Development of a cognitive blood glucose–insulin control strategy design for a nonlinear diabetic patient model

1 Introduction

One of the most important and common chronic diseases in the world is diabetes, sometimes referred to as diabetes mellitus (DM), which is mostly brought on by elevated blood glucose levels. Diabetes can be fatal or significantly impair a person’s quality of life, regardless of gender. In this regard, diabetes is a chronic illness that only develops when blood glucose levels are too high. In particular, the body uses glucose as an energy source, and the hormone of insulin, which is released by the pancreas and regulates the amount of glucose that the cells may use as fuel [1]. In this context, DM is becoming more and more common worldwide, and by 2045, there will likely be over 783 million cases, mostly in low- and middle-income nations [2]. Specifically, about 90% of cases are Type II diabetes mellitus (TIIDM) [3], with the remaining instances being Type I diabetes mellitus (TIDM) and gestational diabetes mellitus (GDM). Serious side effects include nerve damage and cardiovascular problems, and elevated risks of dying young are linked to these disorders. Thus, accurate, timely, and cost-effective blood glucose monitoring and control are essential for those individuals.

In this regard, experts are specifically studying patients with TIDM in great depth. As a result, a lot of research has been conducted to develop several mathematical models of insulin and glucose that, to a certain degree, accurately depict the physiological behaviour of the human body [4]. In essence, the Bergman minimum model was the most significant. Other glucose–insulin models include the nonlinear glucose metabolism model, as described in Kalaimani and Jeyakumar [4] and Patra and Panigrahi [5], and the Lehman-based diabetic patient model, as described in Kralev and Slavov [6].

Furthermore, various insulin controllers were created to function as an artificial pancreas and maintain a patient’s blood glucose levels between 80 and 120 mg/dl to improve the blood glucose tracking error and control the patient’s blood glucose. For example, Saleem and Iqbal [7] suggested a complex-order proportional integral derivative (PID) controller for elevated blood glucose levels in a TID patient model and employed a fractional-order PID controller. However, the disadvantages of these controllers are that the initial values rely on the designer’s experience and the control parameters are adjusted by numerical optimization. Moreover, using a linear Bergman model, Sharma et al. [8] introduced a digital PID controller for diabetic patients’ blood glucose levels. They utilized the Ziegler–Nichols (Z–N) tuning method to determine the control settings. Nevertheless, the Z–N method is not appropriate for investigating and using the global extreme solution of the problem. Therefore, the control system resulted in an overshoot in the patient’s blood glucose response. Additionally, Benzian et al. [9] employed fuzzy logic and fractional-order PID controllers to control the TID patient’s blood glucose levels. They also used various meta-heuristic techniques to adjust the FOPID’s control parameters. Nonetheless, this work’s drawback is that the controllers were only developed for a single patient using the linear Bergman model. In addition, they employed only five rules for the membership function and the trial-and-error approach to determine the input–output fuzzy logic controller’s gain. As a result, the controller produced an insulin control action value that is too fast and suboptimal, which causes the blood glucose level to respond too quickly. To improve the linear blood glucose–insulin model in type I diabetes, Shenbagam et al. [10] suggested a fractional-order controller. They also used a fractional-order PID controller with a genetic algorithm to adjust the five control parameters, which improved the blood glucose tracking error and maintained the patient’s blood glucose levels. Furthermore, a type-2 fuzzy controller was utilized by Yan et al. [11] and Sayed et al. [12] to monitor and stabilize the blood glucose levels in TID for various patient models. In another work, a radial basis function neural network was employed in Barbosa de Farias and Bessa [13] as an intelligent controller for an automated insulin delivery system for a virtual patient model to monitor and regulate the blood glucose level in a matter of days. In a different study, Khaqan et al. [14] used the University of Virginia/Padova metabolic simulator to demonstrate how to estimate the TID patient model. They also created control algorithms that use an intelligent predictive control model with linear and nonlinear controllers to control the blood glucose level for the linear third-order patient model. To stabilize the patient’s blood glucose level and maintain it at a normal physiological level, Dagher and Haggege [15] suggested a PID-PSO controller for the TIDM patient model. This controller uses the PSO algorithm to adjust the optimal control parameters of the adaptive PID controller, allowing it to quickly and optimally determine the insulin-infusion control action injected into the nonlinear Bergman model of the TIDM patients. Additionally, to achieve a fast response and less chattering in an intravenous glucose tolerance test model of a TIDM patient, Babar et al. [16] designed a control glucose–insulin system using a back-stepping-sliding mode controller with three control parameters that were adjusted by the trial-and-error method. The motivation for this work is that DM patients are becoming more and more common worldwide. Particularly, for type 1 patients, the body cannot adequately use the insulin that it generates and requires insulin injections to survive. To tackle this problem, a lot of research has been done to develop several mathematical glucose–insulin control techniques for patient models. Such techniques were proposed to determine the fast and optimal insulin-infusion control action value to enhance the performance of the blood glucose level in the patient model in terms of regulation and stabilization of the blood glucose at the normal physiological level within a suitable time to avoid the hyperglycemia level which, over a longer period, may cause heart disease, vascular disease, blindness, stroke, nerve damage, kidney disease, and amputation. Conversely, the hypoglycemia state may lead to coma, acute seizure, and death [17]. These lengthy complications lowered life expectancy, increased disability, and raised health care expenses for society. Therefore, all of these numbers demonstrate the critical need for research in diabetes prevention and care, as well as the necessity of developing a control system to enhance the control system for these patients and reduce the rate at which the disease’s financial and physiological expenses are increasing [18]. In particular, the problem definition in this study is that patients with TIDM have a difficult condition that basically entails controlling the blood glucose levels to prevent both hyperglycemia and hypoglycemia. Furthermore, controlling and stabilizing the blood glucose level to the normal physiological level in the shortest period is dependent on determining the amount of the insulin-infusion level. By implementing the suggested offline and online cognitive glucose–insulin controller using the grey wolf optimization (GWO) meta-heuristic method, the primary goal of this research is to identify the fastest and best insulin-infusion control action value to improve the blood glucose level performance in various patient models with meal disturbances in terms of regulation and stabilization of the blood glucose at the normal physiological level. This study’s primary contribution involves identifying and adjusting the feedforward neural network controller’s optimal or nearly optimal parameters in addition to the feedback PID-RBF-NN controller’s control gain parameters. This adjustment process will allow for the fast and ideal insulin-infusion control action to be injected into the nonlinear Bergman model of TIDM patients with variable meal disturbances. This process will enable the patient’s blood glucose level to be tracked and stabilized, and it will be kept at a normal physiological level within a suitable time to prevent hyperglycemia and hypoglycaemia. Therefore, the novelty of this research is to design an efficient and robust controller for an artificial pancreas system for blood glucose regulation in TIDM patients.

