A Note On Nonlinear Kalman Filter

In this thesis, we mainly study the kalman filters of discrete linear system and nonlinear system. Furthermore, we discuss their correlative convergence.First of all, we give an overview of the latest research on linear and nonlinear kalman filter, including the research background and the recent situations. By applying the pseudo-innovation theory and other theories, the linear system has been studied thoroughly, and the correlative kalman filter equations has been constructed completely. For nonlinear system, an approach is the extended kalman filter (EKF), which first linearizes the nonlinear system, then discusses its convergence.Secondly, we deduce the kalman filter equations of general linear system by using the pseudo-innovation theory, and obtain the one-step prediction estimation and real filter estimation. Based on the above results, we transfer the general nonlinear system into the corresponding approximate linear system through Tayor Expansion, and give the EKF of the nonlinear system.In the end, we study the convergence of a kind of nonlinear system which has upper triangular structure. It has been proved that under certain conditions such as observer noise, system matrix, observer matrix and initial error covariance matrix being bounded, the difference between the state of the two-dimensional nonlinear system and its prediction estimation is exponentially bounded in mean square sense. For three-dimensional system, the same conclusion has been proved. The conditions above are imposed on the system nonlinear structure only, therefore, the results are convenient to the application both in theory and engineering.