This paper presents an optimal sliding mode cascade control for stabilization of a class of underactuated nonlinear mechanical systems. A discrete-time, nonlinear model predictive control structure is used to optimally select and update the parameters of the sliding mode control surfaces at specified intervals in order to achieve a desired performance objective. The determination of these surface parameters is subject to constraints that arise from the stability conditions imposed by the sliding mode control law and the physical limits on the system such as input saturation. Nominal stability of the optimal cascade control structure is demonstrated and its robust performance is illustrated using an experimental rotary inverted pendulum system.
Issue Section:
Research Papers
References
1.
Mayne
, D. Q.
, Rawlings
, J. B.
, Rao
, C. V.
, and Skokart
, P. O.
, 2000, “Constrained Model Predictive Control: Stability and Optimality
,” Automatica
, 36
, pp. 789
–814
.2.
Magni
, L.
, and Raimondo
, D. M.
, eds.
, F. A.
, 2009, Nonlinear Model Predictive Control: Toward New Challenging Applications
(Lecture Notes in Control and Information Sciences 384), Springer-Verlag
, Berlin/Heidelberg.
3.
Oh
, S.-R.
, and Sun
, J.
, 2010, “Path Following of Underactuated Marine Surface Vessels Using Line-of-Sight Based Model Predictive Control
,” Ocean Eng.
, 37
(2–3
), pp. 289
–295
.4.
Wen
, J.
, Seereeram
, S.
, and Bayard
, D.
, 1997, “Nonlinear Predictive Control Applied to Spacecraft Attitude Control
,” Proceedings of the 1997 American Control Conference (Cat. No.97CH36041)
, Vol. 3
, pp. 1899
–1903
.5.
de Oliveira
, V.
, and Lages
, W.
, 2007, “MPC Applied to Motion Control of an Underactuated Brachiation Robot
,” 2006 IEEE Conference on Emerging Technologies and Factory Automation (IEEE Cat. No.06TH8897).6.
Varga
, A.
, and Lantos
, B.
, 2005, “Model Based Predictive Control of Underactuated Nonlinear Mechatronical Systems
,” Pe
riod. Polytech., Electr. Eng., 49
(1–2
), pp. 123
–140
.7.
Jung
, S.
, and Wen
, J. T.
, 2004, “Nonlinear Model Predictive Control for the Swing-up of a Rotary Inverted Pendulum
,” ASME J. Dyn. Syst., Meas., Control
, 126
(3
), pp. 666
–673
.8.
Utkin
, V. I.
, 1977, “Variable Structure Systems With Sliding Modes
,” IEEE Trans. Autom. Control
, 22
, pp. 212
–222
.9.
Salamci
, M. U.
, Ozgoren
, M. K.
, and Banks
, S. P.
, 2000, “Sliding Mode Control With Optimal Sliding Surfaces for Missile Autopilot Design
,” J. Guid. Control Dyn.
, 23
, pp. 719
–727
.10.
Tokat
, S.
, Eksin
, I.
, Guzelkaya
, M.
, and Soylemez
, M. T.
, 2003, “Design of a Sliding Mode Controller With a Nonlinear Time-Varying Sliding Surface
,” Trans. Inst. Meas. Control
, 25
, pp. 145
–162
.11.
Kim
, K. S.
, and Park
, Y.
, 2004, “Sliding Mode Design via Quadratic Performance Optimization With Pole-Clustering Constraint
,” SIAM J. Control Optim.
, 43
, pp. 670
–684
.12.
Nikkhah
, M.
, Ashrafiuon
, H.
, and Muske
, K.
, 2006, “Optimal Sliding Control for Underactuated Systems
,” Proceedings of the 2006 American Control Conference
, pp. 4688
–4693
.13.
Muske
, K. R.
, Ashrafiuon
, H.
, and Nikkhah
, M.
, 2007, “A Predictive and Sliding Mode Cascade Controller
,” Proceedings of the 2007 American Control Conference
, pp. 4688
–4693
.14.
Bergerman
, M.
, and Xu
, Y.
, 1996, “Robust Joint and Cartesian Control of Underactuated Manipulators
,” ASME J. Dyn. Syst., Meas., Control
, 118
, pp. 557
–565
.15.
Lee
, K.
, Coats
, S.
, and Coverstone-Carroll
, V.
, 1997, “Variable Structure Control Applied to Underactuated Robots
,” Robotica
, 15
, pp. 313
–318
.16.
Su
, C.
, and Stepanenko
, Y.
, 1999, “Adaptive Variable Structure Set-Point Control of Underactuated Robots
,” IEEE Trans. Autom. Control
, 44
, pp. 2090
–2093
.17.
Xu
, R.
, and Ozgüner
, U.
, 2008, “Sliding Mode Control of a Class of Underactuated Systems
,” Automatica
, 44
(1
), pp. 233
–241
.18.
Olfati-Saber
, R.
, 2002, “Normal Form for Underactuated Mechanical Systems With Symmetry
,” IEEE Trans. Autom. Control
, 47
, pp. 305
–308
.19.
Park
, M.
, Chwa
, D.
, and Hong
, S.
