THÈSE DE DOCTORAT Ecole Doctorale « Sciences et ...

S(t k+1 ) = S(t − k+1 ) + τ kC T C,
(4b)
such that, for all k ≥ 0,
λ(t k+1 ) = M(τ k , t k+1 )λ(t − k+1 ),
(4c)
ˆθ(t k+1 ) = ˆθ(t k ) + ∆(t k+1 )(y(t k+1 ) − Cˆx(t − k+1
)), (4d)
with, ∆(t k+1 )=
1+ǫ‖λ(t − k+1 )T C T Σ(t − k+1 )Cλ(t− k+1 )‖λ(t− k+1 )T C T
×Σ(t − k+1 ), and M(τ k, t k+1 ) = ( I n − ρτ k S −1 (t k+1 )C T C ) .
The notation ˆx ∈ R n and ˆθ ∈ R l respectively denote
the estimate of the state, x, and the estimate of the unknown
vector parameter θ, S(t) ∈ R n×n is the so-called
observation gain, λ ∈ R n , Σ ∈ C(R, R p×p ) is symmetric
positive definite for all t, and µ, ǫ ∈ R + , ρ ∈ [1, ∞)
are some design parameters. Note that the existence of
the inverse of matrix S will be ensured in the following.
The initial conditions are denoted ˆx 0 ∈ R n , S 0 > 0
symmetric, λ 0 ∈ R n and ˆθ 0 ∈ R l .
The observer (3a-d)-(4a-d) is composed of a predictor
part (3a-d) and a correction one (4a-d). During
[t k , t k+1 ), the state estimate ˆx has the same dynamics
than system (1) but using the estimate of the unknown
variable, ˆθ, that is kept constant (3d). As commonly
done for high gain observers, the matrix S is defined
as the solution of the differential equation (3b), except
that a term of the form C T C is usually added in order to
ensure S to be positive definite, see for example in [21].
At each sampling instants, observer trajectory is corrected
taking into account the available measure (4a).
Thank to (4b), conditions on the inputs and restriction
on τ, it will be shown that matrix S remains positive
definite. The dynamics of ˆθ are defined by a least mean
square law (4d), that will guarantee the convergence of
the estimate to the true value, under some persistent
excitation conditions. The usefulness of variable λ will
be shown in the stability analysis, like in [23,6].
Remark 1 If the parameter vector θ is equal to zero,
system (3a-d)-(4a-d) has the same form than in [21].
The dynamical equation satisfied by the state observation
error, e = x−ˆx, is, in view of (1), (3a-d) and (4a-d),
for k ≥ 0:
ė(t) = A(u)e(t) + φ(u)˜θ(t k ), t ∈ [t k , t k+1 ),
ǫ
(5a)
e(t k+1 )=(M(τ k , t k+1 )−λ(t k+1 )∆(t k+1 C))e(t − k+1 ), t = t k+1.
(5b)
The following persistent excitation condition is assumed
to hold for system (1) in order to guarantee the identifiability
of vector θ [23,6].
Hypothesis 2 Let λ(·) ∈ R n×s be the matrix solution
of the impulsive ordinary differential equation (3c)-(4c).
The matrix λ(·) is persistently excited so that there exist
k 1 ∈ N, δ > 0 and Σ(·) ∈ C([t 0 , ∞), R p×p ) a timedependent
bounded symmetric positive definite matrix,
∑ k+k1
j=k λT (t − j )CT Σ(t j )Cλ(t − j ) ≥ δI n. (6)
Before stating the main theorem, some preliminary definition
and results are required.
Definition 3 Consider system, for t ∈ [t 0 , ∞):
ẋ(t) = A(u)x(t),
y(t) = Cx(t),
(7)
and the transition matrix Ψ u (·, t 0 ) associated to system
(7), such that Ψ u (t 0 , t 0 ) = I n , the bounded input u is said
to be regularly persistent if there exist t 1 , t 2 ∈ [t 0 , ∞),
α ∈ R + , such that, for all t ≥ t 2 ,
∫ t+t1
t
Ψ T u (s, t 0)C T CΨ u (s, t 0 ) ds ≥ αI n . (8)
As mentioned in Introduction, regularly persistent inputs
are the class of inputs that guarantee the system to
be observable. For more details, see [8,9,14].
Remark 4 In practice, condition (8) can be checked offline,
when u(t) is known for all t ∈ [t 0 , ∞), using numerical
tools for approximating the integral term. Thus, coefficients
t 1 and α are obtained off-line in order to derive
a sampling period that satisfies the bound given in (9).
4.2 Technical results
The following proposition guarantees that matrix S is
positive definite for any t ∈ [t 0 , ∞), under conditions on
the types of inputs applied to the system and τ.
Proposition 5 Let u be a regularly persistent input for
system (1), for all µ ≥ ξ where ξ = 2 sup t≥t0 ‖A(u(t))‖,
if τ ≤ ¯τ, where ¯τ is the unique positive term such that
¯τ =
αe−ξ(t 1 +¯τ)
2ξ‖C T C‖(t , then, for all S(t 1+¯τ) 0) symmetric positive
definite, there exist constants β 1 , β 2 ∈ R + such that, for
t ∈ [t 0 , ∞),
β 1 I n ≤ S(t) ≤ β 2 I n .
The proof of Proposition 5 can be found in [21].
Remark 6 a) Since considered inputs belong to a compact
set ,D, and because A is continuous, ξ is well defined.
b) There always exists such a ¯τ ; indeed writing condition
¯τ =
αe−ξ(t 1 +¯τ)
2ξ‖C T C‖(t as 2ξ ∥ 1+¯τ) C T C ∥ (t1 + ¯τ)¯τ = αe −ξ(t1+¯τ) ,
loosely speaking it is clear that, for positive values of ¯τ,
3