Plasma-Insulin-Cognizant Adaptive Model Predictive Control for Artificial Pancreas Systems

Abstract

An adaptive model predictive control (MPC) algorithm with dynamic adjustments of constraints and objective function weights based on estimates of the plasma insulin concentration (PIC) is proposed for artificial pancreas (AP) systems. A personalized compartment model that translates the infused insulin into estimates of PIC is integrated with a recursive subspace-based system identification to characterize the transient dynamics of glycemic measurements. The system identification approach is able to identify stable, reliable linear time-varying models from closed-loop data. An MPC algorithm using the adaptive models is designed to compute the optimal exogenous insulin delivery for AP systems without requiring any manually-entered meal information. A dynamic safety constraint derived from the estimation of PIC is incorporated in the adaptive MPC to improve the efficacy of the AP and prevent insulin overdosing. Simulation case studies demonstrate the performance of the proposed adaptive MPC algorithm.

1. Introduction

Type 1 diabetes mellitus (T1DM) is a chronic autoimmune disease characterized by the destruction of insulin-producing beta cells and subsequent insulin deficiency. Since insulin is necessary to regulate blood glucose concentration (BGC) levels, exogenous insulin administrated either by multiple daily injections or through an infusion pump is needed for people with T1DM. Automated insulin delivery systems, known as artificial pancreas (AP), consist of a continuous glucose monitoring (CGM) sensor, a continuous subcutaneous insulin infusion pump, and a control algorithm closing the loop between glucose sensing and insulin delivery. Without effective glycemic control, prolonged hyperglycemia (high BGC, defined as BGC > 180 mg/dL) can result in diabetes-related vascular complications and organ failure associated with significant morbidity and mortality [1, 2]. Insulin overdosing is associated with hypoglycemia (low BGC, defined as BGC < 70 mg/dL), which may be fatal. Despite the proven advantages, closed-loop control of BGC using AP remains challenging because of the substantive requirements for: (i) reliable models to accurately describe the glucose-insulin dynamics; (ii) predictive control algorithms that regulate BGC by manipulating exogenous insulin delivery; and (iii) cognizance of insulin constraints to prevent an overdose and improve patient safety.

One of the major challenges in the development of an AP is the complex, nonlinear, and only approximately known biochemical and physiological kinetics and dynamics of glucose-insulin metabolism. Reliable glucose-insulin models for on-line use are not available because of the significant inter- and intra-subject variability in physiology, time-varying delays and nonlinear dynamics caused by both the gradual absorption and utilization of insulin, the transient blood glucose dynamics affecting the CGM sensor measurements, obscure information on the time and amount of disturbances like carbohydrate (meal) consumption, and the lingering effects of previously administered insulin [3–5]. Insulin is infused as either a steady basal infusion or a bolus amount to mitigate large increases in BGC usually administered to counter meal effects. The infrequent basal changes that regulate glucose levels during fasting periods and acute bolus impulses present a challenging configuration for modeling glycemic dynamics. Several compartmental models addressing these challenges are developed to integrate a pharmacokinetic model of insulin action, calculate the time course of plasma and active insulin, and describe the dependence of glucose dynamics on plasma insulin and glucose concentration levels with a pharmacodynamic glucose model. Some of the compartment models filter the administered insulin dose into a derived variable, such as plasma insulin concentration (PIC), to quantify the amount of insulin in the bloodstream, which can better characterize the prolonged diffusion, absorption, and utilization rates of the subcutaneously administered insulin [6, 7].

Although filtering the infused insulin into a derived variable such as PIC has a significant effect on the predictive performance of the model, the application of fixed time-invariant filters may be suboptimal and can diminish the potentially realizable improvement in model accuracy [8, 9]. Despite the capability of compartment models in predicting the expected time course of glucose concentrations in response to variability in the insulin dose, the models are generally nonlinear, relatively complex, and often computationally intractable for identifying individualized models online to enable personalized medicine through real-time predictive control. A more computationally viable alternative is subspace-based state-space system identification, where structured block Hankel matrices are constructed using measured input/output data and computationally efficient projection operations retrieve certain subspaces related to the system matrices. Recent developments in system identification with application to AP systems include a recursive predictor-based subspace identification (PBSID) method that avoids the computationally burdensome procedure of computing or updating singular value decomposition (SVD) factorizations. The PBSID approach is extended to provide stable, time-varying individualized state-space models for glycemic predictions using estimates of the PIC. The adaptive identification allows the realized models to be valid over a diverse range of daily conditions without requiring obscure information on meals, thus avoiding manual meal announcement entries by users [10–13].

Several control algorithms are proposed for AP systems, including proportional-integral-derivative control [14–22], fuzzy logic control [23–25] and model-based predictive controllers such as model predictive control (MPC) [26–33] and generalized predictive control [34–38]. MPC is widely employed in AP systems because of its inherent ability to easily and effectively handle complex systems with control constraints and many input and output variables. MPC algorithms exploit dynamic models of the system in the optimization problem to predict the future evolution of the glucose concentrations over a finite-time horizon to determine the optimal insulin infusion rates with respect to a specified performance index. Furthermore, MPC can explicitly consider the system constraints and multivariable interactions in the optimization problem, and the MPC formulations are not inexorably restricted by the type of model, objective function, or constraints. The improved glucose regulation performance afforded by MPC algorithms is necessary to maintain the desired BGC, and the greater time spent in a safe glycemic range can effectively delay the onset and slow the progression of serious diabetes related complications. To alleviate the adversities of these complications caused by prolonged hyperglycemia, it is desirable to maintain BGC closer to the lower values within a safe glycemic range, though suppressing the BGC towards the lower end of the allowable range increases the probability of hypoglycemic episodes. The risk of hypoglycemic episodes, however, can be attenuated by formulating controllers cognizant of the previously administered unreacted insulin still present in the body.

The amount of previously administered insulin that is readily available to a subject is referred to as the insulin on board (IOB). The IOB is typically determined in infusion pumps through static approximations of the insulin action curves [39, 40]. The time-varying delays induced by the unsteady rates of insulin diffusion, absorption, and utilization, and the diurnal variations in the metabolic states of individuals, have significant effects on the kinetics and dynamics of the metabolic system. Therefore, the insulin decay profiles and action curves used in calculating IOB are not accurate enough over the diverse conditions encountered throughout the day to be reliably used in an AP control system. In contrast to the conventional IOB calculations based on approximated insulin decay curves, accurate estimates of the PIC can be obtained by using CGM data with adaptive observers designed for simultaneous state and parameter estimation. The estimated PIC can be subsequently used to design a predictive control algorithm that is dynamically constrained by the estimated PIC and thus explicitly considers the insulin concentration in the bloodstream as part of the optimal control solution. Incorporating dynamic constraint adjustment in the optimal control problem based on the estimated PIC can prevent insulin overdosing that may lead to hypoglycemia, which can yield a safe and reliable AP even in the presence of significant uncertainty in the system.

Previous studies integrated an adaptive and personalized physiological insulin compartment model (to translate the abrupt bolus and infrequent basal changes into PIC estimates) with a time-varying linear state-space model obtained through the recursive PBSID approach, which provided good prediction accuracy of the glucose trajectory [41, 42]. The identified adaptive models were used to design an adaptive MPC with a feature extraction method using CGM data to detect rapid glycemic deviations from the desired set-point caused by significant disturbances (such as meal consumption) and subsequently adjust the controller constraints. The efficacy of the proposed PIC-cognizant MPC was demonstrated using the University of Virginia/Padova (UVa/Padova) metabolic simulator [43]. This work extends upon these preliminary findings and further compares the PIC-cognizant MPC against the IOB-constrained MPC and a typical MPC without considering information about the active insulin present in the body. Further developing the PIC-cognizant MPC, a behavioral pattern and PIC-cognizant MPC is proposed that uses identified patterns from historical data to modify the basal set-point in anticipation of unannounced meal disturbances. A general flowchart of the proposed method is presented in Fig. 1. Section 2 briefly reviews the nonlinear observer for simultaneous state and parameter estimation and the system identification approach to identify stable linear time-varying models from closed-loop data. The results of the PIC estimation and system identification are leveraged in Section 3 to design an MPC algorithm using the adaptive models to compute the optimal insulin delivery without requiring any user-specified information on the time and amount of carbohydrate consumption. A dynamic safety constraint derived from the estimation of PIC is incorporated in the adaptive MPC algorithm to prevent insulin overdosing. The efficacy of the proposed PIC-cognizant MPC is demonstrated in Section 4 using the UVa/Padova metabolic simulator involving different virtual people with T1DM [44]. The PIC-cognizant MPC and an IOB-constrained MPC are compared, and the improvement in glucose control resulting from basal set-point adjustment based on the identified behavioral patterns is elucidated. Finally, a discussion on the proposed adaptive MPC cognizant of PIC is provided in Section 5, followed by a few concluding remarks in Section 6.

Figure 1:

Flow chart of the proposed artificial pancreas system

2. Preliminaries

In this section, the glucose-insulin dynamic model and the nonlinear observer for PIC estimation are briefly introduced (details in Appendix A), followed by an outline of the PBSID algorithm for the identification of linear, time-varying state-space models (details in Appendix B). Fig. 2 illustrates the proposed modeling and adaptive MPC algorithm. First, a UKF estimates the PIC value using the CGM data and infused insulin information. Then, the PIC estimates and CGM data are used to identify time-varying linear state space model with the recursive PBSID technique. The estimated PIC and the identified state space models form the basis of the adaptive MPC with dynamic constraint adjustment.

Figure 2:

A summary of the developed techniques for the artificial pancreas system

The following notation is used throughout the manuscript: x denotes the states of the nonlinear physiological model, x′ denotes the augmented states of the nonlinear physiological model, $\tilde{x}$ denotes the states of the identified state-space models, and $\overline{x}$ denotes the states of the integrated state-space and physiological compartment model. The notation extends to outputs and input variables as well.

