Abstract
This paper proposes a robust sliding mode-H ∞ control design methodology for a class of nonlinear systems with unmatched parametric uncertainty and external disturbance. The design procedure combines the high robustness of the sliding mode control (SMC) with the H ∞ norm performance. First, based on linear matrix inequalities (LMI) technique and multiple Lyapunov functions approach, the sliding surface design problem is formulated as a H ∞ state-feedback control for a reduced uncertain nonlinear system with polytopic representation. Then, a sliding mode controller that drives the system states to the sliding surface in finite time and maintains a sliding mode is constructed. Finally, a comparative study is done to prove the effectiveness of the results.
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F. A. Faria, M. C. Valentino, and V. A. Oliveira, “A fuzzy Lyapunov function approach for stabilization and control of switched TS fuzzy systems,” Applied Mathematical Modelling, vol. 38, no. 19–20, pp. 4817–4834, 2014. [click]
D. Ye, X. Zhao, and J. Han, “Quantized feedback faulttolerant H ∞ controller design for linear systems with adaptive mechanism,” Optimal Control Applications and Methods, vol. 34, no. 5, pp. 623–634, 2013. [click]
S. Wajdi, S. Anis, and G. Garcia, “Robust sliding mode control approach for systems affected by unmatched uncertainties using H ∞ with pole clustering constraints,” Optimal Control Applications and Methods, vol. 36, no. 6, pp. 919–935, 2015.
W.-J. Liu, K.-K. Shyu, and K.-C. Hsu, “Design of uncertain multi-input systems with state delay and input deadzone nonlinearity via sliding mode control,” International Journal of Control, Automation and Systems, vol. 9, no. 3, pp. 461–469, 2011. [click]
A. Pisano and E. Usai, “Sliding mode control: A survey with applications in math,” Mathematics and Computers in Simulation, vol. 81, no. 5, pp. 954–979, 2011. [click]
A. Polyakov, “Nonlinear feedback design for fixed-time stabilization of linear control systems,” IEEE Transactions on Automatic Control, vol. 57, no. 8, pp. 2106–2110, 2012.
P. V. Suryawanshi, P. D. Shendge, and S. B. Phadke, “Robust sliding mode control for a class of nonlinear systems using inertial delay control,” Nonlinear Dynamics, vol. 78, no. 3, pp. 1921–1932, 2014. [click]
J. Zhang and W. X. Zheng, “Design of adaptive sliding mode controllers for linear systems via output feedback,” IEEE Transactions on Industrial Electronics, vol. 61, no. 7, pp. 3553–3562, 2014.
J.-L. Chang, “Robust dynamic output feedback secondorder sliding mode controller for uncertain systems,” International Journal of Control, Automation and Systems, vol. 11, no. 5, pp. 878–884, 2013. [click]
A. F. de Loza, F. J. Bejarano, and L. Fridman, “Unmatched uncertainties compensation based on high-order sliding mode observation,” International Journal of Robust and Nonlinear Control, vol. 23, no. 7, pp. 754–764, 2013. [click]
S. Mondal and C. Mahanta, “Chattering free adaptive multivariable sliding mode controller for systems with matched and mismatched uncertainty,” ISA Transactions, vol. 52, no. 3, pp. 335–341, 2013.
A. Polyakov and A. Poznyak, “Invariant ellipsoid method for minimization of unmatched disturbances effects in sliding mode control,” Automatica, vol. 47, no. 7, pp. 1450–1454, 2011.
J. M. Andrade-Da Silva, C. Edwards, and S. K. Spurgeon, “Sliding-mode output-feedback control based on LMIs for plants with mismatched uncertainties,” IEEE Transactions on Industrial Electronics, vol. 56, no. 9, pp. 3675–3683, 2009.
J. Yang, S. Li, and X. Yu, “Sliding-mode control for systems with mismatched uncertainties via a disturbance observer,” IEEE Transactions on Industrial Electronics, vol. 60, no. 1, pp. 160–169, 2013.
C. Edwards, “A practical method for the design of sliding mode controllers using linear matrix inequalities,” Automatica, vol. 40, no. 10, pp. 1761–1769, 2004. [click]
H. H. Choi, “Output feedback variable structure control design with an performance bound constraint,” Automatica, vol. 44, no. 9, pp. 2403–2408, 2008.
S. Valilou and M. J. Khosrowjerdi, “Robust sliding mode control design for mismatched uncertain systems with a GH2/H ∞ performance,” Asian Journal of Control, vol. 18, no. 1, pp. 1–9, 2016.
S. Valiloo, M. J. Khosrowjerdi, and M. Salari, “LMI based sliding mode surface design with mixed H2/H ∞ optimization,” Journal of Dynamic Systems, Measurement and Control, vol. 136, no. 1, p. 011016, 2014.
F. Castanos and L. Fridman, “Analysis and design of integral sliding manifolds for systems with unmatched perturbations,” IEEE Transactions on Automatic Control, vol. 51, no. 5, pp. 853–858, 2006.
