Estimation of Nonlinear Dynamic Systems Over Communication Channels

Remote observation of the state trajectory of nonlinear dynamic systems over limited capacity communication channels is studied. It is shown that two extreme cases are possible: Either the system is fully observable or the error in estimation blows up. The key observation is that such behavior is determined by the relationship between the Shannon capacity and the Lyapunov exponents; the well-known characterizing parameters of a communication channel on one side, and a dynamic system from the other side. In particular, it is proved that for nonlinear systems with initial state x(0), the minimum capacity of an additive white Gaussian noise channel required for full observation of the system in the mean square sense is Sigma(i) kappa(i) (x(0)) Delta(i) (x(0)), where Delta(i) (x(0)) s and k(i) (x(0)) s denote distinct Lyapunov exponents and their multiplicity numbers, respectively. Conversely, if the capacity is less than E[Sigma(i) kappa(i) (x(0)) Delta(i) (x(0))], then observation is impossible. In order to show the universality of the result, we obtain the same observability conditions for the digital noiseless channel and the packet erasure channel in sure and almost sure senses, respectively.