Two time scale discrete Kalman filter design for an F-8 aircraft

We consider the stochastic model of an F8 aircraft linearized about some flight condition. The process and observation noise vectors are assumed to be zero mean white Gaussian processes of appropriate intensities. After sampling the system with an allowable sampling period, the dynamics of the aircraft and the observation vector are expressed in the sampled-data form. Our goal is to optimally estimate the states of the aircraft by means of minimizing the mean squared error on the basis of the observed output. Although the optimal solution to the problem can be furnished by a standard Kalman filter, the implementation of this solution requires an estimator which incorporates a full order Riccati equation. Unfortunately, due to the speed and memory limitations of the flight computer, this solution is not practically feasible. Therefore, any reduction in the size of computation is highly desirable if the Kalman filter solution is to be implemented in real time. Moreover, the standard Kalman filter solution is ill-conditioned since the magnitude of the noise covariance matrices are inversely proportional to the small sampling period, making the magnitude of the covariance matrices relatively large compared to those of the system matrices. Consequently, serious numerical difficulties are expected if the filter gain coefficients are to be computed on the basis of the full order Riccati equation. We propose a technique which alleviates both the high dimensionality and the ill-conditioning associated with the problem. Our approach is based on the singular perturbation results.