FUNDAMENTAL MATRIX OF LINEAR CONTINUOUS SYSTEM IN THE PROBLEM OF ESTIMATING ITS TRANSPORT DELAY

3

ИЧЕ

_И

Е

_{ЕХ И}

Е ИЕ

AUTOMATIC CONTROL AND ROBOTICS

62.50: 681.5.01

. . , . . , . .

, 197101, - , , [email protected]

.

-.

, , , ,

« - ».

, ,

, – .

-,

.

-,

.

-.

.

: , , - ,

, .

. (

14.Z50.31.0031).

FUNDAMENTAL MATRIX OF LINEAR CONTINUOUS SYSTEM

IN THE PROBLEM OF ESTIMATING ITS TRANSPORT DELAY

N.A. Dudarenko , N.A. Polinova , A.V. Ushakov

ITMO University, 197101, Saint Petersburg, Russia, [email protected]

Abstract. The paper deals with the problem of quantitative estimation for transport delay of linear continuous systems. The main result is received by means of fundamental matrix of linear differential equations solutions specified in the normal Cauchy form for the cases of SISO and MIMO systems. Fundamental matrix has the dual property. It means that the weight function of the system can be formed as a free motion of systems. Last one is generated by the vector of initial system conditions, which coincides with the matrix input of the system being researched. Thus, using the properties of the system-solving for fundamental matrix has given the possibility to solve the problem of estimating transport linear continuous system delay without the use of derivation procedure in hardware environment and without formation of exogenous Dirac delta function. The paper is illustrated by examples. The obtained results make it possible to solve the problem of modeling the pure delay links using consecutive chain of aperiodic links of the first order with the equal time constants. Modeling results have proved the correctness of obtained computations. Knowledge of transport delay can be used when configuring multi-component technological complexes and in the diagnosis of their possible functional degeneration.

Keywords: fundamental matrix, linear continuous system, Dirac delta function, weight function, transport delay.

Acknowledgements. The work is supported by the Ministry of Education and Science of the Russian Federation (project 14.Z50.31.0031).

, ,

-, ( )

-, ,

50–60-

[1].

,

. ,

,

. В

- .

.

-. – - . В –

-.

-, 70- .

-, [2, 3],

,

, ,

-, ,

-– . Э

-.

.

,

[3–11]

( )t  ( )t  ( ),t (0); ( )t  ( ),t

x Fx Gg x y Cx (1)

, ,

x g y– , ,

- 

 

 

dim x n, dim g dim y m; F G C, ,  , dim

 

F  n n, dim

 

G  n m, dim

 

C  m n.

(1)





( )t  (0), ( ),t t ; ( )t  ( )t

x x x g y Cx . (2)

е , ,

(1), ( )xt x ( )t x ( )t , x ( )t – ,

(0)x 0, x ( )t x x[ (0),g

 

t 0, ],t y ( )t  x ( );t x ( )t – -, ( )gt 0 , x ( )t x g



( ),t x

 

0 0,t



, y ( )t Cx ( )t .

(2) (1) .

1 ( .1). (2) (1),

,

 

 

_{ }

 

 

_{ }

0 0

0 0

( ) ( ) (0) ( , ) ( ) 0 ,

,

( ) ( ) (0) ( , ) ( ) 0

t t

t t

t t

t t

t t t d e e d

t t t d e e d







         

  

      _{ }











F F

F F

x Φ x Φ Gg x Gg

y CΦ x CΦ Gg C x C Gg

(3)

1 (

( )t eFt, ( , )t   ( )t  ( ) eFt

Φ Φ Φ Φ . ■

[3–11]. (1),

- [12]

   

0.

0 0.

t

g t t

t

 



    _



δ

 

t [4, 13]

(3)

 

 

_{ }

_{ }

 

 

_{ }

_{ }

0

0

( ) 0 0

( ) 0 0

t t

t t t

V

t t

t t t

V

t e e d e e

t e e d e e







       

  

       _







F

F F F

F

F F F

x x G x G1

y C x C G C x C G1

. (4)

В (4) 1_Vcol



1_Vi 1,i1,n



 n , .

А (3), (4) (1)

(1):

1. Φ

 

t eFt – ;

2. Φ

 

t, Φ

 

t Φ1

 

 eF t – , Φ

 

t, 0 Φ

 

t ;

3. w

 

t CΦ

 

t GCeFtG – ,



m m



( ) wji

 

t  j

 

t i je t i

F

τd

- 

t

Φ .

-.

[14–16] « » [17–20] , ,

.

-, τd

-, δ- (

-) .

