LMI-Based Determination of the Peak of the Response of Structured Polytopic Linear Systems

This paper addresses the problem of determining the peak of the response to a linear time-invariant (LTI) signal of a linear system whose system matrices are rational functions of an uncertainty vector constrained into a convex bounded polytope. The uncertainty can be time-invariant, bounded-rate time-varying or arbitrarily time-varying. A novel approach based on linear matrix inequalities (LMIs) is proposed for obtaining upper bounds of the sought peak based on the construction of a structured polynomial Lyapunov function in the state and in the uncertainty. A priori and a posteriori conditions for establishing optimality of the obtained upper bounds are also provided. As shown by some numerical examples, which includes the model of an electric circuit, the proposed approach may have significant advantages with respect to the existing methods in terms of conservatism or computational burden.