Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
5.1 Multiple Inputs, Multiple Outputs Case
− y (t − d, z(t − d),s) denotes the output of the system of Subsection 5.1 computed over
the interval s ∈ [t − d; t] with z(t − d) as the initial state,
− y is the measured output of dimension ny.
Let us now denote n∗ = max � �
n1,n2, ..., nny , the size of the largest block, and define the
matrix:
⎛
1/θ
⎜
∆i = ⎜
⎝
n∗−ni 0
0
1/θ
... 0
n∗ .
−(ni−1)
. ..
. ..
. ..
.
0
0 . . . 0 1/θn∗ ⎞
⎟
⎠
(5.2)
−1
and ∆ is given by diag(∆1, ..., ∆ny). The definition of Qθ is the same as before:
Qθ = θ∆ −1 Q∆ −1 .
We need to provide a new definition for the matrix Rθ. To do so, we begin with the matrix:
⎛
θ
⎜
δθ = ⎜
⎝
n∗−n1 0 ... 0
0 θn∗ . −n2 .. .
. .. . .. 0
0 . . . 0 θn∗ ⎞
⎟
⎠
−nny
,
and set:
The initial state of the observer is:
− z(0) ∈ χ ⊂ R n ,
Rθ = 1
θ δθRδθ.
− S(0) ∈ Sn(+), the set of the symmetric positive definite matrices,
− θ(0) = 1.
Remark 48
This definition can also be used for single output systems. Since (ny = 1) ⇒ (n = n1 and
n ∗ = n) ⇒ (n ∗ − n1 = 0). Therefore:
− δθ =1,
− ∆ is the same as in Chapter 3, equation (3.3).
5.1.3 Convergence and Proof
The convergence theorems remain as presented in Chapter 3:
Theorem 49
For any time T ∗ > 0 and any ε ∗ > 0, there exist 0 < d < T ∗ and a function F (θ, Id) such
that for any time t ≥ T ∗ :
�x (t) − z (t)� 2 ≤ ε ∗ e −a (t−T ∗ )
where a>0 is a constant (independent from ε ∗ ).
95