State Space Model Reference Adaptive Control for a Class of Second-Degree Fractional Order Systems with Different Eigenvalues

Year 2024, Volume: 37 Issue: 3, 1300 - 1317, 01.09.2024

Seif Eddine Khelas , Samir Ladaci , Yassine Bensafia

https://doi.org/10.35378/gujs.1287150

Cited By: 1

Abstract

This study proposes an adaptive control synthesis for a class of second-degree fractional order systems with different eigenvalues in the state-space domain. The proposed fractional order adaptive controller is a generalization of the MRAC controller for the class of scalar fractional order systems. In order to control the fractional order plant, an adaptive state space feedback controller is applied based on the error between the system output and a chosen reference model using a fractional adaptation law to make the fractional order plant track the fractional order reference model. We show that the resulting adaptive regulator is able to stabilize the fractional order second degree system with a satisfying performance. A simulation example illustrating these performance properties is provided along with a comparison with a fractional order sliding mode control (FOSMC) to demonstrate the superiority of the proposed control scheme.

Keywords

Fractional order system, MRAC, Second-degree system, Different eigenvalues, Stability analysis

Supporting Institution

Ministry of higher education and scientific research, Algeria

Project Number

PRFU A01L08ES250120200002

Thanks

The authors would like to thank Dr. Zehor Belkhatir for his invaluable assistance in editing the manuscript.

