# Fractional-Order PID Controllers for Temperature Control: A Review

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## Abstract

**:**

## 1. Introduction

## 2. A Developing Trend of Fractional-Order PID Controllers

## 3. Fractional-Order Control Landmarks

- Control system improvement;
- Optimal control time delay contribution;
- Disturbance rejection control contribution.

#### 3.1. Early Fractional Analysis

_{c}is the gain crossover frequency, which is defined at L(jω

_{c}) = 1. The slope of the magnitude curve is defined as parameter γ, which is determined by the fractional slope of −20γ dB/dec; this can be on a log-log scale, with a flattened phase characteristic at the value of −γ π/2 radii for the phase curve of the system [45]. From a researcher’s point of view, properties associated with fractional-order dynamics can be very beneficial in control systems. The phase margin also remains constant, and its role is independent of gain changes in the frequency range. Both the slope and margin of such a curve can be varied in terms of fractional-order and system preservation, which are very important factors for robust control of the system. Figure 2 shows the isodamping properties of the open-loop transfer function (Bode plots), and Figure 3 shows the step response of controlled systems, clearly indicating a flat phase curve at the crossover frequency. Controller parameter variation results in improved robustness of the step response of the system. This transfer function is used to indirectly tune the controller problem and to tune PID controllers [46,47] based on Bode’s function. An example presented in [35] shows the importance and benefits of fractional-order control. Such fundamental systems have potential applications in robotics, such as translation and rotational motor design.

#### 3.2. Concerns about the Optimality of Time-Delay Systems

**Figure 2.**Bode plot comparison of frequency, phase, and magnitude in isodamping [49].

**Figure 3.**Different levels of step responses in isodamping of a control system [49].

#### 3.3. Contributions of Disturbance Rejection in Systems

## 4. Temperature Control Transfer Function

## 5. Fractional-Order PID Controllers in Bioreactors

- Batch;
- Continuous;
- Semi-batch

#### Dragonfly Optimization

- The second factor points to velocity matching among neighborhoods.
- The cohesion factor indicates the attraction of individuals in the center.

Fractional-Order Controller | Control Application | Meta-Heuristic Method | Statistical Evaluation | Comparison PID | Comparison with Optimization Methods | Ref. |
---|---|---|---|---|---|---|

FOPID | Designed for mathematical model | Particle swarm optimization (PSO) | No | No | GA | [55] |

FOPID | Designed as automatic voltage regulator | PSO | No | Yes | No | [56] |

Time-delay | Design for second-order time-delay system | Improved electromagnetism | No | No | GA | [57] |

FOPID | Designed for fractional-order plant | Self-organizing migrating algorithm | No | Yes | No | [58] |

FOPID | Designed for integer-order and fractional-order plant | PSO | No | Real-life | No | [59] |

FOPID | Designed for an invasive real-life analog plant | Invasive weed optimization algorithm | Yes | No | GA & PSO | [60] |

FOPID | Designed for a full-vehicle non-linear activesuspension system | Evolutionary algorithm | No | No | No | [61] |

FOPID | Designed for a first-order system with time delay | Generic algorithm | No | Yes | No | [62] |

FOPID | Designed for a process plant Transfer | Bee colony algorithm | No | No | GA & PSO | [63] |

FOPID | Designed as a voltage regulator | Chaotic ant Swarm | Yes | No | GA & PSO | [64] |

FOPID | Designed for speed control | Artificial bee colony | Yes | Yes | No | [65] |

FOPID (fuzzy) | Designed as an electronic throttle | Fruit fly optimization | No | Yes | No | [66] |

FOPID | Designed as an automatic voltage regulator | PSO | No | Yes | Chaotic Ant Swarm | [67] |

Fractional-Order Controller | Control Application | Statistical Evaluation | Comparison PID | Ref. |
---|---|---|---|---|

FOPID | Designed for control of super-heated steam temperature | No | Yes | [68] |

FOPID | Designed for automatic control for an autotuning module | Yes | No | [69] |

FOPID | Designed the for the safety of medical products | No | No | [70] |

FOPID | Designed for internal temperature control in a roller system | No | Yes | [71] |

FOPID | Designed for steam temperature control of essential oil extraction | Yes | No | [72] |

FOPID | Designed for temperature control assessment | Yes | No | [73] |

FOPID | Designed to cooling wat | No | Yes | [74] |

FOPID | Designed for fuzzy fractional control of a heat exchanger | No | Yes | [75] |

FOPID | Designed for double digital realization | Yes | No | [76] |

FOPID | Designed for anti-windup for temperature control | Yes | No | [77] |

FOPID | Designed as a large-scale dynamic matrix controller | Yes | Yes | [78] |

FOPID | Designed for reheating furnaces with a large time delay | Yes | No | [79] |

FOPID | Designed as a fuzzy PI+PD controller | Yes | Yes | [80] |

## 6. Fractional-Order PID Controller for Ambulance Temperature Control

#### 6.1. Hit and Trial of Genetic Algorithms and Particle Swarm Optimization

#### 6.2. Control Objectives

- (1)
- Abulance temperature control;
- (2)
- Implementation of a FOPID controller an ambulance;
- (3)
- Tuning of PID and fractional-order PID controls with genetic algorithms and particle swarm optimization;
- (4)
- Adjustment of PID and fractional-order PID controls by GA.

