Kalman Filter Riccati Equation for the Prediction, Estimation, and Smoothing Error Covariance Matrices

The classical Riccati equation for the prediction error covariance arises in linear estimation and is derived by the discrete time Kalman filter equations. New Riccati equations for the estimation error covariance as well as for the smoothing error covariance are presented. These equations have the same structure as the classical Riccati equation. The three equations are computationally equivalent. It is pointed out that the new equations can be solved via the solution algorithms for the classical Riccati equation using other well-defined parameters instead of the original Kalman filter parameters. 1. Introduction The classical Riccati equation arises in linear filtering and is associated with time invariant systems described by the following state space equations: for , where is the -dimensional state vector at time , is the -dimensional measurement vector at time , is the system transition matrix, is the output matrix, is the plant noise at time , and is the measurement noise at time . Also, and are Gaussian zero-mean white random processes with covariance matrices and , respectively. Filtering is to use measurements in order to recover information about the state vector. Filtering plays an important role in many fields of science: applications to aerospace industry, chemical process, communication systems design, control, civil engineering, filtering noise from 2-dimensional images, pollution prediction, and power systems are mentioned in [1]. We distinguish three kinds of filtering as follows. Prediction. The aim is to obtain at time information about the state vector at time for some using measurements up till time ; it is clear that prediction is related to the forecasting side of information processing. Estimation. The aim is to recover at time information about the state vector at time using measurements up till time . Smoothing. The aim is to obtain at time information about the state vector at time for some using measurements up till time ; it is clear that smoothing requires delay in producing information about the state vector compared to the estimation case. The discrete time Kalman filter [1] is the most well-known algorithm that solves the filtering problem. Kalman filter uses the measurements up till time in order to produce the (one step) prediction of the state vector and the corresponding prediction error covariance matrix , as well as to produce the estimation of the state vector and the corresponding estimation error covariance matrix . The Kalman filter equations, needed for the computation of the prediction and estimation error