This article is organized as follows. In Section 2, the Bergman minimum model is introduced. Section 3 describes the cognitive glucose–insulin regulation algorithm. Section 4 presents the simulation’s numerical results, and the primary findings from this study are presented in Section 5.

2 The nonlinear diabetic patient model

Figure 1 depicts the general diagram of the insulin–glucagon system [19]. The hormones of insulin and glucagon, which regulate the blood glucose levels, are produced in the pancreas, which releases insulin into the bloodstream to lower the blood glucose levels when they are elevated. To raise blood glucose levels when they are low, glucagon promotes the breakdown of glycogen and the synthesis of glucose from circulating precursors. In our work, we will take a model of the insulin–glucose system based on the Bergman glucose insulin minimum model. In particular, the three important compartments in this model are shown in Figure 2, which describes the relationship among the distant insulin compartment level Ins(t) in mU/l, the plasma glucose compartment level Gp(t) in mg/dl, and the remote insulin Xp(t) in 1/min. Blood glucose, a hormone, and insulin were thought to be stored in two separate compartments and to interact with each other.

Figure 1

The general diagram of the insulin–glucagon system [19].

Figure 2

The compartments of the Bergman glucose–insulin minimal model.

In this context, several research works used the Bergman glucose–insulin basic model, which lacks the biological complexity shown in Sakulrang et al. [19], to study the distribution of insulin and the regulation of blood glucose [20]. The following representation is a description of this model, which is based on nonlinear ordinary differential equations [21]:

This factor will not be included when establishing the transfer function since diabetic patients cannot control their blood sugar levels Y [ Gp ( t ) − h ] +   t = 0 . Instead, a given parameter will be derived based on the assumption of a steady-state condition, as given below [21]:

where Xp(t) is the effect of active insulin in the distant compartment variable in 1/min and Gp(t) is the blood glucose concentration variable in mg/dl. Food(t) is the meal disturbance input variable in mg/dl min, Fb Insulin ( t ) is the controlled insulin-infusion rate variable in mU/l min, and Ins(t) is the blood–insulin concentration variable in mU/l. P ₁ is the glucose effectiveness factor, P ₂ is the delay in insulin actions, P ₃ is the patient parameter, Gb is the basal blood glucose concentration, I _b is the basal blood–insulin concentration, n is the first-order decay rate of plasma insulin, h is the glucose threshold value above which the pancreatic β-cells release insulin, and Y is the rate of the pancreatic β-cells’ release of insulin following the glucose injection with glucose concentration.

Table 1 demonstrates the parameters of the Bergman model equations that describe the normal person and three patients [15,20,21].

To check whether the nonlinear diabetic patient model based on the Bergman minimal mode is stabile in each patient’s parameters, as given in Table 1, a Lyapunov function [22] is proposed as follows:

Clearly, V ( x 1 ( t ) , x 2 ( t ) , x 3 ( t ) ) > 0 if all states ( x 1 ( t ) , x 2 ( t ) x 3 ( t ) ) > 0 , and hence, the function is positive definite in all values of the states.

The nonlinear Bergman minimal mode can be expressed as shown in Eq. (6) based on Eqs. (1)–(4)

where x 1 ( t ) , x 2 ( t ) , and x 3 ( t ) are Gp, Xp, and Ins, respectively. The time derivative of Eq. (5) becomes

By substituting Eq. (6) in (7), the derivative state vector becomes as follows:

Clearly, if all states ( x 1 , x 2 x 3 ) = 0 , V ̇ = 0 , and if all states ( x 1 , x 2 x 3 ) > 0 , V ̇ < 0 . Therefore, the function V ̇ is negative definite in all values of the states and the system is globally asymptotically stable with the four weighting parameters ( p 1 , p 2 , n , p 3 ) of the normal person model.

However, when the parameter p 1 is equal to zero of the Bergman patient model, V ̇ ( x 1 , x 2 , x 3 ) is expressed in Eq. (11) as follows:

The effect of x 1 in Eq. (11) is still apparent, and thus, V ̇ is negative definite in all values of the states, and the system is globally asymptotically stable with the three weighting parameters ( p 2 , n , p 3 ) of the different types of patient models.

3 Cognitive blood glucose–insulin control strategy design

The novelty of this research is to design an efficient and robust controller for the artificial pancreas system for blood glucose regulation in TIDM patients. The general structure of the proposed cognitive blood glucose–insulin control strategy is shown in Figure 3, which demonstrates the five layers for achieving the optimal insulin-infusion level in the nonlinear diabetic patient model to avoid the hyperglycemia and hypoglycemia states and to keep the blood glucose level in the desired normal state.

Figure 3

The five layers of the proposed cognitive blood glucose–insulin control strategy for the artificial pancreas system.

This five-layer structure consists of

Layer #1: The cognitive dataset that represents the attributes of the control system.

Layer #2: The identifier model that represents the different types of nonlinear diabetic patients.

Layer #3: The feedforward neural network controller based on the NARMA-L2 neural network model to determine the maximum insulin-infusion level for each meal.

Layer #4: The feedback PID-RBF-NN controller, which is based on the radial basis function neural network model to find the optimal insulin-infusion value and to keep the blood glucose level in the normal state.

Layer #5: The nonlinear patient Bergman minimal model.

Figure 4 demonstrates the proposed cognitive blood glucose–insulin control strategy for the nonlinear diabetic patient model.

Figure 4

Cognitive blood glucose–insulin control strategy for the nonlinear diabetic patient model.

3.2 The patient identifier model

In general, the identifier model is based on the NARMA-L2 neural network model [23] that will represent the different types of the nonlinear diabetic patient model. Specifically, three different types of the nonlinear Bergman glucose–insulin model are used with the input–output dataset, including insulin Iin(k) in mU/dl and glucose Gp(k) in mg/dl, as well as the dataset in layer #1 to build the proposed nonlinear identifier neural network glucose–insulin model. Therefore, the modelling of the nonlinear glucose–insulin is the primary aim of the proposed cognitive blood glucose–insulin control strategy, which is utilized to provide the preconditions for analysis and will be used in the third layer of the control design. In fact, the purpose of the nonlinear glucose–insulin identification is to find a mathematical model of the different types of the nonlinear Bergman glucose–insulin model whose output corresponds to the output of the nonlinear diabetic patient models. In this regard, neural networks are mathematical models with excellent fault tolerance, adaptiveness, and associative memory capacities that can process data concurrently. To establish the neural network glucose–insulin model, the identifier model of the nonlinear diabetic patient model is constructed using the NARMA-L2 neural network model illustrated in Figure 5.