#### 6.3. Proposed Controller

- (1)
- Proportional gain (K
_{P}) - (2)
- Derivative gain (K
_{D}) - (3)
- Derivative order (µ)
- (4)
- Integral gain (K
_{I}) - (5)
- Complete order (λ)

#### 6.4. Suggested Tuning Methods

#### 6.4.1. Particle Swarm Optimization

#### 6.4.2. Genetic Algorithm (GA)

## 7. Fractional-Order PID Controller for Induction Heating

#### FOPID-ADRC Design Optimized with QPSO

- (1)
- Initial parameters set, position and speed initialized;
- (2)
- Parameters adjusted and passed to fractional-order PID ADRC controller, which runs the control and passed fitness value (output);
- (3)
- Through initial fitness value, the initial global optimal and individual particles are determined;
- (4)
- Update operation and, through the previous step, obtain the fitness value to update and optimize the optimal value. Finally, when the fitness value equals the condition or reaches its extent, the algorithm stops working, and an optimal solution is obtained.

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Zhang, R.; Zou, Q.; Cao, Z.; Gao, F. Design of fractional order modeling based extended non-minimal state space MPC for temperature in an industrial electric heating furnace. J. Process Control
**2017**, 56, 13–22. [Google Scholar] [CrossRef] - Singh, Y.K.; Kumar, J.; Pandey, K.K.; Rohit, K.; Bhargav, A. Temperature Control System and its Control using PID Controller. Int. J. Eng. Res. Technol.
**2016**, 4, 4–6. [Google Scholar] - Kochubei, A.; Luchko, Y. Fractional differential equations. Fract. Differ. Equ.
**2019**, 2013, 1–519. [Google Scholar] [CrossRef] [Green Version] - Monje, C.A.; Chen, Y.; Vinagre, B.M.; Xue, D.; Feliu, V. Fractional Order Systems and Controls; Springer: Berlin/Heidelberg, Germany, 2010. [Google Scholar]
- Chen, Y.; Moore, K. Discretization schemes for fractional-order differentiators and integrators. IEEE Trans. Circuits Syst. I Regul. Pap.
**2002**, 49, 363–367. [Google Scholar] [CrossRef] - Lepik, Ü.; Hein, H. Fractional Calculus. In Haar Wavelets; Springer: Berlin/Heidelberg, Germany, 2014; pp. 107–122. [Google Scholar] [CrossRef]
- Podlubny, I. Geometric and Physical Interpretation of Fractional Integration and Fractional Differentiation. 2001, pp. 1–18. Available online: http://arxiv.org/abs/math/0110241 (accessed on 18 December 2021).
- Ma, C.; Hori, Y. Backlash Vibration Suppression Control of Torsional System by Novel Fractional Order PIDk Controller. IEEE J. Trans. Ind. Appl.
**2004**, 124, 312–317. [Google Scholar] [CrossRef] [Green Version] - Oustaloup, A.; Moreau, X.; Nouillant, M. The CRONE suspension. Control Eng. Pract.
**1996**, 4, 1101–1108. [Google Scholar] [CrossRef] - Biro, O.; Preis, K. On the use of the magnetic vector potential in the finite-element analysis of three-dimensional eddy currents. IEEE Trans. Magn.
**1989**, 25, 3145–3159. [Google Scholar] [CrossRef] - Oustaloup, A. La Commande CRONE: Commande Robuste d’Ordre Non Entier; Hermès: Paris, France, 1991. [Google Scholar]
- Oustaloup, A.; Sabatier, J.; Moreau, X. From fractal robustness to the CRONE approach. ESAIM Proc.
**1998**, 5, 177–192. [Google Scholar] [CrossRef] [Green Version] - El-Khazali, R. Fractional-order PI
^{λ}D^{μ}controller design. Comput. Math. Appl.**2013**, 66, 639–646. [Google Scholar] [CrossRef] - Beschi, M.; Padula, F.; Visioli, A. The generalised isodamping approach for robust fractional PID controllers design. Int. J. Control
**2015**, 90, 1157–1164. [Google Scholar] [CrossRef] - Kumar, A.; Kumar, V. Hybridized ABC-GA optimized fractional order fuzzy pre-compensated FOPID control design for 2-DOF robot manipulator. AEU-Int. J. Electron. Commun.
**2017**, 79, 219–233. [Google Scholar] [CrossRef] - Monje, C.A.; Vinagre, B.M.; Feliu-Batlle, V.; Chen, Y. Tuning and auto-tuning of fractional order controllers for industry applications. Control Eng. Pract.
**2008**, 16, 798–812. [Google Scholar] [CrossRef] [Green Version] - Valério, D.; da Costa, J.S. Tuning of fractional PID controllers with Ziegler–Nichols-type rules. Signal Process.
**2006**, 86, 2771–2784. [Google Scholar] [CrossRef] - Macia, N.F.; Thaler, G.J. Modeling and Control of Dynamic Systems; Thomson Delmar Learning: New York, NY, USA, 2005. [Google Scholar]
- Merrikh-Bayat, F.; Jamshidi, A. Comparing the performance of optimal PID and optimal fractional-order PID controllers applied to the nonlinear boost converter. Presented at the 5th Symposium on Fractional Differentiation and Its Applications (FDA12), Nanjing, China, 17 May 2012. [Google Scholar]
