Open Journal of Applied Sciences, 2013, 3, 421-429
Published Online November 2013 (http://www.scirp.org/journal/ojapps)
http://dx.doi.org/10.4236/ojapps.2013.37052
Open Access OJAppS
Fuzzy Logic Strategy for Solving an Optimal Control
Problem of Glucose and Insulin in Diabetic
Human
Jean Marie Ntaganda
Department of Applied Mathematics, School of Pure and Applied Science, College of Science and Technology,
University of Rwanda (Huye Campus), Butare, Rwanda
Email: jmnta@yahoo.fr, jmntaganda@nur.ac.rw
Received August 17, 2013; revised September 27, 2013; accepted October 7, 2013
Copyright © 2013 Jean Marie Ntaganda. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
This paper aims at the development of an approach integrating the fuzzy logic strategy for a glucose and insulin in dia-
betic human optimal control problem. To test the efficiency of this strategy, the author proposes a numerical compari-
son with the indirect method. The results are in good agreement with experimental data.
Keywords: Fuzzy Logic; Optimal Control; Membership Function; Membership Degree; Glucose; Insulin; Numerical
Simulation
1. Introduction
The blood glucose in human body is controlled by two
key organs: the pancreas and the liver. The key hormones
are insulin and glucagon. In the pancreas, there are clus-
ters of endocrine cells scattered throughout the tissue.
These are the α-cells and the β-cells. The α-cells produce
glucagon and the β-cells produce insulin. The pancreas
secretes these antagonistic hormones into the extracellu-
lar fluid, which then enters the circulatory system and
regulates the concentration of blood glucose. For biolo-
gists, this is known as a simple endocrine pathway. Hu-
man bodies need to maintain glucose concentration level
in a narrow range 70 - 120 mg/dl. If glucose concentra-
tion level is significantly out of the normal range, this
person is considered to have the plasma glucose problem:
Hyperglycemia (greater than 140 mg/dl after an Oral
Glucose Tolerance Test, or greater than 100 mg/dl after a
Fasting Glucose Tolerance Test) or hypoglycemia (less
than 40 mg/dl).
Diabetes mellitus is an endocrine disorder caused by a
deficiency of insulin (Type 1 Diabetes) or a decreased
response to insulin in target tissues (Type 2 Diabetes) [1].
The major long-term effects of diabetes are caused by
hyperglycemia. Prolonged hyperglycemia can cause com-
plications, which may lead to kidney disease, blindness,
loss of limbs, and so on. The hypoglycemia can lead to
dizziness, coma, or even death. Type 1 diabetes was pre-
viously called insulin-dependent diabetes mellitus (IDDM)
or juvenile-onset diabetes. It is an autoimmune disorder
in which the immune system destroys the β-cells of the
pancreas. Previously known as non-insulin dependent dia-
betes (NIDDM) or maturity onset diabetes, Type 2 dia-
betes is essentially a disorder of middle age onwards.
However, with the increase in childhood obesity in recent
years there have been increasing reports of this form of
diabetes in much younger age groups. Therein lies a clue
to the origin of this disorder, because Type 2 diabetes is
closely linked to obesity. As the rates of obesity have
soared over the last twenty years, so in tandem has the
incidence of Type 2 diabetes. The underlying factor lead-
ing to Type 2 diabetes is a failure of the pancreas to pro-
duce sufficient insulin. This may be for one or both of
two reasons. The first is that there are insufficient insulin
producing cells in the pancreas to meet demands. The
second is that there is resistance by the body’s target
cells to the insulin which is produced, thus requiring in-
creasing amounts to ensure effect. Either way the ulti-
mate poverty of effect of insulin is related to over weight
or obesity. One of the main actions of insulin is to help
regulate blood sugar (or to be precise, blood glucose)
levels. It achieves this in part by promoting uptake of
sugar by cells, that is, muscle cells. Insufficient action of
insulin leads to a reduction in sugar uptake and therefore
J. M. NTAGANDA
422
causes an abnormal rise in blood sugar. The result of this
is the symptoms of diabetes (excessive thirst, passing of
excessive amounts of urine, fatigue, frequent skin infec-
tions, e.g., boils, etc,). Type 2 diabetes often runs in
families but even so the ways the patient can prevent its
onset are: keep to a healthy weight, eat a healthy diet and
ensure adequate levels of daily exercise. With regard to
diabetes, regular physical activity aids weight manage-
ment, improves insulin sensitivity and therefore blood
glucose control, aids blood fats control, and improves
your overall sense of well-being.
Since the 1960s, mathematical models have been used
to describe glucose-insulin dynamics [2]. Bergman et al.
[3] proposed a three-compartment minimal model to ana-
lyze the glucose disappearance and insulin sensitivity dur-
ing an intravenous glucose tolerance test. Modifications
have been made to the original minimal model to incor-
porate various physiological effects of glucose and insu-
lin. Cobelli and co-workers [4] developed a revised mini-
mal model in order to separate the effects of glucose
production from utilization. The overestimation of glu-
cose effectiveness and the underestimation of insulin
sensitivity by the minimal model were addressed in yet
another publication by Cobelli et al. [5] where a second
no accessible glucose compartment was added to the
original model. Hovorka and co-workers [6] extended the
original minimal model by adding three glucose and in-
sulin sub-compartments in order to capture absorption,
distribution, and disposal dynamics, respectively. Anir-
ban Roy et al. presented a three compartmental model to
capture the changes in glucose and insulin dynamics due
to exercise [7]. This model incorporates the effects of
physiological exercise into the Bergman minimal model
[3] in order to capture the plasma glucose and insulin
dynamics during, as well as after, periods of mild-to-
moderate exercise.
In this paper we are interested in the role of physical
activity, and how it plays a crucial role in controlling
plasma glucose level and increasing insulin sensitivity in
Type 2 diabetes is highlighted through a bicompartmen-
tal model such that the controls are those of cardiovascu-
lar-respiratory system. Therefore, the formulation of op-
timal control problem is done. There are numerous me-
thods that allow solving this kind of problem. We prefer
to make a comparative study of direct method with an-
other approach based on the fuzzy logic strategy.
This paper is organised as follows. Section 2 presents
the model equations and optimal control problem. A short
description of strategy approach by fuzzy logic for solv-
ing optimal control problems is discussed in this section.
The Section 3 is interested in the application of the direct
approach and the approach integrating the fuzzy logic for
solving an optimal control problem of glucose-insulin in
diabetic human. The numerical simulation is presented in
Section 4. Finally, we present conclusion remarks in Sec-
tion 5.
2. Methods
2.1. Setting of the Problem
Taking account of the physiological properties of glu-
cose-insulin system, we propose a model elaborated in [8]
where we consider a two compartmental model composed
of the liver compartment (LC) and the pancreas com-
partment (PC). The diagram is shown in the Figure 1.
It is well known the arterial pressure leads the
tissues to receive the blood from cardiovascular respira-
tory system whereas the blood comes to cardiovascular
respiratory system from tissues due to arterial pressure

