Proceedings of the 8th International Symposium on Mechatronics and its Applications (ISMA12), Sharjah, UAE, April 10-12, 2012

ISMA12-1

Optimal PID-fuzzy Logic Controller for Type 1 Diabetic Patients

Mohamed Al-Fandi, Yousef Sardahi, Mohammad A. K. Jaradat

Jordan University of Science and Technology

Department of Mechanical Engineering

P.O. Box 3030, Irbid, Jordan majaradat@just.edu.jo

ABSTRACT

In this paper, an optimal PID-FLC (Proportional Integral

Derivative Fuzzy Logic Controller) is proposed. The design of

this system aims to control blood glucose elevation in Type1

diabetic patients. An automated system integrated with a

miniaturized insulin infusion pump and a continuous biosensor

that measures the glucose level was built to replace beta cells in

the pancreas. The main contribution of the paper is that it

introduces an automated insulin delivery system based on a

parallel PID-FLC structure tuned with genetic algorithms. This

control system was compared to an optimal PI-FLC and PD-

FLC as well as a reference model. The results revealed that the

controllers could maintain the glucose level within a normal

range. In addition, the performance of the PI-FLC and the PID-

FLC was very close to that of beta cells in normal individuals.

So, they can be exploited prosperously as control systems to

manage blood glucose concentrations in Type 1 diabetic

patients. In addition, the PID-FLC saved the amount of the

daily delivered insulin. While, its performance was

approximately the same as that of the PI-FLC.

1. INTRODUCTION

Diabetic Mellitus (DM) is a set of lingering metabolic disorders,

which result in hyperglycemia [1]. This occurs when the

pancreatic beta cells become unable to produce insulin or its

secreted amount is not enough as in Type 1 diabetes. In 2001,

Type 1 diabetic patients were estimated to be more than 120

million people around the world [2]. This number is candidate

to reach 300 million by the year 2025 [2]. As the number of the

sufferers increases, the expenses on the health care system will

mount. In addition, this disease causes some risky

complications, such as kidney failure, blindness, nerve damage,

heart attack, and ineffectively of the immune system. [2]. In

order to alleviate these complications, glycemic control was

adopted by many authors. A classical PID controller integrated

with an intravenous enzymatic glucose sensor and an implanted

insulin pump was introduced in [3]. A linear and nonlinear

model predictive control was discussed in [4]. A robust H∞

controller to automate insulin delivery was proposed in [5].

Another algorithm based on an H2 /H∞ control for type I

diabetic patients under intensive care was presented in [6]. A

comparison study between fuzzy logic controllers and

conventional techniques was addressed in [7, 8]. Therein, the

fuzzy systems had better transient characteristics than those of

the conventional ones.

Most of these developed control systems used the simple

Bergman model as a patient model, which does not accurately

describe the glucose-insulin kinetics [10]. In addition, these

controllers focused on keeping the glucose level at a specific

reference point instead of imitating the oscillatory fashion of

glucose and insulin profiles in non-diabetic individuals [10]. A

PI-FLC to control the glucose level in Type 1 diabetic subjects

was investigated in [10]. The objective of the system was to

mimic the glucose-insulin dynamics in healthy persons. The

parameters of the controller including membership functions

and scaling variables were tuned using the genetic algorithm. In

this paper, the PI-FLC was merged with an optimal PD-FLC to

design an optimal PID-FLC. The advantage of this method is

that it reduces the size of the genetic algorithm’s search space

since the PI-FLC and the PD-FLC are optimized separately.

These controllers were implemented on the same glucose-

insulin model. This mathematical model is a set of nonlinear

and delay differential equations that describe glucose-insulin

kinetics in healthy persons.

2. Mathematical model

The simulation trials were conducted on the mathematical

model developed in [11]. In this model, the time change of

glucose concentration (dG/dt) under an external insulin

infusion (Iin) has the form of Eq. (1) while the change in insulin

concentration over time takes the form of Eq. (2).

)1())(())(())(())(()( 253432 tIftIftGftGftGdt

dGin

)2())(()( tIdtIdt

dIiin

This model was solved by using Matlab as shown in Figure 1. In this figure, G(t) is the plasma glucose concentration and I(t)

stands for the insulin concentration. Gin represents the external

glucose intake. The insulin-independent utilization of glucose

by the brain and nerve cells is denoted by f2(G (t)). On the other

side, the insulin-dependent consumption of glucose is

represented by the pair f3(G (t)) f4 (I (t)). f5(I (t)) denotes the

hepatic production of the glucose relying on the insulin level. As in [11], we assumed that the delay of the dependent

utilization, τ3, and that of the glucose production, τ2, were 5 and

15 minutes, respectively. The insulin degradation rate, di, was

assumed to be proportionally related to the insulin concentration

with value of 0.0076 min-1.

