Frequency domain CNN and dissipated energy approach for damage detection in building structures

Abstract

Recent developments tools and techniques for structural health monitoring allow the design of early warning systems for the damage diagnosis and structural assessment. Most methods to damage detection involve vibration data analysis by using identification systems that generally require a mathematical model and much information about the system, such as parameters and states that are mostly unknown. In this paper, a novel frequency domain convolutional neural network (FDCNN) proposed aims to design an identification system for damage detection based on Bouc–Wen hysteretic model. FDCNN, unlike other works, only requires acceleration measurements for damage diagnosis that are very sensitive to environmental noise. In contrast to neural network (NN) and time domain convolutional neural network, FDCNN reduces the computational time required for the learning stage and adds robustness against noise in data. The FDCNN includes random filters in the frequency domain to avoid measurement noise using a spectral pooling operation, which is useful when the system bandwidth is unknown. Incorrect filtering can produce unwanted results, as a shifted and attenuation signal relative to the original. Moreover, FDCNN allows overcoming the parameterization problem in nonlinear systems, which is often difficult to achieve. In order to validate the proposed methodology, a comparison between two different architectures of convolutional neural networks is made, showing that proposed CNN in frequency domain brings better performance in the identification system for damage diagnosis in building structures. Experimental results from reducing scale two-storey building confirm the effectiveness of the proposed.

Appendices

Appendix A: Discrete Fourier transform

Discrete Fourier transform (DFT), denoted by $\mathcal {F}(\cdot )$, is a powerful tool to convert spatial samples into a sequence of complex-valued samples in the frequency domain. Some important properties of DFT are as follows: It is linear and unitary (Cooley et al. 1969), and its inverse transform is given by $\mathcal {F}^{-1}(\cdot )=\mathcal {F}(\cdot )^{*}$ which is the conjugate of the transform itself. This last property is useful during the training stage of CNN. A DFT of n-points is defined as $A=\mathcal {F}(a)$, where $\mathcal {F}$ can be expressed as a matrix F; this matrix is called a DFT matrix and it is constructed as follows:

$$\begin{aligned} F_{n}=\frac{1}{\sqrt{n}}\begin{bmatrix} 1 &{} 1 &{} 1 &{} \dots &{} 1 \\ 1 &{} \omega &{} \omega ^{2} &{} \dots &{} \omega ^{n-1} \\ 1 &{} \omega ^{2} &{} \omega ^{3} &{} \dots &{} \omega ^{2(n-1)} \\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 1 &{} \omega ^{n-1} &{} \omega ^{2(n-1)} &{} \dots &{} \omega ^{(n-1)(n-1)} \end{bmatrix} \end{aligned}$$

where $\omega =\exp ^{\frac{-2\pi \mathrm {i}}{n}}$.

Slight modification is made in order to ensure the DC frequency component in the center row of the matrix.

Remark 7

In frequency analysis, the convolution operation becomes an element-wise product which makes the analysis easier and direct. The convolution operation between $a,b\in \mathfrak {R}^{n}$, using the DFT is:

$$\begin{aligned} \mathcal {F}(a*b)=\mathcal {F}(a)\odot \mathcal {F}(b) \end{aligned}$$

(69)

where $*$ denotes the convolution operation and $\odot$ is an element-wise product. This product reduces the number of operations compared to the convolution stage in TDCNN and it make the training process even faster.

Appendix B: Time domain CNN for modeling time series

Consider an unknown discrete-time nonlinear system

$$\begin{aligned} y(q)=f\left( x(q)\right) .\;\;\;\; x(q+1)=g\left( x(q),u(q)\right) \end{aligned}$$

(70)

where y(q) is the scalar output, x(q) the internal state, u(q) the input, $f(\cdot )$ and $g(\cdot )$ smooth functions, $f,g\in C^{\infty }$ .

A nonlinear autoregressive exogenous (NARX) model for (70) is defined as

$$\begin{aligned} y(q)=\varPhi \left[ \varpi \left( q\right) \right] \end{aligned}$$

(71)

The system dynamics are represented by the unknown nonlinear difference equation $\varPhi$, where

$$\begin{aligned} \varpi \left( q\right) =[y\left( q-1\right) ,\ldots ,y\left( q-n_{y}\right) ,u\left( q\right) ,\ldots ,u\left( q-n_{u}\right) ]^{T} \end{aligned}$$

(72)

y(q) and u(q) within this equation represent measurable output and input for the system, with $n_{y}$ and $n_{u}$ the regression order, respectively, which are unknown.

