随机模态驱动下动态过程贝叶斯递推估计

Control theory and engineering have focused on signal filtering or estimation for long. From the analog filters, digital filters to statistical filters, Kalman filter provides the optimal estimation for linear Gaussian systems and is considered a milestone in modern control theory. From the view of statistical estimation, Bayesian estimation can overcome the limitations of Kalman filter due to its distinct advantages in representing uncertainties and the recursive structure preferred by engineering applications. On the other hand, the existing state estimation methods almost concentrate on the time-driven systems, however, the systems involved both time-evoloving and event-driven mechanisms are encountered commonly in applications. Therefore, it is highly desired to propose some novel estimation theory which can handle mode-driven mechanisms effectively. This project explores the Bayesian estimation of dynamic process driven by stochastic modes based on the analysis of the mechanisms of random mode and the statistics of observational information. By using the main idea of the so-called interacting multiple-model techniques together with Bayesian principle,the estimation theory is developed for the hybrid process with both mode and state. Moreover aim at the complicated dynamics, such as uncertainties, non-linear and time-delays, et al, the recursive algorithms are proposed by considering robustness, estimation accuracy and computational time. The presented project not only has remarkable engineering application worth, but also contains plentiful matter of theory.

信号滤波或估计是控制理论与工程中非常活跃的研究领域，从模拟滤波器、数字滤波器，进而统计滤波器，至Kalman滤波器实现了线性高斯系统的最优状态估计，成为现代控制理论的里程碑。从统计估计的角度，贝叶斯估计独特的不确定性表达能力以及符合工程应用的递推结构，可进一步超越Kalman滤波的限制；另一方面，目前状态估计主要针对基于时间演化的动态系统，而现代工程领域中，存在大量既含时间状态演化，又有事件模态驱动的所谓混杂动态系统，需要新的可以处理模态驱动机制的状态估计理论和方法。本项目研究随机模态驱动下动态过程的贝叶斯估计问题，以随机模态驱动机制分析及观测信息的统计利用为主线，依据贝叶斯法则，吸取交互多模型算法的主要思想，提出模态和状态混杂过程的估计理论；进而针对不确定性、非线性、时滞等复杂动态，围绕递推估计方法的鲁棒性、精度和计算耗时等展开研究。项目具有重要而深刻的理论意义，也有广泛的工程应用价值。

现代工业化和信息化条件下，多模态系统状态估计问题具有很强的实际工程背景，其动态机理的复杂性、信息结构的多样性、以及算法的鲁棒性、精度和效率等要求，构成挑战性课题。贝叶斯理论(Bayesian Theory)考虑随机信号概率密度，利用先验分布与当前的量测来获取过程状态的后验概率分布，由此给出了状态估计一般化法则。目前应用广泛的Kalman滤波依据随机信号的期望、方差等统计特性，给出了状态的最小均方误差估计，其本质是线性系统在高斯噪声下的最优贝叶斯估计。本项目从贝叶斯估计独特的不确定性表达能力以及易于工程应用的递推结构出发，研究具有随机切换模态和时间演化状态的混杂过程的递推估计理论和方法：以交互多模型(Interacting Multiple-Model, IMM)为主要思想，给出模态转移概率(Transition Probabilities, TPs)未知且时变过程的贝叶斯估计方法，实现随机模态递推估计与系统状态递推估计的融合；考虑动态过程的非齐次、非线性、时滞以及时变不确定等特性，提出随机模态驱动的状态估计方法，并应用于故障检测；将滚动时域估计(Moving Horizon Estimation, MHE)作为贝叶斯估计的工程近似，给出随机Markov模态驱动下的非线性系统状态优化估计策略；分析比较Kalman滤波的无限脉冲响应(Infinite Impulse Response, IIR)结构特点，采用有限脉冲响应(Finite Impulse Response, FIR)结构设计滤波器，探讨如何增强递推估计的鲁棒性、保证精度以及减少计算耗时等；针对随机模态驱动下动态过程，设计干扰抑制器、饱和执行器、增益调度控制器以及预测控制器等。本项目以交互近似来避免指数级的计算量，并证明当转移矩阵已知时，所提出的模态和状态混杂过程的估计方法退化为一般IMM估计器；以MHE逼近贝叶斯估计，解决随机模态驱动下的非线性过程的估计问题；以FIR结构替代Kalman的IIR结构，设计更易工程应用的滤波器。这些探索工作在状态估计领域具有很强的科学意义和应用价值。

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