Rate-Cost Tradeoffs in Nonlinear Control

2026-04-22

AI Summary

This study investigates the fundamental trade-off between communication rate and control cost in nonlinear stochastic control systems under finite-rate communication constraints. The authors propose a joint encoding, decoding, and control strategy grounded in the strong functional representation lemma and establish, via non-asymptotic analysis, a logarithmic-order approximation between the minimal achievable communication rate and directed information. The key contribution lies in establishing directed information as the central operational quantity in general nonlinear rate-limited control—a role previously confined to linear-quadratic-Gaussian (LQG) settings and causal source coding—thereby unifying sequential rate-distortion theory and LQG control as special cases within a broader theoretical framework.

Technology Category

• Machine Learning: Information Theory
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

We study the rate-cost tradeoff in rate-limited control of general stochastic control systems, including nonlinear systems, over a finite horizon. At each time step, an encoder observes the state and transmits a description to a controller, which then selects the control action. For an average control-cost threshold $D$, we characterize the minimum achievable communication rate $R_n(D)$ via a nonasymptotic bound: $R_n(D)$ lies within an additive logarithmic gap of the optimal value of a directed-information minimization $F_n(D)$, namely, we show that $F_n(D) \le R_n(D) \le F_n(D)+\log \bigl(F_n(D)+3.4\bigr)+2+\frac{1}{n}$, in bits. This establishes directed information as the operationally relevant quantity governing rate-limited control, thereby broadening its utility beyond its previously established roles in causal source coding and linear quadratic Gaussian (LQG) control to general nonlinear control systems. We prove the upper bound constructively by building an encoding-and-control policy using the strong functional representation lemma at each time step. As special cases of our setting, our framework yields nonasymptotic bounds for sequential (causal) rate-distortion and LQG control.

Problem

• rate-cost tradeoff
• nonlinear control
• stochastic control systems
• rate-limited control
• directed information

Innovation

• rate-cost tradeoff
• directed information
• nonlinear control
• nonasymptotic bound
• strong functional representation lemma