Parameter-dependent Lyapunov function for exact stability analysis of single-parameter dependent LTI systems

In this paper, we propose a class of parameter-dependent Lyapunov functions which can be used to assess the stability properties of linear, time-invariant, single-parameter dependent (LTIPD) systems in a non-conservative manner. It is shown that stability of LTIPD systems is equivalent to the existence of a Lyapunov function of a polynomial type (in terms of the parameter) of known, bounded degree satisfying two matrix inequalities. It is also shown that checking the feasibility of these matrix inequalities over a compact set can be cast as a convex optimization problem.