Applied Computer Science, vol. 16, no. 2, pp. 18–31
Type I Diabetes, Second Order Sliding Mode Control, Chaotic Particle Swarm Optimization, BEM model
Ekhlas H. KARAM*, Eman H. JADOO
DESIGN OF MODIFIED SECOND ORDER
SLIDING MODE CONTROLLER BASED
ON ST ALGORITHM FOR BLOOD GLUCOSE
The type1 of diabetes is a chronic situation characterized by abnormally
high glucose levels in the blood.
Persons with diabetes characterized by
no insulin secretion in the pancreas (ß-cell) which also known as insulin-
dependent diabetic Mellitus (IDDM). In order to keep the levels of glucose
in blood near the normal ranges (70–110mg/dl), the diabetic patients
needed to inject by external insulin from time to time. In this paper,
a Modified Second Order Sliding Mode Controller (MSOSMC) has been
developed to control the concentration of blood glucose levels under a dis-
turbing meal. The parameters of the suggested design controller are
optimized by using chaotic particle swarm optimization (CPSO) technique,
the model which is used to represent the artificial pancreas is a minimal
model for Bergman. The simulation was performed on a MATLAB/SIMULINK
to verify the performance of the suggested controller. The results showed the
effectiveness of the proposed MSOSMC in controlling the behavior of glu-
cose deviation to a sudden rise in blood glucose.
Mustansiriyah University, College of Engineering, Computer Engineering Department,
Palestine Street, 14022, Baghdad, Iraq, firstname.lastname@example.org, email@example.com
Diabetes mellitus is one of the most important chronic diseases which results
from a high blood sugar for a long time due to insufficient insulin generation in
the blood (Bergman, Phillips & Cobelli, 1981). The concentration of glucose in the
bloodstream is naturally regulated by two hormones: insulin and glucagon. Both of
these hormones are secreted by β-cells and α cells in the Langerhans islands of the
pancreas, respectively. The concentration of glucose ranges from 70 to 110 (mg/dL).
Accordingly, there are two states, hyperglycemia (glucose concentration is above the
normal ranges) and hypoglycemia (low glucose concentration than the normal
ranges) (Basher, 2017).
Diabetes is broken down into two major types. The type 1 diabetes mellitus
(T1DM) and Type 2 diabetes mellitus (T2DM) in the first type the patient's body
can’t produce enough insulin and doses of insulin need to be injected into the
human body to control blood glucose levels, while the second type starts with
insulin resistance, a condition in which cells do not respond properly to insulin.
This type of diabetes is a common type and known as noninsulin-dependent
diabetes (Sylvester & Munie, 2017).
In order to prevent the effects of high blood glucose levels the best approach
is to administer insulin during a moment when blood glucose is supposed to rise.
With the Advance of technology, the so-called artificial pancreas emerged its
consists of three main components, glucose sensor, insulin pump and control
techniques to generate the necessary insulin dose based on glucose measurements
(Kaveh & Shtessel, 2006). The block diagram of the closed – loop system for
glucose level control shown in Figure 1.
Fig. 1. Block diagram of closed-loop insulin delivery system
There are several studies that use a closed-loop controller to keep blood glucose
(BG) diabetic concentration within the appropriate range, such as: (Kaveh and
Shtessel, 2006) used higher order sliding mode controller (HOSMC) to regulate
the levels of blood glucose. (Garcia-Gabin, et al., 2009) suggested a sliding mode
predictive control (SMPC) which is the combining sliding mode control technology
with model predictive control (MPC). In a (Abu-Rmileh & Garcia-Gabin, 2011)
used a combination of the robust sliding mode control (SMC) and the Smith predic-
tor (SP) structures. Nasrin et al. suggest a Sliding Mode Control (SMC) based on
Backstepping technique (Parsa, Vali & Ghasemi, 2014). Waqar et al. suggest
a non-linear super twisting control algorithm based on SMC approach has been
addressed for regulation of glucose concentration in blood plasma of type 1 diabetes
patients (Alam, et al., 2018).
In this paper, the MSOSMC is suggested to regulate the levels of blood glu-
cose, the CPSO algorithm was used for tuning the parameters of the controller.