2.1. Adaptive and Personalized PIC Estimator

The Hovorka’s model, a glucose-insulin dynamic model, detailed in Eq. (A.1), is used for designing the PIC estimator [6, 7, 26]. The CGM data provide feedback correction to the UKF for estimating the augmented states of Hovorka’s model, including the PIC. The PIC estimates are used in the identification of a linear time-varying state space model in modifying the controller safety constraints. The PIC estimator can be individualized by appropriately initializing the time-varying parameters (t_max,I(t) and k_e(t)) of the insulin compartmental model (Eqs. (A.1a) to (A.1c)) that have a direct effect on the PIC using partial least squares regression. The demographic information of individuals, such as weight, body mass index, and duration with diabetes, is used to estimate the initial values for the t_max,I(t) and k_e(t) [6, 7]. After initialization, the time-varying parameters and the state variables are estimated on-line using an appropriate observer (UKF). All variables, parameters and the techniques used in Section 2.1 to design the PIC estimator are defined in Appendix A.

2.2. Recursive Subspace-Based System Identification

A stable, reliable, and computationally tractable dynamic model is essential for the design of MPC algorithms for AP systems. In this work, two parameters (t_max,I(t) and k_e(t)) that have a direct effect on the PIC are personalized based on the PLS models. The rest of the parameters are set to their nominal values. We use recursive subspace identification with CGM data and estimated PIC values to constantly update the parameters of a simpler model structure, yielding adaptive and personalized time-varying linear state space model for use in a computationally tractable MPC. The identified model is of the form

$${\tilde{x}}_{k+1}=A_k{\tilde{x}}_k+B_k{\tilde{u}}_k+K_k{e}_k$$ (1a)

$$y_k=C_k{\tilde{x}}_k+e_k$$ (1b)

where ${\tilde{x}}_k\in {ℝ}^{{\tilde{n}}_x}$ denotes the states of the identified model, ${\tilde{u}}_k$ is the estimate of the PIC $\left({\tilde{u}}_k:={\widehat{I}}_k\right)$, (other exogenous input variables may also be added), and y_k is the output CGM measurement. A_k, B_k, and C_k are the system matrices of appropriate dimensions, K_k is the Kalman gain matrix, and $e_k=y_k-{\tilde{y}}_k$ is the deviation between the CGM measurement and model output for feedback to correct the state variables. All variables, parameters and the techniques used in Section 2.2 to identify the model described by Eq. (1) are defined in Appendix B.

Remark 1. The AP control algorithms are typically implemented in computationally prohibitive mobile or embedded devices such as smartphones or insulin infusion pumps. The nonlinear parameterized physiological glucose-insulin dynamic models, such as Hovorka’s model, are computationally intractable for use in the resource-constrained hardware with real-time personalization to account for the inter- and intra-subject variabilities. Therefore, without adapting on-line a significant number of parameters, the nonlinear physiological model Eq. (A.1), or its linearized version along the state trajectory, are not accurate enough for use in BGC prediction of individual subjects. In this work, only the insulin compartment model parameters (t_max,I(t) and k_e(t)) that have a direct effect on the PIC are estimated on-line. The recursive PBSID approach used in this work involves updating all the parameters of a simpler linear model structure, which enables the efficient tracking of the transient glucose-insulin dynamics.

3. Adaptive PIC-Cognizant MPC Algorithm

The insulin compartment model of the glucose-insulin dynamic model is combined with the recursive state space models determined by PBSID algorithm to characterize the time-varying glycemic dynamics. Subsequently, the glycemic and plasma insulin risk indexes and the adaptive MPC formulation leveraging the adaptive identification and PIC estimates are presented.

3.1. Integrating Insulin Compartment Models with Subspace Identification

We combine the data-driven model determined from the PBSID with a physiological insulin compartment model derived from Hovorka’s model [41]. Consider the insulin subsystem from Eq. (A.1) described by the discretized compartment model in Eq. (A.1a)–Eq. (A.1c) as

$$S_{1,k+1}=S_{1,k}+\left(u_k-\frac{S_{1,k}}{t_{\text{max},\text{I},k}}\right)T_s$$ (2a)

$$S_{2,k+2}=S_{2,k+1}+\left(\frac{S_{1,k+1}}{t_{\text{max},\text{l},k}}-\frac{S_{2,k+1}}{t_{\text{max},\text{I},k}}\right)T_s$$ (2b)

$${\tilde{u}}_{k+3}={\tilde{u}}_{k+2}+\left(\frac{S_{2,k+2}}{t_{\text{max},\text{I},k}{V}_{\text{I}}}-k_{\text{e},k}{\tilde{u}}_{k+2}\right)T_s$$ (2c)

where u_k denotes the infused basal and bolus insulin, ${\tilde{u}}_k$ is the PIC, and T_s is the sampling time of 5 minutes. Note that Eq. (A.1b) and Eq. (A.1c) are written with respect to the k + 1 and k + 2 sampling instances. Formulating the insulin compartment as in Eq. (2) means that we can compute ${\tilde{u}}_{k+3}$ (the PIC at sampling time k + 3) knowing u_k (the past information until sampling instance k). As such, the PIC value at the k + 3 sampling instant $({\tilde{u}}_{k+3})$ is affected by the infused basal/bolus insulin (u_k) at the kth sampling instant. Further, considering a delay of order d, we can write

$${\tilde{x}}_{k+1}=A_k{\tilde{x}}_k+B_k{\tilde{u}}_{k-d}+K_k{e}_k$$ (3a)

$$y_k=C_k{\tilde{x}}_k+e_k$$ (3b)

The PIC variable in Eq. (3) is a filtered insulin concentration to handle time-delays and discrete or impulse variations. It is obtained from the compartment model with injected basal and bolus insulin as the input variable. The compartment model in Eqs. (2a) to (2c) is integrated with the recursively identified model Eq. (3) yielding

$${\overline{x}}_{k+1}=L_k{\overline{x}}_k+E_k{u}_k$$ (4a)

$${\overline{y}}_k=F_k{\overline{x}}_k$$ (4b)

where the system matrices of Eq. (3) are used to derive the augmented system model as

$$L_k=\left[\begin{array}{cccc}{A}_k& p_1^I\cdot B_k& p_2^I\cdot B_k& p_3^I\cdot B_k\\ {[\mathrm{0\dots 0}]}_{1\times {\tilde{n}}_x}& p_1^{S_1}& 0& 0\\ {[0\dots 0]}_{1\times {\tilde{n}}_x}& p_1^{S_2}& p_2^{S_2}& 0\\ {[0\dots 0]}_{1\times {\tilde{n}}_x}& p_1^I& p_2^I& p_3^I\end{array}\right],E_k=\left[\begin{array}{c}{p}_4^I\cdot B_k\\ T_s\\ p_3^{S_2}\\ p_5^I\end{array}\right],\text{and}{F}_k=\left[\begin{array}{llll}{C}_k\hfill & 0\hfill & 0\hfill & 0\hfill \end{array}\right]$$

with the new state vector of ${\overline{x}}_k={\left[\begin{array}{llll}{\tilde{x}}_{k+3+d}^T\hfill & {\widehat{S}}_{1,k}\hfill & S_{2,k+1}\hfill & {\tilde{u}}_{k+2}\hfill \end{array}\right]}^T$ and the output as ${\overline{y}}_k={\tilde{y}}_{k+3+d}$. In Table 1, the integrated glycemic model parameters of Eq. (4) are defined. All other parameters of Eq. (4) are detailed in Appendix C.

Table 1:

Nomenclature for the integrated glycemic model

Lk	State (system) matrix of the integrated glycemic model
Ek	Input matrix of the integrated glycemic model
Fk	Output matrix of the integrated glycemic model
${\overline{x}}_k$	States of the integrated glycemic model
${\overline{y}}_k$	Output of the integrated glycemic model
uk	Input of the integrated glycemic model (infused basal and bolus insulin)
Ts	Sampling time of 5 minutes
d	Delay for the effect of PIC on CGM

Remark 2. A common approach to handle the delays and the abrupt changes in the infused insulin is to filter the infused insulin into a newly derived variable that represents the effects of the input. This typically involves using IOB calculations. In this work, we integrate a compartment model with the recursively identified state-space model to translate the infused insulin into estimates of the PIC. The integrated glycemic model has PIC as a state variable and the infused insulin as an input, which allows for direct use of the integrated glycemic model in a predictive controller. Since the model predicts the future dynamic trajectory of the PIC and the CGM output, the predicted future evolution of the PIC can be dynamically constrained over the MPC prediction horizon based on the predicted CGM.

Remark 3. The augmented state vector of Hovorka’s model is estimated using the UKF algorithm with the CGM measurements and the infused insulin information, and the random walk description of the parameters is used to define the covariance matrix of the parameters, which is a tuning parameter of the UKF algorithm. The estimated time-varying parameters of the insulin compartment are then used along with the recursive PBSID approach to construct the integrated glycemic model in Eq. (4). This integrated glycemic model has a simpler linear structure compared to the nonlinear Hovorka’s model, which enables the efficient on-line estimation of the various pharmacokinetic parameters. Since many parameters of the integrated glycemic model are readily updated on-line, the model provides more accurate short-term predictions than the nominal Hovorka’s model when no meal information is considered (i.e. without meal announcements). The integrated glycemic model is then used to design the adaptive MPC.

3.2. Glycemic and Plasma Insulin Risk Indexes

In this subsection, we describe the glycemic and plasma insulin concentration risk indexes for use in the MPC objective function.

3.2.1. Glycemic Risk Index

A glycemic risk index (GRI) is used to determine the weighting matrix for penalizing the deviations of the outputs from their nominal set-point [43, 45]. The time-varying positive semi-definite weighting matrix ${\overline{Q}}_k:=\overline{Q}\left({\overline{y}}_k\right)$ is defined as $\overline{Q}\left({\overline{y}}_k\right):=\alpha \left({\overline{y}}_k\right)\cdot Q^{\text{nom}}$ where Q^nom denotes a nominal weight and

$$\alpha \left({\overline{y}}_k\right):=\{\begin{array}{lll}1.0567\times {10}^{-2}{\left(80-{\overline{y}}_k\right)}^{1.3378}\hfill & \text{ if}{\overline{y}}_k\hfill & \in [50,80]\hfill \\ 0\hfill & \text{ if }{\overline{y}}_k\hfill & \in (80,140]\hfill \\ 0.4607\times {10}^{-2}{\left(-140+{\overline{y}}_k\right)}^{1.0601}\hfill & \text{ if}{\overline{y}}_k\hfill & \in (140,300]\hfill \\ 1\hfill & \text{ otherwise }\hfill & \hfill \end{array}$$ (5)

The glycemic risk index asymmetrically increases the set-point tracking weight when ${\overline{y}}_k$ diverges from the target range. Since hypoglycemic events have serious short-term implications, the set-point penalty increases rapidly in response to hypoglycemic excursions and more gradually in hyperglycemic excursions (Fig. 3).