Figure 5

FH [ − ] and G H[−] of the neural network structure.

For the proposed identification process, the NARMA-L2 model was selected among several traditional neural networks due to its unique advantages. More specifically, strong robustness performance, no output oscillation, favourable dynamic properties, and an increasing degree of the nonlinear glucose–insulin model performance, are provided by the two networks of FH[−] and GH[−], which are raised in the order of the hidden units. The cognitive attributes input A i ( k ) with three additional variable parameters, including Gp(k), Xp(k), and Ins(k) for the NARMA-L2 neural network model, is represented in the following equation:

(13) B i ( k ) = A i ( k ) Gp ( k ) Xp ( k ) Ins ( k − 1 ) .

The proposed NARMA-L2 model of the nonlinear glucose–insulin model can be described in the following equation:

(14) G m ( k + 1 ) = FH [ Gp ( k ) , Xp ( k ) , Ins ( k − 1 ) , Food ( k ) ,   I o ( k ) ,  Go ( k ) , I max ( k ) ] + GH [ Gp ( k ) , Xp ( k ) , Ins ( k − 1 ) , Food ( k ) ,   I o ( k ) ,  Go ( k ) , I max ( k ) ] × Ins ( k ) .

The FH [ − ] and GH [ − ] network weights can be represented as follows: FV _ai represents the weight matrix of FH [ − ] in the hidden layer, FW _b represents the weight matrix of FH [ − ] in the output layer, GV _ai represents the weight matrix of GH [−] in the hidden layer, and GW _b represents the weight matrix of GH [−] in the output layer. To illustrate the calculations in the hidden layer based on the 15 neurons, first, we will sum the net of the weights of FV _ai and GV _ai using the following equations:

(15) FHnet a = ∑ a = 1 n f h FH a i × B i ¯ ,

(16) GHnet a = ∑ a = 1 n f h GH a i × B i ¯ ,

where nfh and ngh represent the hidden nodes’ number, which is equal to 15 nodes.

Second, the neuron outputs of both FH _a and GH _a are calculated as a continuous unipolar sigmoid activation function [24] of the FHnet _a and GHnet _a , as illustrated in the following equations:

(17) FH a = 1 1 + e − FHnet a ,

(18) GH a = 1 1 + e − GHnet a .

Third, to calculate the weighted sum Fnet _o and Gnet _o of the output layers, the following equations are used, respectively.

(19) Fnet o = ∑ a = 1 n f h FW b × FH a ¯ ,

(20) Gnet o = ∑ a = 1 n g h GW b × GH a ¯ .

The one linear neuron passes the sum of both (Fnetfo) and (Gnetgo) through a linear function of slope 1, as shown in Eqs. (21) and (22).

(21) FO = Linear ( Fnet o ) ,

(22) GO = Linear ( Gnet o ) .

The neural network output is the glucose level of the modelling, specifically Gm(k), that can be expressed as given in the following equation:

(23) Gm ( k ) = FO + GO × Ins ( k ) .

As depicted in Figure 6, we use the five strategies to address the challenge of recognizing and simulating the nonlinear glucose–insulin patient model.

Figure 6

The identification and modelling steps for the nonlinear glucose–insulin model [25].

According to the biological description of the nonlinear glucose–insulin patient model, the major physical variable output of the model is the glucose level. Conversely, the cognition attributes in layer #1 are the inputs for the nonlinear glucose–insulin patient model in our work. The structure of the proposed nonlinear glucose–insulin patient identifier model is shown in Figure 7.

Figure 7

The structure of the proposed nonlinear glucose–insulin patient identifier model.

The GWO algorithm is an intelligent algorithm based on the grey wolf predation [26]. Similar to other intelligent algorithms, each grey wolf’s position indicates a feasible response, and the prey indicates the optimal one. Grey wolves are ranked according to the value of their fitness function in an effort to identify the best answer. With three different kinds of grey wolf groups, hierarchical commands can be created.

The grey wolves with the highest fitness function value make up the leader group, often known as the alpha (α) group. The alphas are in charge of making judgments about hunting, waking times, sleeping spots, and other things. Ironically, despite not being the strongest individual in the group, the alpha must be the best pack manager. The wolves in the beta (β) group, the second echelon of leadership, are frequently called co-leaders since they assist the alpha in pack activities and decision-making. The delta (Δ) groups come after them. The probable prey’s location is nearer the wolves α, β, and Δ [27]. The hierarchy of grey wolves during predation is one distinctive feature of GWO. In essence, three primary phases of hunting are conducted to maximize efficiency, including finding the prey, encircling the prey, and attacking the prey. After α groups lead grey wolves to encircle their victim, β and Δ groups assault the prey, and the prey is eventually taken. Because of this process, the approach, which is simple to develop, has few parameters, does not require specialized search parameters, and achieves high convergence performance. At the beginning of the process, a fixed number of grey wolves is used, and their places are selected at random. Each group in the pack will encircle the others according to the following mathematical formuls [26,27]:

(24) D = ∣ C × X p ( iter ) − X ( iter ) ∣ ,

(25) X ( iter + 1 ) = ∣ X p ( iter ) − A × D ∣ .

Particularly, Eq. (24) represents the distance between the grey wolf and its prey. In Eq. (25), iter indicates the current iteration. A and C are coefficient vectors, and Xp and X are the position vectors of the prey and the grey wolf, respectively, representing the formula for the grey wolf’s position update [28]. The formulas used to determine A and C are as follows:

(26) A = 2 a × r 1 − a ,

(27) a = 2 × ( 1 − It Max ) ,

(28) C = 2 × r 2 .

The convergence factor serves as its representation, and the random vectors r ₁ and r ₂ are chosen randomly from the interval (0, 1). The total number of iterations is called It_Max. According to the following equations [29], the grey wolf locations α, β, and Δ (the three temporarily ideal solutions) are averaged to determine the prey position X _p (iter + 1) update, while the remaining locations are ignored for the position update:

(29) X p ( iter + 1 ) = X 1 + X 2 + X 3 3 ,

where

(30) X 1 ( iter ) = X α ( iter ) − A 1 × D α X 2 ( iter ) = X β ( iter ) − A 2 × D β X 3 ( iter ) = X Δ ( iter ) − A 3 × D Δ .

