Ž. marković - metod za sintezu digitalnog zakona upravljanja servo sistemom sa elastično...

7/23/2019 . Markovi - Metod za sintezu digitalnog zakona upravljanja servo sistemom sa elastino spregnutim masama

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______________________________________________________

,,22000033..

::,,....

..::9922//9977

::....,,

....


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1 .................................................. ........................................................... ........................................... 3

1.1 ...................................................... ........................................................... ... 31.2 .................................... 41.3 ...................................................... ........................................................... ............. 8

2 ........................ 10

2.1 ................................................ 102.2 ..................................................... ........................................................... ........... 12

3 ...15

3.1 ......................................................... ........................................................... . 163.2 ................................. 173.3 ...........................18

3.4 ............................. 253.5 ..................................................... ..................... 283.6 ....................................... 303.7 ....................................................... ............................... 333.8 ....................................................... ........... 363.9 ........................................................... . 403.10 ............................................... ..... 44

4 .......................................................... ............................... 48

4.1 - .......................................................... ........... 48

4.2 ........................................................ ......................................... 54

5 ...................................................... ............................... 59

5.1 ...................................................... ......................................... 595.2 ......................................................... ........... 645.3 .......................................................... ............................... 685.4 .......................................................... ............................... 72

6 ....................................... 73

6.1 .......................................................... ............................... 736.2 ....................................................... ........................................................... . 846.3 ......................................... 89

7 ..................................................... ................... 109

7.1 ....................................................... ................................................ 1097.2 - ................ 1167.3

..................................................... ......... 1197.4

SETUP-A .................................................... ......... 137

8 .......................................................... ........................................................... ................... 140

9 ...................................................... ........................................................... ................... 142


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3

1

1.1

, , (pole placement [1], [2] pole assignement technique[3]). .

, , () z , . , . , . . , . , , - . - , . -, -. , - . (Graphical User Interface)

Matlab

6.0.

, , . setup , . .

,

, .


4/145

4

1.2

, . 60%

. , , . 60% , ( ) 30%, 10% [4], [5]. (DC ), . , , . , , () [6]. , () . , [7]. , [8]. ,

[9]. , (DC brushless motor).

, ( ), . Blaschke [10], . Hasse [11], [7], [12].

,

. [13], [14], [5], [15], [16], [17]. , , . , [7], [18].

, [19], [20]. ,


5/145

5

[21]. ,

. , , . :

; (tracking) ; ;

;

, (

), , .

. , ,, , , .

, , . .

, . , :

,, ;

,, ;

;

, ,

, ,

, .


6/145

6

. ; Lin Tsai [22], Bose [23], (. sliding mode control) , . () , . , (PI, PID ), . Haneda Nagao [24] , [25] . [26], [27]. ; .

, . , 100 200Hz. :

,

,

,

( ) .

-

. . , , -. , , , 500 Hz 2kHz. , /. .

, 30 Hz 300 Hz.


7/145

7

-.

[28]-[31].

(10-20 Hz). , - .

,:

1. [32], [33];

2. [31],[34], [35],

3. () [34], [36],[37].

Meschat [38] . [28], [34].

[39], [40]. [41]. , .

, , .

[32], [33], . , [29], [30], [31].

[37] . ,

. , , FIR -. .

[1], [3], [5], [42]. ( ) . [2].

[42] -[44].


8/145

8

, , , .

,

.

1.3

. , ,

. .

. . , , .

, . . , .

, - . , . (Graphical User

Interface) Matlab6.0.

. , , .

, , .


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9

. , -(PI).

, ,

, . , setup-. , .


10/145

10

2

(poleplacement [1], [2] pole assignement technique[3]). . , , .

2.1

:

)()()(

tButAxdt

tdx+= , (2.1)

: x(t) , u(t) , . , :

)()()( kTukTxTkTx +=+ , (2.2)

:

.

,

0

Bdse

e

T

As

AT

=

=

(2.3)

. , . :

)()( kTLxkTu = . (2.4)

u(t), (2.2), ; (2.2) :

)(

0

0

1

)(

0100

0001)(

121

kTukTz

aaaa

TkTz

nn

+

=+

M

L

MMMM

L

L

, (2.5)

:

n

nn

azaz +++

K1

1 , (2.6)


11/145

11

. :

n

nn pzpzzP +++= K11)( . (2.7)

(2.4) :

zLu ~= , (2.8)

[ ]nn apapapL = L2211~

, (2.9)

(2.7). (2.8) :

LxHxLzLu ===~~

, (2.10)

H x z.

. cW cW~

(2.2) (2.5), :

cc HWW =~

. (2.11)

:

=

100

10

1

~ 2

11

1

L

MMM

L

L

n

n

c

a

aa

W . (2.12)

:

1~ = ccWWH . (2.13)

~ (2.5) Cayley-Hamilton- [3] ~ , (2.7):

( )

Iapap

IppP

nn

n

n

nn

)(~

)(

~~

1

11

1

1

++=

=+++=

K

K. (2.14)

(2.9) :

[ ] ( )= ~100~ PL L . (2.15)

1~ = HH , (2.16)

[ ] ( ) [ ] ( )=== HPHHHPHLL 100100~ 1 LL . (2.17), (2.12), :

[ ] [ ]100~100 LL =cW , (2.18)


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12

:

[ ] ( )== PWWWLL ccc11 100

~~L . (2.19)

(2.19) Ackermann-[2], [3].

2.2

, . , () [2], [45]. (2.20):

=+=+

)()(

)()()1(

kCxky

kukxkx. (2.20)

x)

:

)()()1( kukxkx +=+ )) , (2.21)

(2.20). (2.21) (2.20) (2.20) (2.21) , x

)

x (2.20). (2.20) (2.21), x

) (2.20)

. (2.21) :

( ) ( ) ( )[ ]1)()(11 ++=+ kkxCkyKkukkxkkx ))) , (2.22)

. ( )kkx 1+) ( )1+kx . :

xxx )) = . (2.23)

(2.22) (2.20), :

( ) ( ) ( )[ ][ ] ( )1

1)(11

=

==+

kkxKC

kkxCkyKkkxkkx)

)))

(2.24)

(2.24) , . , , (2.20) . (2.22) (2.20) .

