Page 301 - Glucose Monitoring Devices

308 CHAPTER 15 Automated closed-loop insulin delivery

subject can be considered as a batch. Similarities between a chemical batch process
and a particular time window of a subject with T1DM include: following a particular
protocol for meals and diurnal activities (similar to batch recipe) with variations in
amounts, compositions, type, and timings of meals; the quality variables of batch
process product and the performance measure for the quality of glycemic variations
during the batch; and the differences between subjects (intersubject variability), as
well as evolving glucose-insulin dynamics and glycemic disturbances similar to var-
iations in raw material feed and disturbances over the course of the batch. Therefore
the glucose measurements obtained during a batch can be used to adjust the insulin
therapy for the subsequent batch [64].
R2R control algorithms typically update the control law on a time scale of the
entire cycle (i.e., one correction allowed at the end of the batch), thus refining the
control action over the course of multiple cycles until desirable control performance
is obtained. Consider a linear, inputeoutput model of the process for the kth batch
run

y k ¼ Au k þ b k þ e k
where k denotes the batch index, y is the system output, u is the input, A is the
coefficients matrix for relating the inputs to the outputs, b is the bias/drift coefficient
matrix, and e denotes the disturbances. Note that this model represents a batch run,
hence the inputs are a trajectory profile applied to the process, and the outputs are the
measurements collected over the batch run. The target for the process outputs y is
sp
denoted as y .If A and b are known, and A is invertible, then the optimal control
for the system is
sp
u k ¼ A 1 ðy b k Þ
The mismatch between the model and the system and batch-to-batch (run-to-run)
variations are common in practice as systems evolve temporally and disturbances
affect the batch runs. An iterative scheme called exponentially weighted moving
average (EWMA) filter for updating the estimates of the bias coefficient matrix is
proposed as
b k ¼ lðy k 1 Au k 1 Þþð1 lÞb k 1
where 0 < l < 1 is an adjustable tuning parameter for the EWMA filter. Substitut-
ing the optimal control law into the EWMA filter for updating the bias coefficient
matrix
sp
b k ¼ b k 1 þ lðy k 1 y Þ
and hence integrating the optimal control law from the k 1 batch run yields

1 sp
u k ¼ u k 1 þ lA ðy y k 1 Þ
which is the EWMA-type R2R or proportional-type (P-type) R2R control algorithm
due to the proportional contribution of performance measure from the previous
batch. A more general form of R2R is

u k ¼ au k 1 þ r k