References

• [1] Vinagre, B. M., Petras, I., Podlubny, I., Chen, Y.-Q., “Using fractional order adjustment rules and fractional order referencemodels inmodel-reference adaptive control”, Nonlinear Dynamics, 29(1-4):269–279, (2002).
• [2] Ladaci, S., Charef, A., “Commande adaptative à modèle de référence d’ordre fractionnaire d’un bras de robot”, Communication Sciences & Technologie, 1 : 50-52, (2002).
• [3] Ladaci, S., Charef, A., “On fractional adaptive control”, Nonlinear Dynamics, 43(4):365–378, (2006).
• [4] Ladaci, S., Charef, A., “Fractional order adaptive control systems: A survey”, In Mitchell, E.W.,& Murray, S.R. (Eds.), Classification and application of fractals, Nova Science, 261–275,(2012).
• [5] Narendra, K.S., Annaswamy, A.M., “Stable Adaptive Systems”, Prentice Hall, Englewood Cliffs, (1989).
• [6]Murgaš, J., Veselý, V., Hejda, I., “State Space Structures in MRAC”, IFAC Proceedings Volumes, 25(21): 80–83, (1992).
• [7]Mukherjee, D., Raja, G. L., Kundu, P., Ghosh, A., “Design of Optimal Fractional Order Lyapunov Based Model Reference Adaptive Control Scheme for CSTR”, IFAC-PapersOnLine,55(1):436–441, (2022).
• [8]Yan, F., Hou, X., Tian, T., “Fractional-Order Multivariable Adaptive Control Based on a Nonlinear Scalar Update Law”,Mathematics, 10(18):3385, (2022).
• [9] Ladaci, S., Charef, A., Loiseau, J. J., “Robust fractional adaptive control based on the strictly positive realness condition”, International Journal of Applied Mathematics and Computer Science, 19(1):69–76, (2009).
• [10] Shi, B., Yuan, J., Dong, C., “On Fractional Model Reference Adaptive Control”, The Scientific World Journal, 2014, ID 521625, 1–8, (2014).
• [11] Chen, Y., Cheng, S., Wei, Y., Wang, Y., “Indirect model reference adaptive control for a class of linear fractional order systems”, American control conference (ACC), USA, (2016).
• [12] Abedini, M., Nojoumian, M.A., Salarieh, H., Meghdari A., “Model Reference Adaptive Control in Fractional Order Systems Using Discrete-Time Approximation Methods”, Communications in Nonlinear Science and Numerical Simulation, 25(1–3):27-40, (2015).
• [13] Cheng, S., Wei, Y., Chen, Y., Zhou, X., Wang, Y. “Fractional order composite MRAC for MIMO systems based on SDU factorization”, IFAC PapersOnLine, 50(1):7007–7012, (2017).
• [14] Wei, Y., Sun, Z., Hu, Y., Wang, Y.. “On fractional order composite model reference adaptive control”, International Journal of Systems Science, 47(11):1–11, (2015).
• [15] Balaska, H., Ladaci, S., Djouambi, A., Schulte, H., Bourouba, B., “Fractional order tube model reference adaptive control for a class of fractional order linear systems”, International Journal of Applied Mathematics and Computer Science, 30(3): 501–515, (2020).
• [16] Balaska, H., Ladaci, S., Djouambi, A., “Direct fractional order MRAC adaptive control design for a class of fractional order commensurate linear systems”, Journal of Control and Decision, 8(3): 1–15, (2021).
• [17] Bourouba, B., Schulte, H., Ladaci, S., “A novel MRAC-based fractional adaptive control design for a class of fractional order linear systems”, 8th International Conference on Systems and Control, Marrakesh, Morocco. 303–308, (2019).
• [18] Mukherjee, D., Raja, G L., Kundu P., Ghosh, A., “Design of Optimal Fractional Order Lyapunov Based Model Reference Adaptive Control Scheme for CSTR”, IFAC PapersOnLine, 55(1):436–441, (2022).
• [19] Sinha, A.S.C., Kayalar, S., Yurtseven, H.O., “State-space approach to non-linear model reference adaptive control”, International Journal of Systems Science, 23(5): 833-837, (1992).
• [20] Janiszowski, K., Olszewski, M., “State Space Adaptive Control for Nonlinear Systems”, IFAC Postprint Volume, IFAC Symposium, Tokyo, Japan, 145–148, (1994).
• [21] Wu, C., Zhao, J., “State Tracking of MRAC Systems in the Presence of Controller Temporary Failure Based on a Switching Method”, Mathematical Problems in Engineering, 2013, 741216, 1–9, (2013).
• [22] Ladaci, S., “Postoperative blood pressure control using a fractional order adaptive regulator”, 13th international conference on sciences and techniques of automatic control &computer engineering, STA’2012, Monastir, Tunisia, 1–12, (2012).
• [23] Balaska, H., Ladaci, S., Zennir, Y., “Conical tank level supervision using a fractional order model reference adaptive control strategy”, 15th international conference on informatics in control, automation and robotics (ICINCO), Porto, Portugal, 214–221, (2018).
• [24] Tepljakov, A., Alagoz, B. B., Gonzalez, E., Petlenkov, E., Yeroglu, C., “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”, Journal of Circuits, Systems and Computers, 27(11), 1850176, (2018).
