H-Infinity Control With Nearly Symmetric State Matrix

Abstract

In this paper, we give an upper bound on the deviation from H-infinity optimality of a class of controllers as a function of the deviation from symmetry in the state matrix. We further suggest a scalar measure of symmetry which is shown to be directly relevant for estimating nearness to optimality. In connection to this, we give a simple analytical solution to a class of Lyapunov equations for two dimensional state matrices. Finally, we demonstrate how a well-chosen symmetric part for nearly symmetric state matrices may lead not only to near-optimality, but also to controller sparsity, a desirable property for large-scale systems. In the special case that the state matrix is symmetric and Hurwitz, our main result simplifies to give an H-infinity optimal controller with several benefits, a result which has recently appeared in the literature. In this sense, the above is a significant generalization which considers a much wider class of systems, yet allows one to retain the benefits of symmetric state matrices, while offering means of quantifying the effect of this on the H-infinity norm.