- Podlubny, I. Fractional-order system and PI
^{λ}D^{μ}controllers. IEEE Trans. Autom. Control.**1999**, 44, 208–214. [Google Scholar] [CrossRef] - Marinangeli, L.; Alijani, F.; HosseinNia, S.H. Fractional-order positive position feedback compensator for active vibration control of a smart composite plate. J. Sound Vib.
**2018**, 412, 1–16. [Google Scholar] [CrossRef] - Badri, V.; Tavazoei, M.S. Achievable Performance Region for a Fractional-Order Proportional and Derivative Motion Controller. IEEE Trans. Ind. Electron.
**2015**, 62, 7171–7180. [Google Scholar] [CrossRef] - Padula, F.; Visioli, A. On the fragility of fractional-order PID controllers for FOPDT processes. ISA Trans.
**2016**, 60, 228–243. [Google Scholar] [CrossRef] [PubMed] - Tepljakov, A.; Alagoz, B.B.; Yeroglu, C.; Gonzalez, E.; HosseinNia, S.H.; Petlenkov, E. FOPID Controllers and Their Industrial Applications: A Survey of Recent. IFAC-PapersOnLine
**2018**, 51, 25–30. [Google Scholar] [CrossRef] - Čech, M.; Schlegel, M. Generalized robust stability regions for fractional PID controllers. In Proceedings of the 2013 IEEE International Conference on Industrial Technology (ICIT), Cape Town, South Africa, 25–28 February 2013; pp. 76–81. [Google Scholar] [CrossRef]
- Polyakov, A.; Efimov, D.; Perruquetti, W. Robust Stabilization of MIMO Systems in Finite/Fixed Time. Int. J. Robust Nonlinear Control.
**2015**, 26, 69–90. [Google Scholar] [CrossRef] [Green Version] - Schlegel, M.; Medvecová, P. Design of PI Controllers: H∞ Region Approach. IFAC-PapersOnLine
**2018**, 51, 13–17. [Google Scholar] [CrossRef] - Xu, Y.; Bai, W.; Zhao, S.; Zhang, J.; Zhao, Y. Mitigation of forced oscillations using VSC-HVDC supplementary damping control. Electr. Power Syst. Res.
**2020**, 184, 106333. [Google Scholar] [CrossRef] - Chen, Z.; Qin, B.; Sun, M.; Sun, Q. Q-learning-based parameters adaptive algorithm for active disturbance rejection control and its application to ship course control. Neurocomputing
**2020**, 408, 51–63. [Google Scholar] [CrossRef] - Nasirpour, N.; Balochian, S. Optimal design of fractional-order PID controllers for multi-input multi-output (variable air volume) air-conditioning system using particle swarm optimization. Intell. Build. Int.
**2016**, 9, 1–14. [Google Scholar] [CrossRef] - Shahri, M.E.; Balochian, S.; Balochian, H.; Zhang, Y. Design of fractional order PID controllers for time delay systems using differential evolution algorithm. Indian J. Sci. Technol.
**2014**, 7, 1311–1319. [Google Scholar] [CrossRef] - Li, D.; Ding, P.; Gao, Z. Fractional active disturbance rejection control. ISA Trans.
**2016**, 62, 109–119. [Google Scholar] [CrossRef] [PubMed] - Koksal, E. Fractional-order PID and active disturbance rejection control of nonlinear two-mass drive system. IEEE Trans. Ind. Electron.
**2013**, 60, 3806–3813. [Google Scholar] - Alagoz, B.B.; Deniz, F.N.; Keles, C.; Tan, N. Disturbance rejection performance analyses of closed loop control systems by reference to disturbance ratio. ISA Trans.
**2015**, 55, 63–71. [Google Scholar] [CrossRef] [PubMed] - Deniz, F.N.; Keles, C.; Alagoz, B.B.; Tan, N. Design of fractional-order PI controllers for disturbance rejection using RDR measure. In Proceedings of the ICFDA’14 International Conference on Fractional Differentiation and Its Applications 2014, Catania, Italy, 23–25 June 2014; pp. 1–6. [Google Scholar] [CrossRef]
- Héder, M. From NASA to EU: The evolution of the TRL scale in Public Sector Innovation. Innov. J.
