## 4 Control Strategies
### Fuzzy Multimodal Control  
A fuzzy set is defined as a set of objects whose degree of membership is relative to the continuous interval between 0 and 1. Thus, within the operating range, fuzzy membership functions (numeric, graphical, or tabular) are applied to assign fuzzy membership values to the control variable. Trapezoidal, triangular, and Gaussian functions are most commonly used in the literature [46; 47]. Figure 8 shows an example of trapezoidal membership functions. Therefore, fuzzy system inferences involve evaluating rules that describe the dependence between input and output variables. Two concepts are widely used for developing fuzzy systems: fuzzification and defuzzification. Fuzzification involves converting real-world quantities into fuzzy numbers, which can come from sensors or computerized systems, among other sources [47]. The system rules are treated in parallel. Therefore, when an input is provided, different rules with their respective degrees of activation are triggered to infer a control signal. This combination of different rules is called defuzzification [48]. The work aims to develop a nonlinear control system through a multimodel control strategy using local discrete-time PI controllers linked by fuzzy logic. The use of fuzzy logic is because its models are universal approximators of functions in a compact space [49]. Fuzzy models also allow the composition of the global model from multiple local models, which is advantageous for interpretability and controller design. With the development of local controllers for the desired operating points, these can be combined via fuzzy logic so that the application is no longer local, meaning the system is controlled globally. This union favors the application of multimodel hybrid control over a wide operating range, making local PI controllers in discrete time act on the global process plant through the degree of the relevance of local controllers to each operating region [50]. This degree of pertinence is calculated by functions that represent the operating ranges, and, via IF-THEN rules, the output signal, which acts on the global process plant, is determined. To implement the multimodel control strategy, the first step is to obtain the discrete-time PI controllers for each linearized operating point. After obtaining the local controllers, it is necessary to choose the fuzzy rules that will unite all local controllers through the developed fuzzy membership functions, as shown in Figures 9 and 10. Figure 8: Trapezoidal membership functions. Figure 9 shows that each local controller generates its own control signal that acts best close to its operating point. Now, from Figure 10, it can be seen that each local control signal presents a membership function with a corresponding degree of membership \(\mu\), so that  
\[\left\{ \begin{array}{l}u1\text{ will have a membership degree }\mu 1\\ u2\text{ will have a membership degree }\mu 2\\ \vdots\\ un\text{ will have a membership degree }\mu n\end{array} \right.\]  
In this way, depending on the reference position to be tracked, each local controller provides a respective value of the control signal, represented by a function \(u(k)\), so that the local controller's performance is represented by its respective degree of membership \(\mu\), which is used to activate the fuzzy rules. With activation of the fuzzy rules, the calculation of the output signal due to the action of the membership functions is obtained. Based on the response of the system to be controlled and the output of the local models, linear controllers act on each local model in order to correct the response of each model separately. Global control action is obtained by fuzzy inference, which in this case is a combination weighted by the fuzzy mode for the actions of each local PI discrete-time controller. The global control signal excites the motor system to act on the reference position.