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.

翻译：我们研究了有限时域内一般随机控制系统（包括非线性系统）在速率受限控制中的速率-成本权衡。在每个时间步，编码器观测系统状态并将描述传递给控制器，后者据此选择控制动作。针对平均控制成本阈值$D$，我们通过非渐近界刻画了最小可达通信速率$R_n(D)$：$R_n(D)$与定向信息最小化问题$F_n(D)$的最优值之间仅相差一个加性对数项，即我们证明$F_n(D) \le R_n(D) \le F_n(D)+\log \bigl(F_n(D)+3.4\bigr)+2+\frac{1}{n}$（单位为比特）。这一结果确立了定向信息作为速率受限控制中操作相关的核心量，将其应用范围从先前建立的因果信源编码和线性二次高斯（LQG）控制扩展至一般非线性控制系统。我们通过在每个时间步使用强函数表示引理，以构造性方式证明了上界。作为我们框架的特例，该方法为序列（因果）率失真控制和LQG控制提供了非渐近界。