User`s guide

Time Delays
2-53
produces the discrete-time transfer function
Transfer function:
1
z^(–3) * -----------------
z^2 + 0.5 z + 0.2
Sampling time: 0.1
Notice the z^(–3) factor reflecting the three-sampling-period delay on the
input.
Mapping Discrete-Time Delays to Poles at the Origin
Since discrete-time delays are equivalent to additional poles at ,they can
be easily absorbed into the transfer function denominator or the state-space
equations. For example, the transfer function of the delayed integrator
is
You can specify this model either as the first-order transfer function
with a delay of two sampling periods on the input
Ts = 1; % sampling period
H1 = tf(1,[1 –1],Ts,'inputdelay',2)
or directly as a third-order transfer function:
H2 = tf(1,[1 –1 0 0],Ts) % 1/(z^3–z^2)
While these two models are mathematically equivalent, H1 is a more efficient
representation both in terms of storage and subsequent computations.
When necessary, you can map all discrete-time delays to poles at the origin
using the command
delay2z. For example,
H2 = delay2z(H1)
z 0
=
yk 1
+[]
yk
[]
uk 2
–[]+=
Hz()
z
2
–
z 1–
------------=
1 z 1
–()⁄