Convergent upper bounds of peak response of LTI and polytopic LTV systems through LMIs

Abstract

This paper addresses the problem of determining the response peak of dynamical systems, a classical problem in control systems. The paper starts by considering the impulse response of linear time-invariant (LTI) systems. A linear matrix inequality (LMI) condition for establishing upper bounds of the sought peak is proposed by embedding the trajectory onto the level set of a polynomial and by introducing a projection technique for evaluating the extension of this set. This condition is sufficient for any degree of the polynomial, which has to be chosen a priori, and is also necessary whenever this degree is large enough. Hence, the proposed methodology is extended to address the impulse response of polytopic linear time-varying (LTV) systems, in particular, linear systems affected linearly by structured time-varying uncertainty constrained into a polytope. Lastly, generalizations to various responses and specializations to some structures are also presented. As shown by several examples, which include randomly generated systems and physical systems, the proposed conditions may provide significantly less conservative results than the existing LMI methods.