Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
2.6 On Adaptive High-gain Observers
− d(t) is continuously differentiable, and takes its values in a compact subset of R p ,
− both d(t) and d
dtd(t) are bounded,
− v : R + ↦→ R is a measurable function of t and is bounded (i.e. ∃µ >0, |v(t)| ≤ µ), and
− φ is a locally Lipschitz function in x and u, uniformly in d, over the domain of interest.
The Lipschitz constant is independent of d(t).
The observer proposed for such a system comes from earlier works such as [50].
Definition 23
Let us denote by z the estimated state. The observer is
where for i =1, 2
˙z = Az + Bφ(z, d, u) − Hi(Cz − y)
H ′
�
α1,i
i = θi
α2,i
θ2 , ...,
i
αn,i
θn i
The θi’s are small positive parameters such that 0 < θ1 < θ2, and the αi’s are chosen in such
a way that the roots of the polynomial:
have negative real parts.
�
.
s n + α1,is n−1 + α2,is n−2 + ... + αn,i =0
This observer is defined with two values for the high-gain parameter:
− θ1 corresponds to the fast state reconstruction mode,
− θ2 makes the observer much more efficient w.r.t. noise filtering.
The two sets of αi parameters can be chosen such that they are distinct from one another.
The switching scheme between the two values of θ has two main restrictions:
− the value of θ should change whenever an excessively large estimation error is detected,
and
− the value of θ should not change because of overshoots. During an overshoot, the
situation may arise when the estimated trajectory crosses the real trajectory, but has
not yet converged. Switching from θ1 to θ2, in this case, is not desirable.
Definition 24 (Switching scheme)
Let us define δ > 0 and Td > 0, two constant parameters and D =[−δ; δ]. The value of θ
is changed whenever 19 (z1 − y1) exits or enters D. When an overshoot occurs, the estimation
error may enter and exit quickly the domain D: convergence is not achieved yet. The large
value of the high-gain parameter is still needed. Those situations are handled by the use of a
delay timer. Priority is given to the high-gain mode (see Figure 2.2):
19 Matrix C gives us (Cz − y) =(z1 − y1).
28