Aerospace Mechanics

Aerospace Mechanics

Reduced-Order Sliding Mode and Super-Twisting Observers for Nonlinear Electro-Hydraulic Systems: Lyapunov-Based Design and Accuracy–Complexity Analysis

Document Type : Dynamics, Vibrations, and Control

Authors
Hamidreza NasrEsfahani 1
Seyed Ali Zahiripour 2
1 Master's Student, University of Kashan, Kashan, Iran
2 Assistant Professor, University of Kashan, Kashan, Iran
10.47176/MAJ.2025.1564
Abstract
Electrohydraulic systems (EHSs), despite their high power and force density, pose significant challenges for accurate and robust state estimation due to inherent nonlinearities, friction effects, fluid compressibility, and various uncertainties. In this paper, two reduced-order sliding mode observer (SMO) structures are developed, including a first-order sliding mode observer and a high-order sliding mode observer based on the Super-Twisting Algorithm (STA). Within this framework, a Lyapunov-based stability analysis is established for the reduced-order configurations, and sufficient analytical conditions for selecting the observer gains are derived. Furthermore, the high-order sliding mode observer is reformulated into a reduced-order structure, and its stability is rigorously analyzed according to the resulting estimation error dynamics.
The adoption of reduced-order structures leads to decreased computational burden and faster state reconstruction compared to their full-order counterparts. To evaluate performance, the proposed observers are implemented on a five-state electrohydraulic system model under measurement noise, external disturbances, and parametric uncertainties. Their performance is compared with that of an Extended Kalman Filter (EKF) and a derivative-based observer employing a low-pass filter. Simulation results demonstrate that the reduced-order Super-Twisting sliding mode observer achieves superior estimation accuracy, faster convergence, and enhanced robustness against noise and disturbances relative to the other observers considered.

Graphical Abstract

Reduced-Order Sliding Mode and Super-Twisting Observers for Nonlinear Electro-Hydraulic Systems: Lyapunov-Based Design and Accuracy–Complexity Analysis

Highlights

[1]     Zare, A., Jahanshahi, M. Speed Control of Electro-Hydraulic Servo Systems Using a Hybrid Fuzzy Method. Aerospace Mechanics, 2019; 15(1): 39–50. (In Persian) DOR: https://dor.isc.ac/dor/20.1001.1.26455323.1398.15.1.4.1

[2]     Hedrick JK, Rajamani R, Yi K. Observer design for electronic suspension applications. Vehicle System Dynamics. 1994;23(1):413–440. DOI https://doi.org/10.1080/00423119408969068

[3]     An L, Sepehri N. Hydraulic actuator circuit fault detection using extended Kalman filter. Proceedings of the American Control Conference. 2003. DOI https://doi.org/10.1109/ACC.2003.1240505

[4]     Khoshzaban M, Sassani F, Lawrence P. Online state and parameter estimation of an electrohydraulic valve for intelligent monitoring. Proceedings of the IEEE/ASME International Conference on Advanced Intelligent Mechatronics. 1997. DOI https://doi.org/10.1109/AIM.1997.653015

[5]     Sepasi M, Sassani F. On-line fault diagnosis of hydraulic systems using unscented Kalman filter. International Journal of Control, Automation and Systems. 2010;8:149–156. DOI https://doi.org/10.1007/s12555-010-0119-6

[6]     Pan C, Wang Y, Yang SX, Li Z, Xiao J. Compensation function observer-based backstepping sliding-mode control of uncertain electro-hydraulic servo system. Machines. 2024;12(10):719. DOI https://doi.org/10.3390/machines12100719

[7]     Levant A. Sliding order and sliding accuracy in sliding mode control. International Journal of Control. 1993;58(6):1247–1263. DOI https://doi.org/10.1080/00207179308923053

[8]     Wang Y, Zhao J, Zhang H, Wang H. Robust output feedback control for electro-hydraulic servo system with error constraint based on high-order sliding mode observer. Transactions of the Institute of Measurement and Control. 2023. DOI https://doi.org/10.1177/01423312221146225