(2.22). (2.22) , ( )1kkx) 1k , :

( ) ( ) ( ){ }[ ]

[ ] ( )[ ] )()1(11)1(11)()1(11

kKykukkxKCI

kukkxCkyKkukkxkkx

++=

=+++=

)

)))

. (2.25)


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13

(2.25) :

( ) ( ) ( ) ( )11)( == kkxKCkkxkxkkx ))) . (2.26)

( ) KC . :

( ) ( )( ) ( )( ) ( )11

11

)(

=

=

=

kkxCCKI

kkxCKCC

kkxCkkxCky

)

)

))

p , CKI pp ; ICK= : ( ) pCrang = . ( ) ( )kykkxC = , . . p (2.25), .

[46].

2.2.1

2.1. . (2.20) (2.23): xxx

)) = , , :

( )1

)( = kkxLku , (2.27)x

):

( ) ( ) ( )[ ]1)()(11 ++=+ kkxCkyKkukkxkkx ))) . (2.28)

, :

( ) [ ] ( )( ) [ ]( )

=++=+

11

1)(1

kkxKCkkx

kkxLkxLkx. (2.29)

2.1.

n2 . L KC . L

(2.7), KC .


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14

2.2.2

(2.24) . , CK .

L . : (2.20) (2.24) )(zP (2.7). , ; . , , , . z, . , , . , , . : [45]. , : , .


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15

3

. , () z , . , .

,. . .

. , . . , . : , , ( , ,-).


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16

3.1

3.1.. : GR(z)GY(z),

( ) GP(z) G(z).

3.1. : ()()

3.1.() 3.1.(), . . 3.1.() 3.1.() :

( )

( )

( )

( ) .)(

)(

,)(

)(

,)(

)(

,)(

)(

zA

zCzG

zA

zBzG

zR

zSzG

zR

zTzG

P

Y

R

=

=

=

=

(3.1)

uy.

GP(z) , (D/A ), , A/D


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17

antialiasing . GP(z). , . (z)(z). , G(z) .

. u, , .

r y. :

)()()()()()( zyzSzrzTzuzR = . (3.2)

R(R. ).

I

GR(z) GY(z), (3.1), (3.2) :

TR degdeg , (3.3)

SR degdeg . (3.4)

3.2

: GP(z) G(z), (3.2), , . :

( ))(

)(

zA

zBzG

m

mM = , (3.5)

.3.1.() (3.1) (3.2),

:

( ) ( ) ( ) ( )[ ] )()()()( zrzTzBzyzSzBzRzA =+ . (3.6)

(3.5), :

)(

)(

)()()()(

)()(

zA

zB

zSzBzRzA

zTzB

m

m=+

. (3.7)

,

R(z), S(z) T(z) (3.7),:


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18

)()()( zBzTzB m= . (3.8)

)()()()()( zAzSzBzRzA m=+ . (3.9)

, R(z), S(z) T(z)

, : , , , ,

II

GP(z) (3.1) GM(z) (3.5), (3.2) . :

BABA mm degdegdegdeg . (3.10)

3.3 ,X+BY=C

3.2 X Y ,

CYBXA =+ , (3.11)

A, B, C . , (Diophantus246?-330? ..) . 13 260 . : , [47]. , (3.11), , . nnn zyx =+ , (Pierre de

Fermat 1601.-1665. .) x, yz nnn zyx =+ n>2 [48]. 1996. , 120 Andrew J. Wiles.

, (3.11)

(Aryabhatta) , mqC= (3.11) (Bezout) .


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19

3.3.1

, , .

:cybxa =+ , (3.12)

Zcba ,, . () 3.2.

3.1:cab. c | abbc,1, . (b, c) = 1ca. c| a.

: (b, c) = 1, x y bx+cy = 1.a,abx +acy=a. c, ca.

3.2:

(a, b) = 1

c,

ax+by= cx=x0y=y0. , ,

, :

,

,

0

0

tayy

tbxx

=

+=

t.

: je (a, b) = 1, x1y1ax1+by1= 1.x0=cx1y0= cy1ax+by= c,:

1)()( 11 =+ ccybcxa .

xy ax+by= c.:

.

,

00 cbyax

cybxa

=+=+

, :

( ) ( )

( ) ( ).

,0

00

00

yybxxa

yybxxa

=

=+

0b (a, b) = 1, 0xxb btxx = 0 Zt ,

. btxx += 0 . bt 0xx :

( )0yybabt = .

b atyy = 0 . , ,0 tbxx += ,0 tayy = ax+by= c,t.

, (a, b) =gax+by= c,g | c.g | c:


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20

,

,

g

cy

g

bx

g

a

cbyax

=+

=+

, , g

.

(3.12) Brahmagupta [48], . . b. b:

rqba += . (3.13)

ubr :

uqtsqutusqbar === )( . (3.14)

b, b r, b:

11111 qbarrqbab

aq =+=

= , (3.15)

xx,

.0

,

,

,

1

1

11

1

1

1212

1

2

23133231

2

13

122212

1

2

+

+

+

=+=

=

=+=

=

=+=

=

=+=

=

n

n

nnnn

n

n

n

nnnnnnnn

n

nn

q

rrrqr

r

rq

rqrrrrqrr

rq

rqrrrrqrr

rq

rqbrrrqbr

bq

M (3.16)

(3.16) b,u=rn.

(3.12). :

100 =+ ybxa , (3.17)

(3.12) :

cxx = 0 , cyy = 0 . (3.18)

(3.17) . 1ra= 2rb= . :


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21

.1122

1233

4322

3211

+=

+=

+=

+=

nnn

nnnn

rqr

rrqr

rrqr

rrqr

M (3.19)

(3.19) :

1221 = nnn rqr , (3.20)

2331 = nnnn rqrr . (3.21)

(3.21) (3.20), :

23232 )1(1 += nnnnn rqqrq . (3.22)

(3.19), (3.17).

(3.11), . . , ():

, ( ) ()

() ().

, 3.2, :

3.3:A, B C. A BC, CYBXA =+ X = X0 Y = Y0. , , , :

,

,

0

0

QAYY

QBXX

=

+= (3.23)

Q.

3.3: CYBXA =+ :

BX degdeg < , (3.24)

,

AY degdeg < . (3.25)

(3.11). , , .