**2017**, 22, 1–23. [Google Scholar] - Uǧuz, H.; Goyal, A.; Meenpal, T.; Selesnick, I.W.; Baraniuk, R.G.; Kingsbury, N.G.; Haiter Lenin, A.; Mary Vasanthi, S.; Jayasree, T.; Adam, M.; et al. Ce pte d M us pt. J. Phys. Energy.
**2020**, 2, 1–31. [Google Scholar] - Tepljakov, A.; Gonzalez, E.A.; Petlenkov, E.; Belikov, J.; Monje, C.A.; Petráš, I. Incorporation of fractional-order dynamics into an existing PI/PID DC motor control loop. ISA Trans.
**2016**, 60, 262–273. [Google Scholar] [CrossRef] - Dastjerdi, A.A.; Vinagre, B.M.; Chen, Y.; HosseinNia, S.H. Linear fractional order controllers; A survey in the frequency domain. Annu. Rev. Control
**2019**, 47, 51–70. [Google Scholar] [CrossRef] - Tepljakov, A.; Petlenkov, E.; Belikov, J. A flexible MATLAB tool for optimal fractional-order PID controller design subject to specifications. In Proceedings of the 31st Chinese Control Conference, Hefei, China, 25–27 July 2012; pp. 4698–4703. [Google Scholar]
- Morand, A.; Moreau, X.; Melchior, P.; Moze, M. Robust cruise control using CRONE approach. IFAC Proc.
**2013**, 46, 468–473. [Google Scholar] [CrossRef] - Wang, C.; Jin, Y.; Chen, Y. Auto-tuning of FOPI and FO[PI] controllers with iso-damping property. In Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference, Shanghai, China, 15–18 December 2009; pp. 7309–7314. [Google Scholar] [CrossRef]
- Bode, H.W.; Phelps, G. Network Analysis and Feedback Amplifier Design|Hendrik W Bode|Digital Library Bookzz. 1945. Available online: http://bookzz.org/book/565696/cd0c74 (accessed on 22 December 2021).
- Åström, K. Limitations on Control System Performance. Eur. J. Control
**2000**, 6, 2–20. [Google Scholar] [CrossRef] - Nielsen, P. Coastal and Estuarine Processes; World Scientific: Singapore, 2009; pp. 1–360. [Google Scholar] [CrossRef] [Green Version]
- Baños, A.; Cervera, J.; Lanusse, P.; Sabatier, J. Bode optimal loop shaping with CRONE compensators. J. Vib. Control
**2010**, 17, 1964–1974. [Google Scholar] [CrossRef] - Pommier-Budinger, V.; Janat, Y.; Nelson-Gruel, D.; Lanusse, P.; Oustaloup, A. Fractional robust control with iso-damping property. In Proceedings of the 2008 American Control Conference, Seattle, WA, USA, 11–13 June 2008. [Google Scholar]
- Bhambhani, V. Optimal Fractional Order Proportional And Integral Controller For Processes With Random Time Delays. Master’s Thesis, Utah State University, Logan, UT, USA, 2009. [Google Scholar]
- Tepljakov, A.; Alagoz, B.B.; Yeroglu, C.; Gonzalez, E.A.; Hosseinnia, S.H.; Petlenkov, E.; Ates, A.; Cech, M. Towards Industrialization of FOPID Controllers: A Survey on Milestones of Fractional-Order Control and Pathways for Future Developments. IEEE Access
**2021**, 9, 21016–21042. [Google Scholar] [CrossRef] - Bhaskaran, T.; Chen, Y.; Bohannan, G. Practical Tuning of Fractional Order Proportional and Integral Controller (II): Experiments. In Proceedings of the ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Las Vegas, NV, USA, 4–7 September 2007; pp. 1371–1384. [Google Scholar] [CrossRef]
- Nagy, Z.K. Model based control of a yeast fermentation bioreactor using optimally designed artificial neural networks. Chem. Eng. J.
**2007**, 127, 95–109. [Google Scholar] [CrossRef] - Liu, B.; Ding, Y.; Gao, N.; Zhang, X. A bio-system inspired nonline ar intelligent controller with application to bio-reactor system. Neurocomputing
**2015**, 168, 1065–1075. [Google Scholar] [CrossRef] - Tiwari, D.; Pachauri, N.; Rani, A.; Singh, V. Fractional order PID (FOPID) controller based temperature control of bioreactor. In Proceedings of the 2016 International Conference on Electrical, Electronics, and Optimization Techniques (ICEEOT), Chennai, India, 3–5 March 2016; pp. 2968–2973. [Google Scholar] [CrossRef]
- Cheng, J.-K.; Lee, H.-Y.; Huang, H.-P.; Yu, C.-C. Optimal steady-state design of reactive distillation processes using simulated annealing. J. Taiwan Inst. Chem. Eng.
**2009**, 40, 188–196. [Google Scholar] [CrossRef] - Cao, J.-Y.; Cao, B.-G. Design of Fractional Order Controllers Based on Particle Swarm Optimization. In Proceedings of the 2006 1ST IEEE Conference on Industrial Electronics and Applications, Singapore, 24–26 May 2006; pp. 1–6. [Google Scholar] [CrossRef]
- Zamani, M.; Karimi-Ghartemani, M.; Sadati, N.; Parniani, M. Design of a fractional order PID controller for an AVR using particle swarm optimization. Control Eng. Pract.
