An integrated real-time structural damage detection method based on extended Kalman filter and dynamic statistical process control

Abstract

Real-time structural parameter identification and damage detection are of great significance for structural health monitoring systems. The extended Kalman filter has been implemented in many structural damage detection methods due to its capability to estimate structural parameters based on online measurement data. Current research assumes constant structural parameters and uses static statistical process control for damage detection. However, structural parameters are typically slow-changing due to variations such as environmental and operational effects. Hence, false alarms may easily be triggered when the data points falling outside of the static statistical process control range due to the environmental and operational effects. In order to overcome this problem, this article presents a novel real-time structural damage detection method by integrating extended Kalman filter and dynamic statistical process control. Based on historical measurements of damage-sensitive parameters in the state-space model, extended Kalman filter is used to provide real-time estimations of these parameters as well as standard derivations in each time step, which are then used to update the control limits for dynamic statistical process control to detect any abnormality in the selected parameters. The numerical validation is performed on both linear and nonlinear structures, considering different damage scenarios. The simulation results demonstrate high detection accuracy rate and light computational costs of the developed extended Kalman filter–dynamic statistical process control damage detection method and the potential for implementation in structural health monitoring systems for in-service civil structures.

Keywords

extended Kalman filter parameter identification state-space model statistical process control structural damage detection

Introduction

Maintaining civil engineering structures in good conditions utilizing structural health monitoring (SHM) has become an increasingly viable option in recent decades. The goal of SHM is to determine the status of the structure, detect damage, and implement characterization strategies for engineering structures (Yan et al., 2007). Some extreme events, such as earthquake, typhoon, or blast can cause structural damages and lead to structural failure rapidly. During these events, real-time reliable information regarding the condition and the integrity of the structure is invaluable for infrastructure owners and first responders (e.g. policemen, fire fighters, and rescuers) to make rapid decisions (Dyke et al., 2010). Thus, the capabilities of real-time performance evaluation and damage detection in SHM systems are extremely valuable.

Many structural parameter identification methods have been developed in order to estimate system parameters based on the structural response measurements directly, including extended Kalman filter (EKF) (Hoshiya and Saito, 1984; Mariani and Corigliano, 2005), unscented Kalman filter (UKF) (Mariani and Ghisi, 2007; Wu and Smyth, 2007; Chatzis et al., 2015), and particle filter (Chatzi and Smyth, 2009; Eftekhar Azam et al., 2012). In these Kalman filter (KF)-based methods, the dynamic structures are modeled using state space formulation, and include state and measurement equations. These methods are attractive in view of the recursive feature of KF such that they can estimate structural parameters directly, and provide detailed information and understanding about the existence, location, and severity of the structural damage. In the EKF, the state distribution is approximated by a Gaussian random variable, which is then propagated through the first-order linearization of a nonlinear system. The UKF differs from the EKF in the manner of representing Gaussian random variables. The UKF uses an unscented transform to generate a minimal set of carefully chosen sample points to present the state distribution. These sample points capture the means and covariance of the Gaussian random variables, and when propagated through the nonlinear system, captures the posterior mean and covariance accurately to the second order for any nonlinearity. The particle filters can deal with nonlinear systems with non-Gaussian posterior distribution of the state. The concept of the method is that the approximation of the posterior distribution of the state is done through the generation of a large number of samples (weighted particles), using the Monte Carlo method. The drawback is the fact that depending on the problem, a large number of samples may be required, thus making the particle filter analysis computationally expensive. The UKF and particle filter may create a more accurate estimate than the EKF for highly nonlinear systems; however the EKF is widely used in state estimation and structural parameter estimation for civil engineering problems due to its straightforward implementation, high updating frequency, fast convergence speed, and low computational costs (Lei et al., 2012; Liu et al., 2009; Soyoz and Feng, 2008; Yang et al., 2006, 2007; Yin et al., 2013; Zhou et al., 2008). Therefore, the EKF is utilized in this paper.

In the above studies, the structural parameters of the system are assumed to be constant values throughout time. If the identified values of structural parameters deviated from their constant values, then the structural damage was identified. However, this assumption does not consider the influence of varying environmental and operational conditions, which can cause structural parameters to fluctuate and differ from their constant values even when no damage has occurred. The effects of environmental and operational conditions on the variance of structural parameters have been found and reported in many long-term structural monitoring projects (Cross et al., 2013; Peeters et al., 2001; Reynders et al., 2014; Sohn, 2007; Spiridonakos and Chatzi, 2014). Thus EKF for in-service large scale civil structures need to be able to account for these effects.