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22

(3.11). ( ) ( )zAzR =1 ( ) ( )zBzR =2 . :

.0)(

)()()()(

)()()()(

)()()()(

)()()()(

1

111

12

4322

3211

=

+=

+=

+=

+=

+

++

zR

zRzRzQzR

zRzRzQzR

zRzRzQzR

zRzRzQzR

n

nnnn

nnnn

M (3.26)

(3.26) A(z) B(z) Rn(z). A(z) B(z) Rn(z), :

1)( =zRn . (3.27)

(3.26), :

)()()()(1 12 zRzQzRzR nnnn == , (3.28)

,

)()()()( 2131 zRzQzRzR nnnn = . (3.29)

(3.29) (3.28) :

).()()()(

)())()(1()()(1

2233

213

zRzVzRzU

zRzQzQzRzQ

nnnn

nnnnn

+=

++= (3.30)

(3.30), :

)()()()()()()()(1 212211 zBzVzAzUzRzVzRzU +=+= , (3.31)

(3.11):

)()()( 1 zCzUzX = , (3.32)

)()()( 2 zCzVzY = . (3.33)

3.3.2

, z, . .

(3.9), :

)()()()()( zAzSzBzRzA c=+ . (3.34)


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23

(3.34) mA cA .

A(z) B(z):

)(

)(

zA

zB

u

y= , (3.35)

01

1

1 ...)( azazazzA n

n

n ++++= , (3.36)

01

1

1 ...)( bzbzbzbzB n

n

n

n ++++=

, (3.37)

(3.35) , . (z)(z).

(3.34) z:

mkasbra ckkji

ji

kji

ji ,,0, K==+=+=+

, (3.38)

( )BSARm ,degmax= , :

=

+

+

+

cm

cm

c

c

l

p

p

S

n

n

n

nn

n

R

n

n

n

a

a

a

a

s

ss

r

r

r

r

b

b

bb

bbb

b

b

bb

b

a

a

aa

a

a

aa

a

m

1

1

0

1

0

1

1

0

1deg

1

1

01

1

1

01

0

1deg

1

1

01

1

1

01

0

00

0

00

0

0

00

100

0

010

1

0

0

00

1

MM

M

M

M

M

M

444 3444 21 L

LM

MLML

L

LM

MLM

L

L

444 3444 21 L

LM

MLML

L

LM

MLM

L

L

. (3.39)

S:

[ ]11

11

11

0101

11

11

0101

00

00100

00

00

010

1

00

00

0000

++

=

mmn

nn

n

nnn

nn

b

ba

bba

bbbaa

ba

ba

bbaa

ba

LL

LMLM

MLMMLM

LL

LL

LMLMMLMMLM

LL

LL

S (3.40)

(3.39) (Sylvester) .

(z) (z)


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24

, S. (z)(z):

=

=n

i

izzA1

)()( , (3.41)

=

=n

i

in zbzB1

)()( , (3.42)

:

= =

=n

i

n

j

ji

n

n

nn

b1 1

2

)1(

)()()1( S . (3.43)

:

],1[),(,0 njiji , (3.44)

(z)(z), .

(3.39) R(z)S(z)RS:

[ ]T11110110 +

=mllpp

ssssrrrr LLRS (3.45)

R(z)S(z):

cASRS = 1 (3.46)

(3.46) S, . ((3.38) ), :

2degdeg1 ++=+ SRm . (3.47)

),deg(max BSARm= m : ARm deg= BSm deg= . ARm deg= . , (3.47):

,2degdeg1degdeg

,2degdeg1deg

++=++

++=+

SRRA

SRAR

:

1degdeg = AS . (3.48)

(3.34) :

( )

( ) ( ),,degmaxdegdeg

,degdeg

BSABSAAR

ABSAR

cc

c

==

=+

(3.48) Adeg :

( )1deg,degdegmaxdeg = BAAR c . (3.49)


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:

( )

==

.1deg,degdegmaxdeg

,1degdeg

BAAR

AS

c

(3.50)

BSm deg= , :

( )

==

.1deg,degdegmaxdeg

,1degdeg

ABAS

BR

c

(3.51)

(3.50) (3.51) :

1degdegdeg + BAAc . (3.52)

cA , (3.52) ,

:

==

+

.1degdeg

,1degdeg

,1degdegdeg

AS

BR

BAAc

(3.53)

3.3.

3.4

: R(z), S(z) T(z)(3.7):

)(

)(

)()()()(

)()(

zA

zB

zSzBzRzA

zTzB

m

m=+

, (3.7)

:

)()()( zBzTzB m= . (3.8)

)()()()()( zAzSzBzRzA m=+ . (3.9)

(3.7) :

0)()()()( =+ zSzBzRzA . (3.54)

B(z)T(z). (3.54) )(zAm .

(3.9) (3.7) , . .

, . B(z).

B(z)Bm(z), (3.54), , .


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26

, , . z . B(z):

)()()( zBzBzB + = , (3.55)

)(zB , )(zB + z . , )(zB

)(zB + , . )(zB + .

)(zB )()()()( zSzBzRzA + , )(zBm ,

:

)()()( zBzBzB mm = . (3.56)

, )(zBm . )(zB

+ )()()()( zSzBzRzA + , )(zR . :

)()()( zRzBzR = + , (3.57)

(2.33) :

)(

)()(

)]()()()([)(

)()()(

zA

zBzB

zSzBzRzAzB

zTzBzB

m

m

=+

+

+

. (3.58)

, . A(z). , . A(z), z. , A(z) :

)()()( zAzAzA + = , (3.59)

)(zA

(), )(zA+ .

)(zA+ )(zT , :

)()()( zTzAzT = + . (3.60)

)(zA+ )()()()( zSzBzRzA + , )(zS . :

)()()( zSzAzS = + , (3.61)

(3.58) :

)(

)()(

)]()()()([)()(

)()()()(

zA

zBzB

zSzBzRzAzBzA

zTzAzBzB

m

m

=+

++

++

, (3.62)

:


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27

)(

)(

)()()()(

)(

zA

zB

zSzBzRzA

zT

m

m

=+

. (3.63)

)(zAo .