A
P
V
P. The cardiac rhythm

H
and the alveolar venti-
lation
A
V
are the parameters that influence the car-
diovascular-respiratory system to control these pressures.
During the physical effort, the cardiac rhythm is adjusted
via the baroreceptor controls while alveolar ventilation is
adjusted by the respiratory control. The respiratory con-
trol system varies the ventilation rate in response to the
levels of dioxide CO2 and oxygen O2 gases. Conse-
quently, it arises the ventilation rate and cardiac output
influence mutually. It is then obvious that exchanges
between LC and PC are controlled by heart rate
H
and alveolar ventilation
A
V
functions. The mechanism
of this control is not direct and can be represented by
outflow functions between systemic arterial and venous
compartments that depend on heart rate alveolar ventila-
tion (Figure 1).
Figure 1. A schematic diagram of two compartments for
modeling human glucose-insulin. Ql and Qr are left and right
cardiac flow respectively. H is heart rate and denotes
alveolar ventilation. PA and PV represent arterial and ve-
nous pressure respectively.
A
V
Open Access OJAppS
J. M. NTAGANDA 423
These functions represent the mass transfer between
these compartments where the exchanges are represented
by the arrows in the Figure 1. Let us consider that the
parameters
H
and reach their equilibrium values
respectively
A
V
e
H
and , the optimal control problem
during a physical activity can be formulated as follows.
e
A
V
Find
H
, solution of
A
V








max
,
22
12
0
22
12
min ,
d
A
A
HV
Tee
ee
AA
JHV
aGtGa ItI
bHtHbV tVt

 

