INTERVAL ADDITIVE PIECEWISE POLYNOMIAL TIME OPERATION MODEL OF HUMAN-OPERATOR IN A QUASI-FUNCTIONAL ENVIRONMENT

А -Е Е Е А Ы Е , Е А

а т–а е ь 2015 15 № 2 ISSN 2226-1494 http://ntv.ifmo.ru/ SCIENTIFIC AND TECHNICAL JOURNAL OF INFORMATION TECHNOLOGIES, MECHANICS AND OPTICS March–April 2015 Vol. 15 No 2 ISSN 2226-1494 http://ntv.ifmo.ru/en

331:519.7:62.50:681.306

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А : 79214215187@ya.ru

01.12.14, 25.02.15 doi:10.17586/2226-1494-2015-15-2-329-337

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-, . 2015. 15. № 2. . 329–337.

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( 14. Z50.31.0031).

INTERVAL ADDITIVE PIECEWISE POLYNOMIAL TIME OPERATION MODEL

OF HUMAN-OPERATOR IN A QUASI-FUNCTIONAL ENVIRONMENT

M.V. Serzhantovaa, A.V. Ushakova

a

ITMO University, Saint Petersburg, 197101, Russian Federation Corresponding author: 79214215187@ya.ru

Article info

Received 01.12.14, accepted 25.02.15 doi:10.17586/2226-1494-2015-15-2-329-337 Article in Russian

Reference for citation: Serzhantova M.V., Ushakov A.V. Interval additive piecewise polynomial time operation model of human-operator in a quasi-functional environment. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2015, vol.15, no. 2, pp. 329–337. (in Russian)

human-operator activity in the quasi-static environment has given the possibility to analyze and predict functional measures for performance management of human-operator functional activity in manufacturing static environment.

Keywords: human-operator, functional activity, warming-up, tiredness, recreational interval, productivity, piecewise polynomial approximation, additive models with interval parameters.

Acknowledgements. This work was supported by the Government of the Russian Federation, Grant 074-U01 and the Russian Federation Ministry of Education and Science (Project 14. Z50.31.0031)

(Ч )1 ,

. [1–11]

- .

, Ч

-. К ,

-Ч .

Ч

. ,

– , [12],

Ч -

, ,

,

, - ,

-, , , ,

-.

Ч , ,

,

[4, 6, 7]. , , ,

( ) ,

, , . .

-, Ч

. , Ч

[6, 7] .

-Ч

, ,

[13].

А , ,

-,

-, .

-, , ,

-Ч

,

Ч .

: Ч

-,

.

-

-

Ч

Ч , [7]

- . 1.

( . 1)

-Ч : (0) – ; (t₁₁) –

-, ; (t12) –

-Ч ; (t₁₃) –

-1

,

-; (t14) – ( К ),

-; (0–t11) – , ; (t11–t12) –

k; (t13–t14) –

-, ,

« К »



1 



k 0,7k ,

-0, 3

  . ( . 1) [7]

-Ч

-: (0–t11) – 12 (0,2 ); (t11–t12) – 21 (0,35 ); (t12–t13) – 147 (2,45 ). Э

Ч .

. 1. я я Ч

, . 1,

Ч

-. , [7],

 

h t

-, . 2.

. 2. я я

К ( . 2) , «

(t_p) –



1 



k», . 1.

p

t

 , 0,25 0,5 0,75 1,0





1 



k 0,75 k 0,9355 k 0,9844 k 0,9961 k 1

1. я



1 



k

Ч (t_p)

. 1 ,

, ,

,

-,



1 



k h(t)

0

t11 t12 t13 t14

tp

hB(t) 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1

. , Ч

, . 3.

. 3. я я я Ч

( . 3)

. , Ч

-.