**2009**, 17, 1380–1387. [Google Scholar] [CrossRef] - Lee, C.-H.; Chang, F.-K. Fractional-order PID controller optimization via improved electromagnetism-like algorithm. Expert Syst. Appl.
**2010**, 37, 8871–8878. [Google Scholar] [CrossRef] - Abraham, A.; Biswas, A.; Das, S.; Dasgupta, S. Design of fractional order PI
^{λ}D^{μ}controllers with an improved differential evolution. In Proceedings of the 10th annual conference on Genetic and evolutionary computation—GECCO’08, Atlanta, GA, USA, 12–16 July 2008. [Google Scholar] - Maiti, D.; Biswas, S.; Konar, A. Design of a fractional order PID controller using particle Swarm Optimization technique. arXiv
**2008**, arXiv:0810.3776. [Google Scholar] - Kundu, D.; Suresh, K.; Ghosh, S.; Das, S. Designing Fractional-order PI
^{λ}D^{μ}controller using a modified invasive Weed Optimization algortihm. In Proceedings of the 2009 World Congress on Nature & Biologically Inspired Computing (NaBIC), Coimbatore, India, 9–11 December 2009; pp. 1315–1320. [Google Scholar] [CrossRef] - Aldair, A.A.; Wang, W.J. Design of Fractional Order Controller Based on Evolutionary Algorithm for a Full Vehicle Nonlinear Active Suspension Systems. Int. J. Control. Autom.
**2010**, 3, 33–46. [Google Scholar] - Zhang, Y.; Li, J. Fractional-order PID controller tuning based on genetic algorithm. In Proceedings of the 2011 International Conference on Business Management and Electronic Information, Guangzhou, China, 13–15 May 2011. [Google Scholar]
- Rajasekhar, A.; Chaitanya, V.; Das, S. Fractional-Order PI
^{λ}D^{μ}Controller Design Using a Modified Artificial Bee Colony Algorithm. In Swarm, Evolutionary, and Memetic Computing; Springer: Berlin/Heidelberg, Germany, 2011; Volume 7076, pp. 670–678. [Google Scholar] [CrossRef] - Tang, Y.; Cui, M.; Hua, C.; Li, L.; Yang, Y. Optimum design of fractional order PI
^{λ}D^{μ}controller for AVR system using chaotic ant swarm. Expert Syst. Appl.**2012**, 39, 6887–6896. [Google Scholar] [CrossRef] - Rajasekhar, A.; Das, S.; Abraham, A. Fractional Order PID controller design for speed control of chopper fed DC Motor Drive using Artificial Bee Colony algorithm. In Proceedings of the 2013 World Congress on Nature and Biologically Inspired Computing, Fargo, ND, USA, 12–14 August 2013; pp. 259–266. [Google Scholar] [CrossRef] [Green Version]
- Sheng, W.; Bao, Y. Fruit fly optimization algorithm based fractional order fuzzy-PID controller for electronic throttle. Nonlinear Dyn.
**2013**, 73, 611–619. [Google Scholar] [CrossRef] - Wu, Z.; Li, D.; Wang, L. Control of the superheated steam temperature: A comparison study between PID and fractional order PID controller. In Proceeding of the 35th Chinese Control Conference, Chengdu, China, 27–29 July 2016; pp. 10521–10526. [Google Scholar] [CrossRef]
- Merzlikina, E.; Van Va, H.; Farafonov, G. Automatic Control System with an Autotuning Module and a Predictive PID-Algorithm for Thermal Processes. In Proceedings of the 2021 International Conference on Industrial Engineering, Applications and Manufacturing (ICIEAM), Sochi, Russia, 17–21 May 2021; pp. 525–529. [Google Scholar] [CrossRef]
- Mukhtar, A. Ambulance Temperature Control for the Safety of Medical Products using Fractional Order PID Controller Based on Artificial Intelligence Techniques. In Proceedings of the 2020 International Conference on Advances in Computing, Communication & Materials (ICACCM), Dehradun, India, 21–22 August 2020; pp. 205–211. [Google Scholar] [CrossRef]
- Chen, X.; Liu, J.; Sun, X. Self-adaption FOPID controller design for internal temperature in roller system based on data-driven technique. In Proceedings of the 2019 Chinese Control And Decision Conference (CCDC), Nanchang, China, 3–5 June 2019. [Google Scholar]
- Pezol, N.S.; Rahiman, M.H.F.; Adnan, R.; Tajjudin, M. Comparison of the CRONE-1 and FOPID Controllers for Steam Temperature Control of the Essential Oil Extraction Process. In Proceedings of the 2021 IEEE International Conference on Automatic Control & Intelligent Systems (I2CACIS), Shah Alam, Malaysia, 26 June 2021; pp. 253–258. [Google Scholar] [CrossRef]
- Li, R.; Wu, F.; Hou, P.; Zou, H. Performance Assessment of FO-PID Temperature Control System Using a Fractional Order LQG Benchmark. IEEE Access
