Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
2.6 On Adaptive High-gain Observers
presented in Subsections 2.6.1, 2.6.2 and 2.6.3. In the case of Kalman like observers, recall
that the high-gain modification is used to provide a structure to the Q and R matrices.
Therefore adaptation of the high-gain parameter can be understood as an adaptation of the
covariance matrices of the state and measurement noise. This topic has been the object of
quite extensive studies and a large bibliography on the subject is available, both in the linear
and nonlinear cases. Subsection 2.6.4 provides references to such works while Subsections
2.6.5 and 2.6.6 focuses on high-gain constructions specifically.
The adaptation of the high-gain parameter is, as said in Chapter 1, motivated by the
need to combine the antagonistic behaviors of an observer that filters noise efficiently and of
an observer that is able to converge quickly when large perturbations or jumps in the state
are detected. We want to emphasize the usefulness of this approach with three examples:
1. In [64] S. Haugwitz and coworkers describe the model of a chemical reactor coupled
with a highly efficient heat exchanger. The reactor is expected to be used to process
a highly exothermic chemical reaction, and the temperature measured at specific spots
constitutes the multidimensional output variable. Although the inlet concentrations
are supposed to be known and fixed, it may happen that the apparatus meant to blend
the reactants fails. The inlet concentration is then no longer the one expected, thus
provoking a non-measured large perturbation. The authors use, quite successfully, an
extended Kalman filter that takes into account this possibility of failure in the same
spirit as when we estimate the load torque of the DC machine in Chapter 4. An adaptive
high-gain extended Kalman filter can be really efficient for this kind of application by
increasing the performance of the observer at perturbation times.
2. In vehicle navigation, data from an inertial navigation system (INS) — a 3-axis accelerometer
coupled with a gyroscope — and data from a global navigation satellite
systems (GPS, GALILEO, GLONASS) are fusioned. The first type of sensors is very
precise at time 0 but with an error domain that grows with time. Sensors of the second
type have an error domain larger that the one of the INS at time 0 but that is stable.
The purpose here is to know the position as precisely as possible with an error domain
as small as possible. An observer that filters the measurement noise is needed but estimation
error may increase with time because of sudden changes of direction, sudden
changes in the topology of the road or the loss of the GPS signal because of tunnels
or urban canyons. The estimation of the covariance matrices Q and R of Kalman like
filters is the subject of the articles [96, 115] (linear case) or [37] (nonlinear case). The
book of M. S. Greywal [60] provides a solid introduction to this topic.
3. In a refinery, changes of the processed crude oil are perturbations. Starting from the
atmospheric column, the disturbance propagates along the refinery. The speed of propagation
of the disturbance front depends on many parameters (the several processes
have low time constants, crude oil can be retained,...), and is not accurately known. An
EKF 14 like observer is useful when there are no such changes, and an adaptive high-gain
observer would be of use in order to detect the feed change [38, 47, 113].
Finally we want to cite a few techniques used in order to render an observer’s gain adaptive
that we have not considered:
14 Extended Kalman filter.
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