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neutralization process.
2 Controller structure
In this research, the control problem is considered for
systems, which have single control input and single output.
It is known that, PID controller is most widely used in
industry due to its simple control structure and easy design.
The control signal for a system using a conventional PID
controller can be expressed in the time domain as:
where e is the error between desired variable and measured
variable; uPID is the deviated control signal; and Kp, Ki, and
Kd are the proportional, integral, and the derivative gains,
respectively [31–33].
In fuzzy PID controllers, the three parameters of PID
controller (Kp, Ki, and Kd) to be tuned by using fuzzy
tuners. The detailed fuzzy PID scheme is clearly shown in
Figs. 1 and 2.
There are two inputs to the fuzzy controllers: absolute
error (e(t)) and absolute derivative of error (de(t)/dt) and
three outputs.The parameters a and c locate the ‘‘feet’’ of the triangle,
and the parameter b locates the ‘‘peak’’[34, 35].
The fuzzy inference rule used in fuzzy control is ‘‘If
e(t)is Ai and de(t)is Bj; then Kp is Cij; Ki is Dij and Kd is
Eij,’’ where Ai,Bj,Cij,Dij,Eij are fuzzy subsets of inputs
and outputs, and i, j = 1, 2,3,…, n.
Fuzzy control is applied using local inferences, that
means each rule is inferred and the results of the inferences
of individual rules are then aggregated. The most common
inference methods are the max–min method, the max-
product method, and the sum-product method, where the
aggregation operator is denoted by either max or sum and
the fuzzy implication operator is denoted by either min or
prod. Especially, the max–min calculus of fuzzy relations
offers a computationally nice and expressive setting for
constraint propagation [36, 37]. Finally, a defuzzification
method is needed to obtain a crisp output from the aggre-
gated fuzzy result. Popular defuzzification methods include
maximum matching and centroid defuzzification, and the
centroid defuzzification is widely used for fuzzy control
problems [33]. Hence, in this study, the fuzzy reasoning
results of outputs are gained by the aggregation operation
of fuzzy sets of inputs and designed fuzzy rules, where
max–min aggregation method and centroid defuzzification
method are used.
In this paper, fuzzy PID control in three different
chemical processes (level control, temperature control ofunstable continuous stirred tank reactors, and pH control)
compared with classical PID controller.
3 Level control of a two interacting tanks
3.1 Introduction
Level control is one of the most important processes in
Chemical Engineering that has an application in many
processes. In many papers, level is controlled by classic PID
controller or the other types of controllers and acceptable
performance is gained by using them. In this paper, level
control is used for analyzing the fuzzy PID controllers.
Suresh et al. [38] used level control in three non-inter-
acting and interacting tanks for analyzing performance of
fuzzy PID controller. They showed that the performance of
fuzzy PID controller is better than classic controller by
computer simulation. It is important to mention that Suresh
and his colleagues used a relatively simple model for the
simulation of two non-interacting and interacting tanks
(they neglected nonlinear behavior of valves).
AlsoMr. Xu et al. [39] used two cascaded non-interacting
tanks for performance analyzing of fuzzy PID controller.
They simulated the process using first-order model with
delay then designed fuzzy PID control for this process and