Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
with ∆= diag � {1, 1
θ , ..., 1
θ n−1 } � ,
5.2 Continuous-discrete Framework
− the innovation, Ik,d (d ∈ N ∗ ), is computed over the time window [(k − d)δt; kδt] with:
– yk−i denotes the measurement at epoch k − i, and
– ˆyk−i denotes the output of system (5.21) at epoch k − i with z ((k − d)δt) as initial
value at epoch k − d.
Remark 61
In [24] we presented a different version of this observer. In this paper, the adaptation of
the high-gain parameter was determined during the prediction steps via a differential equation.
We changed our strategy with respect to the peculiar manner in which the continuous discrete
version of the observer works.
Recall that the estimation is a sequence of continuous prediction periods followed by discrete
correction steps when a new measurement/observation is available. Consequently, since
innovation is based on the measurements available, it is computed at the correction steps.
Suppose now that θ is adapted via a differential equation. It starts to change during the
prediction step following the computation of a large innovation value, and reaches θ1 after
some time. If alternatively θ is adapted at the end of the correction step, directly after the
computation of innovation, then it reaches θ1 much faster.
We opt for the strategy in which θ is adapted at the end of the correction step 11 .
An advantage brought about by this approach is that now we may remove one of the
assumptions on the adaptation function: “there exists M such that | F
θ 2 | 0 and any ɛ ∗ > 0, there exist
− two real constants µ and θ1,
− d ≥ n − 1 ∈ N ∗ , and
− an adaptation function F(θ, I),
such that for all δt sufficiently small (i.e. 2θ1δt