Statistical process control (SPC) has been widely used to monitor and control processes due to its early detection and prevention capability. The SPC attempts to differentiate additional sources of variation from natural sources in a process using control limits. When the process data deviate from their normal range and trigger the control limits, the excessive variation of data indicates the presence of faults or damage in the system. SPC has been applied to many damage detection problems in the field of SHM, especially in long-term bridge-monitoring research projects (Deraemaeker et al., 2008; Fugate et al., 2001; Kullaa, 2003; Magalhães et al., 2012; Sohn et al., 2000; Zapico-Valle et al., 2011). The above studies all used static SPC methods, in which the control limits are fixed constant values calculated from statistical indicators of historical data. However, the range of the control limits cannot account for the changes in the structural parameters due to environmental and operational variations, which may trigger false alarms when data points fall outside the static control limits but the structure has not been damaged.

To overcome this limitation, the dynamic statistical process control (DSPC) method can be used to continuously adjust control limits and provide adaptive, changing control limits in real time. In order to establish the dynamic control limits for the target parameters, the values of the mean and standard deviation are required to be updated in each time step. The KF provides an ideal approach to establish dynamic control limits for its state variables for engineering damage detection problems (Sun et al., 2012). In each time step, the KF not only updates estimation values for state variables but also generates a state covariance matrix, which stores variance of each state variable in the diagonal elements. Therefore, the combination of EKF and DSPC can achieve both online parameter estimation and dynamic control limits formulation, which has potential to be used for real-time structural damage detection.

Recently, a novel structural damage detection method combining EKF and DSPC has been developed by the authors (Jin et al., 2015a, 2015b). In this article, the EKF-DSPC is further improved with more rigorous derivations and testing scenarios. Moreover, the entire article has been extended with more discussion and insights on the initialization of the state and covariance matrix, noise covariance matrices, and the estimate processes on different structural models. This article aims to develop a real-time EKF-DSPC-based structural damage detection method considering the variation effects that an in-service structure may encounter during operation. Numerical tests are performed to validate the effectiveness of the EKF-DSPC method for identifying structural parameters and detecting the occurrence of damage, including a three-story linear structure and a two-story nonlinear hysteric structure, with multiple common damage scenarios. The testing results show that proposed method has a good performance to identify structural parameters and detect damage with high accuracy and with low computational costs. In section “EKF-based structural parameter identification,” the methodology regarding EKF-based structural parameter identification is reviewed. Section “DSPC” describes the theory of DSPC and summarizes how the two methods are combined for real-time damage detection in this article. In section “Numerical Validation,” applications of the proposed EKF-DSPC method will be presented for a 3-degree-of-freedom (3-DOF) linear structure and a 2-degree-of-freedom (2-DOF) nonlinear hysteretic dynamic system, considering various damage scenarios. Finally, testing results discussion and the benefits of developed damage detection method will be highlighted.

EKF-based structural parameter identification

KF and EKF

The KF provides a recursive solution for linear dynamic systems that can be represented in state-space formulation. Each updated estimate of the state is computed from the previous estimate with new input data only, instead of all previous data points. Thus the KF is less demanding of storage space and is computationally more efficient. The state equation and measurement equation of KF are given by

x_{k + 1} = F_{k} x_{k} + w_{k}

(1)

y_{k} = H_{k} x_{k} + v_{k}

(2)

where x_k is the state vector and y_k is the measurement vector. The process noise w_k and observation noise v_k are independent, zero-mean, Gaussian processes with covariance matrices Q_k and R_k, respectively. The matrices F, H, Q, and R are assumed known and possibly time-varying.

The KF consists of two steps: time update and measurement update. In the time update, the state and covariance propagations are implemented as follows

{\hat{x}}_{k + 1 | k} = F_{k} {\hat{x}}_{k | k}

(3)

P_{k + 1 | k} = F_{k} P_{k | k} {F_{k}}^{'} + Q_{k}

(4)

{\hat{y}}_{k + 1 | k} = H_{k} {\hat{x}}_{k + 1 | k}

(5)

where the prior state ${\hat{x}}_{k | k}$ and state covariance matrix $P_{k | k}$ are propagated to ${\hat{x}}_{k + 1 | k}$ and $P_{k + 1 | k}$ , respectively. The ${\hat{x}}_{k + 1 | k}$ is then used to generate the estimated measurement ${\hat{y}}_{k + 1 | k}$ through H_k.

In the measurement-update step, the filter gain W_k, the posterior state ${\hat{x}}_{k + 1 | k + 1}$ , and the state covariance $P_{k + 1 | k + 1}$ are produced as follows

S_{k + 1} = H_{k + 1} P_{k + 1 | k} {H_{k + 1}}^{'} + R_{k + 1}

(6)

W_{k + 1} = P_{k + 1 | k} {H_{k + 1}}^{'} {S_{k + 1}}^{- 1}

(7)

{\hat{x}}_{k + 1 | k + 1} = {\hat{x}}_{k + 1 | k} + W_{k + 1} (y_{k} - {\hat{y}}_{k + 1 | k})

(8)

P_{k + 1 | k + 1} = P_{k + 1 | k} - W_{k + 1} S_{k + 1} {W_{k + 1}}^{'}

(9)

where S_k₊₁ is the innovation covariance matrix. Note that the roots of the diagonal elements of $P_{k + 1 | k + 1}$ are the standard deviation of each state variable in the state vector.

