Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
2.6 On Adaptive High-gain Observers
− when the state of the system changes (different operating point, modification of the
physical characteristics of the process,...), it becomes more observable. A faster observer
can be considered and is needed to efficiently track the rapidly changing state variables.
The implementation they propose consists of two observers in parallel. Changes in the state
of the system are seen as faults. They are detected via a modified fault detection algorithm
[62]. The original algorithm is decomposed into three steps.
1. The fault input sequence is formed as
wk = α1wk−1 +(zk − zk−1)
where wk is an estimation of the direction of the parameter change. At steady state,
(zk − zk−1) corresponds to the measurement noise and therefore its sign is expected to
change frequently. Depending on the value of α1 so is the case of wk.
2. The fault test sequence is
sk = sign ((zk − zk−1) wk−1) .
− sk negative indicates that the parameter estimate variation is changing direction:
no fault occurs,
− sk positive over many successive tests, indicates that the variation is constant and
a fault is detected.
3. The fault sequence is filtered with the equation:
rk = α2rk−1 + (1 − α2)sk
where α2 determines the speed of fault detection and the rate of false alarms 22 .
This strategy is modified in the three following ways:
1. in order to reduce the influence of small noise components in the signal, the number of
significant figures used in the calculations of (zk − zk−1) is truncated,
2. in order to cope with the situation when (zk − zk−1) = 0, the sequence s is modified as
sk = sign(−10 8 +(zk − zk−1)wk−1).
When the parameter estimations doesn’t change, the sign function remains negative
and no fault is detected,
3. the sequence r(t) is clamped at a minimum value of −0.5, preventing excessively large
drifts at steady state.
22 For high values of α2 we obtain a high pass filter like behavior, and conversely for α2 small.
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