Proportional Integral (PI) Controller for a Process Plant System

Proportional Integral (PI) Controller for a Process Plant System

Musa Idi

Mechatronics and Systems Engineering Department Abubakar Tafawa Balewa University, (ATBU) Bauchi, Nigeria

Mohammed Ahmed

Electrical and Electronics Engineering Department Abubakar Tafawa Balewa University, (ATBU) Bauchi, Nigeria

Borskghinchin Daniel Halilu

Department of Electrical and Electronics Engineering Abubakar Tatari Ali Polytechnic, (ATAP)

Bauchi, Nigeria

Abdulkadir Abubakar Sadiq

Electrical and Electronics Engineering Department Abubakar Tafawa Balewa University, (ATBU) Bauchi, Nigeria

AbstractThis paper proposed Proportional; Integral (PI) control strategies for a process plant system using Cohen and Coon and Hagglund-Astron techniques with the Ziegler-Nichols

and PID controllers were as shown in equations (1), (2), (5) and (3), (4), (6) respectively [3-5].

For PI;

method as a base-line for the tuning of the controllers. The results of the system responses of the designed control schemes

ut K p et Ki etdt

(1)

were successfully simulated using LABVIEW. Comparing the results showed that controller implemented using the Hagglund-

After Laplace transformation it becomes;

Es

Astrom method was the best this is because during the simulation

exercise, the system under control produced an overshoot of 13% and a settling time of 11.4226 seconds. On the other hand, the

system tuned using the Cohen and Coon method has an overshoot of 64% and settling time of 18.9149 seconds which are

U s K p E s Ki s

For the PID;

det

(2)

higher than Hagglund-Astrom and Ziegler-Nichols tuning relations.

u t K p e t Ki e t dt Kd dt

(3)

After Laplace transformation;

KeywordsPI Controller; Hagglund-Astrom Method; Ziegler- Nichols Method; Cohen and Coon method; Process Plant.

U s K p Es Ki

Es K s d

sEs

(4)

1. INTRODUCTION

   In terms of time constants for PI;

   U s 1 Es

   (5)

   Process control involves the regulation of variables in a dynamic system. A process control system maintains a variable in a process at its set-point. A process can be any combination of materials and equipments that produces a desirable result through changes in chemical properties, physical properties or energy. A continuous process produces

   And for PID;

   U s

   K p 1

   K

   1

   p 1

   i

   i

   d

   Es

   (6)

   an uninterrupted flow of product, while a batch process produces an interrupted flow of product. Examples of a process include a home heating system, a dairy processing

   K K p

   i

   i

   and Kd

   K p d

   (7)

   system, petroleum refining process, food processing plant,

   Where: u(t) is the actuating signal, e(t); the error signal, Kp;

   fertilizer production plant and so on. The most common controlled variables in a process include pressure, density, flow rate, temperature, viscosity, colour, hardness, PH, and conductivity [1, 2].

   Several control modes that can be used are the Proportional

   proportional gain, Ki; the integral gain, constant and d ; derivative time constant.

2. METHODOLOGY

   i ; integral time

   (P), Integral (I), Proportional plus Integral (PI), Proportional plus Derivative (PD) and Proportional plus Integral plus Derivative (PID). However, in this study only the PI controller will be used. The primary reason for the integral control is to reduce or eliminate steady state errors, but at the expense of worse transient response. The general forms of PI

   The main purpose of this task is to investigate two PI controllers using the: Cohen and Coon and Hagglund-Astrom PID controller tuning algorithms, in addition to use Ziegler- Nichols tuning relations as a base-line design for the tuning of the proportional and integral gains in the control loop of a process plant and to calculate the PI controller settings using

   their designs and compare their performances. The process plant model used was as shown by equation (8).

   2e0.987s

   Kp, Ki and i are obtained as; 1.353820, 0.701057 and 1.93112 respectively. Therefore, the transfer function of Cohen and Coon PI controller is given as;

   Gs

   2.878s 1

   (8)

   GCC 1.353820

   0.701057

   s

   (13)

   A. Hagglund-Aström Controller

   First, the settings for this type of controller are given in details as shown in table I:

   TABLE I. HAGGLUND-ASTROM PI CONTROLLER SETTINGS

   G(s)	Kp	i
   Kes s	0.35 K	7
   Kes s 1	0.14 0.28 K K	0.33 6.8 10

   From table 1 it is clear that the second row is the one that matches the plant model question as a first order system; and

   C. Ziegler-Nichols PI controller

   This method was suggested as the base-line design to judge the two controller designs which were obtained previously. The reaction curve PID settings for this type of controller were as shown in Table II below:

   Controller Structure	Proportional Gain Kp	Integral Time Constant i	Derivative Time Constant d
   Case (i) P	1 RN L
   Case (ii) PI	0.9 RN L	3L
   Case (iii) PID	1.2 RN L	2L	0.5L

   TABLE II. ZIEGLER-NICHOLS PID TUNING RELATIONS

   the values of K P and i

   were determined as shown below:

   0.987000 , = 2.878000 and K= 2.000000; from

   which Kp, Ki and i

   were obtained as; 0.478227, 0.259775

   and 1.84093 respectively. The PI controller transfer function using this method was therefore as shown in equation (9);

   0.259775

   In this method to get the values of RN and L is by plotting a graph of the unit step input response of the modeled plant or

   plant model as large as possible to obtain an accurate

   GHA 0.478227 s

   (9)

   measurement for Ziegler-Nichols tuning rules. RN

   is the ratio

   1. Cohen and Coon PI controller

      The transfer function of the process plant is of the form as shown in equation (10);

      of the maximum slope of the unit step response to the reference input signal, which is unity in this case and L is the delay time. The slope was found to be 0.7333 and L equal to one second, this makes RN to be 0.733300 and Kp, i and Ki

      Gs

      Kes

      s 1

      (10)

      were 1.227000, 3.00000 and 0.409100 respectively. The transfer function of the Ziegler-Nichols PI controller becomes:

      Then, Kp and i are determined using equations (11) and (12) respectively, with K=2.000000, = 2.878000 and

      0.978000 .

