an investigation of dual stator winding induction machines

( x)
h(
x)
⎛ z h
1 ⎞ ⎛
⎜ ⎟ ⎜
⎜ z L
2 ⎟ ⎜ f
⎜ =
M ⎟ ⎜
⎜ ⎟ ⎜
⎜
n
z ⎟ ⎜ −1
⎝ L f h
n ⎠ ⎝
( x)
⎞
⎟
⎟
⎟ = φ
⎟
⎠
n n
where ( )
( x)
294
(8.27)
φ x : ℜ → ℜ is smooth locally invertible function in a neighborhood of x = 0.
This is known as the controller canonical form, which has the exact same form as the
linear system. The system input is defined as:
[ − a(
x)
v]
( x)
u = +
b
1
(8.28)
where, v is a vector of control signals. Then a linear differential relationship is obtained
and the resulting system equation is given as:
n
L f n
z1
= z&
= v
(8.29)
which is the linear multiple-integrator relation. v can be designed according to the
system requirements.
value
For example, the controller that regulates the output h(
x)
*
y can be designed as:
* ( z1
− y ) − a1z
2 − − an
zn
y = as desired constant
v = −a0
L −1
(8.30)
Similarly, the tracking controller that makes a system output to track a smooth
trajectory y () t
*
n
e
dt
() t
*
can be designed by using the error dynamics e(
t)
y(
t)
− y ( t)
d n
n
n
* ( n)
* ( n)
* ( )
() t − y () t = z − y () t = v − y () t
= .
n
= y
& (8.31)
Therefore, the control signal to make the system track the desired trajectory is
designed as:
v
*
*
* ( n)
() t − a ( z − y ) − a ( z − y ) −L
− a ( z − y )
*( n)
= y
0 1
1 2
n−1
& (8.32)
n