Rate-Cost Tradeoffs in Nonlinear Control
This work addresses the fundamental tradeoff between communication rate and control cost for nonlinear systems, broadening the applicability of directed information beyond previous linear or causal coding contexts.
The paper tackles the problem of characterizing the minimum communication rate needed for rate-limited control of general stochastic systems, including nonlinear ones, over a finite horizon, and shows that the achievable rate lies within an additive logarithmic gap of an optimal directed-information minimization, with a bound expressed as $F_n(D) \\le R_n(D) \\le F_n(D)+\\log \igl(F_n(D)+3.4\igr)+2+\ rac{1}{n}$ in bits.
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.