Computation of piecewise quadratic Lyapunov functions for hybrid systems

Abstract

Several stability conditions for hybrid systems are formulated as LMI problems. This includes the computations of continuous piecewise quadratic Lyapunov functions and multiple Lyapunov functions. The proposed methods are related to classical frequency domain methods, such as the Circle and Popov criteria. Finally, it is pointed out how controller design for hybrid systems can be formulated as an LMI problem. Several examples that highlight the proposed methods are included.

Stability of piecewise affine systems with state-dependent delay, and application to congestion control

“…Following the approach introduced in [12] and [8], and further developped in more recent studies [21], [13], or [18], the stability analysis in this work is based on a Lyapunov function with a piecewise quadratic part.…”

“…2) Introduction of some notions and notations: First, we introduce some elements that will be used to relax the LMI stability conditions, as in the framework of [12].…”

“…with the notations in (12), and π(t) = col{x(t), x(t − τ (t)), 1, x(t −h 1 ), x(t −h 2 )}. Further using the Schur complement ( [4], Section 2.1) and the S-procedure ( [4], Section 2.6.3), we can show that LMI (9) guarantees thaṫ V (t) + 2αV (t) ≤ 0 for all Gξ(t) ∈ X i and i ∈ I.…”

“…(2) This representation motivates us to analyse the stability of congestion control with tools adapted from works about piecewise affine systems such as [12], [8], [13], or [18]. However, the state dependency of the delay, as well as the delayed-state dependency of the switching law constitute new challenges.…”

Controle de sistemas lineares com comutação

“…O artigo (Hespanha, 2005) utiliza extensões do Princípio da Invariância de LaSalle e fornece uma discussão interessante sobre resultados referentes à estabilidade uniforme de sistemas com comutação. Sistemas com comutação a tempo discreto são tratados em (Geromel e Colaneri, 2006b) e resultados sobre a construção de funções de Lyapunov estão disponíveis em (Daafouz e Bernussou, 2001), (Johansson e Rantzer, 1998).…”