Figure 3:

Plot of the glycemic risk index

3.2.2. Plasma Insulin Risk Index

A plasma insulin risk index (PIRI) is defined to manipulate the weighting matrix for penalizing the amount of input actuation (aggressiveness of insulin dosing) depending on the estimated PIC, thus suppressing the infusion rate if sufficient insulin is already present in the bloodstream [43]. The time-varying positive definite weighting matrix ${\overline{R}}_k:=\overline{R}\left({\gamma }_k\right)$ is developed from the PIRI as $\overline{R}\left({\gamma }_k\right):={\gamma }_k\cdot R^{\text{nom}}$, with R^nom as a nominal weight and γ_k defined as

$${\gamma }_k:={\left(\frac{{\tilde{u}}_k}{{\tilde{u}}_{\text{ basal },k}}\right)}^2$$ (6)

where

$${\tilde{u}}_{\text{ basal },k}:=\frac{u_{\text{db},k}}{V_{\text{I}}\cdot {\widehat{k}}_{\text{e},k}}$$ (7)

and u_db is the patient specific (possibly time-varying) basal insulin rate that is known in practice. It is impractical to directly consider PIC estimates to define the parameters of MPC due to the variability among subjects, the normalized value of the PIC is employed. This eliminates the dependency of the PIC estimates to a particular subject by standardizing with the known patient-specific basal PIC value. A plot of the PIRI (Fig. 4) indicates that the penalty weight on the input action increases, and dosing becomes less aggressive, if the estimated PIC is high.

Figure 4:

Plot of the plasma insulin risk index

3.3. Plasma Insulin Concentration Bounds

In the proposed PIC-cognizant MPC, the PIC is a state variable of the derived model and thus the estimated future PIC is dynamically bounded depending on the values of CGM measurements. For instance, if the CGM values are elevated, the bounds on the PIC are increased to ensure sufficient insulin is administered to regulate the BGC. The PIC bounds are determined based on the CGM values as $\mathcal{X}:=\mathcal{X}\left({\overline{y}}_k\right)$, where $\mathcal{X}$ denotes the constraints on the state variables, and in particular defines the lower and upper bounds and a desired target for the normalized PIC through the predicted CGM ${\overline{y}}_k$. Fig. 5 depicts the bounds and the reference target for the normalized PIC as a function of the CGM values. The MPC solution should satisfy the PIC constraints while maintaining the PIC close to the desired value. The nominal PIC bounds can be determined by multiplying the normalized PIC bounds with the basal PIC value. Therefore, appropriate PIC bounds can be determined based on each subject’s basal rate and the CGM values.

Figure 5:

Plot of the plasma insulin concentration bounds

3.4. Adaptive MPC Formulation

We propose a novel adaptive MPC algorithm cognizant of the PIC for computing the optimal insulin infusion rate. It utilizes the glycemic and PIC risk indexes that manipulate the penalty weighting matrices in the objective function. The MPC computes the optimal insulin infusion over a finite horizon using the identified time-varying subspace-based models by solving at each sampling instant (k) the following quadratic programming problem

$$\begin{gathered} {\left\{z_i^{*},m_i^{*}\right\}}_{i=0}^{n_P}:=\underset{m\in \mathcal{M},z\in \mathcal{Z}}{\text{arg}\text{min}}{\mathcal{J}}_{n_P,k}\left({\overline{y}}_k,{\gamma }_k,{\left\{m_i\right\}}_{i=0}^{n_P},{\left\{z_i\right\}}_{i=0}^{n_P}\right) \\ \text{s.t.}\{\begin{array}{l}{z}_{i+1}=L_k{z}_i+E_k{m}_i,\forall i\in {\mathbb{N}}_0^{n_P-1}\hfill \\ q_i=F_k{z}_i,\forall i\in {\mathbb{N}}_0^{n_P}\hfill \\ z_0={\overline{x}}_k\hfill \end{array} \end{gathered}$$ (8)

with the objective function

$${\mathcal{J}}_{n_P,k}\left({\overline{y}}_k,{\gamma }_k,{\left\{m_i\right\}}_{i=0}^{n_P},{\left\{z_i\right\}}_{i=0}^{n_P}\right):=\sum _{i=0}^{n_P}{\left(q_i-r\right)}^T{\overline{Q}}_k\left(q_i-r\right)+m_i^T{\overline{R}}_k{m}_i$$

where $z_k\in {ℝ}^{{\overline{n}}_x}$ and $q_k\in ℝ$ denote the predicted states and output, respectively, for the prediction/control horizon n_P, $m_k\in ℝ$ denotes the constrained input variable, taking values in a nonempty convex set $\mathcal{M}$ with $\mathcal{M}:=\left\{m_k\in ℝ:m_{\text{min},\text{k}}\le m_k\le m_{\text{max},\text{k}}\right\}$, $m_{\text{min},\text{k}}\in ℝ$ and $m_{\text{max},\text{k}}\in ℝ$ denote the lower and upper bounds on the manipulated input, respectively, and r is the target set-point. The index ${\mathbb{N}}_0^{n_P}$ represents all integers in a set as ${\mathbb{N}}_0^{n_P}:=\left\{0,\dots ,n_P\right\}$. The nonempty convex set $\mathcal{Z}$ with $\mathcal{Z}:=\left\{z_k\in {ℝ}^{{\overline{n}}_x}:z_{\text{min},\text{k}}\le z_k\le z_{\text{max},\text{k}}\right\}$, $z_{\text{min},\text{k}}\in {ℝ}^{{\overline{n}}_x}$ and $z_{\text{max},\text{k}}\in {ℝ}^{{\overline{n}}_x}$ denote the lower and upper bounds on the state variables, respectively, with one of the states as the estimated PIC that is constrained through the PIC bounds. The ${\overline{n}}_x$ is the number of states in the integrated glycemic model described by Eq. (4). Furthermore, ${\overline{x}}_k$ provides an initialization of the state vector, ${\overline{Q}}_k\ge 0$, ${\overline{Q}}_k:=\overline{Q}\left({\overline{y}}_k\right)$ is a positive semi-definite symmetric matrix used to penalize the deviations of the outputs from their nominal set-point, and ${\overline{R}}_k>0$, ${\overline{R}}_k:=\overline{R}\left({\gamma }_k\right)$ is a strictly positive definite symmetric matrix to penalize the manipulated input variables. At each iteration, the quadratic programming problem in Eq. (8) is solved, and u_k ≔ m₀ is the optimal solution implemented to infuse insulin over the current sampling interval with the MPC computation repeated at subsequent sampling instances using new CGM measurements, updated states, and newly calculated penalty weights of the objective function.

4. Results

The efficacy of the proposed PIC-cognizant MPC is demonstrated by using the 10 adult subjects of the UVa/Padova metabolic simulator [44] accepted by the U.S. Food and Drug Administration as a substitute to animal trials for preclinical testing of insulin therapies for people with T1DM. Its nonlinear compartmental model is more complex than the model of Hovorka and has 16 states [44].

The subjects are simulated for 12 days with varying times and quantities of meals consumed on each day (Table 2) to evaluate the proposed MPC under challenging realistic scenarios. The meal information is not utilized in the MPC algorithm as the controller is designed to regulate the BGC in the presence of significant disturbances such as unannounced meals. The controller set-point is set to r = 110 mg/dL. In practice, such low glycemic set-points are avoided due to fear of hypoglycemia, though a controller that is aware of the previously administered insulin will moderate aggressive inputs (decrease insulin dosing) when sufficient insulin has been previously administered. The model is designed with a delay of order d = 1 with regards to the effect of PIC on CGM. To illustrate the efficacy of using PIC in AP control systems, we also present the results based on conventional IOB-constrained MPC, where the IOB is used to define the maximum allowable bolus insulin at each sampling time, and an MPC that is not aware of either PIC or IOB information. In addition, the results of an adaptive pattern and PIC-cognizant MPC are provided. In this controller, the behavior patterns of an individual are detected and used to adjust the set-point. For illustration, the pattern and PIC-cognizant MPC adjusts the basal set-point based on the meal consumption times observed in the historical data. The nominal glucose set-point in the controller is r = 110 mg/dL. A retrospective analysis of the previous days is conducted to detect the time periods with meal consumption using a feature extraction method analyzing the recorded CGM data [43]. The average time periods of meal consumption over the historical data are assigned a lower set-point value of r = 80 mg/dL to make the controller more aggressive around those periods.

Table 2:

Meal scenario for 12 days closed-loop experiment using the UVa/Padova metabolic simulator

Meal	Range for values
	Time	Amount (g)
Breakfast	[07:45, 10:10]	[53, 83]
Lunch	[12:30, 14:45]	[44, 77]
Dinner	[17:45, 18:45]	[60, 83]
Snack	[21:30, 23:00]	[17, 51]

In the MPC formulation, the m_min is zero and m_max is defined as

$$m_{\text{max}}=u_{\text{db}}+u_{\text{Bolus},\text{max}}\cdot \frac{{10}^3}{T_s}$$ (9)

where the maximum allowable bolus in units of U is defied as

$$u_{\text{Bolus}}{}_{\text{,max }}:=\{\begin{array}{ll}\text{max}\left(\frac{y_k-r}{\text{CF}},0\right)\hfill & \text{for MPC without insulin constraints}\hfill \\ \text{max}\left(\frac{y_k-r}{\text{CF}}-I_k^{\text{IOB}},0\right)\hfill & \text{for IOB-constrained MPC}\hfill \\ \text{max}\left(\frac{y_k-r}{\text{CF}}-\frac{{\widehat{I}}_k\cdot V_I}{{10}^3},0\right)\hfill & \text{for PIC-cognizant MPC}\hfill \\ \text{max}\left(\frac{y_k-r_l}{\text{CF}}-\frac{{\widehat{I}}_k\cdot V_I}{{10}^3},0\right)\hfill & \text{for pattern and PIC-cognizant MPC}\hfill \end{array}$$ (10)

where CF is the patient-specific correction factor, r_l is the adaptive pattern and PIC-cognizant controller set-point, ${\widehat{I}}_k$ is the estimated PIC, V_I is the insulin distribution volume, and therefore, ${\widehat{I}}_k\cdot V_I$·provides an estimate of the total insulin present in the body. Further, I^IOB represents the available insulin present in the body calculated from different IOB curves based on the CGM measurement y_k using the previous basal and bolus insulin delivery [28, 29, 46]. Note that the only difference between cases mentioned in Eq. (10) is the maximum allowable bolus u_Bolus,max. The PIC bound constraints are personalized to each subject using the patient-specific basal infusion rate. The safety insulin constraints defined for the desired, upper and lower bounds are based on the normalized PIC values that are calculated by dividing the PIC by the basal PIC. The patient-specific basal PIC is the value of the PIC if only basal insulin is administered. The MPC parameters of the proposed formulation are tuned, and the parameter values deemed to provide satisfactory closed-loop results are Q^nom = 100 and R^nom = 1. The prediction horizon for each subject is determined by the time course of insulin action as $n_P={\widehat{t}}_{\text{max},\text{I},\text{k}}/T_s$. We used MATLAB quadratic programming (QP) solver quadprog with the interior-point-convex algorithm to solve QP problem in the MPC. As the CGM and infused insulin data are not normally distributed according to the different normality tests such as Kolmogorov-Smirnov test, median with 25th-75th percentiles of the data are presented in this section.