(1)
subject to








1.3285
0.6253
d,, 0130
d
d,, 025,
d
A
A
GGIfHV G
t
IIGgHV I
t
 
 
(2)
where

0.0346 0.7329
, 0.76040.8627
AA A
fHVVVHH 
  (3)
and

0.3011 0.0262
,0.0191 178.9206
AA A
gHVVHV HH 
 
(4)
with 1 2 1 and 2
b the real constants. The vari-
ables of the mathematical model are glucose
,a,a b
G and
insulin
I
.
2.2. Description of Fuzzy Logic Strategy
Approach
Let us consider the following problem.
Find,

*
01
,, T
N
N
UU U
1N
, that minimizes


TT
01
0
,,
N
kkk k
k
UU xRxUQU

(5)
subject to

1,, 0,,1
,
kkkk
kk
xfxU
kN
xU


(6)
where and are positive defined matrices.
RQ
The problem (5)-(6) can be solved by the dynamic
programming method. This method has a fast conver-
gence, its convergence rate is quadratic and the optimal
solution is often represented as a state of control feed-
back [9]. However, the solution determined by this method
depends on the choice of the initial trajectory and in
some cases this solution is not optimal. It is for this rea-
son that the integration of the fuzzy logic [10] can permit
to determine quickly the optimal solution. We develop a
linearization strategy of the subject system by an ap-
proach based on the fuzzy logic. This approach had been
developed by Takagi-Sugeno [11,12]. The model that has
been introduced in 1985 by Takagi-Sugeno permits to get
some fuzzy linearization regions in the state space [13].
While taking these fuzzy regions as basis, non linear
system is decomposed in a structure multi models which
is composed of several independent linear models [14].
The linearization is made around an operating point con-
tained in these regions.
Let’s consider the set of operating point ,
i
X
1, ,iS
. Different fuzzy approximations of the non-
linear term
NL x can be considered.
1) The approximation of order zero gives:


0i
NL xNLxNL x (7)
2) Using the first order of Taylor expansion series we
obtain:
 


T
1
d.
d
i
ii
x
NL x
NL xNLxNLxxx
x

 


(8)
To improve this approximation, we introduce the fac-
tor of the consequence for fuzzy Takagi-Sugeno system.
This factor permits to minimize the error between the
non linear function and the fuzzy approximation. If
designates this factor, the approximation (8) can be for-
mulated as the following form:




01
T
1
d,
d
with 01.
i
ii
x
NL xNLxNLx
NL X
NL xxx
x
 

 



(9)
If one replaces the term by its value approached
in (6), the linearization around
NL
i
x
leads to
1, ,,
, 1,,;0,,1
kikkikkik
xAxBUCi SkN

(10)
where ,ik
A
and ,ik are square matrix which has N ×
N order and matrix with order.
B
,ik
Therefore, the optimal control problem (5)-(6) becomes
a linear quadratic problem which the feedback control is
given by the following expression [15,16]:
C1N
,,1,,; 0,,1
iki k
UKxi SkN,
  (11)
where

1
TT
,
iiiiiii
K
QBEB BEA
 (12)
is the feedback gain matrix and i discreet Riccati equa-
tion solution of the following form
E

1
TTT T
0.
ii iii iii iiiii
EQA EAA EBRBEBBEA
 
(13)
Open Access OJAppS
J. M. NTAGANDA
Open Access OJAppS
424
It is obvious that the linearization around every oper-
ating point gives the system for which the equations have
the form (10). Because there are operating points, we
have systems which have this form. Therefore, ac-
cording to the relation (11) controls are determined.
The defuzzyfication method [12] permits to determine
only one system and only one control .
S
S
S
k
Then, this transformation gives the following equation:
U
1, 0,,1,
kkk
xAxBUCk N
  (14)
, 0,,,
kk
UKxk N (15)
where