Proceedings of the 8th International Symposium on Mechatronics and its Applications (ISMA12), Sharjah, UAE, April 10-12, 2012

ISMA12-2

Figure 1. glucose-insulin dynamic model.

3. PD-FLC

The block diagram of a control system for Type 1 diabetic

patients is shown in Figure 2. It incorporates a PD-FLC, which

is indicated by the dotted rectangle, and the patient model in

the forward path.

Figure 2. Optimal PD-FLC

This system was designed based on two input-linguistic

variables, the error signal (e) and its rate of change (de/dt). The

error signal is the difference between the glucose level under

the normal model and that under the control system. An

exogenous insulin infusion rate was considered as an output

linguistic variable (Iin,f ). The rapid-acting lispro insulin was

chosen as the injected drug. As in [10], each fuzzy linguistic

variable was divided into seven Gaussian membership

functions and Mamdani type fuzzy logic was chosen as an

inference system.

TABLE 1 FUZZY RULES OF THE PD-FLC

e

ce

CE- NH

CE-NM

CE-NL

CE-Z

CE-PL

CE-PM

CE-PH

E-

NH

NH NH NH NH NM NL Z

E-NM

NH NH NH NM NL Z PL

E-

NL

NH NH NM NL Z PL PM

E-Z NH NM NL Z PL PM PH

E-

PL

NM NL Z PL PM PH PH

E-PM

NL Z PL PM PH PH PH

E-

PH

Z PL PM PH PH PH PH

TABLE 2 LINGUISTIC MEANING

Linguistic variable Meaning

NH Negative High

NM Negative Medium

NL Negative Low

Z Zero

PL Positive Low

PM Positive Medium

PH Positive High

This portioning resulted in 49 fuzzy rules as in Table 1. The

linguistic meaning of each fuzzy term is shown in Table 2.

Under the selected fuzzy inference system, the total output of

the control system (Iin) can be given by [3]:

fininin IGII . (3)

)))*,*((*(,dt

deGCEeGEfuzzdefuzzI fin (4)

Where, defuzz and fuzz are the defuzzification and the

fuzzification operations, respectively. GE, GCE, and GIin are

scaling gains of the error, the change in error, and the fuzzy

system‘s output, respectively.

As shown in Figure 2, the said scaling parameters as

well as those of the membership functions at the input and the

output stage were tuned by the genetic algorithm. In this

research, we ensued the same optimization procedure as in

[11]. The objective function was to minimize the Mean

Absolute Error (MAPE) between the glucose value under the

control system and that under the normal model according to

the following formula:

n

t

elfuzzyel GGGn

MAPE0

modmod /)(1

(5)

Where, n is the simulation time, which goes on for 24 hours.

Gmodel stands for the glucose concentration that is retuned by

the normal model. Gfuzzy denotes the actual glucose level under

the fuzzy controller.

4. PI-FLC

The PI-FLC introduced in [10] was adopted here. The details

about the block diagram of this controller and the optimization

procedure were not repeated here to avoid redundancy.

5. PID-FLC

The success of the fuzzy controllers in controlling nonlinear

plants originates from the nonlinearity existing in their outputs

[12]. In addition, these controllers are simple because they

nearly imitate humans in thinking and their natural language is

easily comprehensible for humans [13]. However, tuning the

parameters of these controllers is not easy since many

parameters need to be selected. When the number of

parameters increases, the search space becomes wide which

lead to non-satisfactory results. To avoid this, the parallel

structure design of the PID controller was adopted as shown in

Figure 3. It is based on uniting the PD-FLC and the PI-FLC.

The output of the controller-the delivered-insulin (mU/min)- is

expressed by the following form:

PIinPDinPIDin III ,,, (6)

Proceedings of the 8th International Symposium on Mechatronics and its Applications (ISMA12), Sharjah, UAE, April 10-12, 2012

ISMA12-3

Where, Iin,PID is the total amount of the insulin delivered by the

PID-FLC. Iin, PD denotes the amount of the insulin delivered by

the PD-FLC. Iin, PI is the amount of insulin released by the PI-

FLC. The parameters λ and β are the perspective participant

rates of the PD and PI, respectively. In order to design an

optimal PID-FLC, λ and β were evaluated by the genetic

algorithm to determine their optimal values in building the

optimal PID FLC.

