Metric and topological entropy bounds on state estimation for stochastic non-linear systems

This paper studies state estimation over noisy channels for stochastic non-linear systems. We consider three estimation objectives, a strong and a weak form of almost sure stability of the estimation error as well as quadratic stability in expectation. For all three objectives, we derive lower bounds on the smallest channel capacity C-0 above which the objective can be achieved with an arbitrarily small error. Lower bounds are obtained via a dynamical systems (through a novel construction of a dynamical system), an information-theoretic and a random dynamical systems approach. The first two approaches show that for a large class of systems, such as additive noise systems, C-0 = infinity, i.e., the estimation objectives cannot be achieved via channels of finite capacity. The random dynamical systems approach is shown to be operationally non-adequate for the problem, since it yields finite lower bounds C-0 under mild assumptions. Finally, we prove that a memoryless noisy channel in general constitutes no obstruction to asymptotic almost sure state estimation with arbitrarily small errors, when there is no noise in the system.