,,
11
11
,
11
11
,
, and ,
SS
iiikiiik
ii
SS
ii ii
ii
SS
i iiki iik
ii
SS
ii ii
ii
xA xB
AB
xx
xC xK
CK
xx















 



0.2671
1
0.9654
2
0.7329
3
0.0346
0.9738
1
0.6989
2
3
0.7604 0.7329
0.0346 0.7604
, 1.49330.7329
0.0346
0.0191 4.6877
0.3011 0.0191
,
ee
fA
ee
fA
eee ee
fA A
e
A
ee
gA
ee e
gA
e
g
CVH
CV H
CfHVHVH
V
CV H
CVHH
CgH

 



 

 

0.0262
0.3011
4.6877
0.30110.0382 .
ee
A
eee
AA
VH
VHV

 

Let us set the following variable change
,
(16)

TT
, and ,
eee e
AA
XGGII UHHVV 

.(18)
Therefore, the system (17) is written as the follows








1.3285
1
12
11223
0.6253
2
21
112 23
d
d
d
d
.
ee
ee
f
fA
ee
ee
ggA
XXGXI
t
CUHCU VC
XXI XG
t
CUHC UVC
 



and where

ii
x
designates membership degree part-
ner to the operating point i
x
.
f
g
(19)
3. Numerical Approaches for Solving the
Optimal Control Problem (1)-(2)
Using explicit Euler scheme on an uniform grid
N
,
the system (19) is approximated by the following (see
Equation (20))
3.1. Fuzzy Logic Strategy
To approximate the optimal control problem (1)-(2), we
propose to use the explicit Euler scheme. The stability of
this scheme constitutes an advantage to approach some
ordinary differential equations.
where max
T
hN
.
To approximate the objective function of the problem
(1), we use the rectangular method. Hence, we obtain
From the function
f
and
g
given by (3) and (4)
respectively, the Taylor expansion around e
H
and
allows the system (2) to become
e
A
V






1.3285
12 3
0.6253
12 3
d
d
d,
d
f
fA f
ggAg
GGI CHCVC
t
IIG CHCVC
t
 
 
(17)


1TT
12
0
,,
N
N
kkkk
k
XUXRXU RUh

(21)
where
T
1, 2,
,
kkk
XXX

. and are the matrix
1
R2
R
ab
11
12
22
00
,
00
RR
ab

 
 
k
f
k
g
The system (20) has two following nonlinear factors:
where












1.3285
1, 11,2,11,22,
1.3285
2,12 3
0.6253
2, 12,1,11,22,
0.6253
1,12 3
1
1
,
e
kkkfkf
ee ee
kffA
e
kkk gkg
ee ee
kggA
XhXhXICUCU
hGXICH CVC
XhXhXGCUCU
hIXGCH CVC
  

 



 


 


(20)
J. M. NTAGANDA 425






1.3285
2
0.6253
1
and .
e
e
NL XXI
NL XXG


The objective is to linearize these terms. This mecha-
nism allows determining the Takagi-Sugeno fuzzy sys-
tem. For this, we apply the fuzzy strategy and we con-
sider the case of health person who exercises most regu-
larly by jogging. We take Ge = 90 mg/dl and Ie = 30
μU/dl. The equilibrium of cardiovascular respiratory pa-
rameters values e
H
and for someone who does
physical activity are given by the table (See [17]).
e
A
V
We consider a universe of discourse
X
which has
two linguistic variables: glucose (GL) and insulin (INS).
Taking account of the physiology, we consider
60,140G and
20, 40I. Therefore, the glucose
(resp. insulin) is low if 60Gmgdl (resp. I = < 20
μU/dl). If (resp. I) is included between 60 and G
140mgdl (resp. 20 and 40Udl
), we suppose that
the glucose (resp. insulin) is normal. While if
(resp. ) we say that the glucose (resp. insulin) is
the highest. Then, GLB (Low glucose), GLN (Normal
glucose) and GLE (the highest glucose) constitute the
terms (fuzzy sets) of the linguistic variable GL. In an
analogous way, INSB (Low insulin), INSN (Normal in-
sulin) and INSE (the highest insulin) are the terms of the
variable linguistic INS.
140G
40
I
According the relation (18) and equilibrium values
given by the Table 1, we have
130, 50X and
215, 5X . During the physical activity, the glucose
(resp. insulin) varies such that we can consider a universe
of discourse
X
where the labels are centered at 30, 10
and 50 (resp. 15, 5 and 5). Then, we suppose that the-
ses centers constitute the operating points values of the
system (20). We designate these points as Mi, 1,i2, 3
,
for the first equation of the system (20) and as
for the second. It is obvious that these points
take the corresponding values in the labels centers of a uni-
verse of discourse
,
i
N
1,2, 3,i
X
[10]. Membership functions asso-
ciated to this labeling are represented in the Figure 2.
To simplify, we consider only the Taylor expansion of
first order around the operating points i
M
and i. We
obtain three systems of the following form (see Equation
(22))
N
Finally, the optimal control problem (1)-(2) can be
formulated as follows.
Find solution of
T
0
,,
N
UU U
 
