Design and Development of Complex-Order PI-PD Controllers: Case Studies on Pressure and Flow Process Control
Abstract
:1. Introduction
- 1.
- Developing a mathematical model that captures the dynamic reaction of a process plant.
- 2.
- Designing an appropriate complex-order PI-PD controller.
- 3.
- Conducting simulations and experiments to compare its performance with conventional PID controllers.
2. Development of Proposed Controller
2.1. Integer and Fractional-Order PID Controllers
2.2. PI-PD Controller
2.3. Fractional-Order PI-PD Controller
2.4. Proposed Complex-Order PI-PD Controller
2.5. Approximation Technique
- Choose the range of and the order N.
- Using MATLAB built-in command frd, compute the frequency response magnitude data of for the chosen range as stated in Equation (7).
- Perform a fit of the frequency response magnitude data for the selected order using the MATLAB built-in command fitmagfrd obtained in the previous step.
- Convert the state-space model obtained in the previous step to a transfer function model.
3. Results and Discussions
3.1. Simulation Study
3.1.1. First-Order Process Model
3.1.2. Second-Order Process Model
3.1.3. Third-Order Process Model
3.2. Experimental Study
3.2.1. Flow Process Model
3.2.2. Pressure Process Model
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Ref. | Controller | Tuning Technique | No. of Parameters | System | Comparing Controllers | Performance Measure |
---|---|---|---|---|---|---|
[2] | PI-PD | Auto tuning | 4 | First and second-order unstable system with time delay | PID | Peak time, IAE, Settling time, Overshoot, Rise time |
[12] | PI-PD | Ziegler–Nichols | 4 | First-order plus dead time (stable and unstable system) | I, PI, and PID | Settling time, Peak overshoot, Peak undershoot (-ve), IAE, ISE, ITAE, ITSE |
[24] | PI-PD | Particle Swarm Optimization (PSO) | 4 | Antilock braking system | Feedback linearization | Friction coefficient (), Settling time, Rise time, Stopping distance (m), Range of torque (N.m) |
[25] | PI-PD | Trial and error | 4 | First and second-order system, higher-order unstable dead time system | PI and PID | Steady state error, Settling time, Peak overshoot |
[26] | PI-PD | Ziegler–Nichols | 4 | Second-order system, temperature control for oil-cooling machines | PID and modified PID | Disturbance rejection |
[20] | PI-PD | Trial and error | 4 | Magnetic levitation system | Feedforward PI-PD | Settling time, Overshoot |
[21] | PI-PD | Trial and error | 4 | Photovoltaic and wind turbine system | IPI, FOPI | Voltage |
[27] | PI-PD | Polynomial curve fitting techniques | 4 | First and second-order system with time delay | PID | Settling time, Overshoot, Jmin |
[28] | Fractional-order PI-PD | Ziegler–Nichols | 6 | Pressure process | PID, FOPID, PI-PD, and FOPI-PD | Rise time, Settling time, IAE Peak overshoot, ISE |
[29] | Fractional-order PI-PD | Trial and error | 5 | Second-order dead time and oscillatory system | PI-PD | Rise time, IAE, Settling time, Peak overshoot |
[16] | Fractional-order PI-PD | PSO and Genetic algorithm | 4 | DC-DC converter | PID, I-PD, and FOPI-PD | Settling time, Overshoot |
[30] | Fractional-order PI-PD | Bode’s integrals | 6 | pH control, distillation column and liquid level plant | FOPI and FOPD | Overshoot, settling time and ITAE |
