Two extensions of topological feedback entropy

Abstract

Topological feedback entropy (TFE) measures the intrinsic rate at which a continuous, fully observed, deterministic control system generates information for controlled set-invariance. In this paper we generalise this notion in two directions; one is to continuous, partially observed systems, and the other is to discontinuous, fully observed systems. In each case we show that the corresponding generalised TFE coincides with the smallest feedback bit rate that allows a form of controlled invariance to be achieved.Keywords Topological entropy · communication-limited control · quantised systems 1 Introduction In 1965, Adler, Konheim, and McAndrew [1] introduced topological entropy as a measure of the fastest rate at which a continuous, discrete-time, dynamical system in a compact space generates initial-state information. Though related to the measuretheoretic notion of Kolmogorov-Sinai entropy (see e.g. [13]), it is a purely deterministic notion and requires only a topology on the state space, not an invariant measure. Subsequently, Bowen [3] and Dinaburg [6] proposed an alternative, metric based definition of topological entropy. This accommodates uniformly continuous dynamics on noncompact spaces and is equivalent to the original definition on compact spaces.These concepts play an important role in dynamical systems but remained largely neglected in control theory. However, the emergence of digitally networked control systems (see e.g. [2]) over the last four decades renewed interest in the information theory of feedback, and in 2004 the techniques of Adler et al. were adapted to introduce the notion of topological feedback entropy (TFE) [11]. Unlike topological

Data Rate of Nonlinear Control Systems and Feedback Entropy

“…In particular, the data rate theorem for the LTFE asserts that the infimal bit rate for local uniform asymptotic stabilization at an equilibrium is given by the LTFE. Proofs of different data rate theorems can be found in Nair et al (2004), Hagihara and Nair (2013), and Kawan (2013).…”

Observation of nonlinear systems via finite capacity channels, Part II: Restoration entropy and its estimates

“…6 (and Rem. 9), q(T, ε) ≤ M 2 cT ∀T ≥ T * and so H(f, K) ≤ c by (14). It remains to let c → R o +, which means that c approaches R o from above.…”

“…As is now known, the issue of the communication bit-rate minimally required for solvability of various control and observation tasks is conceptually and computationally related to the classic concept of the topological entropy (TE) [10,18] and its recent control-oriented analogs [4][5][6][7]14,20,30,36,46,49]. Meanwhile, computation or even estimation of the TE of nonlinear systems has earned the reputation of an intricate matter; this intricacy briskly grows up with the system's dimension.…”

Comprehending Complexity: Data-Rate Constraints in Large-Scale Networks

“…The above model admits "master" systems without an input x i pt 1q φ i rx i ptqs, y i ptq h i rx i ptqs, (11) which influence the peers, being unaffected by them, as well as "slave" systems without an output x i pt 1q φ i rx i ptq, u i ptqs, (12) which are influenced by the peers with no backward effect on them. To embed these cases into (9), it suffices to endow the "master" (11) with a "void" input u i ptq R with no effect on the dynamics of Σ i and put V ij 0 dj to set u i ptq 0 for the sake of definiteness.…”