Robustness of critical bit rates for practical stabilization of networked control systems

2019 IEEE Information Theory Workshop (ITW)

Yüksel

2019

Self Cite

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.

Section: The Noisy Channel Casementioning

confidence: 91%

Stochastic stability of nonlinear dynamical systems under information constraints

2019 IEEE Information Theory Workshop (ITW)

Yüksel

2019

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“…From Lemma 3.3 we know that each v ε is continuous. We would like to interchange the order of the infimum and the limsup in (14). This would be possible if v ε was a subadditive cocycle over θ 1 : U → U.…”

Section: Corollarymentioning

confidence: 99%

Invariance entropy for a class of partially hyperbolic sets

Math. Control Signals Syst.

Silva

2018

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Invariance entropy is a measure for the smallest data rate in a noiseless digital channel above which a controller that only receives state information through this channel is able to render a given subset of the state space invariant. In this paper, we derive a lower bound on the invariance entropy for a class of partially hyperbolic sets. More precisely, we assume that Q is a compact controlled invariant set of a control-affine system whose extended tangent bundle decomposes into two invariant subbundles E + and E 0− with uniform expansion on E + and weak contraction on E 0− . Under the additional assumptions that Q is isolated and that the u-fibers of Q vary lower semicontinuously with the control u, we derive a lower bound on the invariance entropy of Q in terms of relative topological pressure with respect to the unstable determinant. Under the assumption that this bound is tight, our result provides a first quantitative explanation for the fact that the invariance entropy does not only depend on the dynamical complexity on the set of interest.the infimum taken over all invariant probability measures of the control flow that project to P and are supported on L(Q), we are able to recover the lower bound in (1) as a special case. Here we use that for a uniformly hyperbolic set without center bundle, the u-fibers of Q are finite (see [23] for a proof), implying that the measure-theoretic entropy of the bundle RDS ϕ |L(Q) vanishes.As in the uniformly hyperbolic case, examples to which our result can be applied, are provided by invariant systems on flag manifolds on semisimple Lie groups. In the paper at hand, we only discuss a special case, namely systems on projective space induced by bilinear systems on R d , while the more general case is studied in [15].The paper is organized as follows. In Section 2, we precisely formulate our main result, recalling all the concepts involved. The proof is carried out in Section 3 through a series of lemmas and propositions. Section 4 provides a class of examples and Section 5 discusses the contents of the result and relates it to other problems, e.g., submanifold stabilization. Finally, some technical results of independent interest, used in the proof, are collected in Section 6 (Appendix).Notation. We write N = {1, 2, 3, . . .} for the set of positive integers, Z for the set of all integers and R for the set of real numbers. We also write K + := {x ∈ K : x ≥ 0} for K ∈ {Z, R}. For x > 0, log x denotes the natural logarithm of x. If V, W are vector spaces, we write Hom(V, W ) and End(V ) for the spaces of homomorphisms from V to W and endomorphisms on V , respectively.If M is a smooth manifold, we write T x M for the tangent space to M at x ∈ M . If f : M → N is differentiable, Df (x) : T x M → T f (x) N denotes the derivative at x ∈ M . If (M, g) is a Riemannian manifold, we write | · | for the norm in each tangent space T x M , d(·, ·) for the geodesic distance and vol(·) for the Riemannian volume measure on M , respectively. Moreover, exp x denotes the Riemannian exponen...

“…The concept of uniform hyperbolicity for control systems was studied in [7,10,11,13,21]. In [7], controllability and robustness results were proved for chain control sets with a uniformly hyperbolic structure.…”

Section: Introductionmentioning

confidence: 99%

Lyapunov exponents and partial hyperbolicity of chain control sets on flag manifolds

Silva

2019

Isr. J. Math.

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