Linear multivariable control. Algebraic analysis and synthesis methods.

*(English)*Zbl 0751.93002
Chichester etc.: John Wiley & Sons. xi, 369 p. (1991).

In the 1970’s, results and techniques on polynomial matrix descriptions (PMDs) of linear systems started to be developed. By the end of the decade the polynomial matrix approach to linear multivarible feedback systems was established as a rapidly developing area of research. The aim of the book as stated in the preface is to provide a rather detailed account of the basic theory of polynomial and fractional representation methods for algebraic analysis and synthesis of linear multivariable control systems. In the reviewers opinion the authors aim is to fill a gap between the theoretical approaches of system theory and the practical attitudes of control techniques. The contents of the book are organized into seven chapters and two appendices. To hold the attention of the reader many interesting examples and exercises can be found in the text in each chapter. The most relevant bibliography can be found at the end of each chapter.

Chapter one is a mathematical treatise presenting the algebraic structure and properties of rational matrices, rational vector spaces and their associations to the state-space minimal realization theory of proper rational matrices. These subjects provide the necessary background for the rest of the book. Chapter two introduces polynomial matrix model of liner multivariable systems and examines the various definitions of system equivalence, decoupling zeros, (finite) poles and zeros of a multivariable system and its associated transfer function matrix. Chapter three devotes the pole-zero structure at infinity of rational scalar and matrix functions, equivalence of rational matrices at infinity, coprimeness at infinity and the minimal realization theory for polynomial matrices. Also, the concept of column and row properness of polynomial matrices is generalized for general rational matrices. Chapter four deals with the time properties of linear systems, whose dynamics are described by polynomial matrix models. The impulsive behaviour of the solutions of general homogeneous matrix differential equations due to zeros at infinity and the infinite elementary divisors of the associated characteristic matrix is considered. Reachability and controllability of regular and generalized state-space or singular systems are introduced as special cases of a general theory of linear systems. A short survey of proper and stable rational functions and matrices follows in chapter five. Also, the problems of stabilization of a single input single output system by means of dynamic compensation is discussed here. Chapter six presents the investigation of the problem of stabilization of linear multivariable systems by means of feedback compensation. Finally, the algebraic design problems of simultaneous internal stabilization and asymptotic tracking, disturbance suppression and diagonal decoupling by use of dynamic compensation are examined in chapter seven.

The book will be very useful for mathematic and engineering students interested in a modern, if some what difficult, system course, as well as for the experts in control theory. The inserted “Drawings by courtesy of the author” are well chosen.

Chapter one is a mathematical treatise presenting the algebraic structure and properties of rational matrices, rational vector spaces and their associations to the state-space minimal realization theory of proper rational matrices. These subjects provide the necessary background for the rest of the book. Chapter two introduces polynomial matrix model of liner multivariable systems and examines the various definitions of system equivalence, decoupling zeros, (finite) poles and zeros of a multivariable system and its associated transfer function matrix. Chapter three devotes the pole-zero structure at infinity of rational scalar and matrix functions, equivalence of rational matrices at infinity, coprimeness at infinity and the minimal realization theory for polynomial matrices. Also, the concept of column and row properness of polynomial matrices is generalized for general rational matrices. Chapter four deals with the time properties of linear systems, whose dynamics are described by polynomial matrix models. The impulsive behaviour of the solutions of general homogeneous matrix differential equations due to zeros at infinity and the infinite elementary divisors of the associated characteristic matrix is considered. Reachability and controllability of regular and generalized state-space or singular systems are introduced as special cases of a general theory of linear systems. A short survey of proper and stable rational functions and matrices follows in chapter five. Also, the problems of stabilization of a single input single output system by means of dynamic compensation is discussed here. Chapter six presents the investigation of the problem of stabilization of linear multivariable systems by means of feedback compensation. Finally, the algebraic design problems of simultaneous internal stabilization and asymptotic tracking, disturbance suppression and diagonal decoupling by use of dynamic compensation are examined in chapter seven.

The book will be very useful for mathematic and engineering students interested in a modern, if some what difficult, system course, as well as for the experts in control theory. The inserted “Drawings by courtesy of the author” are well chosen.

Reviewer: I.H.H.van de Ven (Eindhoven)

##### MSC:

93-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to systems and control theory |

93B25 | Algebraic methods |

93B50 | Synthesis problems |

93A10 | General systems |

93B52 | Feedback control |

93C05 | Linear systems in control theory |

93C35 | Multivariable systems, multidimensional control systems |