State Feedback Control and Kalman Filtering with MATLAB/Simulink Tutorials

7The Kalman Filter

7.1 Introduction

The Kalman filter is another one of the real‐time computational algorithms for state estimation. The key similarity between state observer from the previous chapters and the Kalman filter is that they both estimate the unknown state variables via the measurement of output variables in combination with knowledge of the process. An observer computes the observer gain vector off‐line for a time‐invariant system. In contrast, the Kalman filter computes the Kalman filter gain in real‐time and recursively, hence it can effectively estimate the state vector for time‐varying systems. The Kalman filter yields optimal gain and state estimate if the process noise and measurement noise are Gaussian and the system is linear.

The chapter begins with an introduction to mathematical models used in the Kalman filter (see Section 7.2.1) followed by a derivation of the filter (see Sections 7.2.2 and 7.2.3). The derivation presented in this section follows an intuitive approach, although there are a number of ways to derive the Kalman filter. After the presentation of the Kalman filter algorithm, in Section 7.2.4, several examples are presented together with MATLAB tutorials. In Section 7.2.5 the compensation of sensor bias and load disturbance is discussed. The commonly encountered scenarios in the Kalman filter applications include multi‐rate sampled data and missing measurements. These important issues are addressed in Section 7.3 with case studies and MATLAB ...