, . , )(zAo )()()()( zSzBzRzA + )(zT . (3.62) (3.63) , :

)()()()()()( zAzAzSzBzRzA om =+ , (3.64)

)()()( zAzBzT om = . (3.65)

,

)()()()()()()()( zAzBzAzAzSzBzRzA om++ =+ , (3.66)

)()()()( zAzAzBzT om+= . (3.67)

)(zAo , )(zR )(zS ,

)(zRf )(zSf ,

, :

)()()( zRzRzR f = , (3.68)

)()()( zSzSzS f = . (3.69)

)(zRf )(zSf ,

, (3.66) (3.67). ,:

)()()( zAzRzA f = , (3.70)

)()()( zBzSzBf

=. (3.71)

:

==+

+

++

).()()()(

),()()()()()()()(

zAzAzBzT

zAzBzAzAzSzBzRzA

om

om , (3.72)

)(zR )(zS , (3.68) (3.69):

)()()( zRzRzR f = , (3.68)

)()()( zSzSzS f = . (3.69)


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28

)(zAo )(zRf )(zSf 3.7.3.8..

3.4.1

:)()()()( zAzAzAzB mo

++ .

, ( )zA+ ( ) 1=+ zA . : )()()( zAzAzB mo

+ .

2.2.1. , :

( ) )det( LzIzAm += , (3.70)

,

( ) )det( KCzIzAo += . (3.71)

( )zAo . ( )zAo )()()()( zSzBzRzA +

)(zT , ( )zAo . . . )()( zAzA mo ( )zAo . , , )()()( zAzAzA oo =

+ .

( )zAo , 3.7.3.8..

3.5

. I II, . (3.3), (3.4) (3.10) (3.2):

TR degdeg , (3.3)

SR degdeg . (3.4)

BABAmm

degdegdegdeg . (3.10)


29/145


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30

, , :

TSR deg1deg1deg +=+= , (3.83)

.

3.6

. ., , , ,

[3]. .

3.4: (3.1):

)(

)()(

zA

zBzGP = , (3.1)

:

)(

)()(

zA

zBzG

o

oo

P = . (3.84)

)(zGP )(zGo

P ,

)(zGM .

)(zG oP , :

)(

)(

)(

)(

)()(

)()()()(

zG

zG

zG

zG

zSzG

zTzGzGzG

Y

R

M

P

M

Po

PP =


31/145

31

)()(

)()()(

zRzA

zSzBzGLG

= ,

:

)()(

)()()(

)()(

)()(

1)(1 zRzA

zAzAzB

zRzA

zSzB

zG mo

LG

=

+=+

+

.

, 3.4.:

)(

)()()(

zB

zBzTzA

m

o

= ,

:

)()()(

)()()(

)()(

)()()()()(1

zRzBzA

zTzAzB

zRzA

zAzTzBzBzG

m

mmLG

=

=++

.

(3.1) (3.5), :

)()(

)()()(1

zRzG

zTzGzG

M

PLG

=+ .

(3.85) ( )( )zRzS

, :

)(1)()(

)()()(

)(

)()(

)(

)(zG

zRzG

zTzGzG

zR

zSzG

zR

zSLG

M

Po

PP +=


32/145

32

(3.88) .

1)( zG oP , .

; ,

(3.85) Tjez = , . .

, (3.85) :

3

1

)()(

)()()( .

4.5.


56/145

56

- III, IV V

III, IVV, 4.1., )(zRf )(zSf ,

)(zR radio button. )(zR )(zS .

4.6.


57/145

57

- VI

VI 4.1. , chek box-, radio button- text box-

VI . , Gm(z) ,

)(zBm )(zAm

)(/)()( zAzBzG mmm = .

4.7.

- VII

VII 4.1. , )(zAo ,

R'(z)-Su SyR(z), S(z)T(z)

, R(z),


58/145

58

optM

. . .

4.8.


59/145

59

5

. , ,.

5.1. (. DC brushless motor).

5.1.

, . , .

5.1

, . () [7]. , .

. .

.


60/145

60

, , , , . ,

[7]. , , . , , . , . , , . . ,

. , .

5.1.1 PWM

PWM . , PWM . ,

. , PWM . , .

PWM , . , , .

5.2. PWM

. *dI *qI * * . (PI) feed-forward *dU

*

qU .

*dU *

qU

*aU ,*

bU *

cU .

e . d-q e

r k . *

aU ,*

bU *

cU .


61/145

61

5.2. PWM

dq, . , , . (VCO) . VCO. VCO . VCO, - . .

. , PWMT , crT

( ),...3,2,1== kTkT PWMcr .

PWMcr TT = . , . , , . , "" .

5.1.2

, . , . ,

q-, 5.3.


62/145

62

5.3. q

. , (), PWM . PWMcr TT = ,

2/crT . . crT .

, . ,PI.

PI . crT :

crcr sT)sT( + 1

1

exp , (5.1)

,

.T

s

)T

s(cr

cr

21

1

2exp

+ (5.2)

,

,sT

(s)Wd

d

+=

1

1 (5.3)

dT , .

. n

.

1

1

1

=

+=

n

i

i

n

)s(

(s)W (5.4)

dd /1=

(5.4) 3 dB. :


63/145

63

.211

=+=

n

i

i)s( (5.5)

n= 2: (5.5) :

,)

j

2(1 21212

=++ (5.6):

2

2

2

1

2

2

2

1

4

2

4

1

2

2

2

1

2

2

2

6)(1

++++==

d

d

. (5.7)

1 2 , :

2

2

2

1

2

2

2

1

2

2

2

1

4

2

4

1

2

2

2

1

0

1

2

6)(lim

21

+

=

++++=

d (5.8)

n= 3: (5.5) :

,)j)( 2(1 3213231212 =+++++ (5.9)

:

2

3

2

2

2

1

11

d

d

++= . (5.10)

n:

.

n

i

i

d

d

=

=

1

2

11 (5.11)

, n :

,s

(s)Wd+

=1

11 (5.12)

d (5.11).

dT

. (5.8) (5.1) (5.2) :

.2

52 22 crcrcrd T)T(TT =+= (5.13)

, :

[ ] .[s]T

.T

Hzfcr

cr

optcr

cr

bw039601ln

21 == (5.14)


64/145

64

, 5 kHz 15 kHz. (5.14) PWMcr TT = PIPWM 198 Hz 594 Hz.

,

.,

,s

(s)Wem

em +=

1

1 (5.15)

em .

em

z-. (5.10) opt :

.sT.T

s crcr

opt

em ][9561ln

3][ == (5.16)

(5.16) PWM . , .

