Complex polytopic Lyapunov functions and componentwise ultimate bounds for switched linear systems: A missing link

This paper deals with switched linear systems with persistent disturbances and under arbitrary switching. For these systems, a systematic componentwise ultimate bound computation method has been previously developed. This method does not employ a Lyapunov function, yet yields a mixture of ellipsoidal and polyhedral sets, which are the type of level sets obtained via complex polytopic Lyapunov functions. In this context, our contribution is as follows: (a) we show that if the aforementioned componentwise method can be applied, then a complex polytopic Lyapunov function exists based on which the same ultimate bound is obtained; (b) we provide a novel algebraic condition for the existence of a complex polytopic Lyapunov function of minimum complexity; (c) we give an example for which a polytopic Lyapunov function exists but the componentwise method cannot be applied. These results serve to establish the precise connection between the two approaches.