非平稳/不稳定线性系统的渐近均值平稳性和随机稳定性的信息通道表征
Characterization of Information Channels for Asymptotic Mean Stationarity and Stochastic Stability of Non-stationary/Unstable Linear Systems
AI总结：
本文针对离散通道（可能有记忆和反馈）上的高斯噪声驱动的不稳定（非平稳）多维线性系统，提出了随机可稳定性的紧的充要条件，涉及递归、渐近均值平稳性、样本路径遍历性和有限二阶矩的存在性。
摘要：
考虑了噪声通信通道上非平稳线性系统的镇定问题。自1970年代以来，信息论和控制论文献中广泛研究了随机稳定源以及不稳定但无噪声或有界噪声系统，并在过去十年中重新引起兴趣。也有关于不稳定/非平稳线性高斯源的非因果和因果编码的研究。本文提出了离散通道（可能有记忆和反馈）上高斯噪声驱动的不稳定（非平稳）可能多维线性系统的随机可稳定性的紧的充要条件。随机稳定性概念包括递归、渐近均值平稳性和样本路径遍历性，以及有限二阶矩的存在性。我们的构造性证明使用了马尔可夫链镇定的随机时间状态相关随机漂移准则。对于渐近均值平稳性（因此样本路径遍历性），对于无记忆通道和一类有记忆通道，通道容量（严格）大于不稳定极点幅度的对数之和是充分的。在温和的技术条件下，该条件也是必要的。提供了由无界噪声驱动的此类系统存在有限平均二阶矩的充分条件。
英文摘要：
Stabilization of non-stationary linear systems over noisy communication channels is considered. Stochastically stable sources, and unstable but noise-free or bounded-noise systems have been extensively studied in information theory and control theory literature since 1970s, with a renewed interest in the past decade. There have also been studies on non-causal and causal coding of unstable/non-stationary linear Gaussian sources. In this paper, tight necessary and sufficient conditions for stochastic stabilizability of unstable (non-stationary) possibly multi-dimensional linear systems driven by Gaussian noise over discrete channels (possibly with memory and feedback) are presented. Stochastic stability notions include recurrence, asymptotic mean stationarity and sample path ergodicity, and the existence of finite second moments. Our constructive proof uses random-time state-dependent stochastic drift criteria for stabilization of Markov chains. For asymptotic mean stationarity (and thus sample path ergodicity), it is sufficient that the capacity of a channel is (strictly) greater than the sum of the logarithms of the unstable pole magnitudes for memoryless channels and a class of channels with memory. This condition is also necessary under a mild technical condition. Sufficient conditions for the existence of finite average second moments for such systems driven by unbounded noise are provided.
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