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Analysis and design of descriptor linear systems. (English) Zbl 1227.93001

Advances in Mechanics and Mathematics 23. Dordrecht: Springer (ISBN 978-1-4419-6396-3/hbk; 978-1-4614-2684-4/pbk; 978-1-4419-6397-0/ebook). xix, 494 p. (2010).
Descriptor systems are otherwise called singular systems, generalized state space systems, semi-state systems, differential algebraic systems, degenerate systems, constrained systems, etc. Descriptor systems have been intensively studied since 1970 and appear in many fields such as power systems, electrical networks etc. Although many articles on descriptor systems have been written until now, there are very few books written on this topic. Among these books, we distinguish the books of L. Dai [Singular control systems. Berlin etc.: Springer-Verlag (1989; Zbl 0669.93034)], S. L. Campbell [Singular systems of differential equations. II. San Francisco-London-Melbourne: Pitman Advanced Publishing Program (1982; Zbl 0482.34008)], P. Kunkel and V. Mehrmann [Differential-algebraic equations. Analysis and numerical solution. Zürich: European Mathematical Society Publishing House (2006; Zbl 1095.34004)]. The book of Duan has as a main reference the book of Dai, but also refers to many papers written for descriptor systems and a lot of the author’s own results on this topic.
Chapter 1 starts with describing different representations of descriptor state systems. Some examples of descriptor systems are presented from different fields, i.e., electrical circuit systems, large scale systems, constrained mechanical systems and robotic systems. At the end of this chapter an outline of the basic problems for descriptor linear system analysis and design is given.
The book is separated into two parts: the first concerns the analysis of descriptor systems, whereas the second one is devoted to the design of descriptor systems.
Part I. Descriptor Linear Systems Analysis. Since the state space representation of a descriptor system is not unique, a number of equivalence relations between descriptor systems have been defined in the past. Chapter 2 introduces the Restricted System Equivalence (RSE) transformation, as well as some properties that remain invariant under this transformation. Based on the RSE transformation, a number of canonical forms are defined for descriptor systems. One of these canonical forms is the Kronecker canonical form that gives rise to information concerning the existence, uniqueness and structure of solutions of descriptor systems. However, such important topics as complete equivalence [A. C. Pugh, G. E. Hayton and P. Fretwell, Int. J. Control 45, 529–548 (1987; Zbl 0623.93013)], strong equivalence [B. D. O. Anderson, W. A. Coppel and D. J. Cullen, J. Aust. Math. Soc., Ser. B 27, 194–222 (1985; Zbl 0594.93015), W. A. Coppel and D. J. Cullen, J. Aust. Math. Soc., Ser. B 27, 223–237 (1985; Zbl 0594.93016)] and fundamental equivalence [G. E. Hayton, P. Fretwell and A. C. Pugh, IEEE Trans. Autom. Control 31, 431–439 (1986; Zbl 0616.93002)] are not considered. The important work of M. Kuijper [First-order representations of linear systems. Boston, MA: Birkhäuser (1994; Zbl 0863.93001)] on descriptor systems is also missing.
Chapter 3 starts with the definition and some criteria for the regularity of descriptor systems which guarantee the existence and uniqueness of solutions. Then, based on the restricted system equivalence it provides two canonical forms for regular descriptor systems, i.e., the standard decomposition form or otherwise the Weierstrass form and the inverse form. The distributional and the classical solution is defined in the sequel. The first one, which exists under any possible initial conditions, is given in terms of generalized functions whereas the second one exists when the initial values are consistent (satisfy the system equations and giving rise to non-impulsive solutions). A reference that is missing from this distinction on the solutions is A. I. G. Vardulakis, E. N. Antoniou and N. Karampetakis [Int. J. Control 72, No. 3, 215–228 (1999; Zbl 1001.93039)]. The definition of the finite and infinite generalized eigenvectors and its relation to the eigenstructure decomposition is discussed in the sequel. Stability with or without impulse-freeness is discussed at the end of this section. Besides the direct criterion for stability, two criteria using generalized Lyapunov equations have also been provided.