**2020**, 8, 116653–116662. [Google Scholar] [CrossRef] - Mei, J.; Li, Z. Fractional order PID control of temperature of supply cooling water. In Proceedings of the 2017 4th International Conference on Information Science and Control Engineering (ICISCE), Changsha, China, 21–23 July 2017. [Google Scholar]
- Aldhaifallah, M. Heat Exchanger Control Using Fuzzy Fractional-Order PID. In Proceedings of the 2019 16th International Multi-Conference on Systems, Signals & Devices (SSD), Istanbul, Turkey, 21–24 March 2019; pp. 73–77. [Google Scholar] [CrossRef]
- Vinagre, B.M.; Petras, I.; Merchan, P.; Dorcak, L. Two digital realizations of fractional controllers: Application to temperature control of a solid. In Proceedings of the 2001 European Control Conference (ECC), Porto, Portugal, 4–7 September 2001. [Google Scholar]
- Pandey, S.; Soni, N.K.; Pandey, R.K. Fractional order integral and derivative (FOID) controller with anti-windup for temperature profile control. In Proceedings of the 2015 2nd International Conference on Computing for Sustainable Global Development (INDIACom), New Delhi, India, 11–13 March 2015. [Google Scholar]
- Teng, Y.; Li, H.; Wu, F. Design of Distributed Fractional Order PID Type Dynamic Matrix Controller for Large-Scale Process Systems. IEEE Access
**2020**, 8, 179754–179771. [Google Scholar] [CrossRef] - Feliu-Batlle, V.; Rivas-Perez, R.; Castillo-García, F.J. Robust fractional-order temperature control of a steel slab reheating furnace with large time delay uncertainty. In Proceedings of the ICFDA’14 International Conference on Fractional Differentiation and Its Applications 2014, Catania, Italy, 23–25 June 2014. [Google Scholar] [CrossRef]
- Tajjudin, M.; Ishak, N.; Rahiman, M.H.F.; Adnan, R. Design of fuzzy fractional-order PI + PD controller. In Proceedings of the 2016 IEEE 12th International Colloquium on Signal Processing & Its Applications (CSPA), Melaka, Malaysia, 4–6 March 2016. [Google Scholar]
- Ahn, H.-S.; Bhambhani, V.; Chen, Y. Fractional-order integral and derivative controller design for temperature profile control. In Proceedings of the 2008 Chinese Control and Decision Conference, Yantai, China, 2–4 July 2008; pp. 4766–4771. [Google Scholar] [CrossRef]
- Yu, W. Simulation and research of PID parameter tuning based on PSO. J. Chongqing Technol. Bus. Univ. (Nat. Sci.)
**2020**, 7, 18–23. [Google Scholar] - Mukhtar, A.; Mukhtar, F. Liquid Level Control Strategy using Fractional Order PID Controller Based on Artificial Intelligence. Int. Res. J. Eng. Technol.
**2020**, 7, 1675–1680. [Google Scholar] - Sondhi, S.; Hote, Y.V. Fractional order PID controller for load frequency control, Energy Conversion and Management. Energy Convers. Manag.
**2014**, 85, 343–353. [Google Scholar] [CrossRef] - Wu, Z.; Zhao, L.; Feng, L. Intelligent vehicle control based on fractional order PID controller. Control. Eng. China
**2011**, 18, 401–404. [Google Scholar] - Padhy, S.; Khadanga, R.K.; Panda, S. A modified GWO technique based fractional order PID controller with derivative filter coefficient for AGC of power systems with plug in electric vehicles. In Proceedings of the 2018 Technologies for Smart-City Energy Security and Power (ICSESP), Bhubaneswar, India, 28–30 March 2018. [Google Scholar] [CrossRef]
- Asl, R.M.; Pourabdollah, E.; Salmani, M. Optimal fractional order PID for a robotic manipulator using colliding bodies design. Soft Comput.
**2017**, 22, 4647–4659. [Google Scholar] [CrossRef] - Haji, V.H.; Monje, C.A. Fractional-order PID control of a chopper-fed DC motor drive using a novel firefly algorithm with dynamic control mechanism. Soft Comput.
**2017**, 22, 6135–6146. [Google Scholar] [CrossRef] - Yeroğlu, C.; Ateş, A. A stochastic multi-parameters divergence method for online auto-tuning of fractional order PID controllers. J. Frankl. Inst.
**2014**, 351, 2411–2429. [Google Scholar] [CrossRef] - Oustaloup, A.; Melchior, P.; Lanusse, P.; Cois, O.; Dancla, F. The CRONE toolbox for Matlab. In Proceedings of CACSD. In In Proceedings of the IEEE International Symposium on Computer-Aided Control System Design, Anchorage, AK, USA, 25–27 September 2000. [Google Scholar] [CrossRef]
- Ateş, A.; Yeroglu, C. Optimal fractional order PID design via Tabu Search based algorithm. ISA Trans.
