Page 439 - Mechanical Engineers' Handbook (Volume 2)

430 Basic Control Systems Design

Figure 49 Block diagrams of a typical digital controller. (a) Diagram showing the components. (b)
Diagram of the s-plane relations. (c) Diagram of the z-plane relations.

C(z) (1 b)(Kz K )
2
1
(58)
R(z) z (K 1 b bK )z b K bK 2
2
2
1
1
where b e 05T .
If the design specifications call for 1 and 1, then the desired s roots are s
1, 1, and the analog PI gains required to achieve these roots are K 3 and K 2.
I
P
Using a sampling time of T 0.1, the z roots must be z e 0.1 , e 0.1 . To achieve these
roots, the denominator of the transfer function (58) must be z 2e 0.1 z e 0.2 . Thus the
2
control gains must be K 2.903 and K 2.717. These values of K and K correspond
1
2
2
1
to K 2.72 and K 1.86, which are close to the PI gains computed for an analog
I
P
controller. If we had used a sampling time smaller than 0.1, say T 0.01, the values of K P
and K computed from K and K would be K 2.97 and K 1.98, which are even closer
P
2
I
1
I