Chapter 2 Linear System Representation
Xmath Control Design Module 2-2 ni.com
Transfer Function System Models
One way of representing continuous-time finite-dimensional linear
time-invariant systems is with the transfer function:
with num(s) and den(s) being polynomials in s. They can be specified either
by their roots or their coefficients. Transfer functions are defined using the
Laplace transform operators for continuous time and the forward shift
operator z for discrete time. Both forms of transfer functions are written
with positive coefficients, each higher order terms having successively
larger coefficients.
Discrete systems are defined analogously, using the z variable instead of s.
Xmath does not automatically perform cancellations of polynomial roots
appearing in both the numerator and the denominator of a transfer function.
If you want to cancel common roots in a transfer function, use the function
cancel( ). For state-space systems, refer to the minimal( ) function.
For more information, refer to the Minimal Realizations section of
Chapter 6, State-Space Design.
To illustrate how you arrive at a particular transfer function, if you have a
system differential equation that takes the form:
(2-1)
Laplace-transforming equation (assuming zero initial conditions) yields:
(2-2)
Collecting terms, you can find the transfer function from U(s) to Y(s), H(s):
(2-3)
The roots of the numerator polynomial are the zeros of the transfer
function, and the roots of the denominator are its poles. In some
circumstances, you might want to construct a transfer function based on
where you know the pole and zero locations to be. For example, you can
Hs() num s()
den s()
------------------=
y
·· 6y
·8y++ 2u
·u–=
s2Ys() 6sY s() 8Ys()++ 2sU s() Us()–=
Ys()
Us()
----------- Hs() 2s1–
s26s8++
--------------------------==