In collaboration with Payame Noor University and the Iranian Society of Instrumentation and Control Engineers

Document Type : Research Article


Department of Mathematics, Payame Noor University (PNU), P.O. Box. 19395-3697, Tehran, Iran.



This paper presents an optimal robust adaptive technique for controlling a certain class of uncertain nonlinear affine systems‎. ‎The proposed approach combines sliding mode control‎, ‎a linear quadratic regulator for optimality, and gradient descent as an adaptive controller‎. ‎ The convergence of the sliding mode control process is proven using two theorems based on the Lyapunov function. Simulation results for pendulum and inverted pendulum systems demonstrate that the proposed method outperforms both the linear quadratic regulator technique and ‎the‎ ‎sliding‎ ‎mode‎ ‎control regarding reduced chattering and improved reaching time‎.


Main Subjects

[1] Artitthang, P., Xu, M., Lin, M., He, Y. (2022). “Robust optimal sliding mode control for the deployment of Coulomb spacecraft formation flying”, Advance in Space Research, 71(1), 439-455.
[2] Bkekri, R., Benamor, A., Alouane, M.A., Fried, G., Messaoud, H.(2018).“Robust adaptive sliding mode control for a human-driven knee joint orthosis”, Industrial Robot, 45(3), 379-389.
[3] Chawla, I., Singla, A. (2021). “Real time stabilization control of a rotary inverted pendulum using LQR based sliding mode controller”, Arabian Journal for Science and Engineering, 46, 2589-2596.
[4] Chairez, I., Utkin, V. (2022). “Electrocardiographically signal simulator based on a sliding mode controlled buck DC-DC power converter”, IFAC-PapersOnLine 55(9), 419-425.
[5] Chen, K. (2018). “Robust optimal adaptive sliding mode control with the disturbance observer for a manipulator robot system”, International Journal of Control, Automation and Systems, 16, 1701-1715.
[6] Das, M. (2016). “Design of optimal sliding mode controller for uncertain systems”, PhD diss.
[7] Ghamgosar, M., Mirhosseini-Alizamini, S.M. (2021). “Design of optimal sliding mode control based on linear matrix inequality for fractional time-varying delay systems”, International Journal of Industrial Electronics, Control and Optimization (IECO), 5(4).
[8] Ghamgosar, M., Mirhosseini-Alizamini, S.M., Dadkhah, M. (2022). “Sliding mode control of a
class of uncertain nonlinear fractional order time-varying delayed systems based on Razumikhin approach”, Computational Methods for Differential Equations, 10(4), 860-875.
[9] Ghamgosar, M., Mirhosseini-Alizamini, S.M., Khaleghizadeh, S. (2022). “Design of sliding mode control based on Razumikhin approach and linear matrix inequality for nonlinear fractional time-varying delay systems”, Journal of Advanced Mathematical Modeling, 12(2), 271-288, (In Persian).
[10] Hunt, T., Krener A. (2010). “Improved patchy solution to the Hamilton-Jacobi-Bellman equations”, 49th IEEE Conference on Decision and Control (CDC),
[11] Jiao, H., Shen, Q. (2020). “Dynamics analysis and vaccination-based sliding mode control of a more generalized SEIR epidemic model”, IEEE Access, 8, 174507-174515.
[12] Irfan, S., Mehmood, A., Razzaq, M.T., Iqbal, J. (2018). “Advanced sliding mode control techniques for inverted Pendulum: Modelling and simulation”, Engineering Science and Technology, an International Journal, 21(4), 753-759.
[13] Khaledi, Gh., Mirhosseini-Alizamini, S.M., Khaleghizadeh, S. (2022). “Sliding mode control design for a class of uncertain time-delay conic nonlinear systems”, Iranian Journal of Science and Technology, Transactions A: Science (46), 583-593.
[14] Kumar, D., Mija, S.J. (2022). “Design and performance evaluation of LQR and optimized sliding mode controllers for a class of under actuated nonlinear systems”, IFAC-PapersOnLine, 55(1), 579-585.
[15] Khalil, H.K. (1996). “Nonlinear system”, Prentice Hall, New Jersey.
[16] Lewis, F.L., Vrabie, D.L., Syrmos, V.L. (2012). “Optimal control”, Third edition, John Wiley & Sons, Inc., Hoboken, NJ.
[17] Liu, J., Wang, X. (2011). “Advanced sliding mode control for mechanical systems”, Springer.
