THÈSE DE DOCTORAT Ecole Doctorale « Sciences et ...

ensured. The work of [12] has been extended to other
classes of systems. In [21], observers for a MIMO class
of state affine systems where the dynamical matrix depends
on the input, have been designed when the inputs
are regularly persistent. In [3], a similar method has
been used for a larger class of systems and applied to
the observation of emulsion copolymerization process.
The observation of a class of systems with output injection
has been treated in [22]. Recently, in [15], a high
gain continuous-discrete observer has been developed
but using constant observation gains.
On the other hand, it appears that for a number of
physical processes some parameters may not be known,
that makes existing results on continuous-discrete time
observers non-applicable. In this paper, the observation
of the class of systems studied in [21] is extended
to the adaptive case. Early developments of adaptive
observers were made in [16,17] for linear systems. Adaptive
observation of nonlinear systems have been investigated
using different techniques that basically rely on
linear adaptive algorithms, through coordinate change
or output injection for instance, see [4,5,10,19,23]. In
[24], a unified interpretation of the latter references has
been proposed that emphasizes their characteristics.
In [6], adaptive observation for state affine systems in
continuous-time is discussed and the work [23] is extended.
In this study, the approach developed in [6]
on extended Kalman filters for a class of MIMO linear
time-variant, is adapted to the sampled measures problem
at the difference that, here, the estimation law is
based on discrete-time adaptive techniques.
After having defined the class of systems considered
and recalled the main objective, a class of adaptive
continuous-discrete observers is designed. Assuming
some persistent excitation conditions and some hypotheses
on the system output sampling times to hold,
the global exponential stability of the observation error
is proved. Finally, a simulation example is performed
which illustrates the design procedure.
2 Nomenclature
First some mathematical notation is introduced. Let
R def
def
= (−∞, ∞), R + = (0, ∞), R + def
0 = [0, ∞) and defined
the Euclidean norm ‖·‖. For p, q, n, m ∈ N, R p×q
represents the set of real matrices of order p × q and
I p ∈ R p×p stands for the identity matrix of order p×p. If
X ⊂ R p×q and Y ⊂ R n×m , C(X, Y) denotes the space of
all continuous functions mapping X → Y. If P ∈ R p×p ,
P > 0 means that P is positive definite. The notation
‖P ‖, for P ∈ R p×q , represents the L 2 -norm of P. For
A : R → R p×q and t ∈ R, the notation A(t − ) denotes
the left limit of A at instant t, if it exists. In all this
study, the initial time is called t 0 ∈ R.
3 Problem statement
The following class of systems is considered, for t ∈
[t k , t k+1 ), k ≥ 0,
ẋ(t) = A(u)x(t) + b(u) + φ(u)θ,
y(t k ) = Cx(t k ),
(1)
where x ∈ R n is the instantaneous state vector, u ∈ D ⊂
R m the input vector ( D is compact), y ∈ R p is the output
vector, θ ∈ R l is a vector of unknown constant parameters,
A ∈ C(R m , R n×n ), b ∈ C(R m , R n ), C ∈ R p×n ,
φ ∈ C(R m , R n×l ) are known, with n, m, p, l ∈ N and
x 0 = x(t 0 ). The notation (t k ) k≥0 represents a strictly
increasing sequence such that lim k→∞ t k = ∞ that models
the sampling times. The maximum sampling step is
denoted τ = max k≥0 (τ k ), where τ k = t k+1 − t k .
Note that the class of systems (1) contains systems of
the form, for k ≥ 0,
ẋ(t) = A(t, u)x(t) + b(t, u) + φ(t, u)θ, t ∈ [t k , t k+1 ),
y(t k ) = Cx(t k ), t = t k+1 ,
(2)
since the dependence on u can be considered like a time
dependence.
The main objective of this paper is to synthesize a global
exponential adaptive observer, as defined in Section 4.3,
for system (1).
4 Observer design and stability study
4.1 Observation structure
The proposed observer can be viewed as an extension to
the adaptive case of the structure developed in [21]. An
auxiliary variable, λ, which plays a key role in the convergence
of parameters estimator, is notably introduced
like in [23,6].
For k ≥ 0, t ∈ [t k , t k+1 ),
˙ˆx(t) = A(u)ˆx(t) + b(u) + φ(u)ˆθ(t k ),
Ṡ(t) = −A(u) T S(t) − S(t)A(u) − µS(t),
for t = t k+1 ,
˙λ(t) = A(u)λ(t) + φ(u),
˙ˆθ(t) = 0,
ˆx(t k+1 ) = ˆx(t − k+1 ) + (λ(t k+1 )∆(t k+1 )
+ρτ k S −1 (t k+1 )C T )(y(t k+1 ) − Cˆx(t − k+1 )),
(3a)
(3b)
(3c)
(3d)
(4a)
2