk+1 − tk )k f

Also note that (4.41) means

tk ≤τ ≤tk+1

(4.59)

4.1 Dynamic Event Triggers with Partial-State Feedback

85

¯ k μ(tk+1 ) ≥ e−cΔt μ(tk ).

(4.60)

  ¯ k ≤ Δtk k f max kμ /k z , kμ /k x , 1 , e−cΔt

(4.61)

1 . k f max kμ /k z , kμ /k x , 1

(4.62)

Thus, we have

i.e., ¯ k ≥ Δtk ecΔt



  ¯ 1 ≥ 1/k f max kμ /k z , kμ /k x , 1 . Then, Δ1 can be chosen such that Δ1 ecΔ We now consider the three cases mentioned in Sect. 3.1: (a) S = Z+ and limk→∞ tk < ∞; (b) S = Z+ and limk→∞ tk = ∞; (c) S = {0, . . . , k ∗ } with k ∗ ∈ Z+ : (1) Suppose that Case (a) happens. Then, we have inf k∈S {tk+1 − tk } = 0, which contradicts with (4.48)  and (4.62). Thus, Case (a) is impossible. (2) In Case (b), we have k∈S [tk , tk+1 ) = [0, ∞), which means that (4.35) holds for all t ∈ [0, ∞).  (3) In Case (c), since tk ∗ +1 = Tmax , we have k∈S [tk , tk+1 ) = [0, Tmax ), and thus (4.35) holds for all t ∈ [0, Tmax ). By the continuation of solutions [62], this implies that x(t) ¯ is defined for all t ∈ [0, ∞), i.e., Tmax = ∞. This ends the proof of Theorem 4.1.



Theorem 4.1 readily solves the infinitely fast sampling problem arising in Example 4.2. Example 4.3 By using the Vz and Vx defined in Example 4.1, we choose γˆ xz (s) = 5s, γˆ zw (s) = 0 and γˆ xw (s) = 5s for s ∈ R+ . According to (4.31), we define V0 (z, x, μ) = max{5Vz (z), Vx (x), 5μ}.

(4.63)

By choosing γ˘ xw (s) = 5s for s ∈ R+ , we have V (z, x, μ) = max{Vz (z), Vx (x)/5, μ}.

(4.64)

Thus, σz (s) = s and σx (s) = s/5 for s ∈ R+ . −1 −1   (s) = s and σx ◦ αx (s) = 2s for s ∈ R+ It can be verified that σz ◦ αz are Lipschitz on compact sets. We choose   Ω(s) = min ∂σz (s)αz (σz−1 (s)), ∂σx (s)αx (σx−1 (s)) = min{s 3 , s/2}

(4.65)

for s ∈ R+ . Then, Ω is positive definite and Lipschitz on compact sets, and satisfies (4.37). Also, Ω(s)/s is non-decreasing for s ∈ (0, ∞).

86

4 Dynamic Event Triggers 1

z x

0.5

system states

0 -0.5 -1 -1.5 -2 -2.5 -3

0

50

100

150

200

250

300

350

time

  Fig. 4.4 The trajectories of the system states z and x with Ω(μ) = min μ3 , μ/2 in Example 4.3

  Fig. 4.5 The control input u with Ω(μ) = min μ3 , μ/2 in Example 4.3

We use a simulation to verify the theoretical results. We choose initial conditions z(0) = −3, x(0) = 1 and μ(0) = 10. Figures 4.4 and 4.5 show the convergence of the states x(t) and z(t), and the control input u(t). The threshold signal μ(t), the sampling error w(t), and the intersampling intervals Δtk = tk+1 − tk are shown in Fig. 4.6. For comparison, we also consider event triggers with exponentially decreasing threshold signals. Figure 4.7 shows the inter-sampling intervals during the control process with Ω(μ) = μ/2 and Ω(μ) = μ/15, respectively. The step-length of the numerical simulation is chosen to be 0.001 for an acceptable simulation accuracy. For the simulation of the event-triggered control system, the step-length actually guarantees strictly positive minimum inter-sampling intervals and infinitely fast sampling

|w| and

4.1 Dynamic Event Triggers with Partial-State Feedback 4 3 2 1

|w|

0 inter-samping intervals

87

10

2

10

0

5

10

15 time

20

25

30

inter-sampling intervals 0.998

0

50

100

150

200

250

300

350

time

Fig. 4.6 The threshold signal  μ(t), the  sampling error w(t), and the inter-sampling intervals Δtk = tk − tk−1 with Ω(μ) = min μ3 , μ/2 in Example 4.3

Fig. 4.7 The inter-sampling intervals Δtk = tk − tk−1 with Ω(μ) = μ/2 and Ω(μ) = μ/15 in Example 4.3, respectively

never happens. However, from Fig. 4.7, we can still observe that the inter-sampling intervals converge to a very small neighborhood of the zero in finite time. A special case is that both the z-subsystem and the x-subsystem are linear and their ISS-Lyapunov functions are in the quadratic form. In this case, Assumption 4.1 can be modified with αz (s) = a z s 2 , αz (s) = a z s 2 , αx (s) = a x s 2 , αx (s) = a x s 2 , γzx (s) = bzx s, γzw (s) = bzw s 2 , γxz (s) = bxz s, γxw (s) = bxw s 2 , αz (s) = az s and αx (s) = ax s for s ∈ R+ with a z , a z , a x , a x , bzx , bzw , bxz , bxw , az , az being positive constants. For the linear case, the small-gain condition (4.19) is equivalent to bxz bzx < 1. Assume that the small-gain condition is satisfied. Then, there exists an  > 0 such that (bxz + )bzx < 1. We choose γˆ xz (s) = (bxz + )s =: bˆ xz s, γˆ xw (s) = (bxw + )s 2 =: bˆ xw s 2 , γˆ zw (s) = (bzw + )s =: bˆ zw s 2 for s ∈ R+ . Then, γ˘ xw can be written in the

88

4 Dynamic Event Triggers

form γ˘ xw (s) = b˘ xw s 2 with b˘ xw being

a positive constant. It can be calculated that z σz (s) = bˆ x s/b˘ xw and σx (s) = s/b˘ xw . Then, the right-hand side of (4.37) equals

min az /bˆ xz , ax b˘ xw s/2. Thus, one can find a positive constant c such that Ω(s) = cs satisfies (4.37).

4.1.3 An Extension In the discussions above, we consider μ(t) to be generated by the system (4.30) with the system dynamics depending solely on μ(t). A more general case is that μ(t) is generated by the system ¯ μ(t) ˙ = −Ω(μ(t), x(t))

(4.66)

with Ω¯ : R+ × Rn → R being an appropriately chosen function and μ(0) > 0. With such modification, the sampling events are triggered based on both μ and x, such that the inter-sampling intervals could be further increased. In this case, the structure of the interconnected system composed of (4.1), (4.2) and (4.66) subject to (4.5) is shown in Fig. 4.8. Under Assumption 4.1, the basic idea is to choose Ω¯ such that • for all μ ∈ R+ and all x ∈ Rn , ¯ Ω(μ, x) ≤ Ω(μ)

(4.67)

where Ω is chosen to satisfy the conditions given in Theorem 4.1; • the system (4.66) is ISS with μ as the state and x as the input, and moreover, ¯ x) ≥ αμ (μ) μ ≥ χμx (|x|) ⇒ Ω(μ,

(4.68)

where αμ is a continuous, positive definite function and χμx is a class K function and satisfies γxw ◦ χμx ◦ α−1 x < Id, γxz (γzw (χμx (α−1 x )))

Fig. 4.8 The structure of the interconnected system composed of (4.1), (4.2) and (4.66)

(4.69)

< Id.

(4.70)

µ

z

x

4.1 Dynamic Event Triggers with Partial-State Feedback

89

Then, the system composed of subsystems (4.1), (4.2) and (4.66) subject to (4.5) can be considered as an interconnection of ISS subsystems, and conditions (4.19), (4.69) and (4.70) form the cyclic-small-gain condition given in Sect. 2.4 for the interconnected system. With μ(t)  generated by (4.66), one can still guarantee the boundedness of μ(t) for all t ∈ k∈S [tk , tk+1 ). And with a similar reasoning as for (4.40), one can find a c¯ > 0 such that μ(t) ˙ ≥ −cμ(t) ¯

(4.71)

 for all t ∈ k∈S [tk , tk+1 ). Then, the validity of (4.66) can be proved in the same way as in the proof of Theorem 4.1. Note that in this case, Lemma 4.2 should also be generalized with (4.36) replaced by ¯ η(t) ˙ = −Ω(η(t), x(t)).

(4.72)

Clearly, with the extension, the decreasing rate of μ could be reduced, and thus the inter-sampling intervals could be increased. One implementation of the Ω¯ satisfying (4.67) and (4.68) is    ¯ Ω(μ, x) = Ω(μ) − χ1 max χ2 ◦ χμx (|x|) − χ2 (μ), 0

(4.73)

where Ω is chosen to satisfy the conditions given in Theorem 4.1, χ1 and χ2 can be any K function. Clearly, with such design, condition (4.68) is satisfied with αμ = Ω.

4.2 An Application to Decentralized Event-Triggered Control This section shows that the dynamic event triggers proposed in Sect. 4.1 can be used in solving the decentralized event-triggered control problem. Indeed, the decentralized controllers only use partial state of the system. We consider the controlled large-scale systems that are transformable into an interconnection of ISS subsystems with the sampling errors as the inputs. The sampling events for each subsystem are triggered by a threshold signal, and the threshold signals for the subsystems are independent with each other to fulfill the requirement of decentralized implementation. By appropriately designing the event triggers, infinitely fast sampling can be avoided for each subsystem and asymptotic regulation is achieved for the large-scale system.

90

4 Dynamic Event Triggers

4.2.1 Problem Formulation We focus on the decentralized event trigger design for a large-scale system composed of N subsystems. Suppose that each the ith subsystem (i = 1, . . . , N ) admits a controller, and the controlled ith subsystem is in the form of x˙i = f i (x, wi )

(4.74)

where xi ∈ Rni is the state, x = [x1T , . . . , x NT ]T ∈ Rn , wi ∈ Rni represents the sampling error of xi , and f i : Rn × Rni → Rni satisfying f i (0, 0) = 0 represents the system dynamics. For a sequence of event-triggering time instants {tki }k∈Si with Si = {0, 1, 2, ...} ⊆ Z+ and t0i = 0, define sampling error wi as i ), k ∈ Si . wi (t) = xi (tki ) − xi (t), t ∈ [tki , tk+1

(4.75)

In this section, the sequence {tki }k∈Si of the ith subsystem is triggered by comparing the sampling error |xi (tki ) − xi (t)| with a continuous and positive threshold signal μi : R+ → R+ . Specifically, the event-triggering time instants are generated by   i = inf t > tki : |xi (tki ) − xi (t)| = μi (t) , k ∈ Si ⊆ Z+ . tk+1

(4.76)

Suppose that x(t) is right maximally defined on [0, Tmax ). Then, it holds that Tmax ≥ sup{tki }

(4.77)

k∈Si

for i = 1, . . . , N . Recall the definition of wi (t) in (4.75). The event trigger defined by (4.76) guarantees that |wi (t)| ≤ μi (t)

(4.78)

for all t ∈ [0, Tmax ). For practical implementation of the event-triggered control law, infinitely fast sampling should be avoided. We aim to develop a new approach to decentralized event-triggered control such that the following two objectives are achievable at the same time. Objective 1: Infinitely fast sampling is avoided. That is, for any specific x(0) and i − tki between the eventany specific μi (0) > 0 for i = 1, . . . , N , the intervals tk+1 triggering time instants for each xi -subsystem (i = 1, . . . , N ) are lower bounded by a positive constant. Objective 2: The closed-loop event-triggered system is forward complete. That is, for any initial state x(0), x(t) is defined for all t ≥ 0, and in addition, x(t) asymptotically converges to the origin.

4.2 An Application to Decentralized Event-Triggered Control

91

In this section, we focus on the event trigger design, and without loss of generality, assume that local feedback control laws have been designed such that each xi -subsystem is input-to-state stable with wi and x j for j = i as the inputs. Assumption 4.2 For i = 1, . . . , N , the xi -subsystem is ISS with an ISS-Lyapunov function Vi : Rni → R+ , which is locally Lipschitz on Rni \{0} and satisfies αi (|xi |) ≤ Vi (xi ) ≤ αi (|xi |) Vi (xi ) ≥

(4.79)

j max{χi (V j (x j )), γi (|wi |)} j =i

⇒∇Vi (xi ) f i (x, wi ) ≤ −θi (Vi (xi ))

a.e.

(4.80)

j

where αi , αi ∈ K∞ , χi , γi ∈ K ∪ {0}, and θi is continuous and positive definite.

4.2.2 Design of Decentralized Event-Triggers With Assumption 4.2 satisfied, the large-scale system (4.74) is ISS with wi for j i = 1, . . . , N as the inputs, if the interconnection gains χi satisfy the cyclic-smallgain condition; see Sect. 2.4.1. If, additionally, the event triggers are designed such that Objective 1 is achieved and each wi (t) asymptotically converges to the origin, then x(t) globally asymptotically converges to the origin. In this subsection, we extend the dynamic event triggers proposed in Sect. 4.1 to decentralized dynamic event triggers. Each threshold signal μi is generated by a dynamic system of the form η˙i (t) = −φi (ηi (t)) μi (t) = ϕi (ηi (t))

(4.81) (4.82)

where ηi ∈ R+ is the state, φi : R+ → R+ is locally Lipschitz and positive definite, and ϕi : R+ → R+ is continuously differentiable on (0, ∞) and of class K∞ . The initial state ηi (0) is chosen to be positive, and thus μi (0) is positive. We first provide technical lemmas on the convergence rate of the closed-loop event-triggered system. Based on the technical lemmas, we appropriately choose the functions φi and ϕi for the event triggers. Technical Lemmas Recall that D + v(t) represents the Dini derivative of a continuous function v : R+ → R. See, e.g., [138] for the definition of the Dini derivative. For convenience of notations, define η = [η1 , . . . , η N ]T . Lemma 4.3 provides a transformation of the ISSLyapunov functions and also an estimate on the convergence rate of the closed-loop event-triggered system. Note that such transformation is based on the equivalence between cyclic-small-gain and gains less than the identity, which is discussed in Sect. 2.4.2.

92

4 Dynamic Event Triggers

Lemma 4.3 Under Assumption 4.2, suppose that the large-scale system composed of (4.74) satisfies the cyclic-small-gain condition (2.68) in Sect. 2.4.1. • For each i = 1, . . . , N , there exists a σi ∈ K∞ being locally Lipschitz on (0, ∞) such that V¯i (xi ) = σi (Vi (xi ))

(4.83)

is an ISS-Lyapunov function of the xi -subsystem, that satisfies

j V¯i (xi ) ≥ max χ¯ i (V¯ j (x j )), γ¯ i (|wi |) j =i

⇒∇ V¯i (xi ) f i (x, wi ) ≤ −θi (V¯i (xi )) a.e.

(4.84)

where χ¯ i ∈ K ∪ {0} satisfies χ¯ i < Id, γ¯ i = σi ◦ γi , and θi is continuous and positive definite. • Consider the large-scale system composed of (4.74), (4.81) and (4.82). Suppose that (4.78) holds for t ∈ [0, Tmax ) for i = 1, . . . , N . By choosing ϕi such that γ¯ i ◦ ϕi < Id for i = 1, . . . , N , the function j

j

V (x, η) = max {V¯i (xi ), ηi }

(4.85)

D + V (x(t), η(t)) ≤ −α(V (x(t), η(t)))

(4.86)

i=1,...,N

satisfies

for all t ∈ [0, Tmax ), where α(s) = min {θi (s), φi (s)}

(4.87)

i=1,...,N

for s ∈ R+ . Proof With the satisfaction of the cyclic-small-gain condition, (4.84) can be proved by directly using the result in Sect. 2.4. It can be checked that V (x, η) is continuously differentiable for almost all (x, η). Recall μi = ϕi (ηi ) defined by (4.82) and the definition of α in (4.87). From (4.84), if |wi | ≤ μi , then j V¯i (xi ) ≥ max{χ¯ i (V¯ j (x j )), γ¯ i ◦ ϕi (ηi )} j =i

⇒ ∇ V¯i (xi ) f i (x, wi ) ≤ −α(V¯i (xi ))

a.e.

(4.88)

The definition of α in (4.87) also implies that η˙i (t) ≤ −α(ηi (t)) for all t ≥ 0.

(4.89)

4.2 An Application to Decentralized Event-Triggered Control

93

We consider the closed-loop event-triggered system composed of (4.74), (4.81) and (4.82) as an interconnected system. Define v(t) = V (x(t), η(t)), vi1 (t) = V¯i (xi (t)), and vi2 (t) = ηi (t). j Consider a specific time instant t ∗ ∈ [0, Tmax ). Since χ¯ i < Id and γ¯ i ◦ ϕi < Id, if vi1 (t ∗ ) = v(t ∗ ), i.e., V¯i (xi (t ∗ )) = V (x(t ∗ ), η(t ∗ )), then there exists a neighborhood Θ of [x T (t ∗ ), η T (t ∗ )]T such that j V¯i ( pi ) ≥ max{χ¯ i (V¯ j ( p j )), γ¯ i ◦ ϕi (qi )} j =i

(4.90)

and ∇ V¯i ( pi ) exists for all [ p1T , . . . , p TN , q1 , . . . , q N ]T ∈ Θ. Due to the continuity of x(t) and η(t), there exists a t  > t ∗ such that j V¯i (xi (t)) ≥ max{χ¯ i (V¯ j (x j (t))), γ¯ i ◦ ϕi (ηi (t))} j =i

(4.91)

for all t ∈ (t ∗ , t  ), which implies ∇ V¯i (xi (t)) f i (x(t), wi (t)) ≤ −α(V¯i (xi (t))) a.e.

(4.92)

for t ∈ (t ∗ , t  ). Then, we have D + vi1 (t ∗ ) ≤ −α(vi1 (t ∗ ))

(4.93)

if vi1 (t ∗ ) = v(t ∗ ). By using the definition of α in (4.87), we also have D + vi2 (t) ≤ −α(vi2 (t)) for all t ≥ 0. Define j

I (t) = {i = 1, . . . , N : vi (t) = v(t), j = 1, 2} J (t) = { j = 1, 2 :

j vi (t)

= v(t), i = 1, . . . , N }

(4.94) (4.95)

for t ∈ [0, Tmax ). By using Lemma 2.9 in [72], we have D + v(t) = max{D + vi (t) : i ∈ I (t), j ∈ J (t)} j

j

≤ max{−α(vi (t)) : i ∈ I (t), j ∈ J (t)} = −α(v(t)) for all t ∈ [0, Tmax ). This ends the proof of Lemma 4.3.

(4.96) 

94

4 Dynamic Event Triggers

Lemma 4.4 Consider two continuous signals ξ1 , ξ2 : [0, Tmax ) → R+ satisfying D + ξ1 (t) ≤ −θξ (ξ1 (t)) D + ξ2 (t) = −θξ (ξ2 (t))

(4.97) (4.98)

for all t ∈ [0, Tmax ), where θξ is a positive definite function and θξ (s)/s is nondecreasing for s ∈ (0, Δ) with constant Δ > 0. If 0 < ξ2 (0) ≤ ξ1 (0) < Δ, then ξ1 (t) ≤ kξ ξ2 (t)

(4.99)

for all t ∈ [0, Tmax ), with kξ = ξ1 (0)/ξ2 (0). Proof Since 0 < ξ2 (0) ≤ ξ1 (0) < Δ, we have kξ ≥ 1. Also, the positive definiteness of θξ implies the decreasing of ξ1 (t) and ξ2 (t), and thus ξ1 (t) < Δ and ξ2 (t) < Δ. With kξ (s)/s assumed to be nondecreasing for s ∈ (0, Δ), we have θξ (ξ1 ) θξ (ξ1 /kξ ) ≥ . ξ1 ξ1 /kξ

(4.100)

Then, by using (4.97), we have D+



ξ1 (t) kξ

=

1 + 1 D ξ1 (t) ≤ − θξ (ξ1 (t)) ≤ −θξ kξ kξ



ξ1 (t) kξ

(4.101)

for all t ∈ [0, Tmax ). With the comparison principle (see, e.g., [138, Lemma 3.4]), (4.98) and (4.101) imply ξ1 (t) ≤ ξ2 (t) kξ for all t ∈ [0, Tmax ). This ends the proof of Lemma 4.4.

(4.102) 

Main Result The main result on decentralized event trigger design is given by Theorem 4.2. Recall the definitions of σi , γ¯ i and θi in Lemma 4.3. Theorem 4.2 Consider the interconnected system composed of (4.74), (4.81) and (4.82) subject to (4.76), with Assumption 4.2 and the cyclic-small-gain condition (2.68) in Sect. 2.4.1 satisfied. The two objectives of decentralized event-triggered control given in Sect. 4.2.1 are achievable if there exists a γ¯ ∈ K∞ such that γ¯ ≥ maxi=1,...,N {γ¯ i } and αi−1 ◦ σi−1 ◦ γ¯ is Lipschitz on compact sets for i = 1, . . . , N . In particular, for i = 1, . . . , N , ϕi and φi can be chosen as follows: • ϕi = ϕ, where ϕ ∈ K∞ is continuously differentiable on (0, ∞) and satisfies that γ¯ ◦ ϕ < Id and αi−1 ◦ σi−1 ◦ ϕ−1 is Lipschitz on compact sets for i = 1, . . . , N ;

4.2 An Application to Decentralized Event-Triggered Control

95

• φi = φ, where φ : R+ → R+ is positive definite and Lipschitz on compact sets and satisfies that φ(s) ≤ mini=1,...,N {θi (s)} for s ∈ R+ , θw is positive definite and Lipschitz on compact sets, and there exists a constant Δ > 0 such that θw (s)/s is nondecreasing for s ∈ (0, Δ), with θw defined by  θw (s) =

∂ϕ(ϕ−1 (s))φ(ϕ−1 (s)) for s > 0; 0 for s = 0.

(4.103)

Proof Note that the condition for Lemma 4.3 is satisfied. Also, the event trigger (4.76) guarantees that (4.78) holds for t ∈ [0, Tmax ). We first analyze the dynamic behavior of μi (t) and ϕ(V (x(t)), η(t)). Recall the definition of V in (4.85). Then, we have ϕ(V (x(t)), η(t)) = max {ϕ(V¯i (xi (t))), ϕ(ηi (t))} i=1,...,N

≥ μi (t)

(4.104)

for all t ∈ [0, Tmax ). We used μi = ϕ(ηi ) for the last inequality. Denote v(t) = V (x(t), η(t)) and w(t) = ϕ(v(t)). Then, with φi = φ and φ(s) ≤ mini=1,...,N {θi (s)} for s ∈ R+ , property (4.86) implies that D + w(t) = ∂ϕ(v(t))D + v(t) ≤ −∂ϕ(v(t))α(v(t)) = −∂ϕ(ϕ−1 (w(t)))α((ϕ−1 (w(t)))) = −∂ϕ(ϕ−1 (w(t)))φ((ϕ−1 (w(t)))) =: −θw (w(t))

(4.105)

for t ∈ [0, Tmax ). The positive definiteness of θw implies that V (x(t), η(t)) ≤ V (x(0), η(0))

(4.106)

for t ∈ [0, Tmax ). Also, from (4.81) and (4.82), one has μ˙ i (t) = −∂ϕ(ϕ−1 (μi (t)))φ(ϕ−1 (μi (t))) = −θw (μi (t)).

(4.107)

Thus, it can be proved that 0 ≤ μi (t) ≤ μi (0) for all t ≥ 0, due to the positive definiteness of θw . Also, since θw is Lipschitz on compact sets, there exists a constant c¯ > 0 such that θw (s) ≤ cs ¯ for 0 ≤ s ≤ ϕ(V (x(0), η(0))). This, together with (4.107), implies

96

4 Dynamic Event Triggers

μ˙ i (t) = −θw (μi (t)) ≥ −cμ ¯ i (t)

(4.108)

along the trajectories of μi with initial state μi (0) ≤ ϕ(V (x(0)), η(0)). Then, a direct application of the comparison principle (see, e.g., [138, Lemma 3.4]) implies that ¯ −t) μi (τ ) ≥ μi (t)e−c(τ

(4.109)

for all τ ≥ t ≥ 0. By using the event trigger (4.76), one has  i )≤ μi (tk+1

i tk+1

tki

| f i (x(τ ), wi (τ ))|dτ

(4.110)

for any k, k + 1 ∈ Si . Now, for i = 1, . . . , N , we prove the existence of a positive lower bound of Δtki = i tk+1 − tki by considering the cases of ϕ(V (x(tki ), η(tki ))) ≥ Δ and ϕ(V (x(tki ), η(tki ))) < Δ, respectively. Case 1: ϕ(V (x(tki ), η(tki ))) ≥ Δ. In this case, given specific x(0), η(0), there exists a 0 ≤ T ∗ < ∞ such that tki ≤ T ∗ . (This can be directly proved by using the convergence property of ϕ(V (x(t), η(t))) given by (4.105)). Property (4.106) means the boundedness of x(t) and η(t) for t ∈ [0, Tmax ), i.e., there exists a finite Δs such that |[x T (t), η T (t)]T | ≤ Δs

(4.111)

for all t ∈ [0, Tmax ). Thus, due to the locally Lipschitz continuity of f i , there exists a Δ f i such that | f i (x(t), wi (t))| ≤ Δ f i

(4.112)

for all t ∈ [0, Tmax ). Then, (4.109) and (4.110) imply μi (0)e

−c(t ¯ ki +Δtki )

 ≤

tki

i tk+1

| f i (x(τ ), wi (τ ))|dτ

i ≤ (tk+1 − tki )Δ f i = Δtki Δ f i ,

(4.113)

¯ k +Δtk ) ≥ μi (0)/Δ f i . i.e., Δtk ec(t ¯ ∗ +Δtki ) ≥ μi (0)/Δ f i . Then, In this case, since tki ≤ T ∗ , it can be proved that Δtki ec(T i ∗ i i c(T ¯ +Δ0 ) = μi (0)/Δ f i , it is concluded that Δtki ≥ Δi0 by choosing Δ0 such that Δ0 e i i i for all the tk satisfying ϕ(V (x(tk ), η(tk ))) ≥ Δ. Case 2: ϕ(V (x(tki ), η(tki ))) < Δ. In this case, we first show the relation between the states x j ( j = 1, . . . , N ) of the subsystems and the threshold signal μi . Recall that w(t) and μi (t) satisfy (4.105) and (4.107), respectively. i

i

4.2 An Application to Decentralized Event-Triggered Control

97

Define kμi = w(0)/μi (0). Then, using Lemma 4.4, we have w(t) ≤ kμi μi (t)

(4.114)

for all t ∈ [0, Tmax ). By using the definition of V (x, η) in (4.85), we have ϕ ◦ σ j (V j (x j (t))) ≤ kμi μi (t)

(4.115)

for j = 1, . . . , N . Since V j (x j (t)) ≥ α j (|x j (t)|), we obtain ϕ ◦ σ j ◦ α j (|x j (t)|) ≤ kμi μi (t)

(4.116)

for j = 1, . . . , N , which implies −1 −1 |x j (t)| ≤ α−1 j ◦ σ j ◦ ϕ (kμi μi (t))

(4.117)

for all t ∈ [0, Tmax ). −1 −1 is Lipschitz on compact sets, there exist positive constants Since α−1 j ◦ σj ◦ ϕ k k j such that x j and μi satisfy |x j (t)| ≤ k ij μi (t)

(4.118)

for all t ∈ [0, Tmax ) for j = 1, . . . , N . By using the locally Lipschitz property of f i , there exists a constant k f i such that | f i (x, wi )| ≤ k f i max {|x j |, |wi |}. j=1,...,N

(4.119)

Then, property (4.110) implies i i ) ≤ (tk+1 − tki )k f i μi (tk+1

max

i tki ≤τ ≤tk+1 ; j=1,...,N

{|x j (τ )|, μi (τ )}

≤ Δtki k f i max {k ij , 1}μi (tki ).

(4.120)

j=1,...,N

Also, note that (4.109) implies i ¯ k ) ≥ e−cΔt μi (tki ). μi (tk+1 i

(4.121)

¯ k It can be proved that e−cΔt ≤ Δtki k f i max j=1,...,N {k ij , 1}, i.e., i

¯ k Δtki ecΔt ≥ i

1 k f i max j=1,...,N {k ij , 1}

.

(4.122)

98

4 Dynamic Event Triggers

¯ 1 Then, by choosing Δi1 such that Δi1 ecΔ = 1/(k f i max j=1,...,N {k ij , 1}), it is concluded that Δtki ≥ Δi1 for all tki satisfying ϕ(V (x(tki ), η(tki ))) < Δ. The achievement of the objective 1 is proved following the discussions in the above two cases. Now, we prove that Tmax = ∞ by considering the following two cases: i

∗

∗

i • There exists an i ∗ ∈ {1, . . . , N } such that Si ∗ = Z+ . Then, since tk+1 − tki has been proved to be lower bounded by a positive constant, Tmax = ∞ follows property (4.77). • Si ⊂ Z+ for all i = 1, . . . , N . Then, there exists a T ∗ < ∞ such that maxk∈Si {tki } ≤ T ∗ < Tmax for all i = 1, . . . , N . In this case, the closed-loop event-triggered system only involves continuous-time dynamics for t ≥ T ∗ . Recall that (4.86) holds for all t ∈ [0, Tmax ). Then, Tmax = ∞ is proved due to causality.

Thus, with the proposed event trigger design, for any x(0) and any μi (0) > 0, (4.86) holds for all [0, ∞). Global asymptotic stability is proved. This ends the proof of Theorem 4.2. 

4.2.3 An Example: Decentralized Event-Triggered Control of a Class of First-Order Nonlinear Systems Section 4.2.2 shows that decentralized event-triggered control can be realized as long as the ISS gains of the subsystems can be appropriately designed. In this subsection, we introduce a refined gain assignment result for decentralized event-triggered control of a class of first-order nonlinear systems. Consider a large-scale system composed of N subsystems, with each ith subsystem (i = 1, . . . , N ) in the form of x˙i = gi (x) + u i

(4.123)

where xi ∈ R is the state, u i ∈ R is the control input, x = [x1T , . . . , x NT ]T , and gi : Lipschitz function representing the dynamics of Rn → Rni is an unknown locally  the xi -subsystem with n = j=1,...,N n j . For each i = 1, . . . , N , it is assumed that j

there exist some known functions ψi ∈ K∞ such that |gi (x)| ≤



j

ψi (|x j |), ∀x.

(4.124)

j=1,...,N

Gain Assignment We propose decentralized control laws in the form of u i = −sgn(xi + wi )κi (|xi + wi |)

(4.125)

4.2 An Application to Decentralized Event-Triggered Control

99

where κi ∈ K∞ is Lipschitz on compact sets, wi ∈ R represents the sampling error of xi , and sgn(r ) takes the sign of r ∈ R, i.e., sgn(r ) = 1 if r > 0, sgn(r ) = 0 if r = 0 and sgn(r ) = −1 if r < 0. Then, each controlled xi -subsystem can be rewritten as x˙i = gi (x) − sgn(xi + wi )κi (|xi + wi |) =: f i (x, wi ).

(4.126) j

To satisfy Assumption 4.2 in Sect. 3.1, we choose κi and the ISS gains χi ( j = 1, . . . , N ) and γi such that 

κi ◦ (Id − γi−1 )(s) ≥

 −1 j j ψi ◦ χi (s) + θi (s)

(4.127)

j=1,...,N

for all s ∈ R+ , where χii = Id. Condition (4.127) can always be satisfied as one may j choose χi ∈ K∞ arbitrarily, choose θi to be any positive definite and continuous function, choose γi ∈ K∞ such (Id − γi−1 ) ∈ K∞ , and choose κi ∈ K∞ large enough. Note that (Id − γi−1 ) ∈ K∞ implies γi > Id. For each xi -subsystem with control law (4.125), we define Vi (xi ) = |xi |

(4.128)

as the ISS-Lyapunov function candidate. It can be directly checked that Vi satisfies (4.79) with αi = αi = Id. j We consider the case in which xi = 0, Vi (xi ) ≥ χi (|x j |) and Vi (xi ) ≥ γi (|wi |). Clearly, Vi (xi ) ≥ γi (|wi |) implies |xi | ≥ |wi | and thus sgn(xi ) = sgn(xi + wi ). Then, ∇Vi (xi ) f i (x, wi ) = ∇Vi (xi )(gi (x) − sgn(xi + wi )κi (|xi + wi |)) = sgn(xi ) (gi (x) − sgn(xi )κi (|xi + wi |)) ≤ |gi (x)| − κi (|xi | − |wi |)  −1  j j ψi ◦ χi (|xi |) − κi ◦ (Id − γi−1 )(|xi |) ≤ j=1,...,N

≤ − θi (|xi |).

(4.129)

To sum up, Assumption 4.2 holds for the subsystem (4.123). Decentralized Event-Trigger Design With Assumption 4.2 satisfied, decentralized event triggers can be designed by using the method in Sect. 4.2.2. j j Since χi ∈ K∞ can be chosen arbitrarily, we choose χi < Id for i = j. Also, we choose γi (s) = ki s and θi (s) = ai s with constants ki > 1 and ai > 0. With such j j treatment, we can choose σi = Id. Then, χ¯ i = χi , γ¯ i = γi and θi = θi . According to Theorem 4.2, we define

100

4 Dynamic Event Triggers

γ(s) ¯ = max {γ¯ i (s)} = max {ki }s i=1,...,N

i=1,...,N

(4.130)

for s ∈ R+ . It is obvious that αi−1 ◦ σi−1 ◦ γ¯ = γ¯ is Lipschitz on compact sets. The condition of Theorem 4.2 is satisfied. As a direct application of Theorem 4.2, the two objectives of event-triggered control can be achieved by choosing ϕi (s) = kϕi s and φi (s) = kφi s, where kϕi = δ mini=1,...,N {1/ki } with 0 < δ < 1 and kφi = mini=1,...,N {ai }. The linearity of ϕi and φi means that the large-scale nonlinear system (4.123) can be globally asymptotically stabilized by decentralized event triggers with exponentially converging threshold signals.

