## 4 Results and Discussions
### System Identification Analysis
#### 4.2.1 Training Setting Evaluation  
The first estimation model is NARX model. In this section, three NARX models are presented and evaluated including SISO, one time-delayed MISO, and two time-delayed MISO models, as shown in Figures 2 and 3. The input parameters are seismic design signal (Figure 5a) and time-delay output of structure behavior (acceleration measurements). The output is a response of the third floor of the building (TMD with mass 1.80 kg (see Figure 6a)), while the maximum response of this floor is higher than the first and second floors. The three models are designed as shown in Figure 2. Table 2 illustrates the evaluation of the NARX model selections. The trial and error process is used to determine the number of parameters that can be used to improve the predicted output signals. Herein, NARX(1) refers to model 1, NARX(2) present model 2 and model 3 is NARX(3) (Figure 2 and Section 2.4). Nine models are presented in Table 2: one and two are SISO models, and three to nine are MISO models. The nonlinear wavelet and sigmoid functions are employed also in this table to evaluate the nonlinear models design. The 5, 10 and 20 neurons are evaluated with SISO model and the \(R^{2}\) are 0.30, 0.33, \(-\)0.018, respectively. Therefore, the 10 neurons are used to design the neural network model. From Table 2, model 8 is the best (NARX(3) with sigmoid function) based on MISO model design with three inputs parameters (seismic signal, one and two output time-delays). The evaluation performance of the models shows that by any increase in the parameters of SISO models the accuracy increased, while the MISO models perform via versa. In addition, the nonlinear functions of the wavelet and sigmoid functions with neural network solution are shown suitable to estimate a good performance of the structural behavior. The sigmoid function performs better than the wavelet with SISO and MISO of the three inputs parameters. The SISO model is the worst. From these results, it is concluded that the simplified MISO-NARX models are suitable to estimate the behavior of controlled structures with high agreement between the observed and the predicted signals (identified) for model 8, as shown in Table 2 and Figure 7. Therefore, model 8 is selected as the best model that can be used to identify the nonlinear behavior of a controlled structure. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Model No.** & **Model** & **Order**\([(\textbf{{u}}_{\textbf{{u}}})\ (\textbf{{u}}_{\textbf{{u}}})\ (\textbf{{u}}_{\textbf{{u}}})]\) & **and Function** & \(\textbf{{R}}^{2}\) & **MAE (g)** & **RMSE (g)** \\ \hline Model 1 & NARX(1) & [(5)(3)(1)]-sigmoid & 0.33 & 0.047 & 0.057 \\ Model 2 & NARX(1) & [(5)(3)(1)]-wavelet & 0.18 & 0.025 & 0.042 \\ Model 3 & NARX(2) & [(2) (2) (1) (0) 1]-sigmoid & 0.99 & \(2.97\times 10^{-4}\) & \(4.35\times 10^{-4}\) \\ Model 4 & NARX(2) & [(2) (2) (1) (0) 1]-wavelet & 0.99 & \(2.80\times 10^{-4}\) & \(4.27\times 10^{-4}\) \\ Model 5 & NARX(2) & [(2) (2) (1) (0) 0]-sigmoid & 0.99 & \(2.95\times 10^{-4}\) & \(4.26\times 10^{-4}\) \\ Model 6 & NARX(3) & [(2) (2) (2) (1) (0) 1]-sigmoid & 0.99 & \(1.48\times 10^{-4}\) & \(2.41\times 10^{-4}\) \\ Model 7 & NARX(3) & [(2) (2) (2) (1) (0) (1) 1]-wavelet & 0.99 & \(5.54\times 10^{-4}\) & \(6.90\times 10^{-4}\) \\ Model 8 & NARX(3) & [(2) (2) (2) (1) (0) (0) 1]-sigmoid & 0.99 & \(1.26\times 10^{-4}\) & \(1.88\times 10^{-4}\) \\ Model 9 & NARX(3) & [(2) (2) (2) (2) (0) (0) 0]-wavelet & 0.99 & \(4.96\times 10^{-4}\) & \(6.09\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Statistical parameter used for the evaluation of Nonlinear AutoRegressive with eXogenous inputs (NARX) models. The second model is called ANFIS model, which is known as the best models used in previous research in controlled structures [1; 12; 24; 25]. The number and type of MF and input time-delayed are main parameters for the ANFIS model as presented in [1]. Table 3 presents the selection evaluation of the numbers used and types of MFs.