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 np, dp, nc, and dc 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 dp is the plant order or plant degree. Similarly,
the degree of dc 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 np is less than
or equal to the degree of dp (which is N) and similarly for nc and dc. 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 funct ion is the transfer function
from the reference input r to the error signal e.
yPu=
uCe=
ery–=
Ps() nps()
dps()
-----------=Cs() ncs()
dcs()
-----------=