Critical Evaluation of Extended Kalman Filtering and Moving-Horizon Estimation

Abstract

State estimators for physical processes often must address different challenges, including nonlinear dynamics, states subject to hard constraints (e.g., nonnegative concentrations), and local optima. In this article, we compare the performance of two such estimators: the extended Kalman filter (EKF) and moving-horizon estimation (MHE). We outline conditions that lead to the formation of multiple optima in the estimator for systems tending to a steady state and propose tests that determine when these conditions hold for chemical reaction networks. Several simulation examples demonstrate estimation failure in the EKF, even in the absence of plantmodel mismatch. We then examine the role that constraints play in determining the performance on these examples of MHE employing local optimization and a "smoothing" update for the arrival cost. This implementation of MHE represents a feasible, on-line alternative to the EKF for industrial practitioners. In each example, the two estimators are given exactly the same information, namely, tuning parameters, model, and measurements; yet MHE consistently provides improved state estimation and greater robustness to both poor guesses of the initial state and tuning parameters in comparison to the EKF. The only price of this improvement is the greater computational expense required to solve the MHE optimization.

Robust Stability of Moving Horizon Estimation Under Bounded Disturbances

“…Here we use the gas-phase irreversible reaction example proposed in [9]: 2A −→ B with the reaction rate r = kc 2 A , k = 0.16. We define the two states x1, x2 as the partial pressures of species A and B, and the measurement as the total pressure.…”

A direct sampling particle filter from approximate conditional density function supported on constrained state space

“…The following gas-phase irreversible reaction of species A to species B occurs in a well mixed, constant volume isothermal batch reactor: (Haseltine & Rawlings, 2003),…”

“…The estimates are slow to converge to the true dynam ics and negative values for the partial pressure of species B are meaningless. Such estimates by EKF are also known to con verge to wrong steady states when the non-negativity constraints are ignored (Haseltine & Rawlings, 2003). Similarly poor perfor mance by the unconstrained EnKF is noted even when sample size is increased (Prakash et al, 2010).…”

Bayesian estimation via sequential Monte Carlo sampling—Constrained dynamic systems

“…MHE form ulates a nonli near programming problem unde r possible constraints, which it solves in overlap pi ng moving windows, usually of a fixed size. The Objective fun ction is usually formu lated to find the least-squares solu tion and the approach has been shown to outperform EKF ( Haseltine & Rawlings, 2005). However, two significant short comings of MHE are that it is not recursive in nature, and it has to rely on multivariate Gaussian or other fixed shape distri butions to represent the prior knowledge or arrival cost at the beginning of each moving window.…”