Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
where:
− X(0) is a random variable,
− X(t) and Y (t) are random processes,
1.3 Contributions
− V (t) and W (t) are two independent Wiener processes, also independent from x(0) (refer
to Appendix C.5).
In this context, Q is the covariance matrix of the state noise, and R is the covariance matrix
of the measurement noise.
We consider again the linear case together with the two assumptions:
1. x(0) is a gaussian random variable, and
2. state and output noises are gaussian white noises.
In this setting the Kalman filter is an estimator of the conditional expectation of the state,
depending on the measurements available so far 6 . When the Q and R matrices of the observer
defined above are set to the Q and R covariance matrices of the system, the noise is optimally
filtered (refer to [43, 45]).
In practice noise characteristics are not properly known. Therefore the matrices Q and
R of the observer are regarded as tuning parameters. They are adjusted in simulation.
In the nonlinear case the analysis in the stochastic setting consists of the computation of
the conditional density of the random process X(t). A solution to this problem is given by an
equation known as the Duncan-Mortensen-Zakaï equation [80, 108]. It is rather complicated
and cannot be solved analytically. Numerically, we make use of several approximations. One
of the methods is called particle filtering (e.g. see [21, 42] for details). When we obfuscate
the stochastic part of the problem, we obtain an observer with an elegant formulation: the
extended Kalman filter. Let us focus on its stochastic properties.
As demonstrated in [101], the extended Kalman filter displays excellent noise rejection
properties. However, proof of the convergence of the estimation error can be obtained only
when the estimated state comes sufficiently close to the real state.
On the other hand, the high-gain extended Kalman filter is proven to globally converge
(in a sense explained in subsequent chapters). However, it behaves poorly from the noise
rejection point of view. Indeed, it has the tendency to amplify noise, thus resulting in a
useless reconstructed signal. This problem is illustrated 7 in Figure 1.3.
1.3 Contributions
This dissertation concentrates on the fusion between the extended Kalman filter and the
high-gain extended Kalman filter by means of an adaptation strategy. Our purpose is to
merge the advantages of both structures:
6
z(t) =E [x(t)/Ft], where Ft = σ (y(s),s∈ [0,t]), i.e. the σ−algebra generated by the output variables for
t ∈ [0; t].
7
These two curves are obtained with the same observers as in Figure 1.2, the values of the tuning parameter
are unchanged. The level of noise added to the output signal is the same in the two simulations.
7