MODELING OF DYNAMIC SYSTEMS WITH MODULATION BY MEANS OF KRONECKER VECTOR-MATRIX REPRESENTATION

А -Е Е Е А Ы Е , Е А е яб ь–к яб ь 2015 15 № 5 ISSN 2226-1494 http://ntv.ifmo.ru/

SCIENTIFIC AND TECHNICAL JOURNAL OF INFORMATION TECHNOLOGIES, MECHANICS AND OPTICS September–October 2015 Vol. 15 No 5 ISSN 2226-1494 http://ntv.ifmo.ru/en

512.643: 62.50

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a

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А : [email protected]

10.04.15, 19.06.15 doi:10.17586/2226-1494-2015-15-5-839-848

–

: А. ., А. .

- // - ,

. 2015. 15. № 5. . 839–848.

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( 074-U01)

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MODELING OF DYNAMIC SYSTEMS WITH MODULATION BY MEANS

OF KRONECKER VECTOR-MATRIX REPRESENTATION

A.S. Vasilyeva, A.V. Ushakova

a

ITMO University, Saint Petersburg, 197101, Russian Federation Corresponding author: [email protected]

Article info

Received 10.04.15, accepted 19.06.15 doi:10.17586/2226-1494-2015-15-5-839-848 Article in Russian

For citation: Vasilyev A.S., Ushakov A.V. Modeling of dynamic systems with modulation by means of Kronecker vector-matrix representation. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2015, vol. 15, no. 5, pp. 839–848.

Abstract

environment modulates signals. The marked system deficiency is partially offset by this paper, which proposes Kronecker vector-matrix representations for purposes of system representation of processes with signal modulation. The main result is vector-matrix representation of processes with modulation with no formal difference from continuous systems. It has been found that abilities of these representations could be effectively used in research of systems with modulation. Obtained model representations of processes with modulation are best adapted to the state-space method. These approaches for counting eigenvalues of Kronecker matrix summaries, that are matrix basis of model representations of processes described by Kronecker vector products, give the possibility to use modal direction in research of dynamics for systems with modulation. It is shown that the use of controllability for eigenvalues of general matrixes applied to Kronecker structures enabled to divide successfully eigenvalue spectrum into directed and not directed components. Obtained findings including design problems for models of dynamic processes with modulation based on the features of Kronecker vector and matrix structures, invariant with respect to the dimension of input-output relations, are applicable in the development of alternate current servo drives.

Keywords

system with modulation, modeling, Kronecker vector-matrix representation, eigenvalues controllability.

Acknowledgements

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

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1 ( .1).К (1) x y, xRn,yRm,

y

x , {x y_i _j; 1, ; 1, }i n j m

,



xy i n j m



col _{i j}; 1, ; 1,

 y

x , xyRnm, (1)

1 ( .1). , xy x

y ,

:

x y y

x   .

2 ( .2). x y ,

-y

x (2) (squeeze)



xy



_s,



xy



_s col



x_iy_i;i1,n



. (2)

2 ( .2). xy

S

1

0T_i 1 0T_{n i} ; 1,

diag _{ }  i n

 _{ }  _

 

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:

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-,

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-, :

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d

x t y t x t y t x t y t x t y t

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 

        ,

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

d

x t y t z t x t y t z t x t y t z t x t y t z t x t y t z t

dt



               .

3 ( .3). К (4)

q p m

n _R

R   

 B

A , (AB) (np mq ),

-





rowA j m i n



col _i,j ; 1, ; 1,



B B

A . (4)

3 ( .3). К

-, : ABBA.

-, ,

,

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4 ( .4).К (5) ARnn BRmm

-)

(AB (nmnm),

B I I A B

A )  _B _A

( , (5)

B A I

I , – , A B

, dim

 

IA dim

 

A , dim

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IB dim

 

B ;  –

.

4 ( .4). A B,

, – (6)

-,





( ) B A Si ,

 

     

A B A I I B A B . (6)

A, B,

(7) :



AB



AB AI_BI I_ABI I_AI_B

Si , , . (7)

К .

К .

1 ( .1). А

B

-, -:







K :det



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0; K Ai Bj;i 1, ;n j 1, ;m k 1,mn



 AB     I A B         . (8)

2 ( .2). А AB

m m n

n _R

R   

 B

A ,

:







l:det

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v

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0; l Ai Bj;i 1, ;n j 1, ;m l 1,mn



 AB   I A B         . (9)

(8) (9) _Ai _Bj – A B.

5 ( .5). А B

A BA (8) ,

B

A BA.

3 ( .3).

-S



PX QZ

 

SP Q



X Z



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- [6, 11, 14].

- ,

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, :

) ( ) ( ); 0 ( ); ( ) ( ) ( )

(t Axt Evt But x yt Cxt

x     , (11)

) ( ) ( ); 0 ( ); ( )

(t Γzt z gt Pz t

z   , (12)

( ) ( ); (0); ( ) ( )

M t  M M t M M t  M M t

z Γ z z g P z . (13)

(11) x – ; x

 

0 –

; v – ; u – ; y

– ; xRn;ν,u,yRm; ,E,B ,C – , ,

; Rnn;E,BRnm;CRmn.

(12) z g –

; zRl;gRm; Γ, – P ; ΓRll; PRml.

(13) z_M gM –

-; zMRh;gMRm;ΓM,PM – ,

h h

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

Γ ; PRmh.

