An Overview
This is a book intended for readers, familiar with the fundamentals of linear system and control theory, who are interested in learning methods for the design of feedback laws for (linear and nonlinear) multivariable systems, in the presence of model unce
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An Overview
1.1 Introduction This is a book intended for readers, familiar with the fundamentals of linear system and control theory, who are interested in learning methods for the design of feedback laws for (linear and nonlinear) multivariable systems, in the presence of model uncertainties. One of the main purposes of the book is to offer a parallel presentation for linear and nonlinear systems. Linear systems are dealt with in Chaps. 2–5, while nonlinear systems are dealt with in Chaps. 6–12. Among the various design options in the problem of handling model uncertainties, the focus of the book is on methods that appeal—in one form or in another—to the so-called “Small-Gain Theorem.” In this respect, it should be stressed that, while some of such methods may require, for their practical implementation, a “high-gain feedback” on selected measured variables, their effectiveness is proven anyway with the aid of the Small-Gain Theorem. Methods of this kind lend themselves to a presentation that is pretty similar for linear and nonlinear systems and this is the viewpoint adopted in the book. While the target of the book are multi-input multi-output (MIMO) systems, for pedagogical reasons in some cases (notably for nonlinear systems) the case of single-input single-output (SISO) systems is handled first in detail. Two major design problems are addressed (both in the presence of model uncertainties): asymptotic stabilization “in the large” (that is, with a “guaranteed region of attraction”) of a given equilibrium point and asymptotic rejection of the effect of exogenous (disturbance and/or commands) inputs on selected regulated outputs. This second problem, via the design of an “internal model” of the exogenous inputs, is reduced to the problem of asymptotic stabilization of a closed invariant set. The book presupposes, from the reader, some familiarity with basic concepts in linear system theory (such as reachability, observability, minimality), elementary control theory (transfer functions, feedback loop analysis), and with the basic concepts associated with the notion of stability of an equilibrium in a nonlinear system. Two Appendices collect some relevant background material in this respect, to the © Springer International Publishing Switzerland 2017 A. Isidori, Lectures in Feedback Design for Multivariable Systems, Advanced Textbooks in Control and Signal Processing, DOI 10.1007/978-3-319-42031-8_1
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1 An Overview
purpose of fixing the appropriate notations and making the presentation—to some extent—self-contained.
1.2 Robust Stabilization of Linear Systems Chapters 2 and 3 of this book deal with the design of control laws by means of which a linear system x˙ = Ax + Bu (1.1) y = Cx can be stabilized in spite of model uncertainties. In this respect, it should be stressed first of all that, while a simple method for stabilizing a system of this form consists in the use of the “observer-based” controller1 x˙ˆ = (A + BF − GC)ˆx + Gy u = F xˆ with F and G such that the matrices A + BF and, respectively, A − G
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