Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
2.5 High-gain Observers
2.5 High-gain Observers
Early descriptions of high-gain observers can be found in multiple references including
J-P. Gauthier et al., [54], F. Deza et al., [46, 47], A. Tornambé, [111], H. K. Khalil et al. and
[50].
The first high-gain observer construction that we present in this section is the Luenberger
style prototype algorithm of [54]. Afterwards we define the high-gain extended Kalman filter.
Its structure is quite similar to that of the extended Kalman filter, which is well known and
well used in engineering [58, 60]. In our case, it has been adapted to the observability normal
form.
High-gain observers are embedded with a structure based on a fixed scalar parameter
(denoted θ) which allows us to prove that the estimation error decays to zero, exponentially
at a rate that depends on the value of θ. When the high-gain parameter is taken equal to 1,
high-gain observers reduce to their initial nonlinear version.
Definition 13
We suppose that all the ai(u) coefficients of the normal form (2.7) are equal to 1. The
classical (or Luenberger) high-gain observer is defined by the equation
dz
dt = Az + b(z, u) − Kθ (Cz − y(t)) (2.8)
where Kθ = ∆K with 11 ∆ = diag �� θ,θ 2 , ...,θ n�� and K is such that (A − KC) is Hurwicz
stable.
It is in fact not important that the ai(u) coefficients equal 1 or any other non zero constant
since a change of coordinates brings us back to the situation where ai = 1.
On the other hand, it is of the utmost importance that the coefficients do not depend on
either u(t) or time:
− the correction gain K is computed off-line (i.e. for u = u ∗ ), (A(u) − KC(u)) may
not remain stable for u �= u ∗ and the convergence of the observer is not guaranteed
anymore.
− in full generality: (A(t) − K(t)C(t) stable ∀t >0), � (the system is stable).
The convergence of the high-gain Luenberger observer is expressed in the following theorem.
Theorem 14
For any a>0, there is a large enough θ > 1 such that ∀(x0,z0) ∈ (χ × χ), we have
for some polynomial k of degree n.
�z(t) − x(t)� 2 ≤ k(a)e −at �z0 − x0�
In Kalman style observers, the correction gain is computed at the same time as the
estimated state, and therefore constantly updated. It is the solution of a Riccati equation
11 Here, diag denotes the square matrix filled with zeros except for the diagonal that is composed of the
adequate vector.
20