1N
1


TT
12
0
min ,kk
kk
Uk
J
XUXRXU RUh

(23)
subject to
1,, ,
, 1,2,3,
kikkikkik
XAXBUCi
  (24)
where
10
, 1,2,3
01
i
h
Ai
h












0.6253 0.6253
12
1.3285 1.3285
12
,
1,2,3
ee
ifi
iee
igi
NI CNI C
Bh
MG CMG C
i



 





f
g
f
g
Let us set






0.6253
123
1.3285
123
,
1, 2, 3.
ee ee
iffA
iee ee
iggA
GNI CHCVC
Ch
IMGCHCVC
i

 



 

100N
and , then the follow-
in
Table 1. Equilibrium of cardratory system
Parameter G (mg/dl) I (μU/dl He (Beats/min) (L/min)
max 10T
g results are
0
found
.90, 1,2,3
00.9
i
Ai



12
3
0.2133 1.95780.15501.4225
,
0.00011 0.00390.000050.0019
0.1256 1.1526
0.00003 0.0013
BB
B







123
4.01425.3775 6.0648
, , .
2.9543 2.9769 2.9851
CCC






iovascular-respi
parameters in the jogging case.
e ee
A
V
Value 90 30 140 15
(22)












1.3285
1, 11,11,22,
1.3285
123
0.6253
2, 12,11,22,
0.6253
123
1
,
1
. 1,2,3
e
kki fkfk
ee ee
iffAf
e
kki gkgk
ee ee
iggAg
XhXhNGCUCU
hGNGCH CV C
XhXhMICUCU
hIMICHCVCi




 



 


 


Open Access OJAppS
J. M. NTAGANDA
426
Figure 2. Triangular membership functions associated to operating points 30, 10 and 50 (resp. 10, 0 and 10) for the linguistic
variable GL (a) (resp. INS (b)) according to the variable change (18). 40 and 5 are the values of entries obtained thanks to the
formula (18).
It is easy to note that the problem (23)-(24) is a linear
quadratic (LQ). Since there are three linear state systems,
the solution leads to three feedback controls of the form
(25)
,, 1,2,3;
kikk
UKXi
where i
K
is a gain feedback.
The implementation can be made in several platforms.
Here we use MATLAB package. If and are
identified matrixes of the second orde
s
(26)
where
1
R
r, we obtain
2
R
12
1.1712 0.00811.2868 0.0055
,
0.0081 5.26310.00555.2631
KK






3.00425.2632

1.39280.0042 .
0
K


The defuzzification transformation allows to obtain
one system. Consequently, for the system(23) thitech-
nique gives the following system
1, 0,,1,
kkk
XAXBUCk N

A
and are matrices and a B22C21
matrix.
In the same way, from the matrixes 1,
K
2,
K
and
3
K
K
the defuzzication process allows to haatrix ve one m
. We propose the following procedure.
The first (second) line of matrixes ,,
iii
A
BC and ,
i
K
1,2, 3,i is defuzzified using the dembgree of meership
2
f
and 3
f
[see the Figure 2(a)] (resp. 2
g
and
3
g
[see
he
the Figure 2(b)]). This ma
reaso
consider the degree of membership of the entry
glle change
thi
nner of procedure is
due to t two following ns.
1) We
ucose (resp. insulin). According to variab (18),
s value is 40 mg/dl [see the Figure 2(a)] (resp.
5Udl
get 1
[see the Figure 2(b)]). After calculations, we
0,
f
20.25
f
and 30.75
f
(θ = (0;
0.2
GL
5; 0.75)) [resp. 1g0,
20.5
g
and 30.5
g
.
0.5 .
linear factor
0;0.5
S
) The non
;
IN
2
NL X






