Adaptive Choice of Process Noise Covariance in Kalman Filter Using Measurement Matrices

For the systems working in unknown environments, the number and quality of measurements may vary depending on the environment. In the Kalman filter, it is common to add fictitious noises to the nominal process and measurement noises to avoid the filter divergence caused by inaccurate system modeling. However, it is difficult to choose an appropriate fictitious noise, especially for the systems whose environment varies with respect to the measurements. A naively chosen fictitious noise often leads to unexpectedly large inflation of state estimation error covariance and even degradation of estimation accuracy, when the number of measurements is less than the number of states. This article proposes a novel method for adding a fictitious noise to the nominal process noise using the measurement matrix at each time step. First, we analyze theoretically the variations of state estimation error covariance and Kalman gain due to a fictitious noise. Next, based on the results, we seek a fictitious noise that minimizes the expected values of measurement residuals at each time step and show that unnecessary inflation of state estimation error covariance can be avoided by choosing the fictitious noise based on the measurement matrix. Experiments on positioning of actual automobiles using global navigation satellite system (GNSS) and inertial navigation system (INS) demonstrate that the proposed method improves the positioning accuracy compared to conventional methods. The proposed method has a novel feature in that it is not adaptive to the measurements themselves, but to the measurement matrix, and can be applied in a computationally effective manner to practical applications.