Blood glucose concentration control for type 1 diabetic patients: a multiple‐model strategy

Abstract

In this study, a multiple‐model strategy is evaluated as an alternative closed‐loop method for subcutaneous insulin delivery in type 1 diabetes. Non‐linearities of the glucose–insulin regulatory system are considered by modelling the system around five different operating points. After conducting some identification experiments in the UVA/Padova metabolic simulator (accepted simulator by the US Food and Drug Administration (FDA)), five transfer functions are obtained for these operating points. Paying attention to some physiological facts, the control objectives such as the required settling time and permissible bounds of overshoots and undershoots are determined for any transfer functions. Then, five PID controllers are tuned to achieve these objectives and a bank of controllers is constructed. To cope with difficulties of the presence of delays in subcutaneous blood glucose (BG) measuring and in administration of insulin, a glucose‐dependent setpoint is considered as the desired trajectory for the BG concentration. The performance of the obtained closed‐loop glucose–insulin regulatory system is investigated on the in silico adult cohort of the UVA/Padova metabolic simulator. The obtained results show that the proposed multiple‐model strategy leads to a closed‐loop mechanism with limited hyperglycemia and no severe hypoglycemia.

1 Introduction

Diabetes is one of the most common endocrine diseases that occurs when the body is unable to produce enough insulin or not to use it properly [1]. Insulin is the most important factor in the regulation of the blood glucose (BG) level in the body which produced by the pancreas. It allows the body to use glucose of carbohydrates in the food for energy supply. According to the report of the International Diabetes Federation (IDF), 425 million people had suffered from diabetes in 2017, which is equivalent to one death every eight seconds [2].

Type 1 diabetes (T1D), also known as insulin‐dependent diabetes mellitus, is caused by an autoimmune disorder when the body immune system attacks the $\beta$ cells in the pancreas, which are responsible for secreting insulin [3, 4]. After a short time, the production of insulin in the body is completely discontinued. Hence, in order keep the BG level within the normal range (70–140 mg/dl), the patients with T1D must be rely on exogenous insulin. The low BG level (<70 mg/dl), which known as hypoglycemia, in a short time can cause dizziness, unconsciousness, coma, and even death and therefore, it must be treated immediately. In addition, high BG levels (>180 mg/dl), which is called hyperglycemia, in the long term can cause chronic diseases including retinopathy, nephropathy, and neuropathy. Currently, there is no definitive treatment for T1D patients and therefore, it cannot be assured that their BG levels have remained permanently in the normal range. The automatic regulation of the BG level in T1D patients is one the most active fields of research. This technology requires at least three components: a sensor that continuously measures the BG levels; a controller which determines the insulin delivery rate to be injected into the body; and an infusion pump to inject the insulin.

To design effective and reliable controllers for an artificial pancreas, many mathematical models were developed in the literature [4, 5]. These models range from simple expressions, that are just related to glucose and insulin subsystems, to very complex mathematical models, which are used in simulators [6]. The most common of these models are Ackerman's linear model, Bergman's non‐linear model which known as minimal model, and Cobelli's model [4]. Based on these models, many methods from control engineering field were designed to regulate the BG concentration of diabetic patients to its normal range (proportional–integral–derivative (PID) control [7, 8], $H_{\mathrm{\infty }}$ control [9, 10], model predictive control (MPC) [11–15], state‐dependent Riccati equation control [16, 17], adaptive control [18, 19], sliding mode control [20], error dynamics shaping [21], and switched linear parameter‐varying (LPV) controller [22]). Comprehensive surveys of the utilised controllers and their corresponding results can be found in [4–6, 23].

