Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
Up to this point, we have already studied the system, i.e., the system is of the normal
form, either naturally or after a change of variables. According to theory, in order to reconstruct
the state of this system, we can use any of the exponentially converging observers of
Chapter 2. However, we wont.
In this chapter, we solve the convergence part of the observability problem. Our goal is
to define an observer that combines the antagonistic behaviors of the extended Kalman filter
(EKF) and the high-gain extended Kalman filter (HG-EKF).
The EKF is extensively used, 1) because of its attractive filtering properties (as explained
in articles such as [101]), and 2) because it actually performs well in practice. However, a
proof of convergence for this algorithm is known only for small initial estimation errors (as
it can be seen in [17, 38] or within the proof of the main theorem below). Additionally, from
a practical point of view, the EKF handles large perturbations with difficulty, as has been
observed in simulations and experiments.
Contrarily, the HG-EKF possesses improved global properties [47]. It converges regardless
of the initial guess and/or independently of large perturbations. On the other hand, it
is rather sensitive with respect to noise.
Recall that the high-gain structure uses a single parameter denoted θ (θ > 1), and referred
to as the high-gain parameter. The HG-EKF does its global job if and only if θ is sufficiently
large. When θ is set to 1, it is formally equivalent to the standard EKF.
The idea here is to make the parameter θ adaptive. Thus,
− when the estimated state is far from the real state, θ is made sufficiently large such
that the observer converges for any initial guess,
− when the estimation is sufficiently close to the real state we allow θ to decrease. Once
this condition is satisfied, the local convergence of the extended Kalman filter is applicable
and the noise is more efficiently smoothed.
It is natural to perform the adaptation under the guise of a differential equation of the
form
˙θ = F(θ, I), (3.1)
where I is some quantity reflecting the amplitude of the estimation error: the smaller I the
smaller the error.
We introduce a simple and natural concept of “innovation” for the quantity I. This innovation
concept is different from the one that is usually used 1 . It allows us to reflect the
estimation error more precisely.
The convergence of this observer is established in the continuous time setting for multiple
inputs, single output systems 2 . This choice is made for the sake of maintaining the simplicity
of the exposure, because a few modifications have to be made in order to cope with multiple
outputs systems. Such modifications are explained in Chapter 5.
1 Most of the time innovation is defined as
− I = y − h(z, u) for the continuous case, with the notations of Definition 15,
− I = y − h(z −
k ,uk) for the discrete case, with the notations of Definition 25.
2 We have the generic case when nu > 1 and the non-generic case when nu = 1, C.f. Chapter 2.
38