SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

Figure 6.7 – Full state feedback (Luenberger observer).
Observation and control of the states are somewhat duals of one another. If the original plant is
completely controllable and observable, all n eigenvalues of the closed-loop system may be
placed arbitrarily by selection of K and all n eigenvalues of the observer may be placed
arbitrarily by selection of L ; this is called a separation principle. In sensorless FOC the motor is
the plant and information about the rotor position is contained in the states. Although it is
possible to control the motor using the full state feedback controller shown, this report uses FOC.
Therefore the estimated state vector is not fed back into any type of controller. Its only purpose is
to produce estimates of the two states that contain information about the rotor position which is
required for FOC. According to Figure 6.7, the ability to arbitrarily set the eigenvalues of the
observer is not dependent on the use of the state feedback controller, thus FOC may be used to
control the plant and the observer can be used without modification. The eigenvalues of the
observer may be placed anywhere in the complex plane (subject to digital system limitations)
according to Equation (6.12).
() s sI
ALC (6.12)
It can be shown that the error dynamics are those of the observer, hence Equation (6.13).
e ( ALC) e
(6.13)
The placement of the observer poles should be sufficiently far into the left-half plane to yield a
good state estimate, but there is the usual tradeoff between bandwidth and sensitivity to noise.
Time has not allowed further reviewing the principles of modern control. References on observers
and pole placement include [45], [46], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58],
[59].
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