The quantitative evaluation of the closed-loop results based on the proposed PIC-cognizant MPC algorithm are presented in Table 3, which gives the percentage of samples in defined glycemic ranges and selected statistics for the glucose measurements. It is readily observed that no hypoglycemia occurs as the BGC is never below 70 mg/dL. Furthermore, the average percentage of time spent in the target range (BGC ∈ [70, 180] mg/dL) and the higher tier of BGC ∈ [180, 250] mg/dL are 66.7% and 31.6%, respectively. The minimum and maximum observed BGC values across all experiments are 79 and 291 ml/dL, respectively. The BGC is thus tightly controlled to be within the safe range for a significant percentage of time. In contrast to the PIC-cognizant MPC, the IOB-constrained MPC causes some severe hypo- and hyperglycemic episodes for a few of the subjects, along with a significant decrease in the average percentage of time spent in the target range (BGC ∈ [70, 180] mg/dL), as shown in Table 4. Overall, the results demonstrate that the PIC-cognizant MPC is able to regulate BGC effectively without requiring meal announcement, as shown by the significant disturbances caused by the diverse timing and amounts of meals, while mitigating severe hypo- and hyperglycemic excursions.

Table 3:

PIC-cognizant MPC results for percentage time spent in different BGC ranges and various statistics for closed-loop experiment using the UVA/Padova metabolic simulator. The Avg values refer to the mean value of each metrics over all subjects.

Subject	Percent of time in range						Statistics				Daily Insulin (U)
	< 55	[55, 70)	[80, 140]	[70, 180)	[180, 250)	> 250	Median	[Q1,Q3]	Min	Max	Total	Basal
1	0.0	0.0	39.9	73.1	26.9	0.0	155	[103, 181]	89.0	226.0	39.5	26.3
2	0.0	0.0	38.4	60.1	38.7	1.2	165	[101, 207]	82.0	261.0	39.6	22.4
3	0.0	0.0	59.3	80.6	19.4	0.0	130	[107, 170]	81.0	224.0	21.0	11.9
4	0.0	0.0	36.9	55.2	43.2	1.6	171	[104, 214]	79.0	263.0	26.2	15.6
5	0.0	0.0	38.2	56.9	36.0	7.1	163	[128, 217]	93.0	291.0	57.0	23.4
6	0.0	0.0	42.4	66.0	33.3	0.7	153	[102, 201]	84.0	256.0	28.6	14.1
7	0.0	0.0	40.4	64.4	30.8	4.7	157	[107, 196]	100.0	277.0	34.4	19.5
8	0.0	0.0	37.5	63.3	35.6	1.1	163	[105, 202]	90.0	257.0	38.8	23.3
9	0.0	0.0	39.1	68.4	31.6	0.0	160	[102, 186]	87.0	236.0	46.0	29.2
10	0.0	0.0	47.0	79.1	20.9	0.0	145	[107, 172]	90.0	235.0	25.8	12.1
Avg	0.0	0.0	41.9	66.7	31.6	1.6	156	[107, 195]	88	253	35.7	19.8

Table 4:

IOB-constrained MPC results for percentage time spent in different BGC ranges and various statistics for closed-loop experiment using the UVA/Padova metabolic simulator. The Avg values refer to the mean value of each metrics over all subjects.

Subject	Percent of time in range						Statistics				Daily Insulin (U)
	< 55	[55, 70)	[80, 140]	[70, 180)	[180, 250)	>250	Median	[Q1, Q3]	Min	Max	Total	Basal
1	0.0	0.0	3.5	31.7	56.0	12.3	206	[167, 236]	100	279	35.9	29.6
2	0.0	0.0	7.7	37.1	43.0	19.9	192	[167, 237]	82	336	36.0	27.2
3	0.0	0.0	3.3	37.1	57.1	5.9	195	[164, 224]	100	285	20.4	16.8
4	0.0	0.0	5.8	24.2	43.9	31.9	207	[181, 265]	83	357	24.5	18.1
5	0.5	1.2	11.2	29.8	40.5	28.0	213	[172, 254]	42	344	55.0	35.3
6	0.3	0.2	8.6	36.2	38.5	24.7	196	[163, 248]	43	334	26.1	19.5
7	0.0	0.0	12.9	39.0	35.5	25.5	191	[160, 251]	100	335	27.6	22.2
8	0.0	0.0	9.0	44.1	33.8	22.1	189	[161, 240]	86	343	34.8	26.4
9	0.0	0.0	4.8	31.2	63.7	5.0	198	[171, 221]	100	290	43.7	33.6
10	2.8	3.3	25.6	57.2	32.2	4.6	166	[123, 188]	40	294	26.5	12.9
Avg	0.4	0.5	9.2	36.8	44.4	18.0	195	[163, 236]	78	320	33.0	24.2

The glycemic trajectory and corresponding insulin dosing decisions computed by the PIC-cognizant MPC and IOB-constrained MPC are shown in the sub-figures (a) and (b) of Figs. 6 and 7, respectively. The glucose values remain within or close to the target range for a significant duration of the experiment with the PIC-cognizant MPC. The basal insulin is sometimes reduced or even completely shutoff, to proactively mitigate hypoglycemia. The reduction in basal insulin is automatically computed by the controller when the PIC is greater than its maximum limit and thus no additional insulin infusion is necessary. The continuation of the basal rate may have resulted in hypoglycemia as a result of overcorrection. The insulin boluses occur in close proximity to the unannounced meals. The PIC-cognizant MPC is capable of effectively regulating the glucose levels and counteracting the unannounced meal disturbances. In contrast, the IOB-constrained MPC results in CGM outputs with greater variability, and with significant glycemic excursions in the hypo- and hyperglycemic ranges, which is attributed to the controller being unable to compute an accurate amount of insulin infusion rate. This deficient performance of the IOB-constrained MPC is caused by the inadequate information on the current and predicted future levels of active insulin present in the body. Fig. 8 shows the variations in the normalized PIC and CGM values for the PIC-cognizant MPC and IOB-constrained MPC, and the PIC-cognizant MPC is able to compute more appropriate insulin infusion rates, which results in PIC values being relatively closer to the desired values than the IOB-constrained MPC.

Figure 6:

Closed-loop CGM output measurements using the UVa/Padova metabolic simulator for 10 adult subjects with (a) PIC-cognizant MPC, (b) IOB-constrained MPC, (c) MPC without insulin constraints, and (d) Pattern and PIC-cognizant MPC

Figure 7:

Closed-loop insulin infusion rates using the UVa/Padova metabolic simulator for 10 adult subjects with (a) PIC-cognizant MPC, (b) IOB-constrained MPC, (c) MPC without insulin constraints, and (d) Pattern and PIC-cognizant MPC

Figure 8:

PIC values based on CGM output measurements fora random select closed-loop experiment with (a) IOB-constrained MPC and (b) PIC-cognizant MPC

The average results for an MPC without insulin constraints are presented in Table 5 (the detailed closed-loop results for each subject can be found in Table D.1), where the controller computes the injected insulin without the IOB or PIC constraints. The quantitative evaluation of the closed-loop results show a poor control performance as all subjects experience severe low BGC. The glycemic trajectory and corresponding insulin dosing decisions computed by the MPC without insulin constraints are shown in the sub-figure (c) of Figs. 6 and 7, respectively. The BGC are low for a significant amount of time due to the overdosing of insulin.

Table 5:

The average of MPC results for percentage time spent in different BGC ranges and various statistics for closed-loop experiment using the UVA/Padova metabolic simulator. Case 1: MPC without insulin constraints, Case 2: IOB-constrained MPC, Case 3: PIC-cognizant MPC, Case 4: Pattern and PIC-cognizant MPC.

Subject	Percent of time in range						Statistics				Daily Insulin (U)
	<55	[55, 70)	[80, 140]	[70, 180)	[180, 250)	>250	Median	[Q1, Q3]	Min	Max	Total	Basal
Case 1	29.5	7.4	28.0	49.4	11.6	2.2	92	[55, 151]	40	287	45.6	6.9
Case 2	0.4	0.5	9.2	36.8	44.4	18.0	195	[163, 236]	78	320	33.0	24.2
Case 3	0.0	0.0	41.9	66.7	31.6	1.6	156	[107, 195]	88	253	35.7	19.8
Case 4	0.0	0.0	44.4	75.3	24.4	0.4	149	[105, 181]	82	240	36.6	17.0

The results for a pattern and PIC-cognizant MPC show further improvement in performance (Table 5). The detailed closed-loop results for each subject can be found in Table D.2. No hypoglycemia occurs and the average percentage of time spent in the target range (BGC ∈ [80, 140] mg/dL) and the higher tier of BGC ∈ [70, 180] mg/dL are 44.4% and 75.3%, respectively. The results also show that the average percentage of time spent in high BGC range (BGC > 250 mg/dL) reduces significantly compared to all other controllers. The minimum and maximum observed BGC values across all experiments are 72 and 294 ml/dL, respectively. The BGC is tightly controlled to be within the safe range for a significant duration of time.

