Fuzzy PD plus I Controller of a CSTR for Temperature Control

2.1 CSTR

A reaction occurring in CSTR is either exothermic or endothermic. Thus, to maintain a constant temperature of the reactor, it needs to be cooled or heated by an external medium. Figure 1 shows the schematic diagram of a CSTR process, where an exothermic reaction was considered. In the figure C_f, T_f, F_f and C, T, F represent the concentration [mol/m³], temperature [K], and flux [m³/sec] at the inlet and outlet of the fluid, respectively; and T_cf, F_cf and T_c, F_c represent the temperature and flux at the inlet and outlet of the coolant, respectively.

Figure 1:

Nonisothermal CSTR process

To simplify the problem, it is assumed that the irreversible reaction is an exothermic reaction, and the reaction is the first order system with respect to the reactant. It is also assumed that the fluid in the reactor is well stirred, the input and output fluxes are identical, and the parameters are constant regardless of the temperature. When the material and energy conservation laws are applied to the CSTR process, the following dimensionless dynamic equations can be obtained [1].

where x₁ and x₂ are the state variables that represent the concentration C and the temperature T, respectively; y and u are the fluid outlet temperature and the coolant temperature, which are output and input of the CSTR, respectively; and d₁ and d₂ are the disturbances. Also, D_a is the Damökhler number, H is the heat of reaction, β is the heat transfer coefficient, V is the volume of the CSTR [m³], γ = E/RT_f, E is the activation energy [cal/mol], and R is the gas constant [cal/mol-K].

In this regard, the x₁, x₂, u, and t in Equation (1) are made to be dimensionless as shown in Equation (2).

Considering the physical limit of the control valve that operates for temperature control in the CSTR process, it is assumed that a saturator expressed as the following nonlinear equations exists between the controller and the CSTR process.

where u_min and u_max are the minimum and maximum values of the saturator, respectively; and u_sat is the output of the saturator or control input.

Thus, the control system that combined the saturator and the CSTR process can be expressed as a block diagram as shown in Figure 2.

Figure 2:

CSTR process with a saturator

When the open loop system in Eq. (1) has the values of D_a= 0.072, γ = 20, H =8, and β=0.3 in a steady state, three equilibrium points (x_eA= [0.144, 0.886]^T, x_eB= [0.447, 2.752]^T, and x_eC = [0.765, 4.705]^T) exist, where x_eA and x_eC are stable equilibrium points and x_eB is an unstable equilibrium point. Figure 3 shows the three equilibrium points on a phase diagram.

Figure 3:

Phase-portrait for three equilibrium points

Figure 4 shows the open loop responses of the CSTR for the various step inputs. As shown in the figure, the response and the gain vary significantly depending on the step input. In particular, when the step input is smaller than 1, a very small difference in the input has a large effect on the esponse. Due to this strong nonlinearity, it is not easy to control the CSTR using an existing linear PID controller.

Figure 4:

Open loop responses of the CSTR for step inputs

3.1 Basic structure of a Fuzzy-PD controller

A fuzzy PD controller generally consists of fuzzification, fuzzy inference by fuzzy rules, and defuzzification, as shown in Figure 5.

Figure 5:

Typical structure of fuzzy PD controller

For the inputs of the fuzzy PD controller, there are two kinds of signals (the error and the variation in the error) as shown in Equation (4); and the output is defined as u(t).

where y_r(t) is the set point, E and ΔE are the linguistic variables for the error and the variation in the error, respectively, K_e and K_ce are the scaling factors for the input signals, and F(·) represents the fuzzy inference.

The fuzzy sets of the linguistic variables for all the input/output of the fuzzy PD controller used in this study were defined by normalization as [-1 1] as follows.

The linguistic labels used to describe the fuzzy sets are ‘negative big’ (NB), ‘negative medium’ (NM), ‘negative small’ (NS), ‘zero’ (Z), ‘positive small’ (PS), ‘positive medium’ (PM), ‘positive big’ (PB). For the membership function, a Gaussian-type fuzzy set was used as shown in Figure 6.

