##
**Essentials of robust control.**
*(English)*
Zbl 0890.93003

Hemel Hempstead: Prentice Hall. xvi, 411 p. (1997).

The book, consisting of a preface, 18 chapters, a bibliography and an index, deals with basic robust and \(\mathcal H_{\infty}\) control theory.

Introduction (Ch. 1), Linear Algebra (Ch. 2) and Linear Systems (Ch. 3) review some basic linear algebra and system theoretic concepts. \({\mathcal H}_2\) and \(\mathcal H_{\infty}\) Spaces (Ch. 4) presents state-space methods for computing real rational \({\mathcal H}_2\) and \(\mathcal H_{\infty}\) transfer matrix norms. Internal Stability (Ch. 5) introduces the feedback structure and discusses its stability. Performance Specifications and Limitations (Ch. 6) considers optimal \({\mathcal H}_2\) and \(\mathcal H_{\infty}\) control and their design limitations. Balanced Model Reduction (Ch. 7) deals with the order reduction of a linear multivariable system using the balanced truncation method. Uncertainty and Robustness (Ch. 8) derives robust stability tests based on the small gain theorem under various modelling assumptions. Linear Fractional Transformation (Ch. 9) introduces the LFT in detail and presents control problems within this framework. \(\mu\) and \(\mu\) Synthesis (Ch. 10) presents robust stability and performance of systems with multiple sources of uncertainties. Controller Parametrization (Ch. 11) characterizes all stabilizing controllers for a given plant in state-space. Algebraic Riccati Equations (Ch. 12) studies the stabilizing solutions of the AREs. \({\mathcal H}_2\) Optimal Control (Ch. 13) and \(\mathcal H_{\infty}\) Control (Ch. 14) treat the LQR problem and simplified and general \(\mathcal H_{\infty}\) control problems. Controller Reduction (Ch. 15) deals with the design of reduced-order controllers by means of controller reduction. \(\mathcal H_{\infty}\) Loop Shaping (Ch. 16) proposes a design technique incorporating loop shaping methods to obtain performance/robust stability tradeoffs, and a particular \(\mathcal H_{\infty}\) minimization problem to guarantee closed-loop stability and a level of robust stability at all frequencies. Gap Metric and \(\nu\)-Gap Metric (Ch. 17) discusses these concepts and considers the controller order reduction within this framework. Miscallaneous Topics (Ch. 18) considers briefly the problems of model validation and the mixed real and complex \(\mu\) analysis and synthesis. The book includes numerous illustrative examples and exercise problems supported by MATLAB commands. This book is an excellent textbook for a graduate course in multivariable control. It will be certainly highly appreciated by practicing control engineers who are interested in applying the state-of-the-art robust control techniques. It grew from the extensively class-tested book by K. Zhou, J. C. Doyle and K. G. Glover [Robust and Optimal Control, Prentice Hall (1996)].

Introduction (Ch. 1), Linear Algebra (Ch. 2) and Linear Systems (Ch. 3) review some basic linear algebra and system theoretic concepts. \({\mathcal H}_2\) and \(\mathcal H_{\infty}\) Spaces (Ch. 4) presents state-space methods for computing real rational \({\mathcal H}_2\) and \(\mathcal H_{\infty}\) transfer matrix norms. Internal Stability (Ch. 5) introduces the feedback structure and discusses its stability. Performance Specifications and Limitations (Ch. 6) considers optimal \({\mathcal H}_2\) and \(\mathcal H_{\infty}\) control and their design limitations. Balanced Model Reduction (Ch. 7) deals with the order reduction of a linear multivariable system using the balanced truncation method. Uncertainty and Robustness (Ch. 8) derives robust stability tests based on the small gain theorem under various modelling assumptions. Linear Fractional Transformation (Ch. 9) introduces the LFT in detail and presents control problems within this framework. \(\mu\) and \(\mu\) Synthesis (Ch. 10) presents robust stability and performance of systems with multiple sources of uncertainties. Controller Parametrization (Ch. 11) characterizes all stabilizing controllers for a given plant in state-space. Algebraic Riccati Equations (Ch. 12) studies the stabilizing solutions of the AREs. \({\mathcal H}_2\) Optimal Control (Ch. 13) and \(\mathcal H_{\infty}\) Control (Ch. 14) treat the LQR problem and simplified and general \(\mathcal H_{\infty}\) control problems. Controller Reduction (Ch. 15) deals with the design of reduced-order controllers by means of controller reduction. \(\mathcal H_{\infty}\) Loop Shaping (Ch. 16) proposes a design technique incorporating loop shaping methods to obtain performance/robust stability tradeoffs, and a particular \(\mathcal H_{\infty}\) minimization problem to guarantee closed-loop stability and a level of robust stability at all frequencies. Gap Metric and \(\nu\)-Gap Metric (Ch. 17) discusses these concepts and considers the controller order reduction within this framework. Miscallaneous Topics (Ch. 18) considers briefly the problems of model validation and the mixed real and complex \(\mu\) analysis and synthesis. The book includes numerous illustrative examples and exercise problems supported by MATLAB commands. This book is an excellent textbook for a graduate course in multivariable control. It will be certainly highly appreciated by practicing control engineers who are interested in applying the state-of-the-art robust control techniques. It grew from the extensively class-tested book by K. Zhou, J. C. Doyle and K. G. Glover [Robust and Optimal Control, Prentice Hall (1996)].

Reviewer: L.Bakule (Praha)

### MSC:

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

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

93B35 | Sensitivity (robustness) |

93D09 | Robust stability |