To evaluate the effect of the PIC estimates on the PIC-cognizant MPC, simulation case studies are conducted where the estimated PIC is replaced with the actual PIC value obtained from the metabolic simulator. Fig. 9 compares the estimated PIC to the actual PIC obtained from the metabolic simulator and the closed-loop control results computed from using the two PIC values. Note that the estimated PIC closely tracks the actual PIC, and thus the PIC-cognizant MPC using the estimated PIC value performs similarly to the MPC using the actual PIC value. As the incorrect specification of the basal insulin rate may also challenge the performance of the PIC-cognizant MPC algorithm, simulation results with incorrect specification of the patient-specific basal insulin rate are conducted (Fig. 10). Note that a ±10% deviation in specifying the patient-specific basal insulin rate can deteriorate the closed-loop performance by increasing or decreasing the optimum insulin infusion. This causes the CGM values to be higher or lower depending on the inaccuracy in the basal insulin value. Despite the changes in the control performance based on the misrepresentation of the basal insulin rate, the CGM is still relatively tightly controlled.

Figure 9:

Comparison of (a) real (computed by the UVa/Padova simulator) and estimated PIC and (b) the closed-loop CGM output measurements for the PIC-cognizant MPC using the real and estimated PIC in the UVa/Padova metabolic simulator for a select adult subject

Figure 10:

Comparison of (a) estimated PIC and (b) the closed-loop CGM output measurements for the PIC-cognizant MPC using the estimated PIC with ±10% deviation in the patient-specific basal insulin rate in UVa/Padova metabolic simulator for a select adult subject

5. Discussion

In this work, a physiological model drawn from the insulin compartment of Hovorka’s model is incorporated with a data-driven model developed using a recursive subspace identification technique. There are several benefits in the integration of compartment models with recursively identified state-space models. The PIC estimates are adaptive and individualized to patients, and thus provide accurate information on the amount of active insulin present in the body. This PIC information can be used for imposing constraints on the insulin dose computation algorithms. Furthermore, in contrast to the discrete basal and acute bolus insulin variations, the estimated PIC from the compartment model readily provides a more appropriate information for model identification. The subspace identification technique with recursive modeling renders the identified models valid over a diverse range of daily activities without requiring onerous and approximate information such as the amount of carbohydrate consumption. The proposed modeling approach with the integrated compartment model can also utilize other additional variables such as biosignals to consider the effects of physical activity on blood glucose concentration.

The PIRI is used to manipulate the penalty weights of the objective function and is specified using the PIC value $({\widehat{I}}_k)$ normalized by the basal PIC $\left({\tilde{u}}_{\text{basal},k}\right)$. The daily basal insulin rate for each patient should be specified appropriately as it affects the weightings of the MPC objective function through the PIRI. Furthermore, the daily basal insulin rate is also used to define the bound constraints for the estimated PIC. The PIC bounds, along with the risk indexes, govern the aggressiveness of the controller. A minimum bound for the PIC is considered in the MPC formulation to enforce the controller to suggest a safe amount of insulin to derive the CGM towards the specified set-point target value in a reasonable amount of time. A maximum bound is considered to avoid giving large doses of insulin that may cause hypoglycemia as a result of overcorrection. A desired PIC value between minimum and maximum bounds for the PIC based on the predicted CGM value is also considered to avoid excess variability in the CGM measurements caused by variations in the PIC values.

The insulin concentration in the bloodstream should be maintained within a safe range. If the PIC decreases to low values (less than the ${\tilde{u}}_{\text{basal}}$ value that characterizes the PIC without disturbances and only steady basal insulin infusion), then the BGC may rise rapidly in response to meal consumption. The rapid increase in the BGC due to carbohydrate consumption and low PIC may cause hyperglycemia, and consequently the controller may suggest a large bolus to derive the high BGC towards the set-point. However, the significant delays in the glucose-insulin dynamics may result in an overcorrection of the high glucose values, which may adversely lead to hypoglycemia. Such abrupt and counteracting behavior should be avoided for effective glucose regulation. One approach to ensure such unfavorable dynamics are avoided is to effectively negotiate the trade-offs between the opposing criteria of the cost function. The glycemic and plasma insulin risk indexes are defined to maintain PIC close to the basal value under normal conditions and thus increase the effectiveness of the AP therapy.

Since bound constraints are considered for the PIC, and as PIC is one of states in the model used in designing the predictive controller, the MPC prediction horizon should be specified sufficiently large to capture the prolonged effects of insulin on the glucose measurements. Specifically, the prediction horizon should be large enough that the peak effect of the administered insulin is evident in the predicted future PIC values. The t_max,I(t) parameter in Hovorka’s model characterizes the time duration of PIC reaching its peak value in response to administered insulin. Therefore, the prediction horizon of the predictive controller should be defined according to the time duration of the insulin subsystem. The adaptive and personalized PIC estimator is able to provide accurate estimates of the insulin present in the bloodstream for direct use in the control algorithm. The presented results are based on an MPC controller without incorporating any additional AP modules like the meal detection and carbohydrate estimation module that automatically recognizes carbohydrate consumption and suggests appropriate boluses [38, 47–49]. Such modules have the potential to further improve the closed-loop performance of the proposed PIC-cognizant MPC for use in safe and reliable AP systems.

The closed-loop stability of the control system is an important criterion to establish. However, establishing stability of an MPC used for glycemic control is not straightforward due to the very complex nature of BGC dynamics in people with T1DM. The time-varying characteristics of the human body and its highly nonlinear behavior, presence of stochastic and unknown disturbances, and uncertain time-varying delays make the problem challenging. For example, the input (infused insulin) affects the glucose dynamics for several hours, which is much longer than the considered length of the MPC prediction horizon for computation reasons. Furthermore, glucose measurements can continue to rise even though a substantial amount of insulin has already been infused because of the slow dynamics in the absorption and utilization of the infused insulin. Additionally, glucose-insulin dynamics vary substantially over time, which necessitates adaptive control schemes. For the AP systems, neither the available physiological models nor the data-driven techniques alone can accurately describe the dynamic behavior of BGC variations because the glucose-insulin dynamics, disturbance effects, and time delays are not well known. The effects of major disturbances to the BGC (such as physical activity, meals, and stress) are not well characterized. Explicitly proving the stability of the closed-loop system for an AP system is a topic of future studies.

6. Conclusions

An adaptive MPC algorithm cognizant of the PIC is proposed for AP systems. A personalized compartment model that translates the abrupt bolus and discrete basal changes into estimates of PIC is integrated with a recursive subspace-based system identification approach to characterize the transient dynamics of glycemic measurements. Subsequent to system identification, an MPC algorithm is designed using the adaptive models to effectively compute the optimal exogenous insulin delivery for AP systems without requiring any user-specified information on the time and amount of carbohydrate consumption. A dynamic safety constraint derived from the estimation of PIC is incorporated in the proposed adaptive MPC algorithm for the efficacy and reliability of the AP system, and to prevent insulin overdosing. The efficiency of the proposed adaptive MPC algorithm cognizant of PIC is demonstrated using simulation case studies.

Highlights.

• An adaptive model predictive control (MPC) with dynamic safety constraint

• Recursive subspace-based system identification to identify stable, high-fidelity linear time-varying models from closed-loop data

• Artificial pancreas system with plasma insulin concentration estimation to prevent insulin overdosing

• Simulation case studies to demonstrate the performance of the proposed adaptive MPC algorithm.

Appendix A -. The glucose-insulin dynamic model of Hovorka

The glucose-insulin dynamic model of Hovorka has different compartments to characterize the blood glucose dynamics, the subcutaneous insulin infusion and the glucose transport from plasma to interstitial tissues. The model equations are

$$\frac{\text{d}{S}_1(t)}{\text{d}t}=u(t)-\frac{S_1(t)}{t_{\text{max},\text{I}}(t)}$$ (A.1a)

$$\frac{\text{d}{S}_2(t)}{\text{d}t}=\frac{S_1(t)}{t_{\text{max},\text{I}}(t)}-\frac{S_2(t)}{t_{\text{max},\text{I}}(t)}$$ (A.1b)

$$\frac{\text{d}I(t)}{\text{d}t}=\frac{S_2(t)}{t_{\text{max},\text{I}}(t)V_{\text{I}}}-k_{\text{e}}(t)I(t)$$ (A.1c)

$$\frac{\text{d}{x}_1(t)}{\text{d}t}=-k_{\text{a},1}{x}_1(t)+k_{\text{b},1}I(t)$$ (A.1d)

$$\frac{\text{d}{x}_2(t)}{\text{d}t}=-k_{\text{a},2}{x}_2(t)+k_{\text{b},2}I(t)$$ (A.1e)

$$\frac{\text{d}{x}_3(t)}{\text{d}t}=-k_{\text{a},3}{x}_3(t)+k_{\text{b},3}I(t)$$ (A.1f)

$$\begin{gathered} \frac{\text{d}{Q}_1(t)}{\text{d}t}=U_{\text{G}}(t)-F_{01}^c(t)-F_R(t)-x_1(t)Q_1(t) \\ +k_{12}{Q}_2(t)+{\text{EGP}}_0\left(1-x_3(t)\right) \end{gathered}$$ (A.1g)

$$\frac{\text{d}{Q}_2(t)}{\text{d}t}=x_1(t)Q_1(t)-\left(k_{12}+x_2(t)\right)Q_2(t)$$ (A.1h)

$$\frac{\text{d}{G}_{\text{sub}}(t)}{\text{d}t}=\frac{1}{\tau }\left(\frac{Q_1(t)}{V_{\text{G}}}-G_{\text{sub}}(t)\right)$$ (A.1i)

where u(t) is the injected insulin (basal and bolus), the two state variables Q₁(t) and Q₂(t) denote the masses of glucose in the accessible and non-accessible compartments, respectively, and state variables S₁(t) and S₂(t) denote the absorption rate of subcutaneously administered insulin. The PIC, I(t), is represented by a first-order differential equation. The insulin action is calculated by using three variables, the influence on transport and distribution (x₁(t)), the utilization and phosphorylation of glucose in adipose tissue (x₂(t)), and the endogenous glucose production in the liver (x₃(t)). The subcutaneous glucose concentration is G_sub(t). The relationship between the BGC and the subcutaneous glucose concentration is considered a first-order dynamic equation. A continuous glucose monitoring (CGM) sensor, a wearable medical device, provides real-time measurement of subcutaneous glucose concentration every 5 minutes. All variables, parameters, units and nominal values used in the model (Eq. (A.1)) are provided in Table A.1.