PID controllers are the most common examples of feedback control algorithms which have been used in many industrial processes. A PID controller or shortly a PID continuously calculates an error value $e(t)$ as the difference between the measured process output and its desired value and applies the control input based on PID terms to bring the measured value back to its desired setpoint. The main reasons for the popularity of the PIDs are (i) availability of many different methods to tune their parameters without applying advanced mathematics; (ii) having acceptable performance; (iii) simplicity of their implementation. Due to these interesting properties, the PID controllers and their extensions were widely used in artificial pancreas [7, 8, 24–27]. For example, Palazzo and Viti simulated a PID controller using Bergman's model [24]. In [25], Chee et al. implemented a step‐wise PID controller on the critically ill patient population. Gupakumaram et al. implemented a PD controller that has a feed‐forward component. Steil et al. implemented a PID controller using an intravenous glucose sensor and an implanted insulin pump [7]. Marchetti et al. [8], simulated a PID controller with a time‐varying desired value for the BG level. Although these controllers led to some remarkable results, they are so sensitive to delay and therefore, delays in glucose sensing and insulin action affect the system performance [22]. Indeed, based on insulin levels, these delays might cause the system to both overshoot and, more importantly, undershoot. In these situations, oscillatory behaviours are observed in the BG concentration and severe hypoglycemia might occur [4, 13]. On the other hand, the presence of delays in subcutaneous BG measuring and administration of insulin is inevitable due to some technological/physiological problems.

In this paper, to use the above‐mentioned advantages of PID controllers in BG concentration control of T1D patients and to tackle the negative effects of the delays, a multiple PID controller is designed. To this end, a bank of five PID controllers is constructed based on some simple linear models of the glucose–insulin system around five different equilibrium points. These models are obtained using system identification techniques based on the result of some tests on average adult patient in UVA/Padova T1D simulator. It should be mentioned that for any operating points (OPs), the control objectives such as the required settling time and permissible bounds of overshoots and undershoots are determined to pay attention to some physiological facts such as the risk of hypoglycemia and severe hyperglycemia. On the other hand, to reduce the negative effects of the delays, a glucose‐dependent setpoint is considered for the BG concentration ($G_{\mathrm{d}}$). Indeed, instead of a constant value for $G_{\mathrm{d}}$ which is generally about 120 mg/dl, a higher value is selected just after the meals. This value is decreased step by step until the BG concentration is regulated to the safe level. Note that the multiple PID controller is just designed for the average adult patient. Nevertheless, in silico experiments in the Food and Drug Administration (FDA)‐approved simulator (the UVA/Padova T1D simulator) show that this bank of PID controllers can be successfully applied to all ten adult patients with very limited hyperglycemia and no hypoglycemia. Comparison of the proposed multiple PID controller performance with two conventional PID controllers demonstrates that our designed controller is so effective. The obtained results are also compared with three recently designed controllers for T1D patients. In closing, it can be claimed that the PID controllers are still powerful enough to be used in an artificial pancreas provided that the proposed extensions are considered in its design procedure. The main contributions of this paper are as follows:

• (i) The presence of non‐linear dynamics is considered to use a multiple‐model strategy.

• (ii) The detrimental effects of time delays are reduced by considering a glucose‐dependent setpoint for the BG concentration.

• (iii) Individualisation of the proposed method can also achieve better performance in the BG concentration control.

The rest of the paper is organised as follows. Section 2 presents the main steps of the multiple‐model controller design. In Section 3, in silico experiments using the UVA/Padova T1D simulator are presented as a first step to validate the proposed multiple‐model approach. Finally, the conclusions of the paper are discussed in Section 4.

2 Methods

2.1 Multiple‐model control

Multiple‐model methods are useful techniques developed to deal with non‐linear dynamics [28]. These methods have become one of the ideal candidates to handle strong non‐linear systems. They can also be used when the system either operates in a wide range of operation space or have large setpoint changes [28]. The main idea of these techniques is to represent a complicated non‐linear system by a combination of local linear subsystems and therefore, linear techniques can be utilised to design a global controller [29]. Therefore, a multiple‐model controller generally consists of two main parts; (i) a bank of controllers which usually contains linear controllers designed based on linear local models of the system; (ii) a decision rule to select one of the controllers in the bank. These methods were successfully applied in a wide variety of applications, such as the boiler‐turbine unit control [30] and the pH neutralisation reactor control [29].

