стр. 66 
e  to s
GC G
+ % TO
% CO
% TO 
+

K
e  to s
t s +1
Branch A error
+
+
Branch B
Figure 81.4 Smith predictor technique.
82 DAHLIN CONTROLLER
Dahlin introduced a method for synthesizing computer feedback controllers [2].
When the process has dead time, the Dahlin synthesis method results in a PID con
troller with an added term that provides deadtime compensation. In fact, the dead
time compensation term is exactly equivalent to the Smith predictor. The basic
advantage of the Dahlin method is that it provides tuning parameters for the PID
part of the controller, while the Smith predictor does not.
A computer controller computes the controller output at regular intervals of time
called sample times. The period of time between samples is called the sample time
T. It is convenient to compute the increment in controller output at each sample
Dm(k) and then add it to the previous controller output m(k  1) to obtain the
updated controller output m(k), where k represents the kth sample. For example, a
computer PI controller computes the controller output in the following manner:
T
Г€ Л˜
Dm(k) = KC ГЌe(k)  e(k  1) + e(k)Л™
tI
ГЋ Лљ (82.1)
m(k) = m(k  1) + Dm(k)
177
DAHLIN CONTROLLER
where e(k) is the error at the kth sample, KC the controller gain, T the sample time,
and tI the integral time.
The Dahlin deadtime compensation controller adds one term to the calculation
of the controller output, as follows:
m(k) = m(k  1) + Dm(k) + (1  q) [m(k  N  1)  m(k  1)] (82.2)
where N is the integer ratio of the dead time to the time constant:
ГЉ to Л†
N = INT (82.3)
Г‹T ВЇ
and q is an adjustable parameter in the range of zero to 1.0 which is related to the
tuning parameter l of the controller synthesis method (see Section 34.2) as follows:
q = e T l (82.4)
The last term of Eq. (82.2) provides deadtime compensation equivalent to the
Smith predictor. Notice that if the dead time is zero, N = 0 and the last term of Eq.
(82.2) vanishes.
The tuning of the controller follows the controller synthesis method of Section
34.2. Since the Dahlin controller compensates for the dead time in the process, the
controller is tuned as if the process had no dead time; that is, use only the process
gain K and time constant t. The formulas of Section 34.2 give us the following
results for the Dahlin controller:
t
KC = tI =t (82.5)
Kl
and the derivative time is zero since the process dead time is taken as zero. If we
were to use the п¬Ѓrst guesses of t from Section 34.2, the initial proportional gain
would be inп¬Ѓnity. This is because, theoretically, if the controller compensates per
fectly for dead time, a very high gain would result in an almost perfect control
without oscillation. In practice, since the process does not normally match the
FOPDT model, a conservative value of the gain should be used. This author rec
ommends a п¬Ѓrst guess of l = 0.1 t.
The following example compares the response of the Dahlin deadtime com
pensation controller to that of a PID controller.
Example 82.1. A step test of the temperature controller of a heat exchanger gives
the following FOPDT parameters:
K = 1%TO %CO t = 0.56min t0 = 0.27min
A computerbased controller with a sample time T = 0.05 min is used to control the
temperature. The CSM method of Section 34.2 results in the following tuning for
a PID controller with t = 0.2(0.27) = 0.054 min:
178 DEADTIME COMPENSATION
KC = 1.73 %CO %TO
t1 = 0.56min
t D = 0.13min
The parameters for the Dahlin deadtime compensation controller with l = 0.056
are
KC = 10.0 %CO %TO
t1 = 0.56min
t D = 0 min
N = INT (0.27 0.05) = 5
q = e (0.05 0.056 ) = 0.41
51
Transmitter Output, %TO
50
49
48
47
0.0 1.0 2.0 3.0 4.0 5.0
Time, minutes
60
58
Controller Output, %CO
56
54
52
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