Page 66 - 2024-Vol20-Issue2

62 | Al-Najari, Hen, Paw, & Marhoon

Fig. 7. Step response Fig. 9. Step response Goriginal

TABLE II.
ZIEGLER NICHOLS CLOSE LOOP PID PARAMETERS

Variable Relation Value
Kcr From drawing 13.8114947
Pcr From drawing
Kp 47.009
Ti 0.6*Kcr 8.28689682
Td 0.5*Pcr
Ki 0.125*Pcr 23.5045
Kd Kp/Ti 5.876125
Kp*Td 0.352566395
48.69484158

Fig. 8. Multi step response as a polynomial model using Pade approximation and hence
control system is designed for it [16]. The first-order pade
1) Ziegler Nichols ZN approximation is:
In this logic, the controller is set as a proportional controller,
and then the process is near the specific operating point of e-? s = 1- ? s (4)
marginal stability. The ‘Kp’ value will increase till the output 2
shows oscillations that are studied. The value of ‘Kp’ cor-
responding to this is called critical gain ‘Kcr’. The period 1 + ? s
of the oscillation ‘Pcr’ is called the critical period. The PID 2
parameters are then calculated using Table II [15].
Where ? is the delay time.
Fig. 7 shows the step response of the original function The delay time of the original function is ? =18.2. There-
using ZN tuning. Fig. 8 shows the Multi-Step Response of fore, the equation (4) will be:
the original function using ZN tuning.
e-18.2s = 1- 18.2 s (5)
2) MATLAB Tuner 1+ 2
The MATLAB Tuner method was used to improve the re-
sponse of the original function. Fig. 9 shows the step response 18.2 s
original function using MATLAB Tuner. Fig. 10 shows the 2
Multi-Step Response of the original function using MATLAB
Tuner. From equations (2) and (5), the approximated transfer
function (Gapproximated) will be:
B. Pade Approximation
As shown in Fig. 9 and Fig. 10, the original function has a -0.8672s + 0.0953 (6)
delay time also the performance of the response is not good. Gapproximated(s) = 58.24s2 + 15.5s + 1
The delay time affects control. To solve this problem, the Pade
approximation method was used to approximate the original Fig. 11 shows the step response of approximated function
function (Goriginal). The FOPDT model of PH is approximated
(Gapproximated) using MATLAB Tuner. Fig. 12 shows the
Multi-Step Response of approximated (Gapproximated) func-
tion using MATLAB Tuner.