The Analysis Of The Unscented Kalman Filter

The Unscented Kaman filter (UKF) is an adaptive filter that is suitable for nonlinear systems. The Kalman filter (KF) was originally proposed by Kalman, which is widely used soon because of the simple structure and the facility of realization. It is unperfect that the KF is only suitable for the linear systems, which lead to the appearance of the Extend Kalman filter (EKF). However, many years of experience in the estimation community has shown that is difficult to implement, difficult to tune and only reliable for systems that are almost linear on time scale of the update. The reason is that simply linearizes all the nonlinear transformations and substitutes Jacobian matrices for the linear transformation in KF equations. The UTembedded in the KF's recursive prediction and update structure is the UKF. In contrastto the EKF, UKF can achieve the second or the third order (Taylor series expansion) for any nonliearity. No explicit Jacobian or Hessian calculations are necessary for the UKF. Remarkably, the computational complexity of the UKF is the same order as that of the EKF. Especially, the UKF needn't the assumption that all transformations are qusi-linear.The algorithms' convergence plays an important role in the application of algorithms. This thesis mainly analyzes the convergence of the UKF algorithm based on existent research results, mainly covering the following aspects: (1) The Unscented Transformation(UT) provides a more direct and explicit mechanism for transforming mean and covariance information. We describe the general UT mechanism along with a variety of special formulations that can be tailored to the specific requirements of different nonlinear filtering and control application. Because the UT offers enough flexibility to allow information beyond mean and covariance to be incorporated into the set of sigma points, it is possible to pick a set that exploits any additional known information about the error distribution associated with an estimation. We introduce the general sigma point selection framework, and give two kinds of efficient methods: a) reducing the computing complexity by decreasing the number of the sigma point; b) scaling the sigma points to influence the effects of the higher order moments, then improve the accuracy of the UT; c) Convergence analysis of the UKF, when used as an observer for nonlinear deterministic discrete-time system, is presented. Based on the UKF that captures the posterior mean and covariance accurately to the second or third order term (in terms of Taylor series expansion) for any form of nonlinearity, sufficient conditions to ensure local asymptotic convergence are established. Furthermore, it is shown that the design of the ar- bitrary matrix plays an important role in the enlarging the domain of attraction and then improving the convergence of UKF significantly. The validity of this proposed improvement. is demonstrated by computer simulation. (2) A non-periodic oscillatory behavior of the UKF when used to filter noisy contaminated chaotic signals is. for the first time, reported in this paper. We show both theoretically and experimentally that the gain of the UKF may not converge or diverge but oscillate aperiodically. More precisely, when a nonlinear system is periodic, the Kalman gain and error covariance of the UKF converge to zero. However, when the system being considered is chaotic, the Kalman gain either converges to a fixed point with a magnitude larger than zero or oscillates aperiodically.