Chapter 4 introduces various types of definitions for controllability and observability. Then, it gives criteria for controllability and observability based on: a) the standard decomposition form, b) directly onto the original systems data, c) the equivalent canonical decomposition form, d) the equivalent inverse form, and e) the equivalent form for derivative feedback. Finally, two problems that are closely related with the concept of controllability and observability are considered: a) system structural decomposition of regular descriptor systems and b) minimal realization of non-proper transfer functions. In my opinion, the evolution of the state in a descriptor system of the form \(E\dot{x}(t)=Ax(t)+Bu(t)\), \(E,A\in \mathbb R^{n\times n}\), \(B\in\mathbb R^{n\times m}\) is actually depending on the initial conditions \(Ex(0-)\) and therefore the notion of observability must be connected with the construction of \(Ex(0-)\) by the knowledge of input and output data in a time interval, something that has not been considered in this chapter. There is also no distinction between the initial values before/after the systems starts i.e. \(x(0-)\) and \(x(0+)\) respectively, that has relations with the existence of the impulsive terms in the solution of the system [Vardulakis et al., loc. cit.].
Part II. Descriptor Linear Systems Design. Chapter 5 considers the regularization problem for descriptor systems, defined as follows: “Given a typical descriptor linear system find a feedback controller of certain type (proportional feedback, derivative feedback or combination of them is studied in this work) for the systems such that the closed loop system is regular.” Necessary and sufficient conditions are established for the proposed regularization problem that depend only on the open-loop system coefficient matrices, and are easily implemented.
Chapter 6 investigates the problem of dynamical order assignment in descriptor systems via full and partial state derivative feedback. It proposes also the solution of the aforementioned problem by proposing all the minimum Frobenius norm solutions. The normalization of descriptor systems, which is a special case of the dynamical order assignment problem (the dynamical order to be assigned equal to the system dimension), is solved by using derivative feedback at the end of the chapter.
Chapter 7 solves the impulse elimination problem. More specifically, it finds a certain type of feedback controller (state-feedback, output-feedback, partial-derivative feedback) for the system such that the closed-loop system is impulse free. Necessary and sufficient conditions are proposed for each case of feedback control and specific solutions have been proposed.
Chapter 8 studies the problems of pole assignment and stabilization for descriptor systems. More specifically, three types of pole assignment have been studied: a) existence of a state-feedback controller that assigns the set of open-loop finite poles to \(\deg \det (sE-A)\) arbitrary prescribed self-conjugate complex numbers, b) existence of a state-feedback controller that assigns the set of open-loop finite poles to rank \(E\) prescribed finite eigenvalues (thus impulsive elimination is also included), c) existence of a state partial-derivative feedback controller that assigns the set of open-loop finite poles to \(n\) (dimension of \(E\)) prescribed finite eigenvalues (thus normalization is also included).
The problem of eigenstructure assignment in descriptor systems via state feedback is considered in Chapter 9, based on the work of the author [Int. J. Control 69, No. 5, 663–694 (1998; Zbl 0949.93029)].
Chapter 10 studies the optimal control problem for descriptor systems with emphases on the infinite time state quadratic regulation problem and the time-optimal control problem. The technique used is first to convert the problem into a corresponding one for normal linear system and then solve it by using well-known techniques for normal linear systems.
Finally, the last chapter studies the problem of observer design for descriptor systems by considering design of several types of observers.
Part III is an appendix that contains mathematical results related to certain chapters of the book.

MSC:

93-02 Research exposition (monographs, survey articles) pertaining to systems and control theory
93B05 Controllability
93B07 Observability
93B10 Canonical structure
93B11 System structure simplification
93B17 Transformations
93B25 Algebraic methods
93B51 Design techniques (robust design, computer-aided design, etc.)
93B52 Feedback control
93B55 Pole and zero placement problems
93B60 Eigenvalue problems
93C05 Linear systems in control theory
93C15 Control/observation systems governed by ordinary differential equations
93C35 Multivariable systems, multidimensional control systems
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
93D15 Stabilization of systems by feedback
93D20 Asymptotic stability in control theory
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