Figure 3. Optimal PID-FLC

When these two participants were passed to the genetic

algorithm, it returned 0.9875 and 0.0132 for the β and λ factors,

respectively. So, Eq. (6) takes the following form:

PIinPDinPIDin III ,,, 9875.00132.0 (7)

It is obvious that the PI-FLC participant rate is bigger than that

of the PD-FLC. This consequence is logical since the PI-FLC

had the smallest MAPE (Mean absolute Percentage Error) of

1.3787%. The resulted PID-FLC recorded the next smallest

error, 1.4136%, while the PD-FLC had the largest error, 2.5938

% as shown in Figure 4 and Table 3. The PID-FLC infused

708.1953 mU/kg/day of insulin while the PI-FLC released

708.8412 mU/kg/day. The PD-FLC recorded the maximum

amount of the delivered insulin, 723.4995 mU/kg/day (see

Table 3). Since their amounts of the external insulin were very

close, the glucose profile under the PI-FLC and the PID-FLC

were very similar. In summary, the PI-FLC was the most

accurate controller but the amount of the infused insulin by the

PID-FLC was the smallest. The PD-FLC had the largest error

and delivered the largest amount of the exogenous insulin. As a

result, its glucose profile were the least accurate.

Figure 4. glucose profile of the three fuzzy controllers

and the reference model

TABLE 3 ERROR AND THE AMOUNT OF THE INSULIN

AFTER OPTIMIZATION

Controller MAPE % Total infused insulin mU/kg/day

PI-FLC 1.3787 708.8412

PID-FLC 1.4136 708.1953

PD-FLC 2.5938 7232.4995

6. ROBUSTNESS

In this section, the robustness of the controllers is tested. The

four robustness tests discussed in [10] were repeated here. In

addition, the response of the controllers under a severe initial

condition, 200 mg/dl, investigated as in [7] was also used.

During these tests, the glucose level was considered normal if it

was within the range 60-140 mg/dl [10].

6.1. Uncertainty in the degradation rate factor

A variation of 20% up the nominal value of di was assumed.

As shown in Figures 5 and Table 4,the smallest MAPE,

11.7188%, was recorded by the PD-FLC. The PI-FLC had the

second smallest error, 14.9950 % while the MAPE of the PID-

FLC was the largest, 15.0234 %. As a response to this

increasing in the insulin degradation, the PD-FLC infused the

largest amount of the external insulin, 860.456 mU/kg/day. The

PID-FLC delivered the next largest amount, 833.381

mU/kg/day while that released by the PI-FLC was the smallest,

832.345 mU/kg/day. Although the PD-FLC controller had the

smallest MAPE, the PI-FLC and PID-FLC economized the

amount of the daily insulin but at the expense of the error. By

looking at Figure 4, we can see that all the three controllers

kept the glucose concentration within the permitted range.

After two hours of the simulation time to the end of the day, the

output of the PI-FLC, PD-FLC and PID-FLC was in the range

95.3157-128.2316 mg/dl, 91.8883-127.6937 mg/dl, and

95.2969-128.1645 mg/dl, respectively. The normal model fell

through retaining the glucose level within the acceptable range.

As a result, the patients with such a clearance rate value will

experience elevated blood glucose concentration or what is

called hyperglycemia. This hyperglycemia appeared because

the injected external insulin is time-dependent with pre-

specified amount (705.25 mU/kg/day). This amount was not

enough to deal with such a condition.

When di was assumed uncertain with a variation of 20% down

its adopted value, the insulin degradation rate was very slow.

The performance of the controllers and the reference model is

shown in Figure 5. As shown in Table 4, the PD-FLC had the

smallest MAPE with a value of 7.7514%. The PID-FLC

showed a little better performance than the PI-FLC. Its MAPE

was 17.2457% while that of the PI-FLC was 17.3114%. The

largest amount of the infused insulin was recorded by the PD-

FLC with a value of 601.791 mU/kg/day while the PI-FLC

released about 601.232601.232 mU/kg/day. The PID-FLC

delivered the smallest amount, 599.597 mU/kg/day. As shown

in Figure 5, after two hours of the simulation time, the PI-FLC

set the glucose in the range 66.2033 -123.4232 mg/dl while the

glucose under the PD-FLC was in the range 82.9347-114.8884

mg/dl. The PID-FLC glucose range was in the range 65.9460-

123.117 mg/dl. All these values are within the acceptable

range. The glucose level under the normal model with this

assumed uncertainty was below 60 mg/dl most of the time.

