State Estimation of Open Dynamical Systems with Slow Inputs: Entropy, Bit Rates, and relation with Switched Systems

Finding the minimal bit rate needed to estimate the state of a dynamical system is a fundamental problem. Several notions of topological entropy have been proposed to solve this problem for closed and switched systems. In this paper, we extend these notions to open nonlinear dynamical systems with slowly-varying inputs to lower bound the bit rate needed to estimate their states. Our entropy definition represents the rate of exponential increase of the number of functions needed to approximate the trajectories of the system up to a specified $\eps$ error. We show that alternative entropy definitions using spanning or separating trajectories bound ours from both sides. On the other hand, we show that the existing definitions of entropy that consider supremum over all $\eps$ or require exponential convergence of estimation error, are not suitable for open systems. Since the actual value of entropy is generally hard to compute, we derive an upper bound instead and compute it for two examples. We show that as the bound on the input variation decreases, we recover a previously known bound on estimation entropy for closed nonlinear systems. For the sake of computing the bound, we present an algorithm that, given sampled and quantized measurements from a trajectory and an input signal up to a time bound $T>0$, constructs a function that approximates the trajectory up to an $\eps$ error. We show that this algorithm can also be used for state estimation if the input signal can indeed be sensed. Finally, we relate the computed bound with a previously known upper bound on the entropy for switched nonlinear systems. We show that a bound on the divergence between the different modes of a switched system is needed to get a meaningful bound on its entropy.