A finite-horizon adaptive Kalman filter for linear systems with unknown disturbances

In this paper, a class of linear systems subject to process disturbances and structured measurement disturbances with unknown time-varying covariances is considered. First, we construct a finite-horizon filter structure to recursively obtain a suit of positive definite matrices and propose the sufficient conditions to ensure the above positive definite matrices to be upper bounds of the unknown covariances of the state estimation, errors, filtering residuals and state prediction errors. Then some parameters are directly determined through simultaneously minimizing such upper bounds, while the other parameters are obtained via optimization through minimizing the upper bound of the covariances of filtering residuals. Furthermore, the parameter optimization is transformed into a convex optimization problem, which can be effectively solved by use of linear matrix inequality (LMI). Hence a finite-horizon adaptive Kalman filter (FHAKF) is proposed. The simulation study is about the joint time-varying time delay and parameter estimation of a nonlinear stochastic system with sensors subject to disturbances with unknown covariances, which shows that the proposed FHAKF has excellent performance and reveals the robustness of the FHAKF against the a priori filter parameters.