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Foundations of Value of Information: A Semantic Metric for Networked Control Systems Tasks

Touraj Soleymani, John S. Baras, Sandra Hirche, Karl H. Johansson

TL;DR

The paper defines the value of information (VoI) as a semantic metric for networked control, linking transmission decisions to regulation performance via a causal rate-regulation tradeoff between two distributed decision makers on a noiseless channel. VoI is computed as the difference in a value function with and without transmitting a sensory measurement, and it depends on the estimation mismatch between encoder and decoder; a Nash equilibrium is shown to exist and to be globally optimal, with a threshold policy on VoI and a certainty-equivalent control law. The authors establish that VoI is a symmetric function of the estimation mismatch, enabling a separable encoder/decoder design and, under equilibrium, linear decoder estimation and no dual effect; they also provide a tractable quadratic VoI approximation with guarantees and illustrate the approach on an inverted pendulum on a cart. The results offer a principled, semantics-driven method to manage information flow in networked control and distributed computation, surpassing traditional information-theoretic or fixed-trigger criteria and enabling efficient, performance-guaranteed control in cyber-physical systems.

Abstract

In this chapter, we present our recent invention, i.e., the notion of the value of information$\unicode{x2014}$a semantic metric that is fundamental for networked control systems tasks. We begin our analysis by formulating a causal tradeoff between the packet rate and the regulation cost, with an encoder and a decoder as two distributed decision makers, and show that the valuation of information is conceivable and quantifiable grounded on this tradeoff. More precisely, we characterize an equilibrium, and quantify the value of information there as the variation in a value function with respect to a piece of sensory measurement that can be communicated from the encoder to the decoder at each time. We prove that, in feedback control of a dynamical process over a noiseless channel, the value of information is a function of the discrepancy between the state estimates at the encoder and the decoder, and that a data packet containing a sensory measurement at each time should be exchanged only if the value of information at that time is nonnegative. Finally, we prove that the characterized equilibrium is in fact globally optimal.

Foundations of Value of Information: A Semantic Metric for Networked Control Systems Tasks

TL;DR

The paper defines the value of information (VoI) as a semantic metric for networked control, linking transmission decisions to regulation performance via a causal rate-regulation tradeoff between two distributed decision makers on a noiseless channel. VoI is computed as the difference in a value function with and without transmitting a sensory measurement, and it depends on the estimation mismatch between encoder and decoder; a Nash equilibrium is shown to exist and to be globally optimal, with a threshold policy on VoI and a certainty-equivalent control law. The authors establish that VoI is a symmetric function of the estimation mismatch, enabling a separable encoder/decoder design and, under equilibrium, linear decoder estimation and no dual effect; they also provide a tractable quadratic VoI approximation with guarantees and illustrate the approach on an inverted pendulum on a cart. The results offer a principled, semantics-driven method to manage information flow in networked control and distributed computation, surpassing traditional information-theoretic or fixed-trigger criteria and enabling efficient, performance-guaranteed control in cyber-physical systems.

Abstract

In this chapter, we present our recent invention, i.e., the notion of the value of informationa semantic metric that is fundamental for networked control systems tasks. We begin our analysis by formulating a causal tradeoff between the packet rate and the regulation cost, with an encoder and a decoder as two distributed decision makers, and show that the valuation of information is conceivable and quantifiable grounded on this tradeoff. More precisely, we characterize an equilibrium, and quantify the value of information there as the variation in a value function with respect to a piece of sensory measurement that can be communicated from the encoder to the decoder at each time. We prove that, in feedback control of a dynamical process over a noiseless channel, the value of information is a function of the discrepancy between the state estimates at the encoder and the decoder, and that a data packet containing a sensory measurement at each time should be exchanged only if the value of information at that time is nonnegative. Finally, we prove that the characterized equilibrium is in fact globally optimal.
Paper Structure (12 sections, 7 theorems, 20 equations, 2 figures)

This paper contains 12 sections, 7 theorems, 20 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Literature Survey
  3. Contributions and Organization
  4. A Tradeoff in the Noiseless Communication Regime
  5. Quantification of the Value of Information
  6. Existence of a Signaling Effect
  7. Existence of a Dual Effect
  8. Emergence of the Value of Information
  9. Global Optimality Analysis
  10. Approximation of the Value of Information
  11. Numerical Example
  12. Conclusion

Key Result

lemma 1

Let $S(k) \succeq 0$ be a matrix obeying the algebraic Riccati equation for $k \in \mathbb{N}_{[0,N]}$ with initial condition $S(N+1) = Q(N+1)$. Then, is equivalent to $\Phi$, where $\theta(k) = \ell(k) \lambda$, $\eta(k) = (u(k) + L(k) x(k) )^T (B(k)^T S(k+1) B(k) + R(k) ) (u(k) + L(k) x(k))$, and $L(k) = (B(k)^T S(k+1) B(k) + R(k))^{-1} B(k)^T$$S(k+1) A(k)$.

Figures (2)

  • Figure 1: The value of information, transmission decision, and control input trajectories. The value of information is scaled by one tenth. The solid lines represent the trajectories under the scheduling policy designed based on the value of information, and the dotted lines represent the trajectories under a periodic scheduling policy.
  • Figure 2: The position, velocity, pitch angle, and pitch rate trajectories. The solid lines represent the trajectories under the scheduling policy designed based on the value of information, and the dotted lines represent the trajectories under a periodic scheduling policy.

Theorems & Definitions (20)

  • remark 1
  • definition 1: Value of Information
  • remark 2
  • lemma 1: stoccontrol
  • definition 2: Value functions
  • lemma 2: stoccontrol
  • lemma 3: voi
  • remark 3
  • definition 3: Dual effect
  • lemma 4: ramesh2013
  • ...and 10 more