Page 811 - The Mechatronics Handbook

0066_Frame_C25 Page 10 Wednesday, January 9, 2002 7:05 PM

In addition to the approach using a z-transform, the ratio of the polynomials in the difference model
(25.6) can be written in a series expansion:

n d j ∞
Gq() = ∑ j=0 d j q ∑ g k q k –
------------------- =
n j
c
∑ j=0 c j q j=0
With the discrete time pulse function u imp (k) as an input, it can be observed that
∞
y imp k() = ∑ g k q d k() = g k
k –
j=0

and it can be concluded that the pulse response y imp (k) equals the coefficients in the series expansion of
the difference equation. Similarly, with the discrete time step function u step (k) as an input, it can be
observed that

∞ k
k –
y imp k() = ∑ g k q u step k() = ∑ g k
j=0 j=0
and it can be concluded that the step response y step (k) values are computed as a finite sum of the coefficients
in the series expansion of the difference equation. The computation of a discrete time pulse response for
a first order discrete time model is given in the following example.
• Consider a first order discrete model given by the difference model

1
Gq() = ------------
q + d

where d indicates the discrete time constant of the system. The series expansion of the difference
model can be computed as follows:

∞
1
Gq() = ----------- = ∑ d j
–
qd
j=0
and it can be seen that the discrete time pulse response

y imp (k) = d k

is an exponential function. For stability the discrete constant d needs to satisfy |d| < 1. Similar as
in the continuous time model it can be observed that the smaller the time constant, the faster the
response. Additionally, the first order discrete time model may exhibit an oscillation in case −1 <
d < 0.

Sinusoid and Frequency Response
So far we have considered transient effects caused by step, pulse, and impulse inputs to investigate the
dynamic properties of a dynamical system. However, periodic inputs occur frequently in practical situ-
ations and the analysis of a dynamic system to periodic inputs and especially sinusoidal inputs can yield
more insight into the behavior of the system.

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