The EKF approach applies the standard KF to nonlinear systems by continually updating a linearization around the previous state estimate through first-order Taylor series expansion. The linearized state matrix F_k is taken as the partial derivative of nonlinear function f(x_k, u_k, k) with respect to x at ${\hat{x}}_{k | k}$ , that is, a Jacobian matrix, while the linearized measurement matrix H_k is obtained as the partial derivative of nonlinear function h(x_k, u_k, k) with respect to x at ${\hat{x}}_{k + 1 | k}$ and are given by

F_{k} = {\frac{\partial f (x_{k}, u_{k}, k)}{\partial x} |}_{x = {\hat{x}}_{k | k}}

(10)

H_{k} = {\frac{\partial h (x_{k}, u_{k}, k)}{\partial x} |}_{x = {\hat{x}}_{k + 1 | k}}

(11)

The recursions of time update and measurement update for EKF follow equations (3)–(9) used in the standard KF. Given the initial values of state vector x₀, the initial state covariance matrix P₀, the process noise covariance matrix Q₀, and the measurement noise covariance matrix R₀, the EKF procedure can be recursively implemented to estimate the structural parameters based on the measurement data of the dynamic system. Due to the nature of the recursion method, these structural parameters can be updated at each time step, which makes real-time structural damage detection possible. More details on KF and EKF can be found in Welch and Bishop (2006).

EKF for linear structures

The equation of motion for an m-DOF linear structure can be represented as

M \overset{\cdot\cdot}{q} (t) + C \overset{\cdot}{q} (t) + Kq (t) = η u (t)

(12)

where M, C, and K represent (m × m) mass matrix, damping matrix, and stiffness matrix, respectively; q(t) is the (m × 1) displacement vector; $\overset{\cdot}{q} (t)$ is the velocity vector; $\overset{\cdot\cdot}{q} (t)$ is the (m × 1) acceleration vector; u(t) is the (m × 1) excitation force vector; and η is the (m × m) excitation influence matrix.

As discussed in section “KF and EKF,” the KF approach gives a simple and efficient way to estimate the state and parameters in any system models. For structural dynamic systems, the states (e.g. displacement and velocity) and parameters (e.g. stiffness and damping) are often required to be identified simultaneously. Therefore, even though the structural model is linear, due to the nonlinear coupling feature between the structural states and parameters, EKF will be used. The state vector for structure model is thus formed as

x' (t) = [\begin{matrix} x (t) \\ α \end{matrix}] = [\begin{matrix} q (t) \\ \overset{\cdot}{q} (t) \\ α \end{matrix}]

(13)

where x(t) is the state of structural dynamic system including displacement q(t) and velocity $\overset{\cdot}{q} (t)$ , and α is the parameter vector to be estimated. The continuous state equation becomes

\begin{matrix} \overset{\cdot}{x}' (t) = [\begin{matrix} \overset{\cdot}{x} (t) \\ \overset{\cdot}{α} \end{matrix}] = [\begin{matrix} \overset{\cdot}{q} (t) \\ \overset{\cdot\cdot}{q} (t) \\ \overset{\cdot}{k} \\ \overset{\cdot}{c} \end{matrix}] \\ = [\begin{matrix} \overset{\cdot}{q} (t) \\ M^{- 1} [η u (t) - (C \overset{\cdot}{q} (t) - Kq (t))] \\ 0 \\ 0 \end{matrix}] \equiv f (x' (t), u (t), t) \end{matrix}

(14)

and the continuous measurement equation becomes

y (t) = h (x' (t), u (t), t)

(15)

By discretizing and linearizing at time t_k (k = 1, 2,…) using first-order Taylor series expansion, the discrete-time space model is formulated as

x'_{k + 1} = F_{k} x'_{k} + w_{k}

(16)

y_{k} = H_{k} x'_{k} + v_{k}

(17)

where

\begin{matrix} F_{k} = {\frac{\partial f ({x'}_{k}, u_{k}, k)}{\partial x'} |}_{x' = {x'}_{k}} \\ = [\begin{matrix} [0] & I & [0] & \dots & [0] \\ - M^{- 1} K & - M^{- 1} C & - M^{- 1} \frac{\partial K}{\partial α_{1}} q (k) - M^{- 1} \frac{\partial C}{α_{1}} \overset{\cdot}{q} (k) & \dots & - M^{- 1} \frac{\partial K}{\partial α_{α}} q (k) - M^{- 1} \frac{\partial C}{α_{α}} \overset{\cdot}{q} (k) \\ [0] & [0] & [0] & \dots & [0] \end{matrix}] \end{matrix}

(18)

When the measurement is the acceleration $\overset{\cdot\cdot}{q} (t)$ , the measurement matrix H_k has the form

\begin{matrix} H_{k} = {\frac{\partial h ({x'}_{k}, u_{k}, k)}{\partial x'} |}_{x' = {x'}_{k}} \\ = [\begin{matrix} - M^{- 1} K & - M^{- 1} C & - M^{- 1} \frac{\partial K}{\partial α} q (k) - M^{- 1} \frac{\partial C}{\partial α} \overset{\cdot}{q} (k) \end{matrix}] \end{matrix}

(19)

The EKF recursion process follows equations (3)–(9) as shown in section “KF and EKF.” The selection of initial state, initial state covariance matrix, process noise, and measurement covariance matrices will be discussed in section “Numerical validation.”

