Real-time simultaneous input-state-parameter estimation with modulated colored noise excitation
Introduction
Structural health monitoring (SHM) using vibration data has received considerable attention for decades [1], [2], [3], [4], [5], [6]. An important aspect in SHM is to estimate the structural health indicators and/or other required quantities, whose values are important for the monitoring algorithm. In general, there are mainly three categories of unknown quantities in SHM applications, namely inputs, structural states and uncertain parameters. Input identification aims to reconstruct the unknown dynamic inputs applied to a structure from its responses by a limited number of sensors [7]. It is motivated by the situations where direct measure of the inputs is not feasible or very difficult, such as dynamic vehicle loads on bridges and dynamic wind loads on tower structures [8], [9], [10]. Structural state estimation is an essential component for the control of adaptive structures [11]. Moreover, reliable state estimation results can be used to reconstruct stress/strain outputs for fatigue damage identification [12]. Since it is not feasible to obtain direct measurements of the entire states for a large structure, reconstruction of the states from the measurements is necessary. Parameter estimation corresponds to indicate the behavioral variability of the structure and the estimation results are always utilized as the indicator to inform the health status of the monitored structure [13], [14], [15].
Simultaneous estimation of the structural states and parameters was the first addressed subproblem. It can be expressed as estimating an augmented state vector which contains the unknown structural states and uncertain model parameters. Various filter-based methods were widely adopted for identifying this augmented state vector, such as extended Kalman filter (EKF) [16], [17], [18], [19], unscented Kalman filter [20] and particle Kalman filter [21].
Joint input-state estimation techniques have been also developed based on the known model parameters of the underlying system. Gillijns and De Moor [22] built a recursive filter approach based on linear minimum-variance unbiased estimation. The input forces were calculated from the innovation by weighted least-squares method and thus the state estimation problem was converted to a conventional Kalman filtering problem for the linear system. The resulting filter in [22] is under the framework of the Kalman filter (KF), except that the actual value of the input was displaced by an optimal estimation. Lourens et al. [23], [24] developed an augmented KF algorithm for joint input-state estimation problem. In this method, the unknown input forces were modeled as random walks and then included in the state vector to form an augmented state vector. Therefore, both the input forces and states can be identified simultaneously by estimating this augmented state vector recursively using the KF. Although methods in [22], [23], [24] provided analytical solutions for joint input-state estimation problems, the predicated input forces and displacement responses using only noisy acceleration measurements encountered low-frequency drifts [23], [24], [25], [26], [27]. This inaccuracy is caused by the integral nature of the inputs and states using only noisy acceleration measurements by filter-based techniques [25]. Azam et al. [25] also utilized random walk models serving for describing the input forces and then developed a dual KF algorithm to predict the unknown input forces and the states of a linear state-space model. In addition, it was revealed that once the covariance matrix of the inputs force in dual KF was tuned properly, the low-frequency drifts in the estimated inputs and displacement responses were disappeared. Therefore, L-curve regularization method was suggested in [25] to calibrate the covariance matrix of the input forces. However, the regularization scheme prohibited real-time identification of unknown inputs and structural states. Naets et al. [26] recommended to add dummy displacement/strain measurements in [23] to alleviate the drift issue. Liu et al. [27] derived recursive solutions for an improved KF with unknown inputs based on the scheme of the conventional KF. Furthermore, data fusion of partially measured displacement and acceleration responses was utilized to tackle the low-frequency drifts in the inputs and displacements estimation. Maes et al. [28] expanded the joint input-state estimation method in [22] to consider the situation where the process noise and measurement noise are correlated. The fused measurements of displacement and acceleration were utilized again in [28] to avoid the drift issue. Sedehi et al. [29] developed a sequential Bayesian estimation of the input forces and states for linear time-invariant systems. In this method, the estimated inputs obtained from the previous time step were disregarded and displaced by a zero vector, while other statistical information about the inputs was held. This prior distribution aimed to resist the detrimental impacts due to erroneous input estimates. By adding this constraint, the low-frequency drifts in the estimated inputs and displacement responses can be significantly alleviated. Li et al. [30] proposed an identification algorithm for estimating ground excitation using modal expansion method and KF approach.