4.2.4 The Linear Case For linear systems, Lyapunov functions in the quadratic form are usually used. In this subsection, we consider a special case of the problem setting in Sect. 4.2.1. Specifically, Assumption 4.2 is modified as follows. Assumption 4.3 For i = 1, . . . , N , the xi -subsystem is ISS with a continuously differentiable ISS-Lyapunov function Vi : Rni → R+ satisfying αi |xi |2 ≤ Vi (xi ) ≤ αi |xi |2 Vi (xi ) ≥

(4.131)

j max{ki V j (x j ), ki |wi |2 } j =i

⇒∇Vi (xi ) f i (x, wi ) ≤ −αi Vi (xi ),

(4.132)

j

where the constants αi , αi , αi > 0 and ki , ki ≥ 0. In particular, based on the equivalence between cyclic-small-gain and gains less than the identity in Lemma 4.3, j assume ki < 1. As a special case of Theorem 4.2, we have the following corollary. Corollary 4.1 Consider the interconnected system composed of (4.74), (4.81) and (4.82) subject to (4.76), with Assumption 4.3 satisfied. The two objectives of decentralized event-triggered control given by Sect. 4.2.1 are achievable if there exists a k > 0 such that k ≥ maxi=1,...,N {ki }. In particular, for i = 1, . . . , N , ϕi and φi can be chosen as follows: √ • ϕi = ϕ, where ϕ is positive constant and satisfies that k · ϕ < 1; • φi = φ, where φ is positive constant and satisfies that φ ≤ mini=1,...,N {αi }. We employ an example to show the effectiveness of the proposed design. Example 4.4 The following model describes the motion of the coupled pendulum shown in Fig. 4.9 [257]:

4.2 An Application to Decentralized Event-Triggered Control

101

m1

m2

x11

x21 k

l a

Fig. 4.9 The coupled inverted pendulums

⎡

⎤ ⎡ 0 x˙11 2 ⎢x˙12 ⎥ ⎢ gl − mkal 2 1 ⎢ ⎥=⎢ ⎣x˙21 ⎦ ⎣ 0 ka 2 x˙22 2 m2l

1 0 0 0

0 ka 2 m1l 2 g l

0 2 − mka2 l 2

⎤ ⎡ ⎤ ⎡ 0 0 x11 ⎢x12 ⎥ ⎢ m1l 2 0⎥ ⎥·⎢ ⎥+⎢ 1 1⎦ ⎣x21 ⎦ ⎣ 0 x22 0 0

⎤ 0   0 ⎥ ⎥ · u1 , 0 ⎦ u2

(4.133)

1 m2l2

where x11 and x22 are the pendulum angles, m 1 and m 2 are the pendulum masses, g is the gravitational acceleration, k is the spring constant, l is the pendulum length, and u 1 and u 2 are the control inputs. Define the state vector xi = [xi1 , xi2 ]T , i = 1, 2. The system (4.133) can be rewritten as x˙i = Ai xi + Bi x3−i + Ci u i ,

(4.134)

where  Ai =

g l

     0 1 0 0 0 2 , Bi = ka 2 , Ci = 1 . − mkai l 2 0 0 mi l 2 mi l 2

(4.135)

In the numerical simulation, consider g/l = 1, m i l 2 = 1 and ka 2 = 0.02. We design a decentralized control law for the xi -subsystem (i = 1, 2) in form of i ), k ∈ Si . u i (t) = −[1.38, 1]xi , t ∈ [tki , tk+1

(4.136)

  1.75 1 Define Vi (xi ) = P xi for xi -subsystem with P = . It can be verified 1 1.5 j that Assumption 4.3 hold with αi = 0.61, αi = 2.63, ki = 0.7, ki = 90 and αi = 0.02. Based on Corollary 4.1, ϕi and φi are chosen as ϕi = 0.05 and φi = 0.02. Then, the event triggers for the subsystems are designed as xiT

102

4 Dynamic Event Triggers 3

x

system state

11

x12

2

x21

1

x

22

0 -1 -2 -3

0

2

4

6

8

10 time

12

14

16

18

20

Fig. 4.10 The state trajectories of the coupled inverted pendulums (4.133) with the event trigger (4.137) in Example 4.4

Fig. 4.11 The control input of the coupled inverted pendulums (4.133) with the event trigger (4.137) in Example 4.4

  i tk+1 = inf t > tki : |xi (tki ) − xi (t)| ≥ μi (t) , k ∈ Si

(4.137)

where μi (t) = 0.05ηi (t), η˙i (t) = −0.02ηi (t) for all t ≥ 0. The simulation result with initial states x1 (0) = [3, −1.5]T , x2 (0) = [−2.5, 2]T and ηi (0) = 0.8 is shown in Figs. 4.10, 4.11 and 4.12. It is verified that the objective of decentralized event-triggered stabilization is achieved for the coupled inverted pendulums (4.133).

4.3 Notes

103

Fig. 4.12 The inter-sampling intervals of the coupled inverted pendulums (4.133) with the event trigger (4.137) in Example 4.4

4.3 Notes The commonly used event trigger is designed such that the sampling error is bounded by a specific threshold signal (which depends on the real-time system state) for convergence of the system state; see, e.g., [86, 148, 165, 245] and Chap. 3. To improve event-triggered control performance, dynamic terms have been introduced to the event triggers. For example, [77, 230] employs an exponentially converging threshold signal for distributed event-triggered control of linear systems. Reference [73] has proposed a Lyapunov-based design for dynamically event-triggered control of nonlinear systems with state feedback. The performance of dynamic event triggers has also been discussed in the framework of hybrid systems [54, 220]. This chapter shows the validity of dynamic event triggers for event-triggered control of nonlinear systems subject to dynamic uncertainties. Due to the nonlinear dynamics, the design is nontrivial. Indeed, an elementary example in this chapter shows that the threshold signal should be carefully chosen, and exponential converging threshold signals that are commonly used for linear systems may not be valid for nonlinear systems. Decentralized control has been widely studied in the controls community, with various engineering applications including power systems, transportation networks, chemical engineering and telecommunication networks. Among the existing results, significant contributions have been made to the development of decentralized nonlinear control theory; see, e.g., [121, 232, 242, 257, 263, 273] and numerous references therein. Recently, the trend of implementing embedded controllers under communication and computation constraints motivates the study of decentralized eventtriggered control. In [55, 65, 67, 137, 271], decentralized event-triggered controllers are designed for large-scale linear systems. The event triggers were designed in the context of decentralized nonlinear control in previous works [54, 188, 190, 247, 278]. In [54, 278], a positive minimum of the inter-sampling intervals is explicitly

104

4 Dynamic Event Triggers

enforced for decentralized event-triggered control of nonlinear systems. In [190], the sampling events of all the subsystems are triggered synchronously. Reference [188] relaxes the requirement of synchronous triggering and does not need continuous-time monitoring of the real-time system state. However, in [188], the event triggers for all subsystems share one common threshold signal, which may limit the decentralized implementation. Reference [230] studies event-based broadcasting for average consensus of multi-agents with linear dynamics. In the problem setting in this chapter, each subsystem of the decentralized eventtriggered control system is controlled by a local controller, using only the feedback information from its corresponding subsystem. That is, there is no real-time information exchange between the controllers. We focus on the fundamental problem of designing decentralized event triggers such that infinitely fast sampling is avoided, and at the same time, global asymptotic regulation is achieved. The problem is solved for a class of large-scale nonlinear systems based on decentralized dynamic event triggers.

Chapter 5

Event-Triggered Input-to-State Stabilization

This chapter takes a step forward toward solving the event-triggered control problem for the nonlinear systems subject to both dynamic uncertainties and external disturbances. To address this problem, an estimation of the influence of the external disturbances is introduced to the event trigger, which, together with the measured system state, determines the threshold signal. It is proved that the new event trigger guarantees the avoidance of infinitely fast sampling, and the closed-loop event-triggered system is input-to-state stable.

5.1 Robust Event-Triggered Control of Nonlinear Systems Under a Global Sector-Bound Condition In this section, we consider the event-triggered control problem for the nonlinear system (3.37) with the control law (3.2). The closed-loop event-triggered system is in the form of (3.38). It is assumed that the system satisfies a global sector-bound condition. d Assumption 5.1 Consider the system (3.38). There exist constants L gx , L w g , Lg ≥ 0 such that d |g(x, w, d)| ≤ L gx |x| + L w g |w| + L g |d|

(5.1)

holds for all x ∈ Rn , w ∈ Rn and d ∈ Rn d . Corresponding to Assumption 3.3, we also assume that the system (3.37) is inputto-state stabilized by the control law (3.2).

© Huazhong University of Science and Technology Press, Wuhan and Springer Nature Singapore Pte Ltd. 2020 T. Liu et al., Robust Event-Triggered Control of Nonlinear Systems, Research on Intelligent Manufacturing, https://doi.org/10.1007/978-981-15-5013-3_5

105

106

5 Event-Triggered Input-to-State Stabilization

Assumption 5.2 The system (3.38) is ISS with w and d as the inputs, and admits a continuously differentiable ISS-Lyapunov function Vx : Rn → R+ satisfying the following conditions: 1. there exist constants αx , αx > 0 such that αx |x|2 ≤ Vx (x) ≤ αx |x|2 , ∀x;

(5.2)

2. there exist constants αx , k xw , k xd > 0 such that   Vx (x) ≥ max k xw |w|2 , k xd |d|2 ⇒∇Vx (x)g(x, w, d) ≤ −αx Vx (x)

(5.3)

for all x, w and d. It should be noted that even if f does not satisfy the global sector bound condition, for some nonlinear systems, such as feedback linearizable systems, g can still be made to satisfy Assumption 5.1 by appropriately choosing the control law κ in (3.2). Our objective is to develop a new event trigger, which fully takes into consideration the external disturbance, such that the following two objectives are achieved: Objective 1: There exists a constant TΔ > 0 such that tk+1 − tk ≥ TΔ

(5.4)

holds for all k ∈ S. Objective 2: The closed-loop event-triggered system (3.38) is ISS with respect to the external disturbance d. In this section, the objectives 1 and 2 are achieved by introducing a variable to the event trigger, which, together with the measured system state, determines the threshold signal. In particular, the event trigger is in the form of tk+1 = inf{t > tk : |w(t)| ≥ max{kwx |x(t)|, kwd ς(t)}},

(5.5)

where kwx and kwd are positive constants to be determined later, and ς = max{ς1 , ς2 }

(5.6)

with ς1 , ς2 : R+ → R+ defined as follows. The signal ς1 is generated by the dynamical system ς˙1 (t) = −ϕ(ς1 (t))

(5.7)

with initial condition ς1 (0) > 0, and ϕ being a positive definite and Lipschitz on compact sets. The signal ς2 : R+ → R+ is the new variable employed to handle the external disturbance d.

5.1 Robust Event-Triggered Control of Nonlinear Systems Under …

107

Here, an asymptotically converging, but positive ς1 is employed to guarantee the positiveness of the threshold signal ς, so that the event trigger (5.5) is well-defined for all x and ς, with no need to exclude the case of x = 0 and ς = 0. In the rest of this section, we study the influence of the external disturbance to the system state, and appropriately choose ς2 to estimate the influence. The basic idea is given by Example 5.1. Example 5.1 Consider a first-order linear system x(t) ˙ = x(t) + u(t) + d(t)

(5.8)

where x ∈ R is the state, u ∈ R is the control input, and d ∈ R represents the external disturbance. The event-triggered control law is designed as u(t) = −2x(tk ), t ∈ [tk , tk+1 ), k ∈ S.

(5.9)

By substituting (5.9) into (5.8), we have x(t) ˙ = x(t) − 2x(tk ) + d(t)

(5.10)

for tk ≤ t < tk+1 , k ∈ S. To handle the unknown external disturbance d, an estimation on the influence of external disturbances ς2 is defined as  0, for t = t0 = 0; (5.11) ς2 (t) = η(t) , for t ∈ (tk , tk+1 ], k ∈ S. t−tk where η(t) =

1 e(t−tk )

max{|x(t) − x(tk )| − (e(t−tk ) − 1)|x(tk )|, 0}.

(5.12)

Then, the system state x satisfies |x(t) − x(tk )| ≤ (e(t−tk ) − 1)|x(tk )| + (t − tk )e(t−tk ) ς2 (t) ≤ (e(t−tk ) − 1)|x(tk )| + (t − tk )e(t−tk ) ς(t) ≤ 2 max{e(t−tk ) − 1, (t − tk )e(t−tk ) } max {|x(tk )|, ς(t)}

(5.13)

for tk < t ≤ tk+1 , k ∈ S. Note that, event trigger (5.5) guarantees that |x(t) − x(tk )| ≤ max{kwx |x(t)|, kwd ς(t)} holds for tk < t ≤ tk+1 , k ∈ S. Also, condition (5.14) is satisfied if

(5.14)

108

5 Event-Triggered Input-to-State Stabilization

 kwx d max {|x(tk )|, ς(t)} , k 1 + kwx w   kwx d |x(t )|, k ς(t) . ≤ max k w 1 + kwx 

|x(t) − x(tk )| ≤ min

(5.15)

This is guaranteed by Lemma B.3. With (5.13) and (5.15) satisfied, it can be proved that tk+1 is not less than the largest t such that 2 max{e(τ −tk ) − 1, (τ − tk )e(τ −tk ) } max {|x(tk )|, ς(τ )}   kwx d max {|x(tk )|, ς(τ )} , k ≤ min 1 + kwx w

(5.16)

holds for all τ ∈ [tk , t]. The definition of ς1 in (5.7) implies that ς is nonzero, and thus max{|x(tk )|, ς} is nonzero. Define TΔ satisfying  2 max e

TΔ

− 1, TΔ e

TΔ





kwx = min , kd 1 + kwx w

 (5.17)

Then, we have inf {tk+1 − tk } ≥ TΔ > 0.

(5.18)

k∈S

For tk ≤ t < tk+1 , k ∈ S, the solution of the system (5.10) with initial condition x(tk ) is x(t) = (2 − e(t−tk ) )x(tk ) +



t

e(t−τ ) d(τ ) dτ ,

(5.19)

tk

and thus, |x(t) − x(tk )| ≤ (e(t−tk ) − 1)|x(tk )| + e(t−tk )



t

|d(τ )| dτ .

(5.20)

tk

Due to the continuity of x(t) with respect to t, (5.20) also holds for t = tk+1 . From (5.11) and (5.20), we have

t ς2 (t) ≤

tk

|d(τ )| dτ t − tk

= avg |d(τ )|.

(5.21)

tk ≤τ ≤t

Here, avg represents the average function and is defined by avg ψ(t) = dt/(b − a).

a≤t≤b

b a

ψ(t)

5.1 Robust Event-Triggered Control of Nonlinear Systems Under …

109

d — d¯ ς2 · · ·

t0

t1

t2

¯ = Fig. 5.1 The signal ς2 for the event trigger, where d(t) (tk , tk+1 ], k ∈ S

d(t)

t

t tk

|d(τ )| dτ /(t − tk ) for t ∈

x(t) ˙ = −x(t) − 2w(t) + d(t)

w(t)

x(t) x d |w(t)| ≤ max{kw |x(t)|, kw ς(t)}

ς2 (t) ς2 (t) ≤ avg |d(τ )|

d(t)

tk ≤τ ≤t

Fig. 5.2 An interconnected system

Figure 5.1 shows the basic idea of estimating the influence of the external disturbance d. By replacing x(tk ) in (5.10) with x(t) + w(t), we have x(t) ˙ = −x(t) − 2w(t) + d(t)

(5.22)

which is ISS with w and d as the inputs. As shown in Fig. 5.2, the closed-loop eventtriggered system can be considered as an interconnection of (5.22) and (5.14). The objective 2 can be achieved by choosing kwx to satisfy the small-gain condition. Figure 5.3 shows the structure of the proposed event-triggered control system. Clearly, if d ≡ 0, then ς2 ≡ 0. In this case, the event trigger (5.5) can be reduced to the existing designs that depends only on the measured state; see Chap. 3. In the case of d = 0, the new variable ς2 is able to estimate the influence of d and is useful in avoiding infinitely fast sampling.

110

5 Event-Triggered Input-to-State Stabilization

x(tk )

u(t) = −2x(tk )

u(t)

ς2 (t)

S

d(t)

x(t) ˙ = x(t) + u(t) + d(t)

E

x(t)

Fig. 5.3 The block diagram of the closed-loop event-triggered system (5.8)–(5.9), where E represents the estimation of the influence of the external disturbance (5.11)

Based on the idea in Example 5.1, this section proposes a new class of event triggers for the system (3.38) without assuming an a priori known upper bound or the convergence of |d|. An estimation of the influence of the external disturbance d is defined as ⎧ for t = t0 = 0; ⎪ ⎨0, ς2 (t) = (t−tk )L1dg E(t,tk ) max{|x(t) − x(tk )| (5.23) ⎪ ⎩ for t ∈ (tk , tk+1 ], k ∈ S. −L g (E(t, tk ) − 1)|x(tk )|, 0}, w

where E(t, tk ) = e(L g +L g )(t−tk ) and L g = L gx /(L gx + L w g ). For practical implementation of (5.23), one might be concerned about the existence of ς2 (t) when t → tk+ . This is guaranteed by the differentiability of |x(t) − x(tk )| − L g (E(t, tk ) − 1)|x(tk )|. Indeed, it can be checked that: a) lim+ max{|x(t) − x

t→tk

x(tk )| − L g (E(t, tk ) − 1)|x(tk )|, 0} = 0 and lim+ (t − tk )L dg E(t, tk ) = 0; b) The right t→tk

derivative of the terms max{|x(t) − x(tk )| − L g (E(t, tk ) − 1)|x(tk )|, 0} and (t − tk )L dg E(t, tk ) at tk exists; c) dtd ((t − tk )L dg E(t, tk )) = 0 when t is close to tk . Then, the existence of lim+ ς2 (t) can be proved by readily applying L’Hôpital’s Rule [107, t→tk

Sect. 4.5]. Theorem 5.1 gives the main result on input-to-state stabilization of the closed-loop event-triggered system (3.38) satisfying the global sector bound condition. Theorem 5.1 Consider the system (3.38). Under Assumptions 5.1 and 5.2, by choosing the constant kwx satisfying  0 < kwx
0 satisfies (5.24) and kwd > 0 is a designable constant. In this case, for t ∈ [tk + TΔ , tk+1 ], k ∈ S, an estimation ς2 can still be made as (5.23). The basic idea of adding TΔ to the event trigger is shown in Fig. 5.4. Theorem 5.2 Consider the closed-loop event-triggered system (3.38). Under Assumptions 5.1 and 5.2, by choosing constant kwx > 0 satisfying (5.24) and constant TΔ satisfying ϕ(TΔ ) = ε

(5.43)

xw with ϕ(TΔ ) = 2 max{L g e L g TΔ − L g , L dg e L g TΔ TΔ }, L g = L gx /(L gx + L w g ), L g = x w x x d d L g + L g and ε = min{kw /(1 + kw ), kw }, and choosing constant kw to be positive, the objectives 1 and 2 are achievable. xw

xw

We use a simulation example to verify the theoretical results. Consider the system given in Example 5.1. We choose kwx = 0.35, kwd = 1 and TΔ = 0.1. The function ϕ defined in (5.7) is chosen as ϕ(s) = s for s ∈ R+ . The simulation result with x(0) = 2 and ς1 (0) = 0.4 shown in Figs. 5.5, 5.6, 5.7 and 5.8 verifies the theoretical result.

5.2 Robust Event-Triggered Control of Nonlinear Systems …

115

4 3

disturbance

2 1 0 -1 -2 -3 -4

0

5

10

15

time

Fig. 5.5 The external disturbance used for the simulation of the event trigger (5.42) in Example 5.1 2.5

system state

2 1.5 1 0.5 0 -0.5

0

5

10

15

time

Fig. 5.6 The state trajectory with the event trigger (5.42) in Example 5.1

5.2 Robust Event-Triggered Control of Nonlinear Systems without a Global Sector-Bound Condition By removing the global sector-bound assumption on g, this section extends the design in Sect. 5.1 to more general nonlinear systems. Assumptions 5.3 and 5.4 are made to replace Assumptions 5.1 and 5.2. Assumption 5.3 Consider the system (3.38). There exists a positive, nondecreasing function L ag : R+ → R+ such that |g(x, w, d)| ≤ L ag (max{|x|, |w|, |d|})(max{|x|, |w|, |d|})

(5.44)

116

5 Event-Triggered Input-to-State Stabilization 0.5 0

control input

-0.5 -1 -1.5 -2 -2.5 -3 -3.5 -4

0

5

10

15

time

Fig. 5.7 The control input with the event trigger (5.42) in Example 5.1

inter-sampling intervals

100

10-1

10

-2

10

-3

10

-4

inter-sampling intervals T =0.1 0

5

10

15

time

Fig. 5.8 The inter-sampling intervals with the event trigger (5.42) in Example 5.1

holds for all x ∈ Rn , w ∈ Rn and d ∈ Rn d . Also, there exist a positive, nondecreasing function L bg : R+ → R+ and a class K∞ function ρ such that |g(x, w,d) − g(x, w, 0)| ≤ L bg (max{|x|, |w|})ρ(|d|)

(5.45)

holds for all x ∈ Rn , w ∈ Rn and d ∈ Rn d . If g is locally Lipschitz and g(0, 0, 0) = 0, then one can always find L ag , L bg and ρ to satisfy Assumption 5.3. Assumption 5.4 The system (3.38) is ISS with w and d as the inputs, and admits an ISS-Lyapunov function Vx : Rn → R+ satisfying the following conditions:

5.2 Robust Event-Triggered Control of Nonlinear Systems …

117

1. there exist αx , αx ∈ K∞ such that αx (|x|) ≤ Vx (x) ≤ αx (|x|), ∀x;

(5.46)

2. there exist γxw and γxd in K∞ such that Vx (x) ≥ max{γxw (|w|), γxd (|d|)} ⇒∇Vx (x)g(x, w, d) ≤ −αx (Vx (x))

(5.47)

for all x, w and d, where αx is a positive definite function. The event trigger proposed in this section is a nonlinear extension of the one in Sect. 5.1: tk+1 = inf{t > tk : |w(t)| ≥ max{γwx (|x(t)|), γwd (ς(t))}},

(5.48)

where γwx and γwd in K∞ with (γwx )−1 and (γwd )−1 being Lipschitz on compact sets are to be designed later, ς is defined in (5.6) with ς1 being generated by (5.7) and ς2 being designed as

ς2 (t) =

⎧ ⎪ ⎨0,

for t = t0 = 0;

1 max{|x(τ ) − x(tk )|ttk (t−tk )L bg (M(t,tk )) ⎪ ⎩ −(t − tk )L ag (M(t, tk ))(M(t, tk )), 0},

(5.49) for t ∈ (tk , tk+1 ], k ∈ S.

where M(t, tk ) := max{|x(τ )|ttk , |w(τ )|ttk } for k ∈ S, and |A(τ )|ab denotes the supremum of any function A over all τ ∈ [a, b] for any real values b ≥ a ≥ 0. With d considered as the input of the controlled x-subsystem, the signal ς2 is designed to estimate its influence. The estimation is based on a continuous comparison between the measured system state and the state of the disturbance-free system. Theorem 5.3 presents the main result on the event-triggered input-to-state stabilization of nonlinear systems without assuming the global sector-bound condition. Theorem 5.3 Under Assumptions 5.3 and 5.4, by choosing function γwx ∈ K∞ with (γwx )−1 being Lipschitz on compact sets and satisfying w x α−1 x ◦ γx ◦ γw < Id,

(5.50)

and choosing γwd ∈ K∞ with (γwd )−1 being Lipschitz on compact sets, the closed-loop event-triggered system (3.38) with event trigger composed of (5.48), (5.6), (5.7) and (5.49) is ISS with d as the input. Proof In accordance with the proof of Theorem 5.1, we still first prove the existence of a positive constant TΔ such that inf k∈S {tk+1 − tk } ≥ TΔ , k ∈ S, and then prove the ISS of the closed-loop event-triggered system (3.38). (a) Existence of a Positive Lower Bound of Inter-Sampling Intervals

118

5 Event-Triggered Input-to-State Stabilization

Define     = inf t > tk : |w(τ )|ttk ≥ max γˆ wx (|x(tk )|), γwd (ς(t)) tk+1

(5.51)

where γˆ wx (s) = γwx ◦ (Id + γwx )−1 (s) for s ∈ R+ .   ≤ tk+1 and tk+1 − tk is lower bounded by a The basic idea is to show that tk+1 positive constant. Indeed, with Lemma B.3, it can be proved that |x(τ ) − x(tk )|ttk ≤     max γˆ wx (|x(tk )|), γwd (ς(t)) implies |x(t) − x(tk )| ≤ max γwx (|x(t)|), γwd (ς(t)) . By also using the definition of w in (3.3), we have  ≤ tk+1 . tk+1

(5.52)

 − tk . Now, we analyze the lower bound of tk+1 From the definition of w in (3.3), it follows that

|x(τ )|ttk = |x(tk ) − w(τ )|ttk ≤ |x(tk )| + |w(τ )|ttk

(5.53)

 , k ∈ S. Then, by also using (5.49), we have for tk ≤ t < tk+1

   |w(τ )|ttk ≤ (t − tk )L ag max |x(tk )| + |w(τ )|ttk , |w(τ )|ttk   × max |x(tk )| + |w(τ )|ttk , |w(τ )|ttk    + (t − tk )L bg max |x(tk )| + |w(τ )|ttk , |w(τ )|ttk ς2 (t)       =: (t − tk ) Lˆ ag max |x(tk )|, |w(τ )|ttk × max |x(tk )|, |w(τ )|ttk     + Lˆ bg max |x(tk )|, |w(τ )|ttk ς2 (t) (5.54)  , k ∈ S. for tk ≤ t < tk+1 With Lemma B.4, for any specific γˆ wx , γwd ∈ K∞ with (γˆ wx )−1 , (γwd )−1 being Lipschitz on compact sets, one can find continuous, positive-valued and nondecreasing functions L˜ ag and L cg such that

     Lˆ ag max |x(tk )|, |w(τ )|ttk × max |x(tk )|, |w(τ )|ttk      ≤ L˜ ag max |x(tk )|, |w(τ )|ttk × max γˆ wx (|x(tk )|), |w(τ )|ttk

(5.55)

and ς2 (t) ≤ L cg (ς2 (t))γwd (ς2 (t)).

(5.56)

By substituting (5.55) and (5.56) into the right-hand side of (5.54), we have

5.2 Robust Event-Triggered Control of Nonlinear Systems …

119

|w(τ )|ttk

      ≤ (t − tk ) L˜ ag max |x(tk )|, |w(τ )|ttk × max γˆ wx (|x(tk )|), γwd (ς2 (t))     + Lˆ bg max |x(tk )|, |w(τ )|ttk × L cg (ς2 (t))γwd (ς2 (t))     =: (t − tk ) L g max |x(tk )|, |w(τ )|ttk , ς2 (t)   × max γˆ wx (|x(tk )|), γwd (ς2 (t)) (5.57)

 , k ∈ S, where L g is a continuous, positive-valued and nondecreasfor tk ≤ t < tk+1 ing function. To replace ς2 with the external disturbance d, the next step is to find the relationship of ς2 and d. Along each trajectory of the system (3.38), we have

|w(τ )|ttk

 τ t    = g(x(τ ), w(τ ), d(τ )) dτ  tk tk t   g(x(τ ), w(τ ), 0) ≤ tk   + g(x(τ ), w(τ ), d(τ )) − g(x(τ ), w(τ ), 0) dτ

(5.58)

for t ≥ tk , k ∈ S. By substituting properties (5.44) and (5.44) in Assumption 5.3 into the right-hand side of (5.58), we have      |w(τ )|ttk ≤(t − tk )L ag max |x(τ )|ttk , |w(τ )|ttk × max |x(τ )|ttk , |w(τ )|ttk t    b t t + L g max |x(τ )|tk , |w(τ )|tk × ρ(|d(τ )|) dτ (5.59) tk

for t ≥ tk , k ∈ S. From (5.49) and (5.59), we have |ς2 (τ )|ttk

  τ  ρ(|d(τ )|) dτ t     tk ≤  ≤ ρ |d(τ )|ttk   τ − tk

(5.60)

tk

 , k ∈ S, and thus, with ς2 (0) = 0 given by (5.49), we have for tk < t ≤ tk+1

  |ς2 (t)| ≤ ρ d [0,t] .

(5.61) 

Under Assumption 5.5, by using [240, Lemma 2.14], one can find ψ, γxw and γxd of class K such that



120

5 Event-Triggered Input-to-State Stabilization

    |x(t)| ≤ ψ(|x(0)|) + max γxw ( w [0,t] ), γxd ( d [0,t] ) =: Δ[0,t]

(5.62)

for 0 ≤ t < Tmax . By substituting (5.61) and (5.62) into (5.57), one can find a continuous, positivevalued and nondecreasing function Lˆ g such that    |w(τ )|ttk ≤(t − tk ) Lˆ g max Δ[0,t] , d [0,t]   × max γˆ wx (|x(tk )|), γwd (ς(t))

(5.63)

 , k ∈ S. Here, due to the continuity of both sides, (5.63) also holds for tk ≤ t ≤ tk+1  for t = tk+1 . With event trigger (5.51), the upper bound of the sampling error w is restricted to satisfy

  |w(τ )|ttk ≤ max γˆ wx (|x(tk )|), γwd (ς(t))

(5.64)

  , k ∈ S. Thus, tk+1 is no less than the largest t such that for tk < t ≤ tk+1

     (τ − tk ) Lˆ g max Δ[0,t] , d [0,t] × max γˆ wx (|x(tk )|), γwd (ς(τ ))   ≤ max γˆ wx (|x(tk )|), γwd (ς(τ ))

(5.65)

holds for all τ ∈ [tk , t]. The definition of ς1 in (5.7) implies that ς is nonzero, and thus  ))} is nonzero. Define max{γˆ wx (|x(tk )|), γwd (ς(tk+1 TΔ∗ =

1    . ˆL g max Δ[0,t  ] , d [0,t  ] k+1 k+1

(5.66)

Then, we have  − tk ≥ TΔ∗ . tk+1

(5.67)

By using (5.52), since Lˆ g is a positive-valued function, the positive lower bound of tk+1 − tk is guaranteed. Figure 5.9 shows the relation betweenthe variables described  by (5.63)–(5.65), where Θ1 (t) := |w(τ )|ttk , Θ2 (t) := max γˆ wx (|x(tk )|), γwd (ς(t))      and Θ3 (t) := (t − tk ) Lˆ g max Δ[0,t] , d [0,t] max γˆ wx (|x(tk )|), γwd (ς(t)) for t ≥ tk , k ∈ S.. (b) ISS Property for the Closed-Loop System The event trigger (5.48) implies that   |w(t)| ≤ max γwx (|x(t)|), γwd (ς(t))

(5.68)

5.2 Robust Event-Triggered Control of Nonlinear Systems …

121

Θ3 (t) Θ1 (t) Θ2 (t) 0

tk+1

∗ tk + T Δ

tk

t

Fig. 5.9 An illustration of relation between Θ1 (t), Θ2 (t) and Θ3 (t)

for t ∈ (tk , tk+1 ], k ∈ S. With the satisfaction of (5.50) and (5.68), (5.47) guarantees   Vx (x(t)) ≥ max γxw ◦ γwd (ς(t)), γxd (|d(t)|)

⇒D + Vx (x(t))g(x(t), w(t), d(t)) ≤ −αx (Vx (x(t)))

(5.69)

for t ∈ (tk , tk+1 ], k ∈ S. From (5.49) and (5.59), we have  ς(t) ≤ max ς1 (t),

t tk

ρ(|d(τ )|) dτ



t − tk   = max ς1 (t), avg ρ(|d(τ )|)

(5.70)

tk ≤τ ≤t

for t ∈ (tk , tk+1 ], k ∈ S. Substituting (5.70) into the right-hand side of (5.69) yields 

 Vx (x(t)) ≥ max

γxd (|d(t)|), γ¯ xz (ς1 (t)), γ¯ xz

 avg ρ(|d(τ )|)

tk ≤τ ≤t

⇒D + Vx (x(t))g(x(t), z(t), w(t), d(t)) ≤ −αx (Vx (x(t)))

(5.71)

for t ∈ (tk , tk+1 ], k ∈ S, where γ¯ xz (s) = γxw ◦ γwd (s) for s ∈ R+ . Now, we consider the three cases discussed in Chap. 3: (1) S = Z+ and limk→∞ tk < ∞; (2) S = Z+ and limk→∞ tk = ∞; (3) S = {0, . . . , k ∗ } with k ∗ ∈ Z+ . (a) Suppose that Case (1) happens. Then, we have inf k∈S {tk+1 − tk } = 0, which contradicts with (5.4).Thus, Case (1) is impossible. (b) In Case (2), we have k∈S [tk , tk+1 ) = [0, ∞), which means that (5.61), (5.62) and (5.71) hold for all t ∈ [0, ∞). (c) In Case (3), S is a finite set {0, . . . , k ∗ } with k ∗ ∈ Z+ . Then, there exists a T ∗ < ∞ such that maxi∈S {tk } ≤ T ∗ < Tmax . In this case, the closed-loop event-triggered

122

5 Event-Triggered Input-to-State Stabilization

system only involves continuous-time dynamics for t ≥ T ∗ . Recall that (5.71) holds for all t ∈ [0, Tmax ). Then, Tmax = ∞ is proved due to causality. Based on the discussions above, with (5.71) satisfied, one can find βx ∈ KL and χςx1 , χdx ∈ K such that   |x(t)| ≤ max βx (|x(0)|, t), χςx1 ( ς1 ∞ ), χdx ( d ∞ )

(5.72)

for all t ≥ 0. The proof is similar to the one of [240, Remark 2.4]. Then, the system composed of (3.38), (5.48), (5.6) and (5.49) is ISS with x as the state and (d, ς1 ) as the inputs. Also, the system composed of(3.38), (5.48), (5.6), (5.7) and (5.49) can be considered as a cascade connection of the ISS system (5.6) and the asymptotically stable system (5.7), and thus it is ISS with (x, ς1 ) as the state and d as the input [138, Lemma 4.7].  We employ a simple example to illustrate the effectiveness of the obtained results in this section. Example 5.2 Consider the nonlinear system x(t) ˙ = x 2 (t) + u(t) + d(t) u(t) = −(9|x(tk )| + 9)x(tk ), t ∈ [tk , tk+1 ), k ∈ S

(5.73) (5.74)

where x ∈ R is the system state, u ∈ R is the control input, and d ∈ R represents the external disturbance. Define Vx (x) = 21 x 2 . It can be verified that Assumptions 5.3 and 5.4 hold with αx (s) = αx (s) = 21 s 2 , γxw (s) = 2s 2 , γxd (s) = s, αx (s) = s, L ag (s) = 37s + 19, L bg (s) ≡ 1 and ρ(s) = s for s ∈ R+ . By using Theorem 5.3, γwx and γwd defined in (5.48) are chosen as γwx (s) = 0.45s and γwd (s) = s for s ∈ R+ , and ϕ defined in (5.7) is chosen as ϕ(s) = s for s ∈ R+ . Two event triggers based on the existing designs are employed for comparison:  = inf{t > tk : |w(k)| ≥ 0.45|x(k)| + }, tk+1 T tk+1

= inf{t ≥

tkT

+ T : |w(k)| ≥ 0.45|x(k)|}.

(5.75) (5.76)

In the numerical simulation, we employ a bounded external disturbance which ultimately converges to zero, to check both the robustness and converging-input converging-state properties of the closed-loop event-triggered systems. Figures 5.10, 5.11, 5.12 and 5.13 show the simulation results of the three event triggers with the same initial state x(0) = 10. It can be checked that most of the sampling intervals of event trigger (5.48) with ς1 (0) = 0.4 is larger than the other two event triggers. Also, event trigger (5.75) with  = 0.015 does not achieve asymptotic convergence even though the disturbance converges. Table 5.1 uses the ultimate bound of x (UB-x), minimum inter-sampling intervals (MISI) and average inter-sampling intervals (AISI) to characterize the performance

5.2 Robust Event-Triggered Control of Nonlinear Systems …

123

2 1.5

disturbance

1 0.5 0 -0.5 -1 -1.5 -2

0

5

10

15

time

Fig. 5.10 The external disturbance used for the simulation of the event triggers (5.48), (5.75) and (5.76) in Example 5.2 0.2

event trigger with

0.15 0.1 system state

1

(0)=0.4

event trigger with =0.015 event trigger with T=0.001

0.05 0 -0.05 -0.1 -0.15 -0.2

0

5

10

15

time

Fig. 5.11 The state trajectories with the event triggers (5.48), (5.75) and (5.76) in Example 5.2

of the event triggers. Regarding both UB-x and AISI, the advantage of the event trigger (5.48) over the event trigger (5.76) is quite clear. Moreover, when x(0) or TΔ is large, event trigger (5.76) may even cause divergence of x. The event trigger (5.48) with ς1 (0) = 0.4 and the event trigger (5.75) with  = 0.015 have the same MISI and AISI, but event trigger (5.75) with  = 0.015 does not achieve asymptotic convergence (UB-x=0.015). The simulation result also shows that larger  leads to larger UB-x.