Due to the non‐linear dynamics of the glucose–insulin regulatory system, it is obvious that a simple linear controller is not able to achieve acceptable performance of an artificial pancreas. Therefore, it is so crucial to consider these non‐linearities in the design procedure of the controller. Nevertheless, such a task needs to know a suitable non‐linear model of the patient's glucose–insulin dynamics. Even if complicated system identification techniques are employed to find these non‐linear dynamics, some other problems still remain. Uncertainties in the obtained model, negative effects of delays in the BG measuring and the administration of insulin, and complexity of the design procedure are some of these problems. In this paper, to consider the non‐linear dynamics of the glucose–insulin regulatory system, a multiple‐model PID (MMPID) controller is designed. Towards this end, the needed linear models for the multiple‐model controller are obtained in the next Section 2.2. Then, in Section 2.3, these models are used to design PID controllers to construct a multiple‐model controller. Finally, in Section 2.4, to reduce negative effects of delays in glucose sensing and insulin action, a glucose‐dependent setpoint is designed for the multiple‐model controller.

2.2 System identification

In formulating the problem of the number of local models, care should be taken in defining the operating spaces. In this paper, five regions are considered. Indeed, the range of the possible BG concentration (50–300 mg/dl) is divided into five regions with OPs $G_i^{\ast }$, $i\in \{1,2,3,4,5\}$. The first one is associated with low BG concentration with OP $G_1^{\ast }=80\text{mg}/\text{dl}$ and the range $R_{\mathrm{l}}=\{G(t):G(t)\le 100\text{mg}/\text{dl}\}$. The second region covers the desired range $R_{\mathrm{d}}=\{G(t):100<G(t)\le 140\text{mg}/\text{dl}\}$ with OP $G_2^{\ast }=120\text{mg}/\text{dl}$. The third region is associated with acceptable range $R_{\mathrm{a}}=\{G(t):140<G(t)\le 170\text{mg}/\text{dl}\}$ with OP $G_3^{\ast }=155\text{mg}/\text{dl}$. The high BG level $R_{\mathrm{h}}=\{G(t):170<G(t)\le 190\text{mg}/\text{dl}\}$ is covered by the fourth region with OP $G_4^{\ast }=180\text{mg}/\text{dl}$. Finally, the fifth region is associated with very high BG concentration with OP $G_5^{\ast }=210\text{mg}/\text{dl}$ and the range $R_{\text{vh}}=\{G(t):G(t)>190\text{mg}/\text{dl}\}$. Corresponding to these five regions, we conduct five experiments to construct a multiple‐model with five different transfer functions. To this end, for any OP, a proper value for the basal insulin dosage ($U_i^{\ast }$ $i=1:5$) added by a zero‐mean exploration noise with small variance is delivered to the average adult patient in the UVA/Padova T1D simulator. Any experiment is conducted under fasting conditions for 120 min with sampling time 5 min. Then, sequences of the applied intravenous insulin rate and the intravenous BG concentration are used as the input/output data to identify a proper transfer function ($H_i^{\ast }(s)$ $i=1:5$) which is valid around the corresponding OP. In this step, MATLAB System Identification Toolbox ${}^{\text{TM}}$ is used to find these transfer functions. Using this toolbox, it is possible to build several types of mathematical models of dynamic systems using the system's input and output signals. In this paper, the least‐squares method is used to obtain a transfer function for any region. It is worth noting that a compromise between the accuracy of the identified models and their complexity should be made. Therefore, in the following, the second‐order transfer functions are utilised. Table 1 reports the obtained results of the identification step.

Table 1.