Table A.1:

Variables, Parameters and Nominal Values for the Hovorka’s Model [26]

Symbol	Definition	Value (unit)
S1(t), S2(t)	Subcutaneously administered insulin absorption	– (mU)
I(t)	Plasma insulin concentration	– (mU L−1)
x1(t)	Effects of insulin on glucose distribution/transport	– (min−1)
x2(t)	Effects of insulin on glucose disposal	– (min−1)
x3(t)	Effects of insulin on endogenous glucose production	– (min−1)
Q1(t)	Masses of glucose in the accessible compartment	– (mmol)
Q2(t)	Masses of glucose in the non-accessible compartment	– (mmol)
GSub(t)	Subcutaneous glucose	– (mmol L−1)
UG(t)	Gut absorption rate	– (mmol min−1)
u(t)	Administration (bolus and basal) of insulin	– (mU min−1)
ke(t)	Insulin elimination from plasma	– (min−1)
tmax,I(t)	Time-to-maximum of absorption of injected insulin	– (min)
k12	Transfer rate	0.066 (min−1)
ka1	Deactivation rate	0.006 (min−1)
ka2	Deactivation rate	0.06 (min−1)
ka3	Deactivation rate	0.03 (min−1)
VI	Insulin distribution volume	0.12 × BW (L)
VG	Glucose distribution volume	0.16 × BW (L)
$S_{IT}^f$	Insulin sensitivity of distribution/transport	51.2 × 10−4(L min−1 mU−1)
$S_{ID}^f$	Insulin sensitivity of disposal	8.2 × 10−4(L min−1 mU−1)
$S_{IE}^f$	Insulin sensitivity of EGP	520 × 10−4(L mU−1)
EGP0	EGP extrapolated to zero insulin concentration	0.0161 (mmol kg−1 min−1)
F01	Non-insulin-dependent glucose flux	0.0097 (mmol kg−1 min−1)
BW	Weight	– (kg)
kb1	$=S_{IT}^f\times k_{a1}$
kb2	$=S_{ID}^f\times k_{a2}$
kb3	$=S_{IE}^f\times k_{a3}$
$F_{01}^c$	$=\{\begin{array}{ll}{F}_{01}\hfill & \text{ if }{G}_{\text{sub}}(t)\ge 4.5\text{ mmol}/\text{L}\hfill \\ F_{01}{G}_{\text{sub}}(t)/4.5\hfill & \text{ otherwise. }\hfill \end{array}$
FR	$=\{\begin{array}{ll}0.003\left(G_{\text{sub}}(t)-9\right)V_G\hfill & \text{ if }{G}_{\text{sub}}(t)\ge 9\text{mmol}/\text{L}\hfill \\ 0\hfill & \text{ otherwise. }\hfill \end{array}$

The nonlinear model, though a good representation of the glucose-insulin dynamics, may not capture the stochastic uncertainties and variations in the glycemic dynamics. To handle these unknown stochastic disturbances, we incorporate additive process and measurement noises with the noise characteristics estimated from real CGM data. Converting the continuous-time nonlinear glucose-insulin model to a discrete-time model through explicit Euler method with a sampling time T_s of 5 minutes, and incorporating uncertainty in the dynamics of the discrete-time system (referred to as process noise) and noise in the measurements, the discretized model can be written in the form

$$x_{k+1}=f\left(x_k,u_k,{\theta }_k\right)+{\omega }_k,{\omega }_k~\mathcal{N}\left(0,Q^{(x)}\right)$$ (A.2a)

$$y_k=g\left(x_k,u_k,{\theta }_k\right)+v_k,v_k~\mathcal{N}\left(0,R^{(y)}\right)$$ (A.2b)

where $x_k\in {ℝ}^{n_x}$, $u_k\in {ℝ}^{n_u}$ and $y_k\in {ℝ}^{n_y}$ denote the vectors of state, input, and output variables, respectively, with the output as the CGM measurements (that is, y_k ≔ G_sub,k + ν_k) and the input variable is the infused insulin. The sampling time T_s is chosen 5 minutes, the regular interval that CGM sensors take glucose measurements. The vector of state variables is

$$x_k={\left[\begin{array}{ccccccccc}{S}_{1,k}& S_{2,k}& I_k& x_{1,k}& x_{2,k}& x_{3,k}& Q_{1,k}& Q_{2,k}& G_{\text{sub},k}\end{array}\right]}^T$$

with ${\omega }_k\in {ℝ}^{n_x}$ and $v_k\in {ℝ}^{n_y}$ denoting the vectors of process and measurement noises, respectively, and ${\theta }_k\in {ℝ}^{n_{\theta }}$ denoting the vector of uncertain time-varying model parameters. In the simultaneous state and parameter estimation approach, the unscented Kalman filter (UKF) algorithm is used, though other nonlinear filtering algorithms may also be used to recursively compute both the state and parameter estimates at each sampling instance [50]. One of the critical states of the model is the PIC, which represents the available insulin in the bloodstream. Quantifying the PIC can enable manipulating the penalty weights of a predictive control algorithm to modulate the controller aggressiveness. Consider the subsystem that describes the subcutaneous insulin infusion and PIC (Eq. (A.1a)–Eq. (A.1c)), including the dynamics of the state variables S₁(t), S₂(t) and I(t). The uncertain parameters t_max,I(t) and k_e(t) have a direct effect on the PIC and are considered as extended states in the model (Eq. (A.1)). Furthermore, as information about the time and quantity of meals is difficult to ascertain and thus considered as unknown disturbances, the gut absorption rate is also included as an extended state in the model (Eq. (A.1)). The purpose of considering the gut appearance rate (U_G(t)) in the PIC estimator is to handle unknown disturbances present in the system, including the effects of unannounced meals (such as main meals or fast-acting carbohydrates for treatment of hypoglycemia) or other uncertainties in the model parameters (such as EGP₀, $F_{01}^c$ and F_R parameters). To achieve the simultaneous estimation using the UKF, the original model is transformed by treating the parameters θ_k ≔ [t_max,I,_k k_e,k U_G,k]^T to be estimated as additional states as follows:

$$x_{k+1}^{\prime }=f^{\prime }\left(x_k^{\prime },u_k\right)+{\omega }_k^{\prime },{\omega }_k^{\prime }~\mathcal{N}\left(0,Q^{(x{)}^{\prime }}\right)$$ (A.3a)

$$y_k^{\prime }=g^{\prime }\left(x_k^{\prime },u_k\right)+v_k^{\prime },v_k^{\prime }~\mathcal{N}\left(0,R^{(y{)}^{\prime }}\right)$$ (A.3b)

where the augmented state vector including the uncertain model parameters is $x_k^{\prime }={\left[\begin{array}{ll}{x}_k^T\hfill & {\theta }_k^T\hfill \end{array}\right]}^T$, ${\omega }_k^{\prime }$ and $v_k^{\prime }$ denote the vectors for the process and observation noises, Q^(x)′ and R^(y)′ denote the covariance matrix and variance of the process and measurement noises, respectively. As the uncertain model parameters do not have an explicit transition function or known dynamics, artificial dynamics are introduced to evolve the uncertain parameters [51]. One way to estimate the uncertain parameters is to consider the unknown parameters as additional state variables with the dynamics ${\theta }_{k+1}={\theta }_k+{\omega }_k^{(\theta )}$, where ω^(θ) is a sequence of independent Gaussian random variables realized from $\mathcal{N}\left(0,Q^{(\theta )}\right)$ and Q^(θ) is the covariance of the normally distributed random stochastic noise in the parameters. Although the random walk dynamics are not explicitly implemented, the assumption of the parameter dynamics enables the design of the UKF for estimating the augmented states [51]. Specifically, Q^(θ) is a tuning parameter that affects the Kalman gain calculation.

Therefore, the new system for simultaneous state and parameter estimation is

$$f^{\prime }\left(x_k^{\prime },u_k\right):=\left[\begin{array}{c}f\left(x_k,u_k,{\theta }_k\right)\\ {\theta }_k\end{array}\right]$$ (A.4a)

$$g^{\prime }\left(x_k^{\prime },u_k\right):=g\left(x_k,u_k,{\theta }_k\right)$$ (A.4b)

Employing the state and parameter estimation approach with Hovorka’s model allows for simultaneously estimating the PIC (I_k), which is a state variable in the model, along with other time-varying parameters. A nonlinear observer for the simultaneous estimation of the state variables and the time-varying parameters can be designed as

$${\widehat{x}}_k^{\prime }={\widehat{x}}_k^{\prime ,-}+K_k^{(\text{f})}\left(y_k-{\widehat{y}}_k^{\prime }\right)$$ (A.5a)

$${\widehat{y}}_k^{\prime }=g^{\prime }\left({\widehat{x}}_k^{\prime ,-}\right)$$ (A.5b)

where ${\widehat{x}}_k^{\prime ,-}=f^{\prime }\left({\widehat{x}}_{k-1}^{\prime },u_{k-1}\right)$ is the prior estimate of the augmented state vector, ${\widehat{x}}_k^{\prime }$ denotes the estimated augmented state vector, $K_k^{(\text{f})}$ is the observer gain obtained by the unscented Kalman filter (UKF) algorithm and ${\widehat{y}}_k^{\prime }$ is the estimated output [7, 50–52]. Bounds on the states and parameters can be similarly augmented as $x_{L,k}^{\prime }\le {\widehat{x}}_k^{\prime }\le x_{U,k}^{\prime }$, where $x_{L,k}^{\prime }:={\left[\begin{array}{cc}{x}_{L,k}^T& {\theta }_{L,k}^T\end{array}\right]}^T$ and $x_{U,k}^{\prime }:={\left[\begin{array}{ll}{x}_{U,k}^T\hfill & {\theta }_{U,k}^T\hfill \end{array}\right]}^T$ denote the lower and upper bounds on the augmented state vector. As the time-varying parameters estimated on-line represent the kinetics and dynamics of insulin action, knowledge of the system is used to construct bounds on the parameter estimates [6, 7]. For instance, the parameters and states cannot have negative values. All variables and parameters are defined in the Table A.2 and the dimensions are n_x = 9, n_u = 1, n_y = 1, and n_θ = 3. In this work, the number of augmented states is n_x + n_θ = 12 and the number of sigma points used in the UKF technique are 2 × 12 + 1 = 25.

