On the PE stabilization of time-varying systems:
open questions and preliminary answers
Antonio Lor´ ıa
†
Antoine Chaillet
†
Gildas Besanc ¸on
⋆
Yacine Chitour
†
Abstract— We address the following question: given a
double integrator and a linear control that stabilizes it
exponentially, is it possible to use the same control input in
the case that the control input is multiplied by a time-varying
term? Such question has many interesting motivations and
generalizations: 1) we can pose the same problem for an
input gain that depends on the state and time; 2) the
stabilization –with the same method– of chains of integrators
of higher order than two is fundamentally more complex
and has applications in the stabilization of driftless systems;
3) the popular backstepping method stabilization method
for systems with non-invertible input terms. The purpose
of this note is two-fold: we present some open questions
that we believe are significant in time-varying stabilization
and present some preliminary answers for a simple, yet
challenging case-study. Our solutions are stated in terms of
persistency of excitation.
I. OPEN QUESTIONS AND MOTIVATIONS
A. Linear time-varying systems
Let us consider the system ˙ x = u with x ∈ R. It is
evident that u = u
∗
with u
∗
= −x stabilizes the system
exponentially. What can be concluded for the integrator
with time-varying gain,
˙ x = g(t)u ? (1)
Which conditions need to be imposed on t → g(t) so
that (1) in closed loop with the same u
∗
be exponentially
stable?
The answer to this question can be found in the literature
on identification and adaptive control. For instance, from
the seminal paper [11] we know that for the system
˙ x = −P (t)x (2)
with x ∈ R
n
, P ≥ 0 piecewise continuous bounded,
and with bounded derivative, it is necessary and sufficient,
for global exponential stability, that P also be persistently
Antonio Lor´ ıa is with CNRS–LSS, Sup´ elec, 3, Rue Joliot Curie,
91192 Gif s/Yvette, France. E-mail: loria@lss.supelec.fr.
Antoine Chaillet is with LSS, Sup´ elec, 3, Rue Joliot Curie, 91192 Gif
s/Yvette, France. E-mail: chaillet@lss.supelec.fr.
Yacine Chitour is with Univerit´ e Paris XII, Orsay, France. E-mail:
chitour@lss.supelec.fr.
Gildas Besanc ¸on is with LAG-ENSIEG, BP 49, St. Martin d’Hˆ eres,
France. E-mail: Gildas.Besancon@inpg.fr
exciting (PE), i.e., that there exist µ> 0 and T> 0 such
that
t+T
t
ξ
⊤
P (τ )ξ ≥ µ (3)
for all unitary vectors ξ ∈ R
n
and all t ≥ 0.
The immediate conclusion for the system of interest,
i.e. (1), is that u
∗
= −x remains a globally exponentially
stabilizing control law if g(·) is non-negative, globally
Lipschitz, locally integrable and PE. An interpretation of
the stabilization mechanism can be given, in this case,
in terms of an “average”. Roughly speaking, one can
dare say that even though it is not the control action
u
∗
that enters the system for each t, this “ideal” control
does drive the system “in average”. For illustration, let
g(t) := sin(t)
2
then, the control action u = − sin(t)
2
x
which, in average
2
corresponds to u = −
1
2
x is tantamount
to applying u
∗
= −x, modulo a gain-scale that only affects
the rate of convergence but not the stabilization property
of u
∗
.
Of course the previous naive thinking relies largely on
the fact that we are dealing with a scalar system. Consider
the higher-order integrator ˙ x
(n)
= u; in the state-space it
takes the form:
˙ x
1
= x
2
(4a)
.
.
.
˙ x
i
= x
i+1
(4b)
.
.
.
˙ x
n
= u. (4c)
It is evident that, for a proper choice of k
i
, we have that
the control u = u
∗
with u
∗
= −
∑
n
i=1
k
i
x
i
renders the
closed-loop system globally exponentially stable. Consider
the following:
Question 1 Does u = g(t)u
∗
with g, ˙ g continuous and
bounded and g PE, stabilize (4) ?
Intuitively one may think that the global exponential sta-
bility (GES) of the closed loop is guaranteed, at least, for
2
We have taken as average of g(t), the function
1
T
T
0
g(t)dt, applied
to sin(t) with T = π.
Proceedings of the
44th IEEE Conference on Decision and Control, and
the European Control Conference 2005
Seville, Spain, December 12-15, 2005
ThA17.1
0-7803-9568-9/05/$20.00 ©2005 IEEE
6847