124

5 Event-Triggered Input-to-State Stabilization 2

event trigger with

(0)=0.4

event trigger with =0.015 event trigger with T=0.001

1 control input

1

0 -1 -2 -3 -4

0

5

10

15

time

Fig. 5.12 The control inputs with the event triggers (5.48), (5.75) and (5.76) in Example 5.2

inter-sampling intervals

10

10

0

event trigger with

(0)=0.4

1

event trigger with =0.015 event trigger with T=0.001

-1

10-2

10-3

10

-4

0

5

10

15

time

Fig. 5.13 The inter-sampling intervals with the event triggers (5.48), (5.75) and (5.76) in Example 5.2

5.3 Robustness Analysis in the Presence of Dynamic Uncertainties Based on the results with full-state feedback in Sects. 5.1 and 5.2, we study robust event-triggered control with partial-state feedback for a class of nonlinear systems by considering the dynamics of the unmeasured state as dynamic uncertainty. Namely, consider

5.3 Robustness Analysis in the Presence of Dynamic Uncertainties

125

Table 5.1 Ultimate bound of x (UB-x), minimum inter-sampling intervals (MISI) and average inter-sampling intervals (AISI) of event-triggered sampling with event triggers (5.48), (5.75) and (5.76), and simulation time 15 s Event trigger UB-x MISI AISI (5.48) ς1 (0) = 0.4, x(0) = 10 (5.48) ς1 (0) = 0.4, x(0) = 20 (5.48) ς1 (0) = 0.4, x(0) = 45 (5.75)  = 0.015, x(0) = 10 (5.75)  = 0.015, x(0) = 20 (5.75)  = 0.015, x(0) = 45 (5.75)  = 0.050, x(0) = 10 (5.75)  = 0.100, x(0) = 10 (5.76) T = 0.001, x(0) = 10 (5.76) T = 0.005, x(0) = 10 (5.76) T = 0.005, x(0) = 45 (5.76) T = 0.018, x(0) = 20 (5.76) T = 0.018, x(0) = 10

≤0.005 ≤0.005 ≤0.005 0.015 0.015 0.015 0.040 0.070 ≤0.005 ≤0.005 ∞ ∞ ≤0.005

0.0048 0.0026 0.0013 0.0048 0.0026 0.0013 0.0048 0.0048 0.0010 0.0050 – – 0.0180

0.064 0.061 0.061 0.064 0.059 0.063 0.140 0.200 0.010 0.017 – – 0.032

z˙ (t) = h(z(t), x(t), d(t)) x(t) ˙ = f (x(t), z(t), u(t), d(t))

(5.77) (5.78)

u(t) = κ(x(tk )), t ∈ [tk , tk+1 ), k ∈ S ⊆ Z+

(5.79)

where [z T , x T ]T with z ∈ Rn z and x ∈ Rn is the state, u ∈ Rm is the control input, d ∈ Rn d represents the unknown external disturbance, h : Rn z × Rn × Rn d → Rn z and f : Rn × Rn z × Rm × Rn d → Rn represent the system dynamics with h(0, 0, 0) = 0 and f (0, 0, 0, 0) = 0, and κ : Rn → Rm represents the feedback control law based on event-triggered data-sampling. In our problem setting, x is available to feedback design, and z cannot be measured. In the system (5.77)–(5.78), the z-subsystem is used to represent unmodeled or unmeasured dynamics, which may cover zero dynamics, observation error dynamics, and sensor processes in practical control systems [122, 143]. The z-subsystem is considered as a perturbation to the x-subsystem. The x-subsystem with z = 0 is usually called the nominal system. With w defined in (3.3), the controlled x-subsystem can be rewritten as x(t) ˙ = f (x(t), z(t), κ(x(t) + w(t)), d(t)) =: g(x(t), z(t), w(t), d(t)). We assume that g is locally Lipschitz in x and z and continuous in w and d.

(5.80)

126

5 Event-Triggered Input-to-State Stabilization

d(t)

z(t) ˙ = h(z(t), x(t), d(t))

z(t)

x(t) x(t) ˙ = g(x(t), z(t), w(t), d(t))

d(t)

x(t) event trigger

x(tk ) +

w(t) −

Fig. 5.14 The block diagram of the closed-loop event-triggered system

The closed-loop event-triggered system can be considered as the interconnection of the unmeasured z-subsystem (5.77), the controlled x-subsystem (5.80) and the event trigger, as shown in Fig. 5.14. Assumption 5.5 is made on the ISS of the z-subsystem and the x-subsystem. Assumption 5.5 Both the z-subsystem (5.77) and the controlled x-subsystem (5.80) are ISS, and admit continuously differentiable ISS-Lyapunov functions Vz : Rn z → R+ and Vx : Rn → R+ such that 1. Vz satisfies the following conditions: (a) there exist αz , αz ∈ K∞ such that for all z, αz (|z|) ≤ Vz (z) ≤ αz (|z|);

(5.81)

(b) there exist γzx , γzd ∈ K∞ such that for all z, x and d, Vz (z) ≥ max{γzx (Vx (x)), γzd (|d|)} ⇒∇Vz (z)h(z, x, d) ≤ −αz (Vz (z))

(5.82)

where αz is a positive definite function. 2. Vx satisfies the following conditions: (a) there exist αx , αx ∈ K∞ such that αx (|x|) ≤ Vx (x) ≤ αx (|x|), ∀x;

(5.83)

(b) there exist γxz , γxw , γxd ∈ K∞ such that Vx (x) ≥ max{γxz (Vz (z)), γxw (|w|), γxd (|d|)} ⇒∇Vx (x)g(x, z, w, d) ≤ −αx (Vx (x))

(5.84)

5.3 Robustness Analysis in the Presence of Dynamic Uncertainties

127

for all x, z, w and d, where αx is a positive definite function. The following standing assumption is made on g defined in (5.80). Assumption 5.6 There exist positive, nondecreasing functions L ag , L bg : R+ → R+ and a class K∞ function ρ such that |g(x, z, w, d)| ≤ L ag (max{|x|, |z|, |w|, |d|}) × max{|x|, |z|, |w|, |d|}

(5.85)

|g(x, z, w, d) − g(x, 0, w, 0)| ≤

(5.86)

L bg (max{|x|, |w|})ρ(max{|z|, |d|})

hold for all x ∈ Rn , z ∈ Rn z , w ∈ Rn and d ∈ Rn d . This section aims to develop an event trigger for the system (5.77)–(5.79) such that the two objectives in Sect. 5.1 are achieved.

5.3.1 Event Trigger Design The event trigger proposed is defined as    tk+1 = inf t > tk : |w(t)| ≥ max γwx (|x(t)|), γwd (ς(t)) ,

(5.87)

where γwx and γwd are class K∞ functions with (γwx )−1 and (γwd )−1 being Lipschitz on compact sets to be designed later, ς is defined in (5.6) with ς1 being generated by (5.7) and ς2 being designed as ⎧ 0, ⎪ ⎪ ⎨

for t = t0 = 0;  t max |x(τ ) − x(tk )|tk ς2 (t) =  ⎪ ⎪ ⎩−(t − t )L a (M(t, t ))(M(t, t )), 0 , for t ∈ (t , t ], k ∈ S k k k k k+1 g 1 (t−tk )L bg (M(t,tk ))

(5.88)

where L ag and L bg are given by Assumption 5.6, respectively, M(t, tk ) := max{|x(τ )|ttk , |w(τ )|ttk }. The closed-loop event-triggered system with the event trigger composed of (5.87)–(5.88) can be considered as an interconnected system, with the interconnection structure shown in Fig. 5.15. The gains γxz , γxw , γxd , γzx and γzd are given by Assumption 5.5, and the designable gains γwx and γwd appear in the event trigger defined by (5.87). Corollary 5.1 Under Assumptions 5.5 and 5.6, suppose that z and d are piece-wise continuous and bounded on the time-line. By choosing γwx ∈ K∞ such that (γwx )−1 is Lipschitz on compact sets and w x α−1 x ◦ γx ◦ γw < Id,

(5.89)

128

5 Event-Triggered Input-to-State Stabilization d γw

γxd γxz

z

d

x

γzx

γzd

x γw

ς2

ς1

w

γxw

d γw

Fig. 5.15 The closed-loop event-triggered system considered as an interconnected system

and choosing γwd ∈ K∞ such that (γwd )−1 is Lipschitz on compact sets, the system composed of (5.80), (5.87), (5.6) and (5.88) is ISS with x as the state and (z, d, ς1 ) as the inputs. Also, the system composed of (5.80), (5.87), (5.6), (5.7) and (5.88) is ISS with (x, ς1 ) as the state and (z, d) as the inputs. Proof The basic idea of the proof of Corollary 5.1 is similar with that of Theorem 5.3 in Sect. 5.2. A sketch of the proof is given here. First, one can find a positive lower bound of the inter-sampling intervals TΔ such that tk+1 − tk ≥ TΔ

(5.90)

for all k ∈ S. At the same time, the ISS of the event-triggered controlled x-subsystem (5.80) is also proved, i.e.,  Vx (x(t)) ≥ max γxz (Vz (z(t))), γxd (|d(t)|), γ¯ xz (ς1 (t)),  γ¯ xz



 avg ρ1 (|z(τ )|)

tk ≤τ ≤t

, γ¯ xz

 avg ρ1 (|d(τ )|)

tk ≤τ ≤t

⇒D + Vx (x(t))g(x(t), z(t), w(t), d(t)) ≤ −αx (Vx (x(t)))

(5.91)

for t ∈ (tk , tk+1 ], k ∈ S, where γ¯ xz (s) = γxw ◦ γwd (s) and ρ1 (s) = 2ρ(s) for s ∈ R+ . Then, we consider the three cases mentioned in Chap. 3: (1) S = Z+ and limk→∞ tk < ∞; (2) S = Z+ and limk→∞ tk = ∞; (3) S = {0, . . . , k ∗ } with k ∗ ∈ Z+ . (a) Suppose that Case (1) happens. Then, we have inf k∈S {tk+1 − tk } = 0, which contradicts with (5.4).Thus, Case (1) is impossible. (b) In Case (2), we have k∈S [tk , tk+1 ) = [0, ∞), which means that (5.91) hold for all t ∈ [0, ∞). (c) In Case (3), S is a finite set {0, . . . , k ∗ } with k ∗ ∈ Z+ . Then, there exists a T ∗ < ∞ such that maxk∈S {tk } ≤ T ∗ < Tmax . In this case, the closed-loop event-triggered system only involves continuous-time dynamics for t ≥ T ∗ . Recall that (5.91) holds for all t ∈ [0, Tmax ). Then, Tmax = ∞ is proved due to causality. Based on the discussions above, with (5.91) satisfied, one can find βx ∈ KL and χςx1 , χzx , χdx ∈ K such that

5.3 Robustness Analysis in the Presence of Dynamic Uncertainties

  |x(t)| ≤ max βx (|x(0)|, t), χςx1 ( ς1 ∞ ), χzx ( z ∞ ), χdx ( d ∞ )

129

(5.92)

for all t ≥ 0. The proof is similar to the one of [240, Remark 2.4]. Then, the system composed of (5.80), (5.87), (5.6) and (5.88) is ISS with x as the state and (z, d, ς1 ) as the inputs. Also, the system composed of (5.80), (5.87), (5.6), (5.7) and (5.88) can be considered as a cascade connection of the ISS system composed of (5.80), (5.87), (5.6) and (5.88) and the asymptotically stable system (5.7), and thus it is ISS  with (x, ς1 ) as the state and (z, d) as the inputs [138, Lemma 4.7]. From (5.71), it can be recognized that average functions are employed to represent the influence of d to the event-triggered controlled x-system. This is caused by the averaging operation used in (5.49) for estimation of the influence of d. Note that the average of d on [tk , t] introduces time delay. Since the time-intervals for averaging are basically the inter-sampling intervals and the inter-sampling intervals may not have a fixed finite upper bound, the time delay caused by the event trigger does not have a fixed finite upper bound.

5.3.2 Small-Gain Synthesis: ISS of the Closed-Loop Event-Triggered System With Corollary 5.1, the closed-loop event-triggered system is transformed into an interconnection of the x-subsystem and the z-subsystem, both of which admit ISS properties. Intuitively, we may employ the nonlinear small-gain theorem for stability analysis. However, the average-type interconnection leads to time delays that depend on the inter-sampling intervals, and may not have a fixed upper bound. An extension of the existing small-gain theorem is needed to address this issue. Theorem 5.4 gives the main result of this section. Theorem 5.4 Under Assumptions 5.5 and 5.6, the system composed of (5.77), (5.80), (5.87), (5.6) and (5.88) is ISS with (z, x) as the state and (d, ς1 ) as the inputs. Moreover, the system composed of (5.77), (5.80), (5.87), (5.6), (5.7) and (5.88) is ISS with (z, x, ς1 ) as the state and d as the input, if the following conditions are satisfied: 1. γxz and γzx are designed to satisfy γxz ◦ γzx < Id;

(5.93)

2. γwx and γwd defined in (5.87) satisfy (5.89) and z −1 ◦ γxw ◦ γwd < Id ρ1 ◦ α−1 z ◦ γx

where ρ1 is defined in (5.91).

(5.94)

130

5 Event-Triggered Input-to-State Stabilization

avg ρ1 (|z(τ )|), t ∈ (tk , tk+1 ], k ∈ S

tk ≤τ ≤t

x ¯

μz x ¯-system

d ς1

Fig. 5.16 The feedback structure for the system composed of (5.77) and (5.80) with the event trigger composed of (5.87) and (5.88)

Proof The closed-loop event-triggered system is considered as an interconnection of the x-subsystem composed of (5.80), (5.87) and (5.88), and the z-subsystem defined by (5.77). Define μz (t) = avg ρ1 (|z(τ )|) and μd (t) = avg ρ1 (|d(τ )|) for tk ≤τ ≤t

tk ≤τ ≤t

t ∈ (tk , tk+1 ], k ∈ S. If μz and μd are considered as external inputs, we can construct an ISS-Lyapunov function candidate for the interconnected system as follows. V (x) ¯ = max {Vx (x), σ(Vz (z))}

(5.95)

where x¯ = [x T , z T ]T and σ ∈ K∞ is continuously differentiable on (0, ∞) satisfying σ(s) > γxz (s) and σ −1 (s) > γzx (s) for s ∈ R+ . Indeed, from (5.82), (5.91) and (5.93), we have   V (x) ¯ ≥ max γ¯ xz (|μz |), γ¯ xz (|μd |), γ¯ xz (ς1 ), χd (|d|) ⇒D + V (x)F(x, ¯ z, w, d) ≤ −α(V (x)) ¯

(5.96a) (5.96b)

where γ¯ xz is defined in (5.91), χd (s) = max{γxd (s), σ ◦ γzd (s)} and α(s) = min{ 21 αx (s), 21 σ (1) (σ −1 (s)) · αz (σ −1 (s))} for s ∈ R+ , and F(x, z, w, d) = [g(x, z, w, d)T , h(z, x, d)T ]T . In fact, the system is a feedback interconnection where x¯ is an input driving μz which, in turn, is an input driving x, ¯ as shown in Fig. 5.16. Then, by defining W (t) = V (x(t)), ¯ we have      z W (t) ≥ max χ avg ρz (W (t)) , γ¯ x avg ρ1 (|d(τ )|) , tk ≤τ ≤t

tk ≤τ ≤t

γ¯ xz (ς1 (t)), χd (|d(t)|) ⇒D + W (t) ≤ −α(W (t))



(5.97a) (5.97b)

for t ∈ (tk , tk+1 ], k ∈ S, where χ ∈ K∞ is continuously differentiable on (0, ∞) sat−1 z −1 (s) for isfying γxw ◦ γwd (s) < χ(s) < γxz ◦ αz ◦ ρ−1 1 (s), and ρz (s) = ρ1 ◦ αz ◦ γx s ∈ R+ .

5.3 Robustness Analysis in the Presence of Dynamic Uncertainties

131

Since the right-hand side of (5.97a) contains W , the implication (5.97a)–(5.97b) does not readily imply the ISS of the closed-loop event-triggered system. Define    (5.98) W ∗ (t) = max W (t), χ avg ρz (W (τ )) tk ≤τ ≤t

for t ∈ (tk , tk+1 ], k ∈ S. To calculate the upper right-hand derivative D + W ∗ of the functional W ∗ , we study the cases of W (t) > χ( avg ρz (W (τ ))), W (t) < tk ≤τ ≤t

χ( avg ρz (W (τ ))) and W (t) = χ( avg ρz (W (τ ))) for t ∈ (tk , tk+1 ], k ∈ S, respectk ≤τ ≤t

tk ≤τ ≤t

tively. Case 1: W (t) > χ( avg ρz (W (τ ))). In this case, W ∗ (t) = W (t). With condition tk ≤τ ≤t

(5.97a)–(5.97b) satisfied, we have D + W ∗ (t) ≤ −α(W ∗ (t))

(5.99)

  whenever W ∗ (t) ≥ max χd (|d(t)|), γ¯ xz (ς1 (t)), γ¯ xz ( avg ρ1 (|d(τ )|)) for t ∈ (tk , tk ≤τ ≤t

tk+1 ], k ∈ S. Case 2: W (t) < χ( avg ρz (W (τ ))). In this case, for t ∈ (tk , tk+1 ], k ∈ S, we have tk ≤τ ≤t

  D + W ∗ (t) = D + χ( avg ρz (W (τ ))) ≤ − tk ≤τ ≤t

1 α0 (W ∗ (t)) t − tk

(5.100)

where α0 (s) = χ(1) (χ−1 (s)) · (Id − ρz ◦ χ)(χ−1 (s)) for s ∈ R+ . Case 3: W (t) = χ( avg ρz (W (τ ))). In this case, we have limh→0+ W (t + h) = tk ≤τ ≤t

limh→0+ χ( avg ρz (W (τ ))). Using exactly the same arguments as in Cases 1 and tk ≤τ ≤t+h

2, it follows that D + W ∗ (t) ≤ −α∗ (W ∗ (t))

(5.101)

  z z whenever W (t) ≥ max χd (|d(t)|), γ¯ x (ς1 (t)), γ¯ x ( avg ρ1 (|d(τ )|)) for t ∈ (tk ,   tk ≤τ ≤t 1 ∗ tk+1 ], k ∈ S, where α (s) = min α(s), t−tk α0 (s) for s ∈ R+ . We can find an arbitrary constant  > 1 such that ∗

α∗ (s) ≥ min



 1 1 1 α(s), α0 (s) =: α∗ (s), t + t + t + 0

132

5 Event-Triggered Input-to-State Stabilization

where α0∗ (s) = min{α(s), α0 (s)} for s ∈ R+ . By combining the above-mentioned three cases, we have    W ∗ (t) ≥ max χd (|d(t)|), γ¯ xz (ς1 (t)), γ¯ xz avg ρ1 (|d(τ )|) tk ≤τ ≤t

⇒D + W ∗ (t) ≤ −

1 α∗ (W ∗ (t)) t + 0

(5.102)

for t ∈ (tk , tk+1 ], k ∈ S. Now we prove that there exists βˆ ∈ KL such that ˆ ∗ (0), t) W ∗ (t) ≤ β(W

(5.103)

  whenever W ∗ (t) ≥ max χd (|d(t)|), γ¯ xz (ς1 (t)), γ¯ xz ( avg ρ1 (|d(τ )|)) for t ∈ (tk , tk ≤τ ≤t

tk+1 ], k ∈ S. A sketch of proof is given here. Let H (t) be the solution of the differential equation 1 α∗ (H (t)), H (0) = W ∗ (0). H˙ (t) = − t + 0

(5.104)

Due to the existence of the time-variable term 1/(t + ) in (5.104), some β¯ ∈ KL ¯ (0), t) for all t ≥ 0 cannot be directly found by extending satisfying H (t) ≤ β(H ¯ We find a existing results given in [159]. In order to find an appropriate function β, sequence in the form of 

H (0), (Id − θ)(H (0)), · · · (Id − θ)[ j] (H (0)) · · ·



(5.105)

where j ∈ Z+ and θ satisfies μα0∗ ◦ (Id − θ)(s) = θ(s) for s ∈ R+ , with 0 < μ < 1. Define  j   Δt j = μ Δtm +  (5.106) m=1

as the time which a point moves along a straight line from point (Id − θ)[ j−1] (H (0)) j  to point (Id − θ)[ j] (H (0)) with speed 1/( Δtm + ) × α0∗ ((Id − θ)[ j] (H (0))). m=1

From (5.105)–(5.106), one can find a β j ∈ KL with j ∈ Z+ between any two adjacent elements in sequence (5.105) as follows:

5.3 Robustness Analysis in the Presence of Dynamic Uncertainties

W ∗ (0)

133

(Id − θ)(W ∗ (0)) (Id − θ)[2] (W ∗ (0))

β1

(Id − θ)[j−1] (W ∗ (0)) (Id − θ)[j] (W ∗ (0))

β2 W ∗ (t) 0 Δt1

βj Δtj

Δt2

Fig. 5.17 The relationship between the sequence defined by (5.105) and the trajectory of W ∗ (t)

β j (H (0), t) =(Id − θ)[ j−1] (H (0))− 1 j 

α0∗ ((Id

 [ j]

− θ) (H (0))) t −

Δtm + 

j−1 

 Δtm

(5.107)

m=0

m=1



j−1 

j 



ˆ t) = β j (s, t) for s ∈ R+ and Δtm , Δtm with Δt0 = 0. Define β(s, for t ∈ m=0 m=1   j−1 j   Δtm , Δtm with j ∈ Z+ . Then, by using the comparison principle t∈ m=0

m=1

[138], we have ˆ (0), t). H (t) ≤ β(H Condition (5.106) implies that Δt j =

μ (1−μ) j

, and thus limm→∞

(5.108) m 

Δt j = ∞.

j=1

ˆ ·) is right maximally defined It can be directly checked that the function β(·, ∗ ∗ ∗ with interval of definition [0, ∞). For 0 ≤ t ≤  T , with T = min{t : W (t) = max χd (|d(t)|), γ¯ xz (ς1 (t)), γ¯ xz ( avg ρ1 (|d(τ )|)) , we have tk ≤τ ≤t

ˆ (0), t). W ∗ (t) ≤ H (t) ≤ β(H

(5.109)

The relationship between the sequence defined by (5.105) and the trajectory of W ∗ (t) is shown in Fig. 5.17. Based on (5.103), one can find β ∈ KL and γd ∈ K such that

134

5 Event-Triggered Input-to-State Stabilization

  |x¯ ∗ (t)| ≤ max β(|x¯ ∗ (0)|, t), γd ( d ∞ )

(5.110)

for all t ≥ 0, with x¯ ∗ = [x¯ T , ς1 ]T . Thus, the closed-loop system is ISS with (z, x, ς1 ) as the state and d as the input. This ends the proof of Theorem 5.4.  We employ an example to show the effectiveness of the proposed design. Example 5.3 Consider the Lorenz system [180]: z˙ 1 = −L 1 z 1 + L 1 x + d1 ,

(5.111)

z˙ 2 = z 1 x − L 2 z 2 + d2 , x˙ = L 3 z 1 − x − z 1 z 2 + u + d3

(5.112) (5.113)

where z = [z 1 , z 2 ]T ∈ R2 is the unmeasured state, x is the measured state, u ∈ R is the control input, and d = [d1 , d2 , d3 ]T ∈ R3 represents the external disturbances, and L 1 , L 2 , L 3 are positive constants. The control law is designed as u(t) = −(a2 |x(tk )|3 + a1 |x(tk )| + a0 )x(tk )

(5.114)

for t ∈ [tk , tk+1 ), k ∈ S, where a2 = 0.305(1 + b)r , a1 = 1.56m 2 + 0.156 and a0 = with m = ((1 + b)r )1/4 /((14/cL 22 )1/4 ), 0 < b, c < 1 and r = 1.25m  L 3 + 1.375 3 2 max 126/(c L 2 ), 3/(cL22 ) .  Define Vz (z) = max σ(0.5z 12 ), 0.5z 22 with σ(s) = 14/(cL 22 s 2 ) for s ∈ R+ and that Vx (x) = 0.5x 2 . It can be verified  Assumptions 5.5 and 5.6 are satisfied with  γzx (s) = r s 2 , γzd (s) = r max s 4 , s 2 , γxz (s) = (s/((1 + b)r ))1/2 , γxw (s) = 12.5s 2 , γxd (s) = 5s, αz (s) = min{∂σ(σ −1 (s))(1 − c)L 1 σ −1 (s), (1 − c)L 2 s}, αx (s) = 0.1s, L ag (s) = 8a2 s 3 + (2a1 + 1)s + a0 + b + 2, L bg ≡ 1 and ρ(s) = s 2 + (1 + L 3 )s for s ∈ R+ . By using Corollary 5.1 and Theorem 5.4, γwx and γwd defined in (5.87) are chosen as x γw (s) = 0.19s and γwd (s) = 0.28s 1/2 ◦ (1 + b)r s 2 ◦ 1.4s 1/2 ◦ min{0.5s 1/2 , s/4 (L 3 + 1)} for s ∈ R+ , and ϕ defined in (5.7) is chosen as ϕ(s) = s for s ∈ R+ . The simulation result with initial state z 1 (0) = −2, z 2 (0) = 4, x(0) = 2 and ς1 (0) = 2 and constants L 1 = 5, L 2 = 3, L 3 = 4, b = 0.1, c = 0.9 is shown in Figs. 5.18, 5.19, 5.20 and 5.21. It is shown that the proposed event trigger is robust with respect to external disturbances and dynamic uncertainties. Also, the convergence of the disturbance implies the convergence of the system state.

5.3.3 A Special Case for Systems Under a Global Sector-Bound Condition If system dynamics h and g are globally sector-bounded, then property (5.1) in Assumption 5.1 can be modified as

5.3 Robustness Analysis in the Presence of Dynamic Uncertainties

135

2

d1

1.5

d

disturbances

1

d

2 3

0.5 0 -0.5 -1 -1.5 -2

0

2

4

6 time

8

10

12

Fig. 5.18 The external disturbances of a Lorenz system in Example 5.3 4

z1 z2

3 system state

x 2 1 0 -1 -2

0

2

4

6 time

8

10

12

Fig. 5.19 The state trajectories of a Lorenz system with the event trigger (5.87) in Example 5.3

d |g(x, z, w, d)| ≤ L gx |x| + L gz |z| + L w g |w| + L g |d|, ∀x, z, w, d.

(5.115)

We still make assumptions on the ISS of the z-subsystem and the input-to-state stabilizability of the x-subsystem. Assumption 5.7 The system (5.77) is ISS with x and d as the inputs, and admits an ISS-Lyapunov function Vz : Rn z → R+ satisfying the following conditions: 1. there exist constants αz , αz > 0 such that αz |z|2 ≤ Vz (z) ≤ αz |z|2 , ∀z; 2. there exist constants αz , k zx and k zd > 0 such that

(5.116)

136

5 Event-Triggered Input-to-State Stabilization

Fig. 5.20 The control input u of a Lorenz system with the event trigger (5.87) in Example 5.3

inter-sampling intervals

100

10-1

10

-2

10

-3

10-4

0

2

4

6 time

8

10

12

Fig. 5.21 The inter-sampling intervals of a Lorenz system with the event trigger (5.87) in Example 5.3

Vz (z) ≥ max{k zx Vx (x), k zd |d|2 } ⇒∇Vz (z)h(z, x, d) ≤ −αz Vz (z)

(5.117)

for all z, x and d. Assumption 5.8 The system (5.80) is ISS with z, w and d as the inputs, and admits an ISS-Lyapunov function Vx : Rn → R+ satisfying the following conditions: 1. there exist constants αx , αx > 0 such that αx |x|2 ≤ Vx (x) ≤ αx |x|2 , ∀x;

(5.118)

5.3 Robustness Analysis in the Presence of Dynamic Uncertainties

137

2. there exist constants αx , k xz , k xw and k xd > 0 such that Vx (x) ≥ max{k xz Vz (z), k xw |w|2 , k xd |d|2 } ⇒∇Vx (x)g(x, z, w, d) ≤ −αx Vx (x)

(5.119)

for all x, z, w and d. In this case, the event trigger is still designed as in (5.42), and the estimation of the influence of the external disturbance d and the unmeasured state z is defined as ς2 (t) =

1 (t − tk )L gzd E(t, tk )   × max |x(t) − x(tk )| − L g (E(t, tk ) − 1)|x(tk )|, 0

(5.120)

for t ∈ [tk + TΔ , tk+1 ] with k ∈ S, where L gzd = 2 max{L gz , L dg }, E(t, tk ) and L g are defined in (5.23), and TΔ is a positive constant satisfying ϕ(TΔ ) = ε

(5.121)

with ϕ(TΔ ) = 2 max{L g e L g TΔ − L g , L gzd e L g TΔ TΔ }, L gxw = L gx + L w g and ε = min{kwx /(1 + kwx ), kwd }. Moreover, the closed-loop event-triggered system is ISS if the linear version of the small-gain condition is satisfied. According to Theorem 5.1, by choosing kwx in (5.42) satisfying (5.24) and choosing TΔ satisfying (5.121), the event-triggered controlled x-subsystem is ISS with z and d as the inputs. Also, a condition for ISS of the closed-loop event-triggered system is still given by using the small-gain idea. xw

xw

Corollary 5.2 Under Assumptions 5.7 and 5.8, the system composed of (5.77), (5.80), (5.87), (5.6) and (5.88) is ISS with (z, x) as the state and (d, ς1 ) as the inputs. Moreover, the system composed of (5.77), (5.80), (5.87), (5.6), (5.7) and (5.88) is ISS with (z, x, ς1 ) as the state and d as the input, if the following conditions are satisfied: 1. k xz and k zx are designed to satisfy k xz · k zx < 1;

(5.122)

2. TΔ , kwx and kwd defined in (5.42) satisfy (5.121), (5.24) and  0 < kwd
0 being the delay time, and fi : Ri × Ri → R and gi : Ri × Ri → R (i = 1, . . . , n) are unknown locally Lipschitz functions. For the sake of simplicity, it is assumed that each subsystem in the system (6.43) has the same time delay. This assumption is relaxed in Sect. 6.2.4. The following assumptions is made on the system dynamics. Assumption 6.2 For i = 1, . . . , n, there exists a constant ci > 0 such that gi (x¯i , z i ) > ci

(6.44)

for all x¯i ∈ Ri and z i ∈ Ri with z i (t) = x¯iθ (t) for all t ≥ 0, and there exists a known ψ fi ∈ K∞ such that | f i (x¯i , z i )| ≤ ψ fi (|x¯i | + |z i |)

(6.45)

for all x¯i ∈ Ri and z i ∈ Ri . This section aims to propose a tool for the design of event-triggered controllers for global asymptotic stabilization of the system (6.43) at the origin. Specifically, a constructive design scheme is developed to design event-triggered controllers taking the form of u(t) = λ(x s (t))

(6.46)

where x s = [x1s , . . . , xns ]T ∈ Rn is the sampled value of x, defined by x s (t) = x(tk ), t ∈ [tk , tk+1 ), k ∈ S ⊆ Z+ ,

(6.47)

where the sequence of sampling times {tk }k∈S with t0 = 0 is generated by an event trigger. For any initial state ξ, it is desired to achieve that 1. x(t) is well-defined over [−θ, ∞), and the closed-loop event-triggered system with x as the state is globally asymptotically stable at the origin, and 2. there exists a positive constant TΔ such that inf k∈S {tk+1 − tk } ≥ TΔ .

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6 Event-Triggered Control of Nonlinear Uncertain Systems …

Controlling time-delayed systems of the form (6.43) has been studied in quite a few existing results; see, e.g., [209, 275] and the references therein. However, event-triggered stabilization of such systems has not been thoroughly investigated. We recognize that the existing work [212] has developed a systematic analysis for semi-global stabilization of nonlinear time-delay systems with event-triggered feedback control, and global asymptotic stability of event-triggered nonlinear time-delay systems is obtained in [57]. Lyapunov–Krasovskii functionals are employed in that result to characterize the stabilization capability of the pre-determined feedback control law. This section studies the event-triggered stabilization problem for the specific class of nonlinear time-delay systems in the strict-feedback form, for which, Lyapunov–Krasovskii functionals may not be readily constructed based on the existing methods. Moreover, the design proposed in this section is able to achieve global robust stabilization in the presence of measurement errors which are not necessarily caused by data sampling. A control law λ that is robust with respect to the sampling error is known to be essential for event-triggered control; see, e.g., [86, 148, 245]. Reference [61] has introduced a steepest descent feedback design to sampled-data control of nonlinear time-delay systems. The design in [61] is based on a known upper bound of the measurement error, and seems not directly applicable to event-triggered control in which the upper bound of the sampling error cannot be predetermined.

6.2.2 Constructive Control Design with Set-Valued Maps We consider control laws with a structure of nested loops: μ1 = λ1 (x1s ) μi = λi (xis − μi−1 ), i = 2, . . . , n − 1 u = λn (xns − μn−1 ).

(6.48) (6.49) (6.50)

We first show how the functions λi for i = 1, . . . , n can be chosen such that the controlled system composed of (6.43) and (6.48)–(6.50) is robustly stable with respect to the sampling error wi = xis − xi , i = 1, . . . , n.

(6.51)

For convenience of discussions, denote w = [w1 , . . . , wn ]T ∈ Rn . During the control law design, we first assume the boundedness of w, i.e., the existence of w[0,∞) , denoted by w∞ . Equivalently, wi [0,∞) , denoted by wi∞ , exists for i = 1, . . . , n. Denote w¯ i∞ = [w1∞ , . . . , wi∞ ]T . Following the approach in Sect. 6.1, one can design a control law such that the closed-loop system is robust to sampling error w. The basic idea of the control design in this subsection is still to transformed the closed-loop system into an interconnec-

6.2 Event-Triggered Control of Nonlinear Time-Delay Systems

155

tion of ISS subsystems, and use refined cyclic-small-gain theorem to guarantee the ISS of the closed-loop system. Specifically, for i = 1, . . . , n, define set-valued maps Λi : Ri × Ri  R as ∗ ): Λi (x¯i , w¯ i∞ ) = {λi (xi + ai wi∞ − μi−1 ∗ ∞ μi−1 ∈ Λi−1 (x¯i−1 , w¯ i−1 ), |ai | ≤ 1},

(6.52)

where λi : R → R is a C 1 function, to be designed later. By default, denote Λ0 (x¯0 , w¯ 0∞ ) = {0}. Specifically, the λi is designed as λi (r ) = −κi (|r |)r

(6.53)

for r ∈ R with κi : R+ → R+ being positive, nondecreasing and C 1 on (0, ∞), to be designed later. For i = 1, . . . , n, define new state variables ∞ ei = d(xi , Λi−1 (x¯i−1 , w¯ i−1 )).

(6.54)

∞ Here, d is defined in (6.15). It can be directly checked that xi − ei ∈ Λi−1 (x¯i−1 , w¯ i−1 ) ∞ for i = 1, . . . , n. Also, for i = 1, . . . , n − 1, μi ∈ Λi (x¯i , w¯ i ) and u ∈ Λn (x¯n , w ∞ ). With the transformation (6.54), when ei = 0, each ei -subsystem can be represented by

e˙i = h i (x¯i , x¯iθ , e¯i+1 ) + gi (x¯i , x¯iθ )(xi+1 − ei+1 ) := Fi (x¯i , x¯iθ , w¯ i∞ , e¯i+1 )

(6.55)

where h i (x¯i , x¯iθ , e¯i+1 ) = gi (x¯i , x¯iθ )ei+1 + f i (x¯i , x¯iθ ) − ((0.5 + 0.5sgn(ei ))∇ max Λi−1 + (0.5 − 0.5sgn(ei ))∇ min Λi−1 )x˙¯i−1 and xi+1 − ei+1 ∈ Λi (x¯i , w¯ i∞ ). Specifically, a detailed derivation for Eq. (6.55) is given as follows. Condition (6.53) implies that each designed control law λi (i = 1, . . . , n) is decreasing and C 1 on (0, ∞). Then, for k = 1, . . . , i − 1 and any specific x¯k , we have ∞ )) and min Λk (x¯k , w¯ k∞ ) = max Λk (x¯k , w¯ k∞ ) = λk (xk − wk∞ − max Λk−1 (x¯k−1 , w¯ k−1 ∞ ∞ λk (xk + wk − min Λk−1 (x¯k−1 , w¯ k−1 )). Moreover, max Λk (x¯k , w¯ k∞ ) and min Λk (x¯k , w¯ k∞ ) are C 1 with respect to x¯k . In the case of ei > 0, the dynamics of ei ∞ ˙ can be rewritten as e˙i = x˙i − ∇ max Λi−1 (x¯i−1 , w¯ i−1 )x¯i−1 = f i (x¯i , x¯iθ ) + gi (x¯i , x¯iθ ) ∞ ˙ θ xi+1 − ∇ max Λi−1 (x¯i−1 , w¯ i−1 )x¯i−1 = gi (x¯i , x¯i )(xi+1 − ei+1 ) + f i (x¯i , x¯iθ ) + gi (x¯i , ∞ ˙ x¯iθ )ei+1 − ∇ max Λi−1 (x¯i−1 , w¯ i−1 )x¯i−1 . The proof for the case of ei < 0 is similar. θ We have e˙i = gi (x¯i , x¯i )(xi+1 − ei+1 ) + f i (x¯i , x¯iθ ) + gi (x¯i , x¯iθ )ei+1 − ∇ min Λi−1 ∞ ˙ (x¯i−1 , w¯ i−1 )x¯i−1 . Thus, when ei = 0, each ei -subsystem can be represented by Eq. (6.55). Proposition 6.1 gives a result on the robustness of the proposed control law with respect to the sampling error. Due to the state-delay terms, the robustness is described by the Ruzumikhin-like Lyapunov function [250]. To simplify the discussions, denote en+1 = 0, e¯i = [e1 , . . . , ei ]T ∈ Ri for i = 1, . . . , n, e = e¯n ∈ Rn and eiθ = ei (t −

156

6 Event-Triggered Control of Nonlinear Uncertain Systems …

θ) ∈ R for i = 1, . . . , n, and denote F(x, x¯nθ , w¯ n∞ , e) = [F1 (x¯1 , x¯1θ , w¯ 1∞ , e¯2 ), . . . , Fn (x¯n , x¯nθ , w¯ n∞ , e¯n+1 )]T : Rn × Rn × Rn × Rn → Rn . Proposition 6.1 Consider the nonlinear time-delay system (6.43) satisfying Assumption 6.2. By appropriately choosing κi defined in (6.53), one can find a locally Lipschitz function V : Rn → R+ such that α(|e|) ≤ V (e) ≤ α(|e|), V (e) ≥ max{γe (V (e)[t−θ,t] ), γw (w ∞ )}

(6.56)

⇒∇V (e)F(x, x¯nθ , w¯ n∞ , e) ≤ −αe (V (e)), a.e. γe < Id,

(6.57) (6.58)

where α, α, γe , γw ∈ K∞ are Lipschitz on compact sets, and αe is a continuous and positive definite function. The proof of Proposition 6.1 is given in Appendix D.7. Proposition 6.1 readily implies the ISS of the closed-loop system with e as the state. For the convenience of event-trigger design based on x, Proposition 6.2 shows the ISS of the same system with w as the input and x as the state. Proposition 6.2 Consider the closed-loop event-triggered system composed of the plant (6.43), the control law (6.46) defined by (6.48)–(6.50) with λi defined by (6.53). By appropriately choosing λi for i = 1, . . . , n, it can be achieved that for any initial state ξ and any w ∞ ,   |x(t)| ≤ max β(ξ[−θ,0] , t), γxw (w ∞ )

(6.59)

holds for all t ≥ 0, where β ∈ KL, and γxw ∈ K∞ is Lipschitz on compact sets. The proof of Proposition 6.2 is given in Appendix D.8.

