Page 303 - Glucose Monitoring Devices

310 CHAPTER 15 Automated closed-loop insulin delivery

where K is the learning gain matrix. The objective of ILC is that
lim e i;k ¼ 0
k/∞
Therefore the input trajectory for the current batch is determined based on the
input trajectory implemented in the previous batch plus the proportional contribution
of tracking error. In certain cases, in practice e iþ1;k 1 can be considered as an
approximate prediction of e iþ1;k , and therefore the P-type ILC is implemented as

sp
u i;k ¼ u i;k 1 þ K y y iþ1;k 1
i
Other types of ILC are proposed for improved robustness and formulations that
employ information from the past batch beyond the current sampling instance for
anticipatory or phase-lead type ILC. Real-time information can be incorporated
into the ILC as
p c
u i;k ¼ u i;k 1 þ r þ r
i;k i;k
where

p p sp
r ¼ K y y i;k 1
i;k i
and
sp
c
r ¼ K c y y i;k 1
i;k
i
p
c
The gains K and K correspond to the information in the previous and current
cycles, respectively. Note that future or anticipatory information can be used to
p c
design r , though future information cannot be used to design r as it is physically
i;k i;k
unrealizable. Another ILC approach integrates feedback control with ILC as
u i;k ¼ u ILC þ u FB
i;k i;k
where u ILC is the input computed by the ILC and u FB is the input computed through
i;k i;k
a real-time feedback controller such as PID control or MPC [65,66].
MPC may also be integrated with ILC to involve future predictions of the current
batch in the control law calculation [67]. Consider an ARX model of the form
1 K 1 K
A q D y i;k ¼ B q D u i d;k þ ε i;k
where a delay of order d is included and the differences between batches are repre-
sented as
K
D y i;k ¼ y i;k y i;k 1
K
D u i;k ¼ u i;k u i;k 1
and ε i;k denotes disturbances or uncertainties. Given that ILC can be expressed as

u i;k ¼ u i;k 1 þ r i;k