1.3285 0.6253
21
resp.
ee
XINL XXG

 
int quatio the
sy
Consi
matr
ch
To aus consider
ervenes only in the first (resp. second) en of
stem (22).
dering these hypothesis, we have the following
ixes.
0.9 00.146
,AB


21.3415
00.90.00004 0.0015
5.58371.3186 0.00513
,.
2.98310.0046 5.2632
CK
 
 






3.2. Direct Approa
pproximate the system (2), let
,1,,
NN
jjN
 (27)
a linear B-splines basis functions on the uniform grid
max , 0,,,
Nk
tk
N
N
kT
 

(28)
such that
N
ik ik
t
N
W
Let us introduce the vector space whose the ba-
Open Access OJAppS
J. M. NTAGANDA 427
sis is .
N
dim N
WW
Let us co
olation o
We have
r and let us take the in-
terp perator
WN
1NN
nside

00,WC T
:
N
N
N
WW


(29)
satisfying
 
,1,
N
kk
ttk

 ,.N (30)
We verify easily that
0
N
N
EE
 

  (31)
0
sup 1.
N
NE
E
W
 
(32)
0
Now, let us set
NN
0
, and
N
Nk
fff
NNNk
kk
k
ggg
N
k
 

(33)  
with




, and ,
k kAk
,
kk
kA
f
fHtV

tg gHtVt
Theref
lowing form
ore, the system () can be ap
solution of the system
proached by the fol-
Find

2
,
NNN
GI W


1.3285
dN
d
N
NN
GI f
t
 
G (34)


0.6253
dN
d
N
NN
P
I
Gg
t
  (35)

,0 ,0
0, 0
NNN
GPII ,
N
(36)
such that
0,0 0
N
N
GG 
 (37)
0,00.
N
N
II 
 (
his approximation, we have the follow-
The tion sequence of the system (34)-(36) co
ve
38)
According to t
sult.
Proposition [8]
solu
ing re
n-
rges uniformly toward the solution of the system (2) on
the interval
max
0, ,T
ximate the
max 0.T
optimTo apal problem (1)-(2), let pro

T
,
us set
x
GI the state vector,

T
000
,
x
GI the initial
state vector,

T
,
eee
x
GI tlibrium state
vector, the control vector and
the equilibrium control v
he wanted equi

T
,A
HV
,
ee
HV

T
e
A
ector;
0
,,,
e
ii i
xx xi
and e
i
designate the components
th
i th
i
of the vector 0
,,
e
x,xx
and .
e
Therefore, the problem (1)-(2) can ta
compact form
ke the following
2
min N
QJ

max
TN
ax


2
22
d,
ee
t xbtt

 
011
ii ij jj
ij

where

T
,
NNN
xxx is solution of the approximated
so
(39)
12
lution (34)-(36).
We must determine

12
,
M
MMM ap- Q


2
MM
an
proximate so
that we can w
(40)
Therefore, wfunction
by
lution of (39) in .It is nec-
essary to noterite

QW

,, 1,2.
M
j
kk
tj

0
M
M
j
k

e can approximate the objective


22
,
11 1
,
Ne Me
ii kijjkj
ki j
t xbt

 
22M
ax

MN
J


 (41)
where max
T
tN
 . The convergence of the discreet o
tive function (41) toward the continuous objective func-
en by the proble
al control problem (1)-(2) is a mini-
mraint. The discreet
tio written as follows.
bjec-
tion givm (39) has been shown in [8].
Finally, the optim
isation problem with const formula-
n of such problem can be
Find
 
11
,MM
M
 solution of





11
T
T
12
min ,
MM
M
MM
N
JtYRYR



 

