如何从被噪声污染的观测数据中得到状态的最优估计被称为滤波问题。根据 状态和观测系统是否线性，滤波问题可以分为线性滤波和非线性滤波，实际中的滤 波问题大多数是非线性的。由于状态的最优估计是基于观测的条件期望，针对非 线性滤波问题，我们一般从两个思路去研究：一个是利用 Duncan-Mortensen-Zakai（DMZ）方程直接求解状态的条件密度函数，从而得到状态的条件期望。此方法一般需要求解 DMZ 方程，例如李代数方法、直接法等；另一个是基于贝叶斯框架研 究滤波算法，例如扩展卡尔曼滤波、粒子滤波等。本论文的工作主要包含两个部分：在第一部分的工作中，我们研究了基于求 解 DMZ 方程的直接法（Direct method）。该方法针对 Yau 滤波系统，求解出其状 态的条件密度函数满足的 DMZ 方程，从而求得状态的条件期望。经典的直接法 是针对某一类限制条件的时不变系统的。在我们的工作中，我们首先将研究对象 从是不变系统推广到时变系统，并消除了对系统的绝大多数约束条件，只保留基 本假设。这部分的工作分为两步实现：首先我们针对带一定限制条件的时变系统， 将其对应的 DMZ 方程的求解通过指数变换转化为对科尔莫戈罗夫前向方程的求 解，再构造变换进一步将其转化为对薛定谔方程的求解，最终将其转化为求解常 微分方程组；第二步我们利用高斯逼近算法，在求解科尔莫戈罗夫前向方程时， 我们将其在每个迭代步的初值分布分解为高斯分布之和，然后将每个高斯分布作 为初值去求解相应的科尔莫戈罗夫前向方程，此时其解可以直接通过求解一系列 常微分方程实现。相比第一步的工作，我们只保留了对系统最基本的假设条件， 即我们将经典的直接法推广到了最一般的时变系统的情形。此外，我们对这两种 直接法进行了严格的收敛性分析。在第二部分的工作中，我们将连续状态-离散观 测的滤波系统通过 Carleman 方法近似为一个双线性系统，然后利用贝叶斯框架针 对该双线性系统构造了一个次最优 (suboptimal) 的滤波算法，我们证明了：该算法 对状态的先验估计和后验估计均是无偏的；先验估计指数收敛于最优估计，即条 件期望；后验估计是关于新息的线性最小均方误差估计。本文提出的所有算法都 用经典算例进行了数值测试，并且与现在常用的滤波算法，例如扩展卡尔曼滤波 算法、粒子滤波算法等，进行了对比，用数值模拟验证了我们提出的算法的有效性。

How to obtain the optimal estimate of the state from measurement corrupted by noises is the goal of filtering problem. According to the linearity of the state and observation system, the filtering problem can be divided into linear and nonlinear filtering, and most filtering problems are nonlinear in real applications. It is known that the optimal mean square estimate of the state is the conditional expectation of the state based on the observation history, and it follows that we can solve the filtering problem once we obtain the conditional probability density function of the state. Usually there are two ways to solve the nonlinear filtering problems: one is to solve the Duncan-Mortensen-Zakai (DMZ) equation which is satisfied by the unnormalized conditional density function, such as Lie algebra method and direct method; the other is based on the Bayesian framework, such as the extended Kalman filter and particle filter.The main work of this dissertation can be divided into two parts. The first part is about the direct method for Yau filtering systems which solves the filtering problems fora class of continuous nonlinear time-varying systems via the Duncan–Mortensen–Zakai(DMZ) equation. Direct method for Yau filtering system has been studied since 1990s and all these results are limited to time-invariant systems. In this part, we extend the direct method so that it is applicable to most general time-varying cases by two steps. In the first step, we consider the filtering system with certain constraints, for which the original DMZ equation is changed into the Kolmogorov forward equation (KFE) by exponential transformations in each time interval, and then, under some assumptions, the KFE can be transformed into a time-varying Schr?odinger equation, which can be solved explicitly by solving a series of ordinary di?erential equations. In the second step, we propose several transformations on the FKE so that it can be solved by means of solving some ordinary di?erential equations if the initial distribution is Gaussian. The corresponding results for any non-Gaussian initial distributions can be obtained via Gaussian approximation. Compared with the work in first step, we need less assumptions and extend the direct method to most general case so that it can treat nearly most general Yau filtering problems under natural assumptions. Furthermore, the convergence of these two direct methods has been analyzed strictly. The second part is about a suboptimal estimation for continuous– discrete bilinear systems. Similar to the Kalman filter, our algorithm includes predictionand updating step. We show rigorously that our algorithm gives an unbiased estimate, the a-priori estimate approaches to the conditional expectation exponentially fast, and the posterior estimate minimizes the conditional variance error in the linear space spanned by the a-priori estimate and the innovation. Our algorithm is also applicable to solve the nonlinear filtering problems. The e?ciencies of all algorithms proposed in this dissertation are tested by classic numerical examples via simulations. The results have been compared with the classic filtering algorithms, such as extended Kalman filter, particle filter and so on. The simulation results show the e?ciencies of our methods.