# Temperature Effects on Vibration-Based Damage Detection of a Reinforced Concrete Slab
## 3 Mathematical Model
### ARX Model  
The three parameters: the auto-regressive order \(a_{n}\), the exogeneous order \(b_{m}\) and the pure time delay between input and output \(d\) are the main features of the auto-regressive with exogenous (ARX) model. The ARX model adopted in this paper consists of an auto-regressive output and an exogeneous input part, which is shown as follows [39]:  
\[y(t)+a_{1}y(t-1)+a_{2}y(t-2)+\cdots+a_{n}y(t-n)=b_{1}u(t-d)+b_{2}u(t-d-1)+ \cdots+b_{m}u(t-d-m+1)+\varepsilon(t) \tag{7}\]  
where \(y(t)\) is the eigenfrequency at time instant \(t\), which is the output; \(u(t-d)\) is the temperature at time instant \(t-d\), which functions as the input; \(\varepsilon(t)\) is the equation error, which functions as the unknown disturbances during the input-output process, such as white noise. The ARX model can be rewritten by z-transformation:  
\[A(z^{-1})y(t)=B(z^{-1})u(t-d)+\varepsilon(t) \tag{8}\]  
where  
\[A(z^{-1})=1+a_{1}z^{-1}+\cdots+a_{n}z^{-n} \tag{9}\]  
\[B(z^{-1})=b_{1}+b_{2}z^{-1}+\cdots+b_{m}z^{-m+1} \tag{10}\]  
When the temperature input vector is \(u=[u(1),u(2),\cdots,u(M)]^{T}\), the output vector of eigenfrequency is \(y=[y(1),y(2),\cdots,y(M)]^{T}\), Equation (7) can be rewritten as:  
\[ \begin{array}{l}y(1)=-a_{1}y(0)\cdots-a_{n}y(1-n)+b_{1}u(1-d)\cdots+b_{m}u( 2-m-d)+\varepsilon(1)\\ y(2)=-a_{1}y(1)\cdots-a_{n}y(2-n)+b_{1}u(2-d)\cdots+b_{m}u(3-m-d)+\varepsilon( 2)\\ \vdots\\ y(M)=-a_{1}y(M-1)\cdots-a_{n}y(M-n)+b_{1}u(M-d)\cdots+b_{m}u(M+1-m-d)+ \varepsilon(M)\end{array} \] (1 To minimize the sum of squares of the equation errors \(\varepsilon(i)\), a criterion based on the least squares method can be defined and the best estimate of the undetermined parameter \(\theta\) is shown as:  
\[\theta=\left[\Phi^{T}\Phi\right]^{-1}\Phi^{T}y \tag{16}\]  
In this paper, the (akaike information criterion) AIC is adopted, which is practical and can be used to check mode order and is defined as:  
\[AIC=\lg\left\{\det\left[\frac{1}{M}\sum_{i=1}^{M}\theta(i,\theta)\theta^{T}(i, \theta)\right]\right\}+\frac{k}{M} \tag{17}\]  
where M is the set number of experimental data; \(\theta\) is vector to be identified; \(k\) is parameter to be identified. The function \(v=aic(H)\), which is based on MATLAB, can be used to calculate the AIC value in the model, if the AIC value is small, n, m, d can be the proper mode order of the system. Equation (12) can be transformed to Equations (18) and (19), which are models with a double input function, the first is the transfer function from input signal \(u(t)\) to output signal which is shown as Equation (20), the second is the transfer function from error signal \(\varepsilon(t)\) to output signal, which can be neglected here. \[y(t)=\frac{B(z^{-1})}{A(z^{-1})}u(t-d)+\frac{\varepsilon(t)}{A(z^{-1})} \tag{18}\]  
\[y(t)=\frac{B(z^{-1})}{A(z^{-1})}z^{-d}u(t)+\frac{\varepsilon(t)}{A(z^{-1})} \tag{19}\]  
\[H(z^{-1})=\frac{b_{1}+b_{2}z^{-1}+\cdots+b_{m}z^{-m+1}}{1+a_{1}z^{-1}+a_{2}z^{ -2}\cdots+a_{n}z^{-n}}z^{-d} \tag{20}\]  
In the paper, three numbers can be determined based on MATLAB, n = 4, m = 2 and d = 1; the transfer functions between temperature and the first four natural frequencies are shown Equations (21)-(24). \[H(f_{1})=\frac{-1.591z^{-1}+1.593z^{-2}}{1-0.3364z^{-1}-0.1731z^{-2}-0.2633z^{ -3}-0.2169z^{-4}} \tag{21}\]  
\[H(f_{2})=\frac{0.8077z^{-1}-0.8219z^{-2}}{1-0.249z^{-1}-0.1884z^{-2}-0.3348z^{ -3}-0.2107z^{-4}} \tag{22}\]  
\[H(f_{3})=\frac{-0.3215z^{-1}+0.3315z^{-2}}{1-0.3122z^{-1}-0.2461z^{-2}-0.2611 z^{-3}-0.1445z^{-4}} \tag{23}\]  
\[H(f_{4})=\frac{-0.02412z^{-1}-0.008753z^{-2}}{1-0.1482z^{-1}-0.3823z^{-2}-0.27 03z^{-3}-0.1906z^{-4}} \tag{24}\]  
Fitting the mean temperature and the first four natural frequencies of the RC slab, the ARX models can be plotted as Figure 12.  
The root-mean-square error can be used to estimate the precision of the model, which can be shown as Equation (25). The less the acquired root mean squared error (RMSE) value, the better the model precision.