[9]     Palli G, Strano S, Terzo M. A novel adaptive-gain technique for high-order sliding-mode observers with application to electro-hydraulic systems. Mechanical Systems and Signal Processing. 2020;144:106875. DOI https://doi.org/10.1016/j.ymssp.2020.106875

[10]   Bahrami R, Celler BG, Savkin AV. Adaptive super-twisting observer for fault reconstruction in electro-hydraulic systems. 2020. DOI https://doi.org/10.48550/arXiv.2002.03411

[11]   M. H. Nguyen and K. K. Ahn, "Extended Sliding Mode Observer-Based Output Feedback Control for Motion Tracking of Electro-Hydrostatic Actuators," Mathematics, vol. 11, no. 20, p. 4324, 2023. DOI https://doi.org/10.3390/math11204324

[12]   Khan Kalantari, S., Mohammadkhani, H. Design of a Robust Nonlinear Autopilot Using Feedback Linearization and an Augmented State Observer for Air-Defense Interceptors. Aerospace Mechanics, 2021; 17(2): 1–12. (In Persian) DOR: https://dor.isc.ac/dor/20.1001.1.26455323.1400.17.2.1.4

[13]   Palli G, Strano S, Terzo M. Sliding-mode observers for state and disturbance estimation in electro-hydraulic systems. Control Engineering Practice. 2018;74:58–70. DOI https://doi.org/10.1016/j.conengprac.2018.02.005

[14]   A. Bonchis, P. I. Corke, and D. C. Rye, "A pressure-based, velocity independent, friction model for asymmetric hydraulic cylinders," in Proc. IEEE Int. Conf. Robotics Autom., 1999, vol. 3, pp. 1746–1751. DOI https://doi.org/10.1109/ROBOT.1999.770361

[15]   H. E. Merritt, *Hydraulic Control Systems*, John Wiley & Sons, 1967.

[16]   L. Mrton, S. Fodor, and N. Sepehri, "A practical method for friction identification in hydraulic actuators," Mechatronics, vol. 21, no. 1, pp. 350–356, 2011. DOI https://doi.org/10.1016/j.mechatronics.2010.08.010

[17]   Chen Y, Ji Y, Guo K. A reduced-order nonlinear sliding mode observer for vehicle slip angle and tyre forces. Vehicle System Dynamics. 2014. DOI https://doi.org/10.1080/00423114.2014.960430

[18]   Alessandri, Angelo, Patrizia Bagnerini, and Roberto Cianci. "State Observation for Lipschitz Nonlinear Dynamical Systems Based on Lyapunov Functions and Functionals." Mathematics 8.9 (2020): 1424. DOI https://doi.org/10.3390/math8091424

[19]   Ruderman, M. Reduced-Order Asymptotic Observer of Nonlinear Friction for Precise Motion Control. Control Engineering Practice, 2026; 168: 106733. DOI: https://doi.org/10.1016/j.conengprac.2025.106733

[20]   Ao, W., Zhang, H., Zhao, N., and Minchala, L. I. Adaptive Neural Security Control for Networked Singular Systems Under Deception Attacks. IEEE Access, 2022; 10: 33230–33237. DOI: https://doi.org/10.1109/ACCESS.2022.3161672

[21]   Levant A. Higher-order sliding modes, differentiation and output-feedback control. International Journal of Control. 2003;76(9–10):924–941. DOI https://doi.org/10.1080/0020717031000099029

[22]   Moreno JA, Osorio M. A Lyapunov approach to second-order sliding mode controllers and observers. Proceedings of the 47th IEEE Conference on Decision and Control. 2008;2856–2861. DOI https://doi.org/10.1109/CDC.2008.4739356

[23]   Ilten E, Demirtas M. Fractional order super-twisting sliding mode observer for sensorless control of induction motor. COMPEL. 2019;38(2):878–892. DOI https://doi.org/10.1108/COMPEL-08-2018-0306

