Page 300 - Glucose Monitoring Devices

Closed-loop glycemic control algorithms 307

where W k is a predefined weight matrix and the selection matrix S can be recursively
updated through the projection approximation subspace tracking method. The esti-
mated state sequence is then employed along with the inputs and measured outputs
to estimate the system matrices A k , B k , C k , D k , and K k by the solution of recursive
least-squares problems. Specifically, after computing an estimate of the state
sequence b x k , two recursive least-squares problems that ensure the stability of the
estimated system are used to determine the state-space matrices, thus yielding the
identified model

x kþ1 ¼ Ax k þ Bu k þ K k e k
y k ¼ Cx k þ Du k þ e k
where K k is the Kalman gain matrix and the e k ¼ y k b y .
k
Adaptive generalized predictive control
Model-based predictive controllers, such as MPC and generalized predictive control
(GPC), are widely used to control a variety of systems with complex dynamics and
long-time delays [39,40]. GPC uses recursively identified time-series models to opti-
mize the cost function.

n 2
n u
X sp 2 X 2
y kþj y l j Du
Jðu; kÞ¼ kþj þ kþj 1
j¼1
j¼n 1
with the predictions for the future outputs made using the recursively updated
model.
1 1
A q y k ¼ B q u k þ g þ ε k
where n 1 and n 2 are the finite minimum and maximum cost horizons and n u is the
control horizon; l is a weighting sequence for the input. Explicit solutions to the
GPC control problem exist for certain cases, and additional constraints on the inputs
can also be included in the optimization problem. Only the first calculated input u k is
applied to the system and in the following sampling instances, the problem solution
is repeated with new model parameters updated using the new measurements made
available.

Run-to-run control
Run-to-run (R2R) control schemes have been adapted from batch industrial pro-
cesses to address the problem of glucose regulation. These techniques have been
successfully implemented in industrial practice to control batch processes with
improvement in control performance based on learning from recent batches. In batch
control, quantitative measures of batch performance from the previous run (such as
the final product quality) are used to determine the input profile trajectory (recipe)
for the next run. The repeated nature of the batch is thus exploited to correct the
future batch based on the performance of the previous batch [63]. To apply R2R con-
trol to the glucose regulation problem, the postprandial period or the daily cycle of a