PARTIAL STABILIZABILITY OF CASCADED SYSTEMS
APPLICATIONS TO PARTIAL ATTITUDE CONTROL
Chaker Jammazi
Laboratoire d’Ing
´
eni
´
erie Math
´
ematique
Ecole Polytechnique de Tunisie. Tunisie
Azgal Abichou
Laboratoire d’Ing
´
eni
´
erie Math
´
ematique
Ecole Polytechnique de Tunisie. Tunisie
Keywords:
Brockett’s Condition, Partial stabilization, Backstepping, Partial Attitude Control.
Abstract:
In this work, the problem of partial stabilization of nonlinear control cascade systems with integrators is con-
sidered. The latter systems present an anomaly, which is the non complete stabilization via continuous pure-
state feedback, this is due to Brockett necessary condition. To cope with this difﬁculty we propose the partial
stabilization. For a given motion of a dynamical system, say x(t, x
0
, t
0
) = (y(t, y
0
, t
0
), z(t, z
0
, t
0
)),
the partial stabilization is the qualitative behavior of the y-component of the motion (i.e the asymptotic
stabilization of the motion with respect to y) and the z-component converges, relative to the initial vector
x(t
0
) = x
0
= (y
0
, z
0
). In the present work, we establish a new results for the adding integrators for partial
stabilization, we show that if the control systems is partially stabilizable, then the augmented cascade system
is partially stabilizable. Two applications are considered. The ﬁrst one is devoted to partial attitude stabiliza-
tion of rigid spacecraft. The second application is intended to the study of underactuated ship. Numerical
simulations are given to illustrate our results.
1 INTRODUCTION
Control problems involving cascaded systems have
attracted considerable attention in the past years. Un-
fortunately many controllable cascaded systems can
not stabilizable by pure state feedback laws this is
due to Brockett (Brockett, 1983) necessary condition.
Several solutions to overcome the limitation imposed
by Brockett condition have been presented in the liter-
ature knowing for example the time-varying method
developed by Morin (Morin et al., 1994). The con-
ception of time-varying feedback laws is an impor-
tant method to solving the stabilization problem, nev-
ertheless, the fact to introduce the time in these feed-
back laws product a oscillation of the system around
his point of equilibrium see for instance Pettersen and
Egeland (Pettersen and Egeland, 1996), (Morin et al.,
1994), (Beji et al., 2004), Pettersen and Nijmeijer (Pe-
tersen and Nijmeijer, 2001).
In this paper, we propose the partial stabilization by
smoothly state feedback laws. Partial stabilizability,
is the asymptotic stability with respect to most of the
system’s state, and the rest converges to same position
which depend to initial conditions.
The aim of the paper is to extend the well known
backstepping theorem to the case of partial stabiliz-
ability of nonlinear control systems. We have shown
that if the original system is partially stabilizable then
the cascade systems with integrators inherits the same
property, to this end we have developed the inver-
sion Lyapunov theorem for the stability with respect
to part. The theoretical result is applied to solving
two problems: The ﬁrst is the partial stabilization of
the rigid spacecraft with two controls, where we have
improve the Zuyev’s (Zuyev, 2001) result that the ve-
locity ω
3
of the third axes converges by using smooth
state feedback laws. The second problem treated is
the attitude of underactuated ship, we have construct
two smooth feedback laws that stabilize asymptoti-
cally ﬁve components and the sixth converges.
A numerical simulations are given to valid our re-
sults.
The paper is structured as follows: The next sec-
tion deals with some mathematical preliminaries. In
particular, the inversion of the Lyapunov theorem of
the stability with respect to part is demonstrated. The
backstepping techniques and partial stabilizability is
treated in section 3. In section 4 we give two appli-
cations for the backstepping result. Issues left for the
future investigation are discussed in the conclusions.
296
Jammazi C. and Abichou A. (2006).
PARTIAL STABILIZABILITY OF CASCADED SYSTEMS APPLICATIONS TO PARTIAL ATTITUDE CONTROL.
In Proceedings of the Third International Conference on Informatics in Control, Automation and Robotics, pages 296-301
DOI: 10.5220/0001205802960301
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