• [25] Navarro-Guerrero, G., Tang, Y., “Fractional-Order Closed-Loop Model Reference Adaptive Control for Anesthesia”, Algorithms 2018, 11:106, (2018).
• [26] Balaska, H., Ladaci, S., Schulte, H., & Djouambi, A., “Adaptive cruise control system for an electric vehicle using a fractional order model reference adaptive strategy”, IFAC-PapersOnLine, 52(13): 194–199, (2019).
• [27] Kang, S., Wu, H., Yang, X., Li, Y., Wang, Y., “Fractional-order robust model reference adaptive control of piezo-actuated active vibration isolation systems using output feedback and multi-objective optimization algorithm”, Journal of Vibration and Control, 26(1–2): 19–35, (2020).
• [28] Khelas, S., Ladaci, S., Bensafia, Y., “Fractional order adaptive MRAC controller for an active suspension system”, Algerian Journal of Signals and Systems, 5(2): 112–117, (2020).
• [29] Bourouba, B., Ladaci, S., Illoul, R., “Robust Fuzzy Adaptive Control with MRAC Configuration for a Class of Fractional Order Uncertain Linear Systems”, International Journal of Robotics and Control Systems, 1(3): 326–337, (2021).
• [30] Ynineb, A. R., Ladaci, S., “MRAC adaptive control design for an F15 aircraft pitch angular motion using Dynamics Inversion and fractional-order filtering”, International Journal of Robotics and Control Systems. 2(2): 240-250, (2022).
• [31] Shi, B., Yuan, J., Dong, C., “On Fractional Model Reference Adaptive Control”, The scientific world journal, 2014: 521625, (2014).
• [32] Aguila-Camacho, N., Gallegos, J.A., “Switched Fractional Order Model Reference Adaptive Control for Unknown Linear Time Invariant Systems”, IFAC-PapersOnLine, 53(2): 3731-3736, (2020).
• [33] Yan, F., Hou, X., Tian, T., “Model reference adaptive control for fractional order systems with matched uncertainty”, 2nd International Conference on Laser, Optics and Optoelectronic Technology (LOPET 2022), Qingdao, China, (2022).
• [34] Cheng, S., Wei, Y., Chen, Y., Wang, Y., Liang, Q., “Fractional-order multivariable composite model reference adaptive control”, International Journal of Adaptive Control and Signal Processing, 31(10): 1467–1480, (2017).
• [35] Balaska, H., Ladaci, “Fractional order output-feedback tube-MRAC design for a class of fractional order transfer functions with unknown parameters”, International Journal of Automation and Control, 17( 3): 287–305, (2023).
• [36] Monje, C.A., Chen, Y., Vinagre, B.M., Xue, D., Feliu-Batlle, V., “Fractional-Order Systems and Controls: Fundamentals and Applications”, Springer Science & Busi-ness Media, (2010).
• [37] Saxena, S., & Hote, Y. V., “Design of robust fractional-order controller using the Bode ideal transfer function approach in IMC paradigm”, Nonlinear Dynamics, 107:983–1001, (2022).
• [38] Djamah, T., Mansouri, R., Bettayeb, M., Djennoune, S., “State space realization of fractional order systems”, Nonlinear Dynamics, 1107(1): 37–42, (2009).
• [39] Duarte-Mermoud, M.A., Aguila-Camacho, N., Gallegos, J.A., Castro-Linares, R., “Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems”, Communications in Nonlinear Science and Numerical Simulation, 22(1–3): 650–659, (2015).
• [40] Li, Y., Chen, Y., Podlubny, I., “Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag–Leffler stability”, Computers & Mathematics with Applications, 59(5):1810–1821, (2010).
• [41] Lancastera, P., Zaballa, I., “Diagonalizable quadratic eigenvalue problems”, Mechanical Systems and Signal Processing, 23(4):1134-1144, (2009).
• [42] Swarnkar P., Jain, S., Nema, R.K., “Effect of Adaptation Gain in Model Reference Adaptive Controlled Second Order System”, Engineering, Technology & Applied Science Research, 1(3):70–75, (2011).
• [43] Gao, W., Hung, J. “Variable structure control of nonlinear systems: A new approach”, IEEE Transactions on Industrial Electronics, 40(1): 45–55, (1993).
• [44] Bartoszewicz, A. “A new reaching law for sliding mode control of continuous time systems with constraints”, Transactions of the Institute of Measurement and Control, 37(4):515-521, (2015).
• [45] Efe, M. Ö. “Fractional order sliding mode control with reaching law approach”, Turkish Journal of Electrical Engineering and Computer Sciences, 18(5): 731-747, (2010).
• [46] Leulmi, M.I., Ladaci, S., Schulte, H. “Fractional Order Model Reference Adaptive Control With Chattering Elimination Algorithm For Wind Turbine Speed Control”. 27th International Conference on Methods and Models in Automation and Robotics (MMAR), IEEE, Międzyzdroje, Poland, (2023).