F. Castaños and L. Fridman, “Dynamic switching surfaces for output sliding mode control: An H ∞ approach,” Automatica, vol. 47, no. 9, pp. 1957–1961, 2011. [click]
J.-L. Chang, “Dynamic output feedback sliding mode control for uncertain mechanical systems without velocity measurements,” ISA Transactions, vol. 49, no. 2, pp. 229–234, 2010. [click]
J. L. Chang, “Dynamic output feedback integral sliding mode control design for uncertain systems,” International Journal of Robust and Nonlinear Control, vol. 22, no. 8, pp. 841–857, 2012. [click]
H. Zhang, X. Liu, J. Wang, and H.-R. Karimi, “Robust sliding mode control with pole placement for a fluid power electrohydraulic actuator (eha) system,” The International Journal of Advanced Manufacturing Technology, vol. 73, no. 5–8, pp. 1095–1104, 2014.
B.-L. Zhang, L. Ma, and Q.-L. Han, “Sliding mode control for offshore steel jacket platforms subject to nonlinear self-excited wave force and external disturbance,” Nonlinear Analysis: Real World Applications, vol. 14, no. 1, pp. 163–178, 2013. [click]
P. Oliveira, R. Oliveira, V. Leite, V. Montagner, and P. Peres, “H ∞ guaranteed cost computation by means of parameter-dependent Lyapunov functions,” Automatica, vol. 40, no. 6, pp. 1053–1061, 2004.
M. Farhoodi and M. Beheshti, “Extended H 2, H ∞ and pole placement LMI characterization for continuous-time systems,” Proc. of IEEE India Conference INDICON, vol. 2, 2008, pp. 536–541.
Y. Xia and Y. Jia, “Robust sliding-mode control for uncertain time-delay systems: an LMI approach,” IEEE Transactions on Automatic Control, vol. 48, no. 6, pp. 1086–1091, 2003.
Y. Xia, J. Chen, G.-P. Liu, L. Wang, and D. Rees, “Robust adaptive sliding mode control for uncertain time-delay systems,” International Journal of Adaptive Control and Signal Processing, vol. 23, no. 9, pp. 863–881, 2009.
G. S. Tombul, S. P. Banks, and N. Akturk, “Sliding mode control for a class of non-affine nonlinear systems,” Nonlinear Analysis: Theory, Methods and Applications, vol. 71, no. 12, pp. 1589–1597, 2009.
S. Chumalee and J. F. Whidborne, “Gain-scheduled H ∞ control via parameter-dependent lyapunov functions,” International Journal of Systems Science, vol. 46, no. 1, pp. 125–138, 2015. [click]
S. Jin and J. Park, “Robust H ∞ filtering for polytopic uncertain systems via convex optimisation,” Control Theory and Applications, IEE Proceedings, vol. 148, no. 1, pp. 55–59, 2001.
M. Zribi, N. Almutairi, M. Abdel-Rohman, and M. Terro, “Nonlinear and robust control schemes for offshore steel jacket platforms,” Nonlinear Dynamics, vol. 35, no. 1, pp. 61–80, 2004. [click]
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Recommended by Associate Editor Young Ik Son under the direction of Editor Jun Hyun Park.
Wajdi Saad was born in Tunisia in 1987. He received his B.S degree in electrical engineering and M.S. diploma degree in automatic control, in 2010 and 2012, respectively from the High School of Sciences and Techniques of Tunis (ESSTT). Currently, he is at the unit of research C3S as Ph.D student in the High National Engineering School of Tunis (ENSIT). His research interests include robust control with sliding mode and photovoltaic systems.
Anis Sellami was born in Tunisia, in 1967. He obtained his Habilitation diploma at Sciences Faculty of Tunis in 2008, and his Ph.D in Electrical Engineering at National School of Engineers of Tunis with LAASCNRS of Toulouse in 1999, and his M.S. in Automatic Control and B.A. in Technical Sciences at Higher School in Technical Teaching, Tunis, Tunisia, in 1993 and 1990 respectively. Since December 2013, he is a professor with the Electrical Engineering Department, High National Engineering School of Tunis (ENSIT). Now, Dr. Anis Sellami is the Director of studies and Vice Director of the ENSIT. He is also member of the executive committee of Tunisian Association of the Numerical Techniques and Automatic control and Tunisian Association of Renewable Energy. His research interests include robust control with sliding mode and photovoltaic systems.
Germain Garcia received the M.S. degree in engineering, the Ph.D degree in automatic control and the Habilitation diploma (HDR), in 1984, 1988 and 1997, respectively from the National Institute of Applied Sciences (INSAT), Toulouse, France. He is currently at the LAAS-CNRS as full professor of INSAT. His research interests include robust control theory, linear matrix inequalities (LMI), constrained control and singularly perturbed models.
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Saad, W., Sellami, A. & Garcia, G. Robust sliding mode-H ∞ control approach for a class of nonlinear systems affected by unmatched uncertainties using a poly-quadratic Lyapunov function. Int. J. Control Autom. Syst. 14, 1464–1474 (2016). https://doi.org/10.1007/s12555-015-0141-9
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DOI: https://doi.org/10.1007/s12555-015-0141-9