Chapter 2
Introduction to SISO Design
Xmath Interactive Control Design Module
2-2
ni.com
The equations describing this system are as follows:
where
y
denotes the plant output or sensor signal
u
denotes the plant input or actuator signal
r
denotes the reference or command input signal
e
denotes the error signal
P
denotes the plant transfer function
C
denotes the controller transfer function
In ICDM, the plant and controller transfer function are required to be
rational, that is, the ratio of two polynomials:
where
n
p
,
d
p
,
n
c
, and
d
c
are polynomials called the plant numerator,
plant denominator, controller numerator, and controller denominator,
respectively. The symbols
n
and
d
are mnemonics for numerator and
denominator. The degree of
d
p
is the plant order or plant degree. Similarly,
the degree of
d
c
is the controller order or controller degree.
The poles and zeros of these transfer functions are the zeros (roots) of the
denominator and numerator polynomials, respectively.
In ICDM,
P
and
C
are required to be proper polynomials; that is, they have
at least as many poles as zeros. In other words, the degree of
n
p
is less than
or equal to the degree of
d
p
(which is
N
) and similarly for
n
c
and
d
c
. In some
situations, the plant and controller are required to be strictly proper, which
means that there are more poles than zeros.
Other important terms include:
•
The loop transfer function
L
is defined as
L = PC
. The loop gain is the
magnitude of the loop transfer function.
•
The sensitivity transfer function is denoted as
S
and given by
S
= 1/(1 +
PC
). The sensitivity transfer function is the transfer function
from the reference input
r
to the error signal
e
.
y
Pu
=
u
Ce
=
e
r y
–
=
P s
( )
n
p
s
( )
d
p
s
( )
-----------
=
C s
( )
n
c
s
( )
d
c
s
( )
-----------
=
