Conventional Kalman Filter Algorithm Research Articles

Maximum likelihood state and parameter estimation via derivatives ofthe V-Lambda filter

Applying the method of maximum likelihood to the problem of parameter and state estimation in linear dynamical systems requires the implementation of a recursive algorithm that consists of a Kalman filter and its derivatives with respect to each of the unknown parameters (the sensitivity equations). Since the conventional Kalman filtering algorithm has been shown to be numerically unstable, a different approach is taken in this paper, which is based on using the K-Lambda square root filtering technique. Equations are developed for the recursive computation of the log-likelihood function gradient (score) and the Fisher information matrix (FIM) in terms of the K-Lambda filter variables, which are the eigenfactors (eigenvalues and eigenvectors) of the estimation error covariance matrix. Based on the singular value decomposition, the recently introduced KLambda filters have been shown to be numerically stable and accurate. Therefore, their usage renders the resulting maximum likelihood scheme numerically robust. Moreover, making the covariance eigenfactors available to the user at all estimation stages, which is an inherent and unique property of the K-Lambda class, adds invaluable insight into the heart of the estimation process.

Robust adaptive Kalman filtering with unknown inputs

A method is proposed to adapt the Kalman filter to the changes in the input forcing functions and the noise statistics. The resulting procedure is stable in the sense that the duration of divergences caused by external disturbances are finite and short and, also, the procedure is robust with respect to impulsive noise (outlier). The input forcing functions are estimated by a running window curve-fitting algorithm, which concurrently provides estimates of the measurement noise covariance matrix and the time instant of any significant change in the input forcing functions. In addition, an independent technique for estimating the process noise covariance matrix is suggested which establishes a negative feedback in the overall adaptive Kalman filter. This procedure is based on the residual characteristics of the standard optimum Kalman filter and a stochastic approximation method. The performance of the proposed method is demonstrated by simulations and compared to the conventional sequential adaptive Kalman filter algorithm. >