Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
2.6 On Adaptive High-gain Observers
say that there exists an integer N ≥ 1 such that
rank ∂
⎛
h(x, uk)
⎜
h(., uk) ◦ f(x, uk)
⎜
∂x ⎜
⎝
.
h(., uk+N−1) ◦ h(., uk+N−2) ◦ h(., uk) ◦ f(x, uk)
⎞
⎟
⎠
|x=x k =n
for all xk ∈ K, and N-tuple of controls (uk, . . . , uk+N−1) ∈ U (where K and U are two
compact subsets of R n and (R nu ) N , respectively),
2. Fk, Hk are uniformly bounded matrices, and F −1
k
Let us define:
1. the time varying matrices α and β by:
exists.
(xk+1 − z k+1/k) = βkFk(xk − zk)
αk+1ek+1 = Hk+1(xk+1 − z k+1/k),
2. the weighting matrices Rk and Qk such that, there exists a parameter 0 < ζ< 1 such
that
�
and that 24
sup |(αk+1)i − 1| ≤
i=1,...,ny
sup |(βk)j| ≤
j=1,...,n
σ(Rk+1)
σ(Hk+1Pk+1/kH ′
k+1 + Rk+1)
�
(1 − ζ)σ(FkPkF ′
k
+ Qk)
σ(F ′
k )σ(Pk)σ(Fk)
Then, the observer (2.18) ensures local asymptotic convergence:
lim
k→∞ (xk − zk) = 0.
In [30] the authors propose to choose Qk = γe ′
k ekIn + δIn where ek = h(z k/k−1) − yk is
the innovation. γ is taken sufficiently large, and δ sufficiently small such that the inequalities
of Assumption (4) are met for all values of ek. Special attention must be given to the fact
that Qk should not be set to a high value when the innovation is small 25 .
The adaptation strategy we propose in Chapter 3 is based on the same kind of strategy
but uses the innovation over a sliding horizon as the quality measurement for the estimation.
By doing this we can link the adaptive scheme to the proof of convergence of the observer,
which is not done for the observer of this section.
24 σ and σ denotes respectively the maximum and minimum singular values.
25 The article of L. Z. Guo and Q. M. Zhu [61], propose a hybrid strategy based on this subsection’s observer.
They use the structure proposed by M. Boutayeb et al together with a neural network approach.
34
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2
.
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2
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