## 3 Fuzzy Logic Control Design
Otherwise, only the ABS control is operating. The next step is to design membership functions for all the inputs and outputs (Figure 5). The linear (triangular) MFs were applied, which are characterized by fast reaction due to the narrow shape as compared to other MFs (exponential, quadratics). The MFs are symmetrical to provide an equal sensitivity for the whole UOD and obtain the whole overlap of the UOD between the MFs. Each variable UOD must have a closed frontier between [_min_, _max_]. For the input variables, the bounds are obtained during the parameterization described in Section 2. The slip input MFs for the ABS are introduced in Figure 5. There are nine MFs in total. The UOD for the front wheels lay between [0.08, 0.22]. This area is accepted according to the operational space obtained in Figure 2 for the front wheels. The only difference between the front and the rear wheels is that the UOD of the slip input for the last ones is accepted in a range [0.07, 0.2], which is also based on the rear slip curves in Figure 2 for the rear wheels. In short, UOD for the slip covers all stable areas for relevant roads. Figure 4: Controller block scheme for a single wheel: ABS FLC—anti-lock braking system fuzzy logic controller, ESP FLC—electronic stability program fuzzy logic controller, 1/s—integrational operation. The \(\upmu_{x}\) MFs are plotted in Figure 5b. The UOD parameterization was obtained during the case study. The \(\upmu_{x}\) operational space is bounded between [10; 40]. The first input of the ESP FLC is the yaw rate. The state consists of nine MFs; UOD is limited in the range between [\(-\)4, 4] and introduced in Figure 5c. The reason behind the range for the UOD is next: when the angular velocity exceeds 4 rad/s and the steering wheel angle change is not conducted, vehicle spin appears and the driver is no longer able to act on lateral control. The second input of the ESP steering wheel angle MFs is shown in Figure 5d. Like the first input, it has nine symmetrically dispelled MFs that are normalized between [\(-\)180, 180]. It is assumed that driver reaction in extreme situation must be limited to half of one full steering wheel turn to each side, left or right, which gives in total 360\({}^{\circ}\). The maximum pressure of the braking system for the studied case is 151 bar. The UOD for the output pressure variables (Figure 5e) is therefore located between [0, 151] and consists of eight MFs. The FLC decides how many bars shall be provided to obtain an optimal slip. Finally, the pressure output for the ESP is obtained in Figure 5f. Likewise, for the ABS the UOD lay between [0, 151]. The ESP has two outputs (Figure 4): brake pressure for the left and the right sides. The decision of which side of the vehicle to control is determined by the rule-base operator. The modus ponens (If-Then) form has been used in this paper for the rule-base design. The multiple input, single output (MISO) form of the linguistic rules for ABS is (taken from [6]):  
\[\textbf{If }u_{1}\text{ is }A^{j}_{1}\textbf{ and }u_{2}\text{ is }A^{k}_{2}\ldots\textbf{ Then }y_{q}\text{ is }B^{p}_{q}, \tag{10}\]  
where \(u_{1}\) and \(u_{2}\) denote the FLC inputs wheel slip and road condition, respectively; \(y_{q}\) denotes the brake pressure; \(A^{j}_{1}\) and \(A^{k}_{2}\) relate to the \(j\)th and \(k\)th linguistic value associated with \(\lambda\) and \(\upmu_{x}\), respectively; and \(B^{p}_{q}\) is the linguistic value of the output braking pressure. Figure 5: Fuzzy logic controller membership functions: (**a**) ABS \(\lambda\) input; (**b**) ABS \(\upmu_{x}\) input; (**c**) ESP \(\psi\) input; (**d**) ESP \(\delta\) input; (**e**) ABS \(p_{ABS}\) output; (**f**) ESP \(p_{ESPI}\) and \(p_{ESP}\) outputs. Linguistic values for the ABS are expressed in Table 3. There are 54 rules for the ABS control in total. The rule base for the ESP is observable in Table 4. It has a multi input, multi output (MIMO) structure. In total, 81 rules are required to control the state.