[18] Mahmoodabadi, M.J., Hadipour-Lakmesari, S. (2021). “Adaptive sliding mode control of HIV-1 infection model”, Informatics in Medicine Unlocked, 25, 100703, DOI: 10.1016/j.imu.2021.100703.
[19] Mahmoodabadi, M.J., Soleymani, T. (2020). “Optimum fuzzy combination of robust decoupled sliding mode and adaptive feedback linearization controllers for uncertain under actuated nonlinear systems”, 64, 241-250.
[20] Mahmoodabadi, M.J., Taherkhorsandi, M., Talebipour, M. (2017). “Adaptive robust PID sliding control of a liquid level system based on multi-objective genetic algorithm optimization”, Control and Cybernetics Journal, 46(3), 227-246.
[21] Muljel, S.D., Nagarale, R.M. (2016). “LQR technique based second order sliding mode control for linear uncertain systems”, International Journal of Computer Applications, 137(7), 23-29.
[22] Nezhadhosein, S., Ghanbari, R., Ghorbani-Mogahdam, Kh. (2022).“A numerical solution for fractional linear quadratic optimal control problems via shifted Legendre polynomials”, International Journal of Applied and Computational Mathematics, 8(4), 1-28.
[23] Nezhadhosein, S., Heydari, A., Ghanbari, R. (2015). “A modified hybrid genetic algorithm for solving nonlinear optimal control problems”, Mathematical Problems in Engineering, Article ID: 139036, 21, DOI: 10.1155/2015/139036.
[24] Ngwako, M.T., Nyandoro, O.T. (2021). “Singular optimal control for a high precision linear DC machine haulage”, IFAC-PapersOnLine, 54(11), 25-30.
[25] Nizar, A., Nouri, A.S. (2012). “A new sliding surface for discrete second order sliding mode control of time delay systems”, Proceedings of the 9th International Multi-Conference on System, Signals and Devices, DOI: 10.1109/SSD.2012.6198049.
[26] Olfati, R. (2002). “Normal forms for under actuated mechanical systems with symmetry”, IEEE Transactions on Automatic Control, 47(2), 305-308.
[27] Ozcan, S., Copur, Eh., Arican, AC., Salamci, MU. (2020). “A modified SDRE-based sub-optimal hypersurface design in SMC”, IFAC-PapersOnLine, 53(2), 6250-6255.
[28] Plestan, F., Shtessel, Y. (2011). “New methodologies for adaptive sliding mode control”, International Journal of Control, Taylor & Francis, 83(9), 1907-1919.
[29] Plestan, F., Glumineau, A., Laghrouche, S. (2008). “A new algorithm for high order sliding mode control”, International Journal of Robust and Nonlinear Control, 18(4-5), 441-453.
[30] Ranjbar, E., Yaghubi, M., Suratgar, A.A.(2020).“Robust adaptive sliding mode control of a MEMS tunable capacitor based on dead-zone method”, AUTOMATIKA, 61(4), 587-601.
[31] Riouali, M., Lahid, F., Elbarrai, I., Namir, A. (2022). “Mathematical modeling of the spread of the COVID’19 with optimal control strategies”, Procedia Computer Science, 203, 481-485.
[32] Sanjeewa, D.A., Parnichkun, M.(2021).“Control of rotary double inverted pendulum system using QR sliding surface based sliding mode controller”, Journal of Control and Decision, 9(1), 89-101.
[33] Slotine, J.E., Li, W. (1991). “Applied nonlinear control”, Prentice Hall, New Jersey.
[34] Soon, C.C., Ghazali, R., Jaafar, H.I. (2017). “Sliding mode controller design with optimized PID sliding surface using particle swarm algorithm”, Procedia Computer Science, 105, 235-239.
[35] Vaseei, S., Zarrabi, M.R.(2020).“An adaptive Lyapunov-based controller for HIV treatment”, Control and Optimization in Applied Mathematics (COAM), 5(2), 1-10.
[36] Wang, J., Liu, L., Li, X. (2020). “Adaptive sliding mode control based on equivalence principle and its application to chaos control in a seven-dimensional power system”, Hindawi, Mathematical Problems in Engineering, Article ID 1565460, DOI: 10.1155/2020/1565460.
[37] Wang, X., Shi, L., Katupitiya, J., Astronautica, A. (2022). “Robust control of a dual-arm space robot for in-orbit screw-driving operation”, Acta Astronautica, 200, 139-148.
[38] Xu, R., Ozgunar, U. (2008). “Sliding mode control of a class of under actuated systems”, Automatica, 44(1), 233-248.
[39] Zhang, D., Cao, L., Tang, S. (2017). “Fractional-order sliding mode control for a class of uncertain nonlinear systems based on LQR”, International Journal of Advanced Robotic Systems, 1-15.