Results of the identification step

BG range	OP, mg/dl	$U^{\ast }$, U/h	Identified transfer function
$R_{\mathrm{l}}$	$G_1^{\ast }=80$	$U_1^{\ast }=7.8$	$H_1^{\ast }(s)={\displaystyle \frac{0.00062}{s^2+0.029s+0.0003}}$
$R_{\mathrm{d}}$	$G_2^{\ast }=120$	$U_2^{\ast }=6.6$	$H_2^{\ast }(s)={\displaystyle \frac{0.0012}{s^2+0.008s+7\times {10}^{-5}}}$
$R_{\mathrm{a}}$	$G_3^{\ast }=155$	$U_3^{\ast }=5.2$	$H_3^{\ast }(s)={\displaystyle \frac{0.0034}{s^2+0.012s+6\times {10}^{-5}}}$
$R_{\mathrm{h}}$	$G_4^{\ast }=180$	$U_4^{\ast }=4.8$	$H_4^{\ast }(s)={\displaystyle \frac{0.0019}{s^2+0.057s+0.0019}}$
$R_{\text{vh}}$	$G_5^{\ast }=210$	$U_5^{\ast }=3.6$	$H_5^{\ast }(s)={\displaystyle \frac{0.0031}{s^2+0.017s+0.0003}}$

It should be noted that these transfer functions are only identified for the average adult patient. Indeed, a bank of PID controllers is just designed based on these transfer functions and then, the multiple‐model controller is applied to all ten adult patients. Nevertheless, it is also possible to individualise this multiple‐model controller which leads to better performance in BG concentration control (see Section 3.4).

The intravenous insulin delivery rate and intravenous glucose measurements are used to take the identification step. In this situation, the negative effects of measurement noises on the identification results are negligible. While this is an invasive method, it is worth noting that these experiments are employed in short periods of time.

2.3 PID controllers design

In this subsection, for the transfer functions represented in Table 1, five PID controllers are designed. The continuous‐time causal form of the PID controller is used in our design as follows:

$$P+I\frac{1}{s}+D\frac{N}{1+s\left(1/N\right)},$$ (1)

where P, I, and D are the PID gains, respectively. N is also the additional low‐pass filtering for the derivative. The robustness of these controllers is so crucial due to two reasons; (i) in our design, unannounced meals (time and CHO amount are unknown to the controller) are considered as external disturbances and therefore, the PID controllers should be tuned in such a way that the effects of meals on the BG regulation are attenuated; (ii) The resulted multiple‐model PID controller is designed based on the data obtained from the average adult patient and is applied to all the other ten patients. In addition to the robustness of the controllers, some other important performance indexes such as the settling time ($t_{\mathrm{s}}$) and the maximum percentage of the overshoot ($M_{\mathrm{o}}$) or undershoot ($m_{\mathrm{u}}$) must be considered. For example, the postprandial hyperglycemia and hypoglycemia are related to $M_{\mathrm{o}}$ and $m_{\mathrm{u}}$, respectively. As another example, the percentage of time spent in a special target depends on the settling time $t_{\mathrm{s}}$. Nevertheless, based on the range of the BG concentration, these control objectives have different weight in our design. For instance, for $G_1^{\ast }=80\text{mg}/\text{dl}$, a fast response without any undershoot is mandatory to avoid hypoglycemia. On the other hand, for the OPs $G_5^{\ast }=210\text{mg}/\text{dl}$, more attention must be paid to the maximum overshoot of the closed‐loop system to avoid any hyperglycemia. Table 2 summarises the importance of the above control objectives in the five regions defined in Section 2.1.

Table 2.