Table A.2:

Nomenclature for the design of PIC estimator

xk	Vector of state variables in Hovorka’s model
f	Hovorka’s model states dynamic function
uk	Input variable in Hovorka’s model
θk	Vector of uncertain time-varying model parameters in Hovorka’s model
wk	Vector of process noises
vk	Measurement noise
Q(x)	Covariance matrix of the process noise
R(y)	Variance of the measurement noise
yk	Output variable in Hovorka’s model
g	Hovorka’s model output dynamic function
nx	Number of states in Hovorka’s model
nu	Number of input in Hovorka’s model
ny	Number of output in Hovorka’s model
nθ	Number of uncertain time-varying model parameters in Hovorka’s model
$x_k^{\prime }$	Augmented state vector including the uncertain model parameters
f′	Augmented Hovorka’s model states dynamic and uncertain model parameters function
$y_k^{\prime }$	Output variable in augmented Hovorka’s model
${\omega }_k^{\prime }$	Augmented process and uncertain model parameters noises
$v_k^{\prime }$	Augmented measurement and uncertain model parameters noises
g′	Augmented output dynamic and uncertain model parameters function
Q(x)′	Covariance matrix of the process and uncertain model parameters noises in the augmented model
R(y)′	Variance of the measurement noise in the augmented model
Q(θ)	Covariance of the normally distributed random stochastic noise in the uncertain model parameters
ω(θ)	A sequence of independent Gaussian random variables
$K_k^{(\text{f})}$	Observer gain
${\widehat{x}}_k^{\prime }$	Posterior estimate of augmented state vector including the uncertain time-varying model parameters
${\widehat{x}}_k^{\prime ,-}$	Prior estimate of augmented state vector including the uncertain time-varying model parameters
${\widehat{y}}_k^{\prime }$	Estimated output variable
Ts	Sampling time
xL,k	Lower bounds of states
xU,k	Upper bounds of states
θL,k	Lower bounds of uncertain time-varying model parameters
θU,k	Upper bounds of uncertain time-varying model parameters
$x_{L,k}^{\prime }$	Lower bounds of augmented states and uncertain time-varying model parameters
$x_{U,k}^{\prime }$	Upper bounds of augmented states and uncertain time-varying model parameters

Appendix B -. The recursive PBSID Technique

The recursive PBSID approach is the identification technique used to track the time-varying system by adapting a linear model [10–13]. In this modeling approach, the stability and fidelity of recursively identified models are ensured by considering constraints for the stability and the correctness of the gain between inputs and output.

Consider a vector autoregressive model with exogenous variables (VARX)

$${\widehat{y}}_{k|k-1}=\sum _{i=0}^p{\vartheta }_{k-i}^u{\tilde{u}}_{k-i}+\sum _{i=1}^p{\vartheta }_{k-i}^y{y}_{k-i}$$ (B.1)

where ${\widehat{y}}_{k|k-1}$ is the predicted CGM output for the kth sampling instance using the past PIC estimates ${\tilde{u}}_k,\dots ,{\tilde{u}}_{k-p}$ and CGM outputs y_k−1, …, y_k−p. The VARX model parameter p is the length of the past window of data considered when predicting the future output. Furthermore, the stacked vector y_k−p,p is defined with respect to the past window of length p as $y_{k-p,p}={\left[y_{k-p}^T{y}_{k-p+1}^T\dots y_{k-1}^T\right]}^T$. Stacked vector ${\tilde{u}}_{k-p,p}$ is also similarly defined. The RLS filters can efficiently estimate the VARX model parameters employed in the PBSID algorithm by tracking the dynamic changes in the data. Nevertheless, measurement noise and unknown disturbances can cause inopportune overfitting and may lead to incorrect signs for the input-output gain terms, which is problematic as the controller designed using the inaccurate model may suggest inappropriate input actions. Although this adverse effect can be mitigated through larger forgetting factors that reduce sensitivity to noise and disturbances, the drawbacks of a sluggish response may be too onerous for certain applications. To overcome the adverse effects of the adaptive filters, the filtering approach is replaced with a constrained convex linear least squares optimization problem. The problem of updating the VARX model parameters is then recast as finding the optimal parameters for the most recently available historical data over a past window p as follows

$$\begin{gathered} {\text{Θ}}_k^{*}:=\underset{{\text{Θ}}_k\in {\text{Ω}}_{\text{Θ}}}{\text{arg min}}{\mathcal{J}}_{p,k} \\ \text{s.t.}\{{\widehat{y}}_{k-j|k-1-j}={\sum }_{i=0}^p{\vartheta }_{k-i}^u{\tilde{u}}_{k-i-j}+{\sum }_{i=1}^p{\vartheta }_{k-i}^y{y}_{k-i-j} \end{gathered}$$ (B.2)

with the objective function

$${\mathcal{J}}_{p,k}:=\sum _{j=0}^p{\Vert y_{k-j}-{\widehat{y}}_{k-j|k-1-j}\Vert }_{Q^{\prime }}^2$$

where Ω_Θ denotes the region of admissible VARX model parameter and ${\text{Θ}}_k\in {ℝ}^{2p+1\times 1}$ is the array of (constrained) VARX model parameters, taking values in a nonempty convex set ${\text{Ω}}_{\text{Θ}}:=\left\{{\text{Θ}}_k\in {ℝ}^{2p+1\times 1}:{\text{Θ}}_{\text{min}}\le {\text{Θ}}_k\le {\text{Θ}}_{\text{max}}\right\}$, and Θ_min and Θ_max denote the lower and upper bounds on the VARX model parameters. Incorporating constraints on the VARX model parameters ensures the sign of the gain relating the inputs to the outputs is correct in the identified state-space model, which improves the prediction accuracy. The computational load of the optimization can be minimized through appropriate initialization by using the solution from the previous sampling time as the initial guess in the current search.

Recognizing that the predicted state ${\tilde{x}}_k$ is given by

$${\tilde{x}}_k={\tilde{A}}^p{\tilde{x}}_{k-p}+\mathcal{L}{\tilde{u}}_{k-p,p}+\mathcal{K}{y}_{k-p,p}$$ (B.3)

where $\mathcal{L}$ and $\mathcal{K}$ denote the extended controllability matrices $\mathcal{L}=\left[{\tilde{A}}^{p-1}\tilde{B}\dots \tilde{A}\tilde{B}\tilde{B}\right]$, $\mathcal{K}=\left[{\tilde{A}}^{p-1}K\dots \tilde{A}KK\right]$, $\tilde{A}=A-KC$ and $\tilde{B}=B-KD$. Since D = 0, it implies that $\tilde{B}=B$. Assuming that the state transition matrix is nilpotent with degree p, that is the contribution of the initial state ${\tilde{x}}_{k-p}$ is negligible for sufficiently large $p\left({\tilde{A}}^p\approx 0\right)$, the predicted state can be expressed as

$${\tilde{x}}_k\approx \mathcal{L}{\tilde{u}}_{k-p,p}+\mathcal{K}{y}_{k-p,p}$$ (B.4)

The past window is hence chosen to be sufficiently large such that the contribution of the initial state can be neglected, which is a well known approximation in closed-loop identification [10, 11, 53]. Pre-multiplying the predicted state by the extended observability matrix Г gives

$$\Gamma {\tilde{x}}_k=\Gamma \mathcal{L}{\tilde{u}}_{k-p,p}+\Gamma \mathcal{K}{y}_{k-p,p}$$ (B.5)

with

$$\Gamma =\left[\begin{array}{c}C\\ C\tilde{A}\\ ⋮\\ C{\tilde{A}}^{f-1}\end{array}\right]$$

The product of the matrices $\Gamma \mathcal{L}$ and $\Gamma \mathcal{K}$ can be constructed from the VARX model coefficient matrices as

$$\Gamma \mathcal{L}=\left[\begin{array}{cccc}{\vartheta }_{k-p}^u& {\vartheta }_{k-p+1}^u& \cdots & {\vartheta }_{k-1}^u\\ 0& {\vartheta }_{k-p}^u& ⋮& {\vartheta }_{k-2}^u\\ \cdots & \cdots & \ddots & ⋮\\ 0& 0& \cdots & {\vartheta }_{k-f}^u\end{array}\right]$$

and

$$\Gamma \mathcal{K}=\left[\begin{array}{cccc}{\vartheta }_{k-p}^y& {\vartheta }_{k-p+1}^y& \cdots & {\vartheta }_{k-1}^y\\ 0& {\vartheta }_{k-p}^y& ⋮& {\vartheta }_{k-2}^y\\ \cdots & \cdots & \ddots & ⋮\\ 0& 0& \cdots & {\vartheta }_{k-f}^y\end{array}\right]$$

where f is the future window length.

