A note on topological feedback entropy and invariance entropy

Colonius, Fritz; Kawan, Christoph; Nair, Girish N.

doi:10.1016/j.sysconle.2013.01.008

Cited by 88 publications

(39 citation statements)

References 16 publications

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“…Since for every reasonable stabilization objective it is necessary to keep certain volumes bounded (or even shrink them to zero), the same ideas as used in the definition of invariance entropy should work universally for stabilization over discrete channels. This intuition was rigorously verified in a number of publications, including [7,9,11,14,22,23].…”

Section: Introductionmentioning

confidence: 70%

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Stochastic stability of nonlinear dynamical systems under information constraints

Kawan

Yüksel

2019

2019 IEEE Information Theory Workshop (ITW)

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We study the following problem: Given a stochastic nonlinear system controlled over a possibly noisy communication channel, what is the largest class of such channels for which there exist coding and control policies so that the closed-loop system is stochastically stable? The stability criterion considered is asymptotic mean stationarity (AMS). We first develop a general method (based on ergodic theory) to derive fundamental lower bounds on information transmission requirements leading to stabilization. Through this method we develop a new notion of entropy which is tailored to derive lower bounds for asymptotic mean stationarity for both noise-free and noisy channels. Our fundamental bounds are consistent, and more refined in comparison with the bounds obtained earlier via information theoretic methods. Moreover, our approach is more versatile in view of the models considered and allows for finer lower bounds when the AMS measure is known to admit further properties such as moment bounds.

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Section: Introductionmentioning

confidence: 70%

“…Colonius & Kawan in [9] introduced the notion of invariance entropy for continuous-time systems for the same stabilization objective. When adapted to the same (discrete-time) setting, the two notions are equivalent, as was shown in [11]. A comprehensive review of these concepts is provided in [22].…”

Section: A Brief Literature Reviewmentioning

confidence: 92%

Stochastic stability of nonlinear dynamical systems under information constraints

Kawan

Yüksel

2019

2019 IEEE Information Theory Workshop (ITW)

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“…More precisely, if there is a periodic trajectory in the interior of the given set K such that the linearization along this trajectory is controllable, and if complete approximate controllability holds on the interior of K (cf. Colonius and Kliemann 2000), then…”

Section: Upper Bounds Under Controllability Assumptionsmentioning

confidence: 95%

Data Rate of Nonlinear Control Systems and Feedback Entropy

Kawan

2014

Encyclopedia of Systems and Control

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Topological feedback entropy is a measure for the smallest information rate in a digital communication channel between the coder and the controller of a control system, above which the control task of rendering a subset of the state space invariant can be solved. It is defined purely in terms of the open-loop system without making reference to a particular coding and control scheme and can also be regarded as a measure for the inherent rate at which the system generates "invariance information."

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“…While the definition in [7] clearly resembles the definition of entropy for dynamical systems in [8] based on open covers, the invariance entropy introduced in [11] is close to the notion of entropy in [9,10] based on spanning sets. Both notions coincide for discrete-time control systems provided that a strong invariance condition holds [12,13].…”

Section: Introductionmentioning

confidence: 99%

Invariance Feedback Entropy of Nondeterministic Control Systems

Rungger

2017

Proceedings of the 20th International Conference on Hybrid Systems: Computation and Control

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We introduce a novel notion of invariance feedback entropy to quantify the state information that is required by any controller that enforces a given subset of the state space to be invariant. We establish a number of elementary properties, e.g. we provide conditions that ensure that the invariance feedback entropy is finite and show for the deterministic case that we recover the well-known notion of entropy for deterministic control systems. We prove the data rate theorem, which shows that the invariance entropy is a tight lower bound of the data rate of any coder-controller that achieves invariance in the closed loop. We analyze uncertain linear control systems and derive a universal lower bound of the invariance feedback entropy. The lower bound depends on the absolute value of the determinant of the system matrix and a ratio involving the volume of the invariant set and the set of uncertainties. Furthermore, we derive a lower bound of the data rate of any static, memoryless coder-controller. Both lower bounds are intimately related and for certain cases it is possible to bound the performance loss due to the restriction to static coder-controllers by 1 bit/time unit. We provide various examples throughout the paper to illustrate and discuss different definitions and results.

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A note on topological feedback entropy and invariance entropy

Cited by 88 publications

References 16 publications

Stochastic stability of nonlinear dynamical systems under information constraints

Stochastic stability of nonlinear dynamical systems under information constraints

Data Rate of Nonlinear Control Systems and Feedback Entropy

Invariance Feedback Entropy of Nondeterministic Control Systems

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