Page 307 - Glucose Monitoring Devices

314 CHAPTER 15 Automated closed-loop insulin delivery

where
2n
X
w i ¼ 1
i¼0
and the covariance of the prior state estimates P x;krk 1 is computed by the weighted
outer product of the transformed points as
2n
X ’ ’ T
P x;kjk 1 ¼ w i X ’ i;kjk 1 X b kjk 1 X ’ i;kjk 1 X b kjk 1
i¼0
. The sigma points are similarly propagated through the measurement function as

’
y ¼ h X ’
i;kjk 1 i;kjk 1
and the estimated prior CGM output b y is approximated by the weighted
krk 1
average of the transformed points as
2n
X
b y w i y
kjk 1 ¼ i;kjk 1
i¼0
as well as the estimated covariance matrices
2n
X T
w i y b y y b y
P y ¼ i;kjk 1 kjk 1 i;kjk 1 kjk 1
i¼0
2n
X ’ T
w i X ’ X b y b y
P xy ¼ i;kjk 1 kjk 1 i;kjk 1 kjk 1
i¼0
’
The Kalman gain K k and posterior updates for the augmented state estimate X b as
kjkj
well as the posterior error covariance matrix P x;krk of the augmented state estimate
are given by the standard Kalman update equations
K k ¼ P xy P 1
y
’ ’
X ¼ X b þ K k y k b y
b
kjk kjk 1 krk 1
P x;krk ¼ P x;krk 1 K k P y K k T
. The UKF algorithm can handle the nonlinear dynamics of the glucose-insulin
model, is robust to noise, and has the ability to compensate for deviations and
converge to the true value of the state variables. The state variables in the
glucose-insulin dynamic models represent a physiological process based on first-
principles, and the state variables should be maintained within a physically realiz-
able range. For example, a negative value for the PIC due to measurement noise
and system uncertainty is not physically possible. Therefore constraints can be
employed in the UKF algorithm to ensure the augmented state estimates