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2010 | Buch

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Analysis and Design of Descriptor Linear Systems

verfasst von: Guang-Ren Duan

Verlag: Springer New York

Buchreihe : Advances in Mechanics and Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Descriptor linear systems theory is an important part in the general field of control systems theory, and has attracted much attention in the last two decades. In spite of the fact that descriptor linear systems theory has been a topic very rich in content, there have been only a few books on this topic. This book provides a systematic introduction to the theory of continuous-time descriptor linear systems and aims to provide a relatively systematic introduction to the basic results in descriptor linear systems theory. The clear representation of materials and a large number of examples make this book easy to understand by a large audience. General readers will find in this book a comprehensive introduction to the theory of descriptive linear systems. Researchers will find a comprehensive description of the most recent results in this theory and students will find a good introduction to some important problems in linear systems theory.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

This chapter gives a brief introduction to descriptor systems. Section 1.1 discusses the state space representation of descriptor systems, with an emphasis on the timeinvariant descriptor linear systems, which will be mainly discussed in this book. Section 1.2 presents examples of descriptor systems, specifically, the electrical circuit systems, the interconnected large-scale systems, the constrained mechanical systems, and the robotic systems. The purpose of this section is to assure the readers about the existence of descriptor systems in various fields. In Sect. 1.3, various feedback laws in descriptor linear systems are first introduced and then, a general introduction to the basic problems involving descriptor linear systems analysis and design is presented. An overview of the book is given in Sect. 1.4 and some notes and comments follow in Sect. 1.5.

Guang-Ren Duan

Descriptor Linear Systems Analysis

Frontmatter

Chapter 2. Equivalence and Solutions of Descriptor Linear Systems

Abstract

Modeling is probably the most basic topic in any system theory. In Sect. 1.1, we introduced the state space representation of descriptor systems which depends very much on the state vector selected. Since there exist different choices of the state vector variables, the state space representation of a given descriptor system is certainly not unique. In normal linear systems theory, this phenomenon has been well described by the concept of algebraic equivalence (see Duan 1996a, 2004a). The purpose of this chapter is to generalize this concept into the case of descriptor linear systems, and based on which to examine some of the basic equivalent structures and the solution of a general descriptor linear system.

Guang-Ren Duan

Chapter 3. Regular Descriptor Linear Systems

Abstract

We have pointed out in Sect. 1.3 that regularity is a very important property for descriptor linear systems. It guarantees the existence and uniqueness of solutions to descriptor linear systems. This chapter studies regular descriptor linear systems and starts in Sect. 3.1 with the definition of regularity of descriptor linear systems and its relation with solutions of descriptor linear systems. The restricted system equivalence for a regular descriptor linear system is introduced in Sect. 3.2. In the other sections of this chapter, some special features of regular descriptor linear systems are introduced. These include the eigenstructure, the state response, the stability as well as the transfer function of regular descriptor linear systems.

Guang-Ren Duan

Chapter 4. Controllability and Observability

Abstract

We know from standard normal linear systems theory that controllability and observability are a pair of very important concepts for linear systems. They provide us with certain structural features of the system as well as some internal properties of the system, and are essential for certain system design problems studied in the later chapters.

Guang-Ren Duan

Descriptor Linear Systems Design

Frontmatter

Chapter 5. Regularization of Descriptor Linear Systems

Abstract

It has been seen from Chapter 3 that regularity is an important property for descriptor linear systems and it offers many nice features, including the existence and uniqueness of solutions to descriptor linear systems. Therefore, for descriptor linear systems which are not regular, finding some feedback controllers such that the closed-loop system is regular is a very basic and important issue in descriptor linear system theory. This issue is termed as regularization of descriptor systems via feedback and is investigated in this chapter.

Guang-Ren Duan

Chapter 6. Dynamical Order Assignment and Normalization

Abstract

It has been demonstrated by classical control that derivative feedback is very essential for improving the stability and the performance of a control system. For descriptor systems, derivative feedback is even more important since it can alter many properties of a descriptor system, which a pure proportional state feedback can not. One of such properties is the dynamical order, and this leads to the problem of dynamical order assignment, which is fundamental in descriptor systems theory.

Guang-Ren Duan

Chapter 7. Impulse Elimination

Abstract

It is seen from Sect. 3.4 that the response of a descriptor linear system may contain impulse terms. These terms may cause saturation of control and may even destroy the system, and hence are not expected to exist. In this chapter, we study the problem of eliminating the impulsive behavior of a descriptor linear system via certain feedback control. As we will soon find out, the problem is closely related with normalization of descriptor linear systems.

Guang-Ren Duan

Chapter 8. Pole Assignment and Stabilization

Abstract

It is wellknown from normal linear systems theory that pole assignment and stabilization are two very basic closely related design problems in linear systems theory. In this chapter, we investigate these two problems for descriptor linear systems.

Guang-Ren Duan

Chapter 9. Eigenstructure Assignment

Abstract

This chapter studies the problem of eigenstructure assignment in descriptor linear systems by state feedback. The complete parametric approach proposed in Duan (1998a) for eigenstructure assignment in S-controllable descriptor linear systems via static state feedback is introduced. General complete parametric expressions in direct closed forms for the right closed-loop eigenvectors associated with the finite closed-loop eigenvalues, and two simple complete parametric solutions for the feedback gain matrix K are established.

Guang-Ren Duan

Chapter 10. Optimal Control

Abstract

This chapter studies optimal control of descriptor linear systems. After a brief introduction given in Sect. 10.1, the basic linear quadratic state regulation problem for descriptor linear systems is investigated in Sect. 10.2. Corresponding to the well-known bang–bang control for normal linear systems, the time-optimal control problem for descriptor linear systems is studied in Sect. 10.3. Some notes and references follow in Sect. 10.4.

Guang-Ren Duan

Chapter 11. Observer Design

Abstract

In this chapter, we consider the problem of observer design for descriptor linear systems, which is involved with reconstructing asymptotically the state vector or a linear combination of the state variables of a descriptor linear system.

Guang-Ren Duan

Appendices

Frontmatter

Appendix A. Some Mathematical Results

Abstract

This appendix presents some mathematical results which are repeatedly used in the book.

Guang-Ren Duan

Appendix B. Rank-Constrained Matrix Matching and Least Square Problems

Abstract

The rank-constrained least square problem seeks the matrix X with an assigned rank p such that the Frobenius norm of AX – B is minimized for some given matrices A and B. This problem reduces to the so-called rank-constrained matrix matching problem when thematrix A is an identity one. It is first shown that the solution to the rank-constrained least square problem can be converted to the solution to the rankconstrained matrix matching problem, and then a complete parametric solution to the rank-constrainedmatrix matching problem is presented. The proposed approach is convenient to use and possesses good numerical reliability since itmainly involves a singular value decomposition. To the best knowledge of the authors, such results are novel in the literature.

Guang-Ren Duan

Appendix C. Generalized Sylvester Matrix Equations

Abstract

This appendix presents the general complete parametric solution to a type of generalized Sylvester matrix equations. The results are essential for solving the problems of eigenstructure assignment (Chapter 9) and parametric observer design (Chap. 11).

Guang-Ren Duan

Backmatter

Titel: Analysis and Design of Descriptor Linear Systems
verfasst von: Guang-Ren Duan
Verlag: Springer New York
Electronic ISBN: 978-1-4419-6397-0
Print ISBN: 978-1-4419-6396-3
DOI: https://doi.org/10.1007/978-1-4419-6397-0

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