论文标题
非线性过滤的双重性
Duality for nonlinear filtering
论文作者
论文摘要
本论文与使用白噪声观察模型的隐藏马尔可夫模型（HMM）的随机过滤问题有关。对于此过滤问题，我们做出了三种类型的原始贡献：（1）随机可观察性的双重可控性表征，（2）双重最小方差的最佳控制随机过滤问题的最佳控制公式，以及（3）使用双重最佳控制配方的滤波器稳定性分析。 对于本论文的首要贡献，建议向后的随机微分方程（BSDE）作为双控制系统。 HMM的可观察性（可检测性）被证明等于双控制系统的可控性（稳定性）。对于线性高斯模型，双重关系减少了线性系统理论中的经典二元性。 第二个贡献是将最小方差估计问题转变为最佳控制问题。约束由双重控制系统给出。最佳解决方案是通过两种方法获得的：（1）通过最大原理的应用和（2）通过最佳值的Martingale表征。最佳解决方案用于得出非线性滤波器。 第三个贡献是通过研究双重最佳控制问题进行滤波器稳定性分析。通过第7章和第8章介绍了两种方法。在第7章中，介绍了有条件的庞加莱不平等（PI）。根据条件PI，获得了各种收敛速率并与文献有关。在第8章中，双重控制系统的稳定性被证明是在某些有限状态空间模型上进行滤波器稳定性的必要条件。
This thesis is concerned with the stochastic filtering problem for a hidden Markov model (HMM) with the white noise observation model. For this filtering problem, we make three types of original contributions: (1) dual controllability characterization of stochastic observability, (2) dual minimum variance optimal control formulation of the stochastic filtering problem, and (3) filter stability analysis using the dual optimal control formulation. For the first contribution of this thesis, a backward stochastic differential equation (BSDE) is proposed as the dual control system. The observability (detectability) of the HMM is shown to be equivalent to the controllability (stabilizability) of the dual control system. For the linear-Gaussian model, the dual relationship reduces to classical duality in linear systems theory. The second contribution is to transform the minimum variance estimation problem into an optimal control problem. The constraint is given by the dual control system. The optimal solution is obtained via two approaches: (1) by an application of maximum principle and (2) by the martingale characterization of the optimal value. The optimal solution is used to derive the nonlinear filter. The third contribution is to carry out filter stability analysis by studying the dual optimal control problem. Two approaches are presented through Chapters 7 and 8. In Chapter 7, conditional Poincaré inequality (PI) is introduced. Based on conditional PI, various convergence rates are obtained and related to literature. In Chapter 8, the stabilizability of the dual control system is shown to be a necessary and sufficient condition for filter stability on certain finite state space model.
