SDP-based analysis and control of the response peak of discrete-time uncertain systems via head/tail partition

This paper addresses the problems of analyzing and controlling the response peak of a discrete-time uncertain linear system. A novel approach is proposed for the analysis problem based on the solution of a semidefinite program (SDP). The novel idea consists of splitting the response into two temporal parts, head and tail, which are analyzed via inspection of the response formula and via quadratic Lyapunov functions polynomially parameterized by the uncertainty. This approach is extended to deal with the control problem by proposing two strategies, one based on a nonconvex search obtained by alternately freezing some decision variables and the other one based on a convex search obtained by constructing a controller-dependent index. The proposed approach has key advantages with respect to the existing methods, which may be conservative, or inapplicable, or computationally prohibitive.