Control Strategy for Time-Delay Systems: Part I: Concepts and Theories

Control Strategy for Time-Delay Systems Part I: Concepts and Theories covers all the important features of real-world practical applications which will be valuable to practicing engineers and specialists, especially given that delays are present in 99% of industrial processes. The book presents the views of the editors on promising research directions and future industrial applications in this area. Although the fundamentals of time-delay systems are discussed, the book focuses on the advanced modeling and control of such systems and will provide the analysis and test (or simulation) results of nearly every technique described.

For this purpose, highly complex models are introduced to ?describe the mentioned new applications, which are characterized by ?time-varying delays with intermittent and stochastic nature, several types of nonlinearities, and the presence ?of different time-scales. Researchers, practitioners, and PhD students will gain insights into the prevailing trends in design and operation of real-time control systems, reviewing the shortcomings and future developments concerning practical system issues, such as standardization, protection, and design.

• Presents an overview of the most recent trends for time-delay systems
• Covers the important features of the real-world practical applications that can be valuable to practicing engineers and specialists
• Provides analysis and simulations results of the techniques described in the book

• Language: English
• Publisher: Academic Press
• Release date: Nov 21, 2020
• ISBN: 9780128206140

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Preface

In a general sense, a system is considered as a time-delay system when the rate of change in system states is related to its previous states. Delays repeatedly appear in different real-world engineering problems. It has been proved that the presence of delays in controlled plants (e.g., communication delays in networked control systems) may have a "vague" feature: they highly threaten the stability of the system and deteriorate the system performance, which imposes great difficulty in the control system design and stability analysis. This necessitates the implementation of particular methodologies and design methods in the control theory to effectively stabilize the time-delayed systems.

The research on control design of time-delay system dates back to the 1960s and has received considerable attention over the two past decades. Over these years, the study of such systems has been reported at many international congresses and organized conferences and published in high-qualified journals of control engineering, control theory mathematics, and numerical problems. It is obvious that any developments in the control theory and numerical schemes, particularly the laws of robust stability and performance evaluation, will greatly help a lot in making progress.

This book is dedicated to present an up-to-date framework in the field of the advanced control strategy for time-delay systems along with an overview of the most recent studies in this context. An attempt is carried out to provide a specific perspective, which is concentrated on the robust design, stability analysis, adaptive control, and synchronization of time-delay systems for graduate students studying robust control schemes and/or their application to systems with delays. The usefulness of the suggested schemes and their effectiveness over other prevalent ones are verified theoretically and described by means of various case-studies. The different chapters of this book introduce the latest advances in the active research field to the readers and acquaint them with a state-of-the-art methodology for investigating time-delay systems.

In Chapter 1 the stability and stabilization of time-delay systems are reviewed with emphasis on complex models of partial differential equations, the Lyapunov-based tools, and systems with input or output delays and state delays.

In Chapter 2 a class of sequential subpredictors (SSP) is provided to solve a stability problem for the systems with a large input delay, where the system states are predicted over a large range of delay times.

Chapter 3 proposes a discrete-time version of adaptive posicast controller (APC) for dealing with plant uncertainties that are not well characterized. To avoid the numerical complexity related to the implementation of time-delay compensation on sampled-data plants, the suggested controller scheme is developed and analyzed in discrete time.

Chapter ) ring and elegant algebraic parlance to stabilize the controller structures with the capability of their further parameterization for time-delayed systems.

Chapter 5 proposes a model predictive control of discrete-time nonlinear systems with unknown time-varying delay via the T-S fuzzy modeling and control approach. In this application, the Lyapunov–Razumikhin function (LRF) is employed such that new stabilization conditions are provided.

In Chapter 6 the observability and observer design for a general class of commensurate time-delayed systems affected by unknown inputs is presented. In this endeavor, sufficient conditions ensuring the observability of the system (a precise definition is given further) is provided, which also allows for recovering the state variables of the system by two different ways.

Chapter 7 addresses a new adaptive-robust time delay control (ARTDC) for tracking control of a class of systems under arbitrarily varying input delay and state-dependent uncertainty.

Chapter 8 discusses the problem of adaptive control of the decentralized model reference for a class of uncertain large-scale systems with constant state and input delays and varying delays in the interconnected terms.

Chapter 9 proposes the Lambert W function (LWF) based approach for design and analysis of linear time-invariant (LTI) multidimensional TDSs with a single time delay.

