Joint input-response estimation for structural systems based on reduced-order models and vibration data from a limited number of sensors

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Abstract

An algorithm is presented for jointly estimating the input and state of a structure from a limited number of acceleration measurements. The algorithm extends an existing joint input-state estimation filter, derived using linear minimum-variance unbiased estimation, to applications in structural dynamics. The filter has the structure of a Kalman filter, except that the true value of the input is replaced by an optimal estimate. No prior information on the dynamic evolution of the input forces is assumed and no regularization is required, permitting online application. The effectiveness and accuracy of the proposed algorithm are demonstrated using data from a numerical cantilever beam example as well as a laboratory experiment on an instrumented steel beam and an in situ experiment on a footbridge.

Highlights

► An existing joint input-state estimator is extended to applications in structural dynamics. ► It allows joint input-state estimation from a limited number of acceleration measurements. ► It can be applied to identify unknown inputs or to predict the response at unmeasured locations. ► Its effectiveness and accuracy are demonstrated using both simulated and experimental data.

Introduction

The problem of estimating the true state of a system, where the term system here encompasses all dynamical systems lending themselves to representation by means of a state and observation/measurement equation, has been studied intensively for many decades. In a civil engineering context, state estimation can be defined as the process of identifying, based on a system model, quantities that allow a complete description of the system state (e.g. displacements, velocities) from vibration response data. The state estimates can be used for a variety of purposes including the prediction of stresses and fatigue loading, real-time structural health monitoring or damage identification, structural control, the determination of response in critical joints, the verification of design calculations, etc. Various state estimators for structural systems behaving both linearly and nonlinearly have been proposed for this purpose, including the well-known Kalman filter and a number of its variants. Surveys can be found in [5], [31].

Examples of state estimation in linear systems include the work by Papadimitriou et al. [24], in which the Kalman filter is used as part of a methodology for estimating the damage accumulation in a structure due to fatigue from output-only vibration measurements at a limited number of locations. Ching and Beck [4] estimated the unknown states of a structure using a Kalman smoother in an application concerning reliability estimation for serviceability limit states. Hernandez and Bernal [17] designed a state estimator for structural dynamic systems based on the assumption that the primary source of uncertainty in the predicted state derives from errors in the matrices of the state-space model. Their estimator distinguishes itself from the related robust Kalman filter (RKF) in that it is derived on deterministic grounds, assumes no process/measurement noise and is significantly simpler to implement. It has been used as part of an iterative scheme for model updating in [15] and has been extended to nonlinear systems in [16], where it was used to estimate the states of a damaged seven-story building from a limited number of acceleration measurements. Smyth and Wu [29] proposed a multi-rate Kalman filter for the fusion of measured displacement and acceleration data sampled at different rates. The filter is designed to circumvent problems related to the integration of accelerometer or the differentiation of displacement data in situations where both these response quantities are available for system monitoring or damage detection. Gao and Lu [11] used the Kalman filter in conjunction with an ARX model for structural damage diagnosis. The linear Kalman filter is also used as part of the subspace identification technique for modal analysis [26] to directly estimate states from measured data without knowledge of the system model.

For nonlinear state estimation and parameter identification in civil engineering the extended Kalman filter (EKF) has been one of the most widely used tools in the past [7], [32]. In recent years, however, many alternative techniques have been presented. Ching et al. [5], [6] compared the performance of the EKF with that of the particle filter (PF), also known as the sequential Monte Carlo method, a Bayesian state estimation method based on stochastic simulation. One of the advantages of the PF in comparison to the EKF is that it is applicable to highly nonlinear systems with non-Gaussian uncertainties. Using strong-motion records obtained during an earthquake in the same seven-story building used in [16], the PF is shown to outperform the EKF in [6]. The performance of the EKF is also compared against that of the unscented Kalman filter (UKF), also applicable to highly nonlinear systems, by Wu and Smyth in a number of numerical examples in [31]. The UKF is later used by Chatzi et al. [3] to investigate the effects of model complexity and parameterization on the quality of the estimation in an experimental application.

