Performance Optimization of Switched Systems: A Model Matching Approach

In this paper we consider a prototypical model matching problem where the various mappings involved are systems that switch arbitrarily among $n$ stable linear time-invariant (LTI) systems. The interest is placed on optimizing the worst-case performance of this model matching system over all possible switchings with either $\ell_{\infty}$ -induced norm or ${\cal H}_{2}$ norm as the performance criterion. This optimization is performed over all Youla-Kucera parameters that switch causally in time among $n$ stable LTI systems. For the particular setup at hand, it is shown that the optimal Youla-Kucera parameter need not depend on the switching trajectory in the cases of partially matched switching and unmatched switching, and that it can be obtained as an LTI solution to an associated standard $\ell_{1}$ or ${\cal H}_{2}$ optimization. In the case of matched switching, two convergent sequences to the optimal solution from above and below are formulated in terms of linear programs and quadratic programs respectively for the $\ell_{\infty}$ -induced and ${\cal H}_{2}$ norm optimizations. An approximate solution with any given precision is possible by finite truncation. Applications of these results to sensitivity minimization, linear switched parameter systems, and cooperative control are provided.