Page 298 - Glucose Monitoring Devices

Closed-loop glycemic control algorithms 305

The projection matrix is multiplied by the black Hankel representation of the sys-
tem to yield
t t
Y i P ¼ G i X i P
U i U i
where Y i P t is computable using numeral algorithms such as LQ factorization. An
U i
efficient implementation of this scheme is the multiinput, multioutput output-error
state-space (MOESP) algorithm, where an estimate of G i is obtained through the
t
dominant left singular vectors of Y i P . Moreover, numerous variations of this
U i t
approach (for instance, multiplying the matrix Y i P with instrumental variables
U i
and/or nonsingular weight matrices before computing the SVD) are proposed to
improve the consistency of the estimate. Four major variants of this method are
PO-MOESP (past outputs MOESP), N4SID (numerical algorithms for subspace
state-space system identification), IVM (instrumental variable method), and CVA
(canonical variate analysis) approach, which differ by the choice of weight matrices
t
T
that pre- and postmultiply the matrix Y i P J , where J is the instrumental variable
U i
matrix. Once an estimate of G i is determined from the dominant left singular vectors
t
T
of W 1 Y i P J W 2 , where W 1 and W 2 denote the weight matrices, a system realiza-
U i
tion can be calculated by retrieving estimates of system matrices A and C from G i ,
while estimates of B and D can be determined by solving a least-squares problem.
This approach yields linear, time-invariant state-space models, while other system
identification techniques are extended to model time-varying systems.
Recursive system identification
The predictor-based subspace identification approach is able to track a time-varying
linear system and can be coupled with a constrained optimization solver to guarantee
the stability of the model [61]. Consider a vector autoregression with exogenous var-
iables (VARX) model
p p
X X y
u
b y kþ1rk ¼ q u k i k 1 q k i k 1
y
i 0 i 1
where b y kþ1rk is the predicted output for the kth sampling instance using the past in-
puts u k ; .; u k p and outputs y k 1 ; .; y k p . The VARX model parameter p is the
length of the past window of data considered when predicting future outputs. The
y
u
coefficient matrices q and q are readily estimated through RLS techniques at
each sampling time. Furthermore, the stacked vector y k p;p is defined with respect
to the past window of length p as
h i T
y k p;p ¼ y T y T .y T
k p k pþ1 k 1
The stacked vector u k p;p is also similarly defined. Furthermore, recognizing that
the predicted state b x k is given by
p
b x k ¼ A b x k p þ Lu k p;p þ Ky k p;p