Rate-Cost Tradeoffs in Nonlinear Control

TL;DR

This paper characterizes the fundamental limits of communication rates needed for controlling nonlinear stochastic systems within a specified cost, establishing directed information as the key metric.

Contribution

It provides a nonasymptotic bound on the rate-cost tradeoff in nonlinear control, extending the role of directed information beyond linear systems.

Findings

Achieves a tight bound within a logarithmic gap for rate-cost tradeoff

Establishes directed information as the core quantity for rate-limited control

Provides bounds applicable to nonlinear, sequential, and LQG control scenarios

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.