N. P. THOMPSON, B. WIDROW AND C. SCHADE 
1209 
corrected by correcting the weights and as these 
weights are in turn used for controlling the 
plant, we are in fact generating an error func- 
tion and using it to correct the system's output. 
The heart of the modeler is a transversal 
filter. The main element in such a filter is a 
tapped delay line. Each tap along the line is 
connected to a weighting element capable of 
weighting the tap's output over a range of plus 
and minus values. 
The output of each weight element is fed into 
a summer and summed with the others. One 
other element enters the summer, this is a 
weighted bias. The output of the summer is 
compared with the output of the system being 
modeled while the input to the delay line is al- 
ways a scaled analog of the input to the system 
being modeled, Figure 4. 
After each delay period, each weight is ad- 
justed according to the Widrow-Hoif^'^ a-LMS 
algorithm : 
a 
(1) 
where 
= g, - X^, W, 
and 
W = the weight coefficient 
a = the convergence factor 
e = the error, i.e., the Model's output sub- 
VALVE 
SETTING 
COMMANDS 
DRUG 
FLOW 
RATE 
ZERO- 
ORDER 
HOLD 
AVERAGE 
BLOOD 
-PRESSURE 
"i-M 
PLANT 
-og(t) 
SAMPLER 
E, = (gi-xTwj) 
ERROR SIGNAL 
USED IN 
ADAPTING WEIGHTS 
BIAS WEIGHT 
stracted from the Plant's output at the j'" 
instant 
Xj = the total input (to the weights) vector at 
the j"' instant 
gj = the j"' sample of the plant's output 
It can be shown that when the input and 
plant are stationary, the weights will converge 
as long as 
0 < a < 2. 
In practice we use a = 0.4. a controls the 
amount of error removed at each update of the 
weights and thus the rate of the model's con- 
vergence. The value must be chosen to assure 
stability in the weight determination on the one 
hand and a reasonable settling time on the 
other. 
Figure 5 shows the adaption of a set of 
weights starting from those of an "average re- 
sponse" and converging to those of the dog 
under control. To save time in the initial set up 
period, we begin with an average set of weights 
for a dog rather than, say, all zeros. 
Having generated this model, one can readily 
predict the response to a given input vector. 
Output = (2) 
The real problem arises in trying to deter- 
mine Xj, the instantaneous input value (i.e., 
the vasopressor I.V. rate) at the j*'' instant. 
Several approaches are being studied. All seem 
oeor 
020r 
18 
1 
_J 1 r-*-^ 
t . i . I 
0 2 
4 
6 8 10 12 14 
16 IS 20 
WEIGHT NUMBER 
Figure 4. — A detail of the transversal filter. 
Figure 5. — A typical adaption sequence of weights. 