Therefore, after estimating the VARX coefficient matrices, the estimated coefficient matrices ${\vartheta }^u$ and ${\vartheta }^y$ can be used to determine all quantities on the right-hand side of Eq. (B.5), and a SVD factorization can be used to readily obtain a low-rank approximation of the state sequence. For recursive identification, a selection matrix $\mathcal{S}$ of appropriate dimensions can be determined such that the basis of the state estimation is consistent at each sampling time as

$${\tilde{x}}_k={\mathcal{S}}_k{\mathcal{W}}_k\left(\Gamma \mathcal{L}{\tilde{u}}_{k-p,p}+\Gamma \mathcal{K}{y}_{k-p,p}\right)$$ (B.6)

where ${\mathcal{W}}_k$ is a pre-defined weight matrix and the selection matrix ${\mathcal{S}}_k$ can be recursively updated through the projection approximation subspace tracking (PAST) method [11]. In this work, the weight matrix W_k is considered to be a fixed identity matrix. The estimated state sequence is then employed along with the inputs and measured outputs to estimate the system matrices by solution of RLS problems. Specifically, after computing an estimate of the state sequence ${\tilde{x}}_k$, two RLS problems that ensure stability of the estimated system are used to determine the state-space matrices, thus yielding the identified model in Eq. (1). A concern regarding the typical recursive identification algorithm is that stochastic disturbances and measurement noise may render the identified models unstable even though the underlying system in inherently stable. Although an optimization problem may find the most appropriate and stable system realization, the solution time may be prohibitive. In this work, the optimization problem is substituted with a simple algorithm that incorporates line search mechanism to reduce the innovation term in the RLS filter in case the identified model becomes unstable. The optimization problems to be solved at each sampling instance for determining the system matrices are

$${\text{Φ}}_k^{(x)*}:=\underset{{\text{Φ}}_k^{(x)}}{\text{min}}{\Vert {\tilde{x}}_k-{\text{Φ}}_k^{(x)}{\psi }_k^{(x)}\Vert }_{F^{\prime }}^2$$ (B.7)

where ${\text{Φ}}_k^{(x)}:={\left[A_k{B}_k{K}_k\right]}^T,{\psi }_k^{(x)}:={\left[\begin{array}{ccc}{\tilde{x}}_{k-1}& {\tilde{u}}_{k-1}& e_{k-1}\end{array}\right]}^T$ as well as

$${\text{Φ}}_k^{(y)*}:=\underset{{\text{Φ}}_k^{(y)}}{\text{min}}{\Vert {\widehat{y}}_k-{\text{Φ}}_k^{(y)}{\psi }_k^{(y)}\Vert }_{H^{\prime }}^2$$ (B.8)

where ${\text{Φ}}_k^{(y)}:=C_k$ and ${\psi }_k^{(y)}:={\tilde{x}}_k$. The solutions to these optimization problems can be updated at each sampling time through recursive least squares filters. If the identified system becomes unstable, the innovation term $e_k^{(x)}:={\tilde{x}}_k-{\text{Φ}}_{k-1}^{(x)}{\psi }_k^{(x)}$ is reduced by multiplication with a scalar $\beta \in [0,1]$ as

$${\text{Φ}}_k^{(x)}:={\text{Φ}}_{k-1}^{(x)}+e_k^{(x)}{\psi }_k^{(x)}{P}_k{\beta }_k$$ (B.9)

where β_k is successively reduced in a line search approach until A_k is stable, with the error covariance matrix denoted P_k. Reducing the innovation term translates to discarding the new information, including disturbance or noise effects, that caused the identified system to become unstable. Note that without knowing the stochastic noise and disturbance characteristics, it is difficult to remove all the effects from the state transition matrix that cause the instability. Therefore, reducing the innovation term in the RLS algorithm until the identified system is stable provides a verifiable stable system realization for implementation. All variables and parameters are defined in the Table B.1 for completeness in which p = 6, f = 2, and ${\tilde{n}}_x=2$.

Table B.1:

Nomenclature for the PBSID technique

Ak	State (system) matrix
Bk	Input matrix
Kk	Kalman gain
Ck	Output matrix
ek	Deviation between the CGM output measurement and the predicted model output
p	Past window
f	Future window
${\tilde{u}}_k$	PIC estimate $({\widehat{I}}_k)$
${\tilde{x}}_k$	States of the identified model
${\widehat{y}}_k$	Predicted output by the VARX model
${\tilde{y}}_k$	Output of the identified model
${\vartheta }_k^u$	VARX model parameters for past PIC estimates
${\vartheta }_k^y$	VARX model parameters for past outputs
${\mathcal{W}}_k$	Weight matrix
${\mathcal{S}}_k$	Selection matrix
Pk	Error covariance matrix
${\tilde{n}}_x$	Number of states in the identified model
Q′	Positive definite matrix weight used in the VARX model optimization problem
F′, H′	Positive definite matrix weight used in the state space model optimization problem
ΩΘ	Region of admissible VARX model parameter
Θk	VARX model parameters
Θmin	Lower bounds of VARX model parameters
Θmax	Upper bounds of VARX model parameters
Γ	Extended observability matrix
$\mathcal{L}$	Extended controllability matrices
${\tilde{u}}_{k-p,p}$	Stacked vector of past PIC estimates
yk–p,p	Stacked vector of past outputs

Appendix C -. The derivations for calculating the parameters of the integrated glycemic model

$$p_1^I=\left(\frac{1}{V_{\text{I}}}\right)\cdot {\left(\frac{T_s}{{\widehat{t}}_{\text{max},\text{I},k}}\right)}^2\cdot \left(1-\left(\frac{T_s}{{\widehat{t}}_{\text{max},\text{I},k}}\right)\right)$$ (C.1)

$$p_2^I=\left(\frac{1}{V_{\text{I}}}\right)\cdot \left(\frac{T_s}{{\widehat{t}}_{\text{max},\text{I},k}}\right)\cdot \left(1-\left(\frac{T_s}{{\widehat{t}}_{\text{max},\text{I},k}}\right)\right)$$ (C.2)

$$p_3^I=1-T_s\cdot {\widehat{k}}_{\text{e},k}$$ (C.3)

$$p_1^{S_1}=p_2^{S_2}=1-\left(\frac{T_s}{{\widehat{t}}_{\text{max},\text{I},k}}\right)$$ (C.4)

$$p_1^{S_2}=\left(\frac{T_s}{{\widehat{t}}_{\text{max},\text{I},k}}\right)\cdot \left(1-\left(\frac{T_s}{{\widehat{t}}_{\text{max},\text{I},k}}\right)\right)$$ (C.5)

$$p_4^I=p_5^I=\left(\frac{T_s}{V_{\text{I}}}\right)\cdot {\left(\frac{T_s}{{\widehat{t}}_{\text{max},\text{I},k}}\right)}^2$$ (C.6)

$$p_3^{S_2}=T_s\cdot \left(\frac{T_s}{{\widehat{t}}_{\text{max},\text{I},k}}\right)$$ (C.7)

In addition to the parameters, the initial value of the state vector in the model Eq. (4) can be determined using known previous information as follows:

$$S_{2,k+1}=\left(\frac{T_s}{{\widehat{t}}_{\text{max},\text{I},k}}\right)\cdot \left({\widehat{S}}_{1,k}-{\widehat{S}}_{2,k}\right)+{\widehat{S}}_{2,k}$$ (C.8)

$${\tilde{u}}_{k+1}=T_s\cdot \left(\frac{{\widehat{S}}_{2,k}}{\left(V_1\cdot {\widehat{t}}_{\text{max},1,k}\right)}-{\widehat{k}}_{e,k}\cdot {\widehat{I}}_k\right)+{\widehat{I}}_k$$ (C.9)

$${\tilde{u}}_{k+2}=T_s\cdot \left(\frac{S_{2,k+1}}{\left(V_{\text{I}}\cdot {\widehat{t}}_{\text{max},\text{I},k}\right)}-{\widehat{k}}_{\text{e},k}\cdot {\tilde{u}}_{k+1}\right)+{\tilde{u}}_{k+1}$$ (C.10)

The values of ${\widehat{S}}_{1,k}$, ${\widehat{S}}_{2,k}$ and ${\widehat{I}}_k$ are obtained from an appropriate state estimation algorithm as in Eq. (A.5).

Appendix D -. Detailed results for MPC without insulin constraints and pattern and PIC-cognizant MPC

Table D.1:

MPC without insulin constraints results for percentage time spent in different BGC ranges and various statistics for closed-loop experiment using the UVA/Padova metabolic simulator. The Avg values refer to the mean value of each metrics over all subjects.

Subject	Percent of time in range						Statistics				Daily Insulin (U)
	<55	[55, 70)	[80, 140]	[70, 180)	[180,250)	>250	Median	[Q1, Q3]	Min	Max	Total	Basal
1	17.2	4.6	43.7	69.9	8.3	0.0	117	[77, 148]	40	236	39.8	7.2
2	20.7	8.5	31.3	59.1	10.3	1.4	102	[61, 156]	40	283	48.7	8.1
3	38.0	13.9	16.4	30.5	16.1	1.5	67	[47, 145]	40	288	37.6	4.1
4	29.9	6.8	25.4	50.2	12.5	0.6	99	[46, 160]	40	278	32.1	6.2
5	37.9	9.3	16.6	27.3	13.9	11.7	76	[40, 183]	40	356	91.3	12.1
6	70.8	2.5	11.9	21.0	3.9	1.8	40	[40, 80]	40	309	16.9	2.6
7	7.7	5.7	42.1	70.1	13.6	2.9	121	[87, 162]	40	312	46.2	8.0
8	18.8	6.2	36.0	62.6	11.9	0.5	115	[70, 157]	40	275	47.7	8.5
9	18.9	9.8	32.8	55.0	16.1	0.2	108	[65, 161]	40	257	58.7	7.8
10	35.4	6.5	23.5	47.8	9.4	0.9	90	[40, 154]	40	278	36.7	4.9
Avg	29.5	7.4	28.0	49.4	11.6	2.2	92	[55, 151]	40	287	45.6	6.9

Table D.2:

Pattern and PIC-cognizant MPC results for percentage time spent in different BGC ranges and various statistics for closed-loop experiment using the UVA/Padova metabolic simulator. The Avg values refer to the mean value of each metrics over all subjects.

Subject	Percent of time in range						Statistics				Daily Insulin (U)
	<55	[55,70)	[80, 140]	[70, 180)	[180,250)	>250	Median	[Q1, Q3]	Min	Max	Total	Basal
1	0.0	0.0	46.1	94.5	5.5	0.0	144	[102, 162]	82	204	40.6	21.6
2	0.0	0.0	40.0	65.3	34.7	0.0	158	[100, 190]	83	241	40.9	19.0
3	0.0	0.0	60.5	83.9	16.1	0.0	123	[105, 166]	72	214	21.2	11.1
4	0.0	0.0	37.5	62.0	38.0	0.0	164	[102, 197]	78	238	26.7	13.0
5	0.0	0.0	39.7	62.4	34.7	2.9	156	[123, 207]	75	294	58.0	21.2
6	0.0	0.0	46.9	72.7	27.3	0.0	146	[101, 183]	83	237	29.2	12.1
7	0.0	0.0	43.7	79.5	20.5	0.0	148	[106, 171]	100	247	37.1	15.3
8	0.0	0.0	39.0	69.3	30.0	0.6	156	[104, 186]	87	274	40.1	20.0
9	0.0	0.0	43.2	82.9	17.1	0.0	149	[102, 172]	84	219	46.6	25.1
10	0.0	0.0	46.9	80.1	19.9	0.0	145	[104, 173]	76	235	26.0	11.5
Avg	0.0	0.0	44.4	75.3	24.4	0.4	149	[105, 181]	82	240	36.6	17.0