The random vectors C ₁, C ₂, and C ₃ characterize the final positions of the solutions, whereas Dα, Dβ, and DΔ in Eq. (30) indicate the distances between α, β, and Δ and other individuals, respectively. Furthermore, their beginning and finishing positions are specified by Eq. (29). When the victim eventually stops moving, the grey wolf attacks to put an end to the search [26,27,28,29]. The fundamental method for creating a process model is to gradually reduce the value of a, which reduces the range of fluctuations in A. Stated otherwise, the corresponding value of A fluctuates in the interval (−a, b) during the iterative process in a manner akin to the linear decrease in the interval [2, 0]. The identifier model based on NRMA-L2 will produce the same actual response of the glucose level after using the neural network’s training procedure, as shown in Figure 7. Then, the grey wolf learning algorithm is applied to reduce the error of the glaucous level model (the neural network model) Gm(k) and the real glaucous level Gp(k), which is nearly equal to 80 mg/dl. Thus, it will be feasible to confirm that the network is correctly learnt and that the model’s glucose level corresponds to the intended glucose level that characterizes the variation between the glaucous level and the glucose level identity model using a testing set. Particularly, the NARMA-L2 neural network is used to generate the series-parallel identification model. It is clear that the network receives the prior inputs and outputs of the glucose level system at each time instant, and that the network’s output provides the prediction error. Nevertheless, this method can only be applied in conjunction with the series-parallel system, in which the inputs to the neural network (identifier) model in the structure are the outputs of the genuine glucose level model. Figure 8 outlines the basic steps of the GWO method for determining and adjusting the weights of the FH[−] and GH[−] neural networks. The mean square error (MSE) is used as the cost function to evaluate each solution in the GWO algorithm [15]:

(31) MSE = ∑ It=1 ITmax 1 K ∑ i = 1 K ( ( Gd ( i ) − Gm ( i ) ) 2 )

where IT_max is the maximum number of iterations and K denotes the maximum number of samples.

Figure 8

The flowchart of updating the weights for the identifier patient model based on the GWO algorithm.

3.4 The feedback controller

The feedback controller’s task is to maintain normal blood glucose levels and determine the ideal insulin infusion value. Based on Gaussian neural networks [30], which have been proven to be effective schemes for learning complex input–output nonlinear mapping and have been employed in the learning and control of nonlinear dynamic systems, the suggested structure of the nonlinear feedback PID-RBF-NN controller is depicted in Figure 9.

Figure 9

The proposed nonlinear feedback PID-RBF-NN controller structure.

The discrete PID controller is described in Eq. (34), which is used to calculate the RBF neural network feedforward algorithm [15]:

(34) Fb Insulin ( k ) = Fb Insulin ( k − 1 ) + K p ( Err ( k ) − Err ( k − 1 ) ) + K i ( Err ( k ) ) + K d ( Err ( k ) − 2 Err ( k − 1 ) + Err ( k − 2 ) ) ,

where K _p, K _i, and K _d are the control gain parameters of the PID controller. Therefore,

(35) Sum 1 ( k ) = Err ( k ) −  Err ( k − 1 ) ) ,

(36) Sum 2 ( k ) = Err ( k ) ,

(37) Sum 3 ( k ) = Err ( k ) − 2 Err ( k − 1 ) + Err ( k − 2 ) .

As its input, the nonlinear feedback PID-RBF-NN controller receives the error signal between the desired and the actual blood glucose levels

(38) RBFnet 1 ( k ) = ( Sum 1 ( k ) − c 1 ) 2 ,

(39) RBFnet 2 ( k ) = ( Sum 2 ( k ) − c 2 ) 2 ,

(40) RBFnet 3 ( k ) = ( Sum 3 ( k ) − c 3 ) 2 ,

where c 1 , c 2 , and c 3 are the centres of the geometric shape of the Gaussian functions of the neurons. The activation function output of the RBF-NN is as follows [13]:

(41) H 1 ( RBFnet ( k ) = α e − RBFnet 1 ( k ) 2 σ 2 ,

(42) H 2 ( RBFnet ( k ) = α e − RBFnet 2 ( k ) 2 σ 2 ,

(43) H 3 ( RBFnet ( k ) = α e − RBFnet 3 ( k ) 2 σ 2 ,

where α and σ are the maximum amplitude of the geometric shape of the Gaussian functions of the neurons and the width of the geometric shape of the Gaussian functions of the neurons, respectively.

The final insulin value in Eq. (44), which serves as the control action Fb Insulin ( k ) and regulates the patient’s blood glucose levels, is the output of the PID-RBF-NN controller

(44) Fb Insulin ( k ) = Fb Insulin ( k − 1 ) + K p × H 1 ( RBFnet ( k ) ) + K i × H 2 ( RBFnet ( k ) ) + K d × H 3 ( RBFnet ( k ) ) .

In order to obtain the optimal or near-optimal insulin control action for the nonlinear Bergman model, which keeps the blood glucose level in the normal state to prevent the hyperglycemia and hypoglycemia states, the control gain parameters of the PID-RBF-NN controller (K _p, K _i, and K _d) can be found offline, and then, adjusted online using the GWO algorithm, as shown in Figure 8.

The GWO method uses the mean square error with the multi-objective function that reduces the error between the desired glucose level and the patient’s glucose level and, at the same time, reduces the value of the insulin control action for each solution:

(45) MSE = ∑ It = 1 IT max 1 K ∑ i = 1 K ( ( Gd ( i ) − Gp ( i ) )   2 + ( ( U max ( i ) − Fb Insulin ( i ) )   2 ) )

where IT_max is the maximum number of iterations and K denotes the maximum number of the samples.

4 Numerical simulation results

In this work, the numerical fourth-order Runge–Kutta (4RK) method based on the MATLAB package with a half-minute sampling time was utilized to implement the cognitive blood glucose–insulin control strategy for the nonlinear diabetic patient model in Figures 3 and 4. This control strategy will achieve the optimal insulin-infusion level for the different types of nonlinear diabetic patient models to avoid the hyperglycemia and hypoglycemia states and to keep the blood glucose level of the patient in the desired normal state. Specifically, five steps are implemented, as illustrated below.

The first step is to determine the cognitive attributes dataset, as given in Table 2, with the blood glucose level of the patient Gp(k), the blood–insulin concentration Ins(k), and the active insulin in the remote compartment performs Xp(k), as an open-loop patient model for a healthy individual as well as for three distinct categories of diabetic patients who are dependent on the glucose starting levels (Go) of (280, 230, 220, and 210) mg/dl, respectively, and without any disturbance meal, as demonstrated in Figure 10a–c. From Figure 10a, the impulse response of the glucose level of the normal person that depends on the initial glucose level of 280 mg/dl reduces the glucose level from 280 mg/dl to the normal state at a value equal to 80 mg/dl during 75 min.