Figure 6:

The membership functions of normalized inputs and output variables

For convenience, the fuzzy rules are summarized in Table 1; and they are interpreted as follows.

Table 1

The fuzzy rule matrix of fuzzy PD controller

For the fuzzy rules, widely used rules based on the experience and knowledge of humans were used, and all the rules consisted of fuzzy relationships. For the fuzzy inference, the Max-Min method was used based on the Mamdani type; and for the defuzzification, the center of gravity method was applied. Figure 7 shows the control surface of the fuzzy PD controller.

Figure 7:

The control surface of Fuzzy PD controller

3.2 Structure of a Fuzzy PD plus I controller

A series of studies using fuzzy technique have recently been conducted, but they perform control by obtaining a linear model at one operating point of CSTR [10][11]. In the present study, a fuzzy PD plus I controller that combined a fuzzy PD controller in parallel with a linear integral controller was used to control the temperature of CSTR, and the performances for the set-point tracking and for the parameter change in the CSTR were improved.

As shown in Figure 8, the fuzzy PD plus I controller can improve the steady-state performance of CSTR temperature through the linear integral controller that is added after the fuzzy inference of the fuzzy PD controller. In Figure 8, K_e and K_ce are the scaling factors for the fuzzy input variables, K_u is the scaling factor for the fuzzy output variable, and K_ie is the gain of the linear integral controller. Appropriate scaling factors for input and output signals enable flexible and adaptive response to the dynamic characteristics of the control target. In this study, they were determined by trial and error.

Figure 8:

Fuzzy PD plus I controller

4.1 Control technique of the comparison target

Chen and Peng [4] recently suggested an adaptive controller for a system with saturator u_min ≤ u(t) ≤ u_max, and the control input can be expressed as Equation (7).

where $$ \tilde{u}\left(t\right)$$ is a hyperbolic tangent function, and is expressed as follows.

where e(t) is the error between the set point and the output, m is the slope, and θ(t) is the bias. $$ \tilde{u}\left(t\right)$$ has a value between -1 and 1, and thus, u(t) is maintained within the limit value of the saturator.

As shown in Equation (8), $$ \tilde{u}\left(t\right)$$ has two adjustable parameters. In particular, the size and sign of the slope m sensitively respond to the characteristics of the controller. Thus, Chen and Peng suggested an algorithm that fixes m and adaptively adjusts only the bias θ(t) depending on the problem.

The tuning algorithm of θ(t) can be expressed as follows.

where η is the learning rate, and it is a positive number. $$ sign\left(\frac{\partial y}{\partial u}\right)$$ is the value of ±1 that is determined by the response direction of the system, and it is obtained from a step response experiment or the physical characteristics of the control target. In this way, the algorithm continuously adjusts the parameters of the controller based on the error between a set point and an actual output. In addition, in Korkmaz et al. [3] that was also used for comparison, the structure of an existing linear PID controller was maintained, and three gains were nonlinearly changed. The nonlinear PID (NPID) controller of Korkmaz is expressed as Equation (10).

where K_p(e), K_i(e), and K_d(e) are nonlinear functions based on the error e and the Gaussian error function, and they play roles similar to the proportional, integral, and derivative gains of a linear PID controller.

where, $$ erf\left(x\right)=\frac{2}{\sqrt{\pi }}{\int }_0^x{e^{-t}}^{2}dt$$

The controller gain parameters a₁, a₂, b, and c in Equation (11) are positive constants, and are appropriately adjusted by a user. Based on this, NPID controller of Korkmaz operates so that the proportional and derivative gains can be increased when the absolute value of the error is large and so that the integral gain can reduce the steady-state deviation when the absolute value of the error is small.