5.2

. , [55]. .

., . , , , . . , . , ., . . ,

.


65/145

65

. , , .

. .

5.2.1

5.4. , .

5.4.

:

( )lmslm

vemm

m K

dt

d

dt

dKM

dt

dJ

=2

2

, (5.17)

( ) llm

vlmsl

l Mdt

d

dt

dKK

dt

dJ

+=

2

2

, (5.18)

dt

d mm

= , (5.19)

dt

d ll

= , (5.20)

:

mJ lJ , ;

vK ();

sK ;

emM ;

lM

m l , ,

m l , .

Mm, m, m

Ml, l, l

Kv, Ks


66/145

66

, ,5.5.

m m

l l .

-- ( ) ( )( )ssK lmv . , vK

. oM lM .

, :

( ) ( ) ( )( ) ( ) ( )( )ssKssKsM lmvlmso += . (5.21)

5.5.

,:

( )( ) ( )( )

( )

( ) ( )( ) ( )sM

KsKJJsJJs

KsK

sMKsKJJsJJs

KsKsJs

l

svlmlm

sv

em

svlmlm

svlm

+++

+

+++

++=

22

22

2

, (5.22)

( )( ) ( )( )

( )

( ) ( )( ) ( )sM

KsKJJsJJs

KsKsJ

sM

KsKJJsJJs

KsKs

l

svlmlm

svm

em

svlmlm

svl

+++

++

+++

+=

22

2

22

. (5.23)

, , :


67/145

67

( )( )

( )

( )

( )( )

( )sMs

K

Ks

JJK

JJ

sKK

sJJ

sM

sK

Ks

JJK

JJ

sK

Ks

K

J

sJJs

l

s

v

lms

lm

s

v

lm

em

s

v

lms

lm

s

v

s

l

lm

m

++

+

+

+

++

+

++

+=

1

11

1

11

2

2

2

, (5.24)

:

( )( )

( )

( ) ( )sM

ss

s

sJJ

sM

ss

ss

sJJs

l

p

p

p

z

z

lm

em

p

p

p

z

z

z

lm

m

++

+

+

++

++

+=

121

12

1

121

121

1

2

2

2

2

2

2

, (5.25)

, :

( ) ( )( )

( )

( )( )

( )sMs

K

Ks

JJK

JJ

sK

Ks

K

J

sJJ

sMs

K

Ks

JJK

JJ

sK

K

sJJs

l

s

v

lms

lm

s

v

s

m

lm

em

s

v

lms

lm

s

v

lm

l

++

+

++

+

++

+

+

+=

1

11

1

1

1

2

2

2

, (5.26)

:

( )( )

( )

( ) ( )sM

ss

ss

sJJ

sM

ss

s

sJJs

l

p

p

p

z

z

z

lm

em

p

p

p

z

z

lm

l

++

++

+

++

++

=

121

121

1

121

1

2

1

2

2

2

2

2

2

, (5.27)

(5.25) (5.27) p z,

p z , :


68/145

68

( )

lm

lmsp

JJ

JJK += , (5.28)

l

sz

J

K= , (5.29)

( )

lms

lmvp

JJK

JJK

4

2 += , (5.30)

ls

vz

JK

K

4

2

= . (5.31)

p z (5.28) (5.29)

[34] ( p) ( z) , :

m

l

z

p

rJ

JR +== 1

. (5.32)

5.3

() .

: () .

5.3.1

, . , , . . , . .

. . , . . , , . .

, . , .


69/145

69

. , capture/compare , . , , .

, , ,, , . , .

5.3.2

. , (EXS)

, . . [25], . up/down 10, 12, 14 16-.

5.6. .

arctg (HF ). . a , ., - . (BVCO).

BVC 16-,

NF

PIHF

VCO

up

down

arctg

16-bitup/down

lsbmsb

sine cosine EXS


70/145

70

. . , ; . , .

[56],

,).s.)(s.(s

)s()(s

NNN

NN

in

out

854342

1142 +++

+= (5.33)

, Ns

BWf ,

.2BW

Nfss

= (5.34)

BWf

, 520 Hz kHz. , :

,1

1)(

ssW

rd

rd+

= (5.35)

rd .

(5.33) . :

.)8.54.3)(4.2(

14)(

++

NN

N

in

out

sss

(5.36)

(5.36) :

,)1)(1(

92.13/14)(

21 ++

NnNn

N

in

out

sss

(5.37)

8.5/4.3,4.2/1, n2n1 == , . (5.8), -

:

7192.0222

1 =+= nnrdn . (5.38)

, (5.34), -

.Hz][

4578.0

Hz][

2s][

BWBW

rdnrdff

=

=

(5.39)


71/145

71

(5.33) kHz1fBW= .

0 1 2 3 4 5

-3

0

0.2

0.4

0.6

0.8

1

1.2

1.4

5.7.

50% . : srd 330= .

5.3.3

, , . N- . ,:

[ ]N

rad2

2 = . (5.40)

, . - , :

,2

2]s/rad[

TT N

== (5.41)

. ,

, , ("") . , , , . .


72/145

72

5.4

, 5.8 , .

5.8.


73/145

73

6 --

. . , , . , . . , -(PI). Simulink-, (*.mdl) .

6.1

. . . , , , . , , ( ), , 5.8., :

:

kWPnom 49,1= , min/3000rnom= , mNTnom = 7,5 , mNT = 24max ,

() 200062,0 mkgJ = ,

: 4=p , 16 , kHzf

bw

1= ,

():


74/145

74

0005.0=em ,

:

200022,0 mkgJ = , radNmsKv /004,0= ,

.1 radNmKs /150= , .2 radNmKs /350= , .3 radNmKs /1400= .

:

200062,0 mkgJ = .

6.1. Simulink-5.8, .

6.1.

, 5: 5.1.(5.15.), , 5.2., 5.3.2., 5.3.3.

.

6.1. , 6.2.

, ; :

,

s

(s)W

em

em

+

=1

1 (6.1)


75/145

75

, 0005.0=em ,

s(s)Wem

0005.01

1

+= . (6.2)

5.22. :

( )( ) ( ) ( )( )svlmlm

svl

em

mmeh

KsKJJsJJs

KsKsJ

sM

ssW

+++++

==22

2

)(

, (6.3)

, 200062,0 mkgJm = ,200084,0 mkgJl = ,

radNmKs /350= radNmsKv /004,0= :

23647

2

511.01084.510208.5

350004.000084.0)(

sss

sssWmeh ++

++= . (6.4)

6.2.