The nonlinear system identification of (71) based on time domain convolutional neural networks (TDCNN) is shown in (73), where ${\hat{y}}_{T}(q)$ is estimation of the real output generated by TDCNN, which is a scalar element.

$$\begin{aligned} {\hat{y}}_{T}(q)=W^{(\ell )\text {T}}\vartheta \end{aligned}$$

(73)

This is fully connected layer with W as synaptic weights vector and $\vartheta$ the stacked output of the last subsample layer of TDCNN.

Two more types of layer are introduced in TDCNN. The first layer in TDCNN is a convolutional one, where two operations are made: convolution and an activation function. The convolution operation is

$$\begin{aligned} \chi _{h}^{(\ell )} = K_{h} * y_{h}^{(\ell -1)} \end{aligned}$$

(74)

$\ell$ represent the actual layer, h-filters per layer are used, and each filter is $K_{h}^{(\ell )}\in R^{f_{\ell }}$. For each element i of $\chi _{h}^{(\ell )}$, the previous operation is equivalent to

$$\begin{aligned} \chi _{i,h}^{(\ell )}=\sum _{a=0}^{f_{\ell }-1}K_{h,a}^{(\ell )}y_{h,i+a}^{(\ell -1)} \end{aligned}$$

(75)

The result of this operation $\chi _{h}^{(\ell )}$ is called the feature map, which contains features properties of the input, and each filter obtains a different feature. These feature maps go through an activation function, different activation functions are used in neural networks for specific tasks and unique properties (Glorot and Bengio 2010), but the one used is in this papers is the rectified linear unit (ReLU). The output of a convolutional layer is defined by (76)

$$\begin{aligned} y_{h}^{(\ell )}=max(0,\chi _{h}^{(\ell )}) \end{aligned}$$

(76)

for the first layer of the CNN, $y_{h}^{(\ell -1)}$ is the input vector

$$\begin{aligned} \hat{\varpi }\left( q\right) =[{\hat{y}}\left( q-1\right) ,\ldots ,{\hat{y}}\left( q-r_{1}\right) ,u\left( q\right) ,\ldots ,u\left( q-r_{2}\right) ]^{T} \end{aligned}$$

(77)

where $r_{1}$ and $r_{2}$ denote the regression order. $r_{1}\ne n_{y}$ and $r_{2}\ne n_{u}$.

After a convolutional layer, a subsample layer is followed; this layer is pretended to be used as data reduction stage, so the strongest response from the filters keeps going through the TDCNN.

In the subsample layers, the operation used is the max-pool, which is defined as

$$\begin{aligned} y_{h}^{(\ell )}=maxpool\left( y_{h}^{(\ell -1)},s_{\ell }\right) \end{aligned}$$

(78)

The input divided in groups of dimension $s_{\ell }$ and from each group the highest values remain. The Shrink depends on the layer where it is applied.

Convolutional and subsample layers can be repeated as many times as the application require in the TDCNN. As mentioned earlier, after the last subsample layer, the outputs of each feature map are stacked to create the vector $\vartheta$

$$\begin{aligned} \vartheta =\left[ y_{1}^{(\ell )T} \; y_{2}^{(\ell )T} \; \cdots ; y_{h}^{(\ell )T} \right] ^{T} \end{aligned}$$

(79)

This helps to manage the last layer in terms of vector and matrices. The complete architecture is shown in Fig. 18.

Fig. 18

Time domain convolutional neural network for system identification

1.1 Training of time domain CNN using backpropagation

The training of the TDCNN’s parameters is realized by the backpropagation algorithm (BPA), which is used to calculated the gradient of the cost function respect each parameter of the TDCNN, propagating it backward through the network to update these parameters. The cost function is used as a measurement of the performance, and the most frequently cost function for identification is the squared error which measures the difference between the real output and the estimated one.

$$\begin{aligned} J(q)=\frac{1}{2}e_{T}^{2}(q) \end{aligned}$$

(80)

$e_{T}(q)$ is the identification error between the TDCNN output and the real output in each instant, i.e., $e_{T}(q)={\hat{y}}_{T}(q)-y(q)$.

The BPA uses the gradient of the cost function with respect to each parameter in the neural network. To calculate the gradient, it uses the chain rule and then each parameter is updated by the delta rule. In the output layer, the weights are updated as follows:

$$\begin{aligned} w_{i}^{(\ell )}(q+1)=w_{i}^{(\ell )}(q)-\eta _{T}\frac{\partial J}{\partial w_{i}^{(\ell )} } \end{aligned}$$

(81)

where $w_{i}^{(\ell )}$ are the elements of the vector $W^{(\ell )}$, $\eta _{T}$ the learning rate defining one for each layer and

$$\begin{aligned} \frac{\partial J}{\partial w_{i}^{(\ell )}}= \frac{\partial J}{\partial e_{T}} \frac{\partial e_{T}}{\partial {\hat{y}}_{T}} \frac{\partial {\hat{y}}_{T}}{\partial w_{i}^{(\ell )}} = e_{T} \vartheta _{i} \end{aligned}$$