To accomplish these objectives Bergman Minimal (BEM) mathematical model
which considered here. The outliner of this paper as follows: The BEM math-
ematical model of blood glucose system presented in section 2. The details of
the SOSMC described in section 3, while the design of the MSOSMC explained
in section 4. And the CPSO algorithm illustrated in section 5. The proposed
controller's analysis and simulation results will be discussed in section 6, while
the final conclusions listed in the last section.
2. BERGMAN GLUCOSE-INSULIN REGULATION MODEL
Specific mathematical models have been suggested to explain the complexi-
ties of diabetes and to compare the interaction between models of glucose and
the delivery of insulin that helps design a diabetes model. Among these models,
the minimal Bergman model, a common reference model in the literature,
approaches the dynamic response of blood glucose concentration in a diabetic to
insulin injections. Bergman model consists of three differential equations as
follows (Sylvester, 2017), (Abu-Rmileh & Garcia-Gabin, 2011):
where: is the plasma glucose concentration in [mg/dL], proportional
to the insulin concentration in the remote compartment [1/min], is the plasma
insulin concentration in [mU/dL], and is injected insulin rate in [mU/min],
are parameters of the model. The term, in the
third equation of this model, serves as an internal regulatory function that formu-
lates insulin secretion in the body, which does not exist in diabetics, the
represent the rate of exogenous insulin. The value of will be significantly
reduced; therefore it can be approximated as zero (Parsa, Vali & Ghasemi,
2014). Which is disturbance signal (meal disturbance) can be modeled by
a decaying exponential function of the following form (Fisher, 1991):
where: represents the absorption rate of the meal, is meal size and
represents the beginning time of meal digestion.
3. SLIDING MODE CONTROLLER DESIGN
SMC is a robust and simple procedure for synthesizing controllers for linear
and nonlinear processes based on the Variable Structure Control (VSC)
The discrete control has high switching frequency, which causes a "chattering-
phenomenon", it considered undesired property that appear in SMC's control
action (Djouima, et al., 2018). There are different methods that have been used
to overcome the chattering phenomena such as replacing the sign(s) by boundary
function like sat(s), using terminal SMC, integral SMC, and other different methods.
One of the most efficient methods to overcome this problem by using Second
Order Sliding Mode Control. There are different SOSMC algorithms, such as
Sub-Optimal (SO), Twisting (TW) and Super-Twisting (ST) algorithm. ST-SMC
does not require the information of ṡ in its formulation and application which is
simpler and preferable (Matraji, Al-Durra & Errouissi, 2018).
The ST-SMC utilized similar design steps as standard SMC. The same
sliding surface as in Eq. (3) is applied and the control laws are stated in Eq. (8).
The sliding surface can be introduced as:
where and is error and derivative of the error respectively, is given
where is the reference input (Basal Value) and is the output signal
The constant is chosen to be positive. The choice of decides the conver-
gence rate of the tracking error.
The ST algorithm is deﬁned by the following control law (Matraji, Al-Durra
& Errouissi, 2018; Levant, 2013):
where and are positive bounded constants. The control law of the super
twisting SOSMC is given by:
4. MODIFIED SLIDING MODE CONTROLLER DESIGN
In this paper, a Modified SOSMC based on super twisting is suggested as
shown in fig. 2, which considered as improvement to the SOSMC, the control
law of the super twisting SOSMC (Eq. (8)) is modified to:
where is nonlinear auxiliary part given by:
where are small positive numbers that will be tuning by (PSO and CPSO)
Fig. 2. Modified SOSMC block diagram
5. CHAOTIC PARTICLE SWARM OPTIMIZATION
The Particle Swarm Optimization algorithms (PSO) is the common evolutionary
techniques. Which is adopt a random sequence for their parameter. The PSO algo-
rithm is initialized with a population of candidate solutions which is called
a particle. N particles are moving around in a D-dimensional search space of the
problem (Amet, Ghanes & Barbot, 2012).
The position of the particle at the iteration is represented by
. The velocity for the particle can be written as
. The best position that has so far been visited by the
particle is represented as = ( ) which is also called pbest.
The global best position attained by the whole swarm is called the global best
(gbest) and represented as () = ( ). The velocity vector at the
iteration is represented as () = (). At the next iteration,
the velocity and position of the particle are calculated according to (11, 12):
where are called acceleration coefficients. is called inertia weight, and
are random value in the range [0, 1]. The parameters and is the
key factors that effected the convergence behavior (Wang, Tan & Liu, 2018).
