Topological Feedback Entropy and Nonlinear Stabilization

Abstract

It is well known in the field of dynamical systems that entropy can be defined rigorously for completely deterministic open-loop systems. However, such definitions have found limited application in engineering, unlike Shannon's statistical entropy. In this paper, it is shown that the problem of communication-limited stabilization is related to the concept of topological entropy, introduced by Adler et al. as a measure of the information rate of a continuous map on a compact topological space. Using similar open cover techniques, the notion of topological feedback entropy (TFE) is defined in this paper and proposed as a measure of the inherent rate at which a map on a noncompact topological space with inputs generates stability information. It is then proven that a topological dynamical plant can be stabilized into a compact set if and only if the data rate in the feedback loop exceeds the TFE of the plant on the set. By taking appropriate limits in a metric space, the concept of local TFE (LTFE) is defined at fixed points of the plant, and it is shown that the plant is locally uniformly asymptotically stabilizable to a fixed point if and only if the data rate exceeds the plant LTFE at the fixed point. For continuously differentiable plants inEuclidean space, real Jordan forms and volume partitioning arguments are then used to derive an expression for LTFE in terms of the unstable eigenvalues of the fixed point Jacobian.

Hyperbolic Chain Control Sets on Flag Manifolds

“…Only recently, with the increasing need of controlling networked systems with limited information flows between different components, entropy notions were established to capture the fundamental bounds for the information rates in a network above which a desired control goal can be achieved (cf. [5,15,17,23]). Hyperbolicity in the control literature is mainly understood either as an undesirable behavior that has to be counteracted by the controller to achieve stability or, in strong contrast, as a vehicle to produce desired trajectories without applying too much control (this is known by the name "control of chaos," cf.…”

Using a single bit to stabilise an n-dimensional quantised non-linear feedforward system

“…Moreover, it is shown in [9] that when R is large enough, any non-linear control system can be 'globally' asymptotically stabilised by the quantised state feedback if they can be stabilised by the true state feedback. The minimality of R required to stabilise a non-linear system is addressed in [10], where a notion of topological feedback entropy (TFE) is introduced. It is proven that a system can be stabilised 'locally' if and only if R exceeds the inherent TFE of that system [10].…”

Robust output feedback stabilization of nonlinear networked systems via a finite data-rate communication channel

“…Recently, there is a growing attention on stabilizing nonlinear systems with feedbacks via finite data-rate or limited capacity communication channels. For discrete-time nonlinear systems, the concept of feedback topological entropy was introduced and a necessary and sufficient data rate for the local stabilization was given in [8]. In [9], the concept of topological entropy was extended to uncertain dynamical systems and adopted to study the robust observability of uncertain nonlinear systems and solvability of the optimal control problem via limited capacity communication channels.…”