A unified Kalman filter

Abstract

When considering problems of linear sequential estimation, two versions of the Kalman filter, the continuous-time version and the discrete-time version, are often used. (A hybrid filter also exists.) In many applications in which the Kalman filter is used, the system to which the filter is applied is a linear continuous-time system, but the Kalman filter is implemented on a digital computer, a discrete-time device. The two general approaches for developing a discrete-time filter for implementation on a digital computer are: (1) approximate the continuous-time system by a discrete-time system (called discretization of the continuous-time system) and develop a filter for the discrete-time approximation; and (2) develop a continuous-time filter for the system and then discretize the continuous-time filter. Generally, the two discrete-time filters will be different, that is, it can be said that discretization and filter generation are not, in general, commutative operations. As a result, any relationship between the discrete-time and continuous-time versions of the filter for the same continuous-time system is often obfuscated. This is particularly true when an attempt is made to generate the continuous-time version of the Kalman filter through a simple limiting process (the sample period going to zero) applied to the discrete-time version. The correct result is, generally, not obtained. In a 1961 research report, Kalman showed that the continuous-time Kalman filter can be obtained from the discrete-time Kalman filter by taking limits as the sample period goes to zero if the white noise process for the continuous-time version is appropriately defined. Using this basic concept, a discrete-time Kalman filter can be developed for a continuous-time system as follows: (1) discretize the continuous-time system using Kalman’s technique; and (2) develop a discrete-time Kalman filter for that discrete-time system. Kalman’s results show that the discrete-time filter generated in this way converges to the appropriate continuous-time filter as the sample period goes to zero. This not only makes the relationship between the two versions of the filter transparent, but also lets the sample period be considered as a design parameter.