Suboptimal l_∞ → l_∞ Control of Switched Linear Models: a Superstability Approach

In this paper we consider the problem of designing an observer-based controller for switched linear systems under arbitrary switching, with guaranteed l_∞ → l_∞ performance. It is shown that, by using the concepts of polyhedral Lyapunov functions and superstability, the problem can be recast into a nonconvex constrained Polynomial Optimization Problem (POP). In turn, this problem can be relaxed into a sequence of convex Semi-Definite Programs (SDPs) by resorting to moments-based polynomial optimization techniques. Further, by exploiting the sparsity pattern exhibited by the problem, we show that the original POP can be equivalently reformulated as an SDP problem with a rank-1 constraint. An efficient algorithm is proposed to solve the later, by seeking low rank solution through re-weighted nuclear norm minimization.