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Sliding Mode Control Stability (Jan 19, 2013)

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• 1. ON THE STABILITY OF SLIDING MODE CONTROL FOR A CLASS OF UNDERACTUATED NONLINEAR SYSTEMSSergey G. Nersesov, Hashem Ashrafiuon, and Parham Ghorbanian A paper from 2010 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 30-July 02, 2010 January 19, 2013
• 2. Dr. Sergey G. Nersesov B.S. and M.S. degrees in aerospace engineering (1997, 1999) M.S. degree in applied mathematics (2003) Ph.D. degree in aerospace engineering (2005)Ass. Prof. at the Department of Mechanical Engineering,Villanova University, Villanova, Dr. Nersesov is a coauthor of the books: - Thermodynamics; A Dynamical Systems Approach (Princeton University Press, 2005) - Impulsive and Hybrid Dynamical Systems; Stability, Dissipativity, and Control (Princeton University Press, 2006).Dr. Hashem Ashrafiuon B.S. degree (1982), an M.S. degree (1984), and a Ph.D. degree (1988) in Mechanical and Aerospace Engineering , State University of New York at Buffalo.Professor at the Department of Mechanical Engineering, Villanova University. Director of Center for Nonlinear Dynamics and Control (CENDAC)Parham Ghorbaniana graduate student at Villanova University.
• 3. I. Problem Setup(^_^)
• 4. II. SMC Design Note: Form (4) Substituting for and form (2) and To satisfy the sliding condition: . Take . We obtain the control law: And finally, we add a sign function: Here η > 0 is a constant parameter indic closed-loop system trajectories reach the (same old..!!)
• 5. III. Closed-loop Systems Note: Eq.(8) and (9) will be used f in the reaching phase. (same old..!!)** Introduce an auxiliary variable…TRICK!! Note: Eq.(10) and (11) will be use in the sliding phase. (new point..!!)
• 6. Remark: On the Sliding Phase
• 7. IV. Stability Analysis for the Reaching Phase
• 8. V. Specialize the Result of Theorem 2.1 to 2DOF UMSsConsider the Euler-Lagrange systems (new point..!!)whose dynamics are given by Given the state and variables:
• 9. VI. Stability Analysis of the Sliding Phase for 2DOF UMSs (new point..!!)
• 10. VII. EXAMPLE: STABILTY ANALYSIS OF AN INVERTED PENDULUMThe equations of motion: (new point..!!)Given the sliding surface: Introduce an auxiliary variable:The SMC law becomes: We get the system dynamics on tThe closed-loop system: Rewrite in the state space: and
• 11. VII. EXAMPLE: STABILTY ANALYSIS OF AN INVERTED PENDULUM (cont)(new point..!!) The Lyapunov derivative along trLyapunov function candidate:
• 12. VII. EXAMPLE: STABILTY ANALYSIS OF AN INVERTED PENDULUM (cont)To find a positive definite and symmetric metrix P (new point..!!)
• 13. VII. EXAMPLE: STABILTY ANALYSIS OF AN INVERTED PENDULUM (cont) Sliding Surface: Domain of Attraction: