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.

[1]  John E. Mottershead,et al.  The sensitivity method in finite element model updating: A tutorial (vol 25, pg 2275, 2010) , 2011 .

[2]  Joel P. Conte,et al.  Bayesian nonlinear structural FE model and seismic input identification for damage assessment of civil structures , 2017 .

[3]  Geert Lombaert,et al.  Joint input-state estimation in structural dynamics , 2016 .

[4]  James L. Beck,et al.  Real-time Reliability Estimation for Serviceability Limit States in Structures with Uncertain Dynamic Excitation and Incomplete Output Data , 2007 .

[5]  Neil D. Lawrence,et al.  Gaussian Process Latent Force Models for Learning and Stochastic Control of Physical Systems , 2017, IEEE Transactions on Automatic Control.

[6]  Thia Kirubarajan,et al.  Estimation with Applications to Tracking and Navigation: Theory, Algorithms and Software , 2001 .

[7]  W. Desmet,et al.  An online coupled state/input/parameter estimation approach for structural dynamics , 2015 .

[8]  Jonathan P. Stewart,et al.  Forced vibration testing of buildings using the linear shaker seismic simulation (LSSS) testing method , 2004 .

[9]  Wen-Hua Chen,et al.  Disturbance observer based control for nonlinear systems , 2004 .

[10]  Costas Papadimitriou,et al.  Bayesian optimal estimation for output‐only nonlinear system and damage identification of civil structures , 2018 .

[11]  Thomas G. Carne,et al.  The Natural Excitation Technique (NExT) for modal parameter extraction from operating wind turbines , 1993 .

[12]  Geert Lombaert,et al.  Verification of joint input-state estimation for force identification by means of in situ measurements on a footbridge , 2016 .

[13]  Bart De Moor,et al.  Unbiased minimum-variance input and state estimation for linear discrete-time systems with direct feedthrough , 2007, Autom..

[14]  Erik I. Verriest.  Optimal filtering for crypto-deterministic systems with application to delay systems with unknown initial data , 2008, 2008 47th IEEE Conference on Decision and Control.

[15]  Mohamed Darouach,et al.  Extension of minimum variance estimation for systems with unknown inputs , 2003, Autom..

[16]  Dapeng Zhu,et al.  Substructure Stiffness and Mass Updating through Minimization of Modal Dynamic Residuals , 2016 .

[17]  Costas Papadimitriou,et al.  Sequential Bayesian estimation of state and input in dynamical systems using output-only measurements , 2019, Mechanical Systems and Signal Processing.

[18]  Yanjun Li,et al.  A general extended Kalman filter for simultaneous estimation of system and unknown inputs , 2016 .

[19]  Yang Wang,et al.  Input Estimation of a Full-Scale Concrete Frame Structure with Experimental Measurements , 2020 .

[20]  Mohammad Valikhani,et al.  Bayesian framework for simultaneous input/state estimation in structural and mechanical systems , 2019, Structural Control and Health Monitoring.

[21]  F. Hemez,et al.  Updating finite element dynamic models using an element-by-element sensitivity methodology , 1993 .

[22]  Wei-Xin Ren,et al.  Damage detection by finite element model updating using modal flexibility residual , 2006 .

[23]  Geert Lombaert,et al.  An augmented Kalman filter for force identification in structural dynamics , 2012 .

[24]  R. DesRoches,et al.  Validation of wireless sensing technology densely instrumented on a full-scale concrete frame structure , 2015 .

[25]  Costas Papadimitriou,et al.  Joint input-response estimation for structural systems based on reduced-order models and vibration data from a limited number of sensors , 2012 .

[26]  Sriram Narasimhan,et al.  A Gaussian process latent force model for joint input-state estimation in linear structural systems , 2019, Mechanical Systems and Signal Processing.

[27]  Peter K. Kitanidis,et al.  Unbiased minimum-variance linear state estimation , 1987, Autom..

[28]  Oral Buyukozturk,et al.  Identification of traffic-induced nodal excitations of truss bridges through heterogeneous data fusion , 2015 .

[29]  Costas Papadimitriou,et al.  Experimental validation of the Kalman-type filters for online and real-time state and input estimation , 2017 .

[30]  C. Papadimitriou,et al.  A dual Kalman filter approach for state estimation via output-only acceleration measurements , 2015 .

[31]  B. Friedland.  Treatment of bias in recursive filtering , 1969 .

[32]  Wim Desmet,et al.  Stable force identification in structural dynamics using Kalman filtering and dummy-measurements , 2015 .

[33]  Satish Nagarajaiah,et al.  A novel unscented Kalman filter for recursive state-input-system identification of nonlinear systems , 2019, Mechanical Systems and Signal Processing.

[34]  Yang Liu,et al.  Statistical Regularization for Identification of Structural Parameters and External Loadings Using State Space Models , 2015, Comput. Aided Civ. Infrastructure Eng..