0t11



, ,

:

 



  





0,11 0 1 1 11 .

h t   t  tt (1)



t11t12



,

k ,

 



11

 



12

 

11





11,12

12 11

1 1

k

h t t t t t t t

t t

     

 . (2)



t₁₂t₁₄



Ч

:

 





 





12,14 1 12 1 14

h t  k tt  tt . (3)

, (t13),

12 13 14

t t t t133 , h13,14

 

t Ч ,

:

 



₁₃

 



₁₃

 

₁₄





13 14 14 ,

13 t t 1t t t t

t t

k t

h     

   

 , (4)

 –

3 , 0



 .

(1)–(4) -

-Ч h0,14

 

t :

 

 

 

 

 



 



 









 







 



 





0,14 0,11 11,12 12,14 13,14

11 12 11 12 14

12 11

13 13 14

14 13 1

1 1 1 1

1

h t h t h t h t h t

t t t t t t t t t t

t t k

t t t t t t

t t

     

 _{         }

 _ 

 

  _ 

_{     } 

 _ 

 

. (5)

, Ч

(1)–(4) ,

-, k



1 



,

 – Ч

-, . 1.

-:

h(t)

k

0



t14t21



: h14,21

 

t  0 1





tt14

 

1 tt21





; (6)



t21t22



:

 



_{      }

_

_

21,22 21 21 22

22 21 1

1 1

k

h t t t t t t t

t t

  

     

 ; (7)



t22t24



: h22,24

 

t     k



1

 



1 tt22

 

1 tt24





. (8)

, (t₂₃),

-22 23 24

t t t t23t143 , h23,24

 

t

-Ч ,

 







 











23, 24 23 23 24

24 23 1

1 k

h t t t t t t t

t t

    

      

 , (9)

(1)–(9) 1



tt_c



– tc [12].

(6)–(9)

-Ч h14,24

 

t :

 

 

 

 

 







 



 









 







 



 





14,24 14,21 21,22 22,24 23,24

21 21 22 22 24

22 21

23 23 24

24 23 1

1 1 1 1

1

1

h t h t h t h t h t

t t t t t t t t t t

t t k

t t t t t t

t t        _{         }  _         _  _{     }   _    . (10)

(5), (10) -

-Ч h0,24

 

t

 

 

 

0,24 0,14 14,24 h t h t h t



 



 









 







 



 





11 12 11 12 14

12 11

13 13 14

14 13 1

1 1 1 1

1

t t t t t t t t t t

t t k

t t t t t t

t t  _{         }  _     _{ }  _{     }   _   







 



 









 







 



 





21 21 22 22 24

22 21

23 23 24

24 23 1

1 1 1 1

1

1

t t t t t t t t t t

t t

k

t t t t t t

t t  _{         }  _          _  _{     }   _    . (11)

( ) Ч , (11)

-. ,

Ч y t

 

.

Ч y t

 

 

0,24

h t (11). ,

-tt,

(11)

:

1. t11 t t11 t t12

 

    

 

11

 

11

 

_

_

 





11 11 _{12 11}

2

0,14 0,14 11,12 12 11

12 11 0,5

0,5

t t t t

t t

t t t t t

k

y t h t dt h t dt t k t t

t t                  





; (12)

2. t12 t t12 t t13

      

 

 





 





 









12 12 13 12 12 12

0,14 0,14 12 11 12,13 12 11

13 11 12

0,5 0,5

0,5 ;

t t t t

t t t t t

t t

y t h t dt k t t h t dt k t t k t

k t t t

                      

3. t13 t t13 t t14       

 

 

 





























13 13 11 13 13 14 13 13 12 11

0,14 0,14 13 13,14 13 11 12 14 13

2

13 14 13 11 12

14 13

0, 5 2

0, 5 1 0, 5 0, 5 0, 5

t t t

t t t t

t t t t t t у у t t t

y t h t d t k t h t d t k t t t k t t

k

t t t k t t t t

t t                  _  _                     





(14)

4. t21 t t21 t t22

      

 









 













 











 





