A Gradient Condition for the Arrival Cost in Moving Horizon Estimation

Within the Moving Horizon Estimation (MHE) formulation, the contributions of measurements outside the horizon are summarized by the arrival cost term. We propose a gradient condition for the arrival cost which ensures that the state estimate does not change if it perfectly explains new measurements. Furthermore, the updated arrival cost locally approximates the arrival cost, which would be obtained by an exact arrival cost update, if the gradient condition is satisfied. Based on the proposed gradient condition, we derive formulas for updating quadratic arrival cost terms from one MHE iteration to the next.