SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

section a state-space model for the motor will be presented; two of the states will jointly contain
to the rotor position information. Using a classical observer, the states can be estimated and the
rotor position can be obtained. First a brief review of the state-space form is presented. Then a
model of the BPMS motor is given. Finally, it will be shown how the said observer fits into a
sensorless FOC control system.
Review of State-Space Structure
In modern control an n th -order linear system is described by a set of (n) simultaneous first-order
differential equations in state-variable form. The inputs to the system are described by the input
vector u ( r 1),
the outputs are described by the output vector y ( m 1),
and the states are
described by the state vector x ( n 1).
The n states are related to one another by the state
equations (Equation 6.6) and are related to the outputs by the output equations (Equation 6.7).
x f( x, u,
t)
(6.6)
y hxu ( , , t)
(6.7)
When the relationships are linear they can be described in vector-matrix format as Equations (6.8)
and (6.9). A is the n n system matrix, B is the n r input matrix, and C is the m n output
matrix.
x AxBu (6.8)
y Cx (6.9)
When the system is so described the relationships can be shown in a time-domain simulation
diagram, Figure 6.3. The eigenvalues are given by the solution to the characteristic equation
(Equation 6.10).
() s sIA
0
(6.10)
Figure 6.3 – State-space representation of linear system.
The system (plant) can be controlled by using output feedback as shown in Figure 6.4, where r is
the setpoint and K is an (r n)
matrix of gains. Output feedback has the limitation that not all
eigenvalues can be placed and the setting of one may cause another to move.
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