Mean stability of switched linear systems

Abstract

A switched linear system is a dynamical system that consists of multiple linear time-invariant systems, called subsystems, and a piecewise constant function, called a switching signal, that orchestrates the switching between the subsystems. For the last two decades switched linear systems have attracted a significant amount of attention due to their wide range of applications.

Stability, which is the first property that every controlled system must have, has been one of the central topics in the study of switched linear systems. In particular mean stability, that roughly speaking requires stability in average, is known to be a rather practical and easy to check stability notion. There are now many conditions available to check the mean stability of linear switched systems with various different stochastic structures. However, there are still some important classes of switched linear systems whose mean stability cannot be checked by available methods.

This thesis gives the characterization of the mean stability of discrete-time switched linear systems with identically and independently distributed system parameters and continuous-time (semi )Markov jump positive linear systems. These conditions can be checked by the eigenvalues of a matrix and also extend the stability conditions obtained in the systems and control theory literature. We also study the mean escape time and the asymptotic behavior of switched Riccati differential equations naturally induced by switched linear systems.