Nonlinear moving horizon state estimation with continuation/generalized minimum residual method

We propose a fast algorithm for nonlinear moving horizon state estimation (MHSE). An optimization problem to be solved at each time step is formulated in a deterministic setting to estimate the unknown state and the unknown disturbance over a finite past, which leads to a nonlinear two-point boundary-value problem (TPBVP) to be solved at each time. Instead of solving the nonlinear TPBVP with an iterative method, the estimate is updated by integrating a differential equation to trace the time-varying solution of the TPBVP, which is a kind of continuation method and needs to solve a linear algebraic equation only once at each sampling time. Moreover, the linear algebraic equation involved in the differential equation is solved efficiently by the generalized minimum-residual method, one of the Krylov subspace methods. The proposed algorithm is evaluated by a numerical simulation and experiment with a hovercraft model the dynamics of which are nonlinear. MHSE by the proposed algorithm generates reasonable 'estimates even when the extended Kalman filter fails, and the proposed algorithm is sufficiently fast for real-time implementation with a sampling period in the order of milliseconds.