H∞ optimization with plant uncertainty and semidefinite programming

Abstract

SUMMARY The fundamental H= problem of control is that of finding the stable frequency response function that best fits worst case frequency domain specifications. This is a non-smooth optimization problem that underlies the frequency domain formulation of the H= problem of control; it is the main optimization problem in qualitative feedback theory for example. It is shown in this article how the fundamental H= optimization problem of control can be naturally treated with modern primal—dual interior point (PDIP) methods. The theory introduced here generalizes and unifies approaches to solving large classes of optimization problems involving matrix-valued functions, a subclass of which are commonly treated with linear matrix inequalities techniques. Also, in this article new optimality conditions for H= optimization problems over matrix-valued functions are proved, and numerical experience on natural (PDIP) algorithms for these problems is reported. In experiments we find the algorithms exhibit (local) quadratic convergence rate in many instances. Finally, H= optimization problems with an uncertainty parameter are considered. It is shown how to apply the theory developed here to obtain optimality conditions and derive algorithms. Numerical tests on simple examples are reported. ( 1998 John Wiley & Sons, Ltd.