EKF for nonlinear structural model

The equation of motion of a hysteretic nonlinear structure subjects to ground excitation can be written as

M \overset{\cdot\cdot}{q} (t) + C \overset{\cdot}{q} (t) + Kz (t) = η u (t)

(20)

where M, C, and K represent mass matrix, damping matrix, and stiffness matrix, respectively; q(t), $\overset{\cdot}{q} (t)$ , and $\overset{\cdot\cdot}{q} (t)$ are the displacement, velocity, and acceleration vector, respectively; u(t) is the excitation force vector; and η is the excitation influence matrix.

The additional vector variable z(t) = [z₁(t) …z_i(t)]^T is the hysteretic component vector, and the hysteretic component can be defined by Bouc–Wen model as

{\overset{\cdot}{z}}_{i} = {\overset{\cdot}{q}}_{i} - β_{i} | {\overset{\cdot}{q}}_{i} | {| z_{i} |}^{α_{i} - 1} z_{i} - γ_{i} {\overset{\cdot}{q}}_{i} {| z_{i} |}^{α_{i}}

(21)

where i represents ith DOF; $\overset{\cdot}{q} (t)$ and z_i are the velocity and hysteretic component of ith DOF, respectively; and α_i, β_i, and γ_i are the hysteretic parameters of the Bouc–Wen model.

The state vector is then

x' (t) = [\begin{matrix} q (t) \\ \overset{\cdot}{q} (t) \\ z \\ k \\ c \\ α \\ β \\ γ \end{matrix}]

(22)

The continuous state equation is thus obtained as follows

\begin{matrix} \overset{\cdot}{x}' (t) = [\begin{matrix} \overset{\cdot}{q} \\ \overset{\cdot\cdot}{q} \\ \overset{\cdot}{z} \\ \overset{\cdot}{k} \\ \overset{\cdot}{c} \\ \overset{\cdot}{α} \\ \overset{\cdot}{β} \\ \overset{\cdot}{γ} \end{matrix}] = [\begin{matrix} \overset{\cdot}{q} (t) \\ M^{- 1} [η u (t) - (Cx (t) + Kz (t))] \\ {\overset{\cdot}{q}}_{i} - β_{i} | {\overset{\cdot}{q}}_{i} | {| z_{i} |}^{α_{i} - 1} z_{i} - γ_{i} {\overset{\cdot}{q}}_{i} {| z_{i} |}^{α_{i}} \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix}] \\ = f (x' (t), u (t), t) \end{matrix}

(23)

Since the measurement is typically the acceleration, the continuous measurement equation becomes

y (t) = M^{- 1} [η u (t) - (Cx (t) + Kz (t))] = h (x' (t), u (t), t)

(24)

The discretization and linearization of the nonlinear system follow the same manner as for the linear method shown in section “EKF for linear structures”. For the formulation of the state-space model, the recursive estimation process follows the standard procedure.

DSPC

The structural parameters for in-service civil structures can be viewed as slow-changing variables, driven by various environmental and operational effects (e.g. temperature, ground and traffic loadings, and system noise). In previous parameter identification research, structural parameters were usually treated as constant values; however, on-site long-term monitoring shows that these structural parameters have significant variances under the above effects (Sohn, 2007), which may cause false alarms. Enabling online monitoring system to detect structural damage while maintaining a low false alarm rate is an important issue in SHM.

SPC can be used to distinguish the abnormal variance from the normal variance of a process. If a structure is in good health without any structural damage, the structural parameters should jump and fall insignificantly within the SPC control limits. However, when structural damage occurs, the key parameter is more likely to jump outside of the range of the control limits, thus the potential damage can be detected. Traditional static SPC uses constant values as upper and lower boundaries of control limits, which can be represented as

[μ - n \times σ, μ + n \times σ]

(25)

where µ is the mean, σ is the standard deviation of the estimated state, and n is a pre-defined integer number to set the confidence level for control limits.

In equation (25), the values of µ and σ are obtained from historical measurements when the structure is operating in good condition. Nevertheless, this approach may not be effective due to the following reasons. If µ and σ are calculated too conservatively, for example, large σ, a large range of SPC control limits may make it difficult to capture any occurrence of damage. If µ and σ are calculated too small, the narrow range of SPC control limits may not be able to update and adjust to varying environmental and operational conditions, thus leading to false alarms.

