Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
2.6 On Adaptive High-gain Observers
disturbances [18]. This section deals with the observer described in [31], and in particular
with the adaptive scheme proposed in [30]. The analysis they propose is set in the discrete
time setting.
First of all, note that the approach we described in the first part of this Chapter, and
the approach followed in the articles cited above are different, in the sense that the systems
considered are not expected to display the same observability property. Indeed in the present
case, the authors only need the system to be N-locally uniformly observable and do not
perform any change of variables. This implies that the class of nonlinear systems for which
the observer is proven to converge is bigger than the one considered in the present work (see
the numerical examples displayed in [30], for example). This observer can be used for systems
that cannot be put into a canonical observability form. The drawback to this approach then,
is that the observer converges locally and asymptotically (i.e. the state error is not upper
bounded by an exponential term).
Definition 26
1. We consider a discrete, nonlinear, system as in Definition 25 where xk ∈ R n , uk ∈ R nu
and yk ∈ R ny . The maps f and h are assumed to be continuously differentiable with
respect to the variable x.
2. The observer is defined as:
�
zk+1/k = f(zk,uk)
Pk+1/k = FkPkF ′
k + Qk
where
and
�
z k+1/k+1 = z k+1/k − Kk+1(h(z k+1/k) − yk+1)
P k+1/k+1 = (In − Kk+1Hk+1)P k+1/k
Kk+1 = P k+1/kH ′
k+1 (Hk+1P k+1/kH ′
k+1 + Rk+1) −1 ,
∂f(x, uk)
Fk = |x=zk
∂x
,
∂h(x, uk)
Hk = |x=z . k+1/k ∂x
(2.18)
This definition is that of a discrete extended Kalman filter. It is completed by a set of
assumptions that appear in the statement of the convergence theorem. Since the extended
Kalman filter is known to converge when the estimated state is very close to the real state,
the following theorem increases the size of this region.
Theorem 27 ([30])
We assume that:
1. the system defined in equation (2.17) is N-locally uniformly rank observable, that is to
33