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Limits of Stability and Stabilization of Time-Delay Systems

A Small-Gain Approach

Authors: Dr. Jing Zhu, Dr. Tian Qi, Dr. Dan Ma, Jie Chen

Publisher: Springer International Publishing

Book Series : Advances in Delays and Dynamics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This authored monograph presents a study on fundamental limits and robustness of stability and stabilization of time-delay systems, with an emphasis on time-varying delay, robust stabilization, and newly emerged areas such as networked control and multi-agent systems. The authors systematically develop an operator-theoretic approach that departs from both the traditional algebraic approach and the currently pervasive LMI solution methods. This approach is built on the classical small-gain theorem, which enables the author to draw upon powerful tools and techniques from robust control theory. The book contains motivating examples and presents mathematical key facts that are required in the subsequent sections. The target audience primarily comprises researchers and professionals in the field of control theory, but the book may also be beneficial for graduate students alike.

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Table of Contents

Frontmatter

Chapter 1. Introduction

Abstract

Physical responses and effects rarely take place instantly upon exerting external signals. It more or less takes time to transport external inputs, such as material, information, or energy from one place to another. This leads to time delay. Time delay prevails in interconnected systems and networks, which may arise from various sources, including physical transport delay, computation delay, and signal transmission delay. In this chapter, we give a number of motivating examples ranging from classical mechanical systems to contemporary biological studies. Along with this introduction, we also provide a concise survey into the literature most relevant to the topics of this book.

Jing Zhu, Tian Qi, Dan Ma, Jie Chen

Chapter 2. Mathematical Background

Abstract

This chapter provides the necessary mathematical background required in our subsequent developments. We review the small-gain theorem and its variants, together with the basics of robust control theory.

Jing Zhu, Tian Qi, Dan Ma, Jie Chen

Chapter 3. Small-Gain Stability Conditions

Abstract

In this chapter, we show that a variety of stability conditions, both existing and new, can be derived for linear systems subject to time-varying delays in a unified manner in the form of scaled small-gain conditions. From a robust control perspective, the development seeks to cast the stability problem as one of robust stability analysis, and the resulting stability conditions are also reminiscent of robust stability bounds typically found in robust control theory. The development is built on the well-known conventional robust stability analysis, requiring essentially no more than a straightforward application of the small-gain theorem. The conditions are conceptually appealing and can be checked using standard robust control toolboxes. Both the \(\mathcal {L}_2\)- and \(\mathcal {L}_\infty \)-stability criteria are presented.

Jing Zhu, Tian Qi, Dan Ma, Jie Chen

Chapter 4. Delay Margin

Abstract

This chapter concerns the robust stabilization of SISO LTI systems subject to unknown delays. The fundamental issue under investigation, referred to as the delay margin problem, addresses the question: What is the largest range of delay such that there exists a feedback controller capable of stabilizing all the plants for delays within that range? Drawing upon analytic interpolation and rational approximation techniques, we develop fundamental bounds on the delay margin. Computational formulas are developed to estimate efficiently the delay margin, within which the delay plant is guaranteed to be stabilizable by a finite-dimensional LTI controller. Analytical bounds are also sought after to show explicitly how plant unstable poles and nonminimum phase zeros may fundamentally limit the range of delay over which a plant can be stabilized.

Jing Zhu, Tian Qi, Dan Ma, Jie Chen

Chapter 5. Stabilization of Linear Systems with Time-Varying Delays

Abstract

In this chapter, we extend the preceding stabilization results to linear systems with time-varying delays. The extension is rendered possible by the operator-theoretic approaches developed in Chaps. 3 and 4: While the small-gain stability conditions are developed in Chap. 3 for systems with time-varying delays, the interpolation approach used in Chap. 4 enables us to address stabilization problems directly based on the small-gain conditions. This insures that the results on the delay margin can be cohesively extended, and in a unified manner. Indeed, it will be seen that bulk of the results in Chap. 4 can be used to address the stabilization of systems with time-varying delays, modulo to some minor modifications. Efficient computational formulas and analytical expressions are also obtained, which incorporate the time-varying delay characteristics of delay range and delay variation rate.

