A new condition and equivalence results for robust stability analysis of rationally time-varying uncertain linear systems

Uncertain systems is a fundamental area of automatic control. This paper addresses robust stability of uncertain linear systems with rational dependence on unknown time-varying parameters constrained in a polytope. For this problem, a new sufficient condition based on the search for a common homogeneous polynomial Lyapunov function is proposed through a particular representation of parameter-dependent polynomials and LMIs. Relationships with existing conditions based on the same class of Lyapunov functions are hence investigated, showing that the proposed condition is either equivalent to or less conservative than existing ones. As a matter of fact, the proposed condition turns out to be also necessary for a class of systems. Some numerical examples illustrate the use of the proposed condition and its benefits.