Entropy and Minimal Data Rates for State Estimation and Model Detection

Liberzon, Daniel; Mitra, Sayan

doi:10.1145/2883817.2883820

Cited by 29 publications

(59 citation statements)

References 16 publications

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“…This result can be viewed as an extension of the data-rate theorem with exponential convergence guarantees. It states that to have exponential convergence of the estimation error and the state, the access rate should be larger than the estimation entropy, the latter concept having been recently introduced in [51]. A similar result for continuous-time systems appears in [40], but only for linear feedback controllers.…”

Section: Necessary Condition On the Access Ratementioning

confidence: 97%

The Value of Timing Information in Event-Triggered Control

Khojasteh

Tallapragada

Cortés

et al. 2020

IEEE Trans. Automat. Contr.

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We study event-triggered control for stabilization of unstable linear plants over rate-limited communication channels subject to unknown, bounded delay. On one hand, the timing of event triggering carries implicit information about the state of the plant. On the other hand, the delay in the communication channel causes information loss, as it makes the state information available at the controller out of date. Combining these two effects, we show a phase transition behavior in the transmission rate required for stabilization using a given event-triggering strategy. For small values of the delay, the timing information carried by the triggering events is substantial, and the system can be stabilized with any positive rate. When the delay exceeds a critical threshold, the timing information alone is not enough to achieve stabilization, and the required rate grows. When the the delay equals the inverse of the entropy rate of the plant, the implicit information carried by the triggering events perfectly compensates the loss of information due to the communication delay, and we recover the rate requirement prescribed by the data-rate theorem. We also provide an explicit construction yielding a sufficient rate for stabilization, as well as results for vector systems. Our results do not rely on any a priori probabilistic model for the delay or the initial conditions. Index TermsData-rate theorem, event-triggered control, control under communication constraints, quantized control I. INTRODUCTIONCyber-physical systems (CPS) are engineering systems that integrate computing, communication, and control. They arise in a wide range of areas such as robotics, energy, civil infrastructure, manufacturing, and transportation [3], [4]. Due to the need for tight integration of different components, requirements and time scales, the modeling, analysis, and design of CPS present new challenges. One key aspect is the presence of finite-rate, digital communication channels in the feedback loop. Data-rate theorems quantify the effect that communication has on stabilization by stating that the communication rate available in the feedback loop should be at least as large as the intrinsic entropy rate of the system (corresponding to the sum of the logarithms of the unstable modes). In this way, the controller can compensate for the expansion of the state occurring during the communication process. Early formulations of data-rate theorems appeared in [5]-[7], followed by the key contributions in [8], [9]. More recent extensions include time-varying rate, Markovian, erasure, additive white and colored Gaussian, and multiplicative noise feedback communication channels [10]-[16], formulations for nonlinear systems [17]-[19], for optimal control [20]-[22], for systems with random parameters [23]-[26], and for switching systems [27]-[29]. Connections with information theory are highlighted in [19], [30]-[33]. Extended surveys of the literature appear in [34], [35] and in the book [36].Another key aspect of CPS to which we pay special attention he...

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Section: Necessary Condition On the Access Ratementioning

confidence: 97%

The Value of Timing Information in Event-Triggered Control

Khojasteh

Tallapragada

Cortés

et al. 2020

IEEE Trans. Automat. Contr.

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“…Some relevant studies that have considered non-linear systems are the following. The papers [21], [38] and [26] studied state estimation for non-linear deterministic systems and noise-free channels. In [21], Liberzon and Mitra characterized the critical data rate C 0 for exponential state estimation with a given exponent α ≥ 0 for a continuous-time system on a compact subset K of its state space.…”

Section: B Literature Reviewmentioning

confidence: 99%

On Optimal Coding of Non-Linear Dynamical Systems

Kawan

Yüksel

2018

IEEE Trans. Inform. Theory

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We consider the problem of zero-delay coding of a dynamical system over a discrete noiseless channel under three estimation criteria concerned with the low-distortion regime. For these three criteria, formulated stochastically in terms of a probability distribution for the initial state, we characterize the smallest channel capacities above which the estimation objectives can be achieved. The results establish further connections between topological and metric entropy of dynamical systems and information theory.

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“…In [18], [19], Liberzon and Mitra characterized the critical data rate C 0 for exponential state estimation with a given exponent α ≥ 0 for a continuous-time system on a compact subset K of its state space. As a measure for C 0 , they introduced a quantity called estimation entropy h est (α, K), which equals the topological entropy on K in case α = 0, but for α > 0 is no longer a purely topological quantity.…”

Section: A Literature Review and Contributionsmentioning

confidence: 99%

Metric and Topological Entropy Bounds for Optimal Coding of Stochastic Dynamical Systems

Kawan

Yüksel

2020

IEEE Trans. Automat. Contr.

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We consider the problem of optimal zero-delay coding and estimation of a stochastic dynamical system over a noisy communication channel under three estimation criteria concerned with the low-distortion regime. The criteria considered are (i) a strong and (ii) a weak form of almost sure stability of the estimation error as well as (ii) quadratic stability in expectation. For all three objectives, we derive lower bounds on the smallest channel capacity C0 above which the objective can be achieved with an arbitrarily small error. We first obtain bounds through a dynamical systems approach by constructing an infinite-dimensional dynamical system and relating the capacity with the topological and the metric entropy of this dynamical system. We also consider information-theoretic and probabilitytheoretic approaches to address the different criteria. Finally, we prove that a memoryless noisy channel in general constitutes no obstruction to asymptotic almost sure state estimation with arbitrarily small errors, when there is no noise in the system. The results provide new solution methods for the criteria introduced (e.g., standard information-theoretic bounds cannot be applied for some of the criteria) and establish further connections between dynamical systems, networked control, and information theory, and especially in the context of nonlinear stochastic systems.f w (x) = f (x, w), f x (w) = f (x, w).so that f w : X → X and f x : W → X.

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Entropy and Minimal Data Rates for State Estimation and Model Detection

Cited by 29 publications

References 16 publications

The Value of Timing Information in Event-Triggered Control

The Value of Timing Information in Event-Triggered Control

On Optimal Coding of Non-Linear Dynamical Systems

Metric and Topological Entropy Bounds for Optimal Coding of Stochastic Dynamical Systems

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