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Goal-Oriented State Information Compression for Linear Dynamical System Control

Li Wang, Chao Zhang, Samson Lasaulce, Lina Bariah, Merouane Debbah

TL;DR

This work addresses the problem of maintaining LQR control performance under rate-limited communication by introducing goal-oriented compression for networked linear dynamical systems. It models compression noise via rate-distortion theory, deriving a convex rate-allocation problem with a closed-form solution that prioritizes earlier time steps in invariant controllers. The approach is validated through simulations, showing gains over uniform-rate schemes and demonstrating adaptability to time-varying dynamics. The results offer practical guidance on when and how to communicate under bandwidth constraints for real-time networked control applications.

Abstract

In this paper, we consider controlled linear dynamical systems in which the controller has only access to a compressed version of the system state. The technical problem we investigate is that of allocating compression resources over time such that the control performance degradation induced by compression is minimized. This can be formulated as an optimization problem to find the optimal resource allocation policy. Under mild assumptions, this optimization problem can be proved to have the same well-known structure as in [1], allowing the optimal resource allocation policy to be determined in closed-form. The obtained insights behind the optimal policy provide clear guidelines on the issue of "when to communicate" and "how to communicate" in dynamical systems with restricted communication resources. The obtained simulation results confirm the efficiency of the proposed allocation policy and illustrate the gain over the widely used uniform rate allocation policy.

Goal-Oriented State Information Compression for Linear Dynamical System Control

TL;DR

This work addresses the problem of maintaining LQR control performance under rate-limited communication by introducing goal-oriented compression for networked linear dynamical systems. It models compression noise via rate-distortion theory, deriving a convex rate-allocation problem with a closed-form solution that prioritizes earlier time steps in invariant controllers. The approach is validated through simulations, showing gains over uniform-rate schemes and demonstrating adaptability to time-varying dynamics. The results offer practical guidance on when and how to communicate under bandwidth constraints for real-time networked control applications.

Abstract

In this paper, we consider controlled linear dynamical systems in which the controller has only access to a compressed version of the system state. The technical problem we investigate is that of allocating compression resources over time such that the control performance degradation induced by compression is minimized. This can be formulated as an optimization problem to find the optimal resource allocation policy. Under mild assumptions, this optimization problem can be proved to have the same well-known structure as in [1], allowing the optimal resource allocation policy to be determined in closed-form. The obtained insights behind the optimal policy provide clear guidelines on the issue of "when to communicate" and "how to communicate" in dynamical systems with restricted communication resources. The obtained simulation results confirm the efficiency of the proposed allocation policy and illustrate the gain over the widely used uniform rate allocation policy.
Paper Structure (10 sections, 4 theorems, 32 equations, 4 figures)

This paper contains 10 sections, 4 theorems, 32 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Perfect transmission
  4. Rate-limited transmission
  5. Analysis of the rate allocation problem
  6. Numerical analysis
  7. Evaluation metric
  8. Time-invariant systems
  9. Time-variant dynamical systems
  10. Conclusion

Key Result

Proposition 1

For a rate-limited invariant dynamical system, the state at time $t+1$ can be derived from the state in a perfect-transmission system as for $0\leq t\leq T$, where $x_{t+1}^{(p)}$ and $x_{t+1}^{(c)}$ represent respectively the state in a perfect-transmitted system, and the state with compressed transmission, at time $t+1$. $\mathrm{sgn(\cdot)}$ is the sign function.

Figures (4)

  • Figure 1: LQR setting with a communication link (the dashed line) from the observer to the controller.
  • Figure 2: Relative LQR costs (\ref{['eq:relative cost']}), The effectiveness of the proposed optimal rate allocation scheme is significantly enhanced when the dynamic process is more influenced by state transitions than by controller inputs (i.e., when the value of $A$ is higher), outperforming the constant-rate scheme more noticeably.
  • Figure 3: Optimal rate allocation for invariant linear dynamic systems with 11 stages, defined with different $(A,B)$ pairs. It demonstrates that when the dynamical process is more influenced by state transitions than by controller inputs (i.e., when A is larger), more transmission resources should be allocated to the earlier states.
  • Figure 4: Optimal rate allocation for invariant linear dynamic systems with 4 stages, where $A$ switches from $A_1$ to $A_2$ at time $t=3$. It indicates that when the influence of state information grows (i.e., when the parameter $A$ increases) in a variant dynamic system, more resources should be allocated to the stages near the point of change.

Theorems & Definitions (7)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Corollary 1