Design of an optimal learning sliding mode control for linear systems

Document Type : Research Article

Authors

1 Department of Mathematics and Computer, Lorestan university, Lorestan, Iran

2 Department of Mathematics and Computer Science, Lorestan university, Lorestan, Iran

Abstract

In this paper, a new learning robust controller based on the sliding mode control method and reinforcement learning approach is designed for a class of SISO linear systems with a relative degree of uncertainty ‎$ r$‎. For this purpose, a hybrid controller that including equivalent controller and learning robust controller, is designed. The proposed controller guarantees asymptotic stability, sliding condition, finite reaching time, elimination of chattering phenomenon and tracking of desired output in an optimally approach. For online approximation of the value function and design of an optimal policy a new robust optimal learning controller is designed. For analytical facilitation and stability analysis, three theorems are proved and a new algorithm is designed. Finally, a simulation example is presented to demonstrate the advantages of the proposed method. The simulation results show the optimality of the controls, the elimination of chattering phenomenon and the tracking of the desired ‎output.‎‎

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References

[1] M. Aghababa, M. Akbari, A chattering-free robust adaptive sliding mode controller for synchronization of two different chaotic systems with unknown uncertainties and external disturbances, J. Appl. Math. Comput. 218 (2012) 5757–5768.
[2] P. An, M. Liu, Y. Wan, F. Lewis, Multi-player H∞ Differential game using on-policy and off-policy reinforcement learning, Sixteenth IEEE International Conference of Control and Automation 1 (2020) 1137–1142.
[3] R. Avram, X. Zhang X, J. Muse, Nonlinear adaptive fault-tolerant quadrotor altitude and attitude tracking with multiple actuator faults, IEEE Trans. Control Syst. Technol. 13 (2018) 701–707.
[4] W. Bai, T. Li, S. Tong, NN reinforcement learning adaptive control for a class of nonstrict-feedback discrete-time systems, IEEE Trans. Cybern. 1 (2020) 4573–4584.
[5] D.P. Bertsekas, Dynamic Programming and Optimal Control, 4th Edition, Athena Scientific, 2012.
[6] S. Chegini, M. Yarahmadi, Quantum sliding mode control via error sliding surface, J. Vib. Control 24 (2018) 5345–5352.
[7] J. Choi, Y. Bakch, Chattering reduction using improved switching functions in the SMC method for battery chargers of PMDs, IEEE Access 99 (2024) 182663–182672.
[8] T. Ciment, S.P. Banks, Nonlinear optimal tracking control with application to super-tankers for autopilot designe, Automatica 40 (2004) 1845–1863.
[9] N. Elseddeq, S. Elghamrawy, A. Eldesouky, Optimized robust learning framework based on big data for forecasting cardiovascular crises, Sci. Rep. 14 (2024) 28224.
[10] K. Guo, H. Zhang, C. Wei, H. Jiang, J. Wang, Novel sliding mode control of the manipulator based on a nonlinear disturbance observer, Sci. Rep. 13 (2024) 30656.
[11] C. Jia, X. Li, H. Wang, Z. Song, Sliding mode-based integral reinforcement learning event triggered control, Int. J. Control Autom. Syst. 23 (2025) 315–331.
[12] B. Kiumarsi, F.L. Lewis, Actor-critic-based optimal tracking for partially unknown nonlinear discrete-time systems, IEEE Trans. Neural Netw. Learn. Syst. 26 (2015) 140–151.
[13] J.Y. Lee, Y. Kim, Hamilton-Jacobi Based Policy-Iteration via Deep Operator Learning, Neurocomputing 646 (2024) 130515.
[14] J. Li, H. Su, Y. Zhang, J. Chu, Chattering free sliding mode control for uncertain discrete time-delay singular systems, Asian J. Cont. 15 (2013) 260–269.
[15] Y. Li, H. Xia, B. Zhao, Policy iteration algorithm based faulttolerant tracking control: an implementation on reconfigurable manipulators, J. Electr. Eng. Technol. 26 (2018) 1740–1751.
[16] T. Niknam, M.H. Khooban, A. Kavousifard, M.R. Soltanpout, An optimal type II fuzzy sliding mode control design for a class of nonlinear systems, Nonlinear Dyn. 75 (2013) 75–83.
[17] H. Pang, Q. Yang, Optimal sliding mode output tracking control for linear systems with uncertainties, International Conference on Machine Learning and Cybernetics 1 (2010) 942–946.
[18] H. Pouyanfar, S. Effati, A. Mansoori, Dynamic adaptation strategies for optimal control in unknown linear time-invariant system, J. Math. Model. 14(1) (2026) 1–17.
[19] R. Shahnazi, Robust exponential concurrent learning adaptive control for systems preceded by dead-zone input nonlinearity, J. Math. Model. 12(2) (2024) 371–385.
[20] J.J.E. Slotine, W. Li, Applied Nonlinear Control, Prentice Hall, 1991.
[21] R. S. Sutton, A. G. Barto, Reinforcement Learning: An Introduction, 2nd Edition, MIT Press, 2018.
[22] H. Toshani, M. Farrokhi, Optimal sliding-mode control of linear systems with uncertainties and input constraints using projection neural network, J. Optim. Control Appl. Methods 39 (2017) 963–980.
[23] D. Vrabie, F. Lewis, Neural network approach to continuous-time direct adaptive optimal control for partially unknown nonlinear systems, Neural Networks 22 (2009) 237–246.
[24] S. Xue, B. Luo, D. Liu, Y. Gao, Integral reinforcement learning-based event-triggered control with input saturation, Neural Networks 131 (2020) 144–153.
[25] Y. Yang, K.G. Vamvoudakis, H. Modares, Y. Yin, D. Wunsch, Hamiltonian-driven hybrid adaptive dynamic programming, J. IEEE Trans. Syst. Man Cybern. Syst. 51 (2020) 6423–6434.
[26] M. Yarahmadi, M. Karbassi, Robust wavelet sliding-mode control via time-variant sliding function, IEICE Trans. Fundam. Electron. Commun. Comput. Sci. 93 (2010) 1181–1189.
[27] B. Zhang, D.W. Ding, Optimal tracking and regulation performance of networked control systems with additive colored Gaussian noise and finite bandwidth, J. Appl. Math. Comput. 501 (2025) 129469.
[28] Y. Zhang, S. Zhang, Y. Wang, D. Liu, Integral sliding mode-based event-triggered nearly optimal tracking control for uncertain nonlinear systems, Int. J. Robust Nonlinear Control 34 (2023) 2325–2332.
Journal of Mathematical Modeling
Volume 14, Issue 1
March 2026
Pages 191-206

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