National Instruments Xmath Interactive Control Design Module ICDM - Manual (Page 58)

Chapter 6 Pole Place Synthesis
© National Instruments Corporation 6-3 Xmath Interactive Control Design Module
where
d
p
(s) = s n + a
1
s n–1 + a
2
s n–2 + ... + a
n
n
p
(s) = b
0
s n + b
1
s n–1 + ... + ab
n
Notice that the order of the plant is n, and allow the possibility that the plant
transfer function is not strictly proper; that is, the plant can have as many
zeros as poles.
Normal Mode
In normal mode, the order (number of poles) of the controller is fixed and
equal to n (the order of the plant), so there are a total of 2n closed-loop
poles. In this case, the 2n degrees of freedom in the closed-loop poles
exactly determine the controller transfer function, which also has 2n
degrees of freedom.
In normal mode, the controller transfer function has order n and is strictly
proper:
C(s) = n
c
(s)/d
c
(s)
where
d
c
(s) = s n + x
1
s n–1 + x
2
s n–2 + ... + x
n
n
c
(s) = y
1
s n–1 + y
2
s n–2 + ... + 2y
n
Therefore, the closed-loop characteristic polynomial has degree 2n:
where
λ
1
, …,
λ
2n
are the closed-loop poles chosen by the user.
χ s ( ) n
c
s ( )n
p
s ( ) d
c
s ( )d
p
s ( )
+ =
s λ
1
–
( ) s λ
2
–
( )… s λ
2n
–
( )
=
s
2n
α
1
s
2n 1 – … α
2
n + + +
( )
=