Page 302 - Glucose Monitoring Devices

Closed-loop glycemic control algorithms 309

where 0 < a < 1 is the forgetting factor and r k the updating law that can be deter-
mined as
sp
r k ¼ Kðy y k 1 Þ
with K as an appropriate gain matrix. It is readily shown that the eigenvalues of the
matrix ½I AKŠ should be within the unit circle for the algorithm to converge, with
the rate of convergence governed by the system matrix A and controller gain K.
Insulin doses can be adjusted using the R2R algorithm by considering the
glucose-insulin data from each postprandial period or day as a batch run. Consider
the manipulated variable u to be the insulin-to-carbohydrate ratio. The insulin
dosage regimen, specifically the insulin-to-carbohydrate ratio, for a batch can be
corrected based on a scalar performance measure y that quantifies the glycemic
excursion during the particular time window. The desired glycemic response trajec-
sp
tory during the batch is denoted y . The performance measure can be the rate of
change of glucose measurements in the postprandial period normalized by the
carbohydrate amount of the meal or a glycemic index summarizing the degree of
glycemic control during the day. The insulin-to-carbohydrate ratio for the next batch
u k is then given by the ratio for the previous batch and the performance measure for
the previous batch as
sp
u k ¼ u k 1 þ Kðy y k 1 Þ
with the gain K as a tuning parameter that determines how aggressive the R2R algo-
rithm is in correcting the insulin-to-carbohydrate ratio based on the previous batch re-
sults. The ability to learn from past periods to correct the insulin therapy regimen for
the subsequent batch is an attractive proposition for improving glycemic control.
Iterative learning control
Iterative learning control (ILC) is an improvement in run-to-run control that uses
more frequent measurements in the form of the error trajectory from the previous
batch to update the control signal for the subsequent batch run [63]. Compared to
the R2R control algorithm of

u k ¼ u k 1 þ r k
where u denotes the input trajectory and r is the updating law, the ILC algorithm is of
the form

u i;k ¼ u i;k 1 þ r i;k
where u i;k is the input trajectory at the ith sampling instant in batch k, u i;k 1 is the
input trajectory from the ith sampling instant of the previous batch, and r i;k is the
updating law. Different choices for the updating law result in different ILC schemes,
such as P-type ILC. Defining the tracking error as
sp
e i;k ¼ y y i;k
i
a simple formulation of ILC is

sp
u i;k ¼ u i;k 1 þ K y y i;k 1
i