To address the above issues, DSPC is adopted in this article to replace the traditional static SPC. Instead of using the pre-defined fixed control limits based on the historical data, the real-time structural damage detection requirements call for an approach to enable the control boundaries to update and adjust to the changing trend of the state parameters. The recursive property of KF and the standard deviation obtained from covariance matrix provide a good solution to update the parameters of DSPC. At each time step, the EKF not only estimates the state variables but also generates state covariance matrix P_k, whose diagonal elements store the variance value of each state variable. The average estimated state variables over the past j points are used as the updated mean values, and the past j standard deviations are also averaged as the updated standard deviation values. In this approach, the control limits of DSPC can be updated in real time using only new measurements.

The EKF assumes a Gaussian random variable to estimate the state. Typically, the “three-sigma Gaussian rule” is widely used in industry to cover the 99.7% probability of all values lying within three standard deviations of the mean in a normal process, which can be empirically treated as “near certainty” (Pukelsheim, 1994; Wheeler and Chambers, 1992; Wiborg et al., 2014). This means that by default, the EKF holds that at every step, the confidence interval in the estimate be equal to the mean ±3 times the standard deviation. The “three-sigma Gaussian rule” will be followed in this article to set n equal to 3, and the range of three-sigma control limits of DSPC is thus described as

[{\bar{μ}}_{α_{i}, k} - 3 {\bar{σ}}_{α_{i}, k}, {\bar{μ}}_{α_{i}, k} + 3 {\bar{σ}}_{α_{i}, k}]

(26)

where ${\bar{μ}}_{α_{i}, k}$ and ${\bar{σ}}_{α_{i}, k}$ are the averaged mean and averaged standard deviation of the state $α_{i}$ at time k, respectively.

The flow chart for the proposed EKF-DSPC structural damage detection method in this article is depicted in Figure 1. As the flow chart shows, starting from a set of initiation for state variables and covariance matrices, the EKF recursively estimates and updates the state variables based on new measurements. Then, in each time step, the DSPC is integrated with EKF to identify the abnormal changes in state variables based on the presented control rules. In this approach, real-time structural damage detection can be achieved with high accuracy in a short period of time.

Figure 1.

Flowchart of EKF-DSPC damage detection method.

Numerical validation

The developed EKF-DSPC structural damage detection method was tested using numerical simulations of dynamic structural systems excited by El Centro earthquake ground motion. Both linear and nonlinear structures with different damage scenarios were considered. Four cases were used to validate the proposed EKF-DSPC method, as shown in Table 1. Simulated acceleration for each story of the structure and the ground excitation were fed into the EKF algorithm to estimate the structural parameters. At the same time, the DSPC was utilized for real-time damage detection purposes. The numerical testing demonstrates the capacity of the developed method to identify the structural parameters online and trigger alarm warnings with high accuracy within a short period of time.

Table 1.

Testing damage scenarios.

Case number	Structure type	Damage type
1	3-DOF linear structure	Stiffness only
2	3-DOF linear structure	Both stiffness and damping
3	2-DOF nonlinear hysteretic structure	Stiffness only
4	2-DOF nonlinear hysteretic structure	Both stiffness and damping

DOF: degree of freedom.

3-DOF linear structure

Consider a 3-DOF linear shear frame structure subject to an earthquake excitation. The equation of motion for the 3-DOF linear structure can be represented as

M \overset{\cdot\cdot}{x} + C \overset{\cdot}{x} + Kx = - ML {\overset{\cdot\cdot}{x}}_{g}

(27)

where $x = [x_{1}; x_{2}; x_{3}]$ , $\overset{\cdot}{x} = [{\overset{\cdot}{x}}_{1}; {\overset{\cdot}{x}}_{2}; {\overset{\cdot}{x}}_{3}]$ , and $\overset{\cdot\cdot}{x} = [{\overset{\cdot\cdot}{x}}_{1}; {\overset{\cdot\cdot}{x}}_{2}; {\overset{\cdot\cdot}{x}}_{3}]$ are the displacement, velocity, and acceleration vector, respectively; ${\overset{\cdot\cdot}{x}}_{g}$ is the acceleration of ground excitation; and L is the ground excitation matrix. The mass matrix M, stiffness matrix K, and damping matrix C are defined as

M = [\begin{matrix} m_{1} & 0 & 0 \\ 0 & m_{2} & 0 \\ 0 & 0 & m_{3} \end{matrix}]

(28)

K = [\begin{matrix} k_{1} + k_{2} & - k_{2} & 0 \\ - k_{2} & k_{2} + k_{3} & - k_{3} \\ 0 & - k_{3} & k_{3} \end{matrix}]

(29)

C = [\begin{matrix} c_{1} + c_{2} & - c_{2} & 0 \\ - c_{2} & c_{2} + c_{3} & - c_{3} \\ 0 & - c_{3} & c_{3} \end{matrix}]

(30)

where mass m₁ = m₂ = m₃ = 1 kg, stiffness k₁ = k₂ = k₃ = 12 kN/m, and damping c₁ = c₂ = c₃ = 0.6 kN s/m.

