Analysis and Synthesis of Singular Systems

By Zhiguang Feng, Jiangrong Li, Peng Shi and 2 more

Analysis and Synthesis of Singular Systems provides a base for further theoretical research and a design guide for engineering applications of singular systems. The book presents recent advances in analysis and synthesis problems, including state-feedback control, static output feedback control, filtering, dissipative control, H8 control, reliable control, sliding mode control and fuzzy control for linear singular systems and nonlinear singular systems. Less conservative and fresh novel techniques, combined with the linear matrix inequality (LMI) technique, the slack matrix method, and the reciprocally convex combination approach are applied to singular systems.

This book will be of interest to academic researchers, postgraduate and undergraduate students working in control theory and singular systems.

• Discusses recent advances in analysis and synthesis problems for linear singular systems and nonlinear singular systems
• Offers a base for further theoretical research as well as a design guide for engineering applications of singular systems
• Presents several necessary and sufficient conditions for delay-free singular systems and some less conservative results for time-delay singular systems

• Language: English
• Publisher: Academic Press
• Release date: Nov 4, 2020
• ISBN: 9780128237403

Zhiguang Feng

Zhiguang Feng received the B.S. degree in automation from Qufu Normal University, Rizhao, China, in 2006, the M.S. degree in Control Science and Engineering from Harbin Institute of Technology, Harbin, China, in 2009, and the Ph.D. Degree in the Department of Mechanical Engineering, The University of Hong Kong, Hong Kong, in 2013. He was a Research Associate in the Department of Mechanical Engineering, University of Hong Kong, Hong Kong, from Oct. 2013 to Feb. 2014. From Mar. 2014 to Apr. 2015, he was a visiting fellow in the School of Computing, Engineering and Mathematics, University of Western Sydney, Australia. He was appointed with Victoria University in Australia as Postdoctoral Research Fellow within the College of Engineering and Science from Oct. 2015 to Mar. 2017.

Book preview

Chapter 1: Introduction

Abstract

The background and present status of development of related research fields are introduced. In addition, the motivation and the research problems are also discussed in this chapter.

Keywords

Singular systems; time-delay systems; singular Markovian jump systems; singular T-S fuzzy systems

1.1 Background

The model of the standard state-space system described by ordinary differential equations (ODEs) is widely used in linear control theory, because the state space approach does not only reveal various properties of the system, but also offers us effective system analysis and synthesis methods. In many practical systems, however, some algebraic constrained laws are imposed on the state components, which lead to the singular system description. A singular system essentially composes of a set of algebraic and differential equations (ADEs), containing information of the static and dynamic constraints of a plant.

Singular systems, also called descriptor systems, semistate space systems and generalized state-space systems, can provide a more natural description of dynamic systems than the standard state-space systems, due to the fact that singular systems can preserve the structure of physical systems more accurately by including nondynamic constraints and impulsive elements. Moreover, because the standard state-space system is a special case of the singular system, the singular system form can describe more practical systems than the standard state-space form. Singular systems have strong application background and are frequently employed to model circuit systems [25,137], economic systems [120], constrained mechanical systems [34,67], aircraft control systems [159] and chemical processes [82]. In this book, linear singular systems and nonlinear singular systems are studied.

Apart from the different form compared with the standard state-space systems, singular systems have fundamental differences as pointed out as follows [34,209]:

•  A singular system may not have a solution. If a solution exists, there may be more than one solutions. This point is different with standard state-space systems, which have a unique solution for any initial conditions.

•  For any initial conditions, the response of a singular system may contain impulse terms and the derivatives of these impulses or noncausal behaviors. Standard state-space systems do not have impulsive or noncausal behaviors.

•  A singular system usually contains three dynamic models: finite dynamic models, infinite models (which lead to the undesired impulsive behaviors), infinite static models. However, a standard state-space system only contains finite dynamic models.

•  The transfer function of a singular system may contain a polynomial matrix, which is not strictly proper, whereas that of a standard state-space system is strictly proper.

Therefore the analysis and the synthesis problems for singular systems are more complicated than those of standard state-space systems, because it is required to consider not only the stability, but also the regularity and nonimpulsiveness (for continuous-time singular systems), or causality (for discrete-time singular systems) characteristics. In sum, investigating singular systems is significant both in practice and theory. The core of studying singular systems basically consists of analysis and synthesis problems such that the singular systems have a desired or satisfactory property, which mainly contains admissibility, performance, and robustness.

