The present study aims to investigate the optimal fractional order PID controller performance in the chaotic system of HIV disease fractional order using the Particle Swarm optimization and Genetic algorithm method. Differential equations were used to represent the chaotic behavior associated with HIV. The optimal fractional order of the PID controller was constructed, and its performance in the chaotic system with HIV fractional order was tested. Optimization methods were used to get PID control coefficients from particle swarm and genetic algorithms. Findings revealed that the equations for the HIV disease model are such that the system’s behavior is greatly influenced by the number of viruses produced by infected cells, such that if the number of viruses generated by infected cells exceeds 202, the disease’s behavior is such that the virus and disease spread. For varying concentrations of viruses, the controller created for this disease does not transmit the disease.
[1] E. Ahmad, A. M. A. El-Sayed, and H. A. A. El-Saka, Equilibrium points, stability and numerical solutions of fractional order predator-prey and rabies models, J. Math. Anal. Appl., 325 (2007), 542–553.
[2] S. Alavi, A. Haghighi, A. Yari, and F. Soltanian, A numerical method for solving fractional optimal control problems using the operational matrix of Mott polynomials, Comput. Methods Differ. Equ., 10(3) (2022), 755– 773.
[3] F. A. Alazabi, M. A. Zohdy, and A. A. Ezzabi, A regulator design for nonlinear HIV-1 infection system, Int. J. Intel. Cont. Sys., 17(1) (2012), 1–6.
[4] M. Arablouye Moghaddam, Y. Edrisi Tabriz, and M. Lakestani, Solving fractional optimal control problems using Genocchi polynomials, Comput. Methods Differ. Equ., 9(1) (2021), 79–93.
[5] N. Bellomo and M. Delitala, The mathematical kinetic, and stochatic game theroy to modelling mutations, pro- gression and immune competition of cancer cells, Phys. Life Rev. 5 (2008), 183–206.
[6] M. Clerc and J. Kennedy, The particle swarm–explosion, stability, and convergence in a multidimensional complex space, IEEE Trans. Evol. Comput., 6(1) (2000), 58–73.
[7] L. Cai, J. Liu, and Y. Chen, Dynamics of an age-structured HIV model with super-infection, Appl. Comput. Math., Int. J., 20(2) (2021), 257–276.
[8] V. Daftardar-Gejji and S. Bhalekar, Chaos in fractional ordered Liu system, Comput. & Math. Appl., 59(3) (2010), 1117–1127.
[9] W. Deng, C. Li, and J. lU, Stability analysis of linear fractional differential system with multiple time delays, Nonlinear Dyn., 48 (2007), 409–416.
[10] Y. Ding, Z. Wang and H. Ye, Optimal control of a fractional-order HIV-immune system with memory, IEEE Transactions On Contril Systems Technology, 20(3) (2012), 763–769.
[11] R. Eberhart and Y. Shi, Comparing inertia weights and constriction factors in particle swarm optimization, In: Proceedings of the 2000 Congress on Evolutionary Computation, Washington DC, 1 (2000), 84–88.
[12] G. B. Fogel, E. S. Liu, M. Salemi, S. L. Lamers, and M. S. McGrath, Evolved Neural Networks for HIV-1 Co- receptor Identification , 2014 IEEE Congress on Evolutionary Computation (CEC), July 6-11, 2014, Beijing, China.
[13] M. Gachpazan, A. H. Borzabadi, and A. V. Kamyad, A measure-theoretical approach for solving discrete optimal control problems, Appl. Math. Comput., 173 (2) (2006), 736–752.
[14] S. Ghasemi, A. Tabesh, and J. Askari, Application of fractional calculus theory to robust controller design for wind turbine generators, IEEE Transactions on Energy Conversion, 29(3) (2014).
[15] A. E. Gohary and I. A. Alwasel, The chaos and optimal control of cancer model with complete unknown parameters, Chaos, Solitons and Fractals, 42 (2009), 2865–2874.
[16] A. S. Hegazi, E. Ahmed, and A. E. Matouk, On chaos control and synchronization of the commensurate fractional order Liu system, Commun. Nonlinear Sci. Num. Simul., 18(5) (2013), 1193–1202.
[17] H. Y. Jia, Z. Q. Chen, and G. Y. Qi, Chaotic characteristics analysis and circuit implementation for a fractional- order system, IEEE Transactions on circuits and systems–I: Regular paper, 61(3) (2014).
[18] N. Kanagaraj and V. N. Jha, (2021), Design of an enhanced fractional order PID controller for a class of second- order system, COMPEL-The Int. J. Comput. Math. Electrical and Electronic Engin., 40(3) (2021), 579–592.
[19] J. Kennedy and R. C. Eberhart, (1995)Particle swarm optimization. In: Proceedings of the 1995 IEEE Interna- tional Conference on Neural Networks, Perth, Australia, 4 (1995), 1942–1948. IEEE Service Center, Piscataway.
[20] M. Khanra, J. Pal, and K. Biswas, Reduced order approximation of MIMO fractional order systems, IEEE J. on Emerging and selected topics in circuits and systems, 3(3) (2013).
[21] X. Liu, L. Hong, and L. Yang, Fractional-order complex T system: bifurcations, chaos control, and synchroniza- tion, Nonlinear Dyn., 75 (2014), 589–602.
[22] X. Liu, L. Hong, L. Yang and D. Tang, Bifurcations of a new fractional-order system with a one-scroll chaotic attractor, Discrete Dynamics in Nature and Society, 2019 (2019), Article ID 8341514, 15 pages.
[23] D. Matiognon, Stability results for fractional differential equations with applications to control processing in: Computational engineering in systems and application multi conference, IMACS IEEE-SMC Proceedings, 2 (1996), 963–968.
