Stabilization of planar non-Markovian switched linear systems with unbounded random delays

Abstract

This paper addresses the problem of designing a control law to stabilize planar (i.e. second order) linear switched autonomous systems when a delay affects the switching instants. The stability of the delayed system is investigated under the assumption that the corresponding deterministic system is stabilized by a switching table. This table is obtained by minimizing a performance metric that is a functional of the norm of the state. It is formally proved that for not necessarily bounded random delays, a linear switched autonomous non-Markovian system is still stable for a sufficiently small value of the multiplier parameter connected to the random delay. We also determine numerically the multiplier parameter value which discriminates stable and unstable behaviors for uniform and normal delay distribution and suggest a method for construction of the switching table based on minimizing the convergence rate.