Control objectives in the five considered regions

BG range	$t_{\mathrm{s}}$	$M_{\mathrm{p}}$	$m_{\mathrm{u}}$
$R_{\mathrm{l}}$	so fast ( ≤ 1 h)	not important	not allowed ($=0\%$)
$R_{\mathrm{d}}$	normal ( ≤ 4 h)	not important	normal ($\le 40\%$)
$R_{\mathrm{a}}$	normal ( ≤ 4 h)	normal ($\le 40\%$)	not important
$R_{\mathrm{h}}$	fast ( ≤ 2 h)	small ($\le 20\%$)	not important
$R_{\text{vh}}$	so fast ( ≤ 1 h)	not allowed ($=0\%$)	not important

There are also some rules of thumb to tune the parameters of the PID controllers with predictable results which can be used to achieve the control objectives summarised in Table 2. In closing, by increasing the proportional gain P, the settling time and the maximum percentage of the overshoot/undershoot are also generally increased [31]. While integral gain I has the same effects on $t_{\mathrm{s}}$, $M_{\mathrm{o}}$, and $m_{\mathrm{u}}$, the derivative part has generally reverse effects on these indexes. Keeping these facts in mind and using the PID Tuner in Simulink/MATLAB ${}^{\circledR }$, it is possible to tune the PID controllers to achieve the control objectives reported in Table 2. The obtained results are represented in Table 3.

Table 3.

Results of the PID controllers design

BG range	P	I	D	N
$R_{\mathrm{l}}$	1.4900	0.0053	190.5706	0.4530
$R_{\mathrm{d}}$	0.9400	0.0013	350.5706	0.5551
$R_{\mathrm{a}}$	1.1100	0.0024	320.7161	0.5519
$R_{\mathrm{h}}$	1.8510	0.0043	168.6012	0.5365
$R_{\text{vh}}$	1.9200	0.0045	138.6127	0.5310

2.4 Glucose‐dependent setpoint design

As mentioned before, there are two major sources of time delays in a closed‐loop glucose–insulin regulatory system; a physiological delay (plasma to interstitial) and a technical one (subcutaneous BG measuring). These delays impose serious challenges in glucose control and if not managed properly, they can cause severe episodes of hypoglycemia [4, 13]. To reduce these negative effects of the delays and have acceptable closed‐loop performance, we consider a glucose‐dependent setpoint for the desired value of the BG concentration. The plot of the considered setpoint $G_{\mathrm{d}}$ as a function of the BG is depicted in Fig. 1. As it can be seen from this figure, instead of a constant value for $G_{\mathrm{d}}$, a higher value is selected when the BG increases due to the meals. In these cases, the error between the BG concentration and its setpoint is not too much and therefore, less insulin is injected to the patient which leads to limited hypoglycemia during next hours. Let us explain this strategy by a numerical example. Assume at $t=t_0$, the BG concentration is 215 mg/dl. If a constant desired trajectory is considered, for example $G_{\mathrm{d}}=120$, the error between $G(t_0)$ and its desired value is 95 mg/dl. However, using the proposed glucose‐dependent setpoint depicted in Fig. 1, this error is 65 mg/dl. Paying attention to the values of the computed errors, less insulin is applied to the patient when the glucose‐dependent setpoint is used and hence, the risk of postprandial hypoglycemia is decreased. According to Fig. 1, by decreasing the BG concentration, its setpoint is also decreased step by step until the BG concentration is regulated to the safe level. On the other hand, for low values of the BG concentration which may cause severe hypoglycemia, this setpoint is increased and as a result, the injected insulin is reduced. In Section 3.3, performance of the closed‐loop system based on this strategy is compared with the system obtained by a constant setpoint. To sum up, it can be said that there are two feedback loops in our designed controller; (i) A feedback to select the setpoint of the BG concentration; (ii) A feedback to select the proper PID controller from the bank of controllers. Configuration of the proposed multiple‐model strategy is depicted in Fig. 2.

Fig. 1.

Graph of the desired value of the BG concentration

Fig. 2.

Configuration of the proposed multiple‐model PID controller with glucose‐dependent setpoint

3 Results

In this section, the closed‐loop performance of the proposed multiple‐model strategy is tested on the in silico adult cohort of the UVA/Padova metabolic simulator. The results are compared with three other recently published works in the field of our study. Positive effects of our main contributions, i.e. the ideas of the multiple‐model control and the glucose‐dependent setpoint, are investigated in Sections 3.2 and 3.3, respectively. Individualisation of the proposed controller is also studied in Section 3.4.