Figure 10

The open-loop responses for a normal person and for different types of diabetic patients in the initial state: (a) the glucose level, (b) the remote insulin level, and (c) the plasma insulin level.

However, for the three different types of diabetic patients, the reduction in the glucose level from the initial levels was very small, and they did not reach the normal glucose level. From Figure 10b, the responses of the remote insulin level of the three different types of diabetic patients are very small values that spread in the blood of the diabetic patients. From Figure 10c, the responses of the plasma insulin level of the three different types of diabetic patients are small values that depend only on the initial plasma insulin level I _o(mU/l) in the blood of the diabetic patients due to the inability or insufficiency of the pancreas to generate insulin.

Figure 11a–c demonstrate the blood glucose level of different types of three patients (Gp₁(k), Gp₂(k), and Gp₃(k)), the blood–insulin concentration (Ins₁(k), Ins₂(k), and Ins₃(k)), and the active insulin in the remote compartment for the three different types of diabetic patients (Xp₁(k), Xp₂(k), and Xp₃(k)) with the same initial glucose levels Go. However, the disturbance meal at breakfast is equal to 15 mg/dl blood glucose level.

Figure 11

The open-loop responses for a normal person and for different types of diabetic patients in the breakfast-meal state: (a) the glucose level, (b) the remote insulin level, and (c) the plasma insulin level.

From Figure 11a, the impulse response of the glucose level of the normal person that depends on the initial glucose level and the breakfast meal reduces the glucose level from 350 mg/dl to the normal state at a value equal to 80 mg/dl during 100 min. However, for the three different types of diabetic patients, the glucose levels increased by more than 350 mg/dl during 10 min with a very small decay in the glucose level from the peak levels of 375 mg/dl, and they did not reach the normal glucose level. Nevertheless, the glucose level of the patient at the hyperglycemia is dangerous.

From Figure 11b, the breakfast meal did not affect the response of the remote insulin level of the three different types of diabetic patients, and they are very small values that spread in the blood of the diabetic patients.

From Figure 11c, the responses of the plasma insulin level of the three different types of diabetic patients are small values that depend only on the initial plasma insulin level I _o(mU/dl) in the blood of the diabetic patients due to the inability or insufficiency of the pancreas to generate insulin.

Figure 12a–c demonstrate the dynamic behaviour of the Bergman glucose insulin minimum model for the three different types of diabetic patients representing the blood glucose level, the blood–insulin concentration, and the active insulin in the remote compartment with the same initial glucose levels Go. However, the disturbance meal at lunch is equal to 20 mg/dl blood glucose level.

Figure 12

The open-loop responses for a normal person and for different types of diabetic patients in the lunch-meal state: (a) the glucose level, (b) the remote insulin level, and (c) the plasma insulin level.

From Figure 12a, the impulse response of the glucose level of the normal person that depends on the initial glucose level and the lunch meal reduces the glucose level from 400 mg/dl to the normal state at a value equal to 80 mg/dl during 110 min.

However, for the three different types of diabetic patients, the glucose level increased by more than 400 mg/dl during 10 min with a very small decay in the glucose level from the peak levels of 425 mg/dl, and they did not reach the normal glucose level. Particularly, the glucose level of the patient at the hyperglycemia level is dangerous.

From Figure 12b, the lunch meal did not affect the response of the remote insulin level of the three different types of diabetic patients, and they are very small values that spread in the blood of the diabetic patients.

From Figure 12c, the responses of the plasma insulin level of the three different types of diabetic patients are small values that depend only on the initial plasma insulin level I _o(mU/l) in the blood of the diabetic patients due to the inability or insufficiency of the pancreas to generate insulin.

Figure 13a–c demonstrate another dataset, which includes adding the disturbance meal at dinner with a blood glucose level equal to 5 mg/dl for the three different types of diabetic patients with the same initial glucose levels Go.

Figure 13

The open-loop responses for a normal person and for different types of diabetic patients in the dinner-meal state: (a) the glucose level, (b) the remote insulin level, and (c) the plasma insulin level.

From Figure 13a, the impulse response of the glucose level of the normal person that depends on the initial glucose level and the dinner meal reduces the glucose level from 280 mg/dl to the normal state at a value equal to 80 mg/dl during 90 min.

However, for the three different types of diabetic patients, the glucose level increased by more than 250 mg/dl during 10 min with a very small decay in the glucose level from the peak levels of 280 mg/dl, and they did not reach the normal glucose level. In particular, the glucose level of the patient at the hyperglycemia level is dangerous.

From Figure 13b, the dinner meal did not affect the response of the remote insulin level of the three different types of diabetic patients, and they are very small values that spread in the blood of diabetic patients.

From Figure 13c, the responses of the plasma insulin level of the three different types of diabetic patients are small values that only depend on the initial plasma insulin level I _o (mU/l) in the blood of the diabetic patients due to the inability or insufficiency of the pancreas to generate insulin.

From Figures 10–13, the responses of the blood glucose levels for the different types of the three patient Bergman models have three real phenomena as follows: they started high, decreased very slowly, and would never return to the normal state, which causes the patient to be in a dangerous state. However, the healthy person curve shows a normal glucose level dropping from a high value to the physiological level, or the normal glucose level (between the upper level of 150 mg/dl and the lower level of 60 mg/dl), in which consuming high blood sugar is not harmful.

Conversely, the glucose value is exceedingly high and well beyond the physiological range, putting the patient in danger mode with hyperglycemia (250 mg/dl) and hypoglycemia (50 mg/dl). In step two, the nonlinear glucose–insulin patient model is displayed in Figure 7 using the structure of the NARMA-L2 neural network described in Figure 5.

Accordingly, the suggested number of nodes in each of the FH[−] and the GH[−] networks, which have three layers, including the input layer, the hidden layer, and the output layer, is as follows: [7:15:1], respectively. The number of nodes in the hidden layer is equal to twice the number of nodes in the input layer plus one. The third step involves learning the identifier model of the nonlinear glucose–insulin patient models using the GWO offline algorithm and then tuning the models online. In order to resolve the numerical problems related to actual values, the signals entering or leaving the neural network (NN) have been treated to lie between (−1) and (+1). Therefore, the scaling functions must be applied at the first layer and the last layer of the neural network terminals (input–output), respectively, such that the ranges of these inputs are as follows: Gp(k) = (50 to 500) mg/dl, Xp(k) = (0 to 0.05) 1/min, Ins(k) = (0 to 700) mU/L, Food(k) = (5 to 25), Io(k) = (50 to 60) mU/L, Go(k) = (210 to 280) mg/dl, and I _max(k) = (0 to 700) mU/L·min⁻¹.