:

23447510

627

511.010613.210237.510604.2

2563.01093.210152.6

)()( ssss

ss

sWsWW mehem +++

++

==

. (6.5)

6.5. sT 0003.0= , :

547.0138.3094.796.7458.4

10791.610836.110628.410373.110182.9)(

2345

98283849

+++++

=

zzzzz

zzzzzG . (6.6)

:

( ) ( )z

z

z

zpKzG nb

14

2

41614

2)(

=

=

, (6.7)

:


76/145

76

zzzzz

zzzzzGzGzG bp ++

+++==

547.0591.2502.4458.3

01118.003024.007622.002262.001512.0)()()(

2345

234

. (6.8)

z

6.8. 6.3 6.4:Bo d e o v e k a r a k te r i s ti k e

Frekvencija (rad/sec)

Faza(deg)

Amplituda(dB)

-60

-40

-20

0

20

40

102

103

104

-450

-360

-270

-180

-90

0

90

6.3.

D i j a g r a m p o l o v a i n u l a

Realna osa

Imaginarnao

sa

-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

6.4.


77/145

77

6.1.1

. z

, .

mJ

oJ ,

:

1.4

om

JJ = ;

2.

3

om

JJ = ;

3.2

om

JJ = ;

4. om JJ = ;

5. om JJ = 2 ;

6.

om JJ = 3 ,

7. om JJ = 4 .

6.5, 6.6 6.7.

odeove karakteristike


Faza(d

eg)

Amplituda(dB)

-60

-40

-20

0

20

40

102

103

104

-450

-360

-270

-180

-90

0

90Jm=4*JoJm=3*JoJm=2*JoJm=JoJo=2*JmJo=3*JmJo=4*Jm

Jm/Jo=4

Jm/Jo=1/4

Jm/Jo=1/4Jm/Jo=4

6.5.


78/145

78


Realna osa

Imaginarnaosa

-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Nule koje se ne menjaju Polovi koji se ne menjaju

Jm/Jo=4

Jm/Jo=1/4

6.6.


Realna osa

Im

aginarnaosa

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Jm/Jo=4

Jm/Jo=1/4

6.7.


79/145

79

. sK ,

:

1.

radNmKs /5,87= ;

2. radNmKs /6,116= ;

3. radNmKs /175= ;

4. radNmKs /350= ;

5.

radNmKs /700= ;

6. radNmKs /1050= ,

7. radNmKs /1400= .

6.8, 6.9, 6.10 6.11.

Bo d e o v e k a r a k t e ri s t i k e


Faza(deg)

Amplituda(dB)

-60

-40

-20

0

20

40

102

103

104

-450

-360

-270

-180

-90

0

90Ks=87,5 Nm/rad

Ks=116,6 Nm/rad

Ks=175 Nm/rad

Ks=350 Nm/rad

Ks=700 Nm/rad

Ks=1050 Nm/rad

Ks=1400 Nm/rad

6.8.


80/145

80


Realna osa

Imaginarnaosa

-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Nula koja se ne menja

Polovi koji se ne menjaju

Ks=87,5 Nm/rad

Ks=1400 Nm/rad

U smeru porasta Ks

U smeru porasta Ks

6.9.


Realna osa

Im

aginarnaosa

0.8 0.85 0.9 0.95 1 1.05 1.1

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Ks=87,5 Nm/rad

Ks=1400 Nm/rad

6.10.


81/145

81

Dijagram polova i nula

Realna osa

Imaginarnaosa

-3.235 -3.23 -3.225 -3.22 -3.215 -3.21 -3.205 -3.2

-6

-4

-2

0

2

4

6

x 10-4

Ks=87,5 Nm/rad Ks=1400 Nm/rad

Smer porasta Ks

6.11.

(5.28) p ,

--. . , 6.12.

(Hz)

114,86118,63125,82

145,29

177,94

229,72

205,47

0

50

100

150

200

250

Jm=Jo/4 Jm=Jo/3 Jm=Jo/2 Jm=Jo Jm=2*Jo Jm=3*Jo Jm=4*Jo

Hz

6.12. p


82/145

82

sK ,

--. sK .

, , , 6.13.

(Hz)

315,30

273,06

222,95

157,65

111,4878,83

90,99

0

50

100

150

200

250

300

350

87,5 116,6 175 350 700 1050 1400

Hz

Nm/rad

6.13. p

sK

, ,

, . .

, :

6.1.

1 -

2 -

3 -

: mJ 2mkg 0.00062 0.00062 0.00062

: lJ 2mkg 0.00208 0.00084 0.00084: ml JJ / . . 3.3548 1.3548 1.3548

: s

K rad

mN 150 350 1400

: p

Hz 89.19 157.65 315.30


83/145

83

2 (6.8):

zzzzz

zzzzzGp ++

+++=

547.0591.2502.4458.3

01118.003024.007622.002262.001512.0)(

2345

234

2 . (6.8)

, :

zzzzz

zzzzzGp ++

+++=

547.0626.2596.4518.3

01121.003005.007813.002224.001515.0)(

2345

234

1 . (6.9)

zzzzz

zzzzzGp ++

+++=

547.0455.2117.4209.3

01076.002989.006705.002299.001454.0)(

2345

234

3 . (6.10)

z(6.8.), (6.9.) (6.10) 6.14 6.15:

-60

-40

-20

0

20

40

60

Amplituda(dB)

101

102

103

104

105

-450

-360

-270

-180

-90

0

90

Faza(deg)

Konfig. br. 1

Konfig. br. 2

Konfig. br. 3

Bodeove karakteristike


6.14.


84/145

84

-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Konfig. br. 1

Konfig. br. 2

Konfig. br. 3

Dijagram polova i nula

Realna osa

Imaginarnaosa

6.15.

6.2

a je , 3. 2, .

I

GP(z) , G(z), (6.16.), 2 :

zzzzzzA ++= 547.0591.2502.4458.3)( 2345 , (6.11)

01118.003024.007622.002262.001512.0)( 234 +++= zzzzzB , (6.12)

( ) 1728.0826.1573.1006.44289.0 234 ++= zzzzzC , (6.13)


85/145

85

sTs 0003.0= .