In all of the above applications, the input forces are assumed either known or broadband, so that they can be modeled as a zero mean stationary white process. In many cases, however, no measurements of the input forces are available or the broadband assumption is violated. The problem of identifying the forces themselves, given that their positions are known, has been studied by Lourens et al. [23].

In this contribution a joint input-state estimation method for linear systems is presented in which no assumptions are made about the dynamic evolution of the unknown excitation. When the positions of the applied forces are known, the algorithm can be used to jointly estimate the states and input forces. Conversely, when the positions of the applied forces are unknown, a set of equivalent forces is identified. In the latter case the points of application of the forces are arbitrarily chosen and equivalent forces, that would produce the same measured response, are identified at all chosen locations. The proposed method is an extension of an existing joint input-state estimation algorithm developed by Gillijns and De Moor [14] for linear systems with direct transmission. Their algorithm, originally proposed for optimal control applications, jointly estimates states and forces which are optimal in a minimum-variance unbiased sense. For applications in structural dynamics it had to be extended to account for numerical instabilities which arise when the number of sensors exceeds the order of the model, i.e. when a large number of sensors is used in conjunction with a reduced-order model constructed from a relatively small number of modes.

The paper has the following structure. First, the state-space formulation for linear dynamic systems is presented in Section 2. The joint input-state estimator of Gillijns and De Moor [14] is briefly introduced in Section 3.1, and extended in Section 3.2. In Section 4, the performance of the algorithm is tested and the need for extending it demonstrated using simulated data from a cantilever beam. Finally, in Sections 5.1 and 5.2, its effectiveness and accuracy are illustrated using, respectively, experimental data from an instrumented steel beam and data collected in situ on a footbridge.

Section snippets

Equations of motion

Consider the continuous-time governing equations of motion for a linear system discretized in spaceMu¨(t)+Cu˙(t)+Ku(t)=f(t)=Sp(t)p(t)where u(t)RnDOF is the vector of displacement, M, C and KRnDOF×nDOF denote the mass, damping and stiffness matrix, respectively, and f(t) is the excitation vector. The excitation is factorized into an input force influence matrix Sp(t)RnDOF×np, and the vector p(t)Rnp representing the np force time histories. Each column of the matrix Sp gives the spatial

Joint input-state estimation

In the field of system control, optimal filtering techniques for linear systems in the presence of unknown inputs have received considerable attention during the last decades [22], [9], [19], [20]. An optimal recursive filter based on the assumption that no prior information about the unknown input is available was first developed by Kitanidis [22] for linear systems without direct transmission. Further work on the topic was later presented by Hsieh [21] and Gillijns and De Moor [13]. The

Simulated example

The effectiveness and applicability of the new algorithm are first illustrated using simulated measurement data in an example where two transient point loads are applied to a cantilever steel beam. In this example, the positions of the forces are assumed known and the forces and states are jointly estimated from simulated acceleration data. The steel beam has a square section with a width of 50.8 mm and a height of 25.4 mm, and is 1 m long. The Young's modulus and density are taken as 210 GPa and

Experimental results

In this section the effectiveness of the proposed algorithm is further illustrated by means of a laboratory experiment on an instrumented steel beam as well as an in situ experiment on a footbridge. In both examples a distinction is made between joint input-state estimation, where the positions of the forces are assumed known and the states as well as the true applied forces are identified, and pure state estimation, where the positions of the forces are assumed unknown and the states are

Conclusions

An algorithm which jointly estimates the input and state of a structure from a limited number of noise-contaminated acceleration measurements was presented. The algorithm was developed by extending an existing joint input-state estimation filter from control system theory to applications in linear structural dynamics. It requires no prior information on the dynamic evolution of the input forces, is easy to implement, and allows online application. Estimation accuracy was tested using both

Acknowledgments

The research presented in this paper has been performed within the framework of the project OT-05-41 “A generic methodology for inverse modeling of dynamic problems in civil and environmental engineering”, funded by the Research Council of the K.U. Leuven. Their financial support is gratefully acknowledged. The first and last authors are members of the K.U. Leuven - BOF PFV/10/002 OPTEC—Optimization in Engineering Center.

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