6.2.3 Main Result of Event-Triggered Stabilization To deal with the time-delay term x¯iθ , we introduce a time-delay threshold signal to the event trigger to avoid infinitely fast sampling. Specifically, the event trigger is designed as: t0 = 0, and

 tk+1 = inf t > tk : |w(t)| ≥ ρ(x[t−θ,t] ), x[t−θ,t] = 0

(6.60)

for k, k + 1 ∈ S, where ρ : R+ → R+ is to be determined later. Theorem 6.4 presents the main result on event-triggered stabilization of the timedelay system (6.43).

6.2 Event-Triggered Control of Nonlinear Time-Delay Systems

157

Theorem 6.4 Consider the closed-loop event-triggered system composed of the plant (6.43), the control law (6.46) in the form of (6.48)–(6.50) with λi defined by (6.53) and the event trigger (6.60). Under Assumption 6.2, with γxw given in Proposition 6.2 chosen to be a of class K∞ and Lipschitz on compact sets, the problem of event-triggered stabilization is solvable by choosing ρ in (6.60) to satisfy the following conditions: (a) ρ is Lipschitz on compact sets and of class K∞ , (b) ρ−1 is Lipschitz on compact sets, and (c) it holds that ρ ◦ γxw < Id.

(6.61)

Proof The properties of γxw guarantee the existence of ρ satisfying properties (a), (b) and (c) in Theorem 6.4. In the rest of the proof, we first prove the state convergence of the closed-loop event-triggered system whenever the signals are well defined, and then prove the existence of a positive inter-sampling interval, so that all the signals in the closed-loop event-triggered system are well defined for all t ≥ 0. (a) Convergence Analysis Suppose that x(·) is right maximally defined for all t ∈ [−θ, Tmax ) with 0 < Tmax ≤ ∞. We first prove that |w(t)| ≤ ρ(x[t−θ,t] )

(6.62)

holds for 0 ≤ t < Tmax . Consider the following two cases. • x[t−θ,t] = 0 for all 0 ≤ t < Tmax . In this case, (6.62) is directly guaranteed by (6.60). • x[t−θ,t] = 0 for some 0 ≤ t < Tmax . In this case, denote t ∗ = min{t : 0 ≤ t < Tmax , x[t−θ,t] = 0}, tk ∗ = max{tk : k ∈ S, tk ≤ t ∗ }.

(6.63) (6.64)

With the event trigger (6.60), we have |w(t)| ≤ ρ(x[t−θ,t] ), i.e., |x(t) − x(tk ∗ )| ≤ ρ(x[t−θ,t] ) for all t ∈ [tk ∗ , t ∗ ). By using the continuity of x(t) and considering the case of t = t ∗ , we have |x(t ∗ ) − x(tk ∗ )| ≤ ρ(x[t ∗ −θ,t ∗ ] ), which implies x(tk ∗ ) = 0. In this case, the system state x and the control input u (given by (6.48)–(6.50) with λi defined by (6.53)) keep zero for all t ∗ ≤ t < Tmax , and the sampling event is never triggered after tk ∗ , that is, S = {0, . . . , k ∗ }. Property (6.62) is proved through the discussions on the two cases above. With (6.59), (6.61) and (6.62) satisfied, from the trajectory-based small-gain theorem for time-delay system [253], there exists a βˆ ∈ KL such that ˆ |x(t)| ≤ β(ξ [−θ,0] , t) for all 0 ≤ t < Tmax . This proves the state convergence property.

(6.65)

158

6 Event-Triggered Control of Nonlinear Uncertain Systems …

(b) Positive Lower Bound of Inter-Sampling Intervals We show that the inter-sampling intervals tk+1 − tk with k, k + 1 ∈ S have a positive lower bound, and all the signals in the closed-loop event-triggered system are defined for all t ≥ 0, i.e., Tmax = ∞. By substituting (6.46) into (6.43) and using (6.51), we have   x˙i (t) = f i x¯i (t), x¯iθ (t) + gi (x¯i (t), x¯iθ (t))xi+1 (t), i = 1, . . . , n − 1,   x˙n (t) = f n x¯n (t), x¯nθ (t) + gn (x¯n (t), x¯nθ (t))λ(x(t) + w(t)). Here, λ(x + w) = λn (xn + wn − μn−1 ) with μi = λi (xi + wi − μi−1 ) for i = 2, . . . , n − 1 and μ1 = λ1 (x1 + w1 ). Then, by using Assumption 6.2 and the definition of λi in (6.53) as well as property (6.62), we have |x(t)| ˙ ≤ L(x[t−θ,t] )x[t−θ,t]

(6.66)

for all t ∈ [tk , tk+1 ), where L is a continuous, positive-valued and nondecreasing function. Recall that ρ satisfies the conditions (a), (b) and (c) in Theorem 6.4. Then, from Lemma B.5, there exists a continuous, positive-valued and nondecreasing function Lˆ such that ˆ L(x[τ −θ,τ ] )x[τ −θ,τ ] ≤ L(x [τ −θ,τ ] )ρ(x[τ −θ,τ ] ) for all τ ∈ [tk , tk+1 ). Thus, tk+1 is not smaller than the largest t that guarantees ˆ (τ − tk )( L(x [τ −θ,τ ] )ρ(x[τ −θ,τ ] )) ≤ ρ(x[τ −θ,τ ] )

(6.67)

for all τ ∈ [tk , t). We prove x[τ −θ,τ ] = 0 for τ ∈ [tk , tk+1 ) by contradiction. If x[τ −θ,τ ] = 0 for τ ≥ tk , then following a similar reasoning as for the second case in the convergence analysis part, it can be shown that x(tk ) = 0 and tk is the last time of sampling. This contradicts with k + 1 ∈ S. Then, (6.65) and (6.67) together imply tk+1 − tk ≥

1 ˆ ˆ L(max{ β(ξ [−θ,0] , 0), ξ[−θ,0] })

.

(6.68)

Since the right-hand side of (6.68) is independent of k, we have inf {tk+1 − tk } ≥ k∈S

1 ˆ ˆ L(max{ β(ξ [−θ,0] , 0), ξ[−θ,0] })

.

(6.69)

This proves the positive lower bound of the inter-sampling intervals. Property (6.69) guarantees the avoidance of Zeno behavior. Since the boundedness of x is guaranteed for all t ∈ [−θ, Tmax ), finite escape time does not appear within the

6.2 Event-Triggered Control of Nonlinear Time-Delay Systems

159

time interval [0, Tmax ), and by the theorem on continuation of solutions [62, Theorem 2 on p. 78], Tmax = ∞. The convergence property is proved by replacing Tmax with ∞ for (6.65). This ends the proof of Theorem 6.4.  Clearly, if the term of ρ(x[t−θ,t] ) is replaced by ρ(|x(t)|), then the proposed event trigger (6.60) is reduced to the standard delay-free form proposed in Chap. 3. In the presence of the time-delay terms, the delay-free event trigger is not able to avoid infinitely fast sampling. Just consider the case in which x(t) = 0 (which means zero threshold signal, i.e., ρ(|x(t)|) = 0), and ξ(t) = 0 (which means nonzero x(t)). ˙ This is the motivation of introducing the time-delay term to the event trigger.

6.2.4 Extension to Systems with Distributed Time-Delays With quite similar techniques, the small-gain design proposed above is also valid for a more general case of distributed time-delays:   x˙i (t) = f i x¯i (t), x¯i,t + gi (x¯i (t), x¯i,t )xi+1 (t), i = 1, . . . , n − 1,   x˙n (t) = f n x¯n (t), x¯n,t + gn (x¯n (t), x¯n,t )u(t)

(6.70)

where the variables except for x¯i,t are with the same meaning as for the system (6.43). Let X i = C i ([−θ, 0]) be the space of all continuous functions from [−θ, 0] to Ri . For a function φ defined on [−θ, a) with constant a > 0, φt (s) = φ(t + s) for all s ∈ [−θ, 0]. For each i = 1, . . . , n, assume that f i : Ri × X i → R and gi : Ri × X i → R satisfy the following properties to guarantee the existence and uniqueness of the solutions: f i is completely continuous (that is, f i is continuous, and the closure of f i (B) is compact for each bounded set B), and f i and gi are Lipschitz on every compact subset of Ri × X i ; see, e.g., [82] for a text of time-delay systems. Accordingly, conditions (6.44) and (6.45) in Assumption 6.2 should be modified as follows: there exist a known positive constant ci and a class K∞ function ψ fi such that gi (x¯i (0), x¯i,0 ) > ci and | f i (x¯i (0), x¯i,0 )| ≤ ψ fi (|x¯i (0)| + x¯i [−θ,0] ) for any continuous x¯i defined on [−θ, 0]. For the system (6.70), the control law (6.48)–(6.50) and the event trigger (6.60) can still be used in the sense that the closed-loop system can still be input-to-state stabilized. Example 6.2 To verify the effectiveness of the proposed design, consider the chemical system composed of two cascade connected reactors [199]: x˙1 = f 1 (x¯1 , x¯1θ ) + g1 x2

(6.71)

f 2 (x¯2 , x¯2θ )

(6.72)

x˙2 =

+ g2 u

with f 1 (x¯1 , x¯1θ ) = −k1 x1 − x1 /D1 − x1θ /D1 + D3 x1θ , f 2 (x¯2 , x¯2θ ) = −k2 x2 − x22 /D2 + R1 x1θ /E 2 − x2 /D2 + R2 x2θ /E 2 + 0.5D4 (x2θ )2 , g1 = (1 − R2 )/E 1 and g2 =

160

6 Event-Triggered Control of Nonlinear Uncertain Systems …

Plant Composed of Cascade Connected Reactors Controller with Nested Loops λ1

xs1

μ1  – + xs2

λ2

x2

ET u Sync

Reactor 2

x1 Reactor 1

ET

Fig. 6.5 The schematic diagram of the closed-loop event-triggered system for cascade connected reactors, where ET represents the event triggers

F/E 2 , where, for i = 1, 2, xi is the composition, Ri represents the recycle flow rate, Di represents the reactor residence time, ki represents the reaction constant, F represents the feed rate, and E i represents reactor volume. In this example, the parameters take value as: Ri = E i = F = ki = 0.5, D1 = D2 = 2, D3 = D4 = 0.4, θ = 1. We evaluate the event-triggered control law (6.48)–(6.50) with λ1 (r ) = −2r and λ2 (r ) = −(20|r | + 10)r for r ∈ R. Direct calculation verifies property (6.59) with γxw (s) = 20s for s ∈ R+ . Based on Theorem 6.4, the event trigger is designed as (6.60) with ρ(s) = 0.05s for s ∈ R+ . The block diagram of the closed-loop eventtriggered two-stage chemical system is shown in Fig. 6.5. Note that the static control law leads to synchronous updates of the state measurement and the control signal, and thus, synchronized event triggers can be installed in both the channels of measurement and control.

x

1

0.4

x2

0.3

state

0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5

0

0.5

1 time

1.5

2

2.5

Fig. 6.6 The state trajectories of the cascade connected reactors with the event trigger (6.60) in Example 6.2

6.2 Event-Triggered Control of Nonlinear Time-Delay Systems

161

14 12

control input

10 8 6 4 2 0 -2 0

0.5

1

1.5

2

2.5

time

Fig. 6.7 The control input of the cascade connected reactors with the event trigger (6.60) in Example 6.2 0

Inter-samping intervals

10

10-1

10-2

-3

10

-4

10

0

0.5

1

1.5

2

2.5

time

Fig. 6.8 The inter-sampling intervals of the cascade connected reactors with the event trigger (6.60) in Example 6.2

The simulation result with initial states ξ1 (t) = (cos(27t) + sin(25t) + sin(29t) + sin(21t))/2 and ξ2 (t) = (cos(23t) + sin(25t) + sin(28t) + sin(21t))/2 for t ∈ [−1, 0] and step size 10−4 is given by Figs. 6.6, 6.7 and 6.8. It is shown that there exists a positive lower bound for the inter-sampling interval tk+1 − tk , and the state x converges to the origin.

162

6 Event-Triggered Control of Nonlinear Uncertain Systems …

6.3 Event-Triggered Output-Feedback Control Chapters 4 and 5 show that the objective of event-triggered stabilization can be achieved even if only partial state is available to feedback. This section shows how the result is used to solve the event-triggered output-feedback control problem for a class of nonlinear uncertain systems.

6.3.1 Problem Formulation Consider nonlinear systems in the output-feedback form, which can be considered as a reduced model of the system (6.1)–(6.2): x˙i = xi+1 + f i (x1 , d), x˙n = u + f n (x1 , d)

i = 1, . . . , n − 1

y = x1

(6.73) (6.74) (6.75)

where y ∈ R is the output, and the other variables are defined as for (6.1) and (6.2). It is assumed that only y is available to feedback control. The following assumption is further made on f 1 , . . . , f n . Assumption 6.3 For each i = 1, . . . , n, there exists a known ψ fi ∈ K∞ being Lipschitz on compact sets such that | f i (x1 , d)| ≤ ψ fi ( [x1 , d T ]T )

(6.76)

holds for all x1 ∈ R and d ∈ Rn d . The objective of this section is to design an event-triggered output-feedback controller for the system (6.73)–(6.75) so that the closed-loop system is in the form of (5.124)–(5.125) and moreover the conditions for event-triggered control are satisfied.

6.3.2 Observer-Based Output-Feedback Controller For convenience of notations, we define ys = y + w

(6.77)

where y s represents the sampled value of y and w is the sampling error. Owing to the output-feedback structure, we design a nonlinear observer for the system (6.73)–(6.75) as

6.3 Event-Triggered Output-Feedback Control

163

ξ˙1 = ξ2 + L 2 ξ1 + ρ1 (ξ1 − y s ) ξ˙i = ξi+1 + L i+1 ξ1 − L i (ξ2 + L 2 ξ1 ), 2 ≤ i ≤ n − 1 ξ˙n = u − L n (ξ2 + L 2 ξ1 )

(6.78) (6.79) (6.80)

where ρ1 : R → R is an odd and strictly decreasing function, and L 2 , . . . , L n are positive constants. In the observer, ξ1 is an estimate of y, and ξi is an estimate of xi − L i y for 2 ≤ i ≤ n. For convenience of discussions, we define observation errors: ζ1 = y − ξ1 , ζi = xi − L i y − ξi , i = 2, . . . , n.

(6.81) (6.82)

Based on the estimation by the observer, we design a nonlinear control law in the following form: e1 = y

(6.83)

e2 = ξ2 − κ1 (e1 − ζ1 ) ei = ξi − κi−1 (ei−1 ), i = 3, . . . , n u = κn (en )

(6.84) (6.85) (6.86)

where κ1 , . . . , κn are continuously differentiable, odd, strictly decreasing and radially unbounded functions. Note that e1 − ζ1 = y − ζ1 = ξ1 . Thus, the control law uses only the state of the observer (ξ1 , . . . , ξn ), and is realizable. The desired system structure is shown in Fig. 6.9, which is in accordance with the general structure given by [17, Fig. 2]. It should be noted that Eq. (6.78) of the observer is constructed to estimate y by using the available y s . The function ρ1 in (6.78) is used to assign an appropriate, probably nonlinear, gain to the observation error system, more precisely, Eq. (6.90), to satisfy the cyclic-small-gain condition. Equations (6.79)–(6.80) are in the same

u

plant

control law

y

ξ1 , . . . , ξn

event trigger

ys

observer

observer-based controller Fig. 6.9 The schematic diagram of the event-triggered output-feedback control system

164

6 Event-Triggered Control of Nonlinear Uncertain Systems …

spirit of the reduced-order observer in [121]. Slightly differently, we use ξ1 instead of the unavailable y in (6.79)–(6.80).

6.3.3 ISS Property of the Subsystems In this subsection, we show that there exist ρ1 , L 2 , . . . , L n , κ1 , . . . , κn to transform the closed-loop system into a network of ISS subsystems. In particular, we prove that the subsystems with ζ1 , ζ¯2 , e1 , . . . , en as the states are ISS, where ζ¯2 = [ζ2 , . . . , ζn ]T . For the subsystems, we define the following ISS-Lyapunov function candidates: Vζ1 (ζ1 ) = |ζ1 |, 1  Vζ¯2 (ζ¯2 ) = ζ¯2T P ζ¯2 2 ,

(6.87)

Vei (ei ) = |ei |, i = 1, . . . , n,

(6.89)

(6.88)

where the symmetric and positive definite matrix P will be determined later. Based on the result in this subsection, in the following subsection, an ISS cyclicsmall-gain method is employed to ultimately transform the closed-loop system into the form of (5.124)–(5.125) with the conditions for event-triggered control satisfied. The ζ1 -Subsystem By taking the derivative of ζ1 , we have ζ˙1 = ρ1 (ζ1 + w) + φ1 (ζ1 , ζ2 , e1 , d)

(6.90)

φ1 (ζ1 , ζ2 , e1 , d) = L 2 ζ1 + ζ2 + f 1 (e1 , d).

(6.91)

where

Under Assumption 6.3, it can be directly verified that there exists a ψφ1 ∈ K∞ being Lipschitz on compact sets such that |φ1 (ζ1 , ζ2 , e1 , d)| ≤ ψφ1 (|[ζ1 , ζ2 , e1 , d T ]T |). The system (6.90) is in the form of the system (C.1), with ζ1 , w and ρ1 (ζ1 + w) ¯ respectively. With Lemma C.1, one can find a corresponding to η, ωm+1 and κ, continuously differentiable ρ1 such that for any constant 0 < c < 1, ζ1 > 0 and any ζ χζ21 , χeζ11 , χdζ1 ∈ K which are Lipschitz on compact sets, the ζ1 -subsystem is ISS with Vζ1 (ζ1 ) as an ISS-Lyapunov function, which satisfies

 ζ d Vζ1 (ζ1 ) ≥ max χζ21 (Vζ¯2 (ζ¯2 )), χeζ11 (Ve1 (e1 )), χw ζ1 (|w|), χζ1 (|d|) ⇒∇Vζ1 (ζ1 ) (ρ1 (ζ1 + w) + φ1 (ζ1 , ζ2 , e1 , d)) ≤ −ζ1 Vζ1 (ζ1 ) for almost all ζ1 , where

(6.92)

6.3 Event-Triggered Output-Feedback Control

165

χw ζ1 (s) =

s c

(6.93)

for s ∈ R+ . The ζ¯ 2 -Subsystem By taking the derivative of ζ¯2 , we can write the ζ¯2 -subsystem as ζ˙¯2 = Aζ¯2 + φ¯ 2 (ζ1 , e1 , d)

(6.94)

where ⎡

−L 2 ⎢ .. In−1 ⎢ . A=⎢ ⎣ −L n−1 −L n 0 · · · 0 ⎤ ⎡ φi2 (ζ1 , e1 , d) ⎥ ⎢ φ¯ 2 (ζ1 , e1 , d) = ⎣ ... ⎦

⎤ ⎥ ⎥ ⎥ ⎦

(6.95)

(6.96)

φn (ζ1 , e1 , d)

with φi (ζ1 , e1 , d) = (L i+1 − L i L 2 )ζ1 − L i f 1 (e1 , d) + f i (e1 , d), i = 2, . . . , n − 1

(6.97)

φn (ζ1 , e1 , d) = −L n L 2 ζ1 − L n f 1 (e1 , d) + f n (e1 , d).

(6.98)

By choosing L 2 , . . . , L n such that A is Hurwitz, there exists a positive definite matrix P = P T ∈ R(n−1)×(n−1) satisfying P A + A T P = −2In−1 . Define Vζ¯0 (ζ¯2 ) = 2 ζ¯T P ζ¯2 . Then, there exist α0 , α0¯ ∈ K∞ such that α0 (|ζ¯2 |) ≤ V 0 (ζ¯2 ) ≤ α0¯ (|ζ¯2 |). ζ¯2

2

ζ¯2

ζ2

With direct calculation, we have

ζ¯2

ζ2

∇Vζ¯0 (ζ¯2 )ζ˙¯2 2

= −2ζ¯2T ζ¯2 + 2ζ¯2T Pi φ¯ 2 (ζ1 , e1 , d) ≤ −ζ¯2T ζ¯2 + |P|2 |φ¯ 2 (ζ1 , e1 , d)|2   1 ζ Vζ¯0 (ζ¯2 ) + |P|2 ψ˘ φ¯1 (|ζ1 |) + ψ˘ φe¯1 (|e1 |) + ψ˘ φd¯ (|d|) , ≤− 2 2 2 λmax (P) 2

(6.99)

ζ where ψ˘φ¯1 , ψ˘ φe¯1 and ψ˘ φd¯ are K∞ functions such that 2

2

2

ζ |φ¯ 2 (ζ1 , e1 )|2 ≤ ψ˘ φ¯1 (|ζ1 |) + ψ˘ φe¯1 (|e1 |) + ψ˘ φd¯ (|d|) 2

2

2

(6.100)

166

6 Event-Triggered Control of Nonlinear Uncertain Systems …

for all ζ1 , e1 ∈ R and d ∈ Rn d . Moreover, under Assumption 6.3, ψ˘ φ¯1 , ψ˘ φe¯1 and ψ˘ φd¯ 2 2 2 ζ ζ can be written as ψ˘ 1 = ψ 1 (s 2 ), ψ˘ e1 = ψ e1 (s 2 ) and ψ˘ d = ψ d (s 2 ) for s ∈ R+ , with ζ

ζ ψφ¯1 , 2

φ¯ 2

ψφe¯1 2

and

ψφd¯ 2

φ¯ 2

φ¯ 2

φ¯ 2

φ¯ 2

φ¯ 2

being Lipschitz on compact sets.

This means that the ζ¯2 -subsystem is ISS with Vζ¯0 as an ISS-Lyapunov function. 2 ζ ζ The ISS gains can be chosen as follows. Define χ˘ 1 = 4λmax (P)|P 2 |ψ˘ 1 , χ˘ e1 = ζ¯2

φ¯ 2

4λmax (P)|P |ψ˘ φe¯1 and χ˘ dζ¯ = 4λmax (P)|P 2 |ψ˘ φd¯ . Then,

ζ¯2

2

2

2

2

 ζ Vζ¯0 (ζ¯2 ) ≥ max χ˘ ζ¯1 (Vζ1 (ζ1 )), χ˘ eζ¯1 (Ve1 (e1 )), χ˘ dζ¯ (|d|) 2 2 2 2   ⇒∇V ¯0 (ζ¯2 ) Aζ¯2 + φ¯ 2 (ζ1 , e1 , d) ≤ −ζ¯ V ¯0 (ζ¯2 ) ζ2

2

ζ2

(6.101)

where ζ¯2 = 1/4λmax (Pi ). ζ Hence, there exist χζ¯1 , χζe¯1 , χdζ¯ ∈ K being Lipschitz on compact sets and a con2 2 2 tinuous, positive definite αζ¯2 such that

 ζ Vζ¯2 (ζ¯2 ) ≥ max χζ¯1 (Vζ1 (ζ1 )), χζe¯1 (Ve1 (e1 )), χdζ¯ (|d|) 2 2 2   ⇒∇Vζ¯ (ζ¯2 ) Aζ¯2 + φ¯ 2 (ζ1 , e1 , d) ≤ −αζ¯ (Vζ¯ (ζ¯2 )) a.e. 2

2

2

(6.102)

The ei -Subsystems (i = 1, . . . , n) It can also be proved that the (e1 , . . . , en )-subsystem can be derived into the form e˙1 = κ1 (e1 − ζ1 ) + ϕ1 (e1 , e2 , ζ2 , d) e˙2 = κ2 (e2 ) + ϕ2 (e1 , e2 , e3 , ζ1 , w)

(6.103) (6.104)

e˙i = κi (ei ) + ϕi (e1 , e2 , . . . , ei+1 , ζ1 , w), i = 3, . . . , n.

(6.105)

For convenience of notations, we denote e˙1 = h 1 (e1 , e2 , ζ1 , ζ2 , d), e˙2 = h 2 (e1 , e2 , e3 , ζ1 , w) and e˙i = h i (e1 , . . . , ei+1 , ζ1 , w) for i = 3, . . . , n. Moreover, under Assumption 6.3, there exist ψϕ1 , . . . , ψφn ∈ K∞ being Lipschitz on compact sets such that |ϕ1 (e1 , e2 , ζ2 , d)| ≤ ψϕ1 (|[e1 , e2 , ζ2 , d T ]T |)

(6.106)

|ϕ2 (e1 , e2 , e3 , ζ1 , w)| ≤ ψϕ2 (|[e1 , e2 , e3 , ζ1 , w] |) T

(6.107)

|ϕi (e1 , e2 , . . . , ei , ζ1 , w)| ≤ ψϕi (|[e1 , e2 , . . . , ei , ζ1 , w] |), i = 3, . . . , n. (6.108) T

With Lemma C.1, we can find continuously differentiable κi for i = 1, . . . , n such that each ei -subsystem is ISS with Vei (ei ) = |ei | as an ISS-Lyapunov function. Specifically, we have

6.3 Event-Triggered Output-Feedback Control

167

 ¯ ¯ Ve1 (e1 ) ≥ max χee21 (Ve2 (e2 )), χζe11 (Vζ1 (ζ1 )), χeζ21 (Vζ¯2 (ζ¯2 )), χeζ21 (Vζ¯2 (|d|) (6.109) ⇒∇Ve1 (e1 )h 1 (e1 , e2 , ζ1 , ζ2 , d) ≤ −e1 Ve1 (e1 ), a.e.  e1  e3 ζ1 w Ve2 (e2 ) ≥ max χe2 (Ve1 (e1 )), χe2 (Ve3 (e3 )), χe2 (Vζ1 (ζ1 )), χe2 (|w|) ⇒∇Ve2 (e2 )h 2 (e1 , e2 , e3 , ζ1 , w) ≤ −e2 Ve2 (e2 ), a.e.

(6.110)

and for i = 3, . . . , n, Vei (ei ) ≥

max

j=1,...,i−1,i+1



χeij (Ve j (e j )), χζe1i (Vζ1 (ζ1 )), χw ei (|w|) e

⇒∇Vei (ei )h i (e1 , . . . , ei+1 , ζ1 , w) ≤ −ei Vei (ei ), a.e.

 (6.111)

where the (·) ’s can be any specified positive constants, χeenn +1 = 0, χζe11 (s) =

s c

(6.112)

for s ∈ R+ , where c can be chosen to be any constant satisfying 0 < c < 1, and the other χ(·) (·) ’s can be any K∞ functions which are Lipschitz on compact sets.

6.3.4 Event-Triggered Output-Feedback Control With the nonlinear observer-based controller, the closed-loop system has been transformed into a network of ISS subsystems with ζ1 , ζ¯2 , e1 , . . . , en as the states. In this subsection, we rewrite the (ζ1 , ζ¯2 , e1 , . . . , en )-system into the form of (5.124)–(5.125) and design an event-triggered controller without infinitely fast sampling. Since only the output y, i.e., e1 , is available to the event trigger, we consider the (ζ1 , ζ¯2 , e2 , . . . , en )-subsystem as the subsystem (5.124) and consider the e1 -subsystem as the subsystem (5.125). For convenience of notations, we define z = [ζ1 , ζ¯2T , e2 , . . . , en ]T and denote z˙ = h(z, e1 , w, d). The main result on event-triggered output-feedback control is given by Theorem 6.5. Theorem 6.5 Under Assumption 6.3, the closed-loop system composed of (6.73)– (6.75), (6.78)–(6.80) and (6.83)–(6.86) can be transformed into an interconnection of two ISS subsystems with the measurement error w caused by data-sampling as the external input. Moreover, asymptotic stabilization can be achieved through eventtriggered control without infinitely fast sampling by using event trigger composed of (5.87), (5.6), (5.7) and (5.88).

168

6 Event-Triggered Control of Nonlinear Uncertain Systems …

Proof By taking the derivation of e1 , we have e˙1 = κ1 (e1 − ζ1 ) + L 2 e1 + e2 + ζ2 + f 1 (e1 , d) := g(e1 , z, d).

(6.113)

It can be verified that Assumption 5.6 is satisfied. We choose the χ(·) (·) ’s such that • the (ζ1 , ζ¯2 , e1 , . . . , en )-system satisfies the cyclic-small-gain condition;  −1 • each χ(·) and the corresponding χ(·) are Lipschitz on compact sets. (·) (·) ∈ K∞ being continuously differentiable on (0, ∞) such that Then, we can find χˆ (·)  −1 (·) (·) (·) are Lipschitz on compact sets and the cyclic-small-gain condition χˆ (·) and χˆ (·) is still satisfied if the χ(·) ˆ (·) (·) ’s are replaced by their corresponding χ (·) ’s. Then, with the Lyapunov-based ISS cyclic-small-gain theorem,   V (z, e1 ) = max Vz (z), Ve1 (e1 )

(6.114)

is an ISS-Lyapunov function of the (z, e1 )-system, where Vz (z) = max

i=2,...,n

  σζ1 (Vζ1 (ζ1 )), σζ¯2 (Vζ¯2 (ζ¯2 )), σei (Vei (ei ))

(6.115)

with the σ(·) ’s being appropriate compositions of the χˆ (·) (·) ’s. By using cyclic-small-gain theorem in Sect. 2.4.1, Vz (z) is an ISS-Lyapunov function of the z-subsystem satisfying   d Vz (z) ≥ max χez1 (Ve1 (e1 )), χw z (|w|), χz (|d|) ⇒∇Vz (z)h(z, e1 , w, d) ≤ −αz (Vz (z)) a.e.

(6.116)

d where χez1 , χw z , χz ∈ K∞ and αz is a continuous, positive definite function. Moreover, e1 χz < Id. Also, by using (6.115) and (6.109), we have

Ve1 (e1 ) ≥ max{χez1 (Vz (z)), χde1 (|d|)} ⇒∇Ve1 (e1 )g(e1 , z, d) ≤ −e1 Ve1 (e1 ) a.e.

(6.117)

where χez1 , χde1 ∈ K∞ and χez1 < Id. Clearly, the interconnection of the z-subsystem and the e1 -subsystem also satisfies the small-gain condition, i.e., χez1 ◦ χez1 < Id.

(6.118)

6.3 Event-Triggered Output-Feedback Control

169

The closed-loop system has been transformed into an interconnection of two ISS subsystems (with z and e1 as the states) satisfying the small-gain condition. Note that all the gain functions and their inverse functions are chosen to be Lipschitz on compact sets. Thus, their compositions are also Lipschitz on compact sets. Then, it can be proved that the conditions (5.131) and (5.132) in Corollary 5.3 are also satisfied. This ends the proof of Theorem 6.5.  If d ≡ 0 in the system (6.73)–(6.75), then we can rewrite the (ζ1 , ζ¯2 , e1 , . . . , en )system in the form of (4.1)–(4.2). We consider the (ζ1 , ζ¯2 , e2 , . . . , en )-subsystem as the subsystem (4.1) and consider the e1 -subsystem as the subsystem (4.2). Theorem 6.6 Under Assumption 6.3, the closed-loop system composed of (6.73)– (6.75), (6.78)–(6.80) and (6.83)–(6.86) can be transformed into an interconnection of two ISS subsystems with the measurement error w caused by data-sampling as the external input. Moreover, if d ≡ 0, then asymptotic stabilization can be achieved through event-triggered control without infinitely fast sampling by using event trigger (4.4) with the threshold signal μ(t) generated by (4.30) or more generally (4.66). Theorem 6.6 can be proved similarly as for Theorem 6.5. We employ an example to demonstrate the design procedure. Example 6.3 Consider the following nonlinear system in the output-feedback form: x˙1 = x2 + 0.1|x1 | + 0.1d, x˙2 = u + 0.01 sin(x1 ),

(6.119) (6.120)

y = x1 ,

(6.121)

where x = [x1 , x2 ]T ∈ R2 is the state, u ∈ R is the control input, y ∈ R is the output, and d ∈ R represents the external disturbance. Following the design procedure in Sect. 6.3.2, an observer is designed as ξ˙1 = ξ2 + 0.2ξ1 − 5(ξ1 − y s ), ξ˙2 = u − 0.2(ξ2 + 0.2ξ1 ).

(6.122) (6.123)

Based on the estimation by the observer, the control law is designed as u = −26(ξ2 + 3.25ξ1 ).

(6.124)

By defining Vζ1 (ζ1 ) = |ζ1 |, Vζ¯2 (ζ¯2 ) = |ζ¯2 |, Ve1 (e1 ) = |e1 | and Ve2 (e2 ) = |e2 |, it ζ¯

can be verified that all the subsystems are ISS with χζ21 (s) = 0.26s, χeζ11 (s) = ζ

0.19s, χdζ1 (s) = s, χζ¯1 (s) = 0.96s, χeζ¯1 (s) = 0.72s, χdζ¯ (s) = 1.2s, χee21 (s) = 0.90s, ζ

ζ¯

2

2

2

ζ

χe11 (s) = 5s, χe21 (s) = 0.90s, χde1 (s) = s, χee12 (s) = 0.90s, χe12 (s) = 0.90s, χw e2 (s) = 5s, ζ1 (s) = 0.1s, ζ¯2 (s) = 0.1s, e1 (s) = 0.1s, and e2 (s) = 0.1s for s ∈ R+ .

170

6 Event-Triggered Control of Nonlinear Uncertain Systems … 5

x

1

4

x2

system state

3 2 1 0 -1 -2

0

2

4

6

8

10 time

12

14

16

18

20

Fig. 6.10 The trajectories of the system states x1 and x2 with the event trigger (5.87) in Example 6.3 3 1 2

observer state

2 1 0 -1 -2

0

2

4

6

8

10 time

12

14

16

18

20

Fig. 6.11 The trajectories of the observer states ξ1 and ξ2 with the event trigger (5.87) in Example 6.3

Define Vz (z) = max{5.1|ζ1 |, |ζ2 |, |e2 |}. It can be verified that χez1 (s) = 0.96s, = 5s, χdz (s) = 5.1s, αz (s) = 0.1s, and χez1 (s) = 0.98s, L ag (s) = 10, L bg (s) = 1 for s ∈ R+ . By using Theorem 6.5, γwx and γwd defined in (5.87) are chosen as γwx (s) = 0.19s and γwd (s) = 0.014s for s ∈ R+ , and ϕ defined in (5.7) is chosen as ϕ(s) = s for s ∈ R+ . The simulation result with initial state x1 (0) = −2, x2 (0) = 2, ξ1 (0) = 0, ξ2 (0) = 0, ς1 (0) = 1 and d(t) = (cos(17t) + sin(15t) + sin(21t) + sin(11t))/5 is shown in Figs. 6.10, 6.11, 6.12 and 6.13. It is shown that the objective of event-triggered output-feedback control is achieved for the system (6.119)–(6.121). χw z (s)

6.3 Event-Triggered Output-Feedback Control

171

25 20

control input

15 10 5 0 -5 -10

0

2

4

6

8

10 time

12

14

16

18

20

16

18

20

Fig. 6.12 The control input with the event trigger (5.87) in Example 6.3

inter-sampling intervals

101

100

-1

10

-2

10

10-3

0

2

4

6

8

10 time

12

14

Fig. 6.13 The inter-sampling intervals with the event trigger (5.87) in Example 6.3

If d ≡ 0 in the system (6.73)–(6.75), then we can rewrite the (ζ1 , ζ¯2 , e1 , . . . , en )system into the form of (4.1)–(4.2). The event-triggered output-feedback control for the system (6.119)–(6.121) can be achieved by using event trigger (4.4) with the threshold signal μ(t) generated by μ(t) ˙ = e−0.09t . The simulation result with initial state x1 (0) = 2, x2 (0) = −1, ξ1 (0) = 0, and ξ2 (0) = 0 is shown in Figs. 6.14, 6.15, 6.16 and 6.17.

172

6 Event-Triggered Control of Nonlinear Uncertain Systems … 2

x

1

1

x2

system state

0 -1 -2 -3 -4 -5

0

5

10

15

20

25 time

30

35

40

45

50

Fig. 6.14 The trajectories of the system states x1 and x2 with the event trigger (4.4) in Example 6.3 2 1

observer state

1

2

0 -1 -2 -3 -4

0

5

10

15

20

25 time

30

35

40

45

50

Fig. 6.15 The trajectories of the observer states ξ1 and ξ2 with the event trigger (4.4) in Example 6.3

6.4 Notes Input-to-state stabilization in the presence of measurement errors plays a crucial role in the designs for event/self-triggered control in the Chaps. 3–5. This could be easily achievable for stabilizable linear systems. However, small converging measurement error may cause the performance of a nonlinear control system to deteriorate, even if the system with no measurement error is asymptotically stable; see, [63, 64] and Examples 1.5 and 1.6. Despite its importance, robust nonlinear control in the presence of measurement errors has not received considerable attention in the present literature. In [64], a controller is designed with set-valued maps and “flattened”

6.4 Notes

173

15 10

control input

5 0 -5 -10 -15 -20 -25

0

5

10

15

20

25 time

30

35

40

45

50

40

45

50

Fig. 6.16 The control input with the event trigger (4.4) in Example 6.3

inter-sampling intervals

101 0

10

-1

10

10-2 -3

10

10-4

0

5

10

15

20

25 time

30

35

Fig. 6.17 The inter-sampling intervals with the event trigger (4.4) in Example 6.3

Lyapunov functions following the backstepping methodology such that the control system is ISS with respect to the measurement disturbances. Reference [117] studies nonlinear systems composed of two subsystems, one is ISS and the other one is input-to-state stabilizable with respect to the measurement disturbance. In [117], the ISS of the control system is guaranteed by the gain assignment technique introduced in [116, 122, 223] and the nonlinear small-gain theorem proposed in [118, 122]. In [147], it is shown that, for general nonlinear control systems, the existence of smooth Lyapunov functions is equivalent to the existence of (possibly discontinuous) feedback stabilizers which are robust with respect to small measurement errors and small additive external disturbances. Discontinuous controllers are developed in [147] for

174

6 Event-Triggered Control of Nonlinear Uncertain Systems …

a class of nonlinear systems such that the closed-loop system is insensitive to small measurement errors. In this chapter, we first introduce a design for event-triggered control of the benchmark nonlinear systems in the strict-feedback form [143]. Motivated by the set-valued map design in [167, 168], we employ set-valued maps to cover the influence of the sampling error, and transform the closed-loop system into a network of ISS subsystems. With the cyclic-small-gain theorem in Sect. 2.4, ISS of the closed-loop system with the sampling error as the input is guaranteed, and the influence of the sampling error is explicitly evaluated. More importantly, it is shown that a nonlinear control law can be designed to satisfy the gain condition to avoid infinitely fast sampling as long as the system has an equilibrium at the origin and the function of the system dynamics is locally Lipschitz. For the nonlinear uncertain systems in the output-feedback form; see [143], the controller composed of an ISS-induced nonlinear observer and a nonlinear control law is developed. By appropriately choosing the parameters, the controlled system can be ultimately transformed into an interconnection of two ISS subsystems with the sampling error of the output as the external input. Following the approaches in Chaps. 4 and 5, the event trigger design is proposed for output-feedback stabilization. In the event-triggered output-feedback control results [17, 84, 154, 195], observers are also employed to reconstruct the plant state by using the sampled data of the outputs. However, these results mainly focus on linear systems. The strict-feedback form in [143] has also been used to model practical time-delay systems, with cascade connected reactors as a typical example [13, 83, 199]. Quite a few results have been developed for stability analysis and control design of uncertain nonlinear time-delay systems; see, e.g., [70, 76, 97, 129, 149, 150, 186, 214, 231, 233, 244, 275, 287, 290] and the references therein. It should be noted that most of the results cannot be directly used to deal with state sampling errors. On the other hand, the existing event-triggered controllers for strict-feedback systems (see, e.g., [165, 269]) have not taken into account time-delays. This chapter also proposes a small-gain-based solution to the event-triggered stabilization problem for a class of nonlinear time-delay systems in the strict-feedback form. The time-delay small-gain theorem [253] is used to guarantee the closed-loop stability. The designs in this chapter are based on a refined gain assignment technique. Indeed, gain assignment is a crucial tool for small-gain based nonlinear control designs [116, 122, 223]. Reference [117, Proposition 4.1] presents a gain assignment technique to guarantee the ISS of the control system with respect to the measurement disturbance and the gain from the measurement disturbance to the corresponding output is assigned to be of class K∞ .

Chapter 7

Event-Triggered Control of Nonholonomic Systems

This chapter studies the event-triggered control problem for nonholonomic systems in the chained form with disturbances and drift uncertain nonlinearities. Both the cases of state-feedback and output-feedback are investigated. To address the effects of nonholonomic constraints in event-triggered control, a new systematic design integrating a state-scaling technique and set-valued maps is proposed. A crucial strategy is to transform the event-triggered control system into an interconnection of multiple input-to-state stable systems, to which the cyclic-small-gain theorem is applied for event-based controller synthesis. It is shown that the cyclic-small-gain based design scheme leads to Zeno-free event-triggered controllers. For the outputfeedback case, a new nonlinear observer is designed to deal with the sampling errors. Interestingly, the obtained results are new even if the plant model is disturbancefree. Both numerical and experimental results validate the efficiency of the proposed cyclic-small-gain-based event-triggered control methodology.

7.1 Event-Triggered State Feedback Control of Uncertain Nonholonomic Systems 7.1.1 Problem Formulation We consider a class of nonholonomic systems subject to disturbances and drift uncertain nonlinearities:

© Huazhong University of Science and Technology Press, Wuhan and Springer Nature Singapore Pte Ltd. 2020 T. Liu et al., Robust Event-Triggered Control of Nonlinear Systems, Research on Intelligent Manufacturing, https://doi.org/10.1007/978-981-15-5013-3_7

175

176

7 Event-Triggered Control of Nonholonomic Systems

x˙0 = d0 (t)u 0 + φ0 (t, x0 ) x˙1 = d1 (t)x2 u 0 + φ1 (t, x0 , x, u 0 ) .. .

(7.1)

x˙n−2 = dn−2 (t)xn−1 u 0 + φn−2 (t, x0 , x, u 0 ) x˙n−1 = dn−1 (t)u + φn−1 (t, x0 , x, u 0 ) where x0 ∈ R and x = [x1 , · · · , xn−1 ]T ∈ Rn−1 are the states, u 0 ∈ R and u ∈ R are the control inputs, and, for each i = 0, 1, . . . , n − 1, the functions di and φi are piece-wise continuous in t and locally Lipschitz with respect to the other arguments. In addition, we assume Assumption 7.1 For each i = 0, . . . , n − 1, there exist positive constants ci1 and ci2 such that ci1 ≤ di (t) ≤ ci2

(7.2)

for all t ≥ 0. Assumption 7.2 There exists a positive constant a0 such that |φ0 (t, x0 )| ≤ a0 |x0 | holds for all (t, x0 ), and for each i = 1, . . . , n − 1, there exists a nonnegative, smooth function φid : R × . . . × R → R+ such that |φi (t, x0 , x, u 0 )| ≤ |[x1 , · · · , xi ]T |φid (x0 , x1 , · · · , xi , u 0 ) holds for all (t, x0 , x, u 0 ). The model (7.1) is known as a perturbed version of the disturbance-free chained form [201] (with di (t) ≡ 1 and φi ≡ 0 for 0 ≤ i ≤ n − 1 in (7.1)). See [109] for examples and discussions on the motivation of the model. It should be noted that, even for the disturbance-free system, few results on event-triggered stabilization have been reported. For the system (7.1), this section aims to design an event-triggered controller to globally asymptotically regulate the states to the origin. Specifically, the eventtriggered controller is expected to be in the form of u 0 (t) = κ0 (x0 (tk ), x(tk )) u(t) = κ(x0 (tk ), x(tk ))

(7.3) (7.4)

for t ∈ [tk , tk+1 ), k ∈ S ⊆ Z+ , with tk being generated by an event trigger such that the following objectives are achieved: for any initial state (x0 (t0 ), x(t0 )),

7.1 Event-Triggered State Feedback Control of Uncertain Nonholonomic Systems

177

(1) the states (x0 (t), x(t)) of the closed-loop system (7.1), (7.3) and (7.4) are welldefined and bounded over [t0 , ∞), and lim x0 (t) = 0,

t→∞

lim x(t) = 0;

t→∞

(2) there exists a TΔ > 0 such that tk+1 − tk ≥ TΔ for all k, k + 1 ∈ S.

7.1.2 Event-Triggered State-Feedback Controller The system (7.1) features a cascade connection of the x0 -subsystem and the xsubsystem, which suggests a two-stage design. Intuitively, one may first design a control law for u 0 to stabilize the x0 -subsystem, and then focus on the stabilization problem of the x-subsystem by considering x0 and u 0 as the external inputs. As for the case of non-event-triggered control, it is critical to retain the controllability of the x-subsystem by avoiding u 0 = 0. In the case of event-triggered control, this is guaranteed by introducing a sector bound to the controller design for u 0 . That is, the x0 -subsystem can be asymptotically stabilized as long as the event-triggered sampling error is within a specific sector bound depending on x0 . Then, the σ-scaling technique [15] is applied to the x-subsystem for an equivalent lower-triangular system, and a recursive design with set-valued virtual control laws is proposed to address the stabilization problem in the presence of the sampling errors. Controller Design for u0 For the x0 -subsystem, consider the following negative-feedback control law: u 0 = −λ0 x0s

(7.5)

where λ0 is a positive design parameter to be determined later, and x0s represents the sampled value of x0 . Define w0 = x0s − x0

(7.6)

as the sampling error. Then, the control law (7.5) can be rewritten as u 0 = −λ0 (x0 + w0 ).

(7.7)

The controlled x0 -subsystem has the following property. Proposition 7.1 Consider the system composed of the x0 -subsystem in (7.1) and the control law (7.5). If for some t0 < Tmax ≤ ∞, w0 satisfies

178

7 Event-Triggered Control of Nonholonomic Systems

|w0 (t)| ≤ δ0 |x0 (t)|

(7.8)

for all t0 ≤ t < Tmax , with constant δ0 satisfying 0 < δ0


a0 , c01 − c02 δ0

(7.10)

then x0 (t) exists for all t0 ≤ t < Tmax and satisfies   x0 (t) ∈ x0 (t0 )e−a1 (t−t0 ) , x0 (t0 )e−a2 (t−t0 )

(7.11)

for all t0 ≤ t < Tmax , with constants a1 = λ0 c02 (1 + δ0 ) + a0 and a2 = λ0 c01 − λ0 c02 δ0 − a0 . With the satisfaction of (7.8)–(7.10), the existence of solutions to the x0 -subsystem for t0 ≤ t < Tmax and the satisfaction of condition (7.11) can be directly proved by using Gronwall–Bellman inequality [138]. A detailed proof of Proposition 7.1 is omitted here due to space limitation. Controller Design for u (A) State Scaling Proposition 7.1 means that if x0 (t0 ) = 0, then x0 (t) = 0 for all t ≥ t0 . Motivated by the σ-scaling technique [15], we consider the following transformation: zi =

xi x0n−i−1

(7.12)

for i = 1, . . . , n − 1. Clearly, (7.12) is discontinuous at x0 = 0. With z i for i = 1, . . . , n − 1 as the new states, the x-subsystem defined by (7.1) is transformed into z˙ i =

(n − i − 1)(d0 u 0 + φ0 ) di u 0 φi z i+1 + n−i−1 − z i , i = 1, . . . , n − 2, x0 x0 x0 (7.13)

z˙ n−1 = dn−1 u + φn−1 (t, x0 , x, u 0 ).

(7.14)

Substituting u 0 given by (7.5) into (7.13) yields z˙ i = λi (t, x0 , w0 )z i+1 + φ¯ i (t, x0 , w0 , x), i = 1, . . . , n − 2, z˙ n−1 = λn−1 (t, x0 , w0 )u + φ¯ n−1 (t, x0 , w0 , x),

(7.15)

7.1 Event-Triggered State Feedback Control of Uncertain Nonholonomic Systems

179

where   w0 , i = 1, . . . , n − 2, λi (t, x0 , w0 ) = −λ0 di (t) 1 + x0 λn−1 (t, x0 , w0 ) = dn−1 (t), (7.16) φi (t, x0 , x, −λ0 (x0 + w0 )) φ¯ i (t, x0 , w0 , x) = x0n−i−1 (n − i − 1)(−λ0 d0 (t)(x0 + w0 ) + φ0 ) − zi , i = 1, . . . , n − 1. x0 Proposition 7.2 gives a property on the transformed z i -subsystems, which is to be used for the recursive design below. Proposition 7.2 Consider the system (7.15). Suppose that Assumptions 7.1 and 7.2 are satisfied, and condition (7.8) holds for all t0 ≤ t < Tmax with some t0 < Tmax ≤ ∞. Then, for each i = 1, . . . , n − 1, there exist positive constants ki1 and ki2 such that ki1 ≤ |λi (t, x0 , w0 )| ≤ ki2

(7.17)

holds for all t0 ≤ t < Tmax and all x0 , w0 , and for each i = 1, . . . , n − 1, there exists a φ¯ id ∈ K∞ being Lipschitz on compact sets such that |φ¯ i (t, x0 , w0 , x)| ≤ φ¯ id ([x0 , z 1 , · · · , z i ]T )

(7.18)

holds for all t0 ≤ t < Tmax and all x0 , w0 , x. The proof of Proposition 7.2 is given in Appendix D.9. The treatment with σ-scaling is crucial for the controller design for u. If x0 (t0 ) = 0, Proposition 7.1 implies that x0 (t) is defined and (7.8) holds for all t0 ≤ t < Tmax , which guarantee the validity of transformation (7.12) for all t0 ≤ t < Tmax . The case of x0 (t0 ) = 0 is addressed in Sect. 7.1.4. (B) Recursive Control Design with Set-Valued Maps The transformed z-system (7.15) is in the lower-triangular form, for which, we propose a recursive design for robust stabilization with respect to the sampling errors. For i = 1, . . . , n − 1, we use z is to represent the sampled value of z i , and define the sampling error as wi = z is − z i .

(7.19)

Also, for convenience of discussions, denote w = [w1 , . . . , wn−1 ]T .

(7.20)

180

7 Event-Triggered Control of Nonholonomic Systems

In the design, it is first assumed that w is piece-wise continuous and bounded for t ∈ [t0 , Tmax ) with some t0 < Tmax ≤ ∞, and denote wT = w [t0 ,Tmax ) , wi T =

wi [t0 ,Tmax ) , and w¯ i T = [w1T , . . . , wi T ]T . Following the approach in Sect. 6.1, we propose a recursive design based on setvalued virtual control laws to deal with the sampling errors. Initial Step: Let e1 = z 1 . Rewrite the e1 -subsystem as e˙1 = λ1 (z 2 − e2 ) + φ¯ 1 (t, x0 , w0 , x) + λ1 e2 , where λ1 := λ1 (t, x0 , w0 ) and e2 is a new state variable to be defined later. Define a set-valued map S1 as S1 (z 1 , w1T ) = {sgn(λ1 )κ1 (z 1 + w1 ) : |w1 | ≤ w1T } where κ1 is a continuously differentiable, odd, strictly decreasing, and radially unbounded function, to be determined later. Define e2 = d(z 2 , S1 (z 1 , w1T )).

(7.21)

Then, we have z 2 − e2 ∈ S1 (z 1 , w1T ). Here, d is defined in (6.15). With a continuously differentiable κ1 , the boundaries of S1 (z 1 , w1T ), i.e., max S1 (z 1 , w1T ) and min S1 (z 1 , w1T ), are continuously differentiable almost everywhere and the derivative of e2 exists almost everywhere. Then, one may use a differential inclusion to represent the dynamics of the e2 -subsystem. As a result, the problem caused by the nondifferentiable sampling error is solved. In the following procedure, the new ei -subsystems (2 ≤ i ≤ n − 1) are derived in a recursive manner and are represented by differential inclusions. Recursive Step: For convenience, denote S0 (¯z 0 , w¯ 0T ) = {0}. For each j = 1, . . . , i, define a set-valued map S j as S j (¯z j ,w¯ j T ) = {sgn(λ j )κ j (z j + w j − p ∗j−1 ) : p ∗j−1 ∈ S j−1 (¯z j−1 , w¯ ( j−1)T ), |w j | ≤ w j T }

(7.22)

where z¯ j = [z 1 , · · · , z j ] and κ j : R → R is a continuously differentiable, odd, strictly decreasing, and radially unbounded function. Specifically, the κi is designed as κi (r ) = −νi (|r |)r

(7.23)

for r ∈ R, with νi : R+ → R+ being positive, nondecreasing and continuously differentiable on (0, ∞) to be determined later. For each k = 1, . . . , i, define ek+1 as

7.1 Event-Triggered State Feedback Control of Uncertain Nonholonomic Systems

ek+1 = d(z k+1 , Sk (¯z k , w¯ kT )).

181

(7.24)

Proposition 7.3 shows that with the set-valued maps defined above, the closedloop system can be transformed into an interconnection of subsystems with ei defined above as the states. Later, it is shown that each ei -subsystem can be input-to-state stabilized. Proposition 7.3 Suppose that x0 = 0. Consider the z-system defined by (7.15) with conditions (7.17) and (7.18) satisfied. With Si defined in (7.22) for i = 1, . . . , n − 1, when ei = 0, the ei -subsystem can be represented by e˙i = λi z i+1 + Φi (t, x0 , w0 , x, z¯ i )

(7.25)

T , x0 ]T |) |Φi (t, x0 , w0 , x, z¯ i )| ≤ ψΦi (|[e¯iT , w¯ (i−1)T

(7.26)

where

with ψΦi ∈ K∞ , e¯i = [e1 , · · · , ei ] and en = 0. If, moreover, the φ¯ k ’s given by (7.18) for k = 1, . . . , i are Lipschitz on compact sets, then one can find a ψΦi ∈ K∞ which is Lipschitz on compact sets. The proof of Proposition 7.3 is given in Appendix D.10. Then, the ei -subsystem can be rewritten as e˙i = λi (z i+1 − ei+1 ) + Φi (t, x0 , w0 , x, z¯ i ) + λi ei+1 ,

(7.27)

z i+1 − ei+1 ∈ Si (¯z i , w¯ i T )

(7.28)

where

according to the definition of ei+1 (7.24). Substituting (7.28) into (7.27), we have e˙i ∈ λi Si (¯z i , w¯ i T ) + Φi (t, x0 , w0 , x, z¯ i ) + λi ei+1 , =: Fi (t, x0 , w0 , x, z¯ i , w¯ i T , ei+1 ).

(7.29)

Final Step: At Step i = n − 1, the true control input u occurs, and thus we can set en = 0. Indeed, the desired controller u can be chosen as follows: p1 = sgn(λ1 )κ1 (z 1s ) pi = sgn(λi )κi (z is − pi−1 ), i = 2, · · · , n − 2 s u = sgn(λn−1 )κn−1 (z n−1 − pn−2 ).

Recall (7.19). It is directly checked that u ∈ Sn−1 (¯z n−1 , wT ).

(7.30) (7.31) (7.32)

182

7 Event-Triggered Control of Nonholonomic Systems

Proposition 7.4 Consider each ei -subsystem defined by (7.29) with Si defined in (7.22) and Φi satisfying (7.26). For any γeeik ∈ K∞ being Lipschitz on compact sets with k = 1, . . . , i − 1, i + 1, any γewi k ∈ K∞ being Lipschitz on compact sets with k = 1, . . . , i − 1, any γexi0 ∈ K∞ being Lipschitz on compact sets and any constant 0 < bi < 1, there exists a continuously differentiable, odd, strictly decreasing, and radially unbounded κi in the form of (7.23) such that Vi (ei ) = |ei | is an ISS-Lyapunov function for the ei -subsystem satisfying  Vi (ei ) ≥ ⇒

max

k=1,...,i−1

max

e

γeeik (Vk (ek )), γeii+1 (Vi+1 (ei+1 )), γewi k (wkT ), γewi i (wi T ), γexi0 (|x0 |)

f i ∈Fi (t,x0 ,w0 ,x,¯z i ,w¯ i T ,ei+1 )



∇Vi f i ≤ −i Vi (ei )

where γewi i (s) = s/bi for s ∈ R+ . The proof of Proposition 7.4 is given in Appendix D.11. Based on Proposition 7.4, we have an interconnection of ISS ei -subsystems. Proposition 7.5 shows that the interconnected system is ISS by appropriately designing the ISS gains of the ei -subsystems to satisfy the nonlinear small-gain condition. Proposition 7.5 For the ei -subsystems defined by (7.29) with Si given by (7.22) for i = 1, . . . , n − 1, by appropriately choosing κi for i = 1, . . . , n − 1, it can be achieved that i ◦ γeei1 < Id γee12 ◦ γee23 ◦ γee34 ◦ · · · ◦ γeei−1 e3 e4 ei γe2 ◦ γe3 ◦ · · · ◦ γei−1 ◦ γeei2 < Id .. . e i ◦ γeii−1 < Id γeei−1

(7.33)

and all the γ’s in (7.33) are Lipschitz on compact sets, for i = 1, . . . , n − 1. Moreover, with such κi for i = 1, . . . , n − 1, there exist βz ∈ KL and γzw , γzx0 ∈ K being Lipschitz on compact sets such that for any z(t0 ), any wT and any continuous and bounded x0 , |z(t)| ≤ max{βz (|z(t0 )|, t), γzw (wT ), γzx0 ( x0 [t0 ,Tmax ) )} holds for all t0 ≤ t < Tmax . The proof of Proposition 7.5 is given in Appendix D.12.

(7.34)

7.1 Event-Triggered State Feedback Control of Uncertain Nonholonomic Systems

183

7.1.3 Main Result Clearly, the event triggers for x0 and z cannot be designed separately because of their interconnection. In fact, the discussions on the z-system is valid only when x0 = 0, which is guaranteed by the sector bound condition (7.8) on the sampling error. Taking advantage of the cascade connection of x0 -subsystem and x-subsystem, we introduce an event trigger which is a combination of two designs: a sector bound design for x0 and a small-gain design for z. For the z-subsystem, x0 is considered as the external input, and to avoid infinitely fast sampling, x0 together with z is introduced to the newly designed threshold signal. Specifically, the event trigger is designed as  tk+1 = inf t > tk :|w0 (t)| ≥ δ0 |x0 (t)| or

|w(t)| ≥ max{ρz (|z(t)|), ρx0 (|x0 (t)|)} ,

(7.35)

where ρz and ρx0 are class K∞ functions with (ρz )−1 and (ρx0 )−1 being Lipschitz on compact sets. The first condition in (7.35) guarantees the satisfaction of condition (7.8) as long as w0 is defined, which can be considered as a static gain interconnection from x0 to w0 . The second condition in (7.35) implies that |w(t)| ≤ max{ρz (|z(t)|), ρx0 (|x0 (t)|)}

(7.36)

as long as w is defined, which can be considered as a static gain interconnection from z and x0 to w. We consider the closed-loop event-triggered system as an interconnected system, with the interconnections described by gains; see Fig. 7.1. In the rest of this subsection, we show that the objective of event-triggered stabilization can be achieved by appropriately choosing the interconnection gains. Theorem 7.1 presents the main result on event-triggered robust stabilization of the nonholonomic system (7.1) with state feedback.

ρx0 ρz w0

x0 δ0

z γzx0

w γzw

Fig. 7.1 The closed-loop event-triggered system considered as an interconnected system: the dashed lines represent the gains that are adjustable by designing the event trigger

184

7 Event-Triggered Control of Nonholonomic Systems

Theorem 7.1 Consider the system (7.1) with event-triggered state-feedback controller composed of (7.5), (7.30)–(7.32) and (7.35). Suppose that x0 (t0 ) = 0. With Assumptions 7.1 and 7.2 satisfied, the objectives of event-triggered robust stabilization is achievable if 1. ρz in (7.35) is of class K∞ with (ρz )−1 being Lipschitz on compact sets and ρz ◦ γzw < Id

(7.37)

where γzw is given in Proposition 7.5 and designed to be Lipschitz on compact sets; 2. ρx0 in (7.35) is of class K∞ with (ρx0 )−1 being Lipschitz on compact sets; 3. δ0 in (7.35) satisfies 0 < δ0
tk : |w0 (t)| ≥ δ0 |x0 (t)|} ,  as the next time instant independently triggered by and define tk+1

 tk = inf t > tk : |w(t)| ≥ max{ρz (|z(t)|), ρx0 (|x0 (t)|)} . Clearly, tk+1 that is generated by the event trigger (7.35) satisfies min{tk , tk }. Our basic idea is to show that there exists a positive lower bound that might depend on x0 (tk ) and z(tk ) for both tk − tk and tk − tk . We first show the lower bound of tk − tk . Define

Ω1 (x0 (tk )) = x0 ∈ R : |x0 − x0 (tk )| ≤

 δ0 |x0 (tk )| , 1 + δ0 Ω2 (x0 (tk )) = {x0 ∈ R : |x0 − x0 (tk )| ≤ δ0 |x0 |} .

(7.42) (7.43)

With Lemma B.3, Ω1 (x0 (tk )) ⊆ Ω2 (x0 (tk )). Moreover, under Assumptions 7.1 and 7.2, we have |d0 u 0 + φ0 (t, x0 )| = | − d0 λ0 x0 (tk ) + φ0 (t, x0 (tk ) − w0 )| a 0 δ0 )|x0 (tk )| ≤ (c02 λ0 + a0 + 1 + δ0 =: C x0 |x0 (tk )|. Then, the minimum time needed for x0 starting at x0 (tk ) to go outside (7.42) can be estimated by TΔ ≥ min

δ0 |x (t )| 1+δ0 0 k

C x0 |x0 (tk )|

, Tmax

= min

δ0 1+δ0

C x0

, Tmax ,

which is well defined and strictly larger than zero for any x0 (tk ) = 0. Since Ω1 (x0 (tk )) ⊆ Ω2 (x0 (tk )) and x0 (t) is continuous on the time-line, the minimum interval needed for the state starting at x0 (tk ) to go outside Ω2 (x0 (tk )) is not less than TΔ . Now, we analyze the lower bound of tk − tk . We also define two sets:

186

7 Event-Triggered Control of Nonholonomic Systems

1 (z(tk )) = z ∈ R : |z − z(tk )| ≤ max{ρz ◦ (Id + ρz )−1 (|z(tk )|), ρx0 (|x0 |)}  =: max{ρ¯z (|z(tk )|), ρx0 (|x0 |)} , (7.44) 2 (z(tk )) = {z ∈ R : |z − z(tk )| ≤ max{ρz (|z|), ρx0 (|x0 |)}}.

(7.45)

Also, with Lemma B.3, we have 1 (z(tk )) ⊆ 2 (z(tk )). With Proposition 7.2 satisfied, one can find a continuous and positive function L such that |˙z | ≤ L(max{|x0 |, |w|, |z(tk )|}) max{|x0 |, |w|, |z(tk )|}.

(7.46)

If z ∈ 1 (z(tk )), then (7.46) satisfies |˙z | ≤L(max{|x0 |, ρ¯z (|z(tk )|), ρx0 (|x0 |), |z(tk )|}) × max{|x0 |, ρ¯z (|z(tk )|), ρx0 (|x0 |), |z(tk )|} ≤L(max{ρˆz (|z(tk )|), ρ¯x0 (|x0 |)}) × max{ρˆz (|z(tk )|), ρ¯x0 (|x0 |)},

(7.47)

where ρˆz (s) = max{ρ¯z (s), s} and ρ¯x0 (s) = max{ρx0 (s), s} for s ∈ R+ . Using the Lemma B.4, for any specific ρ¯z , ρx0 ∈ K∞ with (ρ¯z )−1 and (ρx0 )−1 being Lipschitz on compact sets, one can find continuous, positive-valued and nondecreasing functions Lˆ such that L(max{ρˆz (|z(tk )|), ρ¯x0 (|x0 |)}) max{ρˆz (|z(tk )|), ρ¯x0 (|x0 |)} ˆ ≤ L(max{|z(t k )|, |x 0 |}) max{ρ¯ z (|z(tk )|), ρx0 (|x 0 |)} ˆ ≤ L(max{ z

[t0 ,T ) , x 0 [t0 ,Tmax ) }) × max{ρ¯ z (|z(tk )|), ρx0 (|x 0 |)}. Then, the minimum time needed for z starting at z(tk ) to go outside (7.44) can be estimated by TΔ ≥ min

1 ˆ βˆ Z (|Z (t0 )|, Tmax )) L(

, Tmax .

Since 1 (z(tk )) ⊆ 2 (z(tk )) and z(t) is continuous on the time-line, the minimum interval needed for the state starting at z(tk ) to go outside 2 (z(tk )) is not less than TΔ . From the discussions above, we have tk+1 − tk ≥ TΔ = min{TΔ , TΔ }.

(7.48)

With respect to the possible finite-time accumulation of tk and finite-time divergence of (x0 , x), we consider three cases: 1. S = Z+ and limk→∞ tk < ∞, which means Zeno behavior. 2. S = Z+ and limk→∞ tk = ∞. In this case, (x0 (t), x(t)) is defined on [t0 , ∞).

7.1 Event-Triggered State Feedback Control of Uncertain Nonholonomic Systems

y

187

ω v

(x, y)

θ

x

0 Fig. 7.2 An illustration of the kinematics of a wheeled mobile robot

3. S is a finite set {0, . . . , k ∗ } with k ∗ ∈ Z+ , i.e., there is a finite number of sampling time instants. In this case, tk ∗ < Tmax and we set tk ∗ +1 = Tmax for convenience of discussions. Properties (7.41) and (7.48) together imply that the case of S = Z+ and limk→∞ tk < ∞ is avoided, i.e., Zeno behavior does not happen, and Tmax = ∞, which essentially guarantees the existence of (x0 , z) for all t ≥ t0 . The convergence property is proved readily by replacing Tmax with ∞ for (7.41). This ends the proof of Theorem 7.1.  We employs a benchmark example of wheeled mobile robot to show the effectiveness of the proposed design in this section. Example 7.1 Consider the parking problem for the mobile robot of the unicycle type (as shown in Fig. 7.2) subject to parametric uncertainties. Specifically, the kinematics of the robot are described by [92]: x˙ = p1∗ v cos θ, y˙ = p1∗ v sin θ, θ˙ = p2∗ ω,

(7.49)

where the unknown parameters p1∗ and p2∗ are determined by the radius of the rear wheels and the distance between them, and are assumed to be within a known interval [ pmin , pmax ] with constants 0 < pmin < pmax < ∞. Introduce change of coordinates and feedback: x1 = x sin θ − y cos θ, x2 = x cos θ + y sin θ, x0 = θ, u 0 = ω, and u = v. Then, the system (7.49) can be transformed into x˙0 = p2∗ u 0 , x˙1 = p2∗ x2 u 0 , x˙2 = p1∗ u − p2∗ x1 u 0 , which is in the form of (7.1) with Assumptions 7.1 and 7.2 satisfied.

(7.50)

188

7 Event-Triggered Control of Nonholonomic Systems

Numerical Simulation: For numerical simulation, we consider pmin = 1 and pmax = 1.5. According to Proposition 7.1, we choose λ0 = 0.1 and δ0 = 0.1 and use the control law: u 0 = −0.1x0s .

(7.51)

When x0 (0) = 0, by using the state-scaling transformation (7.12), the (x1 , x2 )subsystem (7.50) is transformed into the (z 1 , z 2 )-subsystem as follows w0 )(z 2 − z 1 ), x0 z˙ 2 = p1∗ u + λ0 p2∗ z 1 x0 (x0 + w0 ).

z˙ 1 = −λ0 p2∗ (1 +

Following the design procedure in Sect. 7.1.2, the control law for u is designed as u = κ2 (z 2s − κ1 (z 1s ))

(7.52)

with κ1 (s) = 2.7s and κ2 (s) = −(0.0056|s|3 + 0.017|s| + 3.42)s for s ∈ R. By defining Vi (ei ) = |ei |, it can be verified that the each ei -subsystem is ISS with γee12 (s) = 0.99s, γee21 (s) = 0.99s, γew11 (s) = 5s, γew21 (s) = 5s, γew22 (s) = 5s, γex20 (s) = 5.88s for s ∈ R+ , and 1 = 2 = 0.1. Direct calculation implies property (7.34) with γzw (s) = 30s and γzx0 (s) = 20s 4 for s ∈ R+ . For event-triggered stabilization of (7.49), the event trigger is designed as  tk+1 = inf t > tk :|w0 (t)| ≥ 0.1|x0 (t)| or |w(t)| ≥ max{0.03|z(t)|, 0.1|x0 (t)|} . (7.53) We also employ a time trigger for comparison: tk+1 = tk + Ts ,

(7.54)

where Ts > 0 is the sampling period. Figures 7.3, 7.4, 7.5, 7.6 and 7.7 show the simulation results of the event trigger (7.53) with initial states x0 = −1, y0 = −1 and θ = 1.2. Event-triggered stabilization is achieved. Moreover, the proposed event-triggered controller with the event trigger (7.53) is robust with respect to the uncertain parameters p1∗ and p2∗ . Figure 7.7 shows that the inter-sampling intervals are lower bounded by a positive constant.  Table 7.1 uses the ultimate bound (UB) of (x 2 + y 2 ), the average inter-sampling intervals (AISI), the minimum inter-sampling intervals (Min-ISI), the maximum inter-sampling intervals (Max-ISI) and the amount of data transferred (ADT) to characterize the performance of the proposed event trigger. Compared with timetriggered control, the proposed event trigger significantly reduces the amount of sampling events.

7.1 Event-Triggered State Feedback Control of Uncertain Nonholonomic Systems

189

1.5

*

p1 *

p2

1

*

p* and p2

1.4 1.3 1.2 1.1 1

0

2

4

6

8

10 time

12

14

16

18

20

Fig. 7.3 Unknown parameters p1∗ and p2∗ in Example 7.1 1.5

x y

x, y and

1 0.5 0 -0.5 -1

0

2

4

6

8

10 time

12

14

16

18

20

Fig. 7.4 The Cartesian coordinates and the heading angle of the kinematic model of the mobile robot in Example 7.1

Experiment: The experimental test-bed is based on a wheeled mobile robot and a motion capture system; see Fig. 7.8. By default, SI units are used. The event-triggered control law (7.51) and (7.52) with event trigger (7.53) design for the mobile robot (7.49) is implemented on a computer which communicates with the robot via a wireless network. The essential sampling frequency used in the experiments is 50 Hz. That is, the triggering condition is periodically verified every 0.02 s. The performance of the proposed event-triggered controller is shown in Figs. 7.9, 7.10. Table 7.2 gives the AISI, Min-ISI, Max-ISI and ADT of the proposed eventtriggered controller, and provides a comparison between event-triggered control and time-triggered control.

