Robust and \(H_\infty\) control.

*(English)*Zbl 0996.93002
Communications and Control Engineering Series. London: Springer. xii, 452 p. (2000).

Due to its theoretical and practical importance, the so-called \(H_\infty \) control problem has drawn considerable attention in the past one and a half decade. The book under review focuses on the \(H_\infty \) control theory for linear time-invariant continuous- and discrete-time systems and related problems such as the \(H_\infty \) almost decoupling problem and the robust and perfect tracking problem. Many of the results presented in this book were originally derived by the author and his co-authors. The material of the book partly overlaps with an earlier book of the author [\(H_\infty \) control and its applications, Springer, London (1998; Zbl 0912.93003)], but substantial new topics are also included.

The book begins with some chapters on background material related to exact problem formulation in time domain, computation-oriented description of canonical forms of matrices and the so-called special coordinate basis. A bilinear and inverse bilinear transformation connecting continuous-time and discrete-time systems is studied in detail, and results on the existence of \(H_\infty \) suboptimal controllers and on the solution to discrete-time Riccati equations.

The central part of the book consists of Chapters 6 to 13, which are devoted to

1) computation of the infimum in \(H_\infty \) optimization problems;

2) derivation of “closed form” parametrization of suboptimal \(H_\infty \) controllers in terms of the \(H_\infty \) norm bound parameter \(\gamma \);

3) solutions to the general \(H_\infty \) almost disturbance decoupling problem;

4) some newly developed results on robust and perfect tracking problems.

For the latter, necessary and sufficient conditions under which the problem is solvable and constructive algorithms to realize the required controller are presented.

The book concludes with some real-life applications: a servo system design for a voice-coil-motor actuator of computer hard disk drives, a piezoelectric actuator control system design and a gyro-stabilized mirror targeting system design are studied.

This book is an excellent research reference and it should be of great value to researchers and engineers practising in the industry.

The book begins with some chapters on background material related to exact problem formulation in time domain, computation-oriented description of canonical forms of matrices and the so-called special coordinate basis. A bilinear and inverse bilinear transformation connecting continuous-time and discrete-time systems is studied in detail, and results on the existence of \(H_\infty \) suboptimal controllers and on the solution to discrete-time Riccati equations.

The central part of the book consists of Chapters 6 to 13, which are devoted to

1) computation of the infimum in \(H_\infty \) optimization problems;

2) derivation of “closed form” parametrization of suboptimal \(H_\infty \) controllers in terms of the \(H_\infty \) norm bound parameter \(\gamma \);

3) solutions to the general \(H_\infty \) almost disturbance decoupling problem;

4) some newly developed results on robust and perfect tracking problems.

For the latter, necessary and sufficient conditions under which the problem is solvable and constructive algorithms to realize the required controller are presented.

The book concludes with some real-life applications: a servo system design for a voice-coil-motor actuator of computer hard disk drives, a piezoelectric actuator control system design and a gyro-stabilized mirror targeting system design are studied.

This book is an excellent research reference and it should be of great value to researchers and engineers practising in the industry.

Reviewer: Éva Gyurkovics (Budapest)

##### MSC:

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

93B36 | \(H^\infty\)-control |