## 4 Numerical Study: A 10-Story Shear Building
### Comparison of Seismic Control Performance  
In order to realize how accurately the MLP and ARX models were able to emulate the LQR, four of the performance indices used by [PERSON] and [PERSON] (1998) were adopted. The first performance index denotes the maximum normalized floor relative displacement which can be expressed as:  
\[J_{1}=\max_{t,i}\left(\frac{|{{\mathbf{x}}_{i}(t)}|}{x^{\max}}\right) \tag{17}\]  
where \(x_{i}(t)\) is the relative displacement of the \(i\)-th floor during the excitation; and \(x^{\max}\) represents the maximum displacement of the uncontrolled shear building. The second performance index is the maximum normalized inter-story drift which can be represented as:  
\[J_{2}=\max_{t,i}\left(\frac{|{{\mathbf{x}}_{i}(t)}/{{\mathbf{h}}_{i}}|}{{{\mathbf{a}}_{i} }^{\max}}\right) \tag{18}\]  
where \(d_{i}(t)\) is the inter-story drift of the \(i\)-th floor during the excitation; \({{\mathbf{h}}_{i}}\) is the story height of the \(i\)-th floor; and \({{\mathbf{d}}_{n}}^{\max}\) represents the maximum normalized inter-story drift of the uncontrolled shear building. The third performance index is the normalized peak absolute acceleration which can be expressed as:  
\[J_{3}=\max_{t,i}\left(\frac{|{{\ddot{{\mathbf{x}}}_{ai}}(t)}|}{{{\ddot{{\mathbf{x}}}_{ ai}}^{\max}}}\right) \tag{19}\]  
where \({{\ddot{{\mathbf{x}}}_{ai}}(t)}\) is the absolute acceleration of the \(i\)-th floor during the excitation; and \({{\ddot{{\mathbf{x}}}_{ai}}^{\max}}\) is the maximum absolute acceleration of the \(i\)-th floor of the uncontrolled shear building. The last performance index considers the maximum control force normalized by the weight of the structure which is formulated as:  
\[J_{4}=\max_{t}\left(\frac{|{{\mathbf{u}}(t)}|}{W}\right) \tag{20}\]  
where \(u(\mathfrak{t})\) is the control force of AMD and \(W\) is the weight of the 10-story shear building. Figure 10: Control simulation model of the ARX (autoregressive with exogenous inputs). Figure 9: Architectures of the exogenous inputs (ARX) neural network: (**a**) series-parallel, and (**b**) parallel. Fourteen earthquake ground acceleration records normalized to a PGA of 2.4 m/s\({}^{2}\) were adopted as the ground excitation to the 10-story shear building with an AMD. The AMD control force was generated by using the LQR state feedback control with optimized weighting matrices using SOS (LQR-SOS), the MLP models with 120, 150, and 200 input acceleration steps (MLP-120, MLP-150, and MLP-200), and the ARX model. As mentioned previously the force capacity of the AMD was assumed \(\pm\)50 kN. The AMD controlled 10-story shear building was subjected to the fourteen earthquakes. Tables 3-7 list the performance indices of each simulation control case. Table 8 compares the average seismic performance of the 10-story shear building subjected to the fourteen earthquakes and the average RMSE of the fourteen excitation cases between the control force generated from LQR-SOS and the neural network controllers. It can be found that the three MLP models emulated the LQR-SOS controller extraordinarily. The seismic responses of the LQR-SOS, and the three MLP models are nearly identical in terms of the four performance indices. The RMSE between the control force generated by the LQR-SOS and the three MLP models is less than 2.0% for each case. On the other hand, the ARX model emulated the LQR-SOS fairly well. The seismic responses of the LQR-SOS, and ARX model are close to each other with respect to the four performance indices. Meanwhile, the RMSE between the LQR-SOS control force and the ARX control force is less than 5.0% for nearly all cases. Generally speaking, the RMSE of the ARX model is approximately two times larger than that of the MPL model.