On the Stability of Sliding Mode Control for a Class of
Underactuated Nonlinear Systems
Sergey G. Nersesov, Hashem Ashrafiuon, and Parham Ghorbanian
Abstract— A system is considered underactuated if the num-
ber of the actuator inputs is less than the number of degrees of
freedom for the system. Sliding mode control for underactuated
systems has been shown to be an effective way to achieve system
stabilization. It involves exponentially stable sliding surfaces so
that when the closed-loop system trajectory reaches the surface
it moves along the surface while converging to the origin.
In this paper, we present a general framework that provides
sufficient conditions for asymptotic stabilization by a sliding
mode controller for a class of underactuated nonlinear systems
with two degrees of freedom. We show that, with the sliding
mode controller presented, the closed-loop system trajectories
reach the sliding surface in finite time. Furthermore, we develop
a constructive methodology to determine exponential stability
of the reduced-order closed-loop system while on the sliding
surface thus ensuring asymptotic stability of the overall closed-
loop system and provide a way to determine an estimate of the
domain of attraction. Finally, we implement this framework on
the example of an inverted pendulum.
I. I NTRODUCTION
Sliding mode control has been shown to be a robust
and effective control approach for stabilization of nonlinear
systems [1]. The approach is based on defining exponentially
stable (sliding) surfaces as a function of the system states
and using the Lyapunov theory to ensure that all closed-
loop system trajectories reach these surfaces in finite time.
Since the closed-loop system dynamics on the surfaces are
exponentially stable, the system trajectories slide along the
surfaces until they reach the origin. Sliding mode controllers
have been successfully developed for a variety of problems
involving underactuated systems [2], [3], [4]. The authors in
[5], [6] introduced a second order sliding mode control ap-
proach with application to the inverted pendulum. The sliding
mode control law proposed for underactuated systems in [4]
categorizes the stabilization problem based on equilibrium
manifold. In their work, the authors present a control law
guaranteeing that all closed-loop system trajectories reach
the proposed sliding surfaces in finite time. However, the
approach in [4] establishes stability of the closed-loop system
dynamics on the sliding surfaces only by linearization and
does not allow to estimate the domain of attraction on the
sliding surface guaranteeing exponential convergence of the
system trajectories.
In this paper, we show finite-time convergence of the
closed-loop system trajectories to a sliding surface using a
Lyapunov function approach presented in [7]. Furthermore,
we develop a methodology to determine a Lyapunov function
for a class of two degree-of-freedom underactuated nonlinear
systems that guarantees exponential stability of the reduced
This research was supported in part by the Office of Naval Research
under Grant N00014-09-1-1195.
The authors are with the Department of Mechanical
Engineering, Villanova University, Villanova, PA 19085-
1681, USA (sergey.nersesov@villanova.edu;
hashem.ashrafiuon@villanova.edu;
parham.ghorbanian@villanova.edu).
order closed-loop system while on the sliding surface. De-
termining such Lyapunov function allows to estimate the
domain of attraction on the sliding surface guaranteeing that
all closed-loop system trajectories entering this domain will
converge exponentially to the origin while sliding along the
surface. Finally, we apply this framework to the inverted
pendulum previously studied in [4] to prove stabilization by a
sliding mode controller and to show various estimates of the
domain of attraction for different sets of the sliding surface
parameters.
II. SLIDING MODE CONTROL FOR UNDERACTUATED
NONLINEAR SYSTEMS
The system is considered underactuated if the number of
actuator inputs m is less than the number of degrees of
freedom n. In this section, we present the sliding mode
control framework for underactuated nonlinear dynamical
systems and develop a general methodology to establish
exponential stability of the closed-loop system during the
sliding phase for two degree-of-freedom Lagrangian systems.
To elucidate this approach, consider a two degree-of-freedom
Lagrangian dynamical system with a generalized position
vector partitioned as q =[q
a
,q
u
]
T
, where q
a
∈ R is the
actuated coordinate, q
u
∈ R is the unactuated coordinate,
and q ∈D⊆ R
2
. Then the equations of motion for this
system can be written as
M
aa
(q(t)) M
au
(q(t))
M
au
(q(t)) M
uu
(q(t))
¨ q
a
(t)
¨ q
u
(t)
=
f
a
(q(t), ˙ q(t)) + u(t)
f
u
(q(t), ˙ q(t))
, (1)
where u(t) ∈ R is the control input, f (q, ˙ q)
[f
a
(q, ˙ q),f
u
(q, ˙ q)]
T
is the vector of Coriolis, centrifugal,
conservative, and non-conservative forces, and ¨ q [¨ q
a
, ¨ q
u
]
T
is the acceleration vector. The inertia matrix is accordingly
partitioned into positive definite elements M
aa
: R
2
→ R
+
and M
uu
: R
2
→ R
+
, and an off-diagonal element M
au
:
R
2
→ R. We can solve (1) for the accelerations as
¨ q
a
(t) = (M
′
aa
(q))
−1
(f
′
a
(q, ˙ q)+ u(t)), (2)
¨ q
u
(t) = (M
′
uu
(q))
−1
[f
′
u
(q, ˙ q)
−M
au
(q)(M
aa
(q))
−1
u(t)], (3)
where
M
′
aa
(q) = M
aa
(q) − M
au
(q)(M
uu
(q))
−1
M
au
(q),
f
′
a
(q, ˙ q) = f
a
(q, ˙ q) − M
au
(q)(M
uu
(q))
−1
f
u
(q, ˙ q),
M
′
uu
(q) = M
uu
(q) − M
au
(q)(M
aa
(q))
−1
M
au
(q),
f
′
u
(q, ˙ q) = f
u
(q, ˙ q) − M
au
(q)(M
aa
(q))
−1
f
a
(q, ˙ q).
We define the sliding surface as a linear combination of
the actuated and unactuated position and velocity variables
s(q, ˙ q) = α
a
˙ q
a
+ λ
a
q
a
+ α
u
˙ q
u
+ λ
u
q
u
= α
a
˙ q
a
+ α
u
˙ q
u
+ s
r
(q), (4)
2010 American Control Conference
Marriott Waterfront, Baltimore, MD, USA
June 30-July 02, 2010
ThB09.4
978-1-4244-7425-7/10/$26.00 ©2010 AACC 3446