Figure 23. Error J versus β and m for G5(s).

Figure 24. The PDβ +I parameters (Ke, Kce, Kie, Ku) versus β and m for G5(s).

5.Conclusion

This chapter has presented several fuzzy control structures with fractional-order derivative and integral actions combined to form a PID-type controller. Comparison with conventional fuzzy and PID controllers is performed to illustrate the effectiveness and robustness of the proposed algorithms. Another distinguished feature of this study is the use of discrete

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10

Figure 25. Parameters ts, tr , tp, ov(%) versus β and m of the step responses of the closedloop system with G5(s) and with a fuzzy PDβ +I controller.

fuzzy fractional-order controllers in the form of series and rational approximations which reveals useful for a digital implementation of these algorithms.

In the first part of chapter, a methodology for tuning fuzzy fractional PID controllers is introduced. This methodology is simple and effective and can be used to replace an existent integer/fractional PID controller in order to enhance system performance. Also, it was demonstrated that, with this approach, the designed controllers are equivalent to the conventional fractional PD/PID controllers by using a linear input-output mapping of the rule base of the fuzzy fractional controller. Moreover, by making the controller nonlinear, the performance of the control system proves to be, in most systems, better than its linear counterpart.

In the second part of chapter, were developed optimal fuzzy fractional PID controllers in which the parameters were tuned through a PSO algorithm. A nonlinear fuzzy control system with saturation in actuator is analyzed. In general, the control strategies presented, give better results than those obtained with conventional integer control structures, showing, once more, its effectiveness in the control of nonlinear systems. An analysis study regarding the influence of order of discrete rational approximations is also provided.

Certainly, the incorporation of fuzzy reasoning into fractional-order controllers will increase the applicability of these controllers.

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