Various mathematical methods can be used in the wide variety of sciences like medical and engineering analysis [1, 2]. These studies embody, but are not limited to molecular studies, signal processing, different types of controllers, image processing and so on [3, 4, 5]. For example, researchers in , proposed a novel method for data compression and designed a compressor for wearable ECG recorder based on their CS algorithm. Diabetes is a metabolic disorder in which blood glucose levels rise above the normal range of 110-170 mg/dl in the fasting blood sugar test. In diabetics, beta cells, which are the source of insulin production in the body, are destroyed and the body cannot control blood glucose levels alone . In this case, blood sugar should be regulated using insulin injection. The best way to regulate the states of a system is to design a proper closed controller for the system. Closed loop controllers get a feedback from output of the system and compare it with the required condition to reach to the desired input. There are wide variety of applications for these types of controllers such as intelligent traffic control [8, 9], biomedical applications  and etc. The closed loop control method works like an artificial pancreas. In this method, the patient's blood glucose concentration is measured immediately by a glucose sensor and the amount of insulin injected is determined by an control algorithm . Finally, this amount of insulin is injected into the patient continuously using an insulin injection pump. Therefore, closed loop control has been considered as the most accurate control strategy in this paper. Blood glucose sensor, insulin injection pump and proper control algorithm are the main components of closed loop control strategy. Figure 1 shows an outline of a closed loop control system for a diabetic patient using an insulin injection pump. Numerous mathematical models for diabetes based on the interaction of insulin and glucose in the body have
Figure 1. Closed-loop control system of a diabetic patient using an insulin injection pump
Common and valid models for diabetes are DDE and ODE. Early models for diabetes could not model the time delay from when blood glucose levels rise to the release of insulin. In some models, in order to obtain fluctuations in insulin secretion, insulin had to be divided into two components, plasma insulin and cellular Ga, which is one of the disadvantages of the proposed model. In this paper, due to the nonlinear behavior of insulin-glucose interaction in the body of type 1 diabetic patient, the Palumbo delayed nonlinear model is used. One of the most important advantages of this model is compliance with the behavior of the patient population according to the IVGTT glucose intravenous injection test [12,13]. Numerous control methods have been used, including methods based on linear and nonlinear predictive control, classical controllers with linearized equations around the equilibrium point, nonlinear slip-fuzzy model controller to regulate model glucose without delay in state variables for diet and some of which are mentioned in references [14,15]. Control methods based on nonlinear models are based on more knowledge of the physiological behavior of the patient community and provide the possibility of presenting different control theory methods in regulating blood glucose. Obviously, the closer the model's behavior is to the nature of the patient's body, the more accurate the resulting control law will be. Providing an appropriate solution for controlling delayed nonlinear models of diabetes is still an important issue. Lack of sensitivity to internal and external disturbances, final accuracy and pervasiveness, as well as limited time convergence, which are the main features of SMC with the human body, make it a good choice for control algorithms, where high accuracy is one of the special importance.
Delayed nonlinear models have received more attention in recent years due to their consistency with the results of diagnostic tests on the patient population. The presence of delays in nonlinear relationships reduces the order of the model, but their analysis will be more complex. In 2007, Palumbo et al. developed a model that includes the τ_g delay and the K_xr index as the plasma insulin concentration index . Plasma glucose concentration (in mM) Plasma insulin concentration (in pM) are considered as the intravenous insulin injection rate (in min / pM) and the controlled signal. Equation (1) represents this delayed nonlinear model.
According to the presentation model, G_0 (τ), I_0 (τ) are the initial values of the patient’s glucose and insulin plasma, which are considered based on the values of and . Introduction of palumbo DDE nonlinear model parameters are as follows:
The nonlinear function , which represents the rate of insulin delivered, is expressed as Equation (2):
Plasma glucose level is in the condition that insulin secretion has reached half of the maximum value [17,19].