**2016**, 60, 109–118. [Google Scholar] [CrossRef] - Ates, A.; Alagoz, B.B.; Yeroglu, C. Master–slave stochastic optimization for model-free controller tuning. Iranian J. Sci. Technol. Trans. Electr. Eng.
**2017**, 41, 153–163. [Google Scholar] [CrossRef] - Ates, A.; Yeroglu, C.; Yuan, J.; Chen, Y.Q.; Hamamci, S.E. Optimization of the FO[PI] controller for MTDS using MAPO with multi objective function. SSRN Electron. J.
**2018**, 1–6. [Google Scholar] [CrossRef] - Bingul, Z.; Karahan, O. Comparison of PID and FOPID controllers tuned by PSO and ABC algorithms for unstable and integrating systems with time delay. Optim. Control Appl. Methods
**2018**, 39, 1431–1450. [Google Scholar] [CrossRef] - Sotner, R.; Jerabek, J.; Kartci, A.; Domansky, O.; Herencsar, N.; Kledrowetz, V.; Alagoz, B.B.; Yeroglu, C. Electronically reconfigurable two-path fractional-order PI/D controller employing constant phase blocks based on bilinear segments using CMOS modified current differencing unit. Microelectron. J.
**2019**, 86, 114–129. [Google Scholar] [CrossRef] - Song, J.; Lin, J.; Wang, L.; Wang, X.; Guo, X. Nonlinear FOPID and Active Disturbance Rejection Hypersonic Vehicle Control Based on DEM Biogeography-Based Optimization. J. Aerosp. Eng.
**2017**, 30, 04017079. [Google Scholar] [CrossRef] - Nie, C.; Yang, H.; Mu, X. Development and trend of vacuum induction melting technology. Vacuum
**2015**, 52, 52–57. [Google Scholar] - Mazumder, A.; Kumar, V.D.; Sethi, S.; Mukherjee, J. Continuous monitoring of temperature of electron beam heated metal evaporation surface using controlled gas purge at viewing port in a vacuum chamber. Vacuum
**2019**, 161, 157–161. [Google Scholar] [CrossRef] - Liu, Z. Research on Temperature Control Technology of Medium Frequency Induction Heating Furnace; Xi’an Shiyou University: Xi’an, China, 2014. [Google Scholar]
- Dequan, S.; Guili, G.; Zhiwei, G.; Peng, X. Application of Expert Fuzzy PID Method for Temperature Control of Heating Furnace. Procedia Eng.
**2012**, 29, 257–261. [Google Scholar] [CrossRef] - Aware, M.V.; Junghare, A.S.; Khubalkar, S.W.; Dhabale, A.; Das, S.; Dive, R. Design of new practical phase shaping circuit using optimal pole–zero interlacing algorithm for fractional order PID controller. Anal. Integr. Circuits Signal Process.
**2017**, 91, 131–145. [Google Scholar] [CrossRef] - Hao, X.; Gu, J.; Chen, N.; Zhang, W.; Zuo, X. 3-D Numerical analysis on heating process of loads within vacuum heat treatment furnace. Appl. Therm. Eng.
**2008**, 28, 1925–1931. [Google Scholar] [CrossRef] - Buliński, P.; Smolka, J.; Golak, S.; Przyłucki, R.; Palacz, M.; Siwiec, G.; Lipart, J.; Białecki, R.; Blacha, L. Numerical and experimental investigation of heat transfer process in electromagnetically driven flow within a vacuum induction furnace. Appl. Therm. Eng.
**2017**, 124, 1003–1013. [Google Scholar] [CrossRef] - Fang, H.; Yuan, X.; Liu, P. Active-disturbance-rejection-control and fractional-order proportional integral derivative hybrid control for hydroturbine speed governor system. Meas. Control.
**2018**, 51, 192–201. [Google Scholar] [CrossRef] [Green Version] - Xue, D.; Zhao, C. Fractional order PID controller design for fractional order system. Control. Theory Appl.
**2007**, 24, 771–776. [Google Scholar] - Li, M.; Li, D.; Wang, J.; Zhao, C. Active disturbance rejection control for fractional-order system. ISA Trans.
**2013**, 52, 365–374. [Google Scholar] [CrossRef] - Han, J. The Technique for Estimating and Compensating the Uncertainties: Active Disturbance Rejection Control Technique; National Defense Industry Press: Beijing, China, 2008. [Google Scholar]
- Chen, Z. Some advances in linear active disturbance rejection control theory and engineering applications. Inf. Control
**2017**, 46, 257–266. [Google Scholar] - Li, J. Active disturbance rejection control: Summary and prospects of research achievements. Control. Theory Appl.
**2017**, 34, 281–294. [Google Scholar]

