Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
2.6 On Adaptive High-gain Observers
Those conditions, imposed on the vector fields fi, appear to be very technical, but are in
fact necessary in order to prove the convergence of the observer. Moreover, the importance
of the functionΓ( u, y) has not to be underestimated as it is of the utmost importance in the
definition of the update law of the high-gain parameter.
Definition 20
The high-gain observer with updated gain is defined by the set of equations:
�
˙z = A(y)x + b(y, z2, .., zn, , u)+θLA(y)K � z1−y
θb �
�
�
˙θ = θ ϕ1(ϕ2 − θ)+ϕ3Γ(u, y) 1 + �n j=2 |ˆx j| vj
��
(2.15)
where L = diag(L b ,L b+1 , ..., L b+n−1 ).
Theorem 21
Consider the system (2.13) and the associated observer (2.15). It is is then possible to
choose the parameters ϕ1, ϕ2, ϕ3 (with ϕ2, ϕ3 high enough) such that for any L(0) ≥ ϕ2 the
estimation error e(t) =z(t) − x(t) is bounded as follows:
|L −1 e(t)| ≤ β1(L(0) −1 e(0),t)+sup s∈[0,t[γ1
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for all t ∈ [0,Tu[. Moreover L statisfies the relation:
⎛�⎛
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�� � �
�
e(0)
⎜�⎜
⎜�⎜
L(t) ≤ 4ϕ2 + β2
,t + sups∈[0,t[γ2 ⎜�⎜
L(0)
⎝�⎝
�
�
δ(s)
ϕ2
α(y(s))δy(s)
δ(s)
ϕ2
α(y(s))δy(s)
Γ(u(s),y(s))
x(s)
where β1 and β2 are KL functions, and γ1, γ2 are functions of class K.
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⎞�⎞
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⎟�⎟
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This work uses a special form of the phase variable representation (2.2) of J-P. Gauthier
et al. and therefore applies to observable systems in the sense of Section (2.2) above. The
observer (2.15) has roughly the same structure as a Luenberger high-gain observer except
for the fact that the correction gain is given as a function of the output error. The update
function is determined by a function that bounds the incremental rate of the vector field
b(y, x, u)(this is the part written in bold in (2.15)).
The strategy adopted here isn’t based on a global Lipschitz vector field, which implies
the off-line search and tuning of the upper bound (or the value) of the high-gain parameter
θ. Instead, the observer tunes itself as a consequence of the adaptation function. However
the function that drives the adaptation may not be that easy to find.
The idea is quite similar to that of E. Bullinger and F. Allgőwer [35], with the difference
being that in their case the adaptation is driven by the output error and in the case of L.
Praly et al. the adaptation is model dependent.
Theorem 21 states 17 that the observer (2.15) together with its adaptation function gives,
at least for bounded solutions, an estimation error converging to a ball centered at the origin
with a radius that depends on the asymptotic L ∞ -norm of the disturbances δ and δy. This
17 The precise and complete theorem appear in the article [16], Theorem 1.
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