3.1 Multiple‐model PID control

All ten in silico adults of the UVA/Padova metabolic simulator are considered under the multiple‐model PID controller designed in Section 2. Continuous glucose monitoring (CGM) is used as the sensor and a generic continuous subcutaneous insulin infusion (CSII) pump is used to inject the insulin. Simulation scenarios with unannounced meals are investigated which are generally the most difficult cases because of the hyperglycemia and controller‐induced hypoglycemia [22]. Again, we should mention that the bank of the PID controllers is designed just based on the identified transfer functions obtained from the results of some tests on average adult patient. However, this multiple‐model controller is applied for all the other ten patients. It does not mean that the same exogenous insulin infusion rates are injected to all the patients. Indeed, the same controller is used for all of them. To compare the performance of the proposed multiple‐model PID controller with the recently published method in [22], two protocols are considered for the meals as their details are represented in Table 4. Hereafter, gCHO stands for grams of carbohydrates. Again, it should be mentioned that the information about the meals is not known to the controller. For these scenarios, the obtained graphs of the BG concentration are depicted in Figs. 3 a and c. The graphs of the corresponding insulin infusion rate are also shown in Figs. 3 b and d. As it can be seen from Figs. 3 a and c, the proposed multiple‐model strategy successfully regulates the BG to the safe level. The control variability grid analysis (CVGA) plots for the duration of the in silico evaluation (three days) are presented in Fig. 4. In addition, the CVGA plots in nighttime (midnight till 6:00 AM) are also presented in this figure, where the white circles and the red squares are related to the first and the second scenarios, respectively. As it can be seen from the plot of the overall CVGA, most of the closed‐loop responses are in the upper B‐zone which is mainly because of the unannounced meals. However, their nighttime counterparts are completely in the A‐zone. For the purpose of comparison, some important indexes are extracted for the proposed method and the method applied in [22]. The results are summarised in Table 5 for both scenarios. For any index, if there is a considerable difference between the controllers, the value of the better controller is specified with bold. From these results and ones depicted in Figs. 3 and 4, it can be concluded that the performance of the proposed multiple‐model controller is remarkable albeit the utilised controller is not individualised.

Table 4.

Meal time‐meal amount – three days scenarios

	Breakfast 1		Lunch 1		Dinner 1		Breakfast 2		Lunch 2		Dinner 2		Breakfast 3		Lunch 3		Dinner 3
	Time	gCHO	Time	gCHO	Time	gCHO	Time	gCHO	Time	gCHO	Time	gCHO	Time	gCHO	Time	gCHO	Time	gCHO
Scn 1	7 AM	50	2 PM	60	8 PM	50	6 AM	50	1 PM	70	7 PM	50	7 AM	50	1 PM	65	9 PM	55
Scn 2	7 AM	50	—	—	8 PM	60	—	—	12 PM	55	9 PM	50	7 AM	50	2 PM	55	8 PM	50

Fig. 3.

Graphs of the BG concentration and insulin infusion rate

(a), (b) BG concentration for in silico adults: scenario 1, (c), (d) BG concentration for in silico adults scenario 2

Fig. 4.

Plots of CVGA for the overall duration of the in silico evaluation and nighttime: the white circles and the red squares are related to the first and the second scenarios, respectively

Table 5.