6.16.

II

GP(z). B(z). 6.4. , B(z), :

.229411.0

,-,192439.0980575.0

,192439.0980575.0

,22777.3

4

3

2

1

=

=+=

=

B

B

B

B

Z

jZ

jZ

Z

(6.14)

(ripple) 1BZ 4BZ . ,

- 2BZ 3BZ z, , , .

B(z) : )()()( zBzBzB + = , )(zB + , )(zB z. :

1)( =+ zB , (6.15)

01118.003024.007622.002262.001512.0)()( 234 +++== zzzzzBzB . (6.16)

III

3.7. , . , )(zR )(zS ,

)(zRf )(zSf , , )()()( zRzRzR f =

)()()( zSzSzS f = .

)(zRf )(zSf .


86/145

86

, ,

)(zR . , , 0)1( =R , . 3.2. ,

, :

11)( = zzRf , (6.17)

1)( =zSf . (6.18)

IV

R, TR degdeg

SR degdeg . , , :

TSR deg1deg1deg +=+= . (6.19)

V

, . :

,0deg,5degdeg === +

BAA (6.20)

,5deg1deg,41degdegdeg =+==== SRATS , (6.21)

5degdeg == AAm , (6.22)

51052451degdeg2degdegdeg =++=++= +BASRA mo . (6.23)

VI

GM(z). mA mB .

, . z-. 6 , (3.118) mA :

( )585.0)( =

=

zezzA

n

T

T

md

s

. (6.24)


87/145

87

nGM(z), . mA ,

. )()( zBzBB mm

= , GM(z), )(zB . )(zBm , ,

, ( ) ( )1/1

= BAK m .

GM(z):

443705.061003.214125.6225.725.4

000288.000078.000196.000058.000039.0

)(

)()(

2345

234

+++++

==zzzzz

zzzz

zA

zBzG

m

mM . (6.25)

6.17.

VII

, )(zAo )(zAo

. , . ,

:)1.0()5.0()7.0()( 2 = zzzzAo . (6.26)


88/145

88

6.18.

)(zR ,)(zS )(zT , )(zR )(zS ,

:

058215.016905.0981306.085814.11)( 2341 ++= zzzzzzR , (6.27)

87592.15211.97422.174635.1436751.4)( 234 ++= zzzzzS , (6.28)

000632.0009401.0035644.0051658.0025829.0)( 234 ++= zzzzzT . (6.29)

6.19. .


89/145

89

6.3

- .

, . 2, : ,. , 2 ( ) ( ), 6.1.1., 6.1. , - (PI

), [26], [27], 2 1(). MATLAB-Simulink, .

6.3.1

. 150 /t=0.2s NmMM nomopt 7.5== , t=0.5s.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

0

1

2

3

4

5

6

7

8Momenat opterecenja

Vreme (s)

Mo

menat(Nm)

Mopt=5.7 Nm

6.20. .

: 1: ( ) )1.0()3.0()5.0()7.0( = zzzzzAo


90/145

90

2: ( ) 22 )1.0()3.0( = zzzAo

3: ( ) 22 )1.0( = zzzAo

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

80

100

120

140

160

180

Vreme (s)

Brzinamotora(ob/min)

Odziv brzine motora - regulator broj 1

referentna brzina 150 ob/min

Polovi opservera Ao su: 0.7, 0.5, 0.3 i 0.1

6.21. 1.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

80

100

120

140

160

Vreme (s)

Brzina(ob

/min)

Odziv brzine motora - regulator broj 2

Polovi opservera Ao su: 0.3, 0.3, 0.1 i 0.1referentna brzina 150 ob/min

6.22. 2.


91/145

91

0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29147.5

148

148.5

149

149.5

150

150.5

Vreme (s)

Brzina(ob/min)

Odziv brzine motora - uvecan detalj

Regulator 2 (polovi opservera Ao su: 0.3, 0.3, 0.1 i 0.1)


Regulator 1 (polovi opservera Ao su: 0.7, 0.5, 0.3 i 0.1)

6.23. 12 .

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

80

100

120

140

160

180

Vreme (s)

Brzina(ob/min)

Odziv brzine motora - regulator 3


Polovi opservera Ao su: 0.1, 0.1, 0 i 0

6.24. 3.


92/145

92

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-2

0

2

4

6

8

10

Vreme (s)

Momenat(Nm)

Odziv momenta motora - regulator 1


Momenat: 5.7 Nm

Ripl momenta: 0.12 Nm

6.25. 1.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-2

0

2

4

6

8

10

Vreme (s)

Momenat(Nm)



Momenat: 5.7 Nm


6.26. 2.


93/145

93

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-10

-5

0

5

10

15

20

25

Vreme (s)

Momenat(Nm)


Polovi opservera Ao su: 0.1, 0.1, 0 i 0

6.27. 3.

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

, z , , 3, , , .

6.3.2

. 150 /t=0.2s NmMM nomopt 7.5== , t=0.5s.


94/145

94

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

80

100

120

140

160

Vreme (s)

Brzina(ob/min)

Odziv brzine motora

referenca brzine 150 ob/min

Jm=0.00062 kgm

Jo =0.00084 kgm

2

2

6.28. .

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

80

100

120

140

160

Vreme (s)

Brzina(ob/min)

Odziv brzine opterecenja

referenca brzine 150 ob/minJm=0.00062 kgm

Jo =0.00084 kgm

2

2

6.29. .


95/145

95

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

80

100

120

140

160

Vreme (s)

Brzina(ob/min)

Odziv brzine motora - povecanje momenta inercije motora za 100%


Jm=0.00124 kgm

Jo =0.00084 kgm

2

2

6.30. 100%.

0.22 0.24 0.26 0.28 0.3 0.32 0.34

135

140

145

150

155

Vreme (s)

Brzina(ob/min)

Uvecan detalj odziva brzine motora -bez promene i sa povecanjem momenta inercije motora za 100%

referenca brzine 150 ob/minJm=0.00124 kgm

Jo =0.00084 kgm

2

2

Jm=0.00062 kgm

Jo =0.00084 kgm

2

2

6.31. 100%.


