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Control of uncertain systems. A linear programming approach. (English) Zbl 0838.93007
Hemel Hempstead: Prentice Hall. xiii, 402 p. (1995).
In many real-world situations, a plant to be controlled is stimulated by a number of unknown disturbances and its parameters are known only with finite accuracy. It leads to the situation in which the system becomes uncertain to a designer and its characteristics are only imprecisely known. One way to attack the synthesis problem in this ase is to improve continuously a knowledge about the system through extensive testing, parameter estimation and identification performed on line during the control process. Such an approach results in the design of adaptive control systems. Alternatively, parameters may be accepted at their a priori level and then a control system is to be designed which should be, in some sense, robust or insensitive. This last point of view is represented in the book. More precisely, the authors attempt to treat in a unified way several techniques of robust control system synthesis including $$l_1$$, $$H_2$$ and $$H_\infty$$ theories and to express the resulting numerical problems in terms of linear programming. This is possible because the authors study the problem of stability and performance robustness for inputs bounded in magnitude and their objective consists in finding bounds for the maximum amplitude of the regulated output in the presence of unknown but bounded-input bounded-output perturbations. The emphasis is on time-domain problems and although the design philosophy is in terms of the input-output theory, all the computations are performed using matrix computations so that the authors use a state representation of the system.
In my opinion, the weak side of the book is that the presentation is entirely for discrete-time systems. Although the motivation behind this is that modern controllers are mostly digital, continuous-time dynamics of plants results in “hybrid” problems, which should be considered especially when problems of robustness are solved. For example, from the robust design point of view, it may be crucial to know what type of discrete approximation is applied and when a discrete-model of the plant is well posed. My second and last criticism to the contents of this book is that the authors ignore completely Zakian’s method of inequalities (developed in the seventies) which at least in its philosophy is similar to the authors’ approach. Nevertheless, the book may be recommended both for graduate students in control theory and its applications as well as for engineers and advanced researches and may be read in several different ways.

##### MSC:
 93-02 Research exposition (monographs, survey articles) pertaining to systems and control theory 90C90 Applications of mathematical programming 93B51 Design techniques (robust design, computer-aided design, etc.) 93D09 Robust stability 93B36 $$H^\infty$$-control