Chapter 10 focuses on the stabilization problem for the class of discrete-time state-delayed systems subjected to saturating actuators and state constraints.

Chapter 11 is concerned with the robust output-feedback delay control synthesis problem for a class of linear systems with time-varying state delay under the integral quadratic constraints (IQCs) framework.

Chapter One: An overview of time-delay control systems

Magdi S. Mahmoud    Distributed Control Research Lab, Systems Engineering Department, KFUPM, Dhahran, Saudi Arabia

Abstract

Delay phenomena are commonly encountered in modeling population dynamics, transportation, propagation, and interactions between coupled dynamics through material, energy, and communication flows. Time-delay systems (TDSs) have been widely investigated in the past decade following the explosive growth of communications and network exchanges. It turns out that time-delays are ubiquitous in technologically-based control systems and possibly degrade the closed-loop system performance. To this end, this chapter addresses the role of delays and surveys major issues pertaining to stability and stabilization of time-delay control systems. In particular, the covered materials include the following:

•  Fundamental analytical results are provided throughout as key lemmas to shed light on the contemporary development in stability of TD systems.

•  The linear matrix inequality (LMI) framework is deployed in deriving delay-dependent stability and stabilization results.

•  Particular emphases are placed on the conservatism and the computational complexity of the results.

•  Several theorems are developed and supported by remarks to emphasize point-by-point technical results.

•  Detailed technical design results are presented and concluded by an outlook of suggestions for future research directions.

Keywords

Time-delay systems; Stability; Stabilization; Delay-dependent stability; Linear matrix inequality

Chapter Outline

Acknowledgement

1.1  Introduction

1.2  Sources of time-delays

1.3  Models and solutions

1.3.1  Retarded systems

1.3.2  Models for linear time-invariant systems

1.4  Stability notion

1.5  Stability theorems

1.6  Stability results for linear delay systems

1.6.1  Constant delay

1.6.2  Time-varying delay

1.6.3  Augmented Lyapunov functional

1.6.4  Triple integral Lyapunov functional

1.6.5  Newton–Leibniz formula

1.6.6  Bounding techniques

1.6.7  Discrete-time systems

1.7  Model transformations

1.8  Delay-dependent stabilization

1.8.1  A class of nonlinear systems

1.9  Resilient delay-dependent control

1.9.1  Continuous-time systems

1.9.2  Descriptor transformation

1.9.3  design

1.9.4  design

1.9.5  design

1.9.6  design

1.9.7  design

1.9.8  design

1.9.9  Numerical simulation 1

1.9.10  Numerical simulation 2

1.9.11  Discrete-time systems

1.9.12  Descriptor model transformation

1.9.13  Robust stability

1.9.14  Nominal feedback stabilization

1.9.15  Resilient feedback stabilization

1.9.16  Numerical simulation 3

1.9.17  Numerical simulation 4

1.9.18  Numerical simulation 5

1.10  The cart-pole system

1.11  Kalman filtering

1.11.1  A class of continuous time-lag systems

1.11.2  Robust Kalman filtering

1.11.3  Numerical simulation 6

1.11.4  Numerical simulation 7

1.12  Networked control systems

1.12.1  State-feedback stabilization

1.12.2  Observer-based feedback stabilization

1.12.3  Lyapunov-based sampled-data stabilization

1.13  Interconnected systems

1.14  Conclusions and future work

References

Acknowledgement

This research work is supported by the deanship of scientific research (DSR) at KFUPM through research project no. IN 161065.

1.1 Introduction

The last few decades have witnessed a rapid development of networked control systems due to their significant advantages, and they have been applied to variant industrial areas, such as unmanned surface vehicles, unmanned space vehicles, smart grids, waste-water treatment processes, Internet-based teleoperation, intelligent transportation systems, and so on. However, time-delays are ubiquitous in networked control systems due to limited network bandwidth and possibly degrade the closed-loop system performance.

The occurrence of time-delay phenomenon appears to prevail many real-world systems and engineering applications. This takes place in either the state, the control input side, or the measurements side. It turns out that delays are strongly involved in challenging areas of communication and information technologies including stabilization of networked controlled systems and high-speed communication networks. In terms of mathematical representation, time-delay systems are often described by functional differential equations (FDEs), which are of infinite dimension [1], [2], [3]. It turns out that a wide variety of dynamical systems can be modeled as time-delay systems [4]. Stability analysis and robust control of time-delay systems (TDSs) are of theoretical and practical importance [5–9]. A great deal of the basic results is reported in [10–18]. Broadly speaking, stability conditions for time-delay systems can be broadly classified into two categories, delay-independent and delay-dependent stability conditions. Much attention was paid to the study of delay-dependent stability conditions as they yield less conservative design results. Recent works presented in [19–35] cover alternative issues pertaining to stability and stabilization of dynamical systems with time-delays.

