Robust LQRs Synthesis for Structured Uncertain Systems: the WDLF and the CI Approaches

Abstract

This paper addresses the design of robust linear quadratic regulators (LQRs) for systems affected polynomially by uncertainty constrained in a semialgebraic set. The problem consists of determining a feedback controller that ensures a desired upper bound on the worst-case value of a quadratic cost. Two linear matrix inequality (LMI) approaches are proposed, the first one based on the construction of a Lyapunov function that weakly depends on the uncertainty, and the second one based on the construction of an index that quantifies the feasibility of different controllers. The proposed approaches have two main advantages with respect to the existing methods, namely, considering not only state-feedback design for polytopic systems but also output-feedback design for systems depending polynomially on the uncertainty, and providing conditions that are not only sufficient but also necessary under some assumptions. These advantages are illustrated through various examples, where it is shown that the existing methods may be more conservative or may be not applicable.