4.2 Data and controller parameters for the simulation

For the data of the CSTR used in the simulation, D_a=0.072, γ= 20, H= 8, β= 0.3, the upper and lower limit values of the saturator were ±5, and the sampling interval was 0.01.

Chen and Peng used η= 0.2 and m= 5 for the temperature control of CSTR with a saturator, and $$ sign\left(\frac{\partial y}{\partial u}\right)$$ was 1 because a CSTR process has positive gain. For the parameters a₁, a₂, b, and c of the NPID controller in Korkmaz et al. [3], 35.87, 21.70, 43.33, and 2.73 were used, respectively.

In addition, the scaling factors of the fuzzy PD plus I controller shown in Figure 8 were selected to be K_e=1.2, K_ce=0.1, K_u=55, and K_ie=1.8 through trial and error.

4.3 Performance comparison

4.3.1 Set-point tracking performance

First, a set-point tracking experiment was conducted considering a control environment that changes the temperature at the outlet of the CSTR coolant. In the experiment, when the process was in a stable equilibrium state (x_eA = [0.144, 0.886]^T), the set point was step-wisely increased to y_r = 2.75, which changed the output y to an unstable equilibrium state x_eB. Then, at t= 10, it was again step-wisely increased to y_r=4.705, which led to a stable equilibrium state x_eC. Figure 9 shows the output y and the saturator output u_sat.

Figure 9:

Set-point tracking responses when set point yr is step-wisely increased from 0.886 to 4.705.

As shown in the figure, both the adaptive controller of Chen and Peng and the NPID controller of korkmaz reached the set point without a steady-state error, but the proposed method approached quickly with a smaller overshoot than those of the other two methods. In particular, for all the controllers, the overshoot of the response was larger when it was changed from x_eB to x_eC compared to when it was changed from x_eA to x_eB; and this is considered because of the intrinsic nonlinearity of the CSTR.

Table 2 summarizes the overshoot (M_p), rising time (t_r=t₉₀-t₁₀), and 2% setting time (t_s) in order to compare the performances of the three methods. In this regard, t₁₀ and t₉₀ refer to the times required for the output to reach 10% and 90% of the set point, respectively. As summarized in the table, the proposed method showed superior performance compared to the other two methods.

Table 2

Comparison of set-point tracking performances when set point yr is step-wisely increased from 0.886 to 4.705.

As for the intrinsic nonlinearity of the CSTR, the response characteristics could vary when the set point y_r is increased or decreased. Figure 10 shows the response where the set point y_r was changed to 2.75 when the process was in an equilibrium state (x_eC = [0.765, 4.705]^T), and the y_r was again stepwisely decreased to 0.886 at t = 10 when the process was at x_eB. Table 3 summarizes the performances of the three methods, similar to Table 2.

Table 3

Comparison of set-point tracking performances when set point yr is decreased from 4.705 to 0.886.

Figure 10:

Set-point tracking responses when set point yr is step-wisely increased from 4.705 to 0.886.

In particular, for the NPID controller of korkmaz, the control input showed significant hunting.

4.3.2 Parameter change performance

The performance of the controller regarding the change in the parameter of the CSTR was examined. As the CSTR was expressed as dimensionless dynamic equations, it was not easy to consider the change in the parameter. Thus, in this study, it was assumed that the heat transfer coefficient β decreased due to the long-term use of the CSTR.

An experiment was first performed assuming a 20% decrease in the nominal value of the heat transfer coefficient β (to 0.24 from 0.3 ) (Figure 11), and an experiment was then performed assuming a 40% decrease to 0.18 (Figure 12). As shown in the figures, the proposed method was more robust to the parameter changes than the other two methods.

Figure 11:

Parameter change responses from –20% for the heat transfer coefficient β

Figure 12:

Parameter change responses from –40% for the heat transfer coefficient β

In particular, when the β was decreased by 40%, the adaptive controller of Chen and the NPID of Korkmaz showed large overshoot.