The N-S component of the El Centro earthquake recorded at the Imperial Valley Irrigation District substation in California on 18 May 1940 was used as the excitation input, as shown in Figure 2. The peak ground acceleration (PGA) was set to 2g. The sampling frequency was 50 Hz for all measured signals. The accelerations of each floor ${\overset{\cdot\cdot}{x}}_{i}$ (i = 1, 2, 3) and the ground excitation ${\overset{\cdot\cdot}{x}}_{g}$ are assumed to be known measurements and used as inputs in the EKF estimation. The unknown parameters to be identified are stiffness and damping of all floors, that is, k_i and c_i (i = 1, 2, 3). Thus the extended state vector is defined as

x' = [x_{1}; x_{2}; x_{3}; {\overset{\cdot}{x}}_{1}; {\overset{\cdot}{x}}_{2}; {\overset{\cdot}{x}}_{3}; k_{1}; k_{2}; k_{3}; c_{1}; c_{2}; c_{3}]

(31)

Figure 2.

Ground acceleration record of El Centro Earthquake.

The initial values for parameters in EKF are defined as follows. The initial values for stiffness k_i and damping c_i were k_i₀ = 15 kN/m and c_i₀ = 0.1 kN s/m (i = 1, 2, 3), and the initial values for displacement x_i and velocity ${\overset{\cdot}{x}}_{i}$ were zero. Thus, the initial state vector was defined as [0; 0; 0; 0; 0; 0; 15; 15; 15; 0.1; 0.1; 0.1]. The corresponding initial error covariance matrix of the extended state vector was a 12 × 12 diagonal matrix as P₀ = diag{1; 1; 1; 1; 1; 1; 10⁵; 10⁵; 10⁵; 10⁵; 10⁵; 10⁵}. The covariance matrix of the process noise w_k was Q₀ = 10⁻⁶I₉, in which I₉ is 9 × 9 identify matrix. The covariance matrix of the measurement noise v_k was selected as R₀ = 10⁻⁶diag{1; 1; 1; 1; 1; 1; 10²; 10²; 10²; 1; 1; 1}. In this article, the state covariance matrix and noise covariance matrices were selected based on the authors’ experience and extensive testing. In all testing cases of the three-story linear structure, the same initial parameters were used.

Based on the EKF and the measurements of the response data, the estimations for all state variables were obtained. As shown in Figure 3, the EKF estimations of the displacements for all three floors are presented as red dashed curves. For comparison, the simulated results of the displacements for all three floors are plotted as blue solid curves. Moreover, in Figure 4 the EKF estimation results for stiffness and damping of each floor are plotted in red dashed curves, while the actual values of the same structural parameters used in the simulation are plotted as blue solid lines. In Figures 3 and 4, both the solid curves and dashed curves almost coincide indicating that the EKF algorithm has an excellent tracking capability to provide high-quality estimations for structural parameters.

Figure 3.

Comparisons of displacement between EKF estimation and measurement.

Figure 4.

Performance of EKF estimation for linear structure.

In order to verify the capability and accuracy of the EKF-DSPC-based damage detection method, different structural damage scenarios are considered in the numerical validation. Two damage cases are presented for the linear structure: (1) damage only on structural stiffness and (2) damage on both stiffness and damping at the same time.

Case 1

k₁ is reduced abruptly from 12 to 6 kN/m at t = 20 s. The parameter estimation and damage detection performance for all six parameters based on the developed EKF-DSPC method is depicted in Figure 5. The testing results indicate that the EKF estimation of the identified parameters (blue solid curves) and the actual values (black dashed curves) almost coincide. When the damage occurs on k₁ at t = 20 s, the sudden drop from 12 to 6 kN/s was captured by EKF rapidly. The effectiveness of DSPC to detect damage on k₁ is verified in Figure 6, which is the zoomed-in process control limits from 19–25 s. The DSPC detected the damage when the EKF estimation of k₁ jumps outside of the control range of DSPC, while the estimation for other identified parameters fell inside DSPC when no structural damage occured. The testing results for damage Case 1 demonstrate that EKF-DSPC method has an excellent tracking capability for the structural parameters during damage events, leading to the online detection of damage on stiffness parameter with high confidence.

Figure 5.

Performance of EKF-DSPC method in Case 1.

Figure 6.

Performance of EKF-DSPC method for k₁ in Case 1.

When using KF-based approaches, the state covariance matrix and noise covariance matrices need to be carefully selected and tuned. The amplitude may also affect the estimate process. As depicted in Figure 5, at the beginning, the earthquake has a much larger amplitude than near the end of the process. As time progresses, the uncertainty seems to grow due to the unnecessarily large noise covariance matrix. During the earlier analysis stage, this level of noise is necessary to enable prediction in the presence of pronounced excitation. To resolve this issue, adaptive noise covariance matrices may be investigated and applied to enhance the robustness and reliability of the developed EKF + DSPC method. In this article, our focuses are the implementation of the integration of these two well-known mechanisms and the applications on the different structural models. The adaptive tuning of noise covariance matrices will be conducted in a future study.

