  Recent research in nonlinear filtering and signal processing has suggested an
efficient derivative-free Extended Kalman filter (EKF) designed for
discrete-time stochastic systems. Such approach, however, has failed to address
the estimation problem for continuous-discrete models. In this paper, we
develop a novel continuous-discrete derivative-free EKF methodology by deriving
the related moment differential equations (MDEs) and sample point differential
equations (SPDEs). Additionally, we derive their Cholesky-based square-root
MDEs and SPDEs and obtain several numerically stable derivative-free EKF
methods. Finally, we propose the MATLAB-oriented implementations for all
continuous-discrete derivative-free EKF algorithms derived. They are easy to
implement because of the built-in fashion of the MATLAB numerical integrators
utilized for solving either the MDEs or SPDEs in use, which are the ordinary
differential equations (ODEs). More importantly, these are accurate
derivative-free EKF implementations because any built-in MATLAB ODE solver
includes the discretization error control that bounds the discretization error
arisen and makes the implementation methods accurate. Besides, this is done in
automatic way and no extra coding is required from users. The new filters are
particularly effective for working with stochastic systems with highly
nonlinear and/or nondifferentiable drift and observation functions, i.e. when
the calculation of Jacobian matrices are either problematical or questionable.
The performance of the novel filtering methods is demonstrated on several
numerical tests.
