Page 440 - Mechanical Engineers' Handbook (Volume 2)

11 Uniquely Digital Algorithms 431

to the analog gain values. This illustrates the earlier claim that analog design methods can
be used when the sampling time is small enough.

Digital Series Compensation
Series compensation can be implemented digitally by applying suitable discrete-time ap-
proximations for the derivative and integral to the model represented by the compensator’s
transfer function G (s). For example, the form of a lead or a lag compensator’s transfer
c
function is
M(s) s c
G (s) K (59)
c
F(s) s d
where m(t) is the actuator command and ƒ(t) is the control signal produced by the main
(PID) controller. The differential equation corresponding to (59) is
˙
˙ m dm K(ƒ cƒ) (60)
Using the simplest approximation for the derivative, Eq. (48), we obtain the following dif-
ference equation that the digital compensator must implement:
dm(t ) K cƒ(t )
m(t ) m(t k 1 ) ƒ(t ) ƒ(t k 1 )
k
k
T k T k
In the z-plane, the equation becomes
M(z) dM(z) K 1 F(z) cF(z)
1 z 1 z (61)
1
T T
The compensator’s pulse transfer function is thus seen to be
1
M(z) K(1 z ) cT
G (z)
c
F(z) 1 z 1 dT
which has the form
z a
G (z) K c (62)
c
z b
where K , a, and b can be expressed in terms of K, c, d, and T if we wish to use analog
c
design methods to design the compensator. When using commercial controllers, the user
might be required to enter the values of the gain, the pole, and the zero of the compensator.
The user must ascertain whether these values should be entered as s-plane values (i.e., K, c,
and d)oras z-plane values (K , a, and b).
c
Note that the digital compensator has the same number of poles and zeros as the analog
compensator. This is a result of the simple approximation used for the derivative. Note that
Eq. (61) shows that when we use this approximation, we can simply replace s in the analog
transfer function with 1 z . Because the integration operation is the inverse of differen-
1
1
tiation, we can replace 1/s with 1/(1 z ) when integration is used. [This is equivalent to
using the rectangular approximation for the integral, and can be verified by finding the pulse
transfer function of the incremental algorithm (47) with K 0.]
P
Some commercial controllers treat the PID algorithm as a series compensator, and the
user is expected to enter the controller’s values, not as PID gains, but as pole and zero
locations in the z-plane. The PID transfer function is