Page 297 - Glucose Monitoring Devices

304 CHAPTER 15 Automated closed-loop insulin delivery

In contrast to the optimization-based approaches, the fundamental operation in the
computationally tractable subspace model identification is a projection, which may
emanate from prudent numerical techniques like singular value decomposition
(SVD) or even QR factorization.
The subspace-based system identification techniques utilize Hankel matrices
constructed from the output measurements and input data [61,62]. To establish these
Hankel matrices, define a vector of stacked output measurements as
T T T T
y kri ¼ y y kþ1 .y kþi 1
k
where i is a user-specified parameter greater than the observability index or, for
simplicity, the system order n. Similarly, define a vector of stacked input variables
as u kri . Through the repeated iterative application of the state-space, it is straightfor-
ward to verify the expression for the stacked quantities:

y kri ¼ G i x k þ F i u kri
where
2 3
C
CA
6 7
6 7
«
G i ¼ 6 7
4 5
i 1
CA
2 3
D 0 / 0
6 7
6 CB D / 0 7
6 7
F i ¼
6 « « 1 « 7
4 5
i 2 i 3
CA B CA B / D
Consider the block Hankel matrix for the outputs

Y i ¼ y 1ri y 2ri . y jri
and similarly U i as a block Hankel matrix of inputs.
Y i ¼ G i X i þ F i U i
where

X i ¼½x 1 x 2 . x j Š
The next step is to estimate the extended observability matrix, followed by
retrieving the system matrices. The basic underlying idea of many common system
identification methods is the orthogonal projection matrix on the null space of U i as
t
P ¼ I U T U i U T 1 U i
U i i i