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Adaptative high-gain extended Kalman filter and applications

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tel-00559107, version 1 - 24 Jan 2011<br />

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4.2 Simulation<br />

Figure 4.10: Effect of the parameters β <strong>and</strong> m on the shape of the sigmoid.<br />

output signal without noise. In the case where the output signal is corrupted by noise<br />

v(t), we have<br />

ymes(t) =y(t − d, x(t − d), τ)+v(τ).<br />

Therefore with x(t − d) =z(t − d):<br />

Id(t) = � t<br />

t−d �y(t − d, x(t − d), τ)+v(τ) − y(t − d, z(t − d), τ)�2 dτ<br />

= � t<br />

t−d �v(τ)�2 dτ �= 0.<br />

(4.7)<br />

We use σ to denote the st<strong>and</strong>ard deviation of v(t). We estimate m2 is the threesigma,<br />

which seams reasonable <strong>and</strong> empirically sound. Then from equation (4.7), we<br />

obtain the relation Id(t) ≤ 9σ 2 d. Therefore, m2 ≈ 9σ 2 d appears to be a reasonable<br />

choice. However, practice demonstrates that m2 computed in this way is over estimated.<br />

Although less common in engineering practice, we advise using a one-sigma rule 10 :<br />

m2 ≈ σ 2 d.<br />

3. Final tuning<br />

All the parameters being set as before, we run a series of simulations with the output<br />

signal corrupted by noise. The parameter load torque is changed suddenly from 0.55 to<br />

2.55 (scenario 2). We modify θ1 in order to improve (shorten) convergence time of the<br />

observer when the system faces perturbations. Overshoots are kept as low as possible.<br />

Remark 46<br />

This methodology can also be applied for a hardware implementation. In the case where<br />

a complete simulator for the process is absent, the observer can be tuned in an open loop:<br />

− when the plant is operating, more or less, in steady state, in order to tune the parameters<br />

related to noise <strong>filter</strong>ing,<br />

− when a perturbation occurs in order to set parameters related to adaptation.<br />

10 In the present example: σ = 2 <strong>and</strong> d = 1, thus m2 = 4.<br />

69

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