Due to the adverse effects of unpredictable environmental disturbances on real control systems, robustness of control performance becomes a substantial asset for control system design. This study introduces a v-domain optimal design scheme for Fractional Order Proportional-Integral-Derivative (FOPID) controllers with adoption of Genetic Algorithm (GA) optimization. The proposed design scheme performs placement of system pole with minimum angle to the first Riemann sheet in order to obtain improved disturbance rejection control performance. In this manner, optimal placement of the minimum angle system pole is conducted by fulfilling a predefined reference to disturbance rate (RDR) design specification. For a computer-aided solution of this optimal design problem, a multi-objective controller design strategy is presented by adopting GA. Illustrative design examples are demonstrated to evaluate performance of designed FOPID controllers.

Centre for Security Information and Advanced Technologies Faculty of Applied Informatics Tomas Bata University in Zlin Zlin Czech Republic

Department of Computer Engineering Inonu University Malatya Turkey

Zobrazit více v PubMed

Podlubny I. Academic Press; San Diego: 1999. Fractional Differential Equations.

Vinagre M.B., Chen Y.Q., Petras I. Two direct Tustin discretization methods for fractional order differentiator/integrator. J Franklin. 2003;I(340):349–362. doi: 10.1016/j.jfranklin.2003.08.001. DOI

Luo Y., Chen Y.Q. Tuning fractional order proportional integral controllers for fractional order systems. J Process Control. 2010;20(7):823–831. doi: 10.1016/j.jprocont.2010.04.011. DOI

Oustaloup A., Mathieu B., Lanusse P. The CRONE control of resonant plants: application to a flexible transmission. Eu J Control. 1995;1:113–121. doi: 10.1016/S0947-3580(95)70014-0. DOI

Xue D., Chen Y.Q. Proceedings of the 4th World Congress on Intelligent Control and Automation. 2002. A Comparative Introduction of Four Fractional Order Controllers; pp. 3228–3235.

Chen Y.Q., Bhaskaran T., Xue D. Practical tuning rule development for fractional order proportional and integral controllers. J Comput Nonlin Dyn. 2008;3(2):214031–214038. doi: 10.1115/1.2833934. DOI

Deniz F.N., Keles C., Alagoz B.B. International Conference on Fractional Differentiation and Its Applications, Catania, Italy. 2014. Design of fractional order PI controllers for disturbance rejection using RDR measure.

Alagoz B.B., Deniz F.N., Keles C. Disturbance rejection performance analyses of closed loop control systems by reference to disturbance ratio. ISA T. 2015;55:63–71. doi: 10.1016/j.isatra.2014.09.013. PubMed DOI

Ates A, Alagoz BB, Yeroglu C, et al. Disturbance rejection FOPID control of rotor by multi-objective BB-BC optimization algorithm. In: ASME/IEEE International Conference on Mechatronic and Embedded Systems and Applications, Ohio, USA; 2017.

Li M., Li D., Wang J. Active disturbance rejection control for fractional order system. ISA T. 2013;52(3):365–374. doi: 10.1016/j.isatra.2013.01.001. PubMed DOI

Monje C.A., Vinagre B.M., Feliu V. Tuning and auto-tuning of fractional order controllers for industry applications. Control Eng Pract. 2008;16(7):798–812. doi: 10.1016/j.conengprac.2007.08.006. DOI

Wang CY, Jin Y, and Chen YQ. Auto-tuning of FOPI and FO [PI] controllers with iso-damping property. In: Proceedings of the 48th IEEE Conference on Decision and Control, 2009 held jointly with the 2009 28th Chinese Control Conference, December 2009, p. 7309–14.

Pan I., Das S. Frequency domain design of fractional order PID controller for AVR system using chaotic multi-objective optimization. Int J Elec Power. 2013;51:106–118. doi: 10.1016/j.ijepes.2013.02.021. DOI

Matignon D. Stability results on fractional differential equations to control processing. In: Processings of Computational Engineering in Systems and Application Multiconference, 1996, 2, p. 963–8.

Hartley T.T., Lorenzo C.F. Dynamics and control of initialized fractional-order systems. Nonlinear Dyn. 2002;29:201–233.

