Stability and Controls Analysis for Delay Systems

By Jinrong Wang, Michal Feckan and Mengmeng Li

Stability and Controls Analysis for Delay Systems is devoted to stability, controllability and iterative learning control (ILC) to delay systems, including first order system, oscillating systems, impulsive systems, fractional systems, difference systems and stochastic systems raised from physics, biology, population dynamics, ecology and economics, currently not presented in other books on conventional fields. Delayed exponential matrix function approach is widely used to derive the representation and stability of the solutions and the controllability. ILC design are also established, which can be regarded as a way to find the control function.

The broad variety of achieved results with rigorous proofs and many numerical examples make this book unique.

• Presents the representation and stability of solutions via the delayed exponential matrix function approach
• Gives useful sufficient conditions to guarantee controllability
• Establishes ILC design and focuses on new systems such as the first order system, oscillating systems, impulsive systems, fractional systems, difference systems and stochastic systems raised from various subjects

• Language: English
• Publisher: Academic Press
• Release date: Nov 26, 2022
• ISBN: 9780323997935

Jinrong Wang

Prof. JinRong Wang has been a Professor of the Department of Mathematics at the School of Mathematics and Statistics at Guizhou University, People’s Republic of China since 2011. He received his master’s degree from Guizhou University in 2006 and PhD from the same university in 2009. He is interested in impulsive differential equations, fractional differential equations, delay differential equations and iterative learning controls. He was recognized as a Highly Cited Researcher in Mathematics in the period of 2015 to 2018.

Book preview

Chapter 1: Introduction

Abstract

This chapter gives the research background and motivation of stability and controllability of delay differential equations.

Keywords

stability; controllability; delay differential equations

It is well known that delay differential equations arise naturally in economics, physics, and control problems. It is an interesting task to develop the idea of Duhamel's principle in classical linear ordinary differential equations (ODEs) to seek the explicit representation of solutions [1] to delay discrete/differential systems. In fact, it is not an easy task to construct a fundamental matrix for linear differential delay systems, even for a simple first order delay system , , with initial condition , , , where A, B are suitable constant matrices. Khusainov and Shuklin in [1] introduced the delayed matrix exponential [1, Definition 0.3] and derived an explicit formula of solutions to such linear differential delay systems with . Diblík and Khusainov [2] adopted the idea to construct the discrete matrix delayed exponential, which was used to derive an explicit formula of solutions to a discrete delay system.

There is rapid development in seeking explicit formulas of solutions to delay differential/discrete equations by introducing various continuous/discrete delayed exponential matrices (see, for example, [3–18]). One of the biggest advantages of continuous/discrete delayed exponential matrices is the possibility to transfer the classical idea of representing the solution of linear ODEs to linear delay differential/discrete equations. This provides a new idea and approach to study the representation of solutions, asymptotic stability, finite time stability, controllability, and iterative learning control (ILC) for various kinds of linear continuous/discrete delay systems, for example, oscillating delay systems, impulsive delay systems, fractional delay systems, and stochastic delay systems.

Stability analysis is one of the most important issues in control systems. In general, the Lyapunov method is used to deal with the asymptotic stability of trivial solutions. However, there exists a stable system which yields undesirable transient performance at one fixed point in practical applications. For this reason, it is necessary to consider the boundedness of system trajectories over a given finite time interval from the engineering point of view instead of determining the long-time asymptotical behavior for the system trajectory from the mathematical point of view. As a result, the concept of finite time stability (the boundedness of system trajectories) is offered to characterize the behavior of dynamical systems.

The finite time stability concept, introduced by Dorato [19], of delay differential equations arises from the fields of multibody mechanics, automatic engines, and physiological systems. Finite time stability means that the system state does not exceed a certain bound for a given finite time interval and seems more appropriate from practical considerations. It is remarkable that Weiss and Infante [20–22] give sufficient conditions for finite time stability of nonlinear systems by introducing the suitable Lyapunov functionals. A solution of an equation or a state of a system is said to be finite time stable if it does not exceed a certain threshold during a fixed finite time interval. Obviously, finite time stability is very different from the classical exponential stability, which deals with equations operated in the whole infinite time interval. Concerning finite time stability, Ulam's stability and stable manifolds of linear systems, impulsive systems and fractional systems, fundamental matrix methods, linear matrix inequalities, algebraic inequalities, and integral inequalities are often used to deal with this issue. For more recent contributions, one can see [23–37].

