ON THE STABILITY OF SLIDING MODE CONTROL FOR A CLASS OF

UNDERACTUATED NONLINEAR SYSTEMS

A paper from 2010 American Control Conference

Marriott Waterfront, Baltimore, MD, USA

June 30-July 02, 2010

Sergey G. Nersesov, Hashem Ashrafiuon, and Parham Ghorbanian

January 19, 2013

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)

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).

Parham Ghorbaniana graduate student at Villanova University.

I. Problem Setup

(^_^)

II. SMC Design

Note:Form (4)

To satisfy the sliding condition: .Take . We obtain the control law:

Substituting for and form (2) and (3) yields

And finally, we add a sign function:Here η > 0 is a constant parameter indicating how fast the closed-loop system trajectories reach the sliding surface .

(same old..!!)

III. Closed-loop Systems

Eq.(8) and (9) will be used for analyzing in the reaching phase.

Note:

Eq.(10) and (11) will be used for analyzingin the sliding phase.

Note:** Introduce an auxiliary variable…TRICK!!

(same old..!!)

(new point..!!)

Remark: On the Sliding Phase

IV. Stability Analysis for the Reaching Phase

Consider the Euler-Lagrange systems whose dynamics are given by

V. Specialize the Result of Theorem 2.1 to 2DOF UMSs

and Given the state variables:

(new point..!!)

VI. Stability Analysis of the Sliding Phase for 2DOF UMSs

(new point..!!)

VII. EXAMPLE: STABILTY ANALYSIS OF AN INVERTED PENDULUM

The equations of motion:

Given the sliding surface:

The SMC law becomes:

The closed-loop system:

We get the system dynamics on the sliding surface:

Introduce an auxiliary variable:

Rewrite in the state space:

(new point..!!)

and

VII. EXAMPLE: STABILTY ANALYSIS OF AN INVERTED PENDULUM (cont)

Lyapunov function candidate:

(new point..!!) The Lyapunov derivative along trajectories:

VII. EXAMPLE: STABILTY ANALYSIS OF AN INVERTED PENDULUM (cont)

To find a positive definite and symmetric metrix P (new point..!!)

VII. EXAMPLE: STABILTY ANALYSIS OF AN INVERTED PENDULUM (cont)

Sliding Surface:

Domain of Attraction: