Sliding Mode Control Design for a Class of Nonlinear Fractional Systems with Application to Glucose-Insulin Systems

[1] A. Ouannas, I. M. Batiha, et al, “Synchroniza- tion of the Glycolysis Reaction-Diffusion Model via Linear Control Law, MDPI, Entropy”, Spe- cial Issue on Advanced Numerical Methods for Differential Equations, Vol. 23, No. 11, pp. 1516, 2021.

[2] M. H. Heydari, M. Razzaghi, “Extended Cheby- shev cardinal wavelets for nonlinear fractional delay optimal control problems”, International Journal of System Science, pp. 1048-1067, 2022.

[3] S. M. Kenneth, and R. Bertram, An Introduction to the Fractional Calculus and Fractional Differ- ential Equations, Wiley-Interscience Publication, US, 1993.

[4] F. Mainardi, “An historical perspective on frac- tional calculus in linear viscoelasticity,” Frac- tional Calculus and Applied Analysis, Vol. 15, pp. 712-717,2012.

[5] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Math- ematical Models, London: Imperial College Press, 2010.

[6] J. Kang, Z. H. Zhu and W. Wang, et al., “Frac- tional order sliding mode control for tethered satellite deployment with disturbances ”, Adv. Space Res., Vol. 59, pp. 263-273, 2017.

[7] R. L. Magin, “Fractional Calculus models of Com- plex dynamics in biological tissues,” Computers and Mathematics with Applications , Vol. 59, pp. 1586-1593, 2010.

[8] Y. Yan and C. Kou, "Stability analysis for a fractional differential model of HIV infection of CD4+ T cells with time delay ", Math.comput. simul, Vol. 82, No. 9, pp. 1572-1585, 2012.

[9] Zh. Wang, X. Huang and H. Shen, “Control of an uncertain fractional order economic system via adaptive sliding mode ”, Neurocomputing, Vol. 83, pp. 83-88, 2012.

[10] M. H. Heydari, M. Razzaghi, “A new class of orthonormal basis functions: application for frac- tional optimal control problems”, International Journal of System Science,pp. 240-252, 2022.

[11] M. H. Heydari, M. Razzaghi, Z. Avazzadeh, “Orthonormal piecewise Bernoulli functions: Appli- cation for optimal control problems generated using fractional integro-differential equations”, Journal of Vibration and Control, 2022.

[12] M. H. Heydari, M. Razzaghi, “A numerical ap- proach for a class of nonlinear optimal control problems with piecewise fractional derivative”, Chaos, Solitons and Fractals, Vol. 152, 2021.

[13] M. U. Saeem, M. Farmand, A. Ahmad, E.U. Haque, and M.O. Ahmad, “A Caputo Fabrizio fractional order model for control of glucose in insulin therapies for diabetes”, Ain Shams Engi- neering Journal, Vol. 11, No. 4, pp. 1309-1316, 2020.

[14] A. Mohammadzadeh and Tufan Kumbasar, “A new fractional order general type-2 fuzzy predic- tive control system and its application for glucose level regulation”, Applied Soft Computing, Vol. 91, 2020.

[15] N. Debbouche, A. O. Almatroud, A. Ouan- nas and I. M. Batiha, “Chaos and coexist- ing attractors in glucose-insulin regulatory sys- tem with incommensurate fractional-order deriva- tives”, Chaos, Solitons & Fractals, Vol. 143, pp. 110575, 2021.

[16] A. Noori, M. A. Sadrnia and M. B. Naghibi- Sistani, “Fault Tolerant control of Blood Glucose concentration using Reinforcement Learning”, International Journal of Industrial Electronics, Control and Optimization, Vol. 3, No. 3, pp. 353-364, 2020.

[17] A. E. Matouk, “Stability Conditions, hyperchaos and Control in a novel fractional order hyper- chaotic system ”, Physics Letters Am Jun, Vol. 373, No. 25, pp. 2166-2173, 2009.

[18] B. Bonilla, M. Rivero, L. Rodriguez-Germa and J.J. Trujillo, “Fractional differential equations as alternative models to nonlinear differential equa- tions ”, Applied Mathematics and Computation , Vol. 187, No. 1, pp. 79-88 , 2007.

[19] A. Pisano and E. Usai, “Sliding mode Control. A Survey with applications in math ”, Mathematicy and Computers in Simulation, Vol. 81, No. 5, pp. 954-979, 2011.

[20] V.I. Utkin, Sliding Modes in Control and opti- mization, New York: Springer-Verlag, 1992.

[21] Mo. Efe and CA. Kasnakoglu, “Fractional adap- tation law for sliding mode Control ”, Interna- tional Journal of Adaptive Control and Signal Processing, Vol. 22 , pp. 968-986, 2008.

[22] A. Si-Ammour, S. Djennoune and M. A. Bet- tayeb, “A sliding mode Control for linear frac- tional Systems with input and State delays ”, Communications in Nonlinear Science and Nu- merical Simulation, Vol. 14, No. 5, pp. 2310 2318, 2009.

[23] Mo. Efe, Fractional order Sliding mode Con- troller design for fractional order dynamic sys- tems. In New Trends in. Nanotechnology and Fractional Calculus Applications, Guvene ZB, Baleanu D, Tenreiro Machado JA. Springer Ver- lag: Dordrect; 463-470, 2010.

[24] S. H. Hossinnian, R. Ghaderi, A. Ranjber,M. Mahmoudian and S. Momani, “Sliding mode Synchronization of an uncertain fractional order chaotic system ”, comput. Math.Appl. , Vol. 59, pp. 1637-1643, 2010.

[25] S. Balochian, A. K. Sedigh and A. Zare, “Vari- able Structur Control of linear time invariant fractional order systems using afinite number of state feedback law ”, Commun. Nonlinear SCi. Number. Simulat. , Vol. 16 , pp. 1433-1442, 2011.

[26] M. S. Tavazoei and M. Haeri, “Synchronization of chaotic fractional order –Systems Via active sliding mode controller ”, Phys., Vol. 387, pp. 57- 70, 2008.

[27] C. Yin, S. Dadras, S.M. Zhong and Y. Q. Chen, “Control of a novel Class of fractional- order Con- trol approach ”, Appl. Mathe. Model , Vol. 37, pp. 2469-2483, 2013.

[28] SW. Wang, Yu. DW and Yu. DL, “Compen- sation for unmatched uncertainty with adaptive RBF network ”, Int J Eng sci., Vol. 3, No. 6 , pp. 801- 804, 2011.

[29] K. S. Kim and Y. Park, “Designing robust slid- ing hyper planes for parametric uncertain systems , a riccati approach ”, Automatica, Vol. 36, No. 7, pp. 1041-1048, 2000.

[30] HC. Han, “Lmi-based sliding Surface design for integral Sliding mode Control of mismatched un- certain systems ”, IEEE Trans Autom Control , Vol. 52, No. 4 , pp. 736-42, 2007.

[31] VI. Utkin and AS. Poznyak, “Adaptive Sliding mode Control ”, Lecture Notes in Control and Information Sciences , pp. 21-53, 2013.

[32] M. O. Efe, “Fractional fuzzy adaptive sliding- mode control of a 2-DoF direct-drive robot arm ”, IEEE Transactions on systems, Man, and cy- bernetics, Vol. 38, No. 6, pp. 1561-1570, 2008.

[33] Y. Wei, PW. Tse, Z. Yao and Y. Wang, “Adaptive backstepping output feedback control for a class of nonlinear fractional order systems ”, Non- linear Dyn, Vol. 86, No. 12 , pp. 1-10, 2016.

[34] H. Li, J. Wang and HK. Lam, “Adaptive sliding mode control for interval type-2 fuzzy systems ”, IEEE Transactions on systems, Man and cyber- netics: systems , Vol. 46, No. 12 , pp. 1-10 ,2016.

[35] H. Delavari, H. Heydarinejad and D. Baleanu, “Adaptive fractional blood glucose regulator based on high-order sliding mode observer ”, IET systems Biology , Vol. 13, No. 2 , pp. 43-54, 2019.

[36] D. Zhang, L. Cao and S. Tang, “Fractional or- der Sliding mode control for a class of uncertain nonlinear systems based on LQR ”, International Journal of Advanced Robotic systems , Vol. 14, No. 2, pp. 1-15, 2017.

[37] L. Chen, R. Wu, Y. He and Y. Chai, “Adaptive sliding mode control for fractional order uncer- tain linear systems with nonlinear disturbances ”, Nonlinear Dyn, Vol. 80 , pp. 51-58, 2015.

[38] B. Jakovljevic, A. Pisano, M. R. Rapaic and E. Usai, “On the sliding-mode control of fractional- order nonlinear uncertain dynamics”, Interna- tional Journal of Robust and Nonlinear Control, Vol. 26, No. 4, 2015.

[39] T. Takamatsu and H. ohmori, “Sliding mode controller design based on backstepping technique for fractional order system ”, SICE Journal of control, Measurment and system Integration, Vol. 9, No. 4, pp. 151-157, 2016

[40] A. Djari, T. Bouden and A. Boulkroune, “De- sign of Fractional order sliding mode controller (FSMC) for a class of fraction order Nonlinear commensurate systems using a Particle swarm op- timization (PSO) Algorithm ”, CEAI, Vol. 16 , No. 3 , pp . 46-55, 2014.

[41] T. Zhan, X. Liu and Sh. Ma, “A new singular system approach to output feedback sliding mode control for fractional order nonlinear systems ”, Journal of the Franklin Institute, Vol. 355, No. 14, pp. 6746-6762, 2018.

[42] Sh. Shi, J. Li and Y. Fang, “Fractional- disturbanceobserver- based sliding mode control for fractional order system with matched and mis- matched disturbances ”, International Journal of control, Automation and systems, Vol. 17, No. 5, pp. 1184-1190, 2019.

[43] Y. Chun, Y. Chen and S. M. Zhong, “LMI based design of a sliding mode controller for a class of uncertain fractional order nonlinear systems ”, Proc. of American control conference, pp. 6511- 6516, 2013.

[44] B. Meng, Zh. Cheng and Zh. Wang, “Adaptive sliding mode control for a class of uncertain non- linear fractional order Hopfield neural networks ”, AIP advances , Vol. 9, 2019.

[45] S. Haghighatnia, H. Toosian Shandiz and A. Alfi, “Conformable Fractional Order Sliding Mode Control for a class of fractional order chaotic systems”, International Journal of Indus- trial Electronics, Control and Optimization, Vol. 2, No.3, pp. 177-188, 2019.

[46] K. Mathiyalagan and G. Sangeetha, “Second- order sliding mode control for nonlinear fractional systems ”, Applied Mathematics and Computa- tion , Vol. 383, 2020.

[47] N. Djeghali, M. Bettayeb and S. Djennoune, “Sliding mode active disturbance rejection con- trol for uncertain nonlinear fractional order sys- tems ”, European Journal of control , Vol. 57, pp. 54-67, 2021.

[48] T. V. Moghaddam, S. K. Yadavar Nikravesh and M. A. Khosravi, “Adaptive constrained sliding mode control of uncertain nonlinear fractional or- der input affine systems ”, Journal of vibration and control, pp. 1-13, 2019.

[49] A. R. Haghighi and R. Ziaratban, “A non- integar sliding mode controller to stabilize frac- tional order nonlinear systems ”, Advances in Dif- ference Equations, 2020.

[50] He. Liu, Ho. Wang, J. Cao, A. Alsaedi and T. Hayat, “Composite learning adaptive slid- ing mode control of fractional order nonlinear systems with actuator faults ”, Journal of the Franklin Institute , Vol. 365, pp. 9580-9599, 2019.

