Robustness analysis of linear systems with time-varying parameters

Abstract

We study the robust stability problem for linear systems with uncertain time-varying parameters. This is done using a multiplier approach in conjunction with parameter-dependent Lyapunov functions. The main result of the paper is an average logarithmic variation rate for the uncertain parameters for robust stability. In the process of doing so, we derive an extended version of the KYP lemma, the parametric KYP lemma, as a general tool to study the robust stability with parameter uncertainty. Using this lemma, we provide conditions under which an affinely parameterized multiplier exists to establish the robust stability of the uncertain system. This type of parametric multiplier then naturally leads to a multi-affine Lyapunov function for robust stability analysis in the state space domain. The multiplier approach used in the paper is very general. In particular, many previous results in the literature on parametric Lyapunov functions and time-varying parameters lead to special multipliers. Another advantage of the proposed multiplier approach is that discrete-time systems can be treated in a similar way.