Page 292 - Glucose Monitoring Devices

Closed-loop glycemic control algorithms 299

subject to:
x 0 ¼ x
x kþ1 ¼ fðx k ; u k Þ; k˛I 0:N 1
k˛I 0:N
y k ¼ gðx k ; u k Þ
hðx k ; u k Þ 0; k˛I 0:N
x k ˛X; u k ˛U; k˛I 0:N
where the objective function JðxÞ is defined by
N
X
sp T sp s T s
Jðx; uÞ¼ ðy k y Þ Q k ðy k y Þþðu k u Þ R k ðu k u Þ
k¼0
with Q k as a (possibly varying) positive semidefinite weighting matrix penalizing the
deviation of the controlled variables from the target set-point y sp and R k as a
(possibly varying) strictly positive definite weighting matrix to penalize the amount
s
of input actions away from a reference input u . The set X denotes that the state vari-
ables are constrained as x min x x max , with x min and x max as the minimum and
maximum values for the state variables. Similarly, the set U denotes that the input
variables are constrained as u min u u max , with u min and u max as the minimum and
maximum values for the input. Therefore the maximum allowable insulin infusion
can be limited based on the estimated insulin on board or the plasma insulin concen-
tration. The optimal insulin infusion sequence fu 0 ; u 1 ; .; u N g is termed feasible for
a given initial state x if the insulin infusion sequence and the corresponding optimal
state sequence fx 0 ; x 1 ; .; x N g computed by the glucose-insulin dynamic model
satisfy the constraints. The mathematical programming problem is solved at each
sampling instance and the first value of the optimal solution (u 0 ) is implemented
to infuse insulin over the current sampling interval. The MPC computation and in-
sulin infusion implementation is repeated at subsequent sampling instances using
new glucose measurements and updated state estimates. Extensions to the MPC
paradigm include explicit MPC and advanced-step MPC algorithms [54e57].
Explicit MPC involves multiparametric programming, where the state of the system
is represented as a vector of parameters so that the optimal solution for all possible
realizations of the state vector can be precomputed as explicit functions to render the
online decisions as expediated function look-ups and evaluations. Advanced-step
MPC uses the prediction of the future state to solve the optimization problem within
the sampling time, and applies a sensitivity-based update to compute the manipu-
lated variable online once the new measurement is available.

Zone model predictive control
In contrast to controlling the glucose values to the desired set-point in conventional
MPC, zone MPC is developed for systems that lack a specific set-point. The
controller objective in zone MPC is to keep the controlled glucose concentrations