21 21 21 21

0,24 14 13 12 11 14,24 14 13 11 12

21,22 14 13 11 12 22 21

0,85 0,15 0,5 1 0,5 0,5 0,5

1 0,5 0,5 0,5 0,5 1 ;

t t

t t

t t t

t

y t k t t t t h t dt k t t t t

h t dt k t t t t t t

                              





(15)

5. t₂₂ t t₂₂  t t₂₃

 

 











 





 











 





 

 







  

22 22 22 22 23 22

0,24 0,24 14 13 11 12 22 21

22,23 14 13 11 12 22 21

14 13 11 12

1 0,5 0,5 0,5 0,5 1

1 0,5 0,5 0,5 0,5 1

1 1 0,5 0,5 0,5 1

t t

t t t t t

t

t t t

y t h t dt k t t t t t t

h t d k t t t t t t

k t k t t t t t

                                              















230,5t21t22



;

(16)

6. t23 t t23 t t24

      

 

 









  











 













 













13 11 23 23 24 23

0,24 0,24 14 13 11 12 23 21 22

23,24 14 13 11 12 24 23 21 22

1 0,5 0,5 0,5 1 0,5

1 0,5 0,5 0,5 1 1 0,5 0,5 0,5 .

t t

t t t t t

t

t t t

y t h t dt k t t t t t t t

h t dt k t t t t k t t t t

                                      





(17)

(12)–(17) k

Ч







11 12 13 14 21 22 23 24



8

arg , , , , , , , 4, , , , 8 1

t

k y k t t t t t t t t t



      . (18)

-

--

-Ч , ,

.

[14] ( Ч)

 

 –

 

 , ,

( ) ,

 

   _ , _. (19)

Ч

 

 [15]

 

         _

 

_ _ , _, (20)

0

 ,

 

 – Ч

( « ») ( « ») . (19),

(20)





0 0, 5 ; 0; 0

               . (21)

(20) (21) Ч, δI

 



Ч

 

 ,

 

I

, (22) Ч

 



 

      δI

 

 , δI

 

   



1



δI

 







. (23)

,

.

: , , ,

. 2.

 

I

δ _ # _% _{0 5 10 15 20 25 30 50}

 

1 I

δ _  _ % 0 5 10 15 20 25 30 50

   





I

δ   _{  }  _ % 0 5 19,8 29,33 38,46 47,05 55,04 80

   



1



I

δ _  _{ } _ % 0 10,25 21 32,25 44,4 56,25 69 125

     



1



I

δ   _{  }  _{ }  _ % 0 15,78 33,3 52,76 61,32 98,49 125,1 262,5 2.

,

-,

(

-).

Ч , ,

(14) (17), ,

– .

-Ч (14) (17) (23).

(14) (17)

 





 





 













 













 





 













 









I 14

0,14 I I 13 I 13

4

11 I 11 12 I 12

1 0, 5 1 δ

1 δ 0, 5 1 δ 1 δ .

0, 5 1 δ 1 δ

t

t

y t k k t t

t t t t



 _{    } 

 

 

 

  _     _

 

    

 

 



  

 

(24)

 





 











 











 









 





 













 









 















 







 













 















 







 













 









I 14 I 13 I 13

0,24 I

8

11 I 11 12 I 12

I 24

I I I 13 I 23

11 I 21 12 I 22

1 0,5 1 δ 0,5 1 δ 1 δ

1 δ

0,5 1 δ 1 δ

1 0,5 1 δ

1 δ 1 1 δ 0,5 1 δ 4 1 δ

0,5 4 1 δ 4 1 δ

t

t t t

y t k k

t t t t

t

k k t t

t t t t



 _{         }

 

   

    

 

 _{    } 

 

     _      



     



  



 



   

 

.

     

(25)

(24) (25),

-Ч

4 8 .

, Ч .

t₂₁ 4 t₁₁; t₂₂ 4 t₁₂; t₂₃ 4 t₁₃, -,

.

