Page 296 - Glucose Monitoring Devices

Closed-loop glycemic control algorithms 303

An innovation term is introduced to update the model coefficients

q k ¼ q k 1 þ k k e k
and the covariance matrix is updated:
1 1 T
k k j P k 1
P k ¼ l P k 1 l k
The RLS algorithm results in the data from the ith preceding sampling instance
i
carrying a weight of l relative to the newest sample, thus discounting the older sam-
ples over time.
The stability of the identified models is an important consideration for control-
relevant models [39,60]. The identified ARX or ARMAX models can be written
in a state-space form as

x kþ1 ¼ Ax k þ Bu k þ Ke k
y k ¼ Cx k þ Du k þ e k
with the state x as a memory of past inputs and outputs and the system matrices
developed through the model parameters. The updates of the model parameters q
can be constrained based on the physiological meanings of the parameters. The opti-
mization problem can also be formulated to ensure the stability of the identified
model as
T T
q k ¼ min ðq k q k 1 Þ P k 1 ðq k q k 1 Þþ e Qe k
k
q k
subject to:
rðAÞ 1
min max
q q k q
where rðAÞ is the spectral radius of the state transition matrix for the state-space rep-
min max
resentation of the ARX/ARMAX model and q and q are the minimum and
maximum of the identified model parameters. The former constraint satisfies the sta-
bility condition of the model and the latter satisfies the physiological properties of
the system.

Subspace-based state-space system identification
Subspace-based state-space system identification techniques are used to determine
the system matrices of a discrete-time state-space model of the form

x kþ1 ¼ Ax k þ Bu k
y k ¼ Cx k þ Du k
where x, y,and u denote the state, input, and output variables, respectively, and A,
B, C,and D. are the system matrices to be determined using inputeoutput data. A
realization of the system matrices can be obtained by solving complicated optimi-
zation problems such as nuclear norm-based structural rank minimization and
maximum likelihood estimation through expectation-maximization algorithms.