The combined input-state-parameter estimation will be much more complicated if the model of the structural system is unknown. In this respect, the joint input-state-parameter estimation problem has attracted much attention quite recently. Zhao et al. [31] proposed a hybrid identification method to estimate the inputs and structural parameters for the concerned problem. In this method, the structural parameters were first calculated based on the minimum modal information and then the unknown inputs were obtained by solving a differentiation equation. Yang et al. [32] and Huang et al. [33] presented adaptive filter techniques for estimating inputs, states and parameters, based on minimizing the quadratic sum-of-square error. Naets et al. [34] again utilized random walks to model the input forces and established a parametric model reduction approach to guarantee the real-time tracking capability of the method. Then, the reduced model was coupled to EKF with augmented states including inputs, structural states and parameters, so an efficient online estimation framework can be achieved. Liu et al. [35] predicted the input forces using the least-squares estimation and then developed a recursive input-state-parameter estimation approach based on the conventional EKF. Ebrahimian et al. [36] developed a sequential maximum a posterior estimation approach, which focused on solving the joint input-parameter estimation problem for nonlinear systems. Lei et al. [37] derived an analytical recursive solution for joint input-state-parameter estimation problem based on unscented Kalman filter. Note that the combined measurements of displacement/strain and acceleration were used in [34], [35], [36], [37] to overcome the typical low-frequency drifts in the estimated inputs and displacements. Dertimanis et al. [38] developed a recursive methodology for joint input-state-parameter estimation problem. A typical KF was utilized to calculate the unknown inputs modeled as random walks and then, based on the calculated inputs, the unscented Kalman filter was employed to predict the structural states and parameters. Moreover, a tuning factor was considered in the random walk model for calibrating the inputs to overcome the drift problem. Castiglione et al. [39] used a time-varying auto-regressive model to describe the unknown inputs and then developed a novel iterative unscented Kalman filter for parameter-input estimation of nonlinear structural systems. Although some existing investigations focused on eliminating the drift problem by using dummy displacement measurements [26], [40] and combined measurements of displacement/strain and acceleration [27], [28], [34], [35], [36], the drift problem remains to be a significant issue for both input-state and input-state-parameter estimations.
In this study, a novel real-time simultaneous input-state-parameter estimation approach with modulated colored noise excitation is presented using only noisy acceleration measurements. The rest of this paper is outlined as follows. Section 2 presents the state-space model of dynamical systems and introduces the concerned problem. Section 3 elaborates the proposed methodology, including the modeling equation for the unknown input forces, the real-time estimation of inputs, structural states and uncertain parameters and the recursive updating the noise covariance matrices in EKF. Section 4 provides the procedure of the proposed algorithm. Section 5 is dedicated to numerical demonstrations. Finally, Section 6 gives the conclusion of the paper.
Section snippets
Problem formulation
Consider a linear dynamical system with degrees of freedom (DOFs) and the continuous-time equation of motion:where , and are the acceleration, velocity and displacement vectors at time , respectively; , and are the mass, damping and stiffness matrices of the system, respectively. The damping matrix and stiffness matrix are parameterized by uncertain structural parameters in ; is the
Modeling of the unknown input forces
The unknown input forces in Eq. (1) are modeled as modulated colored noise given by:where is a zero-mean Gaussian stochastic process with covariance matrix ; is the internal filter state vector; , and are the parameterized state-space model matrices given by:
Summary of the proposed method
The implementation of the proposed real-time simultaneous input-state-parameter estimation algorithm is listed as follows: Compute the one-step-ahead predicated state estimate and its covariance matrix using Eqs. (11), (12), respectively. if thenReal-time simultaneous input-state-parameter estimation 1. Initialization: Set initial values for , , and . 2. Recursive stage: Set for do Compute the updated noise parameter vector using Eq. (21). else
Example 1: Planar truss
A planar truss shown in Fig. 1 is considered in the first example. The information about the truss is listed in Table 1. The first five natural frequencies are , , , and . The damping matrix is taken as: , where and , so the damping ratios of the first two modes are . Six stiffness parameters are used to characterize the stiffness matrix: , where is the nominal stiffness submatrix.
Conclusion
This paper developed a novel approach for real-time simultaneous input-state-parameter estimation with modulated colored noise excitation. The unknown input forces are modeled as modulated colored noise, which is realistic for real environmental excitations. Such modeling can effectively resolve the typical low-frequency drift problem in the estimated inputs and displacement responses for joint input-state-parameter estimation, when only noisy acceleration measurements are used. Moreover, a
CRediT authorship contribution statement
Ke Huang: Methodology, Software, Validation, Formal analysis, Writing – original draft, Writing - review & editing. Ka-Veng Yuen: Conceptualization, Methodology, Resources, Writing - review & editing, Supervision, Project administration, Funding acquisition. Lei Wang: Writing - review & editing, Funding acquisition.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work is funded by The Science and Technology Development Fund, Macau SAR (File no. 019/2016/A1 and SKL-IOTSC-2018-2020), University of Macau (File no. MYRG2018-00048-AAO), Guangdong-Hong Kong-Macau Joint Laboratory Program (Project no. 2020B1212030009), the National Key R&D Program of China (Grant No. 2019YFC1511000) and the National Natural Science Foundation of China (Grant No. 52008037). Their generous support is gratefully acknowledged.
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2023, Mechanical Systems and Signal ProcessingCitation Excerpt :Applications of these methods for identifying seismic excitations and updating nonlinear mechanics-based models have been the subject of recent works by Astroza et al. [22,23]. The parameterization of the input forces through time-varying autoregressive models, Gaussian process models, and colored stochastic processes can help enhance the stability of input estimations [24–26]. Moreover, adaptive calibration of the measurement noise covariance matrix allows obtaining more accurate estimates [27].