Covariance computation in MHE: A nonlinear regression approach

Abstract

Moving horizon estimation (MHE) has emerged as a popular technique for state estimation of nonlinear dynamical systems. A key parameter in MHE is the arrival cost that links the formulation in the current horizon with the past data. Most approaches available in literature use additional filters, such as extended Kalman filter or unscented Kalman filter to obtain the covariance matrix used in the arrival cost. In this work, we propose conceptually simple, alternate ways for obtaining the covariances of the arrival cost. Our approaches are motivated by parameter estimation in nonlinear regression. We view the MHE problem as a nonlinear regression problem with the states being the unknown parameters of the regression problem. The covariance of these estimated parameters can then be computed based on the Jacobian matrix. We also propose a Monte-Carlo based sampling approach for covariance calculation. This approach avoids Jacobian matrix calculation but is computationally intensive. A third approach utilizing approximate Hessian, based on equivalence between nonlinear regression and maximum likelihood estimation, is also proposed. Thus, the proposed approaches are able to compute the arrival cost without requiring any other external filter implementation. The MHE estimation performances obtained with the arrival cost covariances computed using the proposed approaches are compared with extended Kalman filter (EKF) and EKF based MHE implementation on case studies taken from literature. These comparisons demonstrate the utility of the proposed approaches.