      GZN

      s 1.227000 0.409000

      s

      (14)

      K

      (11)

      The Ziegler-Nichols method was used to determine the

      controller parameters K p and Ki which are the proportional

      p K 0.9 12

      gain and integral gain constants respectively, such that the system has good performance.

      30 3

      9

      i

      20

      (12)

3. RESULTS AND DISCUSSION

   Time responses of the closed loop system to a unit step inputs

   with the different controllers are displayed in Fig. 1 and the time response parameters were as shown in Table III below:

   PI Settings	Rise Time (s)	Overshoot (%)	Peak time (s)	Settling Time (s)	Peak Value (s)
   Hagglund -Astrom	2.4828	13.1	6.4712	11.4226	1.13098
   Cohen & Coon	0.7854	66.3	3.1363	18.9149	1.66329
   Ziegler- Nichols	0.9405	33.9	3.1664	11.3576	1.33854

   Fig. 1. Unit step response graph of the plant with the different controllers. TABLE III. PARAMETRIC DATA OF THE THREE CONTROLLERS

   From Table III it can be clearly seen that the system with Hagglund-Astrom controller was the best for the system, because it has smallest overshoot of 13.1%, settling time of 11.4226 seconds and the highest peak time of 6.4712 seconds. On the other hand, using the Cohen and Coon tuning relation, the system has faster response with rise time of 0.7854 seconds, higher overshoot of 66.3% and longer settling time of 18.9149seconds. And for the base line that is using the Ziegler-Nichols tuning method; the system has an overshoot of 33.9%, settling and peak times of 11.3576 and 0.9405 seconds respectively.

   Bode graphs were plotted by using the Labview software which indicate the gain and phase margins of the system for the three different controllers. The plots show the stability as well as determining the form or amount of corrective measure needed for dynamic compensation. The gain margin (GM) is the amount of gain K that can be added to the system to give 0dB [5] which could be read directly from the Bode plot by measuring the vertical distance between curve and the = 1 line at the frequency where the angle of

   =. In addition, the phase margin (PM) is the amount of phase that can be added to a system when the gain is 0dB before the phase reaches [5].

   Fig. 2. Magnitude and Phase Plot for the Hagglund-Astrom Control Method.

   Fig. 3. Magnitude and Phase Plot for the Cohen and Coon Control Method.

   Fig. 4. Magnitude and Phase Plot for the Ziegler_Nichols Control Method.

   The magnitude and phase margin plots for the system with the respective PI controllers were shown in Figs. 2 – 4 while their results for gain margin, phase margin, gain margin and crossover frequency were shown in Table IV.

   TABLE IV. GAIN AND PHASE MARGINS OF THE CONTROL SCHEMES

   PI Controllers	Gain Margin (dB)	Phase Margin (deg)	GM Frequency (Hz)	PM Frequency (Hz)
   Hagglund- Astrom	12.5721	53.6223	1.4663	0.4189
   Cohen & Coon	3.6827	25.1948	1.4826	1.0007
   Ziegler- Nichols	5.4933	42.8767	1.6018	0.8477

   From the magnitude and phase margin plots of respective controllers shown above it can be observed that the gain and phase margins are positive, which further confirms the system stability.

   A 0.1 (10% of unit step) step disturbance was introduced onto the system with the different controllers at the time of 40s (at steady state). The behavior of the system was displayed in Fig. 5.

   Fig. 5. Step responses of the system with 0.1 step disturbance.

   TABLE V. DISTURBANCE RESPONSE PARAMETERS OF THE SYSTEMS

   PI Settings	Overshoot (%)	Peak time (s)	Settling Time (s)	Peak Value
   Hagglund- Astrom	7	3.82	5.76	1.07
   Cohen &Coon	5	2.68	3.44	1.05
   Ziegler- Nichols	5	3.05	4.02	1.05

   From the results above and considering the system responses to the applied disturbance. It can be seen that the system with the Hagglund-Astrom controller rejects the disturbance with an overshoot of 7%, peak time of 3.82s, peak value of 1.07 and settling time of 5.76s while with the Cohen & Coon controller give disturbance rejection with overshoot of 5%, peak time of 2.68s, peak value of 1.05 and settling time of 3.44s. This portrayed that the Cohen & coon method had outperformed the Hagglund-Astrom and Ziegler-Nichols controllers in this regard.

4. CONCLUSION

Cohen and Coon and Hagglund-Astrom tuning algorithms for the control of a process plant were successfully implemented and simulated using Labview software with the Ziegler- Nichols method as a base-line design. The respective PI controller settings were calculated based on the different methods of designs and their performances were compared. Results show that the system was stable using the control schemes with the Hagglund-Astrom controller emerging the best even though the Cohen and Coon shows a slightly better disturbance rejection capability.

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