Feedback data rates for nonlinear systems

This paper poses a simple question: what is the lowest rate, in bits per unit time, at which feedback information can be transmitted in order to stabilise a given dynamical system? Expressions for this fundamental quantity have recently been derived for linear systems, with and without noise. In this work, the case of deterministic, fully observed, continuously differentiable dynamical systems is investigated, under the additional assumptions of controllability to the desired set-point and bounded initial states. By the use of volume-partitioning arguments and local Jordan forms, the infimum feedback data rate is shown to be the base-2 logarithm of the magnitude of the determinant of the open-loop Jacobian on the local unstable subspace, evaluated at the set-point. Connections to the concept of local topological feedback entropy are briefly discussed.