A constrained H∞ smooth optimization technique

Abstract In H∞ optimal control the cost function is the maximum singular value of a transfer function matrix over a frequency range. The optimization is over all stabilizing controllers. In constrained H∞ control the controllers typically have a fixed structure, perhaps conveniently parametrized in terms of a parameter vector. Also, there may be functional constraints involving singular values representing, for example, robustness requirements. Such problems are usually cast as non‐smooth optimization problems. In this paper we consider a general class of constrained H∞ optimization problems and show that these problems can be approximated by a sequence of smooth optimization problems. Thus each of the approximate problems is readily solvable by standard optimization software packages such as those available in the NAG or IMSL library. The proposed approach via smooth optimization is simple in terms of mathematical content, easy to implement and computationally efficient.