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68 Control theory in biomedical engineering

absorption rate is also included as an extended state to be estimated. The
proposed nonlinear observer for the simultaneous estimation of the state
variables and the time-varying parameters can be designed as

x
^ x 0 ¼ ^ 0, + K ðfÞ y k ^ 0 (1a)
y
k k k k
0,
^ y ¼ g ^x (1b)
0
0
k k
0
where ^ 0, is the prior estimate of the augmented state vector, ^ denotes the
x
x
k k
estimated augmented state vector, K ðfÞ is the observer gain obtained by the
k
0
UKF algorithm, g is the augmented output dynamic and uncertain model
0
parameters’ function, and ^ is the estimated output (CGM) (Kola ˚s et al.,
y
k
2009; Dochain, 2003; Hajizadeh et al., 2017a, 2018c). The augmented state
vector includes the states and time-varying parameters of the Hovorka’s
model. The states of the Hovorka’s model include [S 1,k S 2,k I k x 1,k x 2,k x 3,k
T
Q 1,k Q 2,k G sub,k ] (Hovorka et al., 2004). The state variables S 1 (t)and S 2 (t)
describe the absorption rate of subcutaneously administered insulin as basal
and bolus insulin and I(t) represents the PIC in the bloodstream. The Q 1 (t)
and Q 2 (t) describe the glucose masses in the accessible and nonaccessible com-
partments, respectively. The insulin action is computed by using the influence
on transport and distribution (x 1 (t)), the utilization and phosphorylation of
glucose in adipose tissue (x 2 (t)), and the endogenous glucose production in
the liver (x 3 (t)). The subcutaneous glucose concentration is G sub (t). The
T
time-varying parameters include [t max,I, k k e,k U G,k ] . Considering the Hovor-
ka’s insulin compartment model, the two parameters time-to-maximum of
absorption of injected insulin (t max,I (t)) and insulin elimination from plasma
(k e (t)) have a direct effect on the PIC. Furthermore, as information about
meals is difficult to determine, the gut absorption rate (U G, k )is alsoestimated.
2.2 Recursive subspace-based system identification

The proposed recursive system identification technique provides a time-
varying stable state-space model (Hajizadeh et al., 2017b, 2018d). It updates
parameters of state-space matrices online. After integrating the state-space
model with a physiological insulin compartment model (Hajizadeh et al.,
2018b), the final identified glycemic model for use in an AL-MPC becomes:

x k +1 ¼ A k x k + B k u k + d k
(2)
y
k ¼ C k x k
where A k , B k ,and C k are the system matrices, and d k represents unmo-
deled/unmeasured disturbances. For the proposed model, continuous glu-
cose measurement (CGM data) is considered as output, and infused insulin