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Adaptative high-gain extended Kalman filter and applications

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tel-00559107, version 1 - 24 Jan 2011<br />

2.6 On Adaptive High-<strong>gain</strong> Observers<br />

− d(t) is continuously differentiable, <strong>and</strong> takes its values in a compact subset of R p ,<br />

− both d(t) <strong>and</strong> d<br />

dtd(t) are bounded,<br />

− v : R + ↦→ R is a measurable function of t <strong>and</strong> is bounded (i.e. ∃µ >0, |v(t)| ≤ µ), <strong>and</strong><br />

− φ is a locally Lipschitz function in x <strong>and</strong> u, uniformly in d, over the domain of interest.<br />

The Lipschitz constant is independent of d(t).<br />

The observer proposed for such a system comes from earlier works such as [50].<br />

Definition 23<br />

Let us denote by z the estimated state. The observer is<br />

where for i =1, 2<br />

˙z = Az + Bφ(z, d, u) − Hi(Cz − y)<br />

H ′<br />

�<br />

α1,i<br />

i = θi<br />

α2,i<br />

θ2 , ...,<br />

i<br />

αn,i<br />

θn i<br />

The θi’s are small positive parameters such that 0 < θ1 < θ2, <strong>and</strong> the αi’s are chosen in such<br />

a way that the roots of the polynomial:<br />

have negative real parts.<br />

�<br />

.<br />

s n + α1,is n−1 + α2,is n−2 + ... + αn,i =0<br />

This observer is defined with two values for the <strong>high</strong>-<strong>gain</strong> parameter:<br />

− θ1 corresponds to the fast state reconstruction mode,<br />

− θ2 makes the observer much more efficient w.r.t. noise <strong>filter</strong>ing.<br />

The two sets of αi parameters can be chosen such that they are distinct from one another.<br />

The switching scheme between the two values of θ has two main restrictions:<br />

− the value of θ should change whenever an excessively large estimation error is detected,<br />

<strong>and</strong><br />

− the value of θ should not change because of overshoots. During an overshoot, the<br />

situation may arise when the estimated trajectory crosses the real trajectory, but has<br />

not yet converged. Switching from θ1 to θ2, in this case, is not desirable.<br />

Definition 24 (Switching scheme)<br />

Let us define δ > 0 <strong>and</strong> Td > 0, two constant parameters <strong>and</strong> D =[−δ; δ]. The value of θ<br />

is changed whenever 19 (z1 − y1) exits or enters D. When an overshoot occurs, the estimation<br />

error may enter <strong>and</strong> exit quickly the domain D: convergence is not achieved yet. The large<br />

value of the <strong>high</strong>-<strong>gain</strong> parameter is still needed. Those situations are h<strong>and</strong>led by the use of a<br />

delay timer. Priority is given to the <strong>high</strong>-<strong>gain</strong> mode (see Figure 2.2):<br />

19 Matrix C gives us (Cz − y) =(z1 − y1).<br />

28

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