Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
− R is set in order to reflect the measurement noise covariance,
4.2 Simulation
− the diagonal coefficients of Q are made larger for the state variables, which are
unknown or for which the model is less accurate (in our case, this would be the
load torque).
We decided to set R = 1 and attempted several different configurations for Q. Figure
4.7 shows a plot of our estimate of the second state variable (red line) as compared to the
real values (black line) for several simulations. Comparing all the results in the figure,
we consider that the tuning in the third panel (from the top) provided the observer
with optimal noise smoothing properties. The values of Q used in the subsequent
experiments (fourth and fifth panels from the top) do not improve the performance,
while also making the observer really slow.
Contrary to what is normally done (see for example the application part of [38]), we
choose values for the Q matrix, which are much larger for the measured variable than
those that are unknown. By choosing the values in such a way, the observer is retarded
with respect to the observable, which mean that those parameters would converge quite
slowly, as compared to a bad initialization and/or sudden changes in the load torque.
The high-gain mode is meant to cope with those situations.
In order to determine an initial value for θ1, we simulate large disturbances in a
second scenario. The input variable is kept constant 6 (V = 120), and the load torque
is increased from 0.55 to 2.55 at the time step=30. The initial state is still considered
as being equal to the actual system state. Q and R are set to be the values chosen
previously. The high-gain parameter, θ(0) = θ0, is then chosen in order to achieve the
best observer time response during this disturbance. The performance with respect
to noise is ignored. The data corresponding to the estimate of the load torque (third
state variable) are plotted in Figure 4.9. We expect the noise to amplify the overshoot
problem. Thus, we therefore try to select θ0 such that offsets are avoided as much as
possible.
In the definition of the function F0 of the observer (4.5), θ1 has to be set such that
2θ1 = θ0 where θ0 denotes the value we just found.
2. Sigmoid function, innovation and adaptive procedure.
We now consider the fully implemented observer with ˙ θ = F(θ, Id) and θ(0) = 1.
Several parameters can be set in this case regardless of the application:
− β and m: recall that according to Lemma 42 of Chapter 3 the function µ(Id)
should possess the following features:
⎧
⎨
µ(Id) =
⎩
1 ifγ 1 ≤ Id
∈ [0, 1] ifγ 1 ≤ Id < γ0
0 if Id < γ0
6 This not a requirement at al. The input variable may still be considered to be varying when changes in
the load torque are made.
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