The Bergman minimal model is used to design the object slip mode controller in various articles. Lack of sensitivity to internal and external disturbances, unlimited accuracy, resistance, as well as limited convergence time are the main features of the sliding mode that make SMC a suitable choice for algorithms related to the human body, in which accuracy is very important [16,20]. Traditional SMCs have several inherent problems such as rupture in the control signal, which is used to overcome the problems as well as higher accuracy of the higher order slip modeling method. HOSM puts the k-th slider mode variable at a suitable stable source for the controller design:
3.1. HOSM controller design
The system introduced by Bergmann's minimal model is written in the form of state equations:
Where , and are blood plasma glucose concentration (mg / dl), the effect of insulin on blood glucose loss (1/min) and plasma insulin concentration (ml / Uµ), respectively. Stabilization of glucose concentration in the blood of a diabetic patient at the basic level of is an issue of output tracking. Therefore, tracking error is defined as the difference between the concentration level and its essential amount in the blood of a diabetic patient as the following equation:
For the system introduced in Equation (2), the controller must be designed in such a way that the error in the presence of uncertainties, parameter changes and perturbations, the absorption of food is reduced to zero. At first, the relative degree of the system must be determined. Assuming , the relative degree is defined as the number of consecutive derivatives until the control appears in the equation. Thus, the relative degree r-th means that the controller first appears in of the final derivative . Using Equation (3), the control function appears in the equations after three derivations, i.e.:
Since , , and , system (3) has a well-defined relative degree . This causes the controller designed for system (3) to satisfy the relation . For design, the slider variable controller is introduced as (7):
Which is obtained by deriving the following relation:
In the design process, it is assumed that Equation (9) is finite, i.e. . HOSM, which stabilizes at the origin for a limited time, and is considered as the following equation:
It is clear that with the introduction of virtual control, a derivative is added, which leads to an increase in the relative degree of the system from three to four. To calculate , the HOSM derivative is used. The general form for the order nth derivative of a uniform function is as follows:
Increasing the degree from three to four leads to a high frequency switch in the virtual control. While the local control u is continuous so that also the fourth-order quasi-continuous control in Equation (11) can be used instead of the high-order control of Equation (10) [15,21]. The dynamic system is defined as follows:
R is a relative degree and the output of equation is defined as follows:
If , then we have:
In this case, the value of from the input function is changing in intervals as and the changes are checked for each moment, and according to the amount of insulin required, it changes the level of control input slip. In fact, in addition to commanding the slip surface, the proposed slip mode control also provides an optimal based on insulin changes (ISMC) [22,23].
To design a glucose detector system, the variable q(t) is added to the equations of state according to Equation (15).
For quasi-linear representation of the above equations, the matrices and will be as follows:
Assuming the initial conditions of the system are positive, the values and are positive for all times. Therefore, the quasi-linear matrix elements will be continuous and non-zero. On the other hand, the problem is justified by the degree of completeness of the controllability and visibility matrices depending on the situation. The parameters of the nonlinear model of diabetes for the studied patients are based on the fitting of the least squares generalized to the experimental data of the intravenous insulin injection test [24,28]. The values of and can be measured directly, some parameters such as and are fixed and known, and parameters such as , and are estimated for each patient [29,30]. The parameters and are also parameters that are determined based on the conditions of physical stability of each patient based on algebraic relations. In the simulated example, the patient has a body mass index of 50 and indicates a higher than normal level of the patient's blood glucose and the insulin resistance index is . These factors indicate an abnormal insulin secretion rate for a newly diagnosed diabetic patient whose factors such as obesity, inactivity, and genetics have led to a gradual decrease in the patient's insulin secretion rate. This patient also has symptoms of type 2 diabetes if left untreated. The parameters of the patient delayed nonlinear model of Equation (14) are as follows:
The reference glucose signal as a reduction of the initial value of the patient's blood glucose to the normal value of is considered as follows:
Figures 2, 3, and 4 show the convergence of the diabetic patient's blood glucose to the glucose reference signal, the changes in insulin in the patient's blood plasma, and the rate of insulin injection as the control signal provided in the closed-loop improved control system.
Figure 2. Tracing of blood glucose reference with control actions for the patient
Figure 3. Patient insulin injection rate (control signal applied) (u) and patient blood insulin with normal control (I)
Figure 4. Reference blood glucose tracking by applying ISMC control to the patient
Below are the simulation results of the same example with the improved sliding mode control design of ISMC. The simulation results show the ability to properly track optimal glucose for patients based on the optimal insulin injection rate. The amplitude of the control signal or insulin injection rate for the patient is reduced compared to the feedback linearization method (the method used by palumbo), which indicates that the patient reaches the normal state of the body earlier in exchange for a lower injection rate and reduced costs. Also, by comparing Figures 2 and 4, tracking in the improved sliding mode has far better convergence results than in the traditional sliding mode
Figure 5. Patient insulin injection rate (control signal applied) (u) and Patient blood insulin with ISMC control
In this paper, a sliding mode design for a delayed nonlinear model of palumbo diabetes is considered. In this paper, the ISMC method was applied and a continuous and comprehensive controller was designed to maintain blood glucose at the basal level. The example provided by the ode23 function was simulated for delayed problems in MATLAB. In similar studies, only the sliding model controller has been used for the delay model. Simulation results also show that by changing the structure of the sliding mode and optimizing it, delayed biological systems have better controllability. Another advantage of this method is that the controller is resistant to changes in the slip of the object and the control is different for different patients, but eventually the glucose stabilizes to its basic level within a reasonable period of time. The ability to use the proposed method for other similar models and the use of the control method for the studied model are among the suggestions for future projects.
Conflict of interest
The authors certify they have no affiliations with or involvement in any organization or entity with any financial interest or non-financial interest in the subject matter or materials discussed in this manuscript.