Applying the Complex-Step-Based Numerical Integration Within an Extended Kalman Filter

This paper demonstrates the applicability and numerical advantages of using complex time steps for process-model time propagation (integration) within an Extended Kalman filter. Complex-step-based propagation consists of integrating the process model using time steps, Δt ∈ C (rather than Δt ∈ R), in such a manner that the propagated equations still result real. In general, this technique has been shown to increase the achievable accuracy for each order of integration (e.g., a first-order Euler complex integration achieves second-order Euler integration accuracy) and be comparable to standard routines such as Runge-Kutta and ODE45, at appropriate complex-step orders. The results of our numerical experiments show an Extended Kalman filter using complex time-steps allows the use of larger Δt (which reduces the overall number of integrations and resources required) while achieving comparable accuracy to an ODE45-based EKF and more efficient time integration.