190

7 Event-Triggered Control of Nonholonomic Systems

Fig. 7.5 The line velocity and the angular velocity of the kinematic model of the mobile robot in Example 7.1 0.5

parking maneuver

y

0

-0.5

-1

-1.5 -1.5

-1

-0.5 x

0

0.5

Fig. 7.6 Parking maneuver in Example 7.1

7.1.4 A Brief Discussion on x0 (t0 ) = 0 The discussion above only considers the case of x0 (t0 ) = 0. When x0 (t0 ) = 0, an intuitive solution is to first apply some open-loop control action u 0 = u ∗0 = 0

(7.55)

to drive x0 away from zero before closing the loop. Since φ0 is globally Lipschitz, x0 does not escape to infinity in finite time. At a time instant Td > t0 , the control law is switched to (7.5).

7.1 Event-Triggered State Feedback Control of Uncertain Nonholonomic Systems

191

0

inter-sampling intervals

10

10-1

-2

10

0

2

4

6

8

10 time

12

14

16

18

20

Fig. 7.7 The inter-sampling intervals corresponding to the sampling time instants in Example 7.1 Table 7.1 UB, AISI, Min-ISI, Max-ISI and ADT of the controller with event trigger (ET) (7.53) and time trigger (TT) (7.54) Event trigger UB AISI Min-ISI Max-ISI ADT ET, Tm a = 15s TT, Tm = 15s, Ts = 0.02 ET, Tm = 20s TT, Tm = 20s, Ts = 0.02 ET, Tm = 30s TT, Tm = 30s, Ts = 0.02 a Tm

0.038 0.040 0.012 0.013 0.001 0.001

0.183 0.020 0.192 0.020 0.204 0.020

0.108 0.020 0.108 0.020 0.108 0.020

0.270 0.020 0.272 0.020 0.276 0.020

is the running time of the simulation Mobile robot

Motion capture system

WiFi

Fig. 7.8 Block diagram of the experimental platform

82 750 104 1000 147 1500

192

7 Event-Triggered Control of Nonholonomic Systems x y

2

x,y,

1 0 -1 -2 0.5

1

1.5

2

2.5 time

3

3.5

4

4.5

Fig. 7.9 The trajectories of the states of the event-triggered controlled mobile robot in Example 7.1 0.5

parking maneuver

0

y

-0.5 -1 -1.5 -2 -2.5 -1.5

-1

-0.5 x

0

0.5

Fig. 7.10 Parking maneuver of the event-triggered controlled mobile robot in Example 7.1

However, due to the strong nonlinearities in φi , the choice of a constant feedback for u 0 and u may lead to finite escape. To handle this, during [t0 , t0 + Td ], one may apply a feedback control law for u: p1 = κ∗1 (x1s ) pi = κi∗ (xis − pi−1 ), i = 2, · · · , n − 2 s u = κ∗n−1 (xn−1 − pn−2 )

(7.56) (7.57) (7.58)

where the κi∗ with i = 1, · · · , n − 1 are appropriately chosen functions, and xis is the sampled value of xi . Moreover, to guarantee a positive lower bound between the

7.1 Event-Triggered State Feedback Control of Uncertain Nonholonomic Systems

193

Table 7.2 UB, AISI, Min-ISI, Max-ISI and ADT of the controller with event trigger (ET) (7.53) and time trigger (TT) (7.54) Event trigger UB AISI Min-ISI Max-ISI ADT ET (7.53) 0.036 TT (7.54) with 0.034 Ts = 0.02

0.093 0.020

0.020 0.020

1.380 0.020

58 270

inter-sampling intervals, during the time period [t0 , t0 + Td ], an event trigger with a constant threshold signal is employed:  tk+1 = inf t > tk : |X (t) − X (tk )| ≥  ,

(7.59)

where X = [x1 , . . . , xn−1 ]T and  can be any positive constant. This design approach has been applied to the existing works; see, [18]. The control law above makes x0 (t0 + Td ) = 0, and the control laws for u 0 and u are switched to (7.5) and (7.30)–(7.32) at t = t0 + Td , respectively. At the same time, the event trigger is switched to (7.35).

7.2 Event-Triggered Output Feedback Control of Uncertain Nonholonomic Systems 7.2.1 Problem Formulation As shown in the existing literature [71, 109, 178, 258], if di and φi in the system (7.1) satisfy some additional condition, then robust stabilization can still be achieved by dynamic output-feedback with (x0 , x1 ) as the outputs. Based on the state-feedback result, this section is concerned about event-triggered stabilization with output feedback. In this case, a reduced model of the system (7.1) is studied: x˙0 = u 0 + c0 x0 x˙1 = x2 u 0 + φ1 (t, x0 , x, u 0 ) .. .

(7.60)

x˙n−2 = xn−1 u 0 + φn−2 (t, x0 , x, u 0 ) x˙n−1 = u + φn−1 (t, x0 , x, u 0 ) where the variables and functions are with the same meaning as for the system (7.1).

194

7 Event-Triggered Control of Nonholonomic Systems

For the system (7.60), (x0 , x1 ) are considered as the measured outputs, and (x2 , . . . , xn−1 ) are not available to feedback control. The following assumption is further made on φi . Assumption 7.3 For each i = 1, . . . , n − 1, there exists a nonnegative, smooth function ψid such that |φi (t, x0 , x, u 0 )| ≤ |x1 |ψid (x0 , x1 , u 0 ) holds for all (t, x0 , x, u 0 ). For the system (7.60), to achieve stabilization with output feedback, the eventtriggered controller is expected to be in the form of u 0 = υ0 (x0 (tk ), x1 (tk )) ξ˙ = gc (ξ, x0 (tk ), x1 (tk ))

(7.62)

u = υ(ξ, x0 (tk ), x1 (tk ))

(7.63)

(7.61)

where ξ ∈ Rm is the internal state of the dynamic controller. Accordingly, the event trigger should be designed to be able to deal with the internal state ξ of the controller and the unmeasured portion of system state.

7.2.2 Event-Triggered Output-Feedback Controller There are two major difficulties in extending the result of state feedback to output feedback: (a) event-triggered observer design to estimate the unmeasured states with the sampled outputs; (b) output-based event-trigger design to avoid infinitely fast sampling and to guarantee asymptotic convergence. In this section, the problems are solved by developing a novel nonlinear observer which is robust with respect to sampling errors, and by designing a new dynamic event trigger. Some techniques that are similar to the case of state feedback are used without proof. Controller Design for u0 For the case of output-feedback stabilization, the control law for u 0 is still designed as (7.5). Since the system (7.60) is a special case of (7.1), we have a reduced result of Proposition 7.1. Proposition 7.6 Consider the system composed of the x0 -subsystem defined in (7.60) and the control law (7.61). If w0 satisfies |w0 (t)| ≤ μ0 |x0 (t)| for all t0 ≤ t < Tmax , with μ0 satisfying

(7.64)

7.2 Event-Triggered Output Feedback Control of Uncertain Nonholonomic Systems

0 < μ0 < 1,

195

(7.65)

and the λ0 in (7.5) is chosen as λ0 >

|c0 | , 1 − μ0

(7.66)

then x0 (t) exists for all t0 ≤ t < Tmax and satisfies (7.11) for all t0 ≤ t < Tmax with a1 = λ0 + λ0 μ0 + |c0 | and a2 = λ0 − λ0 μ0 − |c0 |. With condition (7.64) satisfied, we can construct an ISS-Lyapunov function for the x0 -subsystem as Vx0 (x0 ) = |x0 |,

(7.67)

∇Vx0 (x0 )x˙0 ≤ −x0 Vx0 (x0 ) a.e.

(7.68)

which satisfies

with x0 = λ0 − |c0 | − λ0 μ0 . Controller Design for u (A) State Scaling We only consider the case of x0 (t0 ) = 0. With the same state scaling (7.12), the x-subsystem in (7.60) is transformed into z˙ i = χ0 (x0 , w0 )z i+1 + χi (x0 , w0 )z i + φˆ i (t, x0 , x, w0 ) z˙ n−1 = u(t) + φˆ n−1 (t, x0 , x, w0 )

(7.69)

where for i = 1, . . . , n − 1,   w0 , χ0 (x0 , w0 ) = −λ0 1 + x0   λ 0 w0 , χi (x0 , w0 ) = (n − i − 1) λ0 − c0 + x0 φi (t, x0 , x, −λ0 (x0 + w0 )) φˆ i (t, x0 , x, w0 ) = . x0n−i−1

(7.70) (7.71) (7.72)

Proposition 7.7 Consider the system (7.69). Suppose that Assumption 7.3 is satisfied, and condition (7.64) holds for all t0 ≤ t < Tmax with some t0 < Tmax ≤ ∞. Then, for each i = 0, . . . , n − 2, there exist positive constants L i1 and L i2 such that L i1 ≤ |χi (x0 , w0 )| ≤ L i2

(7.73)

196

7 Event-Triggered Control of Nonholonomic Systems

holds for all x0 , w0 , and for each i = 1, . . . , n − 1, there exists a φˆ id ∈ K∞ being Lipschitz on compact sets such that |φˆ i (t, x0 , x, w0 )| ≤ φˆ id ([x0 , z 1 ]T )

(7.74)

holds for all t0 ≤ t < Tmax and all x0 , w0 . Proposition 7.7 can be proved similarly as for Proposition 7.2. (B) Observer-based Output-Feedback Controller Consider system (7.69) with x0 and z 1 as the outputs. By taking advantage of the output-feedback structure, we introduce a new nonlinear observer. Instead of the actual outputs x0 and z 1 of the plant, their sampled values x0s and z 1s are used as the inputs of the observer: ξ˙0 = −λ0 x0s + c0 ξ0 ξ ξ ξ˙1 = χ0 (ξ0 , x0s )(ξ2 + L 2 ξ1 ) + χ1 (ξ0 , x0s )ξ1 + ρ(ξ1 − z 1s ) ξ ξ ξ˙i = χ0 (ξ0 , x0s )(ξi+1 + L i+1 ξ1 ) + χi (ξ0 , x0s )ξi   ξ ξ − L i χ0 (ξ0 , x0s )(ξ2 + L 2 ξ1 ) + χ1 (ξ0 , x0s )ξ1

ξ˙n−1

(7.75)

i = 2, . . . , n − 2   ξ ξ = u − L n−1 χ0 (ξ0 , x0s )(ξ2 + L 2 ξ1 ) + χ1 (ξ0 , x0s )ξ1

ξ

where χi (ξ0 , x0s ) := χi (ξ0 , x0s − ξ0 ) for 0 ≤ i ≤ n − 2, ρ : R → R is an odd and strictly decreasing function, and L 2 , . . . , L n−1 are appropriate constants to be determined later. In the observer (7.75), ξ0 is an estimate of x0 , ξ1 is an estimate of z 1 , and ξi is an estimate of z i − L i z 1 for 2 ≤ i ≤ n − 1. For convenience of discussions, we define observation errors: ζ0 = x0 − ξ0 , ζ1 = z 1 − ξ1 ,

(7.76) (7.77)

ζi = z i − L i z 1 − ξi , i = 2, . . . , n − 1.

(7.78)

It should be noted that the ξ0 -subsystem of observer (7.75) is a copy of the controlled x0 -subsystem with the event-triggered feedback control law (7.5). We set ξ0 (t0 ) = x0 (t0 ). Then, it is guaranteed that ζ0 ≡ 0, i.e., ξ0 (t) = x0 (t)

(7.79)

for all t0 ≤ t < Tmax . In this way, x0 is reconstructed by ξ0 using its sampled signal. Based on the estimation by the observer, we introduce a control law in the same form of (7.30)–(7.32):

7.2 Event-Triggered Output Feedback Control of Uncertain Nonholonomic Systems

197

e1 = z 1

(7.80)

e2 = ξ2 − υ1 (e1 − ζ1 ) ei = ξi − υi−1 (ei−1 ), i = 3, . . . , n − 1 u = υn−1 (en−1 )

(7.81) (7.82) (7.83)

where υi ’s for i = 1, · · · , n − 1 are continuously differentiable, odd, monotone and radially unbounded functions. Note that e1 − ζ1 = z 1 − ζ1 = ξ1 . Thus, the control law uses only the state of the observer (ξ1 , . . . , ξn ), and is implementable. Small-Gain Synthesis and Convergence Analysis We consider the closed-loop system composed of the plant (7.69), the observer (7.75) and the control law (7.80)–(7.83) as an interconnection of the ζ1 -subsystem, the (ζ2 , · · · , ζn−1 )-subsystem and the ei -subsystem, and appropriately choose ρ, L 2 , · · · , L n−1 , υ1 , · · · , υn−1 such that each subsystem is ISS. Then, the nonlinear smallgain theorem is employed to analyze the convergence property. For convenience of discussions, denote ζ¯2 = [ζ2 , · · · , ζn−1 ]T . For the subsystems of ζ1 , ζ¯2 and ei , define the following Lyapunov function candidates, respectively: Vζ1 (ζ1 ) = αV (|ζ1 |),

(7.84)

Vζ¯2 (ζ¯2 ) = (ζ¯2T P ζ¯2 ) ,

(7.85)

Vei (ei ) = αV (|ei |), i = 1, . . . , n − 1,

(7.86)

1 2

where αV = s N /N with N ∈ Z+ , and P is a symmetric and positive definite matrix. Both N and P are to be determined later. (A) The ζ¯2 -Subsystem With property (7.79) satisfied, taking the derivative of ζ¯2 leads to ζ˙¯2 = Aζ¯2 + AΔ (x0 , w0 )ζ¯2 + ψζ¯2 (t, ζ1 , e1 , x0 , x, w0 ) where A is a constant matrix satisfying ⎡

L 2 λ0 + b2 ⎢ L 3 λ0 ⎢ ⎢ .. A=⎢ . ⎢ ⎣ L n−2 λ0 L n−1 λ0

−λ0 0 · · · b3 −λ0 · · · .. .. . . . . . 0 0 ··· 0 0 ···

with bi = (n − i − 1)(λ0 − c0 ) for 2 ≤ i ≤ n − 2,

0 0 .. .

0 0 .. .

⎤

⎥ ⎥ ⎥ ⎥ ⎥ bn−2 −λ0 ⎦ 0 0

(7.87)

198

7 Event-Triggered Control of Nonholonomic Systems

AΔ (x0 , w0 ) =

λ 0 w0 ∗ Δ x0

with constant matrix Δ∗ satisfying ⎡

L 2 + n − 3 −1 ⎢ n−4 L3 ⎢ ⎢ . .. ∗ .. Δ =⎢ . ⎢ ⎣ L n−2 0 0 L n−1

0 −1 .. .

··· ··· .. .

0 0 .. .

0 0 .. .

⎤

⎥ ⎥ ⎥ ⎥, ⎥ 0 · · · 1 −1⎦ 0 ··· 0 0

T  and ψζ¯2 (t, ζ1 , e1 , x0 , x, w0 ) =: ψi ζ¯2 , . . . , ψn−1ζ¯2 =: ψζ¯2 with ψi ζ¯2 = L i χi e1 + (L i+1 χ0 − L i L 2 χ0 − L i χ1 )ζ1 + φˆ i − L i φˆ 1 , i = 2, . . . , n − 2 ψn−1ζ¯2 = −L n−1 (χ1 + L 2 χ0 )ζ1 + φˆ n−1 − L (n−1) φˆ 1 . With conditions (7.64), (7.73) and (7.74) satisfied, it can be directly calculated ζ that |AΔ (x0 , w0 )| ≤ μ0 λ0 |Δ∗ |, and there exist ψζ¯ 1 , ψζe¯ 1 , ψζx¯ 0 ∈ K∞ being Lipschitz 2 2 2 on compact sets such that ζ

|ψζ¯2 | ≤ ψζ¯ 1 (|ζ1 |) + ψζe¯ 1 (|e1 |) + ψζx¯ 0 (|x0 |). 2

2

2

(7.88)

By appropriately choosing L 2 , . . . , L n such that A is Hurwitz, there exists a positive definite matrix P = P T ∈ R(n−2)×(n−2) satisfying P A + A T P = −2In−2 . Define V¯ζ¯2 (ζ¯2 ) = ζ¯2T P ζ¯2 . Then, there exist αζ¯2 , αζ¯2 ∈ K∞ such that αζ¯2 (|ζ¯2 |) ≤ V¯ζ¯2 (ζ¯2 ) ≤ αζ¯2 (|ζ¯2 |). With direct calculation, we have ∇ V¯ζ¯2 (ζ¯2 )ζ˙¯2 = −2ζ¯2T ζ¯2 + 2ζ¯2T P AΔ (x0 , w0 )ζ¯2 + 2ζ¯2T Pψζ¯2 3 ≤ − ζ¯2T ζ¯2 + 2μ0 λ0 |P||Δ∗ |ζ¯2T ζ¯2 + 2|P|2 |ψζ¯2 |2 . 2 By choosing μ0 defined in (7.64) satisfying (7.65) and μ0 ≤

1 λ0 |P||Δ∗ |, 4

(7.89)

we have ∇ V¯ζ¯2 (ζ¯2 )ζ˙¯2 ≤ −

  1 ζ Vζ¯2 (ζ¯2 ) + 2|P|2 ψζ¯ 1 (|ζ1 |) + ψζe¯ 1 (|e1 |) + ψζx¯ 0 (|x0 |) . 2 2 2 λmax (P)

7.2 Event-Triggered Output Feedback Control of Uncertain Nonholonomic Systems

199

ζ

Hence, there exist χζ¯1 , χζe¯1 , χζx¯ 0 ∈ K∞ being Lipschitz on compact sets and a 2 2 2 continuous, positive definite ζ¯2 such that  ζ Vζ¯2 (ζ¯2 ) ≥ max χζ¯1 (Vζ1 (ζ1 )), χeζ¯1 (Ve1 (e1 )), χζx¯ 0 (Vx0 (x0 )) 2

2

2

⇒ ∇Vζ¯2 (Aζ¯2 + AΔ ζ¯2 + ψζ¯2 ) ≤ −ζ¯2 (Vζ¯2 (ζ¯2 )), a.e.

(7.90)

(B) The ζ1 -Subsystem and ei -Subsystem By using (7.69) and (7.75)–(7.83), we have the (ζ1 , e1 , . . . , en−1 )-subsystem: ζ˙1 = ρ(ζ1 + w1 ) + (χ1 + L 2 χ0 )ζ1 + χ0 ζ2 + φˆ 1 e˙1 = χ0 υ1 (e1 − ζ1 ) + ψe1 (t, e1 , e2 , ζ2 , x0 , x, w0 ) e˙2 = χ0 υ2 (e2 ) + ψe2 (t, e1 , e2 , e3 , ζ1 , w1 , x0 , x, w0 )

(7.91) (7.92) (7.93)

i) e˙i = χ(a 0 υi (ei ) + ψei (t, e1 , e2 , . . . , ei+1 , ζ1 , ζ2 , x 0 , x, w0 , w1 ),

i = 3, . . . , n − 1

(7.94)

where ai = 1 for i = 3, . . . , n − 2 and an−1 = 0. The systems (7.91), (7.92), (7.93) and (7.94) are in the same form of (7.27). With conditions (7.64), (7.73) and (7.74) satisfied, by using the same techniques as for the proof of Proposition 7.4, we can find continuously differentiable ρ and υi for i = 1, . . . , n − 1 such that the ζ1 -subsystem and each ei -subsystem are ISS with Vζ1 (ζ1 ) = αV (|ζ1 |) and Vei (ei ) = αV (|ei |) as ISS-Lyapunov functions, respectively. Specifically, we have Vζ1 (ζ1 ) ≥ max

ζ¯ χζ21 (Vζ¯2 (ζ¯2 )), χeζ11 (Ve1 (e1 )), x0 1 χw ζ1 (|w1 |), χζ1 (Vx0 (x 0 ))

⇒ ∇Vζ1 (ζ1 )ζ˙1 ≤ −ζ1 Vζ1 (ζ1 ), a.e. Ve1 (e1 ) ≥ max

ζ

χee21 (Ve2 (e2 )), χe11 (Vζ1 (ζ1 )), ζ¯ χe21 (Vζ¯ (ζ¯2 )), χx0 (Vx0 (x0 )) 2

(7.95)

e1

⇒ ∇Ve1 (e1 )e˙1 ≤ −e1 Ve1 (e1 ), a.e.  Ve2 (e2 ) ≥ max

χee12 (Ve1 (e1 )), χee32 (Ve3 (e3 )), ζ x0 1 χe12 (Vζ1 (ζ1 )), χw e2 (|w1 |), χe2 (Vx0 (x 0 ))

⇒ ∇Ve2 (e2 )e˙2 ≤ −e2 Ve2 (e2 ), a.e. and for i = 3, . . . , n − 1,

(7.96) 

(7.97)

200

7 Event-Triggered Control of Nonholonomic Systems

Vei (ei ) ≥

max

j=1,...,i−1,i+1

e

ζ

χeij (Ve j (e j )), χe1i (Vζ1 (ζ1 )), ζ¯ 1 χe2i (Vζ¯2 (ζ¯2 )), χexi0 (Vx0 (x0 )), χw ei (|w1 |)

⇒ ∇Vei (ei )e˙i ≤ −ei Vei (ei ), a.e.

(7.98)

In equations (7.95)–(7.98), the (·) ’s can be any specified positive constants, = 0, and the other χ(·) (·) ’s can be any K∞ functions being Lipschitz on compact sets.

χeenn−1

(C) Small-Gain Synthesis T Denote Y = [ζ1 , ζ¯2 , e1 , · · · , en−1 ]T as the state of the closed-loop system. With the closed-loop system transformed into an interconnection of ISS subsystems, we employ the nonlinear small-gain theorem to guarantee stability. Proposition 7.8 Consider the z-system (7.69) with observer defined by (7.75) and control law defined by (7.5) and (7.80)–(7.83). Suppose that x0 (t0 ) = 0. Choose ρ, L 2 , · · · , L n−1 and υ1 , · · · , υn−1 such that the ISS gains in (7.90) and (7.95)–(7.98) satisfy χii21 ◦ χii32 ◦ · · · ◦ χiir1 < Id

(7.99)

T where r = 2, . . . , n + 1 and i j ∈ {ζ1 , ζ¯2 , e1 , · · · , en−1 }, i j = i j  if j = j  . Then, x0 w there exist βY ∈ KL and γY , γY ∈ K∞ being Lipschitz on compact sets such that for any initial state Y (t0 ) and any piece-wise continuous and bounded w1 and x0 ,

|Y (t)| ≤ max{βY (|Y (t0 )|, t), γYw ( w1 [t0 ,Tmax ) ), γYx0 ( x0 [t0 ,Tmax ) )} holds for all t0 ≤ t < Tmax . Proposition 7.8 can be proved by directly applying the cyclic-small-gain theorem in Sect. 2.4.1.

7.2.3 Main Result Different from the event trigger design for the x-subsystem in Sect. 7.1.3, we introduce a dynamic event trigger to deal with the unmeasured states. Specifically, the sampling times are determined by tk+1 = inf{t > tk : |w0 (t)| ≥ μ0 |x0 (t)| or |w1 (t)| ≥ μ1 (t)},

(7.100)

where μ0 satisfies (7.65) and (7.89) and the threshold signal μ1 is generated by a dynamic system of the form

7.2 Event-Triggered Output Feedback Control of Uncertain Nonholonomic Systems

201

η˙1 (t) = −ν1 (η1 (t))

(7.101)

μ1 (t) = ϕ1 (η1 (t))

(7.102)

where η1 ∈ R+ is the state, ν1 ∈ R+ → R+ is locally Lipschitz and positive definite, and ϕ1 ∈ R+ → R+ is continuously differentiable on (t0 , ∞) and of class K∞ . The initial state η1 (t0 ) is chosen to be positive, and thus μ1 (t0 ) is positive. We will provide technical lemmas on the convergence rate of the closed-loop event-triggered system. Based on the technical lemmas, we appropriately choose the functions ν1 and ϕ1 for the event trigger composed of (7.100)–(7.102). Proposition 7.9 gives transformations of the ISS-Lyapunov functions for the subsystems of ζ1 , ζ¯2 and ei , and constructs a Lyapunov function for the closed-loop event-triggered system by considering η1 as one of the states. Proposition 7.9 Consider that the network of ISS subsystems with [x0 , ζ1 , ζ¯2 , e1 , · · · , en−1 ]T as the states satisfies the cyclic-small-gain condition (7.99). T

(1) There exist x0 , ζ1 ,ζ¯2 and ei for i = 1, · · · , n − 1 being locally Lipschitz on (0, ∞) such that V˜x0 (x0 ) = x0 (Vx0 (x0 )), V˜ζ1 (ζ1 ) = ζ1 (Vζ1 (ζ1 )),

(7.104)

V˜ζ¯2 (ζ¯2 ) = ζ¯2 (Vζ¯2 (ζ¯2 )),

(7.105)

V˜ei (ei ) = ei (Vei (ei )), i = 1, · · · , n − 1

(7.106)

(7.103)

are ISS-Lyapunov functions of the x0 -subsystem, ζ1 -subsystem, ζ¯2 -subsystem and ei -subsystem, respectively. Moreover, (7.90) and (7.95)–(7.98) still hold w1 2) w1 with the ISS-Lyapunov functions (7.104)–(7.106) and χ(s (s1 ) , (s1 ) , χζ1 and χei (s2 ) w w 1 1 w w replaced by χ˜ (s1 ) ∈ K < Id, ˜(s1 ) ∈ P, χ˜ ζ1 = ζ1 ◦ χζ1 and χ˜ ei 1 = ei ◦ χei 1 with s1 , s2 ∈ {x0 , ζ1 , ζ¯2 , e1 , · · · , en−1 }. 1 1 ˜w (2) By choosing ϕ1 such that maxi=2,...,n−1 {χ˜ w ei } ◦ ϕ1 < Id, the function ζ1 , χ V (ς) =

 max

i=1,··· ,n−1

V˜x0 (x0 ),V˜ζ1 (ζ1 ), V˜ζ¯2 (ζ¯2 ), V˜ei (ei ), η1



satisfies D + V (ς(t)) ≤ −α(V (ς(t))),

(7.107)

T where ς = [x0 , ζ1 , ζ¯2 , e1 , · · · , en−1 , η1 ]T and α(s) = mini=1,··· ,n−1 {˜x0 (s), ˜ζ1 (s), ˜ζ¯2 (s), ˜ei (s), ν1 (s)} for s ∈ R+ .

The proof of Proposition 7.9 is in Appendix D.13. Our main result on event-triggered output-feedback stabilization is given by Theorem 7.2.

202

7 Event-Triggered Control of Nonholonomic Systems

Theorem 7.2 Consider the system (7.60) with the output-feedback control law (7.5) and (7.80)–(7.83) and event trigger (7.100)–(7.102). Suppose that x0 (t0 ) = 0. With Assumption 7.3 is satisfied, the objectives of event-triggered robust stabilization is achievable, if the functions ν1 and ϕ1 in (7.102) is chosen as follows: 1 • ϕ1 is continuously differentiable on (0, ∞) and satisfies that maxi=2,...,n−1 {χ˜ w ζ1 , √ −1 −1 −1 −1 −1 −1 −1 w1 −1 χ˜ ei } ◦ ϕ1 < Id, and x0 ◦ ϕ1 , αV ◦ (s1 ) ◦ ϕ1 and αζ¯2 ◦ ζ¯ ◦ ϕ1 are Lip2 schitz on compact sets for s1 ∈ {ζ1 , e1 , · · · , en−1 }. • ν1 is positive definite and Lipschitz on compact sets and satisfies ν1 ≤ mini=1,··· ,n−1 {˜x0 (s), ˜ζ1 (s), ˜ζ¯2 (s), ˜ei (s)} for s ∈ R+ , θν is positive definite and Lipschitz on compact sets, and there exists a constant Δ > 0 such that θν (s)/s is nondecreasing for s ∈ (0, Δ), with θν defined by

θν (s) =

−1 ∂ϕ1 (ϕ−1 1 (s))ν1 (ϕ1 (s)) for s > 0; 0 for s = 0.

Proof As for Theorem 7.1, we can prove the convergence property of the system (7.60) with the output-feedback control law (7.5) and (7.80)–(7.83) and event trigger (7.100)–(7.102) by using the trajectory-based small-gain theorem in Sect. 2.3.1. Now, we prove the existence of a positive lower bound of inter-sampling intervals, which follows along similar lines as the proof of Theorem 7.1. Given specific tk , define tk as the next time instant independently triggered by tk = inf {t > tk : |w0 (t)| ≥ μ0 |x0 (t)|} ,

(7.108)

and define tk as the next time instant independently triggered by tk = inf {t > tk : |w1 (t)| ≥ μ1 (t)} .

(7.109)

Following the proof procedure in Theorem 7.1, one can find a positive constant tΔ such that tk − tk ≥ tΔ . Also, the proof of the lower bound of tk − tk is similar with that of Theorem 4.2. There exists a positive lower bound of the inter-sampling intervals TΔ such that tk − tk > TΔ . Due to space limitation, the analysis of the lower bound of tk − tk is omitted here. From the discussions above, we have tk+1 − tk ≥ tΔ = min{TΔ , TΔ }. Thus, the case of S = Z+ and limk→∞ tk < ∞ is avoided, i.e., Zeno behavior does not happen, and T = ∞, which guarantees the existence of (x0 , x) for all t ≥ t0 , without finite escape time. This ends the proof of Theorem 7.2.  Different from the design of event trigger in Sect. 7.1.3, in this subsection, a dynamic event trigger is employed to guarantee a positive lower bound of the intersampling intervals in the existence of the unmeasured states. One of the previous results can be found in [164], which solves the event-triggered control problem for

7.2 Event-Triggered Output Feedback Control of Uncertain Nonholonomic Systems

203

nonlinear systems composed of two subsystems by designing an asymptotically (not necessarily exponentially) decreasing threshold signal. One of the major differences between this section and the previously published results is that the design of dynamic event trigger for x1 has to take into consideration the influence of x0 . It is worth noting that the dynamic event trigger proposed in Sect. 7.2.3 can also be used to address the event-triggered control problem for the nonholonomic systems (7.1) with state feedback. Interested reader may also consult [73] for a Lyapunov-based design of dynamic event triggers. From the discussions in Sect. 7.1.4, during the time period [t0 , t0 + Td ), we can select the control law (7.55) to drive the x0 away from zero, and apply the observerbased output-feedback controller of the form (7.62)–(7.63).Moreover, following the event trigger design method in Sect. 7.1.4, during the time period [t0 , t0 + Td ], the event trigger is designed as  tk+1 = inf t > tk : |x1 (t) − x1 (tk )| ≥  ,

(7.110)

where  can be any positive constant. When t ≥ Td , we switch the control law u 0 and u into (7.5) and (7.80)–(7.83). At the same time, the event trigger (7.110) is switched into (7.100). Example 7.2 Consider a third-order chained system: x˙0 = u 0 , x˙1 = x2 u 0 + 0.05x1 , x˙2 = u.

(7.111)

We take (x0 , x1 ) as the output. The state x2 is unmeasured (i.e., not available for feedback design). We employ the result in Sect. 7.2.2 to design an event-triggered output-feedback controller. According to Proposition 7.6, we choose λ0 = 0.1 and μ0 = 0.1, and consider u 0 = −0.1x0s .

(7.112)

When x0 (0) = 0, by using the state-scaling transformation (7.12), the (x1 , x2 )subsystem (7.111) is transformed into the (z 1 , z 2 )-subsystem as w0 w0 )z 2 + 0.1(1 + )z 1 + 0.05z 1 , x0 x0 z˙ 2 = u + 2z 1 x0 .

z˙ 1 = −0.1(1 +

Following the design procedure in Sect. 7.2.2, a nonlinear observer is designed as ξ˙0 = −0.1x0s , ξ ξ ξ˙1 = χ0 (ξ2 − 0.1ξ1 ) + χ1 ξ1 + ρ(ξ1 − z 1s ), ξ ξ ξ˙2 = u + 0.1(χ0 (ξ2 − 0.1ξ1 ) + χ1 ξ1 )

204

7 Event-Triggered Control of Nonholonomic Systems ξ

ξ

ξ

where χ0 = −0.1(1 + (ξ0 − x0s )/ξ0 ), χ1 = −χ0 and ρ(r ) = −2r for r ∈ R. Based on the estimation by the observer, we design a nonlinear control law as u = v2 (ξ2 − v1 (ξ1 )),

(7.113)

where v1 = 5.8s and v2 = −40s for s ∈ R. By defining Vζ1 (ζ1 ) = |ζ1 |, Vζ¯2 (ζ¯2 ) = |ζ¯2 |, Ve1 (e1 ) = |e1 | and Ve2 (e2 ) = |e2 |, it can be verified that all the subsystems ζ1 ζ¯ e1 1 are ISS with χζ21 (s) = 0.24s, χeζ11 (s) = 0.19s, χw ζ1 (s) = 5s, χζ¯ (s) = 4s, χζ¯ (s) = 2

ζ¯

ζ

ζ

2

0.67s 2 , χe11 (s) = 5s, χe21 (s) = 0.99s, χee21 (s) = 0.99s, χee12 (s) = 0.99s, χe12 (s) = 1 0.99s, χw e2 (s) = 5s, ζ¯2 (s) = 0.005s, ζ1 (s) = 0.01s, e1 (s) = 0.01s and e2 3

x0

2.5

x

system state

2

x

1 2

1.5 1 0.5 0 -0.5 -1

0

5

10

15

20 time

25

30

35

40

Fig. 7.11 The trajectories of the system states x0 , x1 and x2 in Example 7.2 2 0

1.5

1

obsever state

1

2

0.5 0 -0.5 -1 -1.5 -2

0

5

10

15

20 time

25

30

Fig. 7.12 The trajectories of the observer states ξ0 , ξ1 and ξ2 in Example 7.2

35

40

7.2 Event-Triggered Output Feedback Control of Uncertain Nonholonomic Systems

205

Fig. 7.13 The control inputs u 0 and u in Example 7.2 1

inter-sampling intervals

10

0

10

10-1

10-2

0

5

10

15

20 time

25

30

35

40

Fig. 7.14 The inter-sampling intervals with the event trigger (7.100) in Example 7.2

(s) = 0.01s for s ∈ R+ . According to Proposition 7.9 and Theorem 7.2, ν1 and ϕ1 defined in (7.102) are chosen as ν1 (s) = 0.004s and ϕ1 (s) = 0.04s for s ∈ R+ . Figures 7.11, 7.12, 7.13 and 7.14 show the simulation result with initial states x0 (0) = 2, x1 (0) = 1, x2 (0) = 1, ξ0 (0) = 1, ξ1 (0) = 0, ξ2 (0) = 0, η1 (0) = 2 and μ1 (0) = 0.08.

206

7 Event-Triggered Control of Nonholonomic Systems

7.3 Notes Nonholonomic constraints appear in many mechanical systems [201, 203]. Control of nonholonomic systems has been extensively studied in the past three decades, particularly in the context of nonholonomic mobile robots [34, 52, 101, 110, 119, 120, 161, 183, 197, 215, 218, 243, 264, 279, 289]. According to Brockett [28], nonholonomic systems cannot be stabilized by any smooth or continuous time-invariant state-feedback. Alternatively, discontinuous controllers, time-varying controllers and switching controllers have been developed; see, for example, [15, 109, 142, 252] for the state-feedback stabilization of nonholonomic systems, and [109, 178, 266] for the case where only partial state is available to feedback. For more recent work, see [27, 48, 152, 191, 200] and the references therein. For nonholonomic systems, the existing robust control results are mainly concerned about the external disturbances that directly affect the system dynamics and the drift uncertain nonlinearities [66, 178, 265], and robust stabilization with respect to measurement errors has been little investigated. For the reduced model of unicycles, [198] has studied robust stabilization with heading angle subject to measurement error. An event-triggered tracking controller has been designed and implemented in [218] for a standard model of mobile robots. Reference [243] has developed an eventbased model predictive control strategy for the tracking of nonholonomic mobile robot with coupled input constraint and bounded disturbances. An event-triggered formation tracking problem of nonholonomic mobile robots in a leader’s coordinate frame has been developed by [272] in which the proposed control protocol can be implemented in directed sensor networks based on only relative information detected by onboard sensors of mobile robots. In [79], a distributed consensus-based formation control of networked nonholonomic mobile robots using neural networks in the presence of uncertain robot dynamics with event-based communication is presented. There seems still no systematic tool for robust state-feedback or output-feedback control of nonholonomic systems subject to sampling/measurement errors. This chapter gives a small-gain design for robust event-triggered control of nonholonomic systems, with the ideal chained form as a special case.

