A Lagrange duality characterisation for stability under arbitrary switching in switched positive linear systems

Abstract

ABSTRACTThe present communication is concerned with uniform exponential stability, under arbitrary switching, in discrete-time switched positive linear systems. Lagrange duality is used in order to obtain a new characterisation for uniform exponential stability which is in terms of sets of inequalities involving each of the matrices that represent the modes of the system. These sets of inequalities are shown to generalise the classical linear Lyapunov inequality that characterises, in positive matrices, the property of being Schur. Each solution to these sets of inequalities is shown to provide a representation, in terms of a number of linear functionals, for a common Lyapunov function for the switched positive linear system. A result is further presented which conveys to, a conservative upper bound on the minimum required number of linear functionals (in the above mentioned representation), and also to a method for computing them. Our proof for the aforementioned characterisation is based on another (equivalent) characterisation, in terms of the solvability of a dynamic programming equation associated to the switched positive linear system, which is also reported in the paper. In particular, it is shown that the associated dynamic programming equation has at most one solution. And this solution is shown to be convex, monotonic, positively homogeneous, and it yields a common Lyapunov function for the switched positive linear system.