Page 299 - Glucose Monitoring Devices

306 CHAPTER 15 Automated closed-loop insulin delivery

where L and K denote the extended controllability matrices L ¼ A p 1 B . AB B

and K ¼ A p 1 K . AK K and assuming that the state transition matrix is nilpotent
p
with degree p (A ¼ 0), that is the contribution of the initial state b x k p is negligible
for sufficiently large p, the predicted state can be expressed as
b x k ¼ Lu k p;p þ Ky k p;p
Premultiplying the predicted state by the observability matrix G gives
Gb x k ¼ GLu k p;p þ GKy k p;p
with
C
2 3
CA
6 7
6 7
«
G ¼ 6 7
4 5
CA p 1
The product of the matrices GL and GK can be constructed from the VARX
model coefficient matrices as
u u u
2 q k p q k pþ1 / q
k 1 3
6 u u
0
6 q k p / q 7
6 k 2 7
GL ¼ 6 7
6 « « 1 « 5
4
0 0 u
/ q k f
and
y y y
2 /
q k p q k pþ1 q k 1 3
6
y y
6 0 / 7
6 q k p q k 2 7
6 7
GK ¼
6 1 « 7
6 « « 5
4
0 0 / y
k f
q
where f is the user-specified parameter for the future window length.
Therefore after estimating the VARX coefficient matrices, the estimated coeffi-
u y
cient matrices q and q can be used to determine all quantities on the right-hand side
of the state evolution equation, and an SVD can be used to readily obtain a low-rank
approximation of the state sequence. For recursive identification, a selection matrix
S of appropriate dimensions can be determined such that the basis of the state esti-
mation is consistent at each sampling time as

b x k ¼ S k W k GLu k p;p þ GKy k p;p