In the Chaotic Particle Swarm Optimization algorithms (CPSO) the parameters
and are modified by using logistic map based on the following equation:
where is s a control parameter with a real number from and .
Then introduce a new velocity update as in equations (14).
Important advantages of the chaotic optimization algorithm (COA) are sum-
marized as: easy implementation, short execution time and speed-up of the search.
Observations, however, reveal that the COA also has some problems including:
(i) COA is effective only for small decision spaces; (ii) COA easily converges in
the early stages of the search process. Therefore, hybrid methods have attracted
attention by the researchers (Hadi, 2019) The flowchart that represented this
algorithm illustrated below.
Fig. 3. General flowchart of the CPSO algorithm
6. SIMULATION RESULTS
The results of simulations for BEM model addressed in Eq. (1), parameters of
BEM model are available on table (1), and the suggested controller based on the
CPSO algorithm are offered in this section for a BG levels of 70 mg/dl. The BEM
model response without controller is showen in figure (4). In this paper, the sim-
ulations are carried out dynamically for three patients with the initial conditions
220, 200 and 180mg/dl for patients 1, 2 and 3, respectively. In the simulation,
the meal glucose disturbance that given in Eq. (2) the value of its parameters are
, b , and
Fig. 4. Glucose output of three patients with disturbance
(open-loop glucose regulatory system)
You can note that the glucose value of the normal person is stabilized at the
basal level in the presence of the disturbance (meal), while the patient's glucose
level remains dangerous outside the range. The simulation second part is the pro-
posed controller is applied to the system and the response of a patients in the
presence of the disturbance is tested. To examine the robustness of the control
algorithm to the parameter change, three sets of parameters for three different
patients have been used. The parameters of CPSO algorithm are considered here
as in Table 2.
Tab. 1. Bergman Minimal Model Parameters (Garcia-Gabin, et al.,
2009; Abu-Rmileh & Garcia-Gabin, 2011).
Tab. 2. The parameters of CPSO algorithm
Maximum number of iterations
Number of particles
Inertia weight factor
Chaotic initial value
Table 3 illustrate the optimal parameters for SOSMC and MSOSMC controllers
gotten from the CPSO algorithm.
Tab. 3. Optimal controller parameters
Figures (5 to 10) shows the response of BEM model for three patients after
applying the suggested controllers to regulated the BG level according to Table 3
It can be noticed from simulation results (Figures (5 to 10) and Tables (4 to 6))
of the suggested controllers that the glucose output with these controllers tracks
the desired BG level with small settling time (), study state error (), and the
Mean Absolute Percentage Error () between the glucose value under the
control system and that under the normal model according to the following
where is the duration of simulation, is the glucose value returned by
the reference model, and represents the actual output of the system
under the controller.
Fig. 5. Blood glucose concentration for patient 1 based on the suggested controllers
and CPSO algorithm
Fig. 6. Insulin infusion rate for patient1 based on the suggested controllers and CPSO
Fig. 7. Blood glucose concentration for patient 2 based on the suggested controllers
and CPSO algorithm
Fig. 8. Insulin infusion rate for patient2 based on the suggested controllers
and CPSO algorithm
Fig. 9. Blood glucose concentration for patient 3 based on the suggested controllers
and CPSO algorithm
Fig. 10. Insulin infusion rate for patient3 based on the suggested controllers
and CPSO algorithm
Tab. 4. The simulation result’s evaluation parameters for patient 1
The controller used
Tab. 5. The simulation result’s evaluation parameters for patient 2
The controller used
Tab. 6. The simulation result’s evaluation parameters for patient 3
The controller used
The comparison between controllers is shown in Tables (4 to 6). This tables
illustrates the performance of controllers. The MSOSMC has the best average
performance which satisfies the design requirement.
In this paper, a simple modified second order sliding mode controller has been
suggested based on ST algorithm and CPSO algorithm. The performance analysis
of the suggested control strategy concerning plasma glucose-insulin stabilization
is comprehensively demonstrated by computer simulations. To validate the ro-
bustness of the suggested controller, the diabetic patient is exposed to external
disturbance, that is, a meal. The closed-loop system has been simulated for different
patients with different parameters, in the presence of the food intake disturbance
and it has been shown that the glucose level is stabilized at its basal value (reference
input) in a reasonable amount of time. The effectiveness of the suggested controller
compared with the classical SOSMC are verified by simulation results for three
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