On the necessity of looped-functionals arising in the analysis of pseudo-periodic, sampled-data and hybrid systems

Abstract

Looped-functionals have been shown to be relevant for the analysis of a wide variety of systems. However, the conditions obtained in previous papers on the analysis of sampled-data, impulsive and switched systems have only been shown to be sufficient for the characterisation of their associated discrete-time stability conditions. We prove here that these conditions are also necessary. This result is derived for a wider class of linear systems, referred to as impulsive pseudo-periodic systems, that encompass periodic, impulsive, sampled-data and switched systems as special cases.

Keywords:

• looped-functionals
• periodic systems
• sampled-data systems
• hybrid systems

Notes

1. Note that this sequence does not admit any accumulation point and is unbounded from above.

2. The state is assumed to be right-continuous and the left-limit at t_k is denoted by x(t⁻_k) = lim_s↑0x(t_k + s).