Appendix A

Notions

A.1 Lipschitz Continuity This section gives the definitions of Lipschitz continuity and the related notions that are used in the book. Definition A.1 A function h : X → Y with X ⊆ Rn and Y ⊆ Rm is said to be Lipschitz continuous, or simply Lipschitz, on X , if there exists a constant L h ≥ 0, such that for any x1 , x2 ∈ X , |h(x1 ) − h(x2 )| ≤ L h |x1 − x2 |.

(A.1)

Definition A.2 A function h : X → Y with X ⊆ Rn being open and connected, and Y ⊆ Rm is said to be locally Lipschitz on X , if each x ∈ X has a neighborhood X0 ⊆ X such that h is Lipschitz on X0 . Definition A.3 A function h : X → Y with X ⊆ Rn and Y ⊆ Rm is said to be Lipschitz on compact sets, if h is Lipschitz on every compact set D ⊆ X .

A.2 Comparison Functions The notions of Lyapunov stability and input-to-state stability (ISS) that are used in this book are defined by comparison functions. Here, this section introduces three classes of comparison functions. Definition A.4 A function α : R+ → R+ is said to be positive definite if α(0) = 0 and α(s) > 0 for s > 0. © Huazhong University of Science and Technology Press, Wuhan and Springer Nature Singapore Pte Ltd. 2020 T. Liu et al., Robust Event-Triggered Control of Nonlinear Systems, Research on Intelligent Manufacturing, https://doi.org/10.1007/978-981-15-5013-3

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208

Appendix A: Notions

Definition A.5 A continuous function α : R+ → R+ is said to be a class K function, denoted by α ∈ K, if it is strictly increasing and α(0) = 0; it is said to be a class K∞ function, denoted by α ∈ K∞ , if it is a class K function and satisfies α(s) → ∞ as s → ∞. Definition A.6 A continuous function β : R+ × R+ → R+ is said to be a class KL function, denoted by β ∈ KL, if, for each fixed t ∈ R+ , function β(·, t) is a class K function and, for each fixed s ∈ R+ , function β(s, ·) is decreasing and satisfies limt→∞ β(s, t) = 0.

A.3 Related Notions in Graph Theory This section gives the standard definitions of several notions in graph theory that are used in this book. Definition A.7 A graph G is a collection of points V1 , . . . , Vn and a collection of lines a1 , . . . , am joining all or some of these points. The points are called vertices, and the lines, denoted by the pairs of points they connect, are called links. Definition A.8 A graph G is called a directed graph or simply a digraph if the lines in it have a direction. The lines are called directed links or arcs. Definition A.9 A path in a digraph G is any sequence of arcs where the final vertex of one is the initial vertex of the next one, denoted as the sequence of the vertices it contains. If a path has no repeated vertices, then it is called a simple path. Definition A.10 In a digraph G, if there exists a path leading from vertex i to vertex j, then vertex j is reachable from vertex i. Specifically, any vertex i is reachable from itself. Definition A.11 In a digraph G, the reaching set of a vertex j, denoted by RS( j), is the set of the vertices from which vertex j is reachable. Definition A.12 In a digraph G, a path such that the starting vertex and the ending vertex are the same is called a cycle. If a cycle has no repeated vertices other than the starting and ending vertices, then it is called simple cycle. Definition A.13 A directed tree is a digraph which has no cycle and there exists a vertex from which all the other vertices are reachable. Definition A.14 A spanning tree T of a digraph G is a directed tree formed by all the vertices and some or all of the edges of G.

Appendix B

Technical Lemmas

Lemma B.1 Let β ∈ KL, ρ ∈ K such that ρ < Id, and let μ be a real number in (0, 1]. There exists a βˆ ∈ KL such that for any nonnegative real numbers s and δ, and any nonnegative real function z defined on [0, ∞) and satisfying z(t) ≤ max{β(s, t), ρ(z[μt,∞) ), δ}

(B.1)

for all t ∈ [0, ∞), it holds that ˆ t), δ} z(t) ≤ max{β(s,

(B.2)

for all t ∈ [0, ∞). This kind of technical lemma is motivated by the nonlinear small-gain result originally developed by the authors of [122]; see [122, Lemma A.1]. It should be noted that [122] mainly considers “plus”-type interconnections, while Lemma B.1 is used for the systems with “max”-type interconnections in this book. The major difference is that the signal z(t) in [122, Lemma A.1] satisfies z(t) ≤ β(s, t) + ρ(z[μt,∞) ) + δ ˆ t) + δ instead of (B.1), and the corresponding result is in the form of z(t) ≤ β(s, instead of (B.2). Lemma B.2 Consider χ ∈ K and χi ∈ K ∪ {0} for i = 1, . . . , n. If χ ◦ χi < Id for i = 1, . . . , n, then there exists a χˆ ∈ K∞ such that χˆ > χ, χˆ is continuously differentiable on (0, ∞), and χˆ ◦ χi < Id for i = 1, . . . , n. Proof Define χ¯ (s) = maxi=1,...,n {χi (s)} for all s ≥ 0. Then, χ¯ ∈ K ∪ {0} and χ ◦ χ¯ < Id. Following the proofs of Theorem 3.1 and Lemma A.1 in [118], one can find a χˆ ∈ K∞ such that χˆ > χ, χˆ is continuously differentiable on (0, ∞) and χˆ ◦ χ¯ < Id.  It is easy to verify that χˆ ◦ χi < Id for i = 1, . . . , n.

© Huazhong University of Science and Technology Press, Wuhan and Springer Nature Singapore Pte Ltd. 2020 T. Liu et al., Robust Event-Triggered Control of Nonlinear Systems, Research on Intelligent Manufacturing, https://doi.org/10.1007/978-981-15-5013-3

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210

Appendix B: Technical Lemmas

Lemma B.3 For any a, b ∈ R, if there exist a θ ∈ K and a constant c ≥ 0 such that |a − b| ≤ max{θ ◦ (Id + θ)−1 (|a|), c},

(B.3)

then |a − b| ≤ max{θ(|b|), c}. Proof We first consider the case of θ ◦ (Id + θ)−1 (|a|) ≥ c, which together with (B.3) implies |a − b| ≤ θ ◦ (Id + θ)−1 (|a|).

(B.4)

In this case, |a| − |b| ≤ θ ◦ (Id + θ)−1 (|a|), and thus, (Id − θ ◦ (Id + θ)−1 )(|a|) ≤ |b|. Note that Id − θ ◦ (Id + θ)−1 =(Id + θ) ◦ (Id + θ)−1 − θ ◦ (Id + θ)−1 = (Id + θ)−1 . Then, we have |a| ≤ (Id + θ)(|b|). By using (B.4) again, it can be achieved that |a − b| ≤ θ(|b|). Next, we consider the case of θ ◦ (Id + θ)−1 (|a|) < c. Clearly, from (B.3), it follows that |a − b| ≤ c. Therefore, Lemma B.3 is proved.  Lemma B.4 For any locally Lipschitz function h : Rn 1 × Rn 2 × · · · × Rn m → R p −1 satisfying h(0, . . . , 0) = 0 and any ϕ1 , . . . , ϕm ∈ K∞ with ϕ−1 1 , . . . , ϕm being Lipschitz on compact sets, there exists a continuous, positive, and nondecreasing function L h : R+ → R+ such that  |h(z 1 , . . . , z m )| ≤ L h

 max {|z i |}

i=1,...,m

max {ϕi (|z i |)}

i=1,...,m

(B.5)

for all z, where z = [z 1T , . . . , z mT ]T . Proof For a locally Lipschitz h satisfying h(0, . . . , 0) = 0, one can always find a continuous, positive, and nondecreasing function L h0 : R+ → R+ such that  |h(z 1 , . . . , z m )| ≤ L h0

 max {|z i |}

i=1,...,m

max {|z i |}

i=1,...,m

(B.6)

for all z. Define ϕ(s) ˘ = max {ϕi−1 (s)} i=1,...,m

(B.7)

−1 ˘ is for s ∈ R+ . Then, ϕ˘ ∈ K∞ . Since ϕ−1 1 , . . . , ϕm are Lipschitz on compact sets, ϕ Lipschitz on compact sets. From the definition, one has

Appendix B: Technical Lemmas

 ϕ˘

211

 max {ϕi (|z i |)} = max {ϕ˘ ◦ ϕi (|z i |)}

i=1,...,m

i=1,...,m

≥ max {ϕi−1 ◦ ϕi (|z i |)} i=1,...,m

= max {|z i |}.

(B.8)

i=1,...,m

With the ϕ˘ which is Lipschitz on compact sets, there exists a continuous, positive, and nondecreasing function L ϕ˘ : R+ → R+ such that  ϕ˘



 max {ϕi (|z i |)} ≤ L ϕ˘

i=1,...,m

 max {ϕi (|z i |)}

max {ϕi (|z i |)}.

i=1,...,m

i=1,...,m

(B.9)

Lemma B.4 is proved by substituting (B.8) and (B.9) into (B.6), and defining a continuous, positive, and nondecreasing L h such that  Lh

     max {|z i |} ≥ L h0 max {|z i |} L ϕ˘ max {ϕi (|z i |)}

i=1,...,m

i=1,...,m

i=1,...,m

(B.10) 

for all z.

Lemma B.5 For any continuous, positive-valued and nondecreasing K 0 : R+ → R+ and any ϕ ∈ K∞ being Lipschitz on compact sets, there exists a continuous, positive-valued and nondecreasing K : R+ → R+ such that K 0 (s)s ≤ K (s)ϕ−1 (s)

(B.11)

for all s ∈ R+ . Proof Since ϕ ∈ K∞ and ϕ is locally Lipschitz, there exist continuous, positivevalued and nondecreasing functions K¯ ϕ , Kˆ ϕ : R+ → R+ such that ϕ(s) ≤ K¯ ϕ (s)s = Kˆ ϕ (ϕ(s))s

(B.12)

for all s ∈ R+ . Equivalently, it holds that s ≤ Kˆ ϕ (s)ϕ−1 (s)

(B.13)

K 0 (s)s ≤ K 0 (s) Kˆ ϕ (s)ϕ−1 (s)

(B.14)

for all s ∈ R+ , which implies

for all s ∈ R+ . Lemma B.5 is proved by defining K (s) = K 0 (s) Kˆ ϕ (s) for all  s ∈ R+ .

Appendix C

Gain Assignment

The gain assignment technique, originally proposed in [122], plays an important role in feedback stabilization of interconnected nonlinear systems. It can be considered as a generalization of the celebrated pole placement method from linear time-invariant systems to nonlinear systems. Here, it is shown that the gain assignment method is also crucial for the synthesis of the output-feedback control system to satisfy the ISS gain conditions for solving the event-triggered control problem. Consider a first-order nonlinear system: η˙ = φ(η, ω1 , . . . , ωm ) + κ¯ η s = η + ωm+1

(C.1) (C.2)

where η ∈ R is the state, κ¯ ∈ R is the control input, ω1 , . . . , ωm+1 ∈ R represent external inputs, η s ∈ R is the measurement of η, the nonlinear function φ(η, ω1 , . . . , ωm ) is locally Lipschitz and satisfies |φ(η, ω1 , . . . , ωm )| ≤ ψφ (|[η, ω1 , . . . , ωm ]T |)

(C.3)

with ψφ ∈ K∞ . It can be proved that condition (C.3) implies η

|φ(η, ω1 , . . . , ωm )| ≤ ψφ (|η|) +

m 

ψφωk (|ωk |)

(C.4)

k=1 η

with ψφ , ψφω1 , . . . , ψφωm ∈ K∞ . Lemma C.1 Consider system (C.1). For any specified 0 < c < 1,  > 0 and γηω1 , . . . , γηωm ∈ K∞ , we can find a κ : R → R which is odd and continuously differentiable on (−∞, 0) ∪ (0, ∞) and satisfies

© Huazhong University of Science and Technology Press, Wuhan and Springer Nature Singapore Pte Ltd. 2020 T. Liu et al., Robust Event-Triggered Control of Nonlinear Systems, Research on Intelligent Manufacturing, https://doi.org/10.1007/978-981-15-5013-3

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214

Appendix C: Gain Assignment η

κ((1 − c)s) ≥ ψφ (s) +

m 

 −1  ψφωk ◦ γηωk (s) + s 2 k=1

(C.5)

for all s ≥ 0, such that the closed-loop system with κ¯ = κ(η s ) is ISS with Vη (η) = |η| as an ISS-Lyapunov function, which satisfies   ωk γη (|ωk |) Vη (η) ≥ max k=1,...,m+1   ⇒ ∇Vη (η) φ(η, ω1 , . . . , ωm ) + κ(η s ) ≤ −Vη (η), a.e.

(C.6)

 −1  −1 where γηωm+1 (s) = s/c for s ∈ R+ . Moreover, if γηω1 , . . . , γηωm are Lipschitz on compact sets, then κ can be chosen to be continuously differentiable on (−∞, ∞). By considering Vη (η) = |η| as an ISS-Lyapunov function, Lemma C.1 can be proved based on the proofs of the gain assignment lemmas in [116, 169]. The proof is not provided here due to space limitation.

Appendix D

Proofs

D.1 Proof of Theorem 2.6 The proof is basically a reduced version of the proof for the IOS small-gain theorem given in [122]. We only make slight modifications to handle the difference between the two forms of the ISS property. Pick any specific initial state x(0) and any measurable and locally essentially bounded input u. Step 1–UBIBS: Suppose that the solution x(t) of the interconnected system is defined on [0, Tmax ) with Tmax > 0. Define σi (s) = βi (s, 0) for s ∈ R+ . For i = 1, 2, by using the ISS property (2.37), one has |xi (t)| ≤ max{σi (|xi (0)|), γi(3−i) (x3−i [0,Tmax ) ), γiu (u i ∞ )}

(D.1)

for 0 ≤ t < Tmax , and thus, by taking the supremum of |xi (t)| over [0, Tmax ), we have xi [0,Tmax ) ≤ max{σi (|xi (0)|), γi(3−i) (x3−i [0,Tmax ) ), γiu (u i ∞ )}.

(D.2)

By substituting (D.2) with i replaced by 3 − i in the right-hand side of (D.1), it is achieved that |xi (t)| ≤ max{σi (|xi (0)|), γi(3−i) ◦ σ3−i (|x3−i (0)|), γi(3−i) ◦ γ(3−i)i (xi [0,Tmax ) ), u γi(3−i) ◦ γ3−i (u 3−i ∞ ), γiu (u i ∞ )}.

(D.3)

Define σ¯ i1 (s) = max{σi (s), γi(3−i) ◦ σ3−i (s)},

(D.4)

σ¯ i2 (s) =

(D.5)

max{γiu (s), γi(3−i)

◦

u γ3−i (s)}

for s ∈ R+ . By taking the supremum of xi (t) over [0, Tmax ) and using (D.3), one has © Huazhong University of Science and Technology Press, Wuhan and Springer Nature Singapore Pte Ltd. 2020 T. Liu et al., Robust Event-Triggered Control of Nonlinear Systems, Research on Intelligent Manufacturing, https://doi.org/10.1007/978-981-15-5013-3

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216

Appendix D: Proofs

xi [0,Tmax ) ≤ max{σ¯ i1 (|x(0)|), σ¯ i2 (u∞ ), γi(3−i) ◦ γ(3−i)i (xi [0,Tmax ) )} ≤ max{σ¯ i1 (|x(0)|), σ¯ i2 (u∞ )},

(D.6)

where the small-gain condition (2.38) is used for the last inequality. This means that |xi (t)| is defined on [0, ∞). With the Tmax in (D.6) replaced by ∞, it is achieved that |xi (t)| ≤ max{σ¯ i1 (|x(0)|), σ¯ i2 (u∞ )}

(D.7)

for all t ≥ 0. UBIBS property is proved as property (D.7) holds for any initial state x(0) and any measurable and locally essentially bounded input u. Step 2–ISS: Denote xi∗ = max{σ¯ i1 (|x(0)|), σ¯ i2 (u∞ )} for i = 1, 2. By means of time invariance and causality, (2.37) implies     |xi (t)| ≤ max βi (|xi (t0 )|, t − t0 ), γi(3−i) x3−i [t0 ,t] , γiu (u i ∞ )

(D.8)

for any 0 ≤ t0 ≤ t, and thus, by choosing t0 = t/2, we have     

  u t

t , , γ x , γ  (u  ) |xi (t)| ≤ max βi

xi i(3−i) 3−i [t/2,t] i ∞ i 2 2      t , γi(3−i) x3−i [t/2,t] , γiu (u i ∞ ) (D.9) ≤ max βi xi∗ , 2 for i = 1, 2. By taking the maximum of xi (t) over [t/2, t], it is achieved that τ  τ    



, γi(3−i) x3−i [τ /2,τ ] , γiu (u i ∞ ) xi [t/2,t] ≤ max βi xi

, t/2≤τ ≤t 2 2      u t ∗ (D.10) , γi(3−i) x3−i [t/4,t] , γi (u i ∞ ) ≤ max βi xi , 4 for i = 1, 2. Then, by substituting (D.10) with i replaced by 3 − i into (D.9), one has     t ∗ t ∗ |xi (t)| ≤ max βi xi , , γi(3−i) ◦ β3−i x3−i , , 2 4   γi(3−i) ◦ γ(3−i)i xi [t/4,t] ,  u u γi(3−i) ◦ γ3−i (u 3−i ∞ ) , γi (u i ∞ ) .

(D.11)

Recall the xi∗ = max{σ¯ i1 (|x(0)|), σ¯ i2 (u∞ )} for i = 1, 2. Property (D.11) implies that     |xi (t)| ≤ max β¯i (|x(0)|, t), γi(3−i) ◦ γ(3−i)i xi [t/4,t] , γ¯ iu (u∞ )

(D.12)

Appendix D: Proofs

217

for all t ≥ 0, where     t t , γi(3−i) ◦ β3−i σ¯ (3−i)1 (s), , β¯i (s, t) = max βi σ¯ i1 (s), 2 4 u γ¯ iu (s) = max γiu (s), βi (σ¯ i2 (s), 0) , γi(3−i) ◦ γ3−i (s),    γi(3−i) ◦ β3−i σ¯ (3−i)2 (s), 0 .

(D.13)

(D.14)

Clearly, β¯i ∈ KL, γ¯ iu ∈ K. Then, by using Lemma B.1, there exists a βˆi ∈ KL such that  |xi (t)| ≤ max βˆi (|x(0)|, t), γ¯ iu (u∞ )

(D.15)

for all t ≥ 0, for i = 1, 2. Note that property (D.15) holds for any initial state x(0) and any measurable and locally essentially bounded u. The ISS of the interconnected system is proved.

D.2 Proof of Theorem 2.8 Theorem 2.8 is proved by constructing an ISS-Lyapunov function V for the interconnected system. For γ12 , γ21 ∈ K satisfying the small-gain condition (2.47), we find a σ ∈ K∞ such that it is continuously differentiable on (0, ∞) and satisfies σ > γ21 , σ −1 > γ12 .

(D.16)

This can be achieved because for γ12 , γ21 ∈ K satisfying condition (2.47), there exists a γˆ 12 ∈ K∞ such that γˆ 12 > γ12 and γˆ 12 ◦ γ21 < Id. One can always find a σ ∈ K∞ such that it is continuously differentiable on (0, ∞) and satisfies −1 , γ21 < σ < γˆ 12

(D.17)

which guarantees the satisfaction of (D.16). See [118] for the detailed proof of the existence of such σ. An ISS-Lyapunov function candidate for the interconnected system is defined as V (x) = max {σ(V1 (x1 )), V2 (x2 )} .

(D.18)

Clearly, V is positive definite and radially unbounded. Also, V is continuously differentiable almost everywhere.

218

Appendix D: Proofs

Let f (x, u) = [ f 1T (x, u 1 ), f 2T (x, u 2 )]T . In the following procedure, we prove that there exist a γ ∈ K and a continuous, positive definite α such that V (x) ≥ γ(|u|) ⇒ ∇V (x) f (x, u) ≤ −α(V (x))

(D.19)

for almost all x and all u. For this purpose, define the following sets: A = {(x1 , x2 ) : V2 (x2 ) < σ(V1 (x1 ))}, B = {(x1 , x2 ) : V2 (x2 ) > σ(V1 (x1 ))},

(D.20) (D.21)

O = {(x1 , x2 ) : V2 (x2 ) = σ(V1 (x1 ))}.

(D.22)

For any fixed point p = ( p1 , p2 ) = (0, 0) and a control value v = (v1 , v2 ), consider the following three cases. Case 1: p ∈ A In this case, V (x) = σ(V1 (x1 )) in a neighborhood of p, and consequently ∇V ( p) f ( p, v) =

∂σ(V1 ( p1 )) ∇V1 ( p1 ) f 1 ( p, v1 ). ∂V1 ( p1 )

(D.23)

For p ∈ A, it holds that V2 ( p2 ) < σ(V1 ( p1 )), and based on the definition of σ, V1 ( p1 ) > γ12 (V2 ( p2 )). With (2.46), this implies ∇V1 ( p1 ) f 1 ( p, v1 ) ≤ −α1 (v1 ( p1 ))

(D.24)

whenever V1 ( p1 ) ≥ σ ◦ γ1u (|v1 |). It follows that, for p ∈ A, ∇V ( p) f ( p, v) ≤ −αˆ 1 (V ( p))

(D.25)

whenever V ( p) ≥ γˆ 1u (|v1 |), where γˆ 1u (s) = σ ◦ γ1u (s)

(D.26)

for s ∈ R+ , and αˆ 1 is a continuous and positive definite function such that αˆ 1 (s) ≤ σ d (σ −1 (s))α1 (σ −1 (s))

(D.27)

for s > 0, with σ d (s) = dσ(s)/ds. Case 2: p ∈ B In this case, by using similar arguments as in Case 1, it can be proved that ∇V ( p) f ( p, v) ≤ −αˆ 2 (V ( p))

(D.28)

Appendix D: Proofs

219

whenever V ( p) ≥ γˆ 2u (|v2 |), where αˆ 2 = α2 and γˆ 2u = γ2u . Case 3: p ∈ O First note that it holds for the locally Lipschitz function V that

d

V (ϕ(t)) ∇V ( p) f ( p, v) = dt t=0

(D.29)

for almost all p and all v, with ϕ(t) = [ϕ1T (t), ϕ2T (t)]T being the solution of the initial-value problem ϕ(t) ˙ = f (ϕ(t), v), ϕ(0) = p.

(D.30)

In this case, assume p = ( p1 , p2 ) = (0, 0) and V1 ( p1 ) ≥ γ1u (|v1 |),

(D.31)

γ2u (|v2 |).

(D.32)

V2 ( p2 ) ≥

Then, by using similar arguments as for Cases 1 and 2, one has ∇σ(V1 ( p1 )) f 1 ( p, v1 ) ≤ −αˆ 1 (V ( p))

(D.33)

∇V2 ( p2 ) f 2 ( p, v2 ) ≤ −αˆ 2 (V ( p))

(D.34)

where αˆ 1 and αˆ 2 are continuous and positive definite functions. Note that in this case p1 = 0 and p2 = 0. Then, because of the continuous differentiability of σ, V1 , and V2 , and the continuity of f , there exist neighborhoods X1 of p1 and X2 of p2 such that ∇σ(V1 (x1 )) f 1 (x, v1 ) ≤ −αˆ 1 (V ( p)), ∇V2 (x2 ) f 2 (x, v2 ) ≤ −αˆ 2 (V ( p))

(D.35) (D.36)

for all x ∈ X1 × X2 . Note also that there exists δ > 0 such that ϕ(t) ∈ X1 × X2 for all 0 ≤ t < δ. Now pick Δt ∈ (0, δ). If ϕ(Δt) ∈ A ∪ O, then V (ϕ(Δt)) − V ( p) = σ(V1 (ϕ1 (Δt))) − σ(V1 ( p1 )) 1 ≤ − αˆ 1 (V ( p))Δt. 2

(D.37)

Similarly, if ϕ(Δt) ∈ B ∪ O, then V (ϕ(Δt)) − V ( p) = V2 (ϕ2 (Δt)) − V2 ( p2 ) 1 ≤ − αˆ 2 (V ( p))Δt. 2

(D.38)

220

Appendix D: Proofs

Hence, if V is differentiable at p, then ∇V ( p) f ( p, v) ≤ −α(V ( p))

(D.39)

where α(s) = min{αˆ 1 (s)/2, αˆ 1 (s)/2} for s ∈ R+ . Note that conditions (D.31) and (D.32) can be guaranteed by V ( p) ≥ max{γˆ 1u (|v1 |), γˆ 2u (|v2 |)}.

(D.40)

By combining the three cases, it can be concluded that V ( p) ≥ max{γˆ 1u (|v1 |), γˆ 2u (|v2 |)} ⇒ ∇V ( p) f ( p, v) ≤ −α(V ( p)).

(D.41)

Since V is continuously differentiable almost everywhere, (D.41) holds for almost all p and all v. Property (D.19) is then proved by defining γ(s) = max{γˆ 1u (s), γˆ 2u (s)} for s ∈ R+ . Thus, V is an ISS-Lyapunov function of the interconnected system. Theorem 2.8 is proved.

D.3 More Remarks About Example 4.2 Define υ = x − z and v = z 3 . Then, υ(t) ˙ = x(t) ˙ − z˙ (t) = −υ(t) − w(t) + z 3 (t) ≥ −υ(t) − μ(t) + v(t), for all t ∈



k∈S [tk , tk+1 )

(D.42)

and 5

v(t) ˙ = 3z 2 (t)˙z (t) = −3z 5 (t) = −3v 3 (t)

(D.43)

for all t ≥ 0. We consider the case of v(0) > 0. In this case, v(t) is strictly decreasing and v(t) ≤ v(0) for all t ≥ 0. Then, one can find a c∗ > 0 such that ∗

˘ v(t) ≥ v(0)e−c t =: v(t)

(D.44)

for all t ≥ 0. Moreover, if v(0) < 1, then c∗ can be chosen to be strictly less than one. Define υ ∗ (t) as the solution of the initial value problem ˘ υ˙ ∗ (t) = −υ ∗ (t) − μ(t) + v(t)

(D.45)

Appendix D: Proofs

221

with initial condition υ ∗ (0) = υ(0). Then, a direct application of the comparison principle yields: υ(t) ≥ υ ∗ (t)

(D.46)

 for all t ∈ k∈S [tk , tk+1 ). With μ(t) defined in (4.17) and v(t) ˘ defined above, if c = 1, then ∗



t

e−(t−τ ) (−μ(τ ) + v(τ ˘ )) dτ  t   −c∗ τ dτ eτ −μ(0)e−cτ + v(0)e ˘ = υ(0)e−t + e−t 0   v(0) v(0) −c∗ t μ(0) −ct μ(0) e−t + − e . e − = υ(0) + 1 − c 1 − c∗ 1 − c∗ 1−c

υ (t) = υ(0)e

−t

+

0

(D.47)

Thus, | f (x(t), z(t), w(t))| = | − x(t) − w(t) + z(t)| = |υ(t) + w(t)| ≥ υ(t) − μ(t)   v(0) v(0) −c∗ t μ(0) e−t + − e ≥ υ(0) + 1 − c 1 − c∗ 1 − c∗ μ(0) −ct e − μ(0)e−ct − 1−c ∗ =: m 1 e−t + m 2 e−c t − m 3 e−ct (D.48) for all t ∈ that



k∈S [tk , tk+1 ).

Moreover, there exist z(0) > 0, x(0) > 0, μ(0) > 0 such

m 1 ≥ 0, m 2 > 0, m 2 ≥ 2m 3 , 2c∗ ≤ c.

(D.49)

In this case, it is directly checked that | f (x(t), z(t), w(t))| ≥ υ(t) − μ(t) ≥

m 2 −c∗ t e >0 2

(D.50)

 for all t ∈ k∈S [tk , tk+1 ). Following a similar reasoning, property (D.50) can still be proved in the case of c = 1. Recall υ(t) = x(t) − z(t). With z(0) > 0 and μ(0) > 0, we have z(t) > 0, μ(t) > 0 and thus x(t) > z(t) > 0

(D.51)

222

Appendix D: Proofs

 for all t ∈ k∈S [tk , tk+1 ). With υ(t) − μ(t) > 0 given by (D.50), we also have f (x(t), z(t), w(t)) = −x(t) − w(t) + z(t) = −υ(t) − w(t) ≤ −υ(t) + μ(t) < 0 for all t ∈

(D.52)



k∈S [tk , tk+1 ).

D.4 Proof of Lemma 4.1 We only prove property (4.33). Property (4.34) can be proved similarly. Since σz ∈ K∞ , V¯z is positive definite and radially unbounded. We choose x γ¯ z , γ¯ zw ∈ K∞ such that 

γ˘ xw

−1

◦ γˆ xz ◦ γzx ◦ γ˘ xw ≤ γ¯ zx < Id,  w −1 γˆ z ◦ γzw ≤ γ¯ zw < Id.

(D.53) (D.54)

 −1 The existence of such γ¯ zx and γ¯ zw can be guaranteed as γ˘ xw ◦ γˆ xz ◦ γzx ◦ γ˘ xw <    w −1 −1 ◦ γ˘ xw = Id and γˆ zw ◦ γzw < Id. γ˘ x  −1 ◦ γˆ xz ◦ γzx ◦ γ˘ xw (V¯x (x)). Then, V¯z (z) ≥ γ¯ zx (V¯x (x)) implies V¯z (z) ≥ γ˘ xw  w −1  −1 ◦ γˆ xz (Vz (z)) ≥ γ˘ xw ◦ γˆ xz ◦ By using the definitions of V¯z and V¯x , we have γ˘ x   x w w −1 x w (Vx (x)), which implies Vz (z) ≥ γz (Vx (x)). Also, V¯z (z)≥γ¯ z (|w|) γz ◦ γ˘ x ◦ γ˘ x  −1  −1  w −1 ◦ γˆ xz (Vz (z))≥ γˆ zw ◦ γzw (|w|) and thus Vz (z) ≥ γˆ xz ◦ γ˘ xw ◦ implies γ˘ x  w −1 ◦ γzw (|w|) ≥ γzw (|w|). It is proved that γˆ z V¯z (z) ≥ max{γ¯ zx (V¯x (x)), γ¯ zw (|w|)} ⇒Vz (z) ≥ max{γzx (Vx (x)), γzw (|w|)}.

(D.55)

Property (4.33) can then be proved by using (4.7). This ends the proof of Lemma 4.1.

D.5 Proof of Lemma 4.2 The result of (4.35) can be proved by using a combination of the Lyapunov-based cyclic-small-gain theorem and the comparison principle, and is omitted here. See, e.g., [138, Lemma 3.4], for the comparison principle.

Appendix D: Proofs

223

Suppose that Conditions (A) and (B) are satisfied. Then, with Lemma 4.1, (4.33) and (4.34) hold with α¯ z = α¯ x = Ω

(D.56)

for all [z T , x T , μ]T satisfying V (z, x, μ) < Δ. For convenience of discussions, define v1 (t) = V¯z (z(t)), v2 (t) = V¯x (x(t)), v3 (t) = μ(t), and v(t) = V (z(t), x(t), μ(t)). Then v(t) = max{v1 (t), v2 (t), v3 (t)}. Now, we prove D + v(t) ≤ −Ω(v(t))

(D.57)

 for all T O ≤ t < sup k∈S [tk , tk+1 ), where D + represents the upper right-hand derivative and is defined by D + v(t) = lim sup h→0+

v(t + h) − v(t) . h

(D.58)

 Consider a specific time instant t satisfying T O ≤ t < sup k∈S [tk , tk+1 ). Since γ¯ zx < Id and γ¯ zw < Id, when v1 (t) = v(t) > 0, V¯z (z(t)) = V (z(t), x(t), μ(t)) > 0, there exists a neighborhood Θ of [z T (t), x T (t), μ(t)]T such that V¯z ( pz ) ≥ max{γ¯ zx (V¯x ( px )), γ¯ zw ( pμ )} and ∇Vz ( pz ) exists for all [ pzT , pxT , pμ ] ∈ Θ. Then, due to the continuity of V¯z (z(t)), V¯x (x(t)) with respect to t, there exists a t > t such that V¯z (z(τ )) ≥ max{γ¯ zx (V¯x (x(τ ))), γ¯ zw (|μ(τ )|)} ≥ max{γ¯ zx (V¯x (x(τ ))), γ¯ zw (|w(τ )|)} for all τ ∈ (t, t ), which implies ∇ V¯z (z(τ ))h(z(τ ), x(τ ), w(τ )) ≤ −Ω(V¯z (z(τ )))

(D.59)

for all τ ∈ (t, t ), according to (4.33). Then, we have D + v1 (t) ≤ −Ω(v1 (t)).

(D.60)

Following the same reasoning, if v2 (t) = v(t), then D + v2 (t) ≤ −Ω(v2 (t)).

(D.61)

Note that D + v3 (t) = −Ω(v  3 (t)) holds automatically for v3 (t) = μ(t). For T O ≤ t < sup k∈S [tk , tk+1 ), define I (t) = {i ∈ {1, 2, 3} : vi (t) = v(t)}. Then, by using [72, Lemma 2.9], we have D + v(t) = max{D + vi (t) : i ∈ I (t)} ≤ max{−Ω(vi (t)) : i ∈ I (t)}

224

Appendix D: Proofs

= max{−Ω(vi (t)) : vi (t) = v(t), i = 1, 2, 3} = −Ω(v(t)).

(D.62)

Property (D.57) is proved. Then, by directly applying the comparison principle (see, e.g., [138, Lemma 3.4]), the proof of Lemma 4.2 follows readily.

