## 4 Self-Tuning Fuzzy Control Strategy
### Self-Tuning Hierarchic Fuzzy Logic Control (SHFLC) Strategy  
Because of the uncertainties and nonlinearities in the nature of the earthquake and the isolation system, as well as the discontinuous characteristics of the friction force of PFD, the development of semi-active control methods is a challenging task. Fuzzy logic approach is an effective method to deal with complex nonlinear systems. It can describe relationships between inputs and outputs of a controller using simple verbose statements instead of complicated mathematical terms. Due to its inherent robustness and simplicity, fuzzy logic theory has been widely used to develop controllers for semi-active devices [17; 18; 19]. In the self-tuning hierarchic fuzzy logic control strategy, fuzzy logic controllers (FLC) are used in the design of the sub-level controller and the supervisory controller. In the sub-level FLC, seven Gaussian membership functions were defined for each input variable as shown in Figure 5. The fuzzy sets for the input variables are defined as follows: NL: negative large, NM: negative medium, NS: negative small, ZE: zero, PS: positive small, PM: positive medium, and PL: positive large. Furthermore, the universe of discourse for each input variable ranges from \(-1\) to \(1\). Five Gaussian membership functions were defined to cover the universe of discourse of the output variable voltage that varies from \(0\) to \(1000\) V. The fuzzy sets for the output variables are defined as: VL: very large, L: large, M: medium, S: small, and ZE: zero. After the fuzzification of the input and output variables, fuzzy control rules were established for the sub-level FLC, as shown in Table 3. The control rules are in the form of if-then rules and map the links between the input and output membership functions. The control principles are as follows: (1) if  
Figure 4: Block diagram of the self-tuning fuzzy control strategy. Figure 5: Input and output membership functions for sub-level fuzzy controller: (**a**) input normalized displacement; (b) input normalized velocity; and (**c**) output voltage. the base displacement and velocity have opposite signs (i.e., the isolation system returns to its original position), then the output voltage becomes small in order to ensure the PFD output small damping force; and (2) if the base displacement and velocity have the same sign, then the output voltage becomes large. At the same time, the magnitude of the output is linearly proportional to the magnitude of the input variables. When the displacement and velocity are almost zero or small, the command voltage is about zero, which means that PFD acts as a passive Coulomb damper. The defuzzification of fuzzy control employs the method of centroid to get a crisp output value. The control surface for the sub-level FLC is shown in Figure 6.  
In order to enhance the adaptability of the proposed control strategy under different types of earthquakes, the type and intensity of earthquakes should be distinguished during the design of the supervisory controller. As discussed above, pulse-type near-fault earthquakes possess large acceleration amplitudes and large long duration velocity impulses. Therefore, seismic acceleration \(\ddot{u}_{g}\) and velocity \(\dot{u}_{g}\) are selected as the input variables of the supervisory FLC in order to determine the characteristics of the earthquake. The universe of discourse for input seismic velocity is chosen from 0 to 1.8 m/s. Furthermore, the universe of discourse for input seismic acceleration is chosen from 0 to 4 m/s\({}^{2}\). The outputs of the supervisory FLC are normalization factors \(N_{d}\) and \(N_{v}\) for the displacement and velocity of the base isolation, and normalization factor \(S_{volt}\) for the voltage of the PFD. It should be noted that normalization factors are used to keep the input variables of the sub-level FLC in the range of the universe of discourse. For each input variable, a reasonable normalization factor should be determined.