Extended Kalman Filter for Graph Signals in Nonlinear Dynamic Systems

We consider the problem of recovering random, time-varying graph processes in a nonlinear dynamic system. The Extended Kalman filter (EKF) is a suitable estimator for such dynamics, but its implementation tends to be complex and possibly unstable when tracking high-dimensional graph signals. To tackle this, we propose the graph signal processing (GSP)-EKF, which replaces the Kalman gain in the EKF with a graph filter that aims to minimize the computed prediction error. The resulting structure of the GSP-EKF Kalman gain increases the numerical stability and reduces the computational burden compared with the standard EKF, particularly when dealing with bandlimited graph processes. We show that for a measurement model with orthogonal graph frequencies, the GSP-EKF coincides with the EKF. The GSP-EKF is evaluated for graph signal tracking in power system state estimation. It is shown that in this case, the proposed GSP-EKF 1) attains the EKF under the accurate model; and 2) outperforms the EKF under a model mismatch, while being notably less complex in both cases.