(82)

$\vartheta _{i}$ are the elements of vector $\vartheta$ corresponding to the weight $w_{i}^{(\ell )}$. To previous layer, the gradient, using chain rule, is

$$\begin{aligned} \frac{\partial J}{\partial \vartheta }=\frac{\partial J}{\partial e_{T}} \frac{\partial e_{T}}{\partial {\hat{y}}_{T}} \frac{\partial {\hat{y}}_{T}}{\partial \vartheta } = e_{T}W^{(\ell )} \end{aligned}$$

(83)

For the subsample layer, an reverse operation of maxpool is used to calculate the gradient

$$\begin{aligned} \frac{\partial J}{\partial y^{(\ell -1)}}=up\left( \frac{\partial J}{\partial y^{(\ell )}}\right) \end{aligned}$$

(84)

where $up(\cdot )$ is an operation to increase length of the gradient to match the previous layer and only passing to the positions where the highest response occurs in the forward stage, leaving everything else in zeros. For convolutional layer, the gradient of the cost function with respect to the filters is calculated as

$$\begin{aligned} \frac{\partial J}{\partial K_{h}^{(\ell )}} =y_{h}^{(\ell -1)} * rot180(\delta _{h}^{(\ell )}) \end{aligned}$$

(85)

with $*$ being the convolution operator and

$$\begin{aligned} \delta _{h,i}^{(\ell )}=\frac{\partial J}{\partial y_{h,i}^{(\ell )}}f^{^{\prime } }(\chi _{h,i}^{(\ell )}) \end{aligned}$$

(86)

with $f^{^{\prime }}(\cdot )$ being the derivative of the ReLU operation, that is defined as,

$$\begin{aligned} f^{^{\prime }}(\varOmega )={\left\{ \begin{array}{ll} 1 &{} \text {if }\varOmega >0 \\ 0 &{} \text {otherwise} \end{array}\right. } \end{aligned}$$

In order to update the filters, delta rule is used, therefore

$$\begin{aligned} K_{h}^{(\ell )}(q+1)=K_{h}^{(\ell )}(q)-\eta _{T} \left( y_{h}^{(\ell -1)} * rot180(\delta _{h}^{(\ell )})\right) \end{aligned}$$

(87)

Finally, to backpropagate the gradient to previous layer of a convolutional layer, the equation is

$$\begin{aligned} \frac{\partial J}{\partial y_{h}^{(\ell -1)}}=\delta _{h}^{(\ell )}\odot rot180(K_{h}^{(\ell )}) \end{aligned}$$

(88)

The operator $rot180(\cdot )$ is equivalent to use its parameter from bottom to top, just like a flip over.

Appendix C: Multilayer neural network for system modeling

For comparison, a multilayer perceptron (NN for simplicity) is created. This NN consists of one hidden layer with 35 units, which are paired with activation function $\tanh (\cdot )$; its architecture is shown in Fig. 19.

Fig. 19

Multilayer perceptron architecture for system identification

Consider the nonlinear system to be identify defined in 71 and regard the same input from the CNN described in 35; the output of the units in the hidden layer is defined as:

$$\begin{aligned} X_{NN} = V_{NN}\varpi \end{aligned}$$

(89)

$V_{NN}$ are the synaptic weights in the hidden layers written in matrix form, $X_{NN}$ is the vector output of hidden layer, and each element corresponds to each one of the units in this layer. The output of the NN is

$$\begin{aligned} {\hat{y}}_{NN} = W_{NN}X_{NN} \end{aligned}$$

(90)

where $W_{NN}$ are the synaptic weights in the output layer, dimensions match, so the output is scalar. The training of this NN is done with the backpropagation algorithm. For this matter, the cost function to be minimized is defined as

$$\begin{aligned} J(q)=\frac{1}{2} e_{NN}(q)^2 \end{aligned}$$

(91)

where $e(q)=\left( {\hat{y}}_{NN}(q)-y(q)\right) ^2$ and the update law for the synaptic weights in output and hidden layer is defined with the delta rule, i.e.,

$$\begin{aligned} W_{NN}(q+1)=W_{NN}(q)-\eta _{NN} \frac{\partial J}{\partial W_{NN}} \end{aligned}$$

(92)

and

$$\begin{aligned} V_{NN}(q+1)=V_{NN}(q)-\eta _{NN} \frac{\partial J}{\partial V_{NN}} \end{aligned}$$

(93)

where $\eta _{NN}$ is the learning rate for this NN.