Page 438 - Mechanical Engineers' Handbook (Volume 2)

11 Uniquely Digital Algorithms 429

resulting control law is then converted to discrete-time form, using approximations
for the integral and derivative terms. This method can be successfully applied if the
sampling time is small. The technique is widely used for two reasons. When existing
analog controllers are converted to digital control, the form of the control law and
the values of its associated gains are known to have been satisfactory. Therefore, the
digital version can use the same control law and gain values. Second, because analog
design methods are well established, many engineers prefer to take this route and
then convert the design into a discrete-time equivalent.
2. The performance specifications are given in terms of the desired continuous-time
response and/or desired root locations in the s-plane. From these the corresponding
root locations in the z-plane are found and a discrete control law is designed. This
method avoids the derivative and integral approximation errors that are inherent in
the first method and is the preferred method when the sampling time T is large.
However, the algebraic manipulations are more cumbersome.
The second approach uses the z-transform and pulse transfer functions, which were
outlined in the previous chapter. If we have an analog model of the plant, with its transfer
function G(s), we can obtain its pulse transfer function G(z) by finding the z-transform of
the impulse response g(t) L [G(s)]; that is, G(z) Z[g(t)]. A table of transforms facilitates
1
this process; see Refs. 1 and 2. Figure 49a shows the basic elements of a digital control
system. Figure 49b is an equivalent diagram with the analog transfer functions inserted.
Figure 49c represents the same system in terms of pulse transfer functions. From the diagram
we can find the closed-loop pulse transfer function. It is
C(z) G(z)P(z)
(53)
R(z) 1 G(z)P(z)
The variable z is related to the Laplace variable s by

z e sT (54)
If we know the desired root locations and the sampling time T, we can compute the z roots
from this equation.

Digital PI Control Design
For example, the first-order plant 1/(2s 1) with a zero-order hold has the following pulse
transfer function (Refs. 1 and 2):
1 e 0.5T
P(z) (55)
z e 0.5T
Suppose we use a control algorithm described by the following pulse transfer function:
F(z) Kz K K Kz 1
G(z) 1 2 1 2 (56)
E(z) z 1 1 z 1
The corresponding difference equation that the control computer must implement is

ƒ(t ) ƒ(t k 1 ) Ke(t ) Ke(t k 1 ) (57)
1
k
2
k
where e(t ) r(t ) c(t ). By comparing (57) with (47), it can be seen that this is the digital
k
k
k
equivalent of PI control, where K K and K (K K )/T. Using the form of G(z)
2
1
P
2
I
given by (56), the closed-loop transfer function is