Constantin Volosencu

*"Politehnica" University of Timisoara Romania* 

## **1. Introduction**

After the development of fuzzy logic, an important application of it was developed in control systems and it is known as fuzzy PID controllers. They represent interest in order to be applied in practical applications instead of the linear PID controllers, in the feedback control of a variety of processes, due to their advantages imposed by the non-linear behavior. The design of fuzzy PID controllers remains a challenging area that requires approaches in solving non-linear tuning problems while capturing the effects of noise and process variations. In the literature there are many papers treating this domain, some of them being presented as references in this chapter.

Fuzzy PID controllers may be used as controllers instead of linear PID controller in all classical or modern control system applications. They are converting the error between the measured or controlled variable and the reference variable, into a command, which is applied to the actuator of a process. In practical design it is important to have information about their equivalent input-output transfer characteristics. The main purpose of research is to develop control systems for all kind of processes with a higher efficiency of the energy conversion and better values of the control quality criteria.

What has been accomplished by other researchers is reviewed in some of these references, related to the chapter theme, making a short review of the related work form the last years and other papers. The applications suddenly met in practice of fuzzy logic, as PID fuzzy controllers, are resulted after the introduction of a fuzzy block into the structure of a linear PID controller (Buhler, 1994, Jantzen, 2007). A related tuning method is presented in (Buhler, 1994). That method makes the equivalence between the fuzzy PID controller and a linear control structure with state feedback. Relations for equivalence are derived. In the paper (Moon, 1995) the author proves that a fuzzy logic controller may be designed to have an identical output to a given PI controller. Also, the reciprocal case is proven that a PI controller may be obtained with identical output to a given fuzzy logic controller with specified fuzzy logic operations. A methodology for analytical and optimal design of fuzzy PID controllers based on evaluation approach is given in (Bao-Gang et all, 1999, 2001). The book (Jantzen, 2007) and other papers of the same author present a theory of fuzzy control, in which the fuzzy PID controllers are analyzed. Tuning fuzzy PID controller is starting from a tuned linear PID controller, replacing it with a linear fuzzy controller, making the fuzzy controller nonlinear and then, in the end, making a fine tuning. In the papers (Mohan & Sinha, 2006, 2008), there are presented some mathematical models for the simplest fuzzy PID controllers and an approach to design

Tuning Fuzzy PID Controllers 173

So, the following fuzzy controllers, with dynamics, have, as a central part a fuzzy block FB, an input filter and an output filter. The two filters give the dynamic character of the fuzzy

The fuzzy block does not treat a well-defined mathematical relation (a control algorithm), as a linear controller does, but it is using the inference with many rules, based on linguistic variables. The inference is treated with the operators of the fuzzy logic. The fuzzy block from Fig. 2 has three distinctive parts, in Mamdani type: fuzzyfication, inference and defuzzification. The fuzzy controller is an inertial system, but the fuzzy block is a noninertial system. The fuzzy controller has in the most common case two input variables *x*<sup>1</sup> and *x*2 and one output variable *u*. The input variables are taken from the control system. The inference interface of the fuzzy block releases a treatment by linguistic variables of the input variables, obtained by the filtration of the controller input variables. For the linguistic treatment, a definition with membership functions of the input variable is needed. In the interior of the fuzzy block the linguistic variables are linked by rules that are taking account of the static and dynamic behavior of the control system and also they are taking account of the limitations imposed to the controlled process. In particular, the control system must be stable and it must assure a good amortization. After the inference we obtain fuzzy information for the output variable. The defuzzification is used because, generally, the actuator that follows the controller must be commanded with a crisp value *u*d,. The command variable *u*, furnished by the fuzzy controller, from Fig. 1, is obtained by filtering the defuzzified variable *u*d. The output variable of the controller is the command input for the process. The fuzzification, the inference and the defuzzification bring a nonlinear behavior of the fuzzy block. The nonlinear behavior of the fuzzy block is transmitted also to the fuzzy PID controllers. By an adequate choosing of the input and output filters we may realize different structures of the fuzzy controllers with imposed dynamics, as are the

The basic structure of the fuzzy controllers with dynamics is presented in Fig. 1.

Fig. 1. The block diagram of a fuzzy controller with dynamics

controller. The fuzzy block has the well-known structure, from Fig. 2.

**2. Fuzzy controllers** 

**2.1 Fuzzy controllers with dynamics** 

Fig. 2. The structure of fuzzy block

general PI, PD and PID dynamics.

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, 2009).

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 approach.

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 criteria obtained using the fuzzy PI controller.
