Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
2.6.6 E. Busvelle and J-P. Gauthier
2.6 On Adaptive High-gain Observers
The article [38] of E. Busvelle and J-P Gauthier propose an observer that is high-gain at
time 0 and then decreases toward 1. In a nutshell, the observer evolves from a pure high-gain
mode that ensures convergence to an extended Kalman filter configuration that efficiently
smooths the noise. To achieve this, the high-gain parameter is allowed to decrease and the
convergence is proven. This article is, in some sense, the starting point of the present Ph.D.
work.
Definition 28
The high-gain and non high-gain extended Kalman filter, for a system as defined
in Section 2.4, is given by the three equations:
⎧
⎪⎨
dz
dt = A(u)z + b(z, u) − S(t)−1C ′
R−1 (Cz − y(t))
dS
dt = −(A(u)+b∗ (z, u)) ′
S − S(A(u)+b∗ (z, u)) + C ′
R−1C − SQθS (2.19)
⎪⎩
dθ
dt
= λ(1 − θ)
where Qθ = θ 2 ∆ −1 Q∆ −1 , ∆ = diag( � 1, 1
θ
, ..., ( 1
θ )n−1� ), Q and R as in (2.9), and λ is a
positive parameter.
If θ (0) = 1, then θ (t) ≡ 1 and (2.19) is nothing else than a classical extended Kalman
filter applied in a canonical form of coordinates. Therefore it may not converge, depending
on the initial conditions of the system.
If λ =0and θ (0) = θ0 are large, then θ (t) ≡ θ0 remains large and (2.19) is a high-gain
extended Kalman filter as defined above in Equation (2.9).
The idea in (2.19) is to set θ (0) to a sufficiently large value, and to set λ to a sufficiently
small value such that the observer converges exponentially quickly at the beginning. The
estimated state reaches the vicinity of the real trajectory before θ becomes too small and the
local convergence of the extended Kalman filter guarantees that it will remain close to the
state of the system.
Theorem 29
For any ε ∗ > 0, there exists λ0 such that for all 0 ≤ λ ≤ λ0, for all θ0 large enough, for
all S0 ≥ c Id, for all χ ⊂ R n , χ a compact subset, for all ε0 = z0 − x0,with (z0,x0) ∈ χ 2 , with
x0 ∈ χ the following estimation holds for all t ≥ 0:
Moreover the short term estimate
||ε(t)|| 2 ≤ ||ε (0) || 2 ε ∗ e −at .
||ε(t)|| 2 ≤ ||ε(0)|| 2 θ(t) 2(n−1) e −(a1θ(T )−a2)t
holds for all T>0 and for all 0 ≤ t ≤ T , for all θ0 sufficiently large. The scalars a1 and a2
are positive constants.
This theorem demonstrates that the observer converges for any initial error. Nevertheless
it is clearly not a persistent observer since after some time it is more or less equivalent to
an extended Kalman filter because θ (t) is close to one. In order to make it persistent, the
authors propose to use several such observers, each of them being initialized at different times
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