## 5 Numerical Simulations  
A total of sixteen earthquake records, shown in Table 1, were selected to evaluate the performance of the proposed self-tuning hierarchic fuzzy control strategy. The peak accelerations of the earthquakes are scaled to 0.4 g. A series of time history analyses for the base-isolated structure were performed with MATLAB/Simulink under the mentioned earthquakes. To evaluate the performance of the base-isolated structure for different cases, a total of nine performance indices are employed in this paper, as shown in Table 6[27]. Here, the performance indices \(J_{1}\) through \(J_{5}\) denote peak base shear, peak structural shear, peak base displacement, peak inter-story drift, and peak floor acceleration in the controlled structure normalized by the corresponding values in the uncontrolled structure, respectively; \(J_{7}\) and \(J_{8}\) denote root mean square (RMS) base displacement and floor accelerations of the controlled structure that are normalized likewise, respectively; \(J_{6}\) denotes the peak force generated by piezoelectric friction dampers normalized by the peak base shear in the controlled structure; and \(J_{9}\) denotes energy dissipated by the dampers normalized by the earthquake input energy to the controlled structure. In Table 6, \(V_{0}(t)\) and \(\hat{V}_{0}(t)\) denote the base shear for the controlled and uncontrolled structures, respectively; \(V_{1}(t)\) and \(\hat{V}_{1}(t)\) denote the structure shear for the controlled and uncontrolled structures, respectively; \(x_{b}(t)\) and \(\hat{x}_{b}(t)\) denote the base displacement for the controlled and uncontrolled  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Peak Base Shear** & **Peak Inter-Story Drift** & **RMS Base Displacement** \\ \hline \(J_{1}=\frac{\max_{x_{b}}\|\hat{V}_{0}(t)\|}{\max_{x_{b}}\|\hat{V}_{0}(t)\|}\) & \(J_{4}=\frac{\max_{x_{b}}\|d_{x}(t)\|}{\max_{x_{b}}\|d_{x}(t)\|}\) & \(J_{7}=\frac{\max_{x_{b}}\|\sigma_{x}(t)\|}{\max_{x}\|\hat{\sigma}_{x}(t)\|}\) \\ \hline
**Peak Structural Shear** & **Peak Floor Acceleration** & **RMS Floor Acceleration** \\ \hline \(J_{2}=\frac{\max_{x_{b}}\|\hat{V}_{1}(t)\|}{\max_{x_{b}}\|\hat{V}_{1}(t)\|}\) & \(J_{5}=\frac{\max_{x_{b}}\|a_{x}(t)\|}{\max_{x_{b}}\|d_{x}(t)\|}\) & \(J_{8}=\frac{\max_{x_{b}}\|\sigma_{x}(t)\|}{\max_{x}\|\hat{\sigma}_{x}(t)\|}\) \\ \hline
**Peak Base Displacement** & **Peak Control Force** & **Energy Dissipated by PFD** \\ \hline \(J_{3}=\frac{\max_{x_{b}}\|x_{b}(t)\|}{\max_{x_{b}}\|\hat{x}_{b}(t)\|}\) & \(J_{6}=\frac{\max_{x_{b}}\|f_{d}(t)\|}{\max_{x_{b}}\|f_{d}(t)\|}\) & \(J_{9}=\frac{\int_{0}^{t}f_{d}(t)\dot{x}_{b}(t)dt}{\int_{0}^{t}V_{0}(t)\dot{x}_ {b}(tstructures, respectively; \(d_{f}(t)\) and \(\hat{d}_{f}(t)\) denote the inter-story drift for the controlled and uncontrolled structures, respectively; \(a_{f}(t)\) and \(\hat{a}_{f}(t)\) denote the absolute floor acceleration for the controlled and uncontrolled structures, respectively; \(\hat{f}_{d}(t)\) and \(\hat{f}_{d}(t)\) denote the force generated by the PFD and the base shear in the controlled structures, respectively; \(\sigma_{d}(t)\) and \(\hat{\sigma}_{d}(t)\) denote the RMS base displacement for the controlled and uncontrolled structures, respectively; \(\sigma_{a}(t)\) and \(\hat{\sigma}_{a}(t)\) the RMS absolute floor acceleration for the controlled and uncontrolled structures, respectively; and \(U_{g}(t)\) is vector of the ground accelerations. To evaluate the influence of the pulse effects of the pulse-type near-fault earthquakes (namely, fling-step effect and rupture forward directivity effect) on the base-isolated structure controlled by the SHFLC, the peak inter-story drift, peak floor acceleration and peak base displacement of the SHFLC isolated structure are firstly evaluated. Figure 8 shows the comparison of the average peak inter-story drift and floor acceleration of the SHFLC isolated structure under the four types of earthquake.