Case 2

Parameters k₁ and c₁ were reduced abruptly at t = 20 s, from 12 to 6 kN/m and from 0.6 to 0.4 kN s/m, respectively. Parameter estimation for all stiffness and damping parameters identified by EKF-DSPC are presented in Figure 7. In addition, the detailed view of parameter estimation and damage detection for k₁ and c₁ from 19 to 25 s is presented in Figure 8. The EKF estimation of both k₁ and c₁ jumps outside of the DSPC ranges at t = 20 s; thus, simultaneous damage on both stiffness and damping in linear structure were also detected by EKF-DSPC method successfully.

Figure 7.

Performance of EKF-DSPC method in Case 2.

Figure 8.

Performance of EKF-DSPC method for k₁ and c₁ in Case 2.

As can be noted, the state covariance matrix P does affect the state estimation as well as the control limits for DSPC since the standard deviation is extracted from the P matrix. When damage occurs, the estimate measurement is significantly different from the measurements, thus the state estimation and the state covariance matrix dramatically change at the very early stage as shown in Figure 6. As time progresses, the estimated measurement moves closer to the real measurement, and the state estimation and P become much stable.

Moreover, the developed EKF + DSPC is used for online state estimation and damage detection. Since the recursion process of EKF on each time step only involves the new measurement, the calculations of estimate state, state covariance matrix, and control limits are very fast. Therefore, the developed method is computationally efficient.

2-DOF nonlinear hysteretic shear frame

Civil structures generally exhibit nonlinear hysteretic behavior under damaging events. A numerical study was performed to validate the performance of the EKF-DSPC-based damage detection method for a nonlinear hysteretic model. The goal is to estimate the structure’s parameters and identify the damage by considering the hysteretic behavior exhibited by the structure during an earthquake.

Consider a 2-DOF nonlinear hysteretic shear frame structure, which can be represented by the equations of motion as described in

\begin{matrix} [\begin{matrix} m_{1} & 0 \\ 0 & m_{2} \end{matrix}] [\begin{matrix} {\overset{\cdot\cdot}{x}}_{1} \\ {\overset{\cdot\cdot}{x}}_{2} \end{matrix}] + [\begin{matrix} c_{1} + c_{2} & - c_{2} \\ - c_{2} & c_{2} \end{matrix}] [\begin{matrix} {\overset{\cdot}{x}}_{1} \\ {\overset{\cdot}{x}}_{2} \end{matrix}] \\ + [\begin{matrix} k_{1} + k_{2} & - k_{2} \\ - k_{2} & k_{2} \end{matrix}] [\begin{matrix} z_{1} \\ z_{2} \end{matrix}] = - [\begin{matrix} m_{1} \\ m_{2} \end{matrix}] {\overset{\cdot\cdot}{x}}_{0} \end{matrix}

(32)

in which z₁ and z₂ are the hysteretic components defined by Bouc–Wen model as

{\overset{\cdot}{z}}_{i} = {\overset{\cdot}{x}}_{i} - β_{i} | {\overset{\cdot}{x}}_{i} | {| z_{i} |}^{α - 1} z_{i} - γ_{i} {\overset{\cdot}{x}}_{i} {| z_{i} |}^{α} (i = 1, 2)

(33)

where the mass coefficients m₁ = m₂ = 1 kg, the stiffness coefficients k₁ = k₂ = 15 kN/m, and damping coefficients c₁ = c₂ = 1 kN s/m. The hysteretic parameters α = 1, β₁ = β₂ = 2, and γ₁ = γ₂ = 1.

The El Centro Earthquake with PGA equals to 5g was used as the ground excitation in the numerical testing. The sampling time was 40 s, and the sampling frequency was 100 Hz for all measured signals. The fourth-order Runge–Kutta method (Jameson et al., 1981) was used to implement the numerical simulation to obtain the dynamic responses of the nonlinear hysteretic structure. The measurements of absolute acceleration of each floor ${\overset{\cdot\cdot}{x}}_{1}$ and ${\overset{\cdot\cdot}{x}}_{2}$ and the ground excitation ${\overset{\cdot\cdot}{x}}_{0}$ were used in the EKF-DSPC method to perform system identification and damage detection. The unknown parameters to be identified were the stiffness k_i, damping c_i, restoring force z_i, and hysteretic parameters β_i and γ_i (i = 1, 2). These unknown parameters were appended to the state vector to form the new state vector as

x = [x_{1}; x_{2}; {\overset{\cdot}{x}}_{1}; {\overset{\cdot}{x}}_{2}; z_{1}; z_{2}; k_{1}; k_{2}; c_{1}; c_{2}; β_{1}; β_{2}; γ_{1}; γ_{2}]