Jing Zhu, Tian Qi, Dan Ma, Jie Chen

Chapter 6. Delay Margin Achievable by PID Controllers

Abstract

A time-honored method seemingly of infinite staying power, PID control is favored for its ease of implementation and undoubtedly, has been the most popular means of controlling industrial processes with its unparallel simplicity and unsurpassed effectiveness. This chapter studies the delay margin achievable by LTI controllers of further restricted structure and complexity, i.e., those of PID type. We develop explicit expressions of the exact delay margin and its upper bounds achievable by a PID controller for low-order delay systems, notably the first- and second-order unstable systems. While furnishing the fundamental limits of delay within which a PID controller may robustly stabilize a delay process, our results should also provide useful guidelines in tuning PID parameters and in the analytical design of PID controllers.

Jing Zhu, Tian Qi, Dan Ma, Jie Chen

Chapter 7. Delay Radius of MIMO Delay Systems

Abstract

Built on the development of Chap. 4, in this chapter we consider the stabilization of linear MIMO systems subject to uncertain delays. Based on tangential Nevanlinna–Pick interpolation, we develop in a similar spirit bounding sets in the delay parameter space, in which a MIMO delay plant can be stabilized by a single LTI controller. Estimates on the variation ranges of multiple delays are obtained by solving LMI problems, and further, by computing the radius of delay variations. Both the lower and upper bounds are derived. Analytical bounds show that the directions of the plant unstable poles and nonminimum phase zeros play important roles in restricting the range for stabilizability.

Jing Zhu, Tian Qi, Dan Ma, Jie Chen

Chapter 8. Stabilization of Networked Delay Systems

Abstract

In this chapter, we study the stabilization of networked feedback systems in the presence of stochastic uncertainties and time delays. We model the stochastic uncertainty as a random process in a multiplicative form, and we assess the stability of system based on mean-square criteria. Based on the mean-square small-gain theorem, Theorem 2.5, we develop fundamental conditions of mean-square stabilizability, which ensure that an open-loop unstable system can be stabilized by output feedback. For SISO systems, a general, explicit stabilizability condition is obtained. This condition, both necessary and sufficient, provides a fundamental limit imposed by the system’s unstable poles, nonminimum phase zeros, and time delay. This condition answers to the question: What is the exact largest range of delay such that there exists an output feedback controller mean-square stabilizing all plants under a stochastic multiplicative uncertainty for delays within that range? For MIMO systems, we provide a solution for minimum phase systems possibly containing time delays, in the form of a generalized eigenvalue problem. Limiting cases are also showing how the directions of unstable poles may affect mean-square stabilizability of MIMO minimum phase systems.

Jing Zhu, Tian Qi, Dan Ma, Jie Chen

Chapter 9. Consensus of Multi-agent Systems Under Delay

Abstract

This chapter concerns the consensus problem for continuous-time multi-agent systems (MAS). The network topology is assumed to be fixed, which can be undirected and directed. We assume that the agents’ input is subject to a constant, albeit possibly unknown time delay, and is generated by a distributed dynamic output feedback control protocol. Drawing upon concepts and techniques from robust control theory, notably those concerning gain margin and gain-phase margin optimizations and analytic interpolation, we derive explicit, closed-form conditions for general linear agents to achieve consensus. The results display an explicit dependence of the consensus conditions on the delay value as well as on the agent’s unstable poles and nonminimum phase zeros, showing that delayed communication between agents will generally hinder consensus and impose restrictions on the network topology. We also show that a lower bound on the maximal delay allowable for consensus can be computed by a simple line search method.

Jing Zhu, Tian Qi, Dan Ma, Jie Chen

Backmatter

Title: Limits of Stability and Stabilization of Time-Delay Systems
Authors: Dr. Jing Zhu
Dr. Tian Qi
Dr. Dan Ma
Jie Chen
Copyright Year: 2018
Publisher: Springer International Publishing
Electronic ISBN: 978-3-319-73651-8
Print ISBN: 978-3-319-73650-1
DOI: https://doi.org/10.1007/978-3-319-73651-8