In the learning mode, the responses of the different types of the nonlinear neural network glucose–insulin patient models are shown in Figure 14a and b for two cases, including the breakfast and the lunch meals. These models have a very small error value in the modelling between the Bergman glucose level and the diabetic patient neural network model for 600 patterns as a learning set, and they have good responsiveness of the identifier models with a very good dynamical behaviour during the different cases of disturbance meal inputs (breakfast meal and lunch meal) for the three different types of the nonlinear glucose–insulin patient models.

Figure 14

The response of the different types of the nonlinear neural network glucose–insulin patient models: (a) the breakfast meal learning set and (b) the lunch meal learning set.

The performance of the best convergence curve is depicted in Figure 15. It is based on the offline GWO algorithm for three different types of patient models with 500 iterations and with an agents’ number equal to 32 for the neural networks of FH[−] and GH[−].

Figure 15

The best alpha convergence curve response of the different types of nonlinear neural network glucose–insulin patient models.

In addition, the number of weights is 120. Specifically, the best alpha convergence curve for the optimal updated weights of the identifier neural network is presented in Figure 15.

To verify that these neural network models have remarkable learning, the testing mode in our work is essentially for the nonlinear glucose–insulin patient models to prove that these neural models have been learned in all the excitation and active regions of the real models and eliminated the overlearning problem. This problem means that the network learns on a part of the region and forgets other regions during the learning mode.

However, the meta-heuristic GWO algorithm learning cycle was excellent because this algorithm is very suitable for exploring and exploiting the global extreme solution of the problems.

Figure 16 demonstrates the dynamical behaviour of the nonlinear glucose–insulin patient models during the testing mode.

Figure 16

The responses of the different types of nonlinear glucose–insulin patient models for the dinner meal testing set.

The neural networks glucose level model Gm_1,2,3(k) matched the Bergman glucose level Gp_1,2,3(k) of the three different types of the nonlinear glucose–insulin patient models when we used 600 patterns for the dinner meal. The fourth step involves representing the feedforward neural network controller based on the NARMA-L2 identifier after sufficient learning and testing for the three different types of the nonlinear neural network glucose–insulin patient identifier models. The aim of this step is to determine the maximum value of the insulin level U _max(k) mU/L·min⁻¹ for each meal in the beginning, as shown in Figure 17 with three different patient models.

Figure 17

The response of the maximum insulin level U _max for the three different types of the nonlinear neural network glucose–insulin patient identifier models.

From Figure 17, it is clear that the level value of the insulin for each patient is increased for each sample, and only ten samples are taken for example. These values are generated based on the open-loop system, as shown in Eq. (32) without any closed-loop feedback PID-RBF-NN controller. Thus, the error between Gd(k)−FH[−] was not reduced to zero, which led to an increase in the insulin level.

The fifth step involves representing the closed-loop control system for the nonlinear Bergman model based on the proposed PID-RBF-NN controller with the GWO meta-heuristic method using the multi-objective function. This function will reduce the error between the desired glucose level and the patient’s glucose level, and at the same time, it will reduce the value of the insulin control action for each solution of the control gain parameters for the PID-RBF-NN controller (K _p, K _i, and K _d) to find the best value of the insulin-infusion control action in the transient and the steady-state regions.

The suggested PID-RBF-NN controller settings for the search space areas are displayed for each patient in Table 3. These settings are suitable for exploring and exploiting the global extreme solution with different types of patient models and different types of food meals (breakfast, lunch, and dinner) to find and tune the gain control parameters of the proposed PID-RBF-NN controller.

The response of the suggested closed-loop insulin-infusion PID-RBF-NN controller is shown in Figure 18 when the Bergman diabetic patient model adds the breakfast meal as a disturbance.

Figure 18

The glucose responses for each Bergman patient model based on the closed-loop PID-RBF-NN controller with the breakfast disturbance.

The breakfast disturbance effect was introduced for a duration of ten samples for each patient to show the effectiveness of the insulin-infusion control action. In particular, 15 mg/dl is the suggested breakfast disturbance value. The following points indicate that the suggested overall controller improves the patients’ glucose level response:

The insulin-infusion action stabilizes patient #1’s glucose level, as indicated by the red-colour line, which drops from 360 to 250 mg/dl (the hyperglycemia level) and remains there for 18.5 min. Then, it drops to 150 mg/dl (the upper normal physiological level) and remains there for 45.5 min.

Finally, it drops to 80 mg/dl (the normal physiological level) and remains there for 200 min. In contrast, the glucose level of patient #2, which is represented by the blue-colour line, drops from 358 to 250 mg/dl, the hyperglycemia level, and stabilizes there after 24.5 min. It then drops to 150 mg/dl, the upper normal physiological level, and remains there for 67.5 min before dropping to 80 mg/dl, the normal physiological level. Then, it remains there for 220 min.

The third patient’s glucose level, represented by the yellow-colour line, is finally lowered from 333 to 250 mg/dl, or the upper hyperglycemia level, and then, it stabilized there for 14 min. Next, it drops to 150 mg/dl, or the upper normal physiological level, and remains there for 35 min. After that, it drops to 80 mg/dl, or the normal physiological level, and remains there for 150 min.

It is important to note that at 250 min, the blood glucose levels of the first and second patients precisely reached 80 mg/dl at a steady state. They remain at their typical physiological level.

Table 4 shows the time attributes of the blood glucose level of patients #1, #2, and #3 with breakfast meal disturbances.

As shown in Table 5, the optimal PID-RBF-NN controller parameters are generated for the Bergman diabetic models of patients #1, #2, and #3 with various meal disturbances (breakfast, lunch, and dinner) using the best-proposed values of the GWO algorithm, including 32 agents and 50 iterations.

The suggested cognitive blood glucose control technique is responsible for generating the optimal or nearly optimal insulin control action for each patient model, which decreases the blood glucose levels and keeps them within a reasonable range.

The output response of the insulin-infusion PID-RBF-NN controller during the first ten samples is displayed in Figure 19 when the blood glucose level abruptly rises. The PID-RBF-NN efficiently and rapidly determines the insulin action value for each of the three patients to track the sudden increase in blood glucose levels.

Figure 19

The insulin action of the proposed PID-RBF-NN controller for the three different types of Bergman models with the breakfast disturbance.