Results obtained simulating two scenarios: the MMPID controller and the switched LPV control proposed in [22], where O is overall, N is night, PP is postprandial period which is defined as the 6 h time interval following the start of a meal, $T_{\mathrm{t}}$ is the percentage of time spent in target (70–180 mg/dl), $T_{\mathrm{a}}$ is the percentage of time spent above 180 mg/dl, $T_{\mathrm{b}}$ is the percentage of time spent below 70 mg/dl

	Control strategy	Scenario 1			Scenario 2
		O	N	PP	O	N	PP
mean, mg/dl	MMPID	135	116	163	131	120	164
	LPV	134	101	162	134	114	159
max, mg/dl	MMPID	216	125	216	202	145	202
	LPV	220	113	223	207	153	213
min, mg/dl	MMPID	97	105	101	101	109	108
	LPV	90	98	92	96	98	103
$T_{\mathrm{t}},\%$	MMPID	90.2	99.6	67.4	92.2	100.0	65.3
	LPV	83.3	99.5	67.5	86.7	99.3	71.2
$T_{\mathrm{a}},\%$	MMPID	9.7	0.4	32.5	7.7	0.0	33.6
	LPV	16.5	0.0	32.5	13.3	0.7	28.8
$T_{\mathrm{b}},\%$	MMPID	0.0	0.0	0.0	0.0	0.0	0.0
	LPV	0.2	0.6	0.0	0.0	0.1	0.0

As mentioned in the introduction, MPC techniques were widely used to regulate the BG concentration of diabetic patients to the normal range. Therefore, in the rest of this subsection, the performance of the proposed multiple‐model PID controller is compared with two MPC‐based controllers designed in [32]. For the same scenarios with meal times $\left[\begin{array}{ccccc}7:00& 12:00& 18:30& 7:00& 12:00\end{array}\right]$ and meal amounts $\left[\begin{array}{ccccc}50& 60& 80& 50& 60\end{array}\right]$ gCHO, some important indexes for these three controllers are represented in Table 6. As it can be seen, in many performance indexes, the proposed multiple‐model controller is better than the MPC‐based ones. Nevertheless, since our proposed controller is with unannounced meals, the MPC‐based controllers lead to better values for some indexes just during the postprandial periods.

While bi‐hormonal (insulin and glucagon) artificial pancreas systems are also considered in some studies (see e.g. [33]), due to medical efficacy and technological feasibility, single hormone artificial pancreas systems are usually used where their actuation is only the insulin infusion. In this paper, a single artificial pancreas system is designed and therefore, no insulin is injected by the pump to the patient if the computed control action is negative.

Table 6.

Comparison among the MMPID controller and two MPC techniques proposed in [32], where $T_{\text{tt}}$ is the percentage of time spent in the tight target (80–140 mg/dl) and $T_{\mathrm{h}}$ is the percentage of time spent <50 mg/dl

	Control strategy	O	N	PP
mean, mg/dl	MMPID	134.75	113.81	169.19
	MPC1	140.74	124.42	156.30
	MPC2	139.57	115.96	157.78
$T_{\mathrm{t}},\%$	MMPID	87.35	100.00	58.84
	MPC1	86.33	99.02	76.22
	MPC2	87.64	99.81	77.22
$T_{\text{tt}},\%$	MMPID	70.10	99.93	26.43
	MPC1	52.69	82.07	32.02
	MPC2	59.01	94.69	32.54
$T_{\mathrm{a}},\%$	MMPID	12.65	0.00	41.16
	MPC1	12.45	0.78	23.18
	MPC2	12.31	0.19	22.78
$T_{\mathrm{b}},\%$	MMPID	0.00	0.00	0.00
	MPC1	1.23	0.2	0.6
	MPC2	0.05	0.00	0.00
$T_{\mathrm{h}},\%$	MMPID	0.00	0.00	0.00
	MPC1	0.49	0.01	0.26
	MPC2	0.00	0.00	0.00

3.2 Multiple‐model versus single‐model

In this subsection, positive effects of the utilised system identification technique, i.e. the multiple‐model strategy, are investigated. Towards this end, the first scenario in Table 4 is repeated for all the in silico adult cohort of the UVA/Padova metabolic simulator. However, in the current study, a single PID controller is used to calculate the insulin injection rate. Indeed, we only use the second PID controller designed in Section 2.3. The main reason for this selection is that its corresponding OP $G_2^{\ast }$ is in the tight target (80–140 mg/dl). Fig. 5 shows the graphs of the BG concentration for all the patients. It should be mentioned that to have a fair comparison, the glucose‐dependent setpoint is also used in this simulation. By comparing Figs. 3 a with 5, it can be said that while the multiple‐model strategy is so successful, the performance of the single‐model method is not acceptable and severe hypoglycaemic events are observed in some patients.

