Fourier–Hermite series for stochastic stability analysis of non-linear Kalman filters

Stochastic stability results for the extended Kalman filter and some other non-linear filters have been available for some time now. In this context stochastic stability refers to mean square boundedness of the estimation error. In this article we use Fourier-Hermite series expansion to derive novel stability results for general discrete-time non-linear Kalman filters that can be interpreted as numerical integration rules of Gaussian integrals arising from moment-matching. We also provide an upper bound for the Kalman gain matrix that is not explicitly dependent on the measurement model Jacobian, eliminating thus the need to assume boundedness of this Jacobian. Furthermore, we formulate the system non-linearity assumptions so that it is possible to verify them when the model functions are Lipschitz continuous. We use these results for a priori assessment of the stability of a univariate non-linear filter and verify the results numerically.