Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
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4.2 Simulation
Figure 4.10: Effect of the parameters β and m on the shape of the sigmoid.
output signal without noise. In the case where the output signal is corrupted by noise
v(t), we have
ymes(t) =y(t − d, x(t − d), τ)+v(τ).
Therefore with x(t − d) =z(t − d):
Id(t) = � t
t−d �y(t − d, x(t − d), τ)+v(τ) − y(t − d, z(t − d), τ)�2 dτ
= � t
t−d �v(τ)�2 dτ �= 0.
(4.7)
We use σ to denote the standard deviation of v(t). We estimate m2 is the threesigma,
which seams reasonable and empirically sound. Then from equation (4.7), we
obtain the relation Id(t) ≤ 9σ 2 d. Therefore, m2 ≈ 9σ 2 d appears to be a reasonable
choice. However, practice demonstrates that m2 computed in this way is over estimated.
Although less common in engineering practice, we advise using a one-sigma rule 10 :
m2 ≈ σ 2 d.
3. Final tuning
All the parameters being set as before, we run a series of simulations with the output
signal corrupted by noise. The parameter load torque is changed suddenly from 0.55 to
2.55 (scenario 2). We modify θ1 in order to improve (shorten) convergence time of the
observer when the system faces perturbations. Overshoots are kept as low as possible.
Remark 46
This methodology can also be applied for a hardware implementation. In the case where
a complete simulator for the process is absent, the observer can be tuned in an open loop:
− when the plant is operating, more or less, in steady state, in order to tune the parameters
related to noise filtering,
− when a perturbation occurs in order to set parameters related to adaptation.
10 In the present example: σ = 2 and d = 1, thus m2 = 4.
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