Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
− the machine is fed 54V for 30 seconds
− at t = 30 the input voltage is switched to 42V for another 30 seconds
4.3 real-time Implementation
− the voltage is then raised to 66V and finally shut down after 30 seconds.
At each step, for a period of about 10 seconds, an undetermined frictional force is applied
to the shaft of the motor (those perturbations are applied by hand, which explains why we
operate at such low speeds).
Figure 4.27 shows the estimations provided by the Luenberger observer in an open loop 26 .
The high-gain parameter was set to θ =2.5 and the application was run with a 0.001 seconds
time sampling.
Because of the additional computations needed to dynamically solve the Ricatti equation,
the high-gain Kalman filter is often seen as a very slow observer. Compared to a Luenberger
filter this is indeed the case. Thus, we ran the real-time task at time samplings of 0.01
seconds. On the positive side, this observer is more efficient than a Luenberger when dealing
with systems with a matrix A that depends on the input variable u(t).
The results presented in Figure 4.28 show the estimation of the rotational speed computed
by both an extended Kalman filter (θ = 1) and its high-gain counterpart (θ =2.5). Since
our perturbations are done by hand (and are thus not reproducible), the only way we can
display such information is to run the two observers in parallel. We therefore feel that it
is possible to tune the code of the extended Kalman filter to make it run efficiently with a
sample time of 0.001 seconds. As expected, the non high-gain filter reacts more slowly to the
applied perturbations. As the measurement noise is not large, it is difficult to distinguish the
(bad) influence of a large value of θ on the estimation. This influence can still be noticed in
the time windows [5; 10] and [25; 30].
The adaptive-gain extended Kalman filter is more demanding in terms of computational
time even though this observer runs at the sample time 0.01 seconds. Table 4.2 shows the
values selected for this experiment.
The results of this last experiment are displayed in Figure 4.29. Because this run was
done with no other observer in parallel, a comparison is not easy to make. Still, we can see
that, when dealing with perturbations, the observer has a speed of convergence comparable
to that of the high-gain extended Kalman filter and responds in a nature that is as smooth
as the one of the extended Kalman filter.
If we take a look at Figure 4.30 we see that the parameter θ reacts 9 times during
the experiment: once for every modification plus one extra time corresponding to a bad
initialisation. Those reactions correspond to:
− measured changes of the input voltage,
− non-measured changes in the load torque (i.e. perturbations).
It seems regular that changes occur in the second situation but not in first. In fact, this is
due to modeling errors which means that the innovation may not vanish. This problem is
solved by filtering the innovation 27 in the following way:
26 During perturbations estimation is less precise but stays in a range of 10 to 15% of the real value.
Remember that we had no sensor to measure the torque precisely when the model was calibrated.
27 The convergence result can be proven with such a filtering process. Once the time d ∗ of Theorem 36 has
been set, we have to design the filtering procedure in such a way that I doesn’t vanish in a time less than d ∗ .
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