1. ON THE STABILITY OF SLIDING MODE
CONTROL FOR A CLASS OF
UNDERACTUATED NONLINEAR
SYSTEMS
Sergey 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 Ghorbanian
a 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 UMSs
Consider 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 PENDULUM
The equations of motion:
(new point..!!)
Given the sliding surface: Introduce an auxiliary variable:
The SMC law becomes:
We get the system dynamics on t
The 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 tr
Lyapunov 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: