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This paper aims at using of an approach integrating the fuzzy logic strategy for hypoxemic hypoxia tissue blood carbon dioxide human optimal control problem. To test the efficiency of this strategy, the authors propose a numerical comparison with the direct method by taking the values of determinant parameters of cardiovascular-respiratory system for a 30 years old woman in jogging as her regular physical activity. The results are in good agreement with experimental data.

Hypoxia, or hypoxiation, is a pathological condition related to adequate oxygen supply in human body. The derived adequate oxygen supply can be whole body (generalized hypoxia) or its region (tissue hypoxia). Generalized hypoxia occurs in healthy people when they ascend to high altitude, where it causes altitude sickness leading to potentially fatal complications: high altitude pulmonary edema (HAPE) and high altitude cerebral edema (HACE) [

Hypoxia is also a serious consequence of preterm birth in the neonate. The main cause for this is that the lungs of the human fetus are among the last organs to develop during pregnancy. To assist the lungs to distribute oxygenated blood throughout the body, infants at risk of hypoxia are often placed inside an incubator capable of providing continuous positive airway pressure (also known as a humidicrib).

Hypoxia denotes oxygen deficiency at the mitochondrial sites due to insufficient delivery of oxygen (low

In humans, hypoxia is detected by chemoreceptors in the carotid body. This response does not control ventilation rate at normal

Any physical activity will obviously cause the body to demand more oxygen for normal functioning. The muscles will rob the brain of the marginal amounts of oxygen available in the blood and the time of onset of hypoxic symptoms is shortened.

This paper is organized as follows. Section 2 presents the model equations and optimal control problem. A short description of strategy approach by fuzzy logic for solving optimal control problems is discussed in this section. 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 Section 5.

First of all we focus on the models equations as developed by Guillermo Gutierrez [

From the diagram presented in the

where the variable states

It is known that the human respiratory control system varies the ventilation rate

Consequently, the control of cardiovascular and respiratory system is described via the following two ordinary differential equations respectively.

where the functions

Now let be interested in writing the arterial blood

where

where

By considering the calculation done in [

where

Similarly, blood

where

The respiratory control system aims at keeping

Find

subject to the system (1)-(2) and (3)-(4).

In the relation (7), the positive scalar coefficients

Let us consider

on a regular grid

The functions

where

and the desired final vector

such that it can be written as follows.

where

with

We are looking for

Therefore the cost function (10) becomes

where (12) is determined using rectangular method such that the discretisation is done on a regular grid

Finally, the discrete formulation of optimal problem (7) subject to (1)-(2) and (3)-(4) is written as follows.

where

components of solution of the system (1)-(2) and (3)-(4) associated to

Let us consider the following problem.

Find

subject to

where

The problems (15) and (16) can be solved by the dynamic programming method. This method has a fast convergence, its convergence rate is quadratic and the optimal solution is often represented as a state of control feedback [

Let’s consider the set of operating point

The approximation of order zero gives:

Using the first order of Taylor expansion series we obtain:

To improve this approximation, we introduce the factor 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

If one replaces the term

where

Therefore, the optimal control problems (15) and (16) become a linear quadratic problem which the feedback control is given by the following expression [

where

is the feedback gain matrix and

It is obvious that the linearization around every operating point gives the system for which the equations have the form (20). Because there are

Then, this transformation gives the following equation:

where

and where

To approximate the optimal control problems (7), (1)-(2) and (3)-(4), we propose to use the explicit Euler scheme. The stability of this scheme constitutes an advantage to approach some ordinary differential equations.

The discretisation of the constraints (1)-(2) and (3)-(4) is done using the first order explicit Euler method. From the Equations (1)-(2) and (3)-(4) and taking

Applying the first order explicit Euler’s method, the system (27) is transformed as follow

where

ating point number related to each operating point we have the system

where

Let us set the following variable change

that is

The system (27) can be formulated using the relations (6) and (30) but here we prefer to keep this form. The use of these relations is taken into account in numerical simulation. Therefore, the approximation of objective function (7) is made using the rectangular method and it becomes

where

and where

Find

subject to

To approximate the system (1)-(4), let us consider

a linear B-splines basis functions on the uniform grid

such that

Let us introduce the vector space

1)

2)

Let us consider

satisfying

We verify easily that

Therefore, the system (1)-(4) can be approached by the following form

Find

such that

To approximate the optimal problems (1)-(4), let us set

Therefore, the problems (1) and (2) can take the following compact form

where

We must determine

Therefore, we can approximate the objective function by

where

tion given by the problem (46) has been shown in [

Finally, the optimal control problems (7), (1)-(2) and (3)-(4) are minimisation problems with constraint. The discreet formulation of such problem can be written as follows.

Find

subject to

where

the matrix such that the

Let us consider a hypoxic patient (a 30 year old woman) practicing jogging as physical activity for a period of

take

The operating points associated to those linguistic variables are given in the

Using parameters values from

Variable | Operating Points |
---|---|

[CO_{2}]_{t} | [ |

[CO_{2}]_{v} | [ |

H | [ |

[ |

Variable | W_{1f} | W_{2f} | W_{3f} |
---|---|---|---|

[CO_{2}]_{t} | 0 | 0.4 | 0.6 |

[CO_{2}]_{v} | 0 | 0.27 | 0.73 |

H | 0.3 | 0.7 | 0 |

0.38 | 0.62 | 0 |

Parameter | Value | Parameter | Value |
---|---|---|---|

6 | 38 | ||

K_{v} | 0.05 | 95 | |

pH | 7.35 | 26.5 | |

SaO_{2} | 0.98 | 18.5 | |

0.21 | 47 | ||

P_{ATM} | 760 | RQ | 0.8 |

0.21 | SV | 0.7 | |

10 | 0.0065 | ||

15 | 0.244 | ||

q_{H} | 100 | [O_{2}]_{a} | 0.197 |

q_{V} | 100 | [O_{2}]_{v} | 0.147 |

K | 863 |

The next step of the fuzzy logic strategy at this point is the defuzzification. The formulas used in the defuzzification are illustrated in (26). Now considering our Equation (29), the matrices

Since there are three linear state systems, the solution leads to three feedback controls of the form

where

For solving the optimal control problem (7) subject to the system (1)-(2) and (3)-(4), we take

The controls variation of the cardiovascular respiratory system are represented in

In this work, two numerical approaches have been compared to determine the optimal trajectories of arterial pressures of of carbon dioxide and oxygen as response to controls of cardiovascular-respiratory system subjected to a physical activity. The finding results show that those two used methods are satisfactory and closed. Consequently, the approach integrating the fuzzy logic strategy is very important for the resolution of the optimal control problem. In particular, it gives the optimal trajectories of cardiovascular-respiratory system in the same way it ensures their performance.