, τ_d

(1), j- y_j

 

t i- g t_i

 

. i-

-δ

-

, g ti

 

 

 

t , j-

     

 



, , 0 ; 0 0



j i l l i

y t g t   t g t  _ x  , δ

-, . . .

, .

В τd ,

, δ

-

,

(1). ,

(4) j- i- δ-

-,

 

_{ }

0 0

( ) j t 0 j t j t .

j i i

y t e e e



 F  F  F

x

C x C G C G (5)

(5) .

2 ( .2). y_j



t g t, _i

     

  t ,g t_l 0_{l i} ;x

 

0 0



δ- , i- (1)

(1), j- , ,

 

0  i

x G ,

 

t 

 

t , (0) _i,y_j

 

t  j

 

t y t_j( ) j

 

t _i  jeFt _i

x Fx x G C x C Φ G C G . ■

τ_d

-,

[21] , « – »

- 

 

_{ }

1

1 1

1 1

n

n n

n

T

y s s

s

g s Ts _ s s

 



   

  _{ } _{     }

    

      . (6)

В - (1) (6)

 

 

 

  1 1

1 1

1 0 0 0

0 1 0 0

0 0 0 0

0 ;

0 0 0 1

0 0 0 0

; 1

n

n n

 

 

 

 



 _ 



 



  

      



  



 _ 



 



  

 



  _

 

  _{  }

   _

  

F J I J

0

G C 0

  

     

 

. (7)

, (7)

-   

 

_

_

 

_

_

1 2

2

0

1 2 1 !

0 1 2 !

0 0 0

0 0 0 1

n

n

t t

t t t

t t t n

t t n

e e e e e

t

 

  

 _ 

 

 _ 

 

   _{ }

 

 

 

 

J J

F I

 

     



. (8)

(5) « –

» (6) δ- . (7) (8)

 



 1

_

_



1 !

n

t n t

y t CeFG  t  n e . (9)

 

minarg 0 & 0

d

dy t

t dt

 

 

  _   _

 

 

(9) δ- , τ_d

 



1



_

_

_

_

1 minarg 0 & 0 arg 1 0

n t

d

d t e _n

t n t

dt

 

 



 

           _

 

 

. (10)

(10) ,

-, , (10) n1.

(9) tt_d n1 (10) (9),

 







 1







 1

1 n 1 ! n

d

y t   n  n e  .

В y t

 

d 

-K n2; 3; 5; 10; 20; 40.

n 2 3 5 10 20 40

 

d

y t  0,3679 0,2707 0,1954 0,1318 0,0911 0,0637

K  2,7181 3,6941 5,1177 7,5873 10,9769 15,6986

. я

я n

. 1 g t

 

 

 

t (6)



1



1 1

T     n2; 3; 5; 10; 20; 40. (10).

. 2 g t

 

 

 

t

(6) T 1



 1 1



n2; 3; 5; 10; 20; 40,

. 1 K , .

. 1. y t

 

g t

 

 

 

t (6)

я



1



1 1

T     n2; 3; 5; 10; 20; 40 ( )

0,35 0,3 0,25

0,2 0,15

0,1 0,5

. 2. y t

 

g t

 

 

 

t (6) я T1



 1 1



2; 3; 5; 10; 20; 40

n

. 3

   

1

g t  t (6) T 1



 1 1



n2; 3; 5; 10; 20; 40.

. 3. y t

 

g t

   

1 t (6)

я



1



1 1

T     n2; 3; 5; 10; 20; 40

. 1 . 3 , ,

- 

 

g t   t , ,

   

1

g t  t , . ,

 

 

g t   t .

, . 1, 3,

(10),

.

n – T– .

-,

« – ».

« – »

[23] «

-– », .

0 10 20 30 40 50 60 t, ( )

1

0,8

0,6

0,4

0,2

( ) 1

0,8

0,6

0,4

0,2

References

1. Dralyuk B.N. Sinaiskii G.V. Sistemy Avtomaticheskogo Regulirovaniya Ob"ektov s Transportnym Zapazdyvaniem [Automatic Control System of Objects With Transport Delay]. Moscow, Energiya Publ., 1969, 72 p.

2. Krasovskii N.N. Teoriya Upravleniya Dvizheniem [Theory of Motion Control]. Moscow, Nauka Publ., 1968, 476 p. 3. Zubov V.I. Lektsii po Teorii Upravleniya [Lectures on Control Theory]. 2nd ed. St. Petersburg, Lan' Publ.,

2009, 496 p.

4. Andreev Yu.N. Upravlenie Konechnomernymi Lineinymi Ob"ektami [Control of Finite Linear Objects]. Moscow, Nauka Publ., 1976, 424 p.

