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Stabilization of Furuta Pendulum: A Backstepping Based Hierarchical Sliding Mode Approach with Disturbance Estimation

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Underactuated system, Furuta Pendulum, hierarchical sliding mode, backstepping control, integral action, disturbance estimation

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Stabilization of Furuta Pendulum: A Backstepping Based Hierarchical Sliding Mode Approach with Disturbance Estimation

  1. 1. Stabilization of Furuta Pendulum: A Backstepping Based Hierarchical Sliding Mode Approach with Disturbance Estimation by Shubhobrata Rudra Inspire Research Fellow Electrical Engineering Department Jadavpur University Kolkata
  2. 2. ContentA Few Words on Rotating PendulumAdaptive Backstepping Sliding Mode ControlHierarchical Sliding Mode ControlControl Law for Rotating PendulumSimulation ResultsConclusions
  3. 3. A Few Words on Rotating Pendulum φ u Degree of Freedom: 2 θ No of Control Input: 1State Model of Rotating Pendulum System  q1 p1 q1=θ 2  p1 k 2 tan q 2 k 3 sin q 2 p 1 k 1 u / cos q 2 q2=φ  q2 p2  p2 u
  4. 4. Contd. Standard State Model of Underactuated System  x1 x2  x2 f1 X b1 X u d1 t  x3 x4  x4 f2 X b2 X u d2 t x1 , x2 , x , x4  3 2 f1 X k 2 tan q 2 k 3 sin q 2 p1 f2 X 0 g1 X k 1 sec q 2 g2 X 1
  5. 5. Adaptive Backstepping Sliding Mode Control Define 1st Error variable & its dynamic as: e1 x1 x1d &  e1 x 2  x1d Stabilizing Function: 1 c1 e1 1 1 Control Lyapunov Function (CLF) and its derivative 1 2 1 2 V1 1 e1 2 2 Define 2nd error variable e2 and its derivative as: e2 x2  x1 d and  e2 f1 X b1 X u d1 t 1 d x 1 1 Define first-layer sliding surface s1 and new CLF as 1 s1 e 1 1 e2 and V2 V1 2 s1 2
  6. 6. Contd. Derivative of CLF: V2 2 e1 e 2 c1 e1 s1 1 e2 c1 e1 1 1 f1 X b1 X u d1 t 1 d x 1 Control Input: 1 u1 b1 X 1 e2 c1 e1 1 1 f1 X d 1 M tanh s1 1 d x 1 h1 s1 1 tanh s1 Augmented Lyapunov Function: 1 2  V3 V2 d 1M and d 1 M d1M d 1M 2 1 Adaptation Law:   d1M s 1 1
  7. 7. Hierarchical Sliding Mode Control Control Inputs: 1u1 b1 X 1 e2 c1 e1 1 1 f1 X d 1 M tanh s1 1 d x 1 h1 s1 1 tanh s1 1u2 b2 X 1 e4 c 2 e3 2 2 f2 X d 2 M tanh s 2 2 d x 2 h2 s 2 2 tanh s 2    Adaptation Laws: d1M s 1 1 and d 2M 2 s2 Composite control law: u u1 u2 u sw Define 2nd Layer sliding surface: S s 1 1 2 s2 b X u2 b2 X u1 tanh( S ) K .S Coupling Law: u sw 1 1 2 b 1 1 X 2 b2 X b X u1 b2 X u 2 tanh( S ) KS Composite Control Law: u 1 1 2 b 1 1 X 2 b2 X
  8. 8. Control Law for Rotating Pendulum System . Expression of Control Input for Translational Motion 1  u1 b1 X 1 e2 c1 e1 1 1 f1 X d 1M 1 d x 1 h1 s1 1 tanh s1 Expression of Control Input for Rotational Motion 1  u2 b2 X 2 e4 c 2 e3 2 2 f2 X d 2M 3 d x 2 h2 s 2 2 tanh s 2 Coupling Control Law: b1 X u 2 b2 X u1 tanh( S ) K .S u sw b1 X b2 X Composite Control Law: k1 u1 u2 tanh( S ) K .S cos q 2 u k1 1 cos q 2
  9. 9. Simulation Results Initial Conditions: q1 pi / 3 and p1 0 Shaft Position Shaft Velocity
  10. 10. Contd.Phase Portrait of q1-p1
  11. 11. Contd. Initial Conditions: q2 pi / 6 and p2 0 Pendulum Position Pendulum Velocity
  12. 12. Contd.Phase Portrait of q2-p2
  13. 13. Conclusions Another new method of addressing the stabilization problem for underactuated system. Can easily be extended to address the stabilization problem of other two degree of freedom underactuated mechanical systems. Chattering problem can be reduced with the introduction of second order sliding mode control. Proposed algorithm is applicable for only two-degree of freedom single input systems, research can be pursued to make the control algorithm more generalized such that it will able to address the control problem of any arbitrary underactuated system.
  14. 14. Reference K.J. Astrom, and K. Furuta, “Swing up a pendulum by energy control,” Automatica, 36(2), P- 287–295,2000. V. Sukontanakarn and M. Parnichkun, “Real-time optimal control for rotary inverted pendulum. American Journal of Applied Sciences,” Vol-6, P-1106–1115, 2009. Shailaja Kurode, Asif Chalanga and B. Bandyopadhyay, “Swing-Up and Stabilization of Rotary Inverted Pendulum using Sliding Modes,” Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011. Hera, P.M., Shiriaev, A.S., Freidovich, L.B., and Mettin, U. ‘Orbital Stabilization of a Pre- planned Periodic Motion to Swing up the Furuta Pendulum: Theory and Experiments’, in ICRA’09: Proceedings of the 2009 IEEE International Conference on Robotics and Automation, 12–17 May, IEEE Press, Kobe, Japan, pp. 2971–2976, 2009. W.Wang, J. Yi, D. Zhao, and D. Liu, “Design of a stable sliding-mode controller for a class of second-order underactuated systems,” IEE Proceedings: Control Theory and Applications, vol. 151, no. 6, pp. 683–690, 2004. F. J. Lin, P. H. Shen, and S. P. Hsu, “Adaptive backstepping sliding mode control for linear induction motor drive,” Proc. Inst. Elect. Eng., Electr. Power Appl., vol. 149, no. 3, pp. 184– 194, 2002. S. Sankaranarayanan and F. Khorrami, “Adaptive variable structure control and applications to friction compensation,” in Proc. IEEE CDC Conf. Rec., 1997, pp. 4159–4164. W. Wang, J. Yi, D. Zhao, and D. Liu, “Hierarchical sliding-mode control method for overhead cranes,” Acta Automatica Sinica, vol. 30, no. 5, pp. 784–788, 2004. H. H. Lee, Y. Liang, and S. Del, “A sliding-mode antiswing trajectory control for overhead
  15. 15. Thank You

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