[24]   Mojallizadeh, M. R., Laghrouche, S., Harmouche, M., Olaby, O. A Survey on the Discrete-Time Differentiators in Closed-Loop Control Systems: Experiments on an Electro-Pneumatic System. Control Engineering Practice, 2023; 136: 105546. DOI: https://doi.org/10.1016/j.conengprac.2023.105546

[25]   Sorel, Y., Hawila, I., Cucu-Grosjean, L., Mezouak, M., Clarke, H., Ben Amor, S. Kopernic Dynamic Benchmarks (KDBench): Open-Source Measurement-Based Benchmarks for Probabilistic Real-Time Systems. RTSS@Work 2024 – IEEE Real-Time Systems Symposium Workshops (RTSSW), 2024; pp. 565–572. DOI: https://doi.org/10.1109/RTSS62706.2024.00027

Keywords
Electrohydraulic System Sliding
Mode Observer Reduced
Order Observer Super
Twisting Algorithm State Estimation
Subjects
[1]     Zare, A., Jahanshahi, M. Speed Control of Electro-Hydraulic Servo Systems Using a Hybrid Fuzzy Method. Aerospace Mechanics, 2019; 15(1): 39–50. (In Persian) DOR: https://dor.isc.ac/dor/20.1001.1.26455323.1398.15.1.4.1
[2]     Hedrick JK, Rajamani R, Yi K. Observer design for electronic suspension applications. Vehicle System Dynamics. 1994;23(1):413–440. DOI https://doi.org/10.1080/00423119408969068
[3]     An L, Sepehri N. Hydraulic actuator circuit fault detection using extended Kalman filter. Proceedings of the American Control Conference. 2003. DOI https://doi.org/10.1109/ACC.2003.1240505
[4]     Khoshzaban M, Sassani F, Lawrence P. Online state and parameter estimation of an electrohydraulic valve for intelligent monitoring. Proceedings of the IEEE/ASME International Conference on Advanced Intelligent Mechatronics. 1997. DOI https://doi.org/10.1109/AIM.1997.653015
[5]     Sepasi M, Sassani F. On-line fault diagnosis of hydraulic systems using unscented Kalman filter. International Journal of Control, Automation and Systems. 2010;8:149–156. DOI https://doi.org/10.1007/s12555-010-0119-6
[6]     Pan C, Wang Y, Yang SX, Li Z, Xiao J. Compensation function observer-based backstepping sliding-mode control of uncertain electro-hydraulic servo system. Machines. 2024;12(10):719. DOI https://doi.org/10.3390/machines12100719
[7]     Levant A. Sliding order and sliding accuracy in sliding mode control. International Journal of Control. 1993;58(6):1247–1263. DOI https://doi.org/10.1080/00207179308923053
[8]     Wang Y, Zhao J, Zhang H, Wang H. Robust output feedback control for electro-hydraulic servo system with error constraint based on high-order sliding mode observer. Transactions of the Institute of Measurement and Control. 2023. DOI https://doi.org/10.1177/01423312221146225
[9]     Palli G, Strano S, Terzo M. A novel adaptive-gain technique for high-order sliding-mode observers with application to electro-hydraulic systems. Mechanical Systems and Signal Processing. 2020;144:106875. DOI https://doi.org/10.1016/j.ymssp.2020.106875
[10]   Bahrami R, Celler BG, Savkin AV. Adaptive super-twisting observer for fault reconstruction in electro-hydraulic systems. 2020. DOI https://doi.org/10.48550/arXiv.2002.03411
[11]   M. H. Nguyen and K. K. Ahn, "Extended Sliding Mode Observer-Based Output Feedback Control for Motion Tracking of Electro-Hydrostatic Actuators," Mathematics, vol. 11, no. 20, p. 4324, 2023. DOI https://doi.org/10.3390/math11204324
