Stability analysis of Extended, Cubature and Unscented Kalman Filters for estimating stiff continuous-discrete stochastic systems

This paper studies stiffness and stability properties of Extended Cubature and Unscented Kalman Filters applied to continuous-discrete stochastic systems with stiff dynamic behavior. The main part of these methods relies on numerical integration of Moment Differential Equations (MDEs). Our focus is to understand how the stiffness of MDEs influences performance of the filters. The proposed linear stability analysis shows that the MDEs that havearisen within these methods can enlarge the stiffness of continuous-time stochastic model and, hence, require solvers with advanced stability properties for their effective and accurate treatment. Besides, the proposed nonlinear stability analysis proves that such MDEs may become extremely unstable in simulation intervals. The latter raises the uncertainty of state estimation and results in two interesting implications: (i) the lower-order Extended Kalman Filter outperforms the higher-order Cubature and Unscented Kalman Filters in the accuracy of state estimation of stochastic systems with unstable MDE behavior; (ii) the methods under exploration fail when the stiffness is large enough. Our theoretical consideration is supported by numerical tests with filters based on the MATLAB code ode15s, which is a benchmark solver for stiff ODEs. These are examined on linear and nonlinear stiff continuous-time stochastic models.