Multi-Objective Step Response Shaping via the Fractional-Order Proportional-Integral-Derivative Controller

Abstract

A well-known problem in control system design and analysis is the shaping of the unit step reference response of a system to produce desired transient characteristics for various system references. The necessity of having fast, accurate, and stable control systems for a large number of practical applications has created the need for advanced control methods. In this regard, the development of fractional-order controllers has received considerable attention from the control community. Many papers and books on the topic of fractional-order systems have been published, which also include the usefulness of fractional calculus in the area of controllers. The fractional order proportional integral derivative controller is proven to be versatile, and its design can be obtained for any given target step response. A sufficiently large number of response characteristics, such as performance, phase margin, immunity to plant modeling, and robustness, can be adjusted by means of five tuning parameters. The control strategy of this paper focuses on developing a fractional order proportional integral derivative controller, which aims at overcoming the infeasibility of the controller to satisfy the conflicting goals of go-to speed and settling time in the traditional PID controller. The controller design has two main goals: one is to satisfy system stability, while the other is tuning the overshoot and the settling time. In this direction, the genetic algorithm is implemented. The results are presented through an illustrative example.

Keywords

• PI Controller
• Overshoot
• Step Response
• Stability
• Genetic Algorithm

References

1. H. M. Srivastava, "Fractional-order derivatives and integrals: Introductory overview and recent developments," Kyungpook Mathematical Journal, vol. 60, no. 1, pp. 73-116, 2020.
2. D. Baleanu, Y. Karaca, L. Vázquez, and J. E. Macías-Díaz, "Advanced fractional calculus, differential equations and neural networks: Analysis, modeling and numerical computations," Physica Scripta, vol. 98, no. 11, p. 110201, 2023.
3. C. A. Valentim, J. A. Rabi, and S. A. David, "Fractional mathematical oncology: On the potential of non-integer order calculus applied to interdisciplinary models," Biosystems, vol. 204, p. 104377, 2021.
4. Y. Liu, A. K. Singh, J. Zhao, A. S. Meliopoulos, B. Pal, M. A. bin Mohd Ariff, ... and S. Yu, "Dynamic state estimation for power system control and protection," IEEE Transactions on Power Systems, vol. 36, no. 6, pp. 5909-5921, 2021.
5. H. Jahanshahi, A. Yousefpour, J. M. Munoz-Pacheco, I. Moroz, Z. Wei, and O. Castillo, "A new multi-stable fractional-order four-dimensional system with self-excited and hidden chaotic attractors: Dynamic analysis and adaptive synchronization using a novel fuzzy adaptive sliding mode control method," Applied Soft Computing, vol. 87, p. 105943, 2020.
6. X. Leng, S. Gu, Q. Peng, and B. Du, "Study on a four-dimensional fractional-order system with dissipative and conservative properties," Chaos, Solitons and Fractals, vol. 150, p. 111185, 2021.
7. M. Fiuzy and S. Shamaghdari, "Stability analysis of fractional-order linear system with PID controller in the output feedback structure subject to input saturation," International Journal of Dynamics and Control, vol. 10, no. 2, pp. 511-524, 2022.
8. C. I. Muresan, I. Birs, C. Ionescu, E. H. Dulf, and R. De Keyser, "A review of recent developments in autotuning methods for fractional-order controllers," Fractals and Fractional, vol. 6, no. 1, p. 37, 2022.

9. E. A. Mohamed, E. M. Ahmed, A. Elmelegi, M. Aly, O. Elbaksawi, and A. A. A. Mohamed, "An optimized hybrid fractional order controller for frequency regulation in multi-area power systems," IEEE Access, vol. 8, pp. 213899-213915, 2020.
10. Jankovic, G. Chaudhary, and F. Goia, "Designing the design of experiments (DOE)–An investigation on the influence of different factorial designs on the characterization of complex systems," Energy and Buildings, vol. 250, p. 111298, 2021.
11. X. Rui, J. Zhang, X. Wang, B. Rong, B. He, and Z. Jin, "Multibody system transfer matrix method: the past, the present, and the future," International Journal of Mechanical Systems Dynamics, vol. 2, no. 1, pp. 3-26, 2022.
12. R. Barzegarkhoo, M. Forouzesh, S. S. Lee, F. Blaabjerg, and Y. P. Siwakoti, "Switched-capacitor multilevel inverters: A comprehensive review," IEEE Transactions on Power Electronics, vol. 37, no. 9, pp. 11209-11243, 2022.
13. J. Machowski, Z. Lubosny, J. W. Bialek, and J. R. Bumby, Power System Dynamics: Stability and Control, John Wiley & Sons, 2020.
14. M. Batiha, O. Y. Ababneh, A. A. Al-Nana, W. G. Alshanti, S. Alshorm, and S. Momani, "A numerical implementation of fractional-order PID controllers for autonomous vehicles," Axioms, vol. 12, no. 3, p. 306, 2023.
15. R. Shalaby, M. El-Hossainy, B. Abo-Zalam, and T. A. Mahmoud, "Optimal fractional-order PID controller based on fractional-order actor-critic algorithm," Neural Computing and Applications, vol. 35, no. 3, pp. 2347-2380, 2023.
16. N. A. Ahmed, S. Abdul Rahman, and B. N. Alajmi, "Optimal controller tuning for P&O maximum power point tracking of PV systems using genetic and cuckoo search algorithms," International Transactions on Electrical Energy Systems, vol. 31, no. 10, 2021.
17. C. Yao, Y. Li, M. D. Ansari, M. A. Talab, and A. Verma, "Optimization of industrial process parameter control using improved genetic algorithm for industrial robot," Paladyn, Journal of Behavioral Robotics, vol. 13, no. 1, pp. 67-75, 2022.
18. H. Wu, Z. Hu, and X. Du, "Time-dependent system reliability analysis with second-order reliability method," Journal of Mechanical Design, vol. 143, no. 3, p. 031101, 2021.
19. M. B. Bayram, H. İ. Bülbül, C. Can, and R. Bayindir, "Matlab/GUI based basic design principles of PID controller in AVR," in 4th International Conference on Power Engineering, Energy and Electrical Drives, pp. 1017-1022, 2013.
20. M. H. Lipu, M. A. Hannan, T. F. Karim, A. Hussain, M. H. M. Saad, A. Ayob, ... and T. I. Mahlia, "Intelligent algorithms and control strategies for battery management system in electric vehicles: Progress, challenges and future outlook," Journal of Cleaner Production, vol. 292, p. 126044, 2021.
21. G. Acampora, A. Chiatto, and A. Vitiello, "Genetic algorithms as classical optimizer for the quantum approximate optimization algorithm," Applied Soft Computing, vol. 142, p. 110296, 2023.
22. Dastanpour, S. Ibrahim, R. Mashinchi, and A. Selamat, "Using genetic algorithm to support artificial neural network for intrusion detection system," Journal of Communication and Computer, vol. 11, pp. 143-147, 2014.