Radwan A.G., Soliman A.M., Elwakil A.S. On the stability of linear systems with fractional order elements. Chaos Soliton Fract. 2009;40(5):2317–2328. doi: 10.1016/j.chaos.2007.10.033. DOI

Chen Y.Q., Ahn H.S., Podlubny I. Robust stability check of fractional order linear time invariant systems with interval uncertainties. Signal Process. 2006;86(10):2611–2618. doi: 10.1016/j.sigpro.2006.02.011. DOI

Senol B., Ates A., Alagoz B.B. A numerical investigation for robust stability of fractional order uncertain systems. ISA T. 2014;53(2):189–198. doi: 10.1016/j.isatra.2013.09.004. PubMed DOI

Alagoz B.B. A note on robust stability analysis of fractional order interval systems by minimum argument vertex and edge polynomials. IEEE/CAA J Autom Sin. 2016;3(4):411–421. doi: 10.1109/JAS.2016.7510088. DOI

Alagoz B.B., Yeroglu C., Senol B. Probabilistic robust stabilization of fractional order systems with interval uncertainty. ISA T. 2015;5:101–110. doi: 10.1016/j.isatra.2015.01.003. PubMed DOI

Alagoz B.B. Fractional order linear time invariant system stabilization by brute-force search. T I Meas Control. 2018;40(5):1447–1456. doi: 10.1177/0142331216685391. DOI

Tufenkci S, Senol B, Alagoz BB. Stabilization of Fractional Order PID Controllers for Time-Delay Fractional Order Plants by Using Genetic Algorithm. In: Artificial Intelligence and Data Processing Symposium (IDAP), Malatya, Turkey, 2018, p. 1–4.

Tufenkci S., Senol B., Alagoz B.B. International Artificial Intelligence and Data Processing Symposium (IDAP) IEEE; Malatya, Turkey: 2019. Disturbance Rejection Fractional Order PID Controller Design in v-domain by Particle Swarm Optimization; pp. 1–6.

Hamamci S.E. An algorithm for stabilization of fractional order time delay systems using fractional order PID controllers. IEEE T Automat Contr. 2007;52(10):1964–1969. doi: 10.1109/TAC.2007.906243. DOI

Tan N., Ozguven O.F., Ozyetkin M.M. Robust stability analysis of fractional order interval polynomials. ISA T. 2009;48(2):166–172. doi: 10.1016/j.isatra.2009.01.002. PubMed DOI

Senol B, Yeroglu C. Robust stability analysis of fractional order uncertain polynomials. In: Proceedings of the 5th IFAC Workshop on Fractional Differentiation and its Applications, Nanjing, China, 2012.

Ahn H.S., Chen Y.Q. Necessary and sufficient condition of fractional order interval linear systems. Automatica. 2008;44(11):2985–2988. doi: 10.1016/j.automatica.2008.07.003. DOI

Lu J.G., Chen Y.Q. Robust stability and stabilization of fractional order interval systems with the fractional order α: The 0 < α < 1 case. IEEE T Automat Contr. 2010;55(1):152–158. doi: 10.1109/TAC.2009.2033738. DOI

Farges C., Sabatier J., Moze M. Fractional order polytopic systems: robust stability and stabilization. Adv Differential Equ. 2011;35:1–10. doi: 10.1186/1687-1847-2011-35. DOI

Gao Z. Robust stabilization criterion of fractional order controllers for interval fractional order plants. Automatica. 2015;61:9–17. doi: 10.1016/j.automatica.2015.07.021. DOI

Gao Z. Robust stability criterion for fractional order systems with interval uncertain coefficients and a time-delay. ISA T. 2015;58:76–84. doi: 10.1016/j.isatra.2015.05.019. PubMed DOI

Zheng S., Tang X., Song B. Graphical tuning method of FOPID controllers for fractional order uncertain system achieving robust D-stability. Int J Robust Nonlin. 2015;26(5):1112–1142. doi: 10.1002/rnc.3363. DOI

Zhu Q. Stabilization of stochastic nonlinear delay systems with exogenous disturbances and the event-triggered feedback control. IEEE Trans Autom Control. 2015;64(9):3764–3771.

Hu W., Zhu Q. Some improved razumikhin stability criteria for impulsive stochastic delay differential systems. IEEE Trans Autom Control. 2019;64(12):5207–5213.

Wang B., Zhu Q. Stability analysis of semi-Markov switched stochastic systems. Automatica. 2018;94:72–80.

Alagoz B.B., Ates A., Yeroglu C. Auto-tuning of PID controller according to fractional order reference model approximation for DC rotor control. Mechatronics. 2013;23(7):789–797. doi: 10.1016/j.mechatronics.2013.05.001. DOI

Mousavi Y., Alfi A. A memetic algorithm applied to trajectory control by tuning of fractional order proportional-integral-derivative controllers. Appl Soft Comput. 2015;36:599–617. doi: 10.1016/j.asoc.2015.08.009. DOI

Deniz FN, Keles C, Alagoz BB, et al. Design of fractional order PI controllers for disturbance rejection using RDR measure. In: 2014 International Conference Fractional Differentiation and Its Applications (ICFDA), Catania, Italy, 23-25 June 2014, p. 1–6.