M
(42)
subject to




1.3285
0.6253
d
d
d,
d
N
N
NN
N
N
NN
GGIf
t
I
I
Gg
t
 

where
(43)
is a matrix
12M
M
such that the com-
ponents ,
M
j
k
e those function
N
j
in
N
and Y is ar
the matrix such that the

,ponent is
th
ik com
,
N
e
ik i
x
tx where

T
12
,
NNN
xxx is the solution of
(1)-(2) associated to
N
.
4. Numerical Simulation
Wte cardiovasculartory response
to glucose and insulin dynamic for a 30-year-old trained
e consider the acu respira
Open Access OJAppS
J. M. NTAGANDA
428
women whose mean values are given in Table 1 [17
The solutions of the optimal control problem (23)-(2
and (42)-(43) can be determined in several platform.
n of these solutions is made using M
ges.
To solve the problem (23)-(26) by fuzzy logic strategy
only one program is enough. Using direct approach, the
solutions of the problem (42)-(43) are give
sion of programs based on MATLAB function used in
optimization that is fmincon. This function is a MAT-
rogrws solving
e by A
pproach integratin
and direct aponsequently, t
r the execution of main MATLAB
DIR r. The results are
tel(R) core (TM)2 Duo
fu
].
4)
The
AT-implementatio
LAB packa
n by a succes-
LAB pam which allo minimization prob-
lem with constraints.
In this section, we notHLF, ADIR to designate
respectively the hybrid ag fuzzy logic
proach. Che Table 2 gives us
the results found afte
program for AHLF, Aespectively
obtained using a Processor In
CPU, 2.20 GHZ.
Table 2 shows that the time execution of the main
program to solve the problem (1)-(2) by AHLF is very
small compared to one of ADIR. This argument justifies
the precision of the fuzzy logic strategy.
Considering jogging as physical activity for a 30-year
old trained woman, the variations of the optimal parame-
ters is obtained using the hybrid approach integrating
zzy logic and the direct approach. The results are given
in the curves represented in the Figure 3.
Table 2. Minimal values of the objective function (Jopt) and
the execution time (T) of main program for the resolution of
the optimal control problem (1)-(2) by AHLF, ADIR.
AHLF ADIR
Jopt 10.0524 35.8572
T (Second) 5.8125 20.286
Figure 3. Optimal parameters for a 30 year old woman with
jogging as her physical activity. The curve in solid line
representse wanted valucurve in dasdi-
cates theimal paramee hybrid a inte-
grating the fuzzy logic strategy
For-old womrate and alveolar
ventilation play a crucial ro in the control of the car-
diovascular-respiratory system. Consequently, their sta-
bility ensures the performance of sportsman in general
and of woman in particular. For a woman of 30 years old
where jogging is her regular physical activity, we see that
2.5 minutes after the starting of the exercise the variation
of the heart rate reaches the value close to the equilib-
rium value before having its oscillation around this value
(Figure 3(a)). The solutions from AHLF and ADIR show
that after 5 minutes of the starting time the optimal al-
veolar ventilation (Figure 3(b)) increases to reach the
wanted equilibrium value and the glucose (Figure 3(c))
decreases gradually to reach the wanted value in 10 min-
utes before its oscillation around this value. The use of
these approaches allows also the optimal insulin to in
/j.1365-2362.32.s3.5.x
th
op
e. The hed line in
tter for th
.
pproach
r a 30-yeaan, the heart
le
-
crease and to reach the maximum that is close to the
wanted value at 3th minute before its oscillation around
his value (Figure 3(d)). Comparing the results using
AHLF and ADIR, it is important to see in the Figure 3
that they are much closed.
5. Conclusion Remarks
In this work, we compared two approaches to determine
the optimal trajectories of glucose and insulin as response
to controls of cardiovascular-respiratory system subjected
to a physical activity. The finding results for two used
methods are satisfactory and closed. But the hybrid ap-
proach integrating the fuzzy logic strategy has an advan-
tage over the direct approach in term of time. Conse-
quently, it constitutes an important approach for the
resolution of the optimal control problem. In particular, it
gives the optimal trajectories of glucose-insulin system in
the same way so thet it ensures their performance.
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