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. O PEN QUESTIONS AND MOTIV ATIONS
A. Linear time-varying systems
Let us consider the system ˙x=uwithx∈R.I ti s
evident that u=u∗withu∗=−xstabilizes 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/mapsto→g(t)so
that (1) in closed loop with the sameu∗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)
withx∈Rn,P≥0piecewise continuous bounded,
and with bounded derivative, it is necessary and sufficient ,
for global exponential stability, that Palso 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’H ˆeres,
France. E-mail: Gildas.Besancon@inpg.frexciting (PE), i.e., that there exist µ>0andT>0such
that/integraldisplayt+T
tξ⊤P(τ)ξ≥µ (3)
for all unitary vectors ξ∈Rnand all t≥0.
The immediate conclusion for the system of interest,
i.e.(1), is that u∗=−xremains 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): =s i n ( t)2then, the control action u=−sin(t)2x
which, in average2corresponds to u=−1
2xis tantamount
to applying u∗=−x, modulo a gain-scale that only affects
the rate of convergence but not the stabilization property
ofu∗.
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:
˙x1=x2 (4a)
...
˙xi=xi+1 (4b)
...
˙xn=u. (4c)
It is evident that, for a proper choice of ki, we have that
the control u=u∗withu∗=−/summationtextn
i=1kixirenders the
closed-loop system globally exponentially stable. Consider
the following:
Question 1 Does u=g(t)u∗withg,˙gcontinuous and
bounded and gPE, stabilize (4) ?
Intuitively one may think that the global exponential sta-
bility (GES) of the closed loop is guaranteed, at least, for
2We have taken as average of g(t), the function1
T/integraltextT
0g(t)dt, applied
tosin(t)withT=π.Proceedings of the
44th IEEE Conference on Decision and Control, and
the European Control Conference 2005
Seville, Spain, December 12-15, 2005ThA17.1
0-7803-9568-9/05/$20.00 ©2005 IEEE 6847 a more restricted choice of kiand a particular class of PE
functions g.
While for the case of n=1 the answer is positive, the
general case of n>1is fundamentally different and, in
view of the available tools from the literature of adaptive
control, a proof (or disproof) of the conjecture above is far
from evident.
An extension of Question 1 concerns the analysis of
chain-form systems; more precisely when ˙xi=φ(t)xi+1
for all i<n and˙xn=uwithφ(t)bounded and with
bounded (continuous) derivative. This problem is not of
pure theoretical interest but finds motivations in the control
of nonholonomic systems, both in trajectory tracking and
set-point stabilization. See the thorough discussions in [5],
[7], [13]. For instance, it has been shown in [7] that such
system in closed loop with u=−k1φ(t)x1−k2x2−
k3φ(t)x3−..., and an appropriate choice of the gains
ki, is uniformly globally exponentially stable provided
thatφ(t)is PE, bounded and with bounded derivative.
Implicitly, a similar condition has been used in [13] to show
(non-uniform) asymptotic stability. Further generalizations
following similar guidelines have been obtained for chains
of nonholonomic integrators in [8].
Further natural extensions of these stabilization prob-
lems for chains of integrators concern the stabilization of
systems with drift. Consider the system
˙x=Ax+B(t)u, (5)
where Ais marginally stable, and the system ˙x=Ax+Bu
withBconstant . One may pose the question: under which
conditions on B(t)and possibly on the pair (A,B), does
the control u=u∗withu∗=−Kxsuch that ˙x=(A−
BK)xis GES, stabilize (5) exponentially. In the particular
case that Ais skewsymmetric, the answer seems at hand
in the form of a PE condition on B; for instance, we may
impose that P(t): =B(t)Ksatisfy all the conditions from
[11] and follow a similar reasoning.
Consider now the following problem.
Question 2 LetAbeunstable , is (5) globally exponen-
tially stabilizable by linear time-invariant feedback u∗=
−B(t)⊤Kx, provided that B(t)is PE? If not in general,
is it true at least for particular choices of the gain Kand
PE functions B?
In this note we address Question 2 for the particular case
of the double integrator, cf. Section II-A.B. Nonlinear time-varying systems
1) Nonlinear drift
An interesting nonlinear example that may be viewed
as a generalization of system (5) appears in the control
of spacecrafts with magnetic actuators, where the dynamic
system has the form:
˙ω=S(ω)ω+g(t)u,
where Sdesignates the following matrix:
S=⎡
⎣0ω3−ω2
−ω30ω1
−ω1ω20⎤
⎦
andg(t)is a time-varying matrix which is rank deficient
(i.e.rank{g(t)}<3) for any fixed tbut it is PE. The global
exponential stabilization problem for the spacecraft with
magnetic actuators was solved in [2] using PE arguments.
2) Nonlinear input gain
Let us consider again the simple integrator ˙x=u.
Assume now that the integrator has an input gain that
depends on the state and time, i.e., consider the one-
dimensional driftless system
˙x=φ(t, x)u (6)
where φ:R≥0×R→R≥0is locally Lipschitz in x
uniformly in tand piecewise continuous in t. A natural
question is whether the control input u∗=−x, which
stabilizes the pure integrator exponentially, still stabilizes
the “integrator” (6). Further, is this true for other stabilizing
controls u∗(t, x)?
Following up the results from [10] we obtain that,
when F(t, x): =φ(t, x)u∗(t, x)is continuous and locally
Lipschitz in xuniformly in t, any control input u=
u∗(t, x)that stabilizes ˙x=ualso stabilizes uniformly
asymptotically the system (6) only if F(t, x)is Uδ-PE with
respect to x; we recall the definition of the latter below.
Sufficiency also holds, under certain regularity conditions,
if˙x=F(t, x)is uniformly globally stable (UGS).
For the system ˙x=F(t, x)define x=:col[x1,x2],
correspondingly n=:n1+n2, and the set D1:=
(Rn1\{0})×Rn2.
Definition 1 A function φ(·,·)where t/mapsto→φ(t, x)is locally
integrable, is said to be uniformly δ-persistently exciting
(Uδ-PE) with respect to x1if for each x∈D1there exist
δ>0,T>0andµ>0s.t.∀t∈R≥0,
|z−x|≤δ=⇒/integraldisplayt+T
t|φ(τ,z)|dτ≥µ. (7)
/square
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