随机动力系统最优编码的度量和拓扑熵界
Metric and topological entropy bounds for optimal coding of stochastic dynamical systems
摘要：
我们考虑的问题的最佳零延迟编码和估计的随机动力系统在有噪声的通信信道下的三个估计标准有关的低失真制度。考虑的标准是（i）强和（ii）弱形式的几乎必然稳定的估计误差，以及（ii）二次稳定的期望。对于所有这三个目标，我们推导出最小的信道容量上的下限$C_0$以上的目标可以实现一个任意小的错误。我们首先通过动力系统的方法，通过构建一个无限维的动力系统，并与这个动力系统的拓扑和度量熵的能力得到的界限。我们还考虑信息理论和概率理论的方法来解决不同的标准。最后，我们证明了一个无记忆的有噪信道一般不构成任何障碍，渐近几乎肯定状态估计任意小的误差，当系统中没有噪声。结果为所引入的准则提供了新的求解方法（例如，标准的信息论界限不能应用于某些准则），并建立了动力系统、网络控制和信息论之间的进一步联系，特别是在非线性随机系统的背景下。
英文摘要：
We consider the problem of optimal zero-delay coding and estimation of a stochastic dynamical system over a noisy communication channel under three estimation criteria concerned with the low-distortion regime. The criteria considered are (i) a strong and (ii) a weak form of almost sure stability of the estimation error as well as (ii) quadratic stability in expectation. For all three objectives, we derive lower bounds on the smallest channel capacity $C_0$ above which the objective can be achieved with an arbitrarily small error. We first obtain bounds through a dynamical systems approach by constructing an infinite-dimensional dynamical system and relating the capacity with the topological and the metric entropy of this dynamical system. We also consider information-theoretic and probability-theoretic approaches to address the different criteria. Finally, we prove that a memoryless noisy channel in general constitutes no obstruction to asymptotic almost sure state estimation with arbitrarily small errors, when there is no noise in the system. The results provide new solution methods for the criteria introduced (e.g., standard information-theoretic bounds cannot be applied for some of the criteria) and establish further connections between dynamical systems, networked control, and information theory, and especially in the context of nonlinear stochastic systems.
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