**2. Fuzzy controllers**

172 Introduction to PID Controllers – Theory, Tuning and Application to Frontier Areas

fuzzy PID controllers. The paper (Santos & all, 1096) shows that it is possible to apply the empirical tools to predict the achievable performance of the conventional PID controllers to evaluate the performance of a fuzzy logic controller based on the equivalence between a fuzzy controller and a PI controller. The paper (Yame, 2006) analyses the analytical structure of a simple class of Takagi-Sugeno PI controller with respect to conventional control theory. An example shows an approach to Takagi-Sugeno fuzzy PI controllers tuning. In the paper (Xu & all, 1998) a tuning method based on gain and phase margins has been proposed to determine the weighting coefficients of the fuzzy PI controllers in the frame of a linear plant control. There are presented numerical simulations. Mamdani fuzzy PID controllers are studied in (Ying, 2000). The author has published his theory on tuning fuzzy PID controllers at international conferences and on journals (Volosencu,

This chapter presents some techniques, under unitary vision, to solve the problem of tuning fuzzy PID controllers, developed based on the most general structure of Mamdani type of fuzzy systems, giving some tuning guidelines and recommendations for increasing the quality of the control systems, based on the practical experience of the author. There is given a method in order to make a pseudo-equivalence between the linear PID controllers and the fuzzy PID controllers. Some considerations related to the stability analysis of the control systems based on fuzzy controllers are made. Some methods to design fuzzy PID controllers are there presented. The tuning is made using a graphical-analytical analysis based on the input-output transfer characteristics of the fuzzy block, the linear characteristic of the fuzzy block around the origin and the usage of the gain in origin obtained as an origin limit of the variable gain of the fuzzy block. Transfer functions and equivalence relations between controller's parameters are obtained for the common structures of the PID fuzzy controllers. Some algorithms of equivalence are there presented. The linear PID controllers may be designed based on different methods, for example the modulus or symmetrical criterion, in Kessler's variant. The linear controller may be used for an initial design. Refining calculus and simulations must follow the equivalence algorithm. The author used this equivalence theory in fuzzy control applications as the speed control of electrical drives, with good results. The unitary theory presented in this chapter may be applied to the most general fuzzy PID controllers, based on the general Mamdani structure, which may be developed using all kind of membership functions, rule bases, inference methods and defuzzification methods. A case study of a control system using linear and fuzzy controllers is there also presented. Some advantages of this method are emphasized. Better control quality criteria are demonstrated for control systems using fuzzy controllers tuned, by using the presented

In the second paragraph there are presented some considerations related to the fuzzy controllers with dynamics, the structures of the fuzzy PI, PD and PID controllers. In the third paragraph there are presented: the transfer characteristics of the fuzzy blocks, the principle of linearization, with the main relations for pseudo-equivalence of the PI, PD and PID controllers. A circuit of correction for the fuzzy PI controller, to assure stability, is also presented. In the fourth paragraph there are presented some considerations for internal and external stability assurance. There is also presented a speed fuzzy control system for electrical drives based on a fuzzy PI controller, emphasizing the better control quality

2009).

approach.

criteria obtained using the fuzzy PI controller.
