On The Study Of The Direct Method And Suboptimal Method In Nonlinear Filtering Problems

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 for a class of continuous nonlinear time-varying systems via the Duncan–Mortensen–Zakai(DMZ)equation.Direct method for Yau filtering system has been studied since 1990 s 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 differential equations.In the second step,we propose several transformations on the FKE so that it can be solved by means of solving some ordinary differential 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 prediction and 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 efficiencies 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 efficiencies of our methods.