A Perspective on the Use of Semi-Infinite Optimization in Control System Design

Abstract

Parametric optimization is a powerful tool for the selection of favorable values for design variables, which is being used more and more extensively in various engineering disciplines. In linear control system design, unconstrained parametric optimization has been used for at least twenty years, or more precisely, at least since the introduction of the linear-quadratic regulator problem. However, the fact that parametric optimization was being used was obscured by the derivation of the solution to the linear quadratic regulator problem by means of optimal control theoretic tools, rather than mathematical programming theoretic tools. Nevertheless, it is true that the optimal gain matrix for the linear-quadratic regulator problem is a solution to an unconstrained parametric optimization problem of the form min f(K) (1.1) KR <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> ×R <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> where f (K) is the largest eigenvalue of a symmetric matrix of the form ∫ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sup> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> exp[t(A+BK] <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sup> (Q+K <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sup> RK)exp[t(A+BK]dt (1.2) with Q symmetric and positive semi-definite and R symmetric and positive definite. In keeping with the state of the art in constrained optimization of the sixties (see e.g., [Ath.1, Kwa.1]), the cost function f(K), in the linear quadratic regulator problem, expresses a penalty function approach to the satisfaction of performance requirments. The use of penalty functions in the linear quadratic regulator setting severely limits the designer's ability to satisfy complex time and frequency domain performance requirements. Over the last decade a much more powerful approach has become possible with the development of semi-infinite optimization algorithms for engineering design (see e.g., [Gon.1, Pol.1, Pol.2, Pol.3]).