Page 442 - Mechanical Engineers' Handbook (Volume 2)

12 Hardware and Software for Digital Control 433

1 a
P(z) Kz n (68)
z a
where a exp( T/ ) and n D/T. If we choose D(z) z (n 1) , then with a step command
input, the output c(k) will reach its desired value in n 1 sample times, one more than is
in the dead time D. This is the fastest response possible. From (66) the required controller
transfer function is
1 1 az 1
G(z) (69)
K(1 a)1 z (n 1)
The corresponding difference equation that the control computer must implement is
1
ƒ(t ) ƒ(t k n 1 ) [e(t ) ae(t k 1 )] (70)
k
k
K(1 a)
This algorithm is called a finite-settling-time algorithm because the response reaches its
desired value in a finite, prescribed time. The maximum value of the manipulated variable
required by this algorithm occurs at t 0 and is 1/K(1 a). If this value saturates the
actuator, this method will not work as predicted. Its success depends also on the accuracy
of the plant model.

Dahlin’s Algorithm
This sensitivity to plant modeling errors can be reduced by relaxing the minimum-response-
time requirement. For example, choosing D(z) to have the same form as P(z), namely,
1 a d
D(z) Kz n (71)
d
z a d
we obtain from (66) the following controller transfer function:
K (1 a ) 1 az 1
d
d
G(z) (72)
K(1 a)1 az 1 K (1 a )z (n 1)
d
d
d
3
This is Dahlin’s algorithm. The corresponding difference equation that the control computer
must implement is
ƒ(t ) a ƒ(t k 1 ) K (1 a )ƒ(t k n 1 )
d
k
d
d
K (1 a )
d
d
[e(t ) ae(t )] (73)
K(1 a) k k 1
Normally we would first try setting K K and a a, but since we might not have good
d
d
estimates of K and a, we can use K and a as tuning parameters to adjust the controller’s
d
d
performance. The constant a is related to the time constant of the desired response: a d
d
d
exp( T/ ). Choosing smaller gives faster response.
d
d
Algorithms such as these are often used for system startup, after which the control mode
is switched to PID, which is more capable of handling disturbances.
12 HARDWARE AND SOFTWARE FOR DIGITAL CONTROL
This section provides an overview of the general categories of digital controllers that are
commercially available. This is followed by a summary of the software currently available
for digital control and for control system design.