Polynomial and rational matrices. Applications in dynamical systems theory
It is an excellent book for polynomial and rational matrices and its applications in Dynamical Systems Theory, written by a very well-known scientist in the field of control theory. The book can be used either as a reference for researchers in the field of control theory and circuit theory or for teaching for undergraduates or postgraduates in electrical engineering, electronics, mechatronics and computer engineering. The book is an extended English version of its preceding Polish edition. Apart from the theory, the book includes many illustrative examples that make the theory more clear to the reader. However, the book does not contain enough results as concerns the structure of polynomial and rational matrices at \(s=\infty \) or a specific region \(\Lambda \subseteq \mathbb{C} \cup \left\{ \infty \right\} \), and the respective applications of this specific structure i.e. proper (or proper and stable) solutions of polynomial matrix diophantine equations. The monograph consists of eight chapters, an appendix and a list of references. Chapter 1 is devoted to the structure of polynomial matrices. It covers topics on polynomial matrices such as: elementary operations, division, equivalence, canonical forms, invariants, greatest common divisor, least common multiple, special kind of matrices etc. Chapter 2 starts with some basic definitions and operations of rational functions and matrices. Then, algebraic methods are proposed for the computation of the right (left) matrix fraction description (MFD) and the McMillan canonical form of a rational matrix. By using these methods, the synthesis problem of computing a regulator for a given plant is solved. Chapter 3 starts with a definition of normal rational matrices. Then, it studies the following properties of normal matrices: a) which is the form of the MFD of a normal rational matrix, b) under what conditions the sum, the product and the inverse of normal matrices are normal matrices, c) how a normal matrix can be decomposed into the sum of normal matrices. Finally, it studies the problem of normalisation of a transfer function matrix using state-feedback or output-feedback. Chapter 4 studies the problem of state-space realization of a proper transfer function matrix. It starts with the problem of the existence of a minimal realization for proper transfer function matrices. Then, it is proved that a cyclic realization for a rational proper matrix exists iff the rational matrix is normal, and gives a method for the computation of a cyclic realization. Finally, it proposes a procedure for the computation of the normal transfer function matrix, which is a good approximation of a rational transfer function matrix. Structural stability of cyclic realizations is also studied. Chapter 5 studies singular systems. It starts with properties of singular systems i.e. reachability and cyclicity of feedback systems. Then, it proposes a method for the computation of equivalent standard systems, and gives the decomposition theorem for both standard and singular systems. Electrical circuit examples are given for singular systems. Chapter 6 proposes existence conditions and computational methods for the solution of polynomial, rational and algebraic matrix equations. More specifically, it studies the unilateral (bilateral) polynomial matrix equation \(AX+BY=C\) (\(AX+YB=C\)), the Sylvester equation (and the special case of Lyapunov equation) etc. Chapter 7 deals with perfect and reduced-order observers of singular 1-D and 2-D systems (with or without unknown inputs). It also studies the case of functional observers for singular 1-D systems. Chapter 8 studies structural properties of positive linear systems with delays. Additionally, it proposes solutions to the positive realization problem of a given proper transfer matrix, in terms of multi-variable continuous or discrete time systems with delays in state and control. The Appendix contains some basic definitions and theorems for controllability and observability of linear systems. The book contains a lot of published results of the author.
scientific article; zbMATH DE number 5080872