An information theoretic approach to the modal estimation of switching FIR linear systems

In this paper we tackle the problem of mode estimation in switching systems. From the theoretical point of view, our contribution is twofold: creating a framework that has a clear parallel with a communication paradigm and deriving an analysis of performance. In particular, our work is restricted to the class of systems that randomly switch among a finite alphabet of discrete-time finite impulse response linear operators, therein designated as modes. In our approach, the switching system is viewed as an encoder of the mode, which is interpreted as the message, while a probing signal establishes a random code. Accordingly, the estimator, which knows the code and uses noisy measurements of the output, is constructed as a decoder whose properties can be studied by means of a modification of Shannon's theory. Using a distance function, we define an uncertainty ball where the estimates are guaranteed to lie with probability arbitrarily close to 1. The radius of the uncertainty ball is directly related to the entropy rate of the switching process. It is shown that lower rates lead to smaller uncertainty. Such distance also reflects the informativity of the probing signal (code) and as such can be used as a guide on its choice. The estimator/decoder can be implemented using a low complexity algorithm.