[51] A. Noori, M. A. Sadrnia and M. B. Naghibi- Sistani, “Fault Tolerant control of Blood Glu- cose concentration using Reinforcement Learn- ing”, International Journal of Industrial Electron- ics, Control and Optimization, Vol. 3, No. 3, pp. 353-364, 2020.

[52] Y. Chen, Ch. Tang and M. Roohi, “Design of a model-free adaptive sliding mode control to syn- chronize chaotic fractional-order systems with in- put saturation: An application in secure commu- nications”, Journal of the Franklin Institute, Vol. 358, No. 16, pp. 8109-8137, 2021.

[53] M. Taheri, C. Zhang, Z. R. Berardehi, et al. “No- chatter model-free sliding mode control for syn- chronization of chaotic fractional-order systems with application in image encryption”, Multimed Tools Appl, Vol. 81, pp. 24167- 24197, 2022.

[54] Z. Esfahani, M. Roohi, M. Gheisarnejad, et al, “Optimal Non-Integer Sliding Mode Control for Frequency Regulation in Stand-Alone Mod- ern Power Grids”, MDPI, Applied Sciences, Spe- cial Issue on Microgrids, Vol. 9, No.16, pp. 3411, 2019.

[55] M. Roohi, C. Zhang Y. Chen, “Adaptive model- free synchronization of different fractional-order neural networks with an application in cryptog- raphy”, Nonlinear Dynamics, Vol. 100, pp. 3979- 4001, 2020.

[56] M. Roohi, M. H. Khooban, Z. Esfahani, et al, “A switching sliding mode control technique for chaos suppression of fractional-order complex systems”, Transactions of the Institute of Measurement and Control, Vol. 41, No. 10, pp. 2932-2946, 2019.

[57] I. Boiko, L. Fridman, A. Pisano and E. Usai, “Analysis of Chattering in Systems with Second order sliding modes ”, IEEE Trans Autom. Con- trol , Vol. 52, No. 11, pp. 2085-2102, 2007.

[58] R. Ziaratban, A.R. Haghighi and P. Reihani, “Design of a no-chatter fractional sliding mode control approach for stabilization of non-integer chaotic systems ”, lnt. J. lnd math. Vol. 12, No. 3, pp. 215-223, 2020.

[59] M. R. Soltanpour and M. tt. Khooban, “A parti- cle swarm optimization approach for fuzzy sliding mode control for tracking the robot manipulator ”, Nonlinear Dyn. , Vol. 74, No. 1, pp. 467-478, 2013.

[60] G. Bartolini, A, Pisano and E. Usai, Second order sliding mode control of container cranes , Automatica , Vol. 38, No.10, pp. 1383-1790,2002.

[61] M. S. Asl and M. Javidi, “An improved PC scheme for nonlinear fractional differential equa- tions: error and stability analysis ”, Comput. Appl.Math. , Vol. 324, pp. 101-117,2017.

[62] I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, frac-tional differential equations, to methods of their solution and some of methods of their application , London: Academic press, 1999.

[63] M. N Soorki and M. S Tavazoei, “Adaptive ro- bust Control of fractional swarm systems in the presence of model uncertain ties and external dis- turbances ”, IET Control Theory Applications, Vol. 12, No. 7, pp. 961-969, (2018).

[64] CP. Li and FR. Zhang, “A survey on the sta- bility of fractional differential equations ”, The European Physical Journal Special Topics, Vol. 193, No. 1 , pp. 27-47, 2011.

[65] V. M. Popov, Hyperstability of control system, Springer-Verlag, Berlin, 1973.

[66] R. K. Munje, M. R. Roda and B. E. Kushare, “Speed control of DC Motor Using PI and SMC ”, Proceedings of IPEC IEEE conference, Singa- pore, Vol. 2, pp. 649-656, 2010.

[67] I. N. Doye, H. Voos, M. Darouach and J. G. Schneider, “Static output Feedback H∞ control for a Fractional-order Glucose-insulin system ”, International Journal of control, Automation, and systems, Vol. 13, No. 4, pp. 1-10, 2015.