Convergent LMI relaxations for robust analysis of uncertain linear systems using lifted polynomial parameter-dependent Lyapunov functions

This paper investigates the problems of checking robust stability and evaluating robust ℋ2ℋ2 performance of uncertain continuous-time linear systems with time-invariant parameters lying in polytopic domains. The novelty is the ability to check robust stability by constructing a particular parameter-dependent Lyapunov function which is a polynomial function of the uncertain system matrices, as opposed to a general polynomial function of the uncertain parameter. The degree of the polynomial is tied to a certain integer κκ. The existence of such Lyapunov function can be proved by solving parameter-dependent Linear Matrix Inequalities (LMIs), which are guaranteed to be solvable for a sufficiently large yet finite value of κκ whenever the system is robustly stable. Extensions to guaranteed ℋ2ℋ2 cost computation are also provided. Numerical aspects concerning the programming and the evaluations of the proposed tests are discussed and illustrated by examples.