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Recursive Quantization for $\mathcal{L}_2$ Stabilization of a Finite Capacity Stochastic Control Loop with Intermittent State Observations

Shrija Karmakar, Ritwik Kumar Layek

TL;DR

This work addresses the challenge of stabilizing an unstable discrete-time stochastic plant when the state is observed through a finite-capacity channel subject to intermittent observations. It develops rigorous necessary and sufficient conditions on intermittence parameters for L2 stability under Bernoulli and Markov switching, for both scalar and vector plants, and introduces recursive quantization schemes that adapt to channel capacity to achieve stabilization. The results cover infinite capacity as well as finite capacity scenarios, providing explicit thresholds and per-dimension quantizer update rules, and are complemented by constructive algorithms and illustrative examples. The findings have practical implications for networked control where bandwidth and reliability constraints interact with control performance, and they point to open problems such as ensuring spectral-norm based stability for vector systems and extending to output feedback and robust settings.

Abstract

The problem of $\mathcal{L}_2$ stabilization of a state feedback stochastic control loop is investigated under different constraints. The discrete time linear time invariant (LTI) open loop plant is chosen to be unstable. The additive white Gaussian noise is assumed to be stationary. The link between the plant and the controller is assumed to be a finite capacity stationary channel, which puts a constraint on the bit rate of the transmission. Moreover, the state of the plant is observed only intermittently keeping the loop open some of the time. In this manuscript both scalar and vector plants under Bernoulli and Markov intermittence models are investigated. Novel bounds on intermittence parameters are obtained to ensure $\mathcal{L}_2$ stability. Moreover, novel recursive quantization algorithms are developed to implement the stabilization scheme under all the constraints. Suitable illustrative examples are provided to elucidate the main results.

Recursive Quantization for $\mathcal{L}_2$ Stabilization of a Finite Capacity Stochastic Control Loop with Intermittent State Observations

TL;DR

This work addresses the challenge of stabilizing an unstable discrete-time stochastic plant when the state is observed through a finite-capacity channel subject to intermittent observations. It develops rigorous necessary and sufficient conditions on intermittence parameters for L2 stability under Bernoulli and Markov switching, for both scalar and vector plants, and introduces recursive quantization schemes that adapt to channel capacity to achieve stabilization. The results cover infinite capacity as well as finite capacity scenarios, providing explicit thresholds and per-dimension quantizer update rules, and are complemented by constructive algorithms and illustrative examples. The findings have practical implications for networked control where bandwidth and reliability constraints interact with control performance, and they point to open problems such as ensuring spectral-norm based stability for vector systems and extending to output feedback and robust settings.

Abstract

The problem of stabilization of a state feedback stochastic control loop is investigated under different constraints. The discrete time linear time invariant (LTI) open loop plant is chosen to be unstable. The additive white Gaussian noise is assumed to be stationary. The link between the plant and the controller is assumed to be a finite capacity stationary channel, which puts a constraint on the bit rate of the transmission. Moreover, the state of the plant is observed only intermittently keeping the loop open some of the time. In this manuscript both scalar and vector plants under Bernoulli and Markov intermittence models are investigated. Novel bounds on intermittence parameters are obtained to ensure stability. Moreover, novel recursive quantization algorithms are developed to implement the stabilization scheme under all the constraints. Suitable illustrative examples are provided to elucidate the main results.
Paper Structure (36 sections, 18 theorems, 67 equations, 11 figures, 2 algorithms)

This paper contains 36 sections, 18 theorems, 67 equations, 11 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Organization of the paper
  4. Notations
  5. Problem Formulation
  6. Main Results
  7. Bernoulli intermittence model
  8. Scalar Plant
  9. Vector Plant
  10. Markov intermittence model
  11. Scalar Plant
  12. Vector Plant
  13. Algorithms
  14. Examples
  15. Bernoulli intermittence
  16. ...and 21 more sections

Key Result

Lemma 1

For the scalar plant described in section scalar_bernoulli, in absence of quantization (i.e., assuming $\mathcal{C}\rightarrow\infty$), the conditional state variance at the $(k+1)^{th}$ time step is given by: , and the (expected) state variance at the $(k+1)^{th}$ time step is given by: Here, ${\pmb \xi}_k = \alpha {\pmb \gamma}_k + (\alpha - bl)(1 - {\pmb \gamma}_k)$ , $l$ is the infinite hori

Figures (11)

  • Figure 1: The finite capacity LQG loop with intermittent observations
  • Figure 2: Upper bounds of $p$ for $\mathcal{L}_2$ stability of the system under Bernoulli intermittence
  • Figure 3: State variance simulation and the theoretical asymptotic bound for scalar plant under Bernoulli intermittence
  • Figure 4: Stable loop state trajectory under Bernoulli intermittence
  • Figure 5: Unstable loop state trajectory under Bernoulli intermittence
  • ...and 6 more figures

Theorems & Definitions (18)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 4
  • Lemma 5
  • Lemma 6
  • Theorem 7
  • Lemma 8
  • Theorem 9
  • Lemma 10
  • ...and 8 more