The maximum insulin-infusion control action values for patients #1, #2, and #3 are 304, 378, and 311 mU/L·min⁻¹, respectively.

Figure 20 shows the maximum level value of the insulin for each patient, which is a decreased value for each sample, where only ten samples were taken as an example. These values are generated based on the closed-loop feedback PID-RBF-NN controller.

Figure 20

The response of the maximum insulin level U _max for the three different types of the nonlinear neural network glucose–insulin patient identifier models with the breakfast disturbance.

Hence, the error between Gd(k) and FH[−] reduces to zero, which leads to a decrease in the U _max insulin level to almost zero, which will reduce the insulin control action to zero.

To determine the optimal control gain settings of the PID-RBF-NN controller, Figure 21 shows the response of the best alpha convergence for the three patients.

Figure 21

The best convergence curve response of the PID-RBF-NN controller for the three different types of Bergman patient models with the breakfast disturbance.

The distant insulin level for each patient, which is a representation of the whole-body insulin level, is displayed in Figure 22. Figure 23 displays the plasma insulin level for each subject and illustrates how quickly the amount spreads throughout the body in a 300-min period.

Figure 22

The distant insulin level for each patient with the breakfast disturbance.

Figure 23

The plasma insulin level for the three different types of the Bergman patient models with the breakfast disturbance.

Figure 24 illustrates how the proposed closed-loop insulin-infusion PID-RBF-NN controller reacts when the lunch meal is added as a disturbance in the Bergman diabetic patient model.

Figure 24

The glucose level responses for each Bergman patient model based on the closed-loop PID-RBF-NN controller with the lunch disturbance.

The lunch disturbance effect was introduced for a duration of ten samples for each patient to show the effectiveness of the insulin-infusion control action. In particular, 20 mg/dl is the suggested lunch disturbance value. The following points indicate that the suggested overall controller improves the patients’ glucose level response:

The insulin-infusion action stabilizes patient #1’s glucose level, as indicated by the red-colour line, which drops from 407 to 250 mg/dl (the hyperglycemia level) and remains there for 22.5 min. Then, it drops to 150 mg/dl (the upper normal physiological level) and remains there for 50 min.

Finally, it drops to 80 mg/dl (the normal physiological level) and remains there for 200 min.

In contrast, the glucose level of patient #2, which is represented by the blue-colour line, drops from 409 to 250 mg/dl, the hyperglycemia level, and stabilizes there after 33 min. It then drops to 150 mg/dl, the upper normal physiological level, and remains there for 73.5 min before dropping to 80 mg/dl, the normal physiological level. It remains there for 230 min.

The third patient’s glucose level, represented by the yellow-colour line, is finally lowered from 382 to 250 mg/dl, or the upper hyperglycemia level, and it stabilized there for 17.5 min. Next, it drops to 150 mg/dl, or the upper normal physiological level, and remains there for 40 min. After that, it drops to 80 mg/dl, or the normal physiological level, and remains there for 150 min.

It is important to note that at 250 min, the blood glucose levels of the first and second patients precisely reached 80 mg/dl at a steady state. They remain at their typical physiological level.

Table 6 shows the time attributes of the blood glucose level of patients #1, #2, and #3 with lunch meal disturbances.

The output response of the insulin-infusion PID-RBF-NN controller during the first ten samples is displayed in Figure 25 when the blood glucose level abruptly rises. The PID-RBF-NN efficiently and rapidly determines the insulin action value for each of the three patients to track the sudden increase in blood glucose levels. The maximum insulin-infusion control action values for patients #1, #2, and #3 are 321, 283, and 281 mU/L·min⁻¹, respectively.

Figure 25

The insulin action of the proposed PID-RBF-NN controller for the three different types of the Bergman models with the lunch disturbance.

Figure 26 shows the maximum level value of the insulin for each patient, which is a decreased value for each sample, where only ten samples were taken as an example. These values are generated based on the closed-loop feedback PID-RBF-NN controller. Thus, the error between Gd(k) and FH[−] reduces to zero, which leads to a decrease in the U _max insulin level to approximately zero, which will reduce the insulin control action to zero.

Figure 26

The response of the maximum insulin level U _max for the three different types of the nonlinear neural network glucose–insulin patient identifier models with the lunch disturbance.

To determine the optimal control gain settings of the PID-RBF-NN controller, Figure 27 shows the response of the best alpha convergence for the three patients.

Figure 27

The best convergence curve response of the PID-RBF-NN controller for the three different types of Bergman patient models with the lunch disturbance.

The distant insulin level for each patient, which is a representation of the whole-body insulin level, is displayed in Figure 28.

Figure 28

The distant insulin level for each patient with the lunch disturbance.

Figure 29 displays the plasma insulin level for each subject and illustrates how quickly the amount spreads throughout the body in a 300-min period.

Figure 29

The plasma insulin level for the three different types of Bergman patient models with the lunch disturbance.

The suggested closed-loop insulin-infusion PID-RBF-NN controller’s response to the addition of dinner as a perturbation in the Bergman diabetic patient model is shown in Figure 30. To demonstrate the efficacy of the insulin-infusion control action, a dinner disturbance effect was created for ten samples per subject. Specifically, the recommended dinner disturbance value is 5 mg/dl. The recommended overall controller enhances the patients’ glucose level response in the following ways:

Figure 30

The glucose level responses for each Bergman patient model based on the closed-loop PID-RBF-NN controller with the dinner disturbance.

The glucose level of patient #1 is the red-colour line, which rises from 271 to 250 mg/dl (the hyperglycemia level) and stays there for 9 min. Then, it falls to 150 mg/dl (the upper normal physiological level) and stays there for 37.5 min. Finally, it falls to 80 mg/dl (the normal physiological level) and stays there for 160 min. This shows that the insulin-infusion action stabilizes patient #1’s glucose level.

Patient #2’s glucose level, shown by the blue-colour line, on the other hand, falls from 265 to 250 mg/dl, the hyperglycemia level, and then stabilizes there after 10 min. Next, it falls to the upper normal physiological level of 150 mg/dl and stays there for 53.5 min. After that, it falls to the normal physiological level of 80 mg/dl and stays there for 235 min.

Finally, the third patient’s glucose level, shown by the yellow line, drops from the upper hyperglycemia level of 250 to 245 mg/dl. It then falls to the top normal physiological level of 150 mg/dl, where it stays for 26.5 min. It then falls to the usual physiological level of 80 mg/dl, where it stays for 150 min.

It should be noted that the first and second patients’ blood glucose levels reached 80 mg/dl at the steady state at 250 min. Their physiological state is still normal.

