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

HERNÁN HAIMOVICH; ALEXIS J. VALLARELLA; MARÍA M. SERON

Córdoba

Workshop; 2015 XVI Workshop on Information Processing and Control (RPIC); 2015

This paper deals with switched linear systems withpersistent disturbances and under arbitrary switching. For thesesystems, a systematic componentwise ultimate bound computa-tion method has been previously developed. This method does notemploy a Lyapunov function, yet yields a mixture of ellipsoidaland polyhedral sets, which are the type of level sets obtainedvia complex polytopic Lyapunov functions. In this context, ourcontribution is as follows: (a) we show that if the aforementionedcomponentwise method can be applied, then a complex polytopicLyapunov function exists based on which the same ultimatebound is obtained; (b) we provide a novel algebraic conditionfor the existence of a complex polytopic Lyapunov functionof minimum complexity; (c) we give an example for whicha polytopic Lyapunov function exists but the componentwisemethod cannot be applied. These results serve to establish theprecise connection between the two approaches.