Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
5.1 Multiple Inputs, Multiple Outputs Case
The matrices A(u) and C(u) are given by:
⎛
⎞
A1(u) 0 . . . 0
⎜ 0 A2(u) . . . 0 ⎟
A = ⎜
⎝
.
. .. . ..
⎟
. ⎠
0 . . . 0 Any(u)
, Ai(u)
⎛
0 α
⎜
= ⎜
⎝
2 i . . . 0
0 . ⎞
. .
. ..
⎟
0 ⎟
.
. .. . ..
⎟ ni α ⎠
i
0 . . . 0 0
and the matrix C(u) is the generalization:
⎛
C1(u)
⎜ 0
C = ⎜
⎝ 0
0
C2(u)
...
...
...
. ..
0
0
0
⎞
⎟
⎠
0 0 ... Cny(u)
, Ci = � α1 i (u) 0 ... 0 � .
Finally the vector field b(x, u) is defined as:
⎛ ⎞
b1(x, u)
⎜
b(x, u) = ⎜ b2(x, u) ⎟
⎝ ... ⎠
bny(x, u)
, bi(x,
⎛
b
⎜
u) = ⎜
⎝
1 i (x1i ,u)
b2 i (x1i ,x2i ,u)
...
b ni
i (x,u)
⎞
⎟
⎠ .
Remark 47
The very last component of each element bi(., .) of the vector field b is allowed to depend
on the full state. As one can see, the linear part is a Brunovsky form of observability: this
is clearly a generalization of system (3.2). Nevertheless not every observable system can be
transformed into this form.
5.1.2 Definition of the Observer
In order to preserve the convergence result, apply a few modifications to the observer. There
are mainly three points to consider
− the definition of innovation is adapted, rather trivially, to a multidimensional output
space;
− the matrix∆ , generated with the parameter θ is not the generalization one would
imagine initially;
− and the definition of the matrix Rθ must remain compatible with the matrix∆.
Let us first recall the equations of the observer:
⎧
⎨
⎩
dz
dt = A(u)z + b(z, u) − S−1C ′
dS
dt
(Cz − y(t))
= −(A(u)+b∗ (z, u)) ′
S − S(A (u)+b∗ (z, u)) + C ′
R −1
dθ
dt
θ C − SQθS
= F(θ, Id (t))
where Q and R are symmetric positive definite matrices of dimension (n × n) and (ny × ny)
respectively. The innovation at time t is:
� t
Id(t) = �y(s) − y (t − d, z(t − d),s) � 2 ny R ds
where:
t−d
R −1
θ
94