The primary objective of this chapter is to wider familiarize the readers with TDSs and provide a systematic treatment of modern ideas and techniques for researchers. It presents a guided tour from basic classical results to recent developments on Lyapunov-based analysis and design with applications to the attractive topics of network-based control and interconnected time-delay control systems.

The primary objective of this chapter is to

•  wider familiarize the readers with TDSs,

•  provide a systematic treatment of modern ideas and techniques for researchers,

•  present a guided tour from basic classical results to recent developments on Lyapunov-based analysis and design, and

•  present applications to the attractive topics of resilient delay-dependent stability and stabilization, network-based control, and interconnected TDSs.

Essentially, it provides an overview on the mathematical modeling and technical progress of stability and stabilization of TDSs. Particular emphases will be placed on issues concerned with the conservatism and the computational complexity of the results. For simplicity in exposition, the discussions are limited to linear or linearizable systems. Some methods and techniques used to derive stability conditions for time-delay systems are reviewed. Several future research directions on this topic are also discussed.

The remaining part of the chapter is organized as follows: In Section 1.2, many applications are presented to illuminate the sources of delay phenomena. Section 1.3 is devoted to the models and solutions of time-delay systems. After recalling the notion of stability of TDS in Section 1.4, a comprehensive theoretical study of the methods of Lyapunov are provided in Section 1.5 to establish pertinent theorems leading to stability conditions. Specializing to linear class of TDS, Section 1.6 includes stability analysis under different delay patterns where the ensuing results/conditions are formulated as linear matrix inequalities for both continuous-time and discrete-time systems. This provides a convenient computational tool for control research and investigations. In Section 1.7, we present delay-dependent stability results using model transformations. In Section 1.8, we move to study the stabilization problem of TDSs and establish appropriate design techniques. A typical application of TDSs is the cart-pole systems presented in Section 1.9. Complete results of resilient delay-dependent stability and stabilization are developed in Section 1.10. On a different research line, we provide in Section 1.11 a concise treatment of Kalman filtering for continuous and discrete TDSs. Finally, Section 1.12 is devoted to brief account of networked control systems followed by a class of interconnected TDSs in Section 1.13. Pertinent concluding remarks are outlined in Section 1.14. Several numerical simulations are provided throughout the chapter.

Notations: denote the nto denote, respectively, the transpose, the inverse, the minimum eigenvalue, the maximum eigenvalue of any square matrix Wstands for a symmetrical and positive-definite (negative-definite) matrix W; I ; δH denotes the first difference of H. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations. In symmetric block matrices, we use the symbol • to represent the terms induced by symmetry. Sometimes the arguments of a function will be omitted when no confusion can arise.

The following facts are provided in [6].

Fact 1.1

, we have the following matrix inequality:

Fact 1.2

, it follows that

Lemma 1.1

Finsler's lemma, [18]

Let , , and of rank . The following statements are equivalent:

, ;

;

;

.

Lemma 1.2

For given two vectors and and matrix defined over a prescribed interval Ω, it follows for any matrices , , and , we have the following inequality:

subject to

1.2 Sources of time-delays

There are many applications where time-delay phenomena appear quite naturally. This includes, but not limited to:

A.  Automotive: Combustion model (ignition delay), electromechanical brakes (actuator delay).

B.  Electrical networks.

C.  Epidemics: Understanding the dynamics of biological processes and epidemics is a challenge for health workers engaged in managing treatment strategies. The underlying mechanisms can be revealed by considering epidemics and diseases as dynamical processes, for which the hematology dynamics can be modeled by

(1.1)

which formulates the circulating cell populations in one compartment, where y represents the circulating cell population, λ is the cell-loss rate, and the monotone function F (describing a feedback mechanism) denotes the flux of cells from the previous compartment. The delay τ represents the average length of time required to go through the compartment.

D.  Glucose-insulin model: represent the levels of plasma glycemia and insulinemia. Then

(1.2)

is the apparent delay with which the pancreas varies secondary insulin release in response to varying plasma glucose concentrations, and f is the nonlinear function that models the Insulin Delivery Rate.