Fig. 5.

Graphs of the BG concentration when a single‐model PID controller is used

3.3 Effects of the glucose‐dependent setpoint

As mentioned before, the main reason for using a glucose‐dependent setpoint for the BG concentration is to reduce the negative effects of the delays. To investigate this issue, for all the in silico adult cohort of the UVA/Padova metabolic simulator, the first scenario in Table 4 is repeated. Nevertheless, a constant setpoint $G_{\mathrm{d}}=130(\text{mg}/\text{dl})$ is used as the desired value of the BG concentration. Note that in the current study, the utilised controller is the multiple‐model PID controller designed in Section 2.3. Graphs of the BG concentration for all the patients are depicted in Fig. 6. By comparing this figure with Fig. 3 a, it can be seen that while the glucose‐dependent setpoint strategy is successful, the observed hypoglycemia resulted by using the constant setpoint is not acceptable. One might think that the hypoglycemia could be limited by increasing the value of the constant setpoint. However, in such a case, long‐term hyperglycemia could occur.

Fig. 6.

Graphs of the BG concentration when the constant setpoint G_d  = 130 (mg/dl) is used

3.4 Individualisation of the multiple‐model PID controller

In Fig. 3 a, the closed‐loop BG of the adult patient # 6 under the previous multiple‐model PID controller is depicted with the dashed line. From this figure, it can be seen that the second overshoot of this patient is high and therefore, in this subsection, a multiple‐model PID controller is designed for this patient. To this end, the system identification step is repeated based on the data obtained from the tests on this patient. Then, the PID bank is redesigned using the updated transfer functions. For brevity, we only present the closed‐loop BG of this patient when he/she is under its individualised controller and the first scenario in Table 4 is considered. The obtained results are depicted in Fig. 7, where the solid line shows the patient's BG concentration and the dashed line shows the CGM measurements. This figure demonstrates that the individualisation of the controller leads to 41 mg/dl reduction of the maximum overshoot of the BG. In addition, while the measured BG has a large delay and spoiled with noise, the closed‐loop response is completely acceptable. Note that it is possible to repeat this individualisation procedure for all the other patients.

Fig. 7.

Results of individualisation of the proposed control strategy for the patient # 6

4 Conclusions

In this paper, a closed‐loop method for subcutaneous insulin delivery in T1D has been proposed based on the multiple‐model strategy. Two major issues have been considered in our control design; (i) Non‐linear dynamics of the glucose–insulin regulatory system; (ii) Delays in subcutaneous BG measuring and in the administration of insulin. The first issue has been addressed using a multiple‐model strategy by identifying the system around five OPs. It should be noted that this identification step has been just done on average adult patient in UVA/Padova T1D simulator. A bank of PID controllers has been constructed based on the identified transfer functions. For any OP, different control objectives have been followed paying attention to some physiological constraints such as the risk of hypoglycemia. For the second issue, a glucose‐dependent set‐point has been considered to reduce the negative effects of delays. The performance of the proposed multiple‐model strategy has been evaluated in silico and the obtained results have been compared with some other closed‐loop methods for insulin delivery in T1D. In silico results have been promising in terms of mean glucose, time in target, and the number of patients in the critical zone of the CVGA. Individualisation of the proposed method has been also investigated to achieve better performance. The promising results are so encouraging and the proposed artificial pancreas system is worthy to be tested in‐vivo.