Dwell time analysis of deterministic and stochastic switched systems

Abstract

This paper collects a number of recent results on stability and L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> gain of switched linear systems (both deterministic and stochastic) under a dwell time constraint. The switching signal orchestrates the commutations between linear systems (in the deterministic case) or Markov jump linear systems (in the stochastic case). In the latter case, the switching affects both the dynamics of the underlying systems and the associated transition probability matrices. The main focus is on the computation of the minimum dwell time ensuring stability and an upper bound of the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> gain, in the deterministic case, or stochastic stability (both in the mean square sense and almost sure sense), in the stochastic case. Being the minimum dwell time very hardly computable, viable procedures are proposed for a computation of an upper bound through Kronecker calculus, standard H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> theory and coupled Lyapunov inequalities.