E.  Heat exchanger: distributed delay due to conduction in a tube.

F.  Hydraulic networks: the transport phenomenon of water is modeled as a varying time-delay

G.  Intelligent building: time-delay due to wireless transmission of sensor data

H.  Manufacturing process: The metal cutting process on a lathe can be described as

(1.3)

The study of this model is critical in understanding the regenerative chattering phenomenon.

I.  Marine robotics: transport delay due to sonar measurement of depth.

J.  Neutral delay systems: Arise, for instance, in the analysis of the coupling between transmission lines and population dynamics: evolution of forests. The model is based on a refinement of the delay-free logistic (or the Pearl–Verhulst equation), where effects as soil depletion and erosion have been introduced:

(1.4)

where x is the population, r is the intrinsic growth rate, and K is the environmental carrying capacity.

K.  Population dynamics: Predator–prey model based on the Volterra model with predator (y) and prey (x) populations (t is the life time of prey):

(1.5)

1.3 Models and solutions

A general model of TDSs can be expressed as

(1.6)

where

(1.7)

1.3.1 Retarded systems

, or ϕ may involve bounded jumps at some discontinuity instants). The nature of the solution (and of its initial value) distinguishes FDE from ODE.

Definition 1.1

[3]

A function x of the retarded functional differential equation (RDE)

(1.8)

, and x(t) satisfies is a solution of (1.8) with initial value φ at σ .

, and x is a solution of .

The interested reader is referred to [3] for further useful discussions.

1.3.2 Models for linear time-invariant systems

In the linear time-invariant case (LTI), the corresponding general time-delay model is

(1.9)

(1.10)

where

is a constant instantaneous matrix;

, represent discrete-delay phenomena;

;

account for the neutral part;

are input matrices;

.

Note that in delayed parts as well. The particular case of (1.9)–(1.10)

(1.11)

has been investigated extensively in the literature.

1.4 Stability notion

We further recall the following stability notion for time-delay system (1.5).

Definition 1.2

, then the trivial solution of time-delay system (1.5) is called stable.

The following properties are readily recognized:

•  If the trivial solution of time-delay system (1.5) is stable and if δ , then the trivial solution of time-delay system (1.5) is called uniformly stable.

•  If the trivial solution of time-delay system , then the trivial solution of time-delay system (1.5) is called asymptotically stable.

•  If the trivial solution of time-delay system , then the trivial solution of time-delay system (1.5) is called uniformly asymptotically stable.

•  If the trivial solution of time-delay system can be an arbitrarily large finite number, then the trivial solution of time-delay system (1.5) is called globally (uniformly) asymptotically stable

1.5 Stability theorems

In the study of stability analysis of time-delay systems, the methods of Lyapunov functions and Lyapunov–Krasovskii functionals play important roles. Two Lyapunov methods are often used:

A.  Lyapunov–Krasovskii functional (LKF) method,

B.  Lyapunov–Razumikhin function (LRF) method.

It is significant to observe that the LKF method deals with functionals that essentially have scalar values, whereas the LRF method involves only functions rather than functionals.

In this section, we review these two methods; see [6] for details.

Consider the following time-delay system

(1.12)

where

,

,

.

be the solution of (1.12) at time t .

The Lyapunov–Krasovskii stability method is provided by the following theorem.

Theorem 1.1

Let , and let be continuous nondecreasing functions with and and for . Suppose that there exists a continuous functional such that

;

,

where

Then the trivial solution of time-delay system (1.5) is uniformly stable. If for , then the trivial solution of time-delay system (1.5) is uniformly asymptotically stable. Additionally, if , then the trivial solution of time-delay system (1.5) is globally uniformly asymptotically stable.

are quite effective in the derivation of the stability conditions. This will in turn requires the modification of the conditions in Theorem 1.1. See [8] for details.

The Lyapunov–Razumikhin stability method is provided by the following theorem.

Theorem 1.2

Let , let be continuous nondecreasing functions with and and for , and let v be strictly increasing. Suppose that there exists a continuous functional such that

;

if

where

Then the trivial solution of time-delay system (1.5) is uniformly stable. If for , then there exits a continuous nondecreasing function for . If the foregoing condition 2) is strengthened to if

then the trivial solution of time-delay system (1.5) is uniformly asymptotically stable. Additionally, if , then the trivial solution of time-delay system (1.5) is globally uniformly asymptotically stable.

The following Halanay result [7] also plays an important role in the stability analysis of time-delay