##
**Multivariable control. A graph-theoretic approach.**
*(English)*
Zbl 0682.93006

Mathematical Research 41. Berlin (GDR): Akademie-Verlag (ISBN 3-05-500452-3). 274 p. (1988).

This book is a nice introduction to the graph theoretic approach to the analysis and synthesis of linear time-invariant multivariable control processes of the form \(x'=Ax+Bu\), \(y=Cx\).

A direct graph can be associated to the system by the following rules: each (input, output or state) variable corresponds to one vertex, and edges (connections) between pairs of vertices depend on the structure of matrices \(A, B, C\). Actually, according to the physical experience, it is not required that entries of \(A, B, C\) are exactly known. However, in general one knows whether a certain entry is zero or not (i.e., whether a certain connection in the graph is active).

This description of systems allows a structural analysis which is of great aid, especially in the case of large scale systems. Several classical concepts and problems of systems theory can be reviewed and fruitfully investigated in this framework. In particular, the author presents a number of results about controllability, observability, state feedback, and output feedback pole placement, disturbance rejection.

The book contains also an appendix where some elements of graph theory are given.

A direct graph can be associated to the system by the following rules: each (input, output or state) variable corresponds to one vertex, and edges (connections) between pairs of vertices depend on the structure of matrices \(A, B, C\). Actually, according to the physical experience, it is not required that entries of \(A, B, C\) are exactly known. However, in general one knows whether a certain entry is zero or not (i.e., whether a certain connection in the graph is active).

This description of systems allows a structural analysis which is of great aid, especially in the case of large scale systems. Several classical concepts and problems of systems theory can be reviewed and fruitfully investigated in this framework. In particular, the author presents a number of results about controllability, observability, state feedback, and output feedback pole placement, disturbance rejection.

The book contains also an appendix where some elements of graph theory are given.

Reviewer: A.Bacciotti

### MSC:

93Bxx | Controllability, observability, and system structure |

93-02 | Research exposition (monographs, survey articles) pertaining to systems and control theory |

93C05 | Linear systems in control theory |

93C35 | Multivariable systems, multidimensional control systems |

93A15 | Large-scale systems |