D.6 Proof of Lemma 6.1 We prove (6.17) with φi : Ri × Ri+1 × R × Rn d  R satisfying T ∞T , w¯ i−1 , d T ]T |) |φi | ≤ ψφi (|[e¯i+1

(D.63)

∞ ) for with ψφi ∈ K∞ . From the definition of ei , one has xi − ei ∈ Si−1 (x¯i−1 , w¯ i−1 i = 2, . . . , n. With e1 = x1 , one has

e˙1 = x2 + f 1 (x¯1 , d) = x2 − e2 + f 1 (x¯1 , d) + e2 =: x2 − e2 + φ1 (x¯1 , e¯2 , d)

(D.64)

where x2 − e2 ∈ S1 (x¯1 , w¯ 1∞ ). With Assumption 6.1 satisfied, there exists a ψφ1 ∈ K∞ being Lipschitz on compact sets such that |φ1 | ≤ ψφ1 (|[e¯2T , w¯ 0∞ , d T ]T |). Suppose that for each i = 1, . . . , k − 1, the ei -subsystem can be represented in the form of (6.17) with property (D.63) satisfied. We prove that the ek -subsystem can be represented in the form of (6.17) and satisfies property (D.63) with i = k. Consider the recursive definition of Si in (6.14). The strictly decreasing property of κi implies that with specific x¯i , max Si (x¯i , w¯ i∞ ) = κi (xi − wi∞ − max ∞ ∞ )) and min Si (x¯i , w¯ i∞ ) = κi (xi + wi∞ − min Si−1 (x¯i−1 , w¯ i−1 )) for Si−1 (x¯i−1 , w¯ i−1 ∞ i = 1, . . . , n, where by default, S0 (x¯0 , w¯ 0 ) = {0}. The continuously differentiability of κi implies the continuously differentiability of max Si (x¯i , w¯ i∞ ) with respect to x¯i . We only consider the case where ek > 0. The proof for the case of ek < 0 is similar. In the case of ek > 0, by using its definition, one has ∞ ) ∂ max Sk−1 (x¯k−1 , w¯ k−1 x˙¯k−1 ∂ x¯k−1 ∞ ) ∂ max Sk−1 (x¯k−1 , w¯ k−1 = xk+1 − ek+1 + f i (x¯k , d) − x˙¯k−1 + ek+1 ∂ x¯k−1 ∞ ∈ Sk (x¯k , w¯ k∞ ) + φk (x¯k , w¯ k−1 , ek+1 , d). (D.65)

e˙k = x˙k −

Appendix D: Proofs

225 ∂ min S

(x¯

,w¯ ∞ )

k−1 k−1 k−1 ˙ For the case of ei < 0, we have e˙k = x˙k − x¯k−1 = xk+1 − ek+1 + ∂ x¯k−1 ∞  ∂ min Sk−1 (x¯k−1 ,w¯ k−1 ) ∞ ,e f i (x¯k , d) − x¯˙k−1 + ek+1 . Then, φk (x¯k , w¯ k−1 k+1 , d) = f k ( x¯ k , d) −

∂ x¯k−1

∞ ) ∞ )   ∂ max Sk−1 (x¯k−1 ,w¯ k−1 ∂ min Sk−1 (x¯k−1 ,w¯ k−1 x˙¯k−1 + ek+1 . + (0.5 − 0.5sgn(ek )) (0.5 + 0.5sgn(ek )) ∂ x¯k−1 ∂ x¯k−1

The satisfaction of condition (D.63) can be directly proved by using the definitions of κi and Si for i = 1, . . . , k − 1 and Assumption 6.1. Notice that in the case of k = n, xk+1 = u k and ek+1 = 0. Now, we prove (6.18). With property (D.63) satisfied, one can find ψφe1i , . . . , e w , ψφwi1 , . . . , ψφii−1 , ψφdi ∈ K∞ being Lipschitz on compact sets such that it holds ψφi+1 i i+1 ek  wk ¯ k∞ ) + ψφdi (|d|). that |φi | ≤ k=1 ψφi (|ek |) + i−1 k=1 ψφi (w Each κi is chosen to be in the form of κi (r ) = −νi (|r |)r , where νi : R+ → R+ is positive, nondecreasing and continuously differentiable on (0, ∞), and satisfies (1 − ci )νi ((1 − ci )s)s ≥ ψφeii (s) +



 −1 ψφeki ◦ γeeik (s)

k=1,...,i−1,i+1

+

i−1 

 −1  −1 ψφwik ◦ γewi k (s) + ψφdi ◦ γedi (s) + i s

(D.66)

k=1

for s ∈ R+ , with constant i > 0. For any constant 0 < ci < 1, i > 0, any ψφe1i , . . . ,  e −1 e wi−1 w1 d ψφi+1 , ψ , . . . , ψ , ψ ∈ K being Lipschitz on compact sets, any γe k ∈ ∞ φ φ φ i i i i  wi −1 K∞ for k = 1, . . . , i − 1, i + 1 being Lipschitz on compact sets, any γei k ∈  d −1 ∈ K∞ K∞ being Lipschitz on compact sets for k = 1, . . . , i − 1 and any γei being Lipschitz on compact sets, the existence of such νi can be guaranteed by [116, Lemma 1]. Recall the definition Vi (ei ) = |ei | for i = 1, . . . , n. Consider the case of Vi (ei ) ≥

max

k=1,...,i−1

 e γeeik (Vk (ek )), γeii+1 (Vi+1 (ei+1 )), . γewi k (wk∞ ), γewi i (wi∞ ), γedi (d ∞ )

(D.67)

 −1 In this case, it holds that |ek | ≤ γeeik (|ei |) for k = 1, . . . , i − 1, i + 1, wk∞ ≤  −1  w −1 (|ei |) for k = 1, . . . , i − 1, wi∞ ≤ ci |ei | and d ∞ ≤ γedi (|ei |). γei k ∞ ), it holds that |xi − With the definition of ei , for any pi−1 ∈ Si−1 (x¯i−1 , w¯ i−1 pi−1 | ≥ |ei | and sgn(xi − pi−1 ) = sgn(ei ). With wi∞ ≤ ci |ei |, for any ∞ ), and any |ai | ≤ 1, one has |xi − pi−1 + ai wi∞ | ≥ (1 − ci )|ei | pi−1 ∈ Si−1 (x¯i−1 , w¯ i−1 and sgn(xi − pi−1 + ai wi∞ ) = sgn(ei ). Thus, for any pi ∈ Si (x¯i , w¯ i∞ ), by using the definition of κi , one has | pi | ≥ κi ((1 − ci )|ei |) and sgn( pi ) = sgn(ei ). Then, for any pi ∈ Si (x¯i , w¯ i∞ ), it holds that

226

Appendix D: Proofs

−sgn(ei )( pi + φi ) ≤ − sgn(ei ) pi + |φi | = − | pi | + |φi | ≤ − κi ((1 − ci )|ei |) +

i+1 

ψφeki (|ek |) +

k=1

i−1 

ψφwik (w¯ k∞ ) + ψφdi (d ∞ )

k=1

≤ (1 − ci )νi ((1 − ci )|ei |)|ei | +

i+1 

 −1 ψφeki ◦ γeeik (|ei |)

k=1

 −1 ∞ + ψφdi ◦ γedi (d ) +

i−1 

 −1 ψφwik ◦ γewi k (|ei |)

k=1

≤ − i |ei | = −i Vi (ei )

(D.68)

Property (6.18) is proved as when ei = 0.

D.7 Proof of Proposition 6.1 We first prove that by appropriately choosing λi , the controlled system is transformed into an interconnected system composed of ISS subsystems with ISS gains designable, and then employ the nonlinear small-gain theorem to guarantee the stability of the interconnected system. ISS of the ei -Subsystems Condition (6.53) implies that the designed control law λi is decreasing and C 1 on (0, ∞). Then, for any specific x¯i , max Λi (x¯i , w¯ i∞ ) = λi (xi − wi∞ − max ∞ ∞ )) and min Λi (x¯i , w¯ i∞ ) = λi (xi + wi∞ − min Λi−1 (x¯i−1 , w¯ i−1 )). Λi−1 (x¯i−1 , w¯ i−1 ∞ ∞ 1 Moreover, max Λi (x¯i , w¯ i ) and min Λi (x¯i , w¯ i ) are C with respect to x¯i . Recall the definition of h i in (6.55). Assumption 6.2 guarantees   |h i (x¯i , x¯iθ , e¯i+1 )| ≤ ψh i |x¯i | + |x¯iθ | + |ei+1 | ,

(D.69)

where ψh i is Lipschitz on compact sets and of class K∞ . For k = 1, . . . , i − 1, the definition of ek+1 in (6.15) implies that min Λk ≤ xk+1 − ek+1 ≤ max Λk , and thus there exist class K∞ functions ψxk+1 being Lipschitz on compact sets for k = 1, . . . , i − 1 such that |xk+1 | ≤ ψxk+1 (|e¯k+1 | + w¯ k∞ ).

(D.70)

θ θ | ≤ ψxk+1 (|e¯k+1 | + w¯ k∞ ) holds. Based on (D.69) Property (D.70) implies that |xk+1 e w and (D.70), there existclass K∞ functions ψhe1i , . . . , ψhi+1 , ψhwi1 , . . . , ψh ii−1 and ψˆ he1i , i

Appendix D: Proofs

227

. . . , ψˆ heii which are Lipschitz on compact sets such that



h i (x¯i , x¯ θ , e¯i+1 )

i

≤

i+1  k=1

ψheki (|ek |) +

i−1 

ψhwik (wk∞ ) +

k=1

i 

ψˆ heki (|ekθ |).

(D.71)

k=1

There exists a κi which is positive, nondecreasing and C 1 on (0, ∞) such that ci (1 − bi )κi ((1 − bi )s)s ≥ ψheii (s) +



 −1 ψheki ◦ γeeik (s)

k=1,...,i−1,i+1

+

i−1  k=1

 −1 ψhwik ◦ γewi k (s) +

i 

 −1 (s) + i s ψˆ heki ◦ γˆ eeik

(D.72)

k=1

for s ∈ R+ , with constant i > 0, where ci is defined in (6.44), bi is a positive constant (·) ∈ K∞ is C 1 on (0, ∞) and their and satisfies 0 < bi < 1, and each designable γ(·) inverse functions are Lipschitz on compact sets. With κi satisfying (D.72) and Vi (ei ) = |ei |, we consider ⎫ ⎧ ek ei+1 ⎨ γei (Vk (ek )), γei (Vi+1 (ei+1 )), ⎬ γewk (wk∞ ), b1i wi∞ , Vi (ei ) ≥ max . k=1,...,i−1 ⎩ ei ⎭ γˆ eik (Vk (ekθ )), γˆ eeii (Vi (eiθ ))

(D.73)

 −1  −1 In this case, we have wk∞ ≤ γewi k (|ei |) for k = 1, . . . , i − 1, |ek | ≤ γeeik  −1 (|ei |) for k = 1, . . . , i − 1, i + 1, |ekθ | ≤ γˆ eeik (|ei |) for k = 1, . . . , i and wi∞ ≤ bi |ei |. ∞ ), With the definition of ei in (6.15), when ei = 0, for any μi−1 ∈ Λi−1 (x¯i−1 , w¯ i−1 it holds that |xi − μi−1 | ≥ |ei | and sgn(xi − μi−1 ) = sgn(ei ), which means sgn(xi − ei − μi−1 ) = sgn(ei ). Thus, for i = 1, . . . , n, each Λi (x¯i , w¯ i∞ ) defined by (6.52) can be equivalently represented by Λi (x¯i , w¯ i∞ ) = {λi (eid ) : |ai | ≤ 1},

(D.74)

where eid = ei + ai wi∞ + sgn(ei )|wi0 | with wi0 = xi − ei − μi−1 and μi−1 ∈ ∞ Λi−1 (x¯i−1 , w¯ i−1 ). With property wi∞ ≤ bi |ei | satisfied, when ei = 0, we have sgn(eid ) = sgn(ei ), |eid | ≥ (1 − bi )|ei |.

(D.75)

Specifically, property (D.75) is proved as follows. We first consider the case of ei > 0. In this case, according to the definition of ai in (D.74), wi∞ in below equation (6.51) and bi in (D.72), we have eid = ei + ai wi∞ + |wi0 | > ei > (1 − bi )ei > 0, and thus sgn(eid ) = sgn(ei ) holds. For the case of ei < 0, property wi∞ ≤ bi |ei |

228

Appendix D: Proofs

implies that eid = ei + ai wi∞ − |wi0 | ≤ (1 − ai bi )ei − |wi0 | < 0, and thus |eid | ≥ (1 − ai bi )|ei | + |wi0 | > (1 − bi )|ei |. Thus, (D.75) holds. From (6.53), (6.55) and (D.71)–(D.75), we have ∇Vi (ei )e˙i = sgn(ei )(h i (x¯i , x¯iθ , e¯i+1 ) + gi (x¯i , x¯iθ )(xi+1 − ei+1 ))

  −1 ≤ ψheii (|ei |) + ψheki ◦ γeeik (|ei |) k=1,...,i−1,i+1

+

i−1 

i   −1  −1 ψhwik ◦ γewi k (|ei |)+ ψˆ heki ◦ γˆ eeik (|ei |)

k=1

k=1

− ci (1 − bi )κi ((1 − bi )|ei |)|ei |



≤ −i |ei | = −i Vi (ei ).

(D.76)

Nonlinear Small-Gain Synthesis e

e

e

Define γ¯ eeik = max{γeeik , γˆ eeik }, γ¯ eeii = γˆ eeii and γ¯ eii+1 = γeii+1 , where γeeik , γeii+1 , γˆ eeik and γˆ eeii with 1 ≤ k ≤ i − 1 are defined in (D.73). By appropriately choosing κi , we can (·) such that design the ISS gains γ¯ (·) i ◦ γ¯ eei1 < Id γ¯ ee12 ◦ γ¯ ee23 ◦ γ¯ ee34 ◦ · · · ◦ γ¯ eei−1 e3 e4 ei γ¯ e2 ◦ γ¯ e3 ◦ · · · ◦ γ¯ ei−1 ◦ γ¯ eei2 < Id .. . e i ◦ γ¯ eii−1 < Id γ¯ eei−1 γ¯ eeii < Id

(D.77)

for i = 1, . . . , n. Denote υi = Vi (eiθ ). The ei -subsystem is considered as a state-delay-free system, that is, υk for k = 1, . . . , i are considered as external inputs. Condition (D.77) basically guarantees the satisfaction of the cyclic-small-gain condition for the interconnection of the ei -subsystems. Based on the Lyapunov formulation of the cyclicsmall-gain theorem, we construct an ISS-Lyapunov function as V (e) = max {σi (Vi (ei ))} i=1,...,n

(D.78)

with σ1 (s) = s σi (s) =

γee12

(D.79) ◦ ··· ◦

for s ∈ R+ which satisfies (6.56) and

i γeei−1 (s),

i = 2, . . . , n

(D.80)

Appendix D: Proofs

229

V (e) ≥ ϑ ⇒ ∇V (e)F ≤ −αe (V (e)), a.e.

(D.81)

where ϑ = max {σi ( max {γewi k (wk∞ )}), σi ( max {γˆ eeik (|υk |)})}, i=1,...,n k=1,...,i k=1,...,i √ α(s) = min σi (s/ n), i=1,...,n

α(s) = max σi (s), i=1,...,n  1 −1 σi (σi (s)) · i σi−1 (s) αe (s) = min i=1,...,n 3 for s ∈ R+ , with σi (s) = dσi (s)/ds. Recall the definition of υk . Condition (D.81) still holds with |υk | replaced by Vk (ek )[t−θ,t] . From (D.78), it can be directly proved that Vk (ek )[t−θ,t] ≤ σk−1 (V (e)[t−θ,t] ). Define  −1 ek (D.82) γe (s) = max σi ( max {γˆ ei ◦ σk (s)}) i=1,...,n k=1,...,i  (D.83) γw (s) = max σi ( max {γewi k (s)}) i=1,...,n

k=1,...,i

for s ∈ R+ . Then, property (D.81) can be reformulated as   V (e) ≥ max γe (V (e)[t−θ,t] ), γw (w ∞ ) ⇒∇V (e)F ≤ −αe (V (e)), a.e.

(D.84)

This proves properties (6.56) and (6.57). Now, we prove property (6.58). Consider the following two cases. • γe = γˆ ee11 . By using condition (D.77), we can prove γe < Id. • γe = σi ◦ γˆ eeik ◦ σk−1 with 2 ≤ i ≤ n and 1 ≤ k ≤ i. Recall the definition of σ(·) in (D.79) and (D.80). We have i k ◦ γˆ eeik ◦ (γee12 ◦ · · · ◦ γeek−1 )−1 σi ◦ γˆ eeik ◦ σk−1 = γee12 ◦ · · · ◦ γeei−1 k i = γee12 ◦ · · · γeek−1 ◦ γeekk+1 ◦ · · · ◦ γeei−1 ◦ γˆ eeik k ◦ (γee12 ◦ · · · ◦ γeek−1 )−1 k k := γee12 ◦ · · · γeek−1 ◦ γk ◦ (γee12 ◦ · · · ◦ γeek−1 )−1 .

(D.85)

With (D.77) satisfied, it can be directly proved that γk < Id, and thus, σi ◦ γˆ eeik ◦ σk−1 < Id. Property (6.58) can be proved through the discussions on the two cases above.

230

Appendix D: Proofs e

Since the ISS gains γeii+1 ∈ K∞ defined in (D.73) for i = 1, . . . , n − 1 are C 1 on (0, ∞) and their inverse functions are Lipschitz on compact sets, α, α, γe , γw ∈ K∞ are Lipschitz on compact sets, and αe is a continuous and positive definite function. This ends the proof of Proposition 6.1.

D.8 Proof of Proposition 6.2 With [240, Lemma 2.14] and [159, Lemma 4.4], property (6.57) implies that there exists a βe ∈ KL satisfying βe (s, 0) = s such that   |V (e(t))| ≤ max βe (V (e(0)), t) , γe (V (e)[−θ,t] ), γw (w ∞ )

(D.86)

With γe < Id satisfied, inspired by the small-gain theorem for time-delay systems [253, Theorem 1], one can find a βˆe ∈ KL such that    |e(t)| ≤ max βˆe ξe [−θ,0] , t , γˆ w (w ∞ )

(D.87)

where e(s) = ξe (s) for s ∈ [−θ, 0], γˆ w (s) = α−1 ◦ γw (s) for s ∈ R+ . The definitions of e1 , . . . , en in (6.54) imply that |e| ≤ αex (|x|) and |x| ≤ max{αex (|e|), αwx (w ∞ )}, where αex , αex and αwx are of class K∞ and Lipschitz on compact sets. Then, by using (D.87), we have |x(t)| ≤ max{β(ξ[−θ,0] , t), γxw (w ∞ )}

(D.88)

where β(s, ·) = αex ◦ βˆe (αex (s), ·) and γxw (s) = max{αex ◦ γˆ w (s), αwx (s)} for s ∈ R+ . From the definitions above, β is of class KL, and γxw is of class K∞ and Lipschitz on compact sets. This ends the proof of Proposition 6.2.

D.9 Proof of Proposition 7.2 In view of (7.2), (7.8), (7.9) and (7.10), for any initial instant t0 ≥ 0 and any initial condition x0 (t0 ) = 0 ∈ R, we have 1 − δ0 ≤ 1 + wx00 ≤ 1 + δ0 , and thus for each 1 ≤ i ≤ n − 2, λ0 ci1 (1 − δ0 ) ≤ | − λ0 di (t)(1 + wx00 )| ≤ λ0 ci2 (1 + δ0 ). Moreover, with Assumption 7.1 and (7.16) satisfied, we have cn−11 ≤ |λn−1 (t, x0 , w0 )| ≤ cn−12 . Then, there exist two positive constants ki1 and ki2 such that (7.17) holds. From (7.12), Assumptions 7.1 and 7.2, we have

Appendix D: Proofs

231

|φ¯ i (t,x0 , w0 , x)| ≤

|[x1 , . . . , xi ]T | |x0n−i−1 |

φid (x0 , . . . , xi , −λ0 (x0 + w0 ))

+ (λ0 c02 (1 + δ0 ) + a0 )(n − i − 1)|z i | ≤ |[x0i−1 z 1 , . . . , z i ]T |φid (x0 , x0n−2 z 1 , . . . , x0n−i−1 z i , −λ0 (x0 + w0 )) + (λ0 c02 (1 + δ0 ) + a0 )(n − i − 1)|z i |. (D.89) An upper bound of |φ¯ i (t, x0 , w0 , x)| in (D.89) implies that there exists a known ¯ φi ∈ K∞ such that (7.18) holds for all x0 , z 1 , . . . , z i . This ends the proof of Proposition 7.2.

D.10 Proof of Proposition 7.3 For simplicity, we use Sk and φ¯ i instead of Sk (¯z k , w¯ kT ) and φ¯ i (t, x0 , w0 , x) for k = 1, . . . , i − 1, respectively. We only consider the case of ei > 0. The proof for the case of ei < 0 is similar. Consider the recursive definition of Sk defined in (7.22). For k = 1, . . . , i − 1, we have max Sk = sgn(λk )κk (z k − bk1 max Sk−1 + bk2 min Sk−1 − bk3 w¯ kT ),

(D.90)

min Sk = sgn(λk )κk (z k − bk1 min Sk−1 + bk2 max Sk−1 + bk3 w¯ kT ),

(D.91)

where bk1 = (sgn(λk ) + 1)/2, bk2 = (sgn(λk ) − 1)/2 and bk3 = sgn(λk ). With conditions (7.2), (7.8) and (7.9) satisfied, it can be directly proved that the functions sgn(λi )’s are constant function. The continuous differentiability of the κk ’s implies the continuous differentiability of max Sk with respect to z k , max Sk−1 and min Sk−1 for k = 1, . . . , i − 1. Using the property of composition of continuously differentiable functions, we can see max Si−1 and min Si−1 are continuously differentiable with respect to z¯ i−1 and thus ∇ max Si−1 and ∇ min Si−1 are continuous with respect to z¯ i−1 . In the case of ei > 0, the dynamics of ei can be rewritten as e˙i = z˙ i − ∇ max Si−1 z˙¯ i−1 = λi z i+1 + φ¯ i − ∇ max Si−1 z˙¯ i−1 := λi z i+1 + Φi (t, x0 , w0 , x, z¯ i ).

(D.92)

Note that z˙¯ i−1 = [λ1 z 2 + φ¯ 1 , . . . , λi−1 z i + φ¯ i−1 ]T . With (7.2), (7.17) and (7.18) satisfied, one can find a ψΦ∗ i ∈ K∞ such that |Φi (t, x0 , w0 , x, z¯ i )| ≤ ψΦ∗ i (|¯z iT , x0 |T ).

(D.93)

232

Appendix D: Proofs

To prove (7.26), for each k = 1, . . . , i − 1, we look for a ψzk+1 ∈ K∞ such that T T T , w¯ kT ] |). For k = 1, . . . , i − 1, from the definitions of ek+1 in |z k+1 | ≤ ψzk+1 (|[e¯k+1 (7.24), we can observe min Sk ≤ z k+1 − ek+1 ≤ max Sk and thus |z k+1 | ≤ max{| max Sk |, | min Sk |} + |ek+1 |.

(D.94)

For each z k+1 (k = 1, . . . , i − 1), using repeatedly (D.94) and (D.90)–(D.91), one T T T , w¯ kT ] |). This, together with can find a ψzk+1 ∈ K∞ such that |z k+1 | ≤ ψzk+1 (|[e¯k+1 (D.93), leads to the satisfaction of (7.26). If the φ¯ k ’s defined in (7.18) for k = 1, . . . , i are Lipschitz on compact sets, then all the class K∞ functions determining ψΦi are Lipschitz on compact sets, and one can find a ψΦi ∈ K∞ which is Lipschitz on compact sets. This ends the proof of Proposition 7.3.

D.11 Proof of Proposition 7.4 With the definition of ei , when ei = 0, for any pi−1 ∈ Si−1 (¯z i−1 , w¯ (i−1)T ), it holds that |z i − pi−1 | ≥ |ei | and sgn(z i − pi−1 ) = sgn(ei ), which implies sgn(z i − ei − pi−1 ) = sgn(ei ). Thus, for i = 1, . . . , n − 1, each Si (¯z i , w¯ i T ) in the form of (7.22) can be rewritten as Si (¯z i , w¯ i T ) = {sgn(λi )κi (eim ) : |wi | ≤ wi T },

(D.95)

where eim = ei + wi + sgn(ei )|wi0 | with wi0 = z i − ei − pi−1 and pi−1 ∈ Si−1 (¯z i−1 , w¯ (i−1)T ). Moreover, with (7.17) and (7.26) satisfied, one can find ψΦe1i , . . . , e w ψΦi+1 , ψΦwi1 , . . . , ψΦii−1 ,ψΦx0i ∈ K∞ being Lipschitz on compact sets such that i |Φi + λi ei+1 | ≤

i+1 

ψΦeki (|ek |) +

k=1

i−1 

ψΦwik (wkT ) + ψΦx0i (|x0 |)

k=1

We choose νi defined in (7.23) satisfying ki1 (1 − bi )νi ((1 − bi )s)s ≥ ψΦei i (s) +



 −1 ψΦeki ◦ γeeik (s)

k=1,...,i−1,i+1

+

i−1 

 −1  −1 ψΦwik ◦ γewi k (s) + ψΦx0i ◦ γexi0 (s) + i s

k=1

for s ∈ R+ , with constant i > 0, where ki1 is defined in Proposition 7.2 and each (·) (·) −1 ∈ K∞ and the corresponding (γ(·) ) are Lipschitz on compact sets. γ(·) With Vi (ei ) = |ei |, we consider the case of

Appendix D: Proofs

233

Vi (ei ) ≥

max

k=1,...,i−1

 e γeeik (Vk (ek )), γeii+1 (Vi+1 (ei+1 )), . γewi k (wkT ), b1i wi T , γexi0 (|x0 |)

In this case, we have  −1 (|ei |), k = 1, . . . , i, |wkT | ≤ γewi k  ek −1 |ek | ≤ γei (|ei |), k = 1, . . . , i − 1, i + 1,  x0 −1 |x0 | ≤ γei (|ei |),

(D.98)

|wi T | ≤ bi |ei |.

(D.99)

(D.96) (D.97)

Recall the definition of eim in (D.95). With 0 < bi < 1 and property (D.99), when ei = 0, we have sgn(eim ) = sgn(ei ), |eim | ≥ (1 − bi )|ei |.

(D.100)

Using (D.96)–(D.100), we have ∇Vi (ei )e˙i ≤

ψΦei i (|ei |) +



 −1 ψΦeki ◦ γeeik (|ei |)

k=1,...,i−1,i+1

+

i−1 

 −1  −1 ψΦwik ◦ γewi k (|ei |) + ψΦx0i ◦ γexi0 (|ei |)

k=1

− ki1 (1 − bi )νi ((1 − bi )|ei |)|ei |



≤ −i |ei | = −i Vi (ei ). This ends the proof of Proposition 7.4.

D.12 Proof of Proposition 7.5 With the cyclic-small-gain condition (7.33) satisfied, by using the technique in [162], we construct an ISS-Lyapunov function for the closed-loop system as V (e) = max {σi (Vi (ei ))}

(D.101)

i=1,...,n

e

i for i = 2, . . . , n − 1 where γekk+1 ∈ K∞ with σ1 = Id and σi = γee12 ◦ γee23 ◦ · · · ◦ γeei−1 for k = 1, . . . , n − 2 are Lipschitz on compact sets. Then, σi and σi−1 for i = 1, . . . , n − 1 are Lipschitz on compact sets. From the definition of V in (D.101), one has

234

Appendix D: Proofs

α(|e|) ≤ V (e) ≤ α(|e|)

(D.102)

√ where α(s) = mini=1,...,n σi (s/ n − 1) and α(s) = maxi=1,...,n σi (s) for s ∈ R+ . ek+1 With the γek chosen above, both α and α are of class K∞ and Lipschitz on compact sets. The influence of the sampling error wi for i = 1, . . . , n can be described by     wk  x0 ϑ = max σi max γei (wkT ) , σi ◦γei (x0 [t0 ,T ) ) . i=1,...,n

k=1,...,i

According to the Lyapunov-based cyclic-small-gain theorem [162], it holds that V (e) ≥ ϑ ⇒ max ∇V (e) f ≤ −α(V (e)), a.e. f ∈F

(D.103)

where F := [F1 (t, x0 , w0 , x, z¯ 1 , w¯ 1T , e1 ), . . . , Fn−1 (t, x0 , w0 , x, z¯ n−1 , wT , en )]T . Note that en = 0.     and γex0 (s) = maxi=1,...,n Define γew (s) = maxi=1,...,n σi maxk=1,...,i γewi k (s)   x0 w x0 w σi ◦ γei (s) for s ∈ R+ . Then, γe , γe ∈ K∞ , and γe is locally Lipschitz. With property (D.103), there exists a βe ∈ KL such that V (e(t)) ≤ max

βe (V (e(t0 )), t), γew (|wT |), γex0 (x0 [t0 ,T ) )



for all t0 ≤ t < Tmax , which together with (D.102) implies |e(t)| ≤max

α−1 ◦βe (α(|e(t0 )|), t) , α−1 ◦γew (|wT |), α−1 ◦γex0 (x0 [t0 ,T ) )

 (D.104)

for all t0 ≤ t < Tmax . According to the definitions of e1 , . . . , en−1 , it can be observed that if wi T ≥ 0 for i = 1, . . . , n − 1, then |ei | ≤ |z i − κi−1 (ei−1 )| ≤ |z i | + |κi−1 (ei−1 )|. One can find an αez ∈ K∞ such that |e| ≤ αez (|z|).

(D.105)

Also, one can find αez , αw z ∈ K∞ such that |z| ≤ max{αez (|e|), αw z (wT )}. By substituting (D.105) and (D.106) into (D.104), one achieves |z(t)| ≤ max{βz (|z(t0 )|, t), γzw (wT ), γzx0 (x0 [t0 ,Tmax ) )}, where βz ∈ KL and γzw , γzx0 ∈ K. This ends the proof of Proposition 7.5.

(D.106)

Appendix D: Proofs

235

D.13 Proof of Proposition 7.9 By choosing the χ(·) (·) ’s in (7.90) and (7.95)–(7.98) satisfying the cyclic-small-gain condition [162], property (1) in Proposition 7.9 can be proved by directly using [170]. The event trigger defined in (7.100) implies that |w1 | ≤ μ1 . With μ1 = ϕ1 (η1 ) defined by (7.102), we have |w1 | ≤ ϕ1 (η1 ). Define ϑ(t) = V (ς(t)), ϑ0 (t) = V˜x0 (x0 (t)), ϑ1 (t) = V˜ζ1 (ζ1 (t)), ϑ2 (t) = V˜ζ¯ 2

ζ¯ 1 (ζ¯2 (t)), ϑi+2 (t) = V˜ei (ei ) and ϑn+2 (t) = η1 (t). Since χ˜ ζ21 , χ˜ eζ11 , χ˜ ζx10 < Id and χ˜ w ζ1 ◦ T ¯ ϕ1 < Id, if ϑ1 (tT ) = ϑ(tT ), then there exists a neighborhood Θ of [ζ1 , ζ2 , e1 , x0 , η1 ]T such that  ¯  ζ2 ˜ e1 ˜ χ ˜ ( V ( p )), χ ˜ ( V ( p )), ¯ 2 e1 3 ζ2 ζ1 ζ1 V˜ζ1 ( p1 ) ≥ max . 1 χ˜ w ˜ ζx10 (V˜x0 ( p4 )) ζ1 ◦ ϕ1 (q1 ), χ

and ∇ V˜ζ1 ( p1 ) exists for all [ p1 , p2T , p3 , p4 , q1 ]T ∈ Θ. Due to the continuity of [ζ1 , ζ¯2T , e1 , x0 , η1 ]T , there exists a tT > tT such that  V˜ζ1 (ζ1 (t)) ≥ max

ζ¯ χ˜ ζ21 (V˜ζ¯2 (ζ¯2 (t))), χ˜ eζ11 (V˜e1 (e1 (t))), 1 χ˜ w ˜ ζx10 (V˜x0 (x0 (t))) ζ1 ◦ ϕ1 (η1 (t)), χ

 .

for all t ∈ (tT , tT ), which implies ∇ V˜ζ1 (ζ1 (t))ζ˙1 (t) ≤ −˜ζ1 (V˜ζ1 (ζ1 (t))) for t ∈ (tT , tT ). Thus, if ϑ1 (tT ) = ϑ(tT ), then D + ϑ1 (tT ) ≤ −α(ϑ1 (tT )). From the discussions above, if ϑi (tT ) = ϑ(tT ) for i ∈ {0, 2, . . . , n + 1}, then D + ϑi (tT ) ≤ −α(ϑi (tT )). By using the definition of α in (7.107), we also have D + ϑn+2 (t) ≤ −α(ϑn+2 (t)) for all t ≥ 0. Define I (t) = {i = 0, 1, . . . , n + 2 : ϑi (t) = ϑ(t)}. By using [72, Lemma 2.9], we have D + ϑ(t) = max{D + ϑi (t) : i ∈ I (t)} ≤ max{−α(ϑi (t)) : i ∈ I (t)} = −α(ϑ(t)). This ends the proof of Proposition 7.9.

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© Huazhong University of Science and Technology Press, Wuhan and Springer Nature Singapore Pte Ltd. 2020 T. Liu et al., Robust Event-Triggered Control of Nonlinear Systems, Research on Intelligent Manufacturing, https://doi.org/10.1007/978-981-15-5013-3

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Index

A Asymptotic gain, 22, 24

B Boundedness, 77, 96

C Cascade connected reactors, 159 Class KL function, 24, 208 Class K function, 208 Class K∞ function, 208 Closed-loop event-triggered system, 44 Comparison principle, 82, 84, 94, 96, 112, 222, 224 Continuation of solution, 85, 113, 159 Controller discontinuous, 193, 206 switching, 206 time-varying, 206 Converse Lyapunov theorem, 21 Coordinate transformation, 19 Coupled pendulum, 100

D Decentralized control, 89 Digraph, 208 path, 208 reachability, 208 reaching set, 208 simple cycle, 208 spanning tree, 208

tree, 208 Dini derivative, 91 Dynamic uncertainty, 74, 105, 124

E Equilibrium, 19 Euler approximation, 65 Event trigger decentralized, 91 dynamic, 8, 73, 81, 91, 106, 117, 127, 137 positive offset, 10, 54 Robust, 43 static, 6, 46, 59 synchronized, 160 time-delay, 156 time-regularized, 11 Event-triggered control, 3, 43 constructive design, 154 decentralized, 89 of discrete-time nonlinear systems, 58, 64, 68 of linear systems, 87, 100, 107 of nonholonomic systems, 187, 203 of nonlinear systems, 115, 122, 134, 149 output feedback, 162, 193 Robust, 105 state feedback, 143, 175 subject to dynamic uncertainties, 138 subject to external disturbances, 53, 138 with output feedback, 169 with partial state feedback, 73, 75, 85 Event-triggered observer, 162, 196

© Huazhong University of Science and Technology Press, Wuhan and Springer Nature Singapore Pte Ltd. 2020 T. Liu et al., Robust Event-Triggered Control of Nonlinear Systems, Research on Intelligent Manufacturing, https://doi.org/10.1007/978-981-15-5013-3

251

252 Event-triggered stabilization decentralized, 94 External disturbances, 55, 105

F Finite escape time, 43, 158, 190, 202 Finite-time accumulation, 43 Forward completeness, 24, 29, 43, 78

G Gain assignment, 98, 173, 213 Gain digraph, 32 Gain margin, 67 Global convergence, 20 Graph, 208 directed, see Digraph Gronwall–Bellman inequality, 178

H Hurwitz, 22, 23, 49, 165

I Infinitely fast sampling, 45, 46, 77, 79, 80 Input-to-output practical stability, 29 Input-to-State Stability (ISS) Lyapunov formulation, 26 max-type formulation, 22 plus-type formulation, 22 Interconnected system, 27 ISS gain, 22 Lyapunov-based, 25 ISS-Lyapunov function, 5, 7, 25, 29, 32, 50, 53, 74, 80, 91, 106, 116, 126, 135, 138 construction, 33, 34, 36, 79, 92, 130, 147, 168, 217 dissipation form, 25 gain margin form, 25

L L’Hôpital’s Rule, 110 Linear systems, 7, 9, 22, 49, 53 Lipschitz locally, 19, 43, 207 on compact sets, 46, 47, 52, 54, 179, 182, 184, 196, 200, 202, 207 Lipschitz continuity, 207 Local essential boundedness, 22 Lorenz system, 134

Index Lyapunov direct method, 20 Lyapunov function, 20, 21 strict, 20 Lyapunov’s second theorem, 20

M Measurement feedback control, 15 Mechanical systems, 206 Motion capture system, 189

N Negative-feedback control law, 177 Nonholonomic mobile robot, 206 Nonholonomic systems cascade, 177 in the chained form, 175, 203 in the output-feedback form, 193 subject to disturbances, 175 subject to drift uncertain nonlinearities, 175 Nonlinear small-gain condition, 28, 45, 52, 55, 57, 58, 200 Lyapunov-based, 54 Nonlinear small-gain design, 43, 44, 89, 183 Lyapunov-based, 45, 59, 61, 76, 94, 117, 127, 129, 137, 139, 168 of event-triggered controllers, 144 trajectory-based, 51, 57 Nonlinear small-gain theorem, 19, 27–29 cyclic, 31, 33, 34 Lyapunov-based, 29, 34 trajectory-based, 27 Nonlinear systems, 3, 19, 21, 50 cascade, 28 continuous-time, 31 discrete-time, 31, 58 hybrid, 31 in the lower-triangular form, 143 in the output-feedback form, 162 interconnected, 27, 28, 43 large-scale, 29, 31 with distributed time-delays, 159 with time delays, 152

O Observer-based controller, 163, 194

P Positive definite function, 207

Index R Rademacher’s theorem, 35 Robust control, 44 Robustness, 45 Ruzumikhin–Lyapunov function, 155 S Sampled-data control law, 4, 7, 9 Sampling error, 5, 44, 177, 179 Sector-bound condition, 105, 134, 177, 194 Self trigger, 49, 55 Self-triggered control, 49, 55, 149 Set-valued map based design, 144, 179 σ-scaling technique, 178 Simple path, 34 Small-gain theorem classical, 39 nonlinear, 19, 27–29, 31, 33, 34 Solution of dynamic systems, 19 Stability, 19, 43 asymptotic, 20 global, 20 global asymptotic, 20 input-to-output, 28, see Input-to-output stability (IOS) input-to-state, 19, 21, 27, 45, 51, 55, 58, 147, see Input-to-state stability (ISS) Lyapunov, 19

253 Robust, 26, see Robust stability, 45, 54, 57 sufficient conditions, 20 weakly robust, 26, see Weakly robust stability Stabilization, 19 asymptotic, 5, 7, 10, 46, 52 event-triggered, 43, 45, 50, 60, 62, 105, 115, 129, 167, 183, 202 global, 3, 5 input-to-state, 7, 105, 147, 156 output feedback, 167 robust, 60, 129 semi-global, 3, 5

U Unboundedness Observability (UO), 28 Uniformly Bounded-Input Bounded-State property (UBIBS), 24

W Wheeled mobile robot, 187

Z Zeno behavior, 8, 44