Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
5.2 Continuous-discrete Framework
preserved provided that the sample time δt is small enough. (Both examples and counterexamples
can be found in this article (see also [14] on the same topic)).
Let us state the main theorem of [15].
Theorem 59
Assume that a nonlinear system is observable for every input u(.) and uniformly infinitesimally
observable 9 , then for all M>0, there exists a δ0 > 0 such that the associated
δ−sampled system is observable for all δ ≤ δ0 and all M, D−bounded input u δ .
5.2.1 System Definition
Let us consider the continuous-discrete version of the multiple input, single output system of
Chapter 3 (Equation 3.2):
�
dx
dt = A(u(t))x + b(x(t),u(t))
(5.21)
yk = Cxk
where
− δt is the constant sampling time of the measurement procedure,
− x (t) ∈ χ ⊂ R n , χ compact, and xk = x(kδt), k ∈ N,
− u(t) ∈ Uadm ⊂ R nu bounded, and uk = u(kδt), k ∈ N,
− y (t) ∈ R, and yk = y(kδt), k ∈ N.
The matrices A (u) and C (u) are defined by:
⎛
0
⎜
A(u) = ⎜
.
⎝
a2 (u)
0
0
a3 (u)
. ..
· · ·
. ..
. ..
0
⎞
0
⎟
. ⎟
0 ⎟
an (u) ⎠
0 · · · 0
C = � 1 0 · · · 0 �
with 0