Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
5.2 Continuous-discrete Framework
Remark 60
Notice that the matrix C is defined differently than in the previous systems. Details lie in
Appendix B, Remark 94 in particular.
5.2.2 Observer Definition
In the continuous-discrete setting the observer is defined by:
1. a set of prediction equations for t ∈ [(k − 1)δt,kδt[,
2. a set of correction equations at times t = kδt.
In the following we use the notations:
− z(t) is the estimated state for all t ∈](k − 1)δt,kδt[,
− zk(−) is the estimated state at the end of a prediction period,
− zk(+) is the estimated state after a correction step (i.e. at the beginning of a new
prediction period).
The prediction equations for t ∈ [kδt, (k + 1)δt[, with initial values zk−1(+), Sk−1(+)
are
�
˙z
˙S
=
=
A(u)z + b(z, u)
− (A(u)+b∗ (z, u)) ′
S − S (A(u)+b∗ (z, u)) − SQθS
(5.22)
where S0 is a symmetric definite positive matrix taken inside a compact subset of the form
aId ≤ S0 ≤ bId.
The correction equations are:
⎧
zk(+) = zk(−) − Sk(+) −1C ′
where
⎪⎨
⎪⎩
Sk(+) = Sk(−)+C ′
�i=d
r −1
θ Cδt
Ik,d = �yk−i − ˆyk−i�
i=0
2
θk = F (θk−1, Ik,d)
− x0 and z0 belongs to χ, a compact subset of R n ,
− θ(0) = θ0 = 1,
− r and Q are symmetric definite positive matrices 10 :
– Qθ = θ∆ −1 Q∆ −1 ,
– rθ = 1
θ r,
r −1
θ δt(Czk(−) − y)
10 r is written in capital letters to emphasize the fact that the system is single output.
105
(5.23)