(34)

The initial values for the parameters in the EKF were defined as follows. The initial values for stiffness k_i and damping c_i were k_i₀ = 20 kN/m and c_i₀ = 0.5 kN s/m (i = 1, 2), and the initial values for hysteretic parameters β_i and γ_i were β_i₀ = 1.5 and γ_i₀ = 0.5 (i = 1, 2). In addition, the initial values for displacement x_i, velocity ${\overset{\cdot}{x}}_{i}$ , and restoring force z_i were zero, that is, x_i₀ = 0, ${\overset{\cdot}{x}}_{i 0} = 0$ , and Z_i₀ = 0 (i = 1, 2). Thus, the initial state vector is defined as [0; 0; 0; 0; 0; 0; 20; 20; 0.5; 0.5; 1.5; 1.5; 0.5; 0.5]^T. The initial error covariance matrix of the extended state vector is a 14 × 14 diagonal matrix P₀ = diag{1; 1; 1; 1; 1; 1; 10⁵; 10⁵; 10⁵; 10⁵; 10⁵; 10⁵; 10⁵; 10⁵}. The covariance matrix of the process noise w_k is selected as Q₀ = 10⁻⁶ diag{1; 1; 1; 1; 1; 1; 10²; 10²; 1; 1; 1; 1; 1; 1}. The covariance matrix of the measurement noise v_k was selected as R₀ = 10⁻⁶I₂. In all testing cases of the two-story nonlinear hysteretic structure, the same above initial parameters will be used.

Before any structural damage was introduced, the EKF estimation for all state variables for the nonlinear structure were obtained and verified by comparing them with simulated measurement data. The comparison of hysteresis loops (displacement vs restoring force) between EKF estimation and measurement are presented in Figure 9. Moreover, the EKF estimations for the eight structural parameters are presented in Figure 10. In both Figures 9 and 10, the curves of EKF estimation all coincide with the curves of measurement, which indicates the accuracy of EKF to be used for parameter estimation for nonlinear hysteretic structures.

Figure 9.

Comparison of hysteresis loops between EKF estimation and measurement.

Figure 10.

Performance of EKF estimation for nonlinear hysteretic structure.

Two structural damage scenarios were considered in the numerical validation for the nonlinear hysteretic structure: (1) damage only on structural stiffness and (2) damage on both stiffness and damping at the same time, as presented in following Cases 3 and 4, respectively.

Case 3

Parameter k₁ was reduced abruptly from 15 to 10 kN/m at t = 20 s. The parameter estimation and damage detection performance based on EKF-DSPC method is depicted in Figure 11. In addition, the details of damage detection for k₁ from 18 to 23 s are presented in Figure 12. The EKF estimation of k₁ jumped outside of the DSPC ranges at t = 20 s; thus, the damage on stiffness in nonlinear structure was also detected by EKF-DSPC method successfully.

Figure 11.

Performance of EKF-DSPC method in Case 3.

Figure 12.

Performance of EKF-DSPC method for k₁ in Case 3.

Compared with Case 1 for linear case, when moving to the nonlinear case, the algorithm may need longer time to get converged and the oscillation in the early stage is more severe.

Case 4

Parameters k₁ and c₁ were reduced abruptly from 15 to 10 kN/m and from 1 to 0.5 kN s/m at t = 20 s, respectively. Parameter estimation and the details of the damage detection by EKF-DSPC are presented in Figures 13 and 14. As can be observed, when the damage occurs on k₁ and c₁ concurrently at t = 20 s, the sudden drop of both parameters can be captured by EKF within rapid time. The effectiveness of DSPC is verified by the fact that at t = 20 s, only parameters k₁ and c₁ run out of the DSPC ranges, while other parameters are still within the DSPC range.

Figure 13.

Performance of EKF-DSPC method in Case 4.

Figure 14.

Performance of EKF-DSPC method for k₁ and c₁ in Case 4.

Conclusion

In this article, a real-time structural damage detection method for civil structures was developed based on the combination of EKF and DSPC. Based on the acceleration measurements, the damage-sensitive parameters involved in the state-space model of structural dynamic systems can be estimated by the EKF algorithm. The EKF produces real-time estimation means and standard deviations for the identified structural parameters to form the dynamic control limits to detect any abnormalities in the selected parameters. The developed EKF-DSPC damage detection method was validated using simulation response data of a three-story linear structure and a two-story nonlinear hysteretic structure under earthquake excitation considering different damage scenarios, and the results demonstrated a fast convergence rate, high damage detection accuracy, and light computational costs. The EKF-DSPC method can be easily replicated in other structural damage detection problems for both linear and nonlinear structures. Moreover, since the EKF is a well-developed methodology that does not require large computation costs, and the DSPC is capable of handling system variations caused by operational and environmental effects, the EKF-DSPC method has the good potential to be implemented in real-time SHM systems for in-service civil structures.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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