Sequential non-linear least-square estimation for damage identification of structures
Introduction
An important goal of any structural health monitoring system is to ensure the reliability and safety of the structure in addition to the minimization of the life-cycle cost. An early detection of structural damages is critical for the decision making of repair and replacement maintenance in order to guarantee a specified structural reliability. Consequently, the structural damage detection, based on vibration data measured from the structural health monitoring system, has received considerable attention recently (e.g., [1], [2]).
Time domain analysis techniques, including the least-square estimation (LSE) (e.g., [3], [4], [5]) and the extended Kalman filter (EKF) (e.g., [6], [7], [8], [9], [10], [11], [12]), have been used for the identification of structural parameters. In practice, acceleration responses are usually measured on-line, whereas the velocity responses can be obtained through a single numerical integration. For the displacement response, however, a double numerical integration from the acceleration response results in a significant numerical drift that is also magnified seriously when a damage occurs (e.g., [13]), and it is difficult to remove the drift on-line. Hence, the application of the LSE approach for the on-line damage identification of structures may require the displacement (in addition to the acceleration) measurements, which may not be practical. Under certain conditions for some specific models of non-linear hysteretic structures, the displacement measurement may not be needed, and hence the on-line damage identification is possible (e.g., [3], [4], [5]) as described in [13].
With only the measurements of acceleration responses, the on-line system identification and damage detection are possible based on the EKF approach. This is because in the EKF approach, both the constant -parametric vector and the -state vector are treated as unknown quantities to be estimated. In this approach, an unknown extended state vector , consisting of and , is introduced, and the resulting state equation for is highly non-linear. The non-linear equation is then linearized and the Kalman filter approach is applied. Unfortunately, some poles corresponding to the unknown of the linearized state equation lie on the imaginary axis, such that the solutions (estimates) may easily become unstable. Further, due to the linearization, the solutions may not converge if the initial guesses of the parametric values are outside the region of convergence. Likewise, the dimension of the extended state vector is quite large, and hence the computational efforts required for estimating is quite involved.
In order to remove all the drawbacks of LSE and EKF approaches for the on-line system identification, a new approach, referred to as the sequential non-linear least-square estimation (SNLSE), is proposed in this paper. Here, we only need the acceleration response measurements, and consider the -parametric vector and the -state vector as unknown quantities to be estimated. Unlike the EKF approach, we estimate and separately and sequentially. In the first step, the state vector is assumed to be given and the estimate of the parametric vector is obtained by minimizing an objective function (sum square errors). In the second step, the estimate is considered to be a function of the unknown state vector, i.e., , and the objective function (sum square errors) is a highly non-linear function of . Then, the non-linear objective function is linearized to become a quadratic form of , and the estimate of the state vector is obtained by minimizing the resulting quadratic objective function. This proposed approach is referred to as the SNLSE, and the solutions thus obtained and presented in this paper are not available in the previous literature.
The new approach described above is for the on-line identification of constant structural parameters. When structural damages occur, some parameters will vary, such as the stiffness degradation, and the structural parameters become time-varying. To account for the parametric variations, the methods of constant forgetting factor (e.g., [8]), variable forgetting (or fading) factor (e.g., [9]), and variable trace (e.g., [3]) have been proposed with limited success (e.g., [5]). Recently, an adaptive tracking technique has been proposed for the LSE [5] and EKF [11], [12] approaches, which has been demonstrated to be capable of identifying the structural damages on-line. Such an adaptive tracking technique will be implemented in the proposed SNLSE approach to track the structural damage. Simulation results for both linear and non-linear structures will be presented to demonstrate the efficiency of the proposed adaptive SNLSE in tracking the structural damage.
Section snippets
Sequential non-linear LSE
The equation of motion of a -DOF non-linear structure can be expressed asin which mass matrix; -displacement vector; -damping force vector; -stiffness force vector; ; and . The acceleration responses and the excitation forces are measured, and the unknowns to be identified are the state vector , including displacement and velocity vectors, and
Adaptive non-linear LSE
The recursive solution in Eqs. (5)–(7) is derived based on the premise of constant parametric vector . Here, the adaptive tracking technique proposed in [5] will be implemented in the new SNLSE solutions derived above in order to identify time-varying parameters of the structures for detecting the damages as follows.
To track the variation of each parameter, say the th element of , the estimation error is proposed to be expressed by ,
Determination of adaptive factor matrix
The adaptive factor matrix is proposed to be determined by adapting the current input data at as follows. Letin which -residual error vector and -predicted error vector. Substituting Eq. (26) into Eq. (29) and taking an expectation, one obtainsin which Eq. (30) has been used. The covariance matrix of the predicted error on the right-hand side of Eq. (31) can be estimated and is
Numerical examples
To demonstrate the accuracy and effectiveness of the proposed new SNLSE approach for parametric identifications and damage detection of structures, the Phase I ASCE structural health monitoring benchmark building, two 2-DOF non-linear building models and one 5-DOF non-linear hysteretic building model subject to either external forces or earthquake excitations will be considered. In these examples, the absolute accelerations of each floor and the external excitations are measured. All the
Conclusion
A new innovative approach, referred to as the sequential non-linear least-square estimation (SNLSE), has been proposed to identify the system parameters on-line for structural health monitoring. In this approach, acceleration responses and external excitations are measured, whereas the structural parameters and response state vector are estimated similar to the extended Kalman filter (EKF) approach. Analytical recursive solutions for the unknown structural parameters and that for the unknown
Acknowledgments
This research is supported by US National Science Foundation NSF-CMS-0140710.
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2020, International Journal of Non-Linear MechanicsCitation Excerpt :In the past decades, lots of vibration based structural damage identification approaches have been proposed aiming at identifying structural stiffness employed to describe structural damage in either time or frequency domains [1–3]. In time domain identification approaches, the least square estimation (LSE) based methods and the Kalman filter (KF) based methods are widely employed to identify structural parameters such as stiffness to evaluate the severity of damages [4–9]. Because it would be infeasible to measure dynamic responses at each degree-of-freedom (DOF) of an engineering structure to be identified, the KF methods [10], including extended Kalman filter (EKF) and unscented Kalman filter (UKF) [11,12], have been formulated to perform such identifications using a limited number of output responses and input information.