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( )t col g t{ _j( )g_Mj( ); 1, }t j m

ν . (14)

, g(t)

(14) (К ) g( )t g_M( )t

, S (3) К :

 

t  



( )t  M( )t



ν S g g . (15)

) (t

g g_M(t)(12) (13), (15)

-:

)) ( ) ( )( (

)) ( )

( ( )) ( ) ( ( )

(t S gt g_M t S Pz t P_Mz_M t S P P_M z t z_M t

v           . (16)

(16) v(t)

-) ( ) (t z_M t

z  .

К z(t)z_M(t)

,

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

( )( ( ) ( )); (0) (0).

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M M M M M M

M M M

d

t t t t t t t t

dt

t t t t t t

t t                          

z z z z z z z z

Γz z z Γ z Γ I I Γ z z

Γ Γ z z z z

  (17) ,         M z z x z ~ ₍₁₈₎ .

1 ( .1). (11)

(14), (12) (13),

-  - 

 



T



M T

T

t t

t) ~~() ~ (); ~(0) 0, 0 0

(

~ _{Ax Bu x x z z}

x     , (19)

), ( ~ ~ ) ( ); ( ~ ~ )

(t C_Xxt yt C_Yxt

x   (20)

(19), (20)                     0 ~ ; 0 ) ( ~ B B Γ Γ P P ES A A M M          

, (21)

] 0 [ ~ ]; 0 [ ~ 

 C C

I

Cx  X y  . □ (22)

.

(18) (11)

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(17). ■

0 ) (t 

u (19), (20)

 

~ ~

 

0; (t) ~ exp

 

~ ~

 

0; (t) ~ exp

 

~ ~

 

0 exp

(t)

~ _{A x x C A x y C A x}

x  t  x t  y t . (23)

, (19), (20)

(11) , .

,

(21),

. (21) _М 0, (23) –

   

 



T



M T T 0 0 , 0 ) 0 (

~_{x  x z z , (0) 1}

M 

z , (21)

 

_{ }

 

                     VM M 1 z x x 0 Γ P P ES A A 0 0 0 ~ , 0 ) ( ~           , (24)



q j m



col _j

VM  1; 1,

1 .

, . . ,

( - ) g_M(t) ( ) g(t),

. A~ (21)

. ,

 

 A

-A ΓΓ_M

,



 

A  

  

A  Γ ΓM



.

, ,

(12),

[14], [15]

(11) (13).

(12)

-- ,

(11)

.

:

1. (19), (20) ;

2. (19), (20) (21), (22)

(11)

;

3. (21)

(20)

-.

 ,

-:

1.   М p,

2.   М p,

3.     p М ,

4.   M ,

 





arg W j 0, 05



    .

 

2 ₂ 2 2

p

p p

W s

s s

 

    (25)

p

  ,  1/.

- (11)





2 2

0 1 0

; ; 1 0

2

p

p

p

   

   

   

    _ _

 

A E B C .

, (19), (20) (21), (22)

2 2 2

0 1 0 0 0 _{1 0 1}

0 1 0

2 0 _{; ; ;}

0 0 0

0 0 0 0

0 0 0

0 0 0 0

T T

p p p p

x y

M

M

    _{   }

    _{   }

          

_{ } _{ }  

   



    _{   }

 _  _ _{    }

 

A B C C , (26)

2 2 2

0 1 0 0 0 0 0 _{1 0 1}

0 1 0

2 0 0 0

0 0 0

0 0 0 0 _; 0 _{; ;}

0 0 0

0 0 0 0 0

0 0 0

0 0 0 0 0

0 0 0

0 0 0 0 0

p p p p

T T

M

x y

M

M

M

    _{   }

    _{   }

       _{   }

 _{ }    _{   }

   

  _{ } _{ }

 _{ }    _{   }

    _{   }

 

    _{   }

 _{ }  _ _{     }

 

A B C C . (27)

(26)

, (27)

 M .

. 1 (25),

(19), (20),

. 1.

. 2.

:

1 –

4

М



  ; 2 –   _М ; 3 –  _М 1,5; 4 –   _М 16

. 3.

( . 2)

. 2, 3

-. «1»,

-- M

 

 W j

 

 [16],

y(t)

1,5

1

0,5

0

–0,5

–1

0 1 2 3 4 t,

1,5

1

0,5

0

–0,5

–1

–1,5

–2

y(t)

0 2 4 6 8 10 12 14 16 18 t,

1 2 3 4

1,5

1

0,5

0

–0,5

–1

–1,5

–2

y(t)

0 2 4 6 8 10 12 14 16 18 t,

 

M  , .

«2», M

 

 ,

M

 

 . «3»,

 

M  ,

. «4»,

, .

. 4 4 . 2 . 3,   _М 16 ,

-.

. 4.

-   _М 16

(1 – ; 2 – )

. 5–7

-16 М

  , M

 

 ,

-, .

. 5.

0, 25

  

, , , .

,

.

-.

-0,02

0,015

0,01

0,005

0

–0,005

y(t)

0 1 2 3 4 5 t,

1

2

y(t)

0 5 10 15 t,

3

2

1

0

–1

–2

–3

. 6.

  

. 7.

1, 5   

,

-- ,

. .

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Ва ь А С – , , - , 197101,

, [email protected]

У а А а В а – , , ,

, - , 197101, ,

[email protected]

Andrey S. Vasilyev – postgraduate, ITMO University, Saint Petersburg, 197101, Russian

Federation, [email protected]