Simultaneous input-state estimation with direct feedthrough based on a unifying MMSE framework with experimental validation

Abstract

This paper addresses the simultaneous input-state estimation problem for discrete-time linear stochastic systems with unknown input. A unifying minimum-mean-square-error (MMSE) estimation framework is presented for a comprehensive characterization and direct comparison among a few popular input estimation approaches. In this work, the dynamical model of the input is assumed to be unknown and the input is treated as a random variable at each time step. To account for the unknown input dynamics, we propose an input estimator that adopts a white Gaussian input model with a finite covariance, short-named as the FIC (finite input covariance) estimator. Theoretical formulation of the FIC estimator is first compared with two other estimators, one using a Gaussian random walk input model combined with augmented Kalman filter (AKF) and another one using a deterministic input model with weighted least squares (WLS) estimation. Based on the unifying MMSE framework presented in this paper, it is proved that when the input covariance of the FIC estimator approaches infinity and the feedthrough matrix has full-column rank, the estimator is equivalent to the well-known WLS estimator.The FIC estimator is validated and compared with the AKF and WLS estimator using simulated measurements from 2-story shear structure and experimental measurements from a full-scale concrete frame. The 2-story shear structure is excited by two different types of input and the corresponding acceleration responses are used to compare estimator performance. With only acceleration measurements, the FIC estimator eliminates a low-frequency drift error in the estimated input and states with a tight estimation confidence interval. Detailed discussion on the effect of estimator covariances on input estimation is also provided. A priori knowledge of the statistical property of the unknown input can provide insights in the tuning of FIC estimator covariances. In addition, field acceleration measurements from a full-scale concrete frame under shaker excitation are used to compare the estimation results and validate the proposed FIC estimator. The FIC estimator is shown to provide better estimates with tighter estimation confidence interval in comparison to the AKF and WLS estimators.