[31] | Fractional-order PI-PD | Stability boundary locus and the weighted geometrical centre | 6 | pH control and multiple dead time process models | Different PI-PD control designs | Statistical analysis |
[15] | Cascade PI-PD | Stochastic fractal search and Pattern search algorithm | 4 | Plug-in Electric Vehicles | PI and PID | Settling time, IAE, ISE, ITAE, ITSE |
[13] | Fractional-order cascaded PI-PD | Graphical method | 6 | Cart-Inverted Pendulum, second and third-order linear time-invariant unstable system | PI λ-PDμ, -, PIλDμ | Settling time, Overshoot, Phase margin (deg), Bandwidth (rad/s), Delay margin () |
[17] | Modified PI-PD | Auto tuning | 7 | DC motor, automatic voltage regulator | PID, modified PID, and modified PI-PD | Settling time, Overshoot, IAE |
[18] | Modified PI-PD | Ziegler–Nichols | 7 | Aircraft pitch angle and disc position control | PID | Settling time, Overshoot, IAE, Gain and Phase margin |
[32] | Improved PI-PD | Trial and error | 6 | FOPTD DC motor model | - | Steady state error |
[21] | Modified PI-PD Smith predictor | Auto tuning | 4 | Multiple dead time process models | Smith predictor PI and Smith predictor PID | Settling time, Overshoot |
[22] | Optimal PI-PD | Cuckoo search algorithm | 4 | DC motor, third-order system transfer function | I-PD | Settling time, ISE |
Controllers | OS (%) | Settling Time (, Seconds) | Rise Time (, Seconds) | |||||||
---|---|---|---|---|---|---|---|---|---|---|
PID | 17.5 | 4.12 | 9.46 | - | - | - | - | 0.7406 | 98.3306 | 66.4899 |
PI-PD | 17.5 | 4.12 | 9.46 | - | - | - | - | 3.7814 | 117.7727 | 32.3616 |
FOPID | 17.5 | 4.12 | 9.46 | 0.98 | 0.02 | - | - | 0 | 86.9470 | 30.8176 |
FOPI-FOPD | 17.5 | 4.12 | 9.46 | 0.98 | 0.02 | - | - | 0 | 138.7361 | 68.7531 |
COPI-PD | 17.5 | 4.12 | 9.46 | 0.98 | - | 0.01 | - | 3.7814 | 117.7864 | 32.3687 |
PI-COPD | 17.5 | 4.12 | 9.46 | - | 0.02 | - | 0.01 | 3.7642 | 117.6919 | 31.7754 |
COPI-FOPD | 17.5 | 4.12 | 9.46 | 0.98 | 0.02 | 0.01 | - | 0.0318 | 139.1953 | 69.9659 |
FOPI-COPD | 17.5 | 4.12 | 9.46 | 0.98 | 0.02 | - | 0.01 | 0 | 138.5352 | 68.5506 |
COPI-COPD | 17.5 | 4.12 | 9.46 | 0.98 | 0.02 | 0.01 | 0.01 | 0.0014 | 138.9981 | 69.7455 |
Controllers | OS (%) | Settling Time (, Seconds) | Rise Time (, Seconds) | |||||||
---|---|---|---|---|---|---|---|---|---|---|
PID | 17.5 | 4.12 | 9.46 | - | - | - | - | 3.663 × 10 | 116.8035 | 14.1159 |
PI-PD | 17.5 | 4.12 | 9.46 | - | - | - | - | 0.9225 | 105.4154 | 34.6860 |
FOPID | 17.5 | 4.12 | 9.46 | 0.98 | 0.02 | - | - | 0 | 113.8499 | 30.5718 |
FOPI-FOPD | 17.5 | 4.12 | 9.46 | 0.98 | 0.02 | - | - | 0.0547 | 146.5290 | 93.6863 |
COPI-PD | 17.5 | 4.12 | 9.46 | 0.98 | - | 0.01 | - | 0.9242 | 105.4284 | 34.6943 |
PI-COPD | 17.5 | 4.12 | 9.46 | - | 0.02 | - | 0.01 | 0.8785 | 105.2664 | 33.0961 |
COPI-FOPD | 17.5 | 4.12 | 9.46 | 0.98 | 0.02 | 0.01 | - | 0.1751 | 147.0024 | 94.8372 |
FOPI-COPD | 17.5 | 4.12 | 9.46 | 0.98 | 0.02 | - | 0.01 | 0 | 148.1435 | 95.0625 |
COPI-COPD | 17.5 | 4.12 | 9.46 | 0.98 | 0.02 | 0.01 | 0.01 | 0.0076 | 148.6575 | 96.3397 |
Controllers | OS (%) | Settling Time (, Seconds) | Rise Time (, Seconds) | |||||||
---|---|---|---|---|---|---|---|---|---|---|
