Nonlinear filtering with exogenous Kalman filter and double Kalman filter

Abstract

We propose two variants of the Linearized Kalman Filter (LKF). The model linearization is made about an auxiliary state estimate that can be seen as an exogenous input to the LFK, and the resulting two-stage estimator is called an exogenous Kalman filter (XKF). Since the linearized model of the LKF does not depend on the estimate from the LKF, it follows from stability theory of cascades that the stability properties of the XKF are inherent from the auxiliary state estimator, since the nominal LKF is globally exponentially stable. In some cases, the use of nonlinear transforms and immersions can be used to render the nonlinear dynamics into a globally valid linear time-varying (LTV) form that can be used for design of the auxiliary state estimator. Even though the noise and disturbances may be influenced in a non-favorable way by such transforms, they allow the auxiliary state estimator to be designed using a time-varying Kalman filter, leading to a two-stage estimator called the Double Kalman Filter (DKF). The DKF is inherently globally exponentially stable for nonlinear systems that can be transformed into uniformly observable LTV systems. The XKF and DKF overcome the potential instability of the extended Kalman Filter (EKF) and similar algorithms that can result from inaccurate initialization, by applying a feed-forward/cascade structure instead of the feedback inherent in the EKF linearization. The method is illustrated using examples.