Fundamental performance limitations via entropy estimates with hidden Markov models

This paper is concerned with entropy based fundamental limitation results for the disturbance rejection problem in a scalar dynamical system. We propose a hidden Markov model (HMM) framework for entropy calculations. Using methods from ergodic theory, we obtain explicit entropy bounds for both the classical Bode problem (with linear dynamics) as well as certain cases of nonlinear dynamics. An important message of this paper is that the limitations are due to fundamental issues pertaining to estimation as opposed to the stabilization control problem.