The analysis problem in the control-theoretic context is to establish conditions under which a system is guaranteed to have these properties. Admissibility of singular systems is a property as important as the stability of standard state-space systems. A singular system is said to be admissible if it is asymptotically stable, regular, and impulse-free (for continuous-time singular systems) or causal (for discrete-time singular systems). The existence and uniqueness of solution to a singular system can be guaranteed by the regularity. Nonimpulsiveness for a continuous-time singular system means there is not impulsive behavior with the consistent initial conditions, whereas causality for a discrete-time singular system means that the states of a system in the past do not depend on the state in the future. Asymptotic stability guarantees that the state of a singular system approaches the equilibrium when time goes to infinity.

performance represents the maximum gain of the system, which characterizes the worst-case norm of the regulated outputs over all exogenous inputs with bounded energy. By designing the loop gain of the system to be less than unity, the closed-loop system is guaranteed to be asymptotically stable. Positive real property is widely used in adaptive control, absolute stability, and robust stable analysis. The stability of a closed-loop system can be realized by passivity control such that the phase lag of the system is less than 180 degrees and positive real control theory, but also provides a more flexible and less conservative robust control design as it allows a better trade-off between the gain and phase performances. Based on an input-output energy-related consideration, dissipativity theory has generalized many independent theorems or lemmas, for example, the passivity theorem, bounded real lemma (BRL), Kalman–Yakbovich–Popov lemma (KYPL) and the circle criterion, and provides a unified framework for the analysis and design of control systems [114]. Dissipative systems are very useful for a wide range of fields, such as system, circuit, network, and control theory [12], [147]. It gives strong links amongst physics, systems theory, and control engineering. Since its introduction, the theory of dissipativity has attracted extensive attention in system control, such as for nonlinear systems [242] and for linear systems [130].

Time delays are sources of instability and poor performance of a dynamical system. They always exist in many dynamical systems [89]. Consequently, many stability results and controller design approaches of delay systems have been reported in the literature. Singular time-delay systems are in essence delay differential equations coupled with functional equations, and thus the robust stability problem for singular systems is much more complicated than that for state-space systems, because it requires to consider not only stability robustness, but also regularity and causality (absence of impulses), which may affect the stability of the system. The problems arising from singular time-delay systems are significant both in theory and in practice. A considerable number of studies have been devoted to singular time-delay systems, such as the results on continuous-time systems [156,226], discrete-time cases [88,233], and the references therein. It is worth noticing that various methods were developed to obtain less conservative results. The free-weighting matrices approach [53,231], Jensen inequality method [89], the reciprocally convex combination approach [48], Wirtinger inequality [149], and delay partitioning method [4] are used in many papers. In this paper, the delay partitioning and the reciprocally convex combination approaches have been used to reduce conservative of results for bilinear system with time-varying delay.

On the other hand, with respect to the uncertainties, the method of sliding mode control (SMC) has fast response and good robustness as competitive advantages. Sliding mode control, which is based on the theory of variable structure systems, has been widely applied to robust control of nonlinear systems. The sliding mode control employs a discontinuous control law to drive the state trajectory toward a specified sliding surface and maintain its motion along the sliding surface in the state space [232]. The dynamic performance of the sliding mode control system has been confirmed as an effective robust control approach with respect to system uncertainties and unknown disturbance when the system trajectories belong to predetermined sliding surface.

The synthesis problem in the control-theoretic context is to design a controller or a filter such that the closed-loop system has a desired or satisfactory behavior. Generally speaking, an effective way to synthesize controllers/filters investigation is often based on some performance-based criteria, under which the controlled/filtered system has the desired properties. Then, a controller or a filter will be designed to guarantee the closed-loop system or the filtering error system to satisfy the criterion. For filter design problem, among various filter design methods, Kalman filtering approach is one of the most popular way, which estimates the state vector by optimizing the covariance of the estimation error and passive filtering, has also been an effective approach. Designed filters may be classified into full-order filters and reduced-order filters. When the order of the filter equals the order of the plant, it is called full-order filter; when it is lower, it is called a reduced-order filter. When considering some large-scale systems, a full-order filter may be not suitable to apply, and the reduced-order filter is a better