-

-

Ч

0, 3

 

0, 0156

1 0, 9844

 

  

( )  t 45 0, 75 , (18)

 

1 0,1451

k  .

-Ч

 





I

δ  0,1 10% .

1. Ч :

 

1

11 12 13 14

22 23 23 24

0,1451 ; 0, 2 ; 0, 55 ; 3, 0 ; 4, 0 ;

4, 2 ; 4, 55 ; 7, 0 ; 8, 0 .

k t t t t

t t t t



    

   

 

y t 1 . 4,

. 3.

. 4. y t

 

Ч :

я (1); я я (2);

я я (3)

2. Ч ,

, ,

:

 









 





 





 













 





 













 





 









1

I 11 I 11

12 I 12 13 I 13

21 11 I 11 22 12 I 12

23 13 I 13

1 δ 0,1 0,1306 ; 1 δ 0,1 0, 22 ;

1 δ 0,1 0, 605 ; 1 δ 0,1 2, 7 ;

4 1 δ 0,1 4, 22 ; 4 1 δ 0,1 4, 605 ;

4 1 δ 0,1 6, 7 .

k k t t

t t t t

t t t t t t

t t t



           

           

               

       

 

y t 2 . 4,

. 3.

3. Ч ,

, ,

:

 









 

 













 





 













 





 













 





1 1 I

11 I 11 12 I 12

13 I 13 21 11 I 11

22 12 I 12 23 13 I 13

1 δ 0,1 0,1596 ;

1 δ 0,1 0,18 ; 1 δ 0,1 0, 495 ;

1 δ 0,1 3, 3 ; 4 1 δ 0,1 4,18 ;

4 1 δ 0,1 4, 495 ; 4 1 δ 0,1 7, 3 .

k k

t t t t

t t t t t

t t t t t t

 

 

      _{ }

 

           

             

               

 

y t 3 . 4,

. 3.

y(t)

1,2

1

0,8

0,6

0,4

0,2

0 1 2 3 4 5 6 7 8 9 10 t

3

1

Ч _Ч k t11 t12 t13 t21 t22 t23 y t

 

_t8

1 0,1451 0,2 0,55 3,0 4,2 4,55 7,0 1

2 0,1306 0,22 0,605 2,7 4,22 4,605 6,7 0,8

3 0,1596 0,18 0,495 3,3 4,18 4,495 7,3 1,26

3. y t

 

Ч

-- ,

-, -

-,

.

References

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2. Baron S., Kleinman D.L., Miller D.C., Levision W.H., Elkind J.I. Application of optimal control theory to prediction of human performance in a complex task. Proc. 5th Annual NASA-University Conference on Man-ual Control. Cambridge, 1969, pp. 367–387.

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12. Besekerskii V.A., Popov E.P. Teoriya Sistem Avtomaticheskogo Regulirovaniya [The Theory of Automatic Control Systems]. St. Petersburg, Professiya Publ., 2003, 752 p.

13. Franks L. Signal Theory. Prentice-Hall, NJ, 1969, 318 p.

14. Kalmykov S.A., Shokin Yu.I., Yuldashev Z.Kh. Metody Interval'nogo Analiza [Methods of Interval Analysis]. Novosibirsk, Nauka Publ., 1986, 222 p.

15. Dudarenko N.A., Polyakova M.V., Ushakov A.V. Formirovanie interval'nykh vektorno-matrichnykh model'nykh predstavlenii antropokomponentov-operatorov v sostave slozhnykh dinamicheskikh sistem [Formation of the interval vector-matrix modeling representations of anthropocomponents-operators as a part of complex dynamic systems]. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2010, no. 6 (70), pp. 32–36.

С а ваМа яВя ав в а – , , ,

-, 197101-, , 12noch@mail.ru

У а вА а В а в – , , ,

, - , 197101, ,

79214215187@ya.ru

Maya V. Serzhantova – PhD, Associate professor, ITMO University, Saint Petersburg, 197101, Russian Federation, 12noch@mail.ru