Mathematical descriptions of large-scale systems often include a set of parameters whose values cannot be predicted with any precision. Ensuring stability in the presence of such uncertainties is a fundamental theoretical problem, which has lead to the development of numerous robust control strategies. Given the wealth of existing results in this field, it is fair to say that even a superficial overview would clearly exceed the scope of this book. In view of that, we will focus our attention on two specific issues that are related to parametric stability – the existence of moving equilibria and the design of parametricallydependent control laws. In the following sections we will consider these problems in some detail, and will show how they can be resolved in the framework of linear matrix inequalities. For the sake of clarity, we will examine only smaller systems, with the understanding that this approach can be easily extended to large-scale problems along the lines proposed in Chaps. 2 and 3.
We begin with several brief observations regarding the existence and stability of equilibria in nonlinear dynamic systems. In the analysis of such systems, it is common practice to treat these two properties separately. Establishing the existence of an equilibrium usually involves the solution of a system of nonlinear algebraic equations, using standard numerical techniques such as Newton’s method (e.g., Ortega and Rheinboldt, 1970). Once the equilibrium of interest is computed, it is translated to the origin by an appropriate change of variables. Stability properties and feedback design are normally considered only after such a transformation has been performed. This two-step methodology has been widely applied to systems that contain parametric uncertainties, and virtually all control schemes developed along these lines implicitly assume that the equilibrium remains fixed at the origin for the entire range of parameter values (e.g., Šiljak, 1969, 1989; Ackermann, 1993; Barmish, 1994; Bhattacharya et al., 1995).
Linear Matrix Inequality Parametric Stability Reference Input Gain Schedule Schedule Variable
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