The concept of controllability was first introduced by Kalman in the early 1960s. Some very important conclusions about the behavior of linear and nonlinear dynamical systems have been obtained. In the field of engineering technology, there are many real controlled systems with obvious delay effects, such as vehicle active suspension systems, rocket engine combustion systems, and other mechanical systems. The delay effect leads to the evolution of the system state with time depending on not only the current state of the system, but also the state of the system for a certain period of time in the past. Time-delay systems are one of the main tools to study delay effects. Thus, the situation will be much more complicated for time-independent delay differential controlled systems, or even for linear differential systems with pure delay with one input. In fact, in contrast to the theory of classical linear differential/difference controlled systems, when driving these delay systems to rest, one is required to control not only the value of the state at the final time, but also the memory accumulated with an aftereffect. In addition, the representation of the solution is not easy to characterize without knowing the fundamental matrix of a homogeneous delay differential system. As a result, various classes of control methods for delay differential systems were considered and different variants of controllability were developed in the past decades.

Khusainov and Shuklin [38] initially studied relative controllability of linear differential systems with pure delay via the associated delayed matrix exponential. Thereafter, Khusainov et al. [5] studied the Cauchy problem for a second order linear differential equation with pure delay. Next, in [4], relative controllability of linear discrete systems with a single constant delay was initially studied using the so-called discrete delayed matrix exponential, where an equivalent condition was stated for an initial value problem to have a control function. In addition, one of the control functions was found. Moreover, Pospíšil et al. [39] extended the study of controllability to linear discrete pure delay systems with constant coefficients and multiple control functions and derived a representation of solution in the form of a matrix polynomial using the Z-transform [18] to a system of nonhomogeneous linear difference equations with any finite number of constant delays and linear parts given by pairwise permutable matrices. Various criteria of relative controllability for linear discrete pure delay systems are presented and the associated control functions are also constructed.

In 1978, Uchiyama [40] proposed the concept of ILC from the viewpoint of human learning ability to deal with robotic systems. In the past decades, various types of iterative updating laws with uniform trial lengths have been widely used to deal with the issues in robotics, process control, and biological systems. A large amount of literature on ILC has been published for various types of systems such as discrete time systems, fractional differential systems, impulsive hybrid systems, distributed parameter systems, and networked stochastic systems. After reviewing the previous literature, we observe a restriction that the length of process operation and the desired trajectory are invariant in different iterations. Obviously, this condition may limit the application range of ILC. In fact, the case of operation terminating early does exist [41], for example, the functional electrical stimulation for upper limb movement [42]. This case strongly motivated us to consider ILC under randomly iteration-varying length environments. For more recent contributions on this hot topic, one can refer to [43–55] and reference therein. It is remarkable that the ILC problem of a delay spring–mass–damper system, which has received much attention, is very important in many practical mechanical systems [56].

ILC, which has become an important topic in modern data-driven control, is proposed in [57,58]. Note that ILC problems for linear discrete delayed controlled systems have been studied extensively by transferring to analyze a constructed Roesser model. For more related contributions, one can refer to [59–69] and reference therein.

This monograph is devoted to the rapidly developing research area of the stability, controllability, and ILC theory of delay difference/differential controlled systems. Such basic theory should be the starting point for further research concerning the optimal control, numerical analysis, and applications of other types of delay systems. The book is divided into seven chapters. Chapter 1 gives the introduction. In Chapter 2, we present the finite time stability, controllability, and ILC for first order delay differential systems. Chapter 3 is devoted to the study of the same issues in Chapter 2 for second order oscillating delay systems. In Chapter 4, we study asymptotical stability, finite time stability, and controllability for impulsive delay systems. In Chapter 5, we extend some content in Chapter 1 to fractional delay systems by introducing impulsive delay fundamental matrices. Chapter 6 is devoted to establishing the same issues as in Chapter 2 for difference delay systems. Chapter 7 further establishes the related controllability results for first order and second order stochastic delay systems.

Key features of this book are stated here. We present the representation and stability of solutions via the delayed exponential matrix function approach. We give useful sufficient conditions to guarantee controllability. We discuss ILC design and focus on new systems such as first order systems, oscillating systems, impulsive systems, fractional systems, difference systems, and stochastic systems occurring in various fields.

This book is useful for researchers and graduate students for research, seminars, and advanced graduate courses in pure and applied mathematics, engineering, and related disciplines.

References

[1] D.Ya. Khusainov, G.V. Shuklin, Linear autonomous time-delay system with permutation matrices solving, Stud. Univ. Žilina Math. Ser.