, 2006, “Decoupling Control of a Class of Underactuated Mechanical Systems Based on Sliding Mode Control
,” Proceedings of the SICE-ICASE International Joint Conference
, pp. 806
–810
.20.
Martinez
, R.
, Alvarez
, J.
, and Orlov
, Y.
, 2008, “Hybrid Sliding-Mode-Based Control of Underactuated Systems With Dry Friction
,” IEEE Trans. Ind. Electr.
, 55
(11
), pp. 3998
–4003
.21.
Lee
, S. H.
, Park
, J. B.
, and Choi
, Y. H.
, 2009, “Finite-Time Control of Nonlinear Underactuated Systems Using Terminal Sliding Surface
,” Proceedings of the IEEE International Symposium on Industrial Electronics
, pp. 626
–631
.22.
Sankaranarayanan
, V.
, and Mahindrakar
, A.
, 2009, “Control of a Class of Underactuated Mechanical Systems Using Sliding Modes
,” IEEE Trans. Rob.
, 25
(2
), pp. 459
–467
.23.
Brockett
, R. W.
, 1983, “Asymptotic Stability and Feedback Linearization
,” Diff. Geom. Cont. Theory
, pp. 181
–191
.24.
Riachy
, S.
, Orlov
, Y.
, Floquet
, T.
, Santiesteban
, R.
, and Richard
, J.-P.
, 2008, “Second-Order Sliding Mode Control of Underactuated Mechanical Systems I: Local Stabilization With Application to an Inverted Pendulum
,” Int. J. Robust Nonlinear Control
, 18
(4–5
), pp. 529
–543
.25.
Santiesteban
, R.
, Floquet
, T.
, Orlov
, Y.
, Riachy
, S.
, and Richard
, J.-P.
, 2008, “Second-Order Sliding Mode Control of Underactuated Mechanical Systems II: Orbital Stabilization of an Inverted Pendulum With Application to Swing Up/Balancing Control
,” Int. J. Robust Nonlinear Control
, 18
(4–5
), pp. 544
–556
.26.
Ashrafiuon
, H.
, and Erwin
, R. S.
, 2008, “Sliding Mode Control of Underactuated Multibody Systems and Its Application to Shape Change Control
,” Int. J. Control
, 81
, pp. 1849
–1858
.27.
Avis
, J. M.
, Nersesov
, S. G.
, Nathan
, R.
, Ashrafiuon
, H.
, and Muske
, K. R.
, 2010, “A Comparison Study of Nonlinear Control Techniques for the RTAC System
,” Nonlinear Anal.: Real World Appl.
, 11
(4
), pp. 2647
–2658
.28.
Garcia-Gabin
, W.
, and Camacho
, E.
, 2003, “Sliding Mode Model Based Predictive Control for Non Minimum Phase Systems
,” Proceedings of the European Control Conference
, pp. 957
–962
.29.
Perez
, M.
, Jimenez
, E.
, and Camacho
, E.
, 2010, “Design of an Explicit Constrained Predictive Sliding Mode Controller
,” IET Control Theory Appl.
, 4
(4
), pp. 552
–562
.30.
Zhou
, J.-S.
, Liu
, Z.-Y.
, and Pei
, R.
, 2001, “A New Nonlinear Model Predictive Control Scheme for Discrete-Time System Based on Sliding Mode Control
,” Proceedings of the American Control Conference
, Vol. 4
, pp. 3079–308
4.31.
Slotine
, J. -J. E. E.
, and Li
, W.
, 1991, Applied Nonlinear Control
, Prentice-Hall
, Englewood Cliffs, NJ.
32.
Khalil
, H. K.
, 2002, Nonlinear Systems, 3rd ed.
, Prentice-Hall, Upper Saddle River
, NJ
.33.
Bhat
, S. P.
, and Bernstein
, D. S.
, 2003, “Nontangency-Based Lyapunov Tests for Convergence and Stability in Systems Having a Continuum of Equilibria
,” SIAM J. Control Opt.
, 42
(5
), pp. 1745
–1775
.34.
Spong
, M. W.
, 1997, “Underactuated Mechanical Systems
,” Control Problems in Robotics and Automation
, B.
Siciliano
and K.
Valavanis
, eds., Springer-Verlag
, London
.35.
Nersesov
, S. G.
, Ashrafiuon
, H.
, and Ghorbanian
, P.
, 2010, “On the Stability of Sliding Mode Control for a Class of Underactuated Nonlinear Systems
,” Proceedings of the 2010 American Control Conference
, pp. 3446
–3551
.36.
Bryson
, A. E.
, and Ho
, Y.-C.
, 1975, Applied Optimal Control, Revised ed.
, Taylor & Francis
, Levittown, PA.
37.
Costa
, E. F.
, and do Val
, J. B. R.
, 2009, “Uniform Approximation of Infinite Horizon Control Problems for Nonlinear Systems and Stability of the Approximating Controls
,” IEEE Trans. Autom. Control
, 54
(4
), pp. 881
–886
.38.
Aosta
, J. A.
, 2010, “Furuta’s Pendulum: A Conservative Nonlinear Model for Theory and Practise
,” Math. Probl. Eng.
, 2010, Article ID 742894.Copyright © 2012
by American Society of Mechanical Engineers
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