Alagoz Baris Baykant, Deniz Furkan Nur, Keles Cemal, Tan Nusret. Disturbance rejection performance analyses of closed loop control systems by reference to disturbance ratio. ISA Trans. 2015;55:63–71. doi: 10.1016/j.isatra.2014.09.013. PubMed DOI

Alagoz B.B., Tan N., Deniz F.N. Implicit disturbance rejection performance analysis of closed loop control systems according to communication channel limitations. IET Control Theory A. 2015;9(17):2522–2531. doi: 10.1049/iet-cta.2015.0175. DOI

Tepljakov A., Alagoz B.B., Gonzalez E. Model reference adaptive control scheme for retuning method-based fractional order PID control with disturbance rejection applied to closed-loop control of a magnetic levitation system. J Circuit Syst Comp. 2018;1850176 doi: 10.1142/S0218126618501761. DOI

Ates A, Alagoz BB, Yeroglu C, et al. Disturbance rejection FOPID control of rotor by multi-objective bb-bc optimization algorithm. In: ASME 2017 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, 13th ASME/IEEE International Conference on Mechatronic and Embedded Systems and Applications Cleveland, Ohio, USA, Vol. 9, 6–9 August 2017.

Ozbey N., Yeroglu C., Alagoz B.B. A set-point filter type 2DOF fractional order PID control system design scheme for improved disturbance rejection control. ICFDA. 2018 doi: 10.2139/ssrn.3273677. DOI

Jin Q., Shi Y., Liu Q. Graphical robust PID tuning for disturbance rejection satisfying multiple objectives. Chem Eng Commun. 2018;205(12):1701–1711. doi: 10.1080/00986445.2018.1469014. DOI

Chen YQ, Petras I, Xue D. Fractional Order Control - A Tutorial. In: American Control Conference, Missouri, USA, 2009, p. 1397–411.

Petras I. Stability of Fractional order systems with rational orders: A Survey. Fract Calc Appl Anal. 2009;12:269–298.

Alagoz B.B. Hurwitz stability analysis of fractional order LTI systems according to principal characteristic equations. ISA T. 2017;70:7–15. doi: 10.1016/j.isatra.2017.06.005. PubMed DOI

Biswas A., Das S., Abraham A. Design of fractional order PIλDμ controllers with an improved differential evolution. Eng Appl Artif Intel. 2009;22(2):343–350. doi: 10.1016/j.engappai.2008.06.003. DOI

Shahri E.S.A., Alfi A., Machado J.A.T. Fractional fixed-structure H∞ controller design using Augmented Lagrangian Particle Swarm Optimization with Fractional Order Velocity. Appl Soft Comput. 2019;77:688–695. doi: 10.1016/j.asoc.2019.01.037. DOI

Holland J.H. University of Michigan Press; 1975. Adaptation in Natural and Artifcial Systems.

Holland J.H. Genetic Algorithms. Sci Am. 1992;267:66–73. https://www.jstor.org/stable/24939139

Goldberg D.E. Addison-Wesley; 1989. Genetic Algorithms in Search, Optimization & Machine Learning.

Francq P. Genetic Algorithms; 2012. http://www. otletinstitute.org/wikics/Genetic_Algorithms.html.

Das S., Saha S., Das S.A. Gupta, On the selection of tuning methodology of FOPID controllers for the control of higher order processes. ISA T. 2011;50:376–388. doi: 10.1016/j.isatra.2011.02.003. PubMed DOI

Zhang Y, Li J. Fractional order PID controller tuning based on genetic algorithm. In: 2011 International Conference on Business Management and Electronic Information, IEEE, Vol. 3, May 2011, p. 764–7.

Xue D. FOTF toolbox, Mathworks. https://www.mathworks.com/matlabcentral/ fileexchange/60874-fotf-toolbox.

Zhu Q., Wang H. Output feedback stabilization of stochastic feedforward systems with unknown control coefficients and unknown output function. Automatica. 2018;87:166–175.

Wang H., Zhu Q. Adaptive output feedback control of stochastic nonholonomic systems with nonlinear parameterization. Automatica. 2018;98:247–255.

Wang H., Zhu Q. Finite-time stabilization of high-order stochastic nonlinear systems in strict-feedback form. Automatica. 2015;54:284–291.