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Some other methods of controlling have been imposed
on this process and have gained acceptable answers but
fuzzy PID control and fuzzy control have been imposed
rarely.
Khaksar et al. [44] used a fuzzy self-tuning PID con-
troller for temperature control of mentioned process and
showed that its performance is better than classic PID
controller.
In this section, we consider temperature control problem
of a CSTR in which an unstable first-order exothermic
reaction takes place. A fuzzy PID controller is used in this
section.
4.2 Modeling equations
Consider a CSTR in which an exothermic first-order
reaction takes place (A ? B). The material and energy
balances based on the assumptions of constant volume
inside the reactor, perfect mixing and constant physical
properties allow obtaining the dynamic model. The
differential equations can be written in dimensionless form
as follows [41].where x1,x2,x3, and qc are the dimensionless concentra-
tion, reactor temperature, cooling-jacket temperature, and
cooling-jacket flow rate, respectively. The main objective
is controlling the reactor temperature with manipulating the
coolant flow rate (qc). The values of the parameters are
given in Table 5.
CSTR exhibits ignition/extinction behavior for this set
of parameters. Here, we study the system at two operating
points, one unstable (Op1) and one stable (Op2).
The unsteady-state values of the three states and of the
manipulated input are (Op1):
x1s ¼ 0:4737; x2s ¼ 3:1702; x3s ¼0:2968; qcs
¼ 1:482225
The high-temperature steady state (Op2) is open loop stable
with the following steady-state values [44, 45]:
x1s ¼ 0:2028; x2s ¼ 5; x3s ¼ 0:4079; qcs ¼ 0:97854.3 Simulation results
In this section, trial-and-error method is used to find the
controller parameters of classic PID. Afterward, these
parameters are optimized and resulted parameters are
shown in the Table 6.
In fuzzy PID, fuzzy tuner must calculate the controller
parameters. This tuner has two inputs: error e(t) and
derivative of error d(t), and three outputs: Kp, Ki and Kd.
For the input and output variables (e,de, Kp, Ki, and Kd),
five membership functions Z, VS, S, M, and B are used.
They are Z, zero; VS, very small; S, small; M, medium;
and B, big. The triangle membership function for inputs
and outputs is shown in Fig. 7.
The fuzzy inference rules are shown in Table 7.The proposed fuzzy PID-type controller has been com-
pared with the classic PID controller. The closed loop
responses of CSTR are shown in Fig. 8.
The results show that the fuzzy PID controller has better
performance than PID controller.
5 pH control system
pH control is vital in some processes and design of it must
be accurate for these processes [46, 47]. Different methods
have been investigated for designing of linear and nonlin-
ear pH processes control in different papers; for instance,
control of total model [48], control of inner model [49],
control of same reactions, and fuzzy PI control in basis of
scheduled gain [50–53].
Regarding high nonlinearity of pH process, performance
of classic PID controller is inappropriate and we must find
better controllers such as fuzzy controller and predictive
controller for pH process.
Most of the research on pH control in last 2 decades are
about using fuzzy controller or combining of it with con-
trollers on basis of model prediction [54–56]. In another
work [48], there is a review on controller’s types that
imposed to pH neutralization process.
According to the papers on pH neutralization process
control, fuzzy PID controller is a relatively appropriate