Polytope Lyapunov functions for stable and for stabilizable LSS

We present a new approach for constructing polytope Lyapunov functions for continuous-time linear switching systems (LSS). This allows us to decide the stability of LSS and to compute the Lyapunov exponent with a good precision in relatively high dimensions. The same technique is also extended for stabilizability of positive systems by evaluating a polytope concave Lyapunov function ("antinorm") in the cone. The method is based on a suitable discretization of the underlying continuous system and provides both a lower and an upper bound for the Lyapunov exponent. The absolute error in the Lyapunov exponent computation is estimated from above and proved to be linear in the dwell time. The practical efficiency of the new method is demonstrated in several examples and in the list of numerical experiments with randomly generated matrices of dimensions up to $10$ (for general linear systems) and up to $100$ (for positive systems). The development of the method is based on several theoretical results proved in the paper: the existence of monotone invariant norms and antinorms for positively irreducible systems, the equivalence of all contractive norms for stable systems and the linear convergence theorem

Polyhedral Lyapunov Functions with Fixed Complexity

Polyhedral Lyapunov functions can approximate any norm arbitrarily well. Because of this, they are used to study the stability of linear time varying and linear parameter varying systems without being conservative. However, the computational cost associated with using them grows unbounded as the size of their representation increases. Finding them is also a hard computational problem. Here we present an algorithm that attempts to find polyhedral functions while keeping the size of the representation fixed, to limit computational costs. We do this by measuring the gap from contraction for a given polyhedral set. The solution is then used to find perturbations on the polyhedral set that reduce the contraction gap. The process is repeated until a valid polyhedral Lyapunov function is obtained. The approach is rooted in linear programming. This leads to a flexible method capable of handling additional linear constraints and objectives, and enables the use of the algorithm for control synthesis