Worst-case topological entropy and minimal data rate for state estimation of switched linear systems

In this paper, we study the problem of estimating the state of a switched linear system (SLS), when the observation of the system is subject to communication constraints. We introduce the concept of worst-case topological entropy of such systems, and we show that this quantity is equal to the minimal data rate (number of bits per second) required for the state estimation of the system under arbitrary switching. Furthermore, we provide a closed-form expression for the worst-case topological entropy of switched linear systems, showing that its evaluation reduces to the computation of the joint spectral radius (JSR) of some lifted switched linear system obtained from the original one by using tools from multilinear algebra, and thus can benefit from well-established algorithms for the stability analysis of switched linear systems. Finally, drawing on this expression, we describe a practical coder-decoder that estimates the state of the system and operates at a data rate arbitrarily close to the worst-case topological entropy.