Transfer Function Representations
Control System Toolbox™ software supports transfer functions that are continuous-time or discrete-time, and SISO or MIMO. You can also have time delays in your transfer function representation.
A SISO continuous-time transfer function is expressed as the ratio:
of polynomials N(s) and D(s), called the numerator and denominator polynomials, respectively.
You can represent linear systems as transfer functions in polynomial or factorized (zero-pole-gain) form. For example, the polynomial-form transfer function:
can be rewritten in factorized form as:
tf model object represents transfer functions in
polynomial form. The
zpk model object represents transfer
functions in factorized form.
MIMO transfer functions are arrays of SISO transfer functions. For example:
is a one-input, two output transfer function.
Commands for Creating Transfer Functions
Use the commands described in the following table to create transfer functions.
Create Transfer Function Using Numerator and Denominator Coefficients
This example shows how to create continuous-time single-input, single-output
(SISO) transfer functions from their numerator and denominator coefficients using
Create the transfer function :
num = [1 0]; den = [1 3 2]; G = tf(num,den);
den are the numerator and
denominator polynomial coefficients in descending powers of s.
den = [1 3 2] represents the denominator polynomial s2 + 3s + 2.
G is a
tf model object, which is a data
container for representing transfer functions in polynomial form.
Alternatively, you can specify the transfer function G(s) as an expression in s:
Create a transfer function model for the variable s.
s = tf('s');
Specify G(s) as a ratio of polynomials in s.
G = s/(s^2 + 3*s + 2);
Create Transfer Function Model Using Zeros, Poles, and Gain
This example shows how to create single-input, single-output (SISO) transfer
functions in factored form using
Create the factored transfer function :
Z = ; P = [-1-1i -1+1i -2]; K = 5; G = zpk(Z,P,K);
P are the zeros and poles (the roots
of the numerator and denominator, respectively).
K is the gain of
the factored form. For example, G(s) has a
real pole at s = –2 and a pair of complex poles at
s = –1 ± i. The
P = [-1-1i -1+1i -2] specifies these pole
G is a
zpk model object, which is a data
container for representing transfer functions in zero-pole-gain (factorized) form.