This means that a patient whose body consumes the insulin

slowly will suffer from hypoglycemia if he undergoes insulin

0 500 1000 150080

90

100

110

120

130

140

150

simulation time

glu

ocse m

g/d

l

reference model

PI-FLC

PD-FLC

PID-FLC

Proceedings of the 8th International Symposium on Mechatronics and its Applications (ISMA12), Sharjah, UAE, April 10-12, 2012

ISMA12-4

treatment under the normal model with its pre-specified amount

of insulin. This hypoglycemia hurts the brain function which

results in coma [10]. As before, it can be observed that the

performance of the PI-FLC and that of PID-FLC are very close.

To sum up, the PD-FLC controller showed better performance

than the others while the PI-FLC and the PID-FLC delivered

less amount of the exogenous insulin. The three controllers

maintained the glucose normal while the reference model

failed.

Figure 5. glucose profile when di =0.00912

TABLE 4 ERROR AND INSULIN AMOUNT OF THE

CONTROLLERS UNDER (UNCERTAINTY TEST)

Controller MAPE

20%up

MAPE

20% down

Insulin

20%up

Insulin

20%down

PI-FLC 14.995 17.3114 832.345 601.2317

PID-FLC 15.0234 17.2457 833.381 599.5973

PD-FLC 11.7188 7.7514 860.4559 601.7913

Figure 6. glucose profile when di =0.00608

6.2. Error in sensor measurements

The sensor’s performance may be affected by patient

conditions [8]. To simulate this case, a white Gaussian

noise with variance of 0.1 and zero mean was assumed.

The measured glucose, Gmeasured, has the form of Eq. (8).

NGG fuzzymeasured (8) (6)

Where, Gfuzzy is the glucose value under the fuzzy

controllers and N is the white Gaussian noise. As shown

in Table 5, the PI-FLC was the most robust controller in

this test. It had a MAPE value of 4.4765%. As known, the

derivative controller is very sensitive to noise. So, the PD

recorded the largest MAPE with a value of 9.2793%,

while the PID-FLC had the second smallest error of

5.6695%. The total amount of the daily-injected insulin

was 742.7466 mU/kg/day in the case of the PD-FLC. The

PI-FLC secreted the next largest amount, 716.6236

mU/kg/day. The PID-FLC injected the smallest amount,

710.9893 mU/kg/day. The glucose profile is shown in

Figure 7. It can be noticed that the PI-FC still more close

to the normal model than the others. As a conclusion, the

PI-FLC was the most effective controller while the PID-

FLC was more effective in saving the infused insulin than

the other approaches.

Figure 7.glucose profile with white noise added as

disturbance to the system output

TABLE 5 NOISE IN THE SENSOR MEASUREMENTS

Controller MAPE (%) Insulin Amount

PI-FLC 4.4765 716.6236

PID-FLC 5.6695 710.9893

PD-FLC 9.2793 742.7466

6.3. Severe initial condition

Figure 8 shows the performance of the controllers as response

to high glucose concentration, 200 mg/dl. As shown in Table

6, the PI-FLC was the fastest in correcting the high glucose

level. It took about 89.835 minute to bring the glucose back to

its accepted range. The PID-FLC was the second fastest

controller. About 90.04 minute was taken to return the glucose

to its permitted range. The PD-FLC was the slowest, 122.5

minute. It was even slower than the reference model.

Figure 8. glucose profiles of the controllers and the

normal model under severe initial condition.

0 500 1000 150050

100

150

200

250

simulation time

glu

ocse m

g/d

l

uncertain reference model

PI-FLC

PD-FLC

PID-FLC

0 500 1000 15000

50

100

150

200

simulation time

glu

ocse m

g/d

l

uncertain reference model

PI-FLC

PD-FLC

PID-FLC

0 500 1000 150060

80

100

120

140

160

simulation time

glu

ocs

e m

g/d

l

reference model

PI-FLC

PD-FLC

PID-FLC

0 500 1000 150060

80

100

120

140

160

180

200

simulation time

glu

ocse m

g/d

l

reference model

PI-FLC

PD-FLC

PID-FLC

Proceedings of the 8th International Symposium on Mechatronics and its Applications (ISMA12), Sharjah, UAE, April 10-12, 2012

