Moving horizon estimation based on the constrained nonlinear gradient descent method^*

This paper proposes a constrained nonlinear gradient descent (NGD) algorithm specifically designed for moving horizon estimation (MHE) in discrete-time nonlinear systems. In the first step, the original MHE formulation is simplified by removing the process and measurement noise from the optimization variables, leaving only the system states to be estimated. This reformulation enables the use of NGD method, which offers unique advantages in terms of computational efficiency, implementation complexity, and practicality. While MHE shares similarities with model predictive control (MPC), the key distinction lies in the time-varying nature of the arrival cost. Therefore, we employ an extended Kalman filter (EKF)-based strategy to update the arrival cost matrix. To ensure the convergence and computational efficiency of the NGD algorithm, we adaptively update the step size through a backtracking line search (BLS) algorithm. In addition, we discuss the projection operator and various implementation details. The proposed algorithm is validated using a simulation example of an intelligent vehicle.