Table 7 shows the time attributes of the blood glucose level of patients #1, #2, and #3 with dinner meal disturbances.

When the blood glucose level suddenly rises, Figure 31 shows the output response of the insulin-infusion PID-RBF-NN controller for the first ten samples. To monitor the abrupt rise in blood glucose levels, the PID-RBF-NN quickly and effectively calculates the insulin action value for each of the three patients. Patients #1, #2, and #3 have maximum insulin-infusion control action values of 309, 354, and 219 mU/L·min⁻¹, respectively.

Figure 31

The insulin action of the proposed PID-RBF-NN controller for the three different types of Bergman models with the dinner disturbance.

For instance, only ten samples were collected, and Figure 32 displays the maximum level of insulin for each patient, which is reduced for each sample. Since the closed-loop feedback PID-RBF-NN controller is used to generate these values, the error between Gd(k) and FH[−] falls to zero, lowering the U _max insulin level to approximately zero and lowering the insulin control action to zero.

Figure 32

The response of the maximum insulin level U _max for the three different types of the nonlinear neural network glucose–insulin patient identifier models with the dinner disturbance.

Figure 33 displays the response of the best alpha convergence for the three patients to identify the PID-RBF-NN controller’s ideal control gain settings. Figure 34 shows each patient’s remote insulin level, which is a representation of the insulin level across the body.

Figure 33

The best convergence curve response of the PID-RBF-NN controller for the three different types of Bergman patient models with the dinner-meal disturbance.

Figure 34

The distant insulin level for each patient with the dinner disturbance.

Each subject’s plasma insulin level is shown in Figure 35, which also shows how quickly the amount circulates throughout the body in a 300-min period.

Figure 35

The plasma insulin level for the three different types of Bergman patient models with the dinner disturbance.

To validate the effectiveness of the optimization algorithm (GWO) for tuning the parameters of the cognitive PID-RBF-NN controller in this work, we compared the simulation results of the proposed cognitive controller with the results of other types of controllers taken from [9,11], and [15], as shown in Table 8 in terms of reaching the blood glucose level at a normal physiological level at a minimum time and for showing the time enhancement percentage by using Eq. (46):

Using only five rules for the membership function and the trial-and-error method to determine the gain in the input–output fuzzy logic controller, the fractional-order PID and the fuzzy logic controllers in Benzian et al. [9] were constructed for the linear Bergman model and only for the first patient. As a result, the controller produces an insulin control action value that is too quick and suboptimal, which causes the blood glucose level to respond too quickly.

The suggested controller, on the other hand, employs the nonlinear Bergman model, the feedforward neural controller, and the feedback PID-RBF-NN with the GWO heuristic method. The controller has produced optimal or nearly optimal insulin control action based on the best parameters found by the optimization algorithm, which results in bringing the blood glucose level down to a normal physiological level without overshooting or response oscillation. When compared to the fractional-order PID and the fuzzy logic controller algorithms, the comparison findings demonstrated that the PID-RBF-NN with the GWO algorithm improved the time to attain the blood glucose level in a normal condition for patient #1 by 10% [9].

In Yan et al. [11], the type-2 fuzzy controller was created only for the first patient using the linear patient Bergman model. The four control gains in the control law were obtained through a process of trial and error, which results in a minor oscillation in the blood glucose level response as the controller produces a rapid and suboptimal value of the insulin control action. The proposed cognitive controller, which consists of two controllers, namely the feedforward neural controller and the feedback PID-RBF-NN controller with the heuristic GWO method, works with a nonlinear patient Bergman model. Based on the best parameters identified by the optimization algorithms, the controller generates an optimal or nearly optimal insulin control action, bringing the blood glucose level to a normal physiological level without oscillating or overshooting. When compared to the type-2 fuzzy controller algorithm, the comparison findings demonstrated that the PID-RBF-NN with the GWO algorithm improved the time to attain the blood glucose level in normal conditions for patient #1 by 25% [11].

The offline PSO approach was utilized to determine the three control parameters of the PID-PSO controller in [15], which was constructed for the linear Bergman model. As a result, the insulin control action produced by the controller is fast, but the spike action is high. When compared to the PID-PSO controller algorithm, the comparison findings demonstrated that the PID-RBF-NN with the GWO algorithm improved the time to attain the blood glucose level in normal conditions for patient #1 by 6% [15].

The results of the simulation show that the proposed cognitive glucose–insulin controller with the GWO algorithm can produce the best insulin control action. This action enables the nonlinear patient Bergman model to track the necessary blood glucose level with the least amount of tracking error and to achieve optimal performance without oscillation in the output blood glucose levels of the different patient types.

5 Conclusions

Using a feedforward neural controller and a feedback PID-RBF-NN controller with the GWO algorithm, this study demonstrated the design and simulation of an offline and online cognitive glucose–insulin control strategy for blood glucose level monitoring and control in three distinct nonlinear Bergman patient models with various disturbance meals. Three different patient models were employed as a nonlinear model to solve the problem statement to monitor and stabilize the blood glucose level response in diabetic patients by determining the optimal insulin-infusion level and keeping the blood glucose level at the normal physiological level. The following problems can be effectively resolved by using the glucose–insulin management technique based on the suggested five layers:

• The blood glucose level is efficiently monitored and maintained at a normal physiological level of 60–120 mg/dl without oscillation at the goal level of 80 mg/dl.

• An ideal or almost ideal smooth value of the insulin-infusion control action was generated to enhance the blood glucose level response in diabetes patients without reaching the saturation condition.

• A high tracking precision of the measured blood glucose level is achieved by the proposed controller, which is based on the PID-RBF-NN with the GWO algorithm with offline and online tuning control settings that provide smooth insulin action without a large spike or a saturation state-based feedforward NARMA-L2 identifier.

• The maximum tracking error level for monitoring blood glucose approaches zero at intervals longer than 220 min.

• By comparing the proposed controller with the fractional-order PID controller, the proposed controller enhanced the time by 10% to reach the blood glucose level at a normal physiological level with the lunch disturbance. Moreover, compared to the type-2 fuzzy control algorithm, the proposed controller enhanced the time by 25% to reach the blood glucose level at a normal physiological level with the lunch disturbance.

• Finally, compared to the PID-PSO control algorithm, the proposed controller enhanced the time by 6% to reach the blood glucose level at a normal physiological level with the lunch disturbance.

To create an artificial pancreas, the suggested glucose–insulin control strategy based on offline and online PID-RBF-NN controllers with the GWO algorithm will be experimentally implemented in future works utilizing an FPGA development board with an insulin pump device.