**Figure 1.**Block diagram of PID controllers [2].

**Figure 4.**Step response of open-loop experimental data [2].

**Figure 5.**Flow diagram of a bioreactor tank [53].

**Figure 9.**Ambulance output comparison plot of PID Controller and ga in terms of time and temperature.

**Figure 11.**Ambulance output comparison plot with PSO, along with PID, in terms of time and temperature.

**Figure 20.**Simplified block diagram of induction heating [101].

**Figure 21.**Block diagram of ADRC structural control system [102].

**Figure 22.**Flow Diagram of QPSO parameter optimization algorithm [102].

**Figure 23.**Frequency graph of temperature order monitor control effects (four cases) [102].

Sr.# | Gains | PID | PID+GA | PID + PSO |
---|---|---|---|---|

1 | ${K}_{D}$ | 18.81 | 0.50 | 60.61 |

2 | ${K}_{f}$ | 2.26 | 1.50 | 1.19 |

3 | ${K}_{p}$ | 8.59 | 5.00 | 6.99 |

Gains | GA | PSO | Hit & Trial |
---|---|---|---|

${K}_{p}$ | 6.02 | 7.98 | 11.79 |

${K}_{I}$ | 2.001 | 2.09 | 2.39 |

λ | 0.85 | 0.89 | 0.99 |

${K}_{D}$ | 1.002 | 33.62 | 34.62 |

µ | 0.85 | 0.89 | 0.79 |

Sr.# | Time Specification | PID | PID + PSO | FOPID + PSO | PID + GA | FOPID + GA | FOPID |
---|---|---|---|---|---|---|---|

1. | ISE | $1.2\text{}\times \text{}{10}^{4}$ | 9846 | 5437 | $1.16\text{}\times \text{}{10}^{4}$ | 6821 | 9878 |

2. | Peak Value | 48.6 | 43.1 | 39.1 | 46.5 | 40.2 | 41.5 |

3. | IAE | 625 | 531 | 360 | 590 | 498 | 675 |

4. | Rise Time | 5.8 | 7.2 | 2.5 | 6.5 | 3.5 | 4.4 |

5. | Setting Time | 64.2 | 50.8 | 47 | 55 | 48.8 | 48.8 |

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**MDPI and ACS Style**

Jamil, A.A.; Tu, W.F.; Ali, S.W.; Terriche, Y.; Guerrero, J.M.
Fractional-Order PID Controllers for Temperature Control: A Review. *Energies* **2022**, *15*, 3800.
https://doi.org/10.3390/en15103800

**AMA Style**

Jamil AA, Tu WF, Ali SW, Terriche Y, Guerrero JM.
Fractional-Order PID Controllers for Temperature Control: A Review. *Energies*. 2022; 15(10):3800.
https://doi.org/10.3390/en15103800

**Chicago/Turabian Style**

Jamil, Adeel Ahmad, Wen Fu Tu, Syed Wajhat Ali, Yacine Terriche, and Josep M. Guerrero.
2022. "Fractional-Order PID Controllers for Temperature Control: A Review" *Energies* 15, no. 10: 3800.
https://doi.org/10.3390/en15103800