An LMI characterization of polynomial parameter-dependent Lyapunov functions for robust stability

This paper investigates the robust stability of continuous-time, time-invariant linear uncertain systems in polytopic domains. Robust stability is checked by constructing a quadratic parameter-dependent Lyapunov function. The matrix associated with this quadratic Lyapunov function is a polynomial function of the uncertain parameters, expressed as a particular polynomial matrix involving powers of the dynamic matrix of the system and one symmetric matrix to be determined. The degree of this polynomial matrix function is arbitrary. Finsler’s Lemma is used to lift the obtained stability conditions into a larger space in which sufficient stability tests can be developed in the form of Linear Matrix Inequalities (LMIs), which must be verified at the vertices of the uncertainty polytopic domain. Examples illustrate the method, which is compared with similar results in the literature by means of random numerical experiments.