Introduction

So far we have only studied the discrete Kalman filter because that is the filter that is usually implemented in real-world applications. However, there is also a continuous version of the Kalman filter. Although the continuous Kalman filter is usually not implemented, it can be used as an aid in better understanding the properties of the discrete Kalman filter. For small sampling times the discrete Kalman filter in fact becomes a continuous Kalman filter. Because the continuous Kalman filter does not require the derivation of a fundamental matrix for either the Riccati equations or filtering equations, the continuous filter can also be used as a check of the discrete filter. In addition, we shall see that with the continuous polynomial Kalman filter it is possible, under steady-state conditions, to derive transfer functions that exactly represent the filter. These transfer functions cannot only be used to better understand the operation of the Kalman filter (i.e., both continuous and discrete) but can also aid with such mundane issues as helping to choose the correct amount of process noise to use.

First, we will present the theoretical differential equations for the continuous Riccati equations and Kalman filter. Next, we will derive different-order continuous polynomial Kalman filters. We will demonstrate for the case of zero process noise that the Kalman gains and covariance matrix predictions of the continuous polynomial Kalman filter exactly match the formulas we have already derived for the discrete polynomial Kalman filter. We will then show that when process noise...