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TABLE 6 SEVER GLUCOSE TEST

Controller Time

PI-FLC 89.835

PID-FLC 90.04

PD-FLC 122.5

6.4. Changing the glucose intake profile

The glucose intake profile introduced in [10] was

used here to compare the controllers’ performance under

this test. Table 6 shows the MAPE and the amount of the

externally infused insulin under this test. As shown , the

PD-FLC had the largest error and delivered the maximum

amount of the insulin. The error between the glucose

profile under the PID-FLC and that under the undisturbed

normal model was the smallest. Its released insulin

amount was also the smallest. The PI-FLC recorded the

second smallest error and the second lowest amount of

the external insulin. The glucose profile of these

techniques as well as the disturbed reference model is

shown in Figure 9. At the starting of the simulation time,

the glucose concentration reached 145.5494, 144.6585,

and 144.6608 under the PD-FLC, the PI-FLC and the

PID-FLC, respectively. After 2 hours of simulation, the

PD-FLC recorded the highest glucose level of 131.7177

mg/dl. The PI-FLC had the next maximum glucose value

of 127.9845. The PID-FLC output reached 127.8682

mg/dl. The low limits of these controllers during this time

were 90.7963 mg/dl, 86.6861 mg/dl, and 86.4651 mg/dl,

respectively. All these values are within the permitted

range. In addition, it can be observed that the PI-FLC and

the PID-FLC behave in the same fashion. In general, all

the techniques could keep the glucose level normal most

of the time TABLE 7 CHANGE GLUCOSE INTAKE PROFILE.

Controller MAPE (%) Insulin Amount

PI-FLC 12.7602 747.638

PID-FLC 11.9346 746.994

PD-FLC 16.7494 893.1328

Figure 9. glucose profiles of the controllers and the

reference model, change glucose intake profile

6.5. Unexpected glucose intake

This part tests controllers’ under a sudden glucose intake

for five minutes at a local peak level of glucose [10]. An

amount of glucose intake equals to 100 mg/min was

added at time from 768 min to 772 min. The change in

the plasma glucose under this test has the following form:

))(())(())((

))(()()(

25343

2exp

tIftIftGf

tGftGtGdt

dGectedunin

(7)

Both of the reference model and the controllers exhibited a

good performance under this test as shown in Figure10.

Overall, in the disturbance interval, glucose was below 132

mg/dl, which is within the normal range. The PI-FLC, PID

FLC and disturbed reference model had approximately the

same glucose peak value. The PD-FLC kept the glucose level

lower than that of the other methods because its glucose peaks

were smaller before adding this disturbance. After 795 minutes,

the glucose value under the PI-FLC, PID-FLC and model was

lower than that of the PD-FLC.

Figure 10. glucose profiles of the controllers and the

reference model, unexpected glucose intake.

7. CONCLUSION

In this study, an optimal PID-FLC based on the

parallel augmentation of a PI-FLC and a PD-FLC was

proposed. These controllers were implemented on and

compared to a glucose-insulin dynamic model of healthy

persons. To compare the performance of the three

controllers, they were subjected to a series of tests. In the

uncertainty test, the PD-FLC showed better performance

than the others did. The PI-FLC and the PID-FLC kept

the glucose normal and they saved the amount of the

infused insulin but on the expense of the error. The

normal model failed to retain the glucose level normal

under this test. The PI-FLC was the most robust and the

fastest controller when the noise in the sensor

measurements and the severe initial condition were

assumed. Unlike the PI-FLC, the PD-FLC had the largest

error when the noise was assumed and the slowest

correction time of the high glucose level. In these two

tests, the performance of the PID-FLC was very close to

that of the PI-FLC, but not better. However, the PID-FLC

had the smallest error and delivered the lowest amount of

0 500 1000 150080

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170

simulation time

glu

ocse m

g/d

l

disturbed reference model

PI-FLC

PD-FLC

PID-FLC

720 740 760 780 800

100

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PI-FLC

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PID-FLC

Proceedings of the 8th International Symposium on Mechatronics and its Applications (ISMA12), Sharjah, UAE, April 10-12, 2012

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insulin when the input glucose profile was changed. In

this test, the PI-FLC came in the second rank after the

PID-FLC with respect to error and insulin amount. All

approaches showed acceptable performance and when an

unexpected amount of glucose was added. Over all, the

PI-FLC and the PID-FLC seemed to be more robust in

most of the tests. Hence, they are good candidates as

control systems to regulate blood concentrations in Type

1 diabetic subjects.

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