N. P. THOMPSON, B. WIDROW AND C. SCHADE 
1211 
in the controller show that it is possible to com- 
pute an optimum control path with a given set 
of constraints. This makes use of Rosen's gra- 
dient projection method.^ The upshot of this is 
that such optimum control leads to solutions for 
the optimal drug rates which can be fitted 
within exponentially decaying envelopes cen- 
tered about the steady state input values. Thus 
equation 4 is still used, but now the input values 
Xi are held to Xj where Xj is limited maximally 
to the maximal drug rate allowed or the value 
dictated by the upper border to the exponential 
envelope whichever is least and minimally to 
the minimal drug rate allowed or the value dic- 
tated by the lower border of the exponential en- 
velope whichever is greatest. Such control we 
have come to call phase III control. 
In mathematical representation : 
[ UBi, Xi > UBi 
Xi = Xi LBi < Xi < UBi 
I LBi, Xi <C LBi 
N 
ri - JWu Xi-k+i 
K = 2 
UBi = 
LBi = 
UCi = 
LCi = 
where 
where Xi = - 
Wi 
= min[di, UCi] 
= max [Ci, Lci] 
= Xs3 + vexp [-t|;(i-j)] 
= Xs. - vexp [-i//(i-j)] 
di = upper limit of I.V. rate (input) 
Ci = lower limit of I.V. rate (input) 
ip = 1/NL In v/8 
where 
L 
N 
= time between weights 
- number of weights 
jx A max | X^s — Xi | allowed at time i 
8 A max | Xss — Xi | allowed at time i 
RESULTS 
J 
T 
1 1 r 
5 6 7 
TIME (minutes) 
Figure 6. — A typical run start up. The predicted blood 
pressure (+H — |-) converges on the actual blood pres- 
sure ( ). 
Figure 6 shows a typical example of such a 
start up. Note how the predicted blood pressure 
value converges on the actual value. Figure 7 is 
later on after the model has converged and the 
blood pressure is being held at an average of 
160 mm Hg using a vasopressor (Levophed) in 
the I.V. This is higher than the dog's normal 
blood pressure and was used to allow the model 
to converge. At about two minutes the affect of 
Arfonad becomes apparent. While still on man- 
ual control the average blood pressure command 
is set for 100 mm Hg and the I.V. rate in- 
SICK DOG 
20 
0 
10* 
I 
40 
20 
ARFONAD INJECTED "ff 
MANUAL CONTROL 
AUTOMATIC CONTROL 
— r- 
10 
~~1 
12 
TIME (minutes) 
A run is begun under open loop conditions. 
That is, the physician commands the drug rate. 
However, the model "observes" these activities. 
Figure 7. — The initiation of automatic control and the 
system's response to the administration of Arfonad. 
This is phase I control. 