PID | 2.4532 | 4.8789 | 0.3314 | - | - | - | - | 22.1842 | 97.8430 | 17.0763 |
PI-PD | 2.4532 | 4.8789 | 0.3314 | - | - | - | - | 6.7384 | 88.5320 | 15.3694 |
FOPID | 2.4532 | 4.8789 | 0.3314 | 0.98 | 0.02 | - | - | 54.8152 | 93.8416 | 11.3073 |
FOPI-FOPD | 2.4532 | 4.8789 | 0.3314 | 0.98 | 0.02 | - | - | 15.8640 | 94.9228 | 12.0943 |
COPI-PD | 2.4532 | 4.8789 | 0.3314 | 0.98 | - | 0.01 | - | 6.6980 | 88.5421 | 15.3753 |
PI-COPD | 2.4532 | 4.8789 | 0.3314 | - | 0.02 | - | 0.01 | 7.3362 | 88.0580 | 15.3340 |
COPI-FOPD | 2.4532 | 4.8789 | 0.3314 | 0.98 | 0.02 | 0.01 | - | 16.5639 | 94.7933 | 12.0374 |
FOPI-COPD | 2.4532 | 4.8789 | 0.3314 | 0.98 | 0.02 | - | 0.01 | 15.8146 | 94.8964 | 12.0969 |
COPI-COPD | 2.4532 | 4.8789 | 0.3314 | 0.98 | 0.02 | 0.01 | 0.01 | 16.5141 | 94.7665 | 12.04 |
Controllers | OS (%) | Settling Time (, Seconds) | Rise Time (, Seconds) | |||||||
---|---|---|---|---|---|---|---|---|---|---|
PID | 1 | 1 | 0.1 | - | - | - | - | 6.1045 × | 237.6946 | 78.3350 |
PI-PD | 1 | 1 | 0.1 | - | - | - | - | 0 | 240.7698 | 80.6822 |
FOPID | 1 | 1 | 0.1 | 0.98 | 0.02 | - | - | 1.2636 × | 239.3677 | 79.5915 |
FOPI-FOPD | 1 | 1 | 0.1 | 0.98 | 0.02 | - | - | 0 | 241.8704 | 81.9465 |
COPI-PD | 1 | 1 | 0.1 | 0.98 | - | 0.01 | - | 1.2549 × | 239.3861 | 79.6018 |
PI-COPD | 1 | 1 | 0.1 | - | 0.02 | - | 0.01 | 1.2648 × | 239.3679 | 79.5916 |
COPI-FOPD | 1 | 1 | 0.1 | 0.98 | 0.02 | 0.01 | - | 2.6519 × | 241.5152 | 82.1655 |
FOPI-COPD | 1 | 1 | 0.1 | 0.98 | 0.02 | - | 0.01 | 0 | 241.8741 | 81.9517 |
COPI-COPD | 1 | 1 | 0.1 | 0.98 | 0.02 | 0.01 | 0.01 | 2.3946 × | 241.5194 | 82.1699 |
Controllers | OS (%) | Settling Time (, Seconds) | Rise Time (, Seconds) | |||||||
---|---|---|---|---|---|---|---|---|---|---|
PID | 0.5 | 0.5 | −0.01 | - | - | - | - | 6.5841 | 163.0095 | 18.5776 |
PI-PD | 0.5 | 0.5 | −0.01 | - | - | - | - | 0.3905 | 175.9587 | 38.7020 |
FOPID | 0.5 | 0.5 | −0.01 | 0.98 | 0.02 | - | - | 4.0610 | 162.1264 | 25.0510 |
FOPI-FOPD | 0.5 | 0.5 | −0.01 | 0.98 | 0.02 | - | - | 0.2565 | 178.9595 | 40.4344 |
COPI-PD | 0.5 | 0.5 | −0.01 | 0.98 | - | 0.01 | - | 0.2876 | 176.0390 | 38.7466 |
PI-COPD | 0.5 | 0.5 | −0.01 | - | 0.02 | - | 0.01 | 7.5495 × | 177.5542 | 83.0930 |
COPI-FOPD | 0.5 | 0.5 | −0.01 | 0.98 | 0.02 | 0.01 | - | 0 | 180.4267 | 86.5732 |
FOPI-COPD | 0.5 | 0.5 | −0.01 | 0.98 | 0.02 | - | 0.01 | 0 | 180.2877 | 84.9259 |
COPI-COPD | 0.5 | 0.5 | −0.01 | 0.98 | 0.02 | 0.01 | 0.01 | 2.1894 × | 180.4000 | 86.4863 |
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Share and Cite
Bin Roslan, M.N.; Bingi, K.; Devan, P.A.M.; Ibrahim, R. Design and Development of Complex-Order PI-PD Controllers: Case Studies on Pressure and Flow Process Control. Appl. Syst. Innov. 2024, 7, 33. https://doi.org/10.3390/asi7030033
Bin Roslan MN, Bingi K, Devan PAM, Ibrahim R. Design and Development of Complex-Order PI-PD Controllers: Case Studies on Pressure and Flow Process Control. Applied System Innovation. 2024; 7(3):33. https://doi.org/10.3390/asi7030033
Chicago/Turabian StyleBin Roslan, Muhammad Najmi, Kishore Bingi, P. Arun Mozhi Devan, and Rosdiazli Ibrahim. 2024. "Design and Development of Complex-Order PI-PD Controllers: Case Studies on Pressure and Flow Process Control" Applied System Innovation 7, no. 3: 33. https://doi.org/10.3390/asi7030033