[12]   Khan Kalantari, S., Mohammadkhani, H. Design of a Robust Nonlinear Autopilot Using Feedback Linearization and an Augmented State Observer for Air-Defense Interceptors. Aerospace Mechanics, 2021; 17(2): 1–12. (In Persian) DOR: https://dor.isc.ac/dor/20.1001.1.26455323.1400.17.2.1.4
[13]   Palli G, Strano S, Terzo M. Sliding-mode observers for state and disturbance estimation in electro-hydraulic systems. Control Engineering Practice. 2018;74:58–70. DOI https://doi.org/10.1016/j.conengprac.2018.02.005
[14]   A. Bonchis, P. I. Corke, and D. C. Rye, "A pressure-based, velocity independent, friction model for asymmetric hydraulic cylinders," in Proc. IEEE Int. Conf. Robotics Autom., 1999, vol. 3, pp. 1746–1751. DOI https://doi.org/10.1109/ROBOT.1999.770361
[15]   H. E. Merritt, *Hydraulic Control Systems*, John Wiley & Sons, 1967.
[16]   L. Mrton, S. Fodor, and N. Sepehri, "A practical method for friction identification in hydraulic actuators," Mechatronics, vol. 21, no. 1, pp. 350–356, 2011. DOI https://doi.org/10.1016/j.mechatronics.2010.08.010
[17]   Chen Y, Ji Y, Guo K. A reduced-order nonlinear sliding mode observer for vehicle slip angle and tyre forces. Vehicle System Dynamics. 2014. DOI https://doi.org/10.1080/00423114.2014.960430
[18]   Alessandri, Angelo, Patrizia Bagnerini, and Roberto Cianci. "State Observation for Lipschitz Nonlinear Dynamical Systems Based on Lyapunov Functions and Functionals." Mathematics 8.9 (2020): 1424. DOI https://doi.org/10.3390/math8091424
[19]   Ruderman, M. Reduced-Order Asymptotic Observer of Nonlinear Friction for Precise Motion Control. Control Engineering Practice, 2026; 168: 106733. DOI: https://doi.org/10.1016/j.conengprac.2025.106733
[20]   Ao, W., Zhang, H., Zhao, N., and Minchala, L. I. Adaptive Neural Security Control for Networked Singular Systems Under Deception Attacks. IEEE Access, 2022; 10: 33230–33237. DOI: https://doi.org/10.1109/ACCESS.2022.3161672
[21]   Levant A. Higher-order sliding modes, differentiation and output-feedback control. International Journal of Control. 2003;76(9–10):924–941. DOI https://doi.org/10.1080/0020717031000099029
[22]   Moreno JA, Osorio M. A Lyapunov approach to second-order sliding mode controllers and observers. Proceedings of the 47th IEEE Conference on Decision and Control. 2008;2856–2861. DOI https://doi.org/10.1109/CDC.2008.4739356
[23]   Ilten E, Demirtas M. Fractional order super-twisting sliding mode observer for sensorless control of induction motor. COMPEL. 2019;38(2):878–892. DOI https://doi.org/10.1108/COMPEL-08-2018-0306
[24]   Mojallizadeh, M. R., Laghrouche, S., Harmouche, M., Olaby, O. A Survey on the Discrete-Time Differentiators in Closed-Loop Control Systems: Experiments on an Electro-Pneumatic System. Control Engineering Practice, 2023; 136: 105546. DOI: https://doi.org/10.1016/j.conengprac.2023.105546
[25]   Sorel, Y., Hawila, I., Cucu-Grosjean, L., Mezouak, M., Clarke, H., Ben Amor, S. Kopernic Dynamic Benchmarks (KDBench): Open-Source Measurement-Based Benchmarks for Probabilistic Real-Time Systems. RTSS@Work 2024 – IEEE Real-Time Systems Symposium Workshops (RTSSW), 2024; pp. 565–572. DOI: https://doi.org/10.1109/RTSS62706.2024.00027
Aerospace Mechanics
Volume 22, Issue 2 - Serial Number 84
Summer
Summer 2026
Pages 61-78

  • Receive Date 24 March 2026
  • Revise Date 19 May 2026
  • Accept Date 09 June 2026
  • Publish Date 23 July 2026