While information theory has been introduced to characterize the fundamental limitations of control and filtering for a few decades, the existing information-theoretic methods are indirect and cumbersome for analyzing the limitations of continuous-time systems. To answer this challenge, we lift the information-theoretic analysis to continuous function spaces by the I-MMSE relationships. Continuous-time control and filtering systems are modeled into the additive Gaussian channels with and without feedback, and the total information rate is identified as a control and filtering trade-off metric and calculated from the estimation error of channel inputs. Fundamental constraints for this trade-off metric are first derived in a general setup and then used to capture the limitations of various control and filtering systems subject to linear and nonlinear plant models. For linear scenarios, we show that the total information rate quantifies the performance limits, such as the minimum entropy cost and the lowest achievable mean-square estimation error, in the time domain. For nonlinear systems, we provide a direct method to calculate and interpret the total information rate and its lower bound by the Stratonovich-Kushner equation.

翻译：尽管信息论被引入用于刻画控制与滤波基本局限性已有数十年，但现有信息论方法对于分析连续时间系统的局限性而言是间接且繁琐的。为应对这一挑战，我们借助I-MMSE关系将信息论分析提升至连续函数空间。连续时间控制与滤波系统被建模为带反馈与不带反馈的加性高斯信道，总信息率被确立为控制与滤波的权衡度量，并通过信道输入的估计误差计算得出。首先在一般性框架下推导了该权衡度量的基本约束条件，进而利用这些约束来刻画受线性和非线性被控对象模型约束的各种控制与滤波系统的局限性。对于线性场景，我们证明总信息率在时域上量化了性能极限，例如最小熵代价和最低可达均方估计误差。对于非线性系统，我们提供了一种通过Stratonovich-Kushner方程计算并解释总信息率及其下界的直接方法。