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Stability and robustness of multivariable feedback systems. (English) Zbl 0552.93002
The MIT Press Series in Signal Processing, Optimization, and Control, 3. Cambridge, Massachusetts - London: The MIT Press. XI, 171 p. (1980).
This book, based on the author’s Ph. D. thesis, focusses on stability problems of nonlinear, multiloop feedback systems. It is a welcome contribution to an area currently receiving considerable attention by researchers, namely, the design of robust feedback systems that yield satisfactory performance in the face of modelling uncertainties and approximations (linearization, neglecting fast dynamics or weak coupling effects, insufficient knowledge of the model). The main theoretical results of the book lead to an effective methodology for the analysis of robustness and sensitivity of multivariable feedback systems. The book has five chapters and seven appendices for proofs. Chapter 1 gives the motivation for this research monograph and presents the main results as well as previous works on related subjects. Chapter 2 forms the theoretical foundation of the book. The central result is an abstract and powerful stability theorem. In essence, the theorem shows that a multiloop feedback system is closed-loop stable if there exists a topological separation (into two disjoint sets) of the function space on which the system’s dynamical input-output relations are defined. These relations may be specified, for example, in frequency domain by transfer function matrices or in time domain by possibly nonlinear state equations. In the former case, the topological separation condition leads to multiloop generalizations of circle and Popov frequency criteria; in the latter case, the Lyapunov stability theory emerges as a special case of the general problem. A methodology, based on this stability theory, is proposed for multiloop system robustness and stability margin analysis. Chapters 3 and 4 consider the implications of the new theory with regard to the stability margins of modern multivariable linear-quadratic- Gaussian (LQG) optimal estimators and controllers for continuous-time and discrete-time (or sampled-data) systems, respectively. The design- specific stability margins are characterized in terms of the system matrices, quadratic performance-index weighting matrices, and the optimal solution of the corresponding Riccati equation. These margins are characterized as a convex set of nonlinear dynamical deviations between the design model and the actual plant that can be tolerated without inducing instability. The established output-feedback separating-type property for linear-quadratic state-feedback controllers and constant gain Kalman filters leads to a powerful technique, based on the LQG optimal methodology, for the synthesis of robust dynamical output- feedback compensators for nonlinear systems. Chapter 5 summarizes some of author’s conclusions and discusses the potential applications of the theory developed in the book.
Reviewer: A.Varga

##### MSC:
 93-02 Research exposition (monographs, survey articles) pertaining to systems and control theory 93B35 Sensitivity (robustness) 93C35 Multivariable systems, multidimensional control systems 93C10 Nonlinear systems in control theory 93C55 Discrete-time control/observation systems 93C57 Sampled-data control/observation systems 93C99 Model systems in control theory 93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, $$L^p, l^p$$, etc.) in control theory 93D10 Popov-type stability of feedback systems 93D15 Stabilization of systems by feedback 93D25 Input-output approaches in control theory 93E11 Filtering in stochastic control theory 93E20 Optimal stochastic control