5. Arnol'd V.I. Obyknovennye Differentsial'nye Uravneniya [Ordinary Differential Equations]. 4th ed. Moscow, MTsNMO Publ., 2012, 380 p.

6. Bellman R. Introduction to Matrix Analysis. NY, M Graw-Hill, 1960, 328 p.

7. Gantmakher F.R. Teoriya Matrits [Matrix Theory]. Moscow, Nauka Publ., 1973, 575 p.

8. Pontryagin L.S. Obyknovennye Differentsial'nye Uravneniya [Ordinary Differential Equations]. Izhevsk, Regulyarnaya i Khaoticheskaya Dinamika Publ., 2001, 400 p.

9. Anderson B.D.O., Moore J.B. Linear Optimal Control. Prentice-Hall, 1971, 413 p.

10. Zadeh L.A., Desoer C.A. Linear System Theory: The State Space Approach. NY, McGraw-Hill, 1963, 628 p. 11. Dudarenko N., Slita O., Ushakov A. Sovremennaya Teoriya Mnogomernogo Upravleniya: Apparat

Prostranstva Sostoyanii [The Modern Theory of Multivariable Control: The Unit of the State Space]. Saar-brucken: LAP LAMBERT Academic Publishing, 2011, 418 p.

12. Dirak P.A.M. The Principles of Quantum Mechanics. 4th ed. Oxford University Press, 1958, 324 p.

13. Matematicheskaya Entsiklopediya [Encyclopedia of Mathematics]. Moscow, Sovetskaya Entsiklopediya Publ., 1979, vol. 2, 1103 p.

14. Popov E.P. Dinamika Sistem Avtomaticheskogo Regulirovaniya [Dynamics of Automatic Control Systems]. Moscow, Gostekhizdat Publ., 1954, 800 p.

15. Rotach V.Ya. Raschet Dinamicheskikh Promyshlennykh Avtomaticheskikh Sistem Regulirovaniya [Calcula-tion of Dynamic Industrial Automatic Control Systems]. oscow, Energiya Publ., 1973, 440 p.

16. Shavrov A.A. Kompensator Transportnogo Zapazdyvaniya v Sistemakh Avtomaticheskogo Upravleniya [Com-pensator of Transport Delay in Automatic Control Systems]. Vestnik RGAZU, Agroinzheneriya, 2004, p. 52. 17. Tang G.-Y., Fu P.-L. Suboptimal control approach of linear time-delay systems. Proc. 14th World Congress

of IFAC. China, 1999, pp. 99–103.

18. Cao Y.-Y., Lam J., Sun Y.-X. Robust control for uncertain systems with time-delay and jump parameters.

Proc. 14th World Congress of IFAC. China, 1999, pp. 191–196.

19. Jankovic M. Control of nonlinear systems with time delay. Proc. 42nd IEEE Conference on Decision and Control. Maui, USA, 2003, vol. 5, pp. 4545–4550. doi: 10.1109/CDC.2003.1272267

20. Kharitonov V.L., Niculescu S.-I., Moreno J., Michiels W. Static output feedback stabilization: necessary conditions for multiple delay controllers. IEEE Transaction on Automatic Control, 2005, vol. 50, no. 1, pp. 82–86. doi: 10.1109/TAC.2004.841137

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

22. Akunov T.A., Dudarenko N.A., Polinova N.A., Ushakov A.V. Issledovanie kolebatel'nosti protsessov v aperiodicheskikh nepreryvnykh sistemakh, porozhdaemoi faktorom kratnosti sobstvennykh chisel [Process oscillativity study in aperiodic continuous systems, generated by eigenvalues multiplication factor]. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2013, no. 3 (85), pp. 55–61. 23. Dudarenko N., Ushakov A. Analiz Mnogomernykh Dinamicheskikh Sistem: Tekhnologiya Kontrolya

Vyrozhdeniya [Analysis of Multi-Dimensional Dynamic Systems: Technology of Degeneration Control]. Saarbrucken: LAP LAMBERT Academic Publishing, 2012, 232 p.

Д а На а я

А а в а – , , [email protected], , , , 197101,

-П ва Н а

А а в а – [email protected], , 197101, - , ,

У а в А а

В а в – , , [email protected], , , , 197101,

-Natalia A. Dudarenko – PhD, Associate professor, Associate professor, ITMO University, 197101, Saint Petersburg, Russia, [email protected]

Nina A. Polinova – postgraduate, ITMO University, 197101, Saint Petersburg, Russia, [email protected]

Anatoly V. Ushakov – D.Sc., full professor, ITMO University, 197101, Saint Petersburg, Russia, [email protected]

П я к е а 21.05.14