A convex condition for robust stability analysis via polyhedral Lyapunov functions

In this paper we study the robustness analysis problem for linear continuous-time systems subject to parametric time-varying uncertainties making use of piecewise linear (polyhedral) Lyapunov functions. A given class of Lyapunov functions is said to be "universal" for the uncertain system under consideration if the robust stability of the system is equivalent to the existence of a Lyapunov function belonging to the class. In the literature it has been shown that the class of polyhedral functions is universal, while, for instance, the class of quadratic functions is not. This fact justifies the effort of developing efficient algorithms for the construction of polyhedral Lyapunov functions. In this context, we provide a low computational cost procedure, based on a novel convex condition, for the construction of a polyhedral Lyapunov function. In the section on the numerical examples, we consider some benchmark problems for the robust stability analysis and we show that the proposed low computational cost approach, though only sufficient, is less conservative than all the other approaches presented so far in the literature.