[24] S. Z. Moussavi, M. Alasvandi, S. Javadi, and E. Morad, PMDC motor speed control optimization by implementing ANFIS and MRAC, Int. J. Cont. Sci. Engin., 4(1) (2014), 1–8.
[25] E. L. Mazzaferri and S. M. Jhiang, Long-term impact of initial surgical and medical therapy on papillary and follicular thyroid cancer, The American Journal of Medicine, 97(5) (1994), 418–428.
[26] F. Merrikh-Bayat, More details on analysis of fractional-order Lotka-Volterra equation, The 5th IFAC symposium on fractional differentiation and its applications (FDA12), 14-17 May 2012, Hohai University, Nanjing, China, arXiv:1401.0103.
[27] S. Micael, N. M. Couceiro, F. Ferreira, and J. A. Tenreiro Machado, Control optimization of a robotic bird, Computer Science, (2009).
[28] K. Moaddy, A. Freihat, M. Al-Smadi, et al., Numerical investigation for handling fractional-order Rabi- novich–Fabrikant model using the multistep approach, Soft Comput., 22 (2018), 773–782.
[29] I. Petras, A note on the fractional-order Volta’s system, Commun. Nonlinear Sci. Num. Simul., 15(2) (2010), 384–393.
[30] A. Razminia, V. J. Majd, and D. Baleanu, Chaotic incommensurate fractional order R¨ossler system: active control and synchronization, Adv. Differ. Equ., 15 (2011).
[31] S. Sayyad Delshad, M. Mostafa Asheghan, and M. Hamidi Beheshti, Synchronization of N-coupled incommensurate fractional-order chaotic systems with ring connection, Commun. Nonlinear Sci. Num. Simul., 16(9) (2009), 3815– 3824.
[32] M. Shahiri, R. Ghaderi, A. Ranjbar N. S. H. Hosseinnia, and S. Momani, Chaotic fractional-order Coullet system: Synchronization and control approach, Commun. Nonlinear Sci. Num. Simul., 15(3) (2010), 665–674.
[33] Y. Shi and R. Eberhart, A modified particle swarm optimizer. In: Evolutionary Computation Proceedings, IEEE World Congress on Computational Intelligence, The 1998 IEEE International Conference on, Anchorage, Alaska, (1998), 69–73.
[34] M. S. Tavazoei and M. Haeri, A necessary condition for double scroll attractor existence in fractional-order systems, Phys. Let. A, 367(1–2) (2007), 102–113.
[35] M. S. Tavazoei and M. Haeri (2008), Chaotic attractors in incommensurate fractional order systems, Phys. D, 237(20) (2008), 2628–2637.
[36] M. S. Tavazoei and M. Haeri, Limitations of frequency domain approximation for detecting chaos in fractional order systems, Nonlinear Anal.: Theo., Meth. & Appl., 69(4) (2008),1299–1320.
[37] M. S. Tavazoei, M. Haeri, M. Attari, S. Bolouki, and M. Siami, More details on analysis of fractional-order Van der Pol oscillator, J. Vibration Cont., 15(6) (2009), 803–819.
[38] M. S. Tavazoei, M. Tavakoli-Kakhki, and F. Bizzarri, Nonlinear fractional-order circuits and systems: motivation, A brief overview, and some future directions, IEEE Open Journal of Circuits and Systems, 1, 220–232.
[39] J. Wang Gary, G. Yen Marios, and M. Polycarpou, 9th International symposium on neural networks, Shenyang, China, July 11-14, 2012. Proceedings, Part I.
[40] Z. Wang, X. Huang, and G. Shi,(2011)Analysis of nonlinear dynamics and chaos in a fractional order financial system with time delay, Comput. & Math. Appl., 62(3) (2011), 1531–1539.
[41] H. Zarei, A. V. Kamyad, and S. Effati, Multiobjective optimal control ofHIV dynamics, Math. Problems Engin., 2010 (2010), Article ID 568315, 29 pages.
[42] H. Zarei, A. V. Kamyad and M. H. Farahi, Optimal control of HIV dynamic using embedding method, Computa- tional and Mathematical Methods in Medicine, Hindawi Publishing Corporation, 2011.
[43] H. Zarei, A. Vahidian Kamyad, and A. Akbar Heydari, Fuzzy modeling and control of HIV infection, Computa- tional and Mathematical Methods in Medicine, 2012 (2012), Article ID 893474, 17 pages.
[44] C. Zeng and Q. Yang, A fractional order HIV internal viral dynamics model, CMES, 59(1) (2010), 65–77.
Mohammadi, S., & Hejazi, R. (2023). Optimal fractional order PID controller performance in chaotic system of HIV disease: particle swarm and genetic algorithms optimization method. Computational Methods for Differential Equations, 11(2), 207-224. doi: 10.22034/cmde.2022.51132.2127
MLA
Shaban Mohammadi; Reza Hejazi. "Optimal fractional order PID controller performance in chaotic system of HIV disease: particle swarm and genetic algorithms optimization method". Computational Methods for Differential Equations, 11, 2, 2023, 207-224. doi: 10.22034/cmde.2022.51132.2127
HARVARD
Mohammadi, S., Hejazi, R. (2023). 'Optimal fractional order PID controller performance in chaotic system of HIV disease: particle swarm and genetic algorithms optimization method', Computational Methods for Differential Equations, 11(2), pp. 207-224. doi: 10.22034/cmde.2022.51132.2127
VANCOUVER
Mohammadi, S., Hejazi, R. Optimal fractional order PID controller performance in chaotic system of HIV disease: particle swarm and genetic algorithms optimization method. Computational Methods for Differential Equations, 2023; 11(2): 207-224. doi: 10.22034/cmde.2022.51132.2127
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