Global Uniqueness Tests and Performance Bounds for \boldmath$H^\infty$ Optima

Abstract

Optimization of sup norm--type performance functions over the space of $H^\infty$ functions is central to the subject of $H^\infty$ design, that is, design where stability is the key constraint. Problems with large amounts of plant uncertainty are often highly nonconvex and therefore may have many solutions. In this paper, even for highly nonconvex problems, we give a test one can perform, once a local optimum f* has been computed, to see if it is a global optimum. The uniqueness phenomena we discovered uses $H^\infty$ properties heavily and are considerably stronger than what occurs in other types of general optimization. Also, even when f* may not be a global optimum we give a way to use it to bound the best performance possible. Uniqueness results are valuable for assuring an engineer that a local optimum obtained in a computer run is in fact a true global optimum. This can save a practitioner a lot of time and anguish in that it replaces the usual process of initializing an optimization run many times to see if it always goes to the same local optimum; and even after vast numbers of experiments never being sure. One of the least intuitive properties of SISO (single input, single output) control is that a (local) optimum for a carefully set up $H^\infty$ problem (cf. Theorem 9.4.1 in [J. W. Helton and O. Merino, Classical Control Using $H^\infty$ Methods: Theory, Optimization, and Design, SIAM, Philadelphia, 1998], [J. W. Helton and D. E. Marshall, Indiana Univ. Math. J., 39 (1990), pp. 157--184]) even with large amounts of plant uncertainty is unique. Such problems are quite nonconvex, so the fact is surprising. While the result is false in general for MIMO (multiple input, multiple output) control (cf. [J. W. Helton and O. Merino, Michigan Math. J., 41 (1994), pp. 285--287]), in this note we are describing MIMO situations where uniqueness holds. The setting in this paper is simultaneous (Pareto) optimization of several competing performances $\Gamma_1, \dots , \Gamma_\ell$ and we obtain uniqueness results for its solutions.