A parametric sensitivity method for Arrival Cost design in nonlinear Moving Horizon Estimation

Moving Horizon Estimation (MHE) is an optimization based state estimation algorithm where a fixed number of past measurements in a moving time horizon are used to infer the states. The strengths of MHE are the capability of directly incorporating nonlinear model equations without approximations, and the ease of incorporating physical knowledge like conservation of mass or limits on noise as inequality constraints. When a new measurement is available the oldest measurement in the horizon is discarded to make room for a new one. All discarded measurements are summarized in a term known as the Arrival Cost, which also acts as a prior for the initial state in the MHE. In this work we introduce a novel methodology for calculating the Arrival Cost based on parametric nonlinear programming sensitivities. This is done by interpreting the MHE as an approximation of the Full Information (FI) estimator, where all available measurements are used to estimate the states, by formulating an Ideal Arrival Cost such that the MHE and FI estimator becomes identical. This Ideal Arrival Cost is a parametric optimization problem, and the sensitivity of the optimal solution manifold of this problem is used to approximate the Ideal Arrival Cost. Our method incorporates inequality constraints elegantly into the Arrival Cost, and we show that the proposed method introduces a smaller approximation error of the Ideal Arrival Cost compared to similar methods in literature. In a distillation column simulation example we show that the approximation error of the Full Information Estimate by the MHE with the new Arrival Cost method is reduced to 0.35% from 8.06% by using the Extended Kalman Filter (EKF) as the Arrival Cost or 5.66% by using the Smoothed EKF (SEKF).