96/145

96

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-4000

-3500

-3000

-2500

-2000

-1500

-1000

-500

0

500

1000

Vreme (s)

Brzina(ob/min)

Odziv brzine motora sa smanjenjem momenta inercije motora za 50%


Jm=0.00031 kgm

Jo =0.00084 kgm

2

2

6.32. 50%.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-2

-1

0

1

2

3

4

5

6

7

Vreme (s)

Moment(Nm)

Odziv elektromagnetnog momenta

Jm=0.00062 kgm

Jo =0.00084 kgm

2

2

Moment: 5.7 Nm


6.33. .


97/145

97

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-2

-1

0

1

2

3

4

5

6

7

8

Vreme (s)

Moment(Nm)

Odziv elektromagnetnog momenta motora pri povecanju momenta inercije motora za 100 %

Jm=0.00124 kgm

Jo =0.00084 kgm

2

2

Moment: 5.7 Nm


6.34. 100%..

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-20

-15

-10

-5

0

5

10

15

20

25

Vreme (s)

Momenat(Nm)

Odziv elektromagnetnog momenta motora pri smanjenju momenta inercije motora za 50%

Jm=0.00031 kgm

Jo =0.00084 kgm

2

2

6.35. 50%..


98/145

98

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

80

100

120

140

160

Vreme (s)


Odziv brzine motora pri povecanju momenta inercije opterecenja od 100%

Jm=0.00062 kgm

Jo =0.00168 kgm

2

2


6.36. 100%.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

80

100

120

140

160

Vreme (s)

Br

zinamotora(ob/min)

Odziv brzine motora pri smanjenju momenta inercije opterecenja od 50%

Jm=0.00062 kgm

Jo =0.00042 kgm

2

2


6.37. 50%.


99/145

99

0.22 0.24 0.26 0.28 0.3 0.32

130

135

140

145

150

155

Vreme (s)


Uvecani detalj - odziv brzine motora za razlicite promene momenta inercijeopterecenja: pri povecanju od 100% i smanjenju od 50%

Jm=0.00062 kgm

Jo =0.00168 kgm

2

2

Jm=0.00062 kgm

Jo =0.00042 kgm

Jm=0.00062 kgm

Jo =0.00084 kgm

2

2

2

2


6.38. , 100% 50%.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

0

1

2

3

4

5

6

7

Vreme (s)

Momenat(Nm)

Odziv elektromagnetskog momenta motora pri povecanju momenta inercije opterecenja od 100%

Jm=0.00062 kgm

Jo =0.00148 kgm

2

2

Momenat: 5.7 Nm


6.39. 100%.


100/145

100

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

0

1

2

3

4

5

6

7

Vreme (s)

Momenat(Nm)

Odziv elektromagnetskog momenta motora pri smanjenju momenta inercije opterecenja od 50%

Jm=0.00062 kgm

Jo =0.00042 kgm

2

2

Momenat: 5.7 Nm


6.40. 50%.

. , , ( ). , , , . , , , , , .

. , , , , . , .

6.3.3

.


101/145

101

150 /t=0.2s NmMM nomopt 7.5== ,

t=0.5s.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

80

100

120

140

160

Vreme (s)

Brzina(ob/min)

Odziv brzine motora pri promeni koeficijenta krutosti osovine

Ks=1000 Nms/rad

6.41. .

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-20

0

20

40

60

80

100

120

140

160

Vreme (s)

Brzina(o

b/min)

Odziv brzine motora pri promeni koeficijenta krutosti osovine

Ks=150 Nms/rad

6.42. .


102/145

102

0.5 0.52 0.54 0.56 0.58 0.6115

120

125

130

135

140

145

150

155

Vreme (s)

Brzina(ob/min)

Uvecan detalj odziva brzine motora pri promeni koeficijenta krutosti osovine

Ks=1000 Nms/rad

Ks=350 Nms/rad

Ks=150 Nms/rad

6.43.,.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

0

1

2

3

4

5

6

7

Vreme (s)

Momenat(Nm)

Odziv elektromagnetskog momenta motora pri promeni koeficijenta krutosti osovineKs=1000 Nms/rad

Momenat: 5.7 Nm


6.44. .


103/145

103

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

0

1

2

3

4

5

6

7

8

Vreme (s)

Momenat(Nm)

Odziv elektromagnetskog momenta motora pri promeni koeficijenta krutosti osovine

Ks=150 Nms/rad

Momenat: 5.7 Nm


6.45..

. , , . 6.41. , -.

6.3.4

. 2 ( ) ( ), 6.1.1., 6.1. ,, 150 / t=0.2 s NmMM nomopt 7.5== , t=0.5s.


104/145

104

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

80

100

120

140

160

180

Vreme (s)

Brzina(ob/min)

Odziv brzine motora pri prelasku konfiguracije broj 2 u konfiguraciju broj 1


6.46.21.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-2000

-1500

-1000

-500

0

Vreme (s)

Brzina(ob/min)

Odziv brzine motora pri prelasku konfiguracije broj 2 u konfiguraciju broj 3


6.47.23.


105/145

105

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

0

1

2

3

4

5

6

7

8

Vreme (s)

Momenat(Nm)

Odziv elektromagnetnog momenta motora pri prelasku konfiguracije broj 2 u konfiguraciju broj 1

Momenat: 5.7 Nm


6.48.21.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-20

-15

-10

-5

0

5

10

15

20

25

Vreme (s)

Momenat(Nm)

Odziv elektromagnetnog momenta motora pri prelasku konfiguracije broj 2 u konfiguraciju broj 3

6.49.23.


106/145

106

2 1 . , ,

(). , 2 3, , ,,. .

6.3.5 PI

- (PI)

, [26], [27]. : 2 1. , , 150 /t=0.2s NmMM nomopt 7.5== , t=0.5s.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

80

100

120

140

160

180

Vreme (s)

Brzina(ob/min)

Uporedni prikaz odziva brzine motora za sistem sapolinomijalnim regulatorom i sistem sa PI regulatorom

Polinomijalni

regulator

Ripl brzine 0.1 ob/min

PI regulator

Ripl brzine 10 ob/min

6.50.PI .


107/145

107

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-6

-4

-2

0

2

4

6

8

10

12

14

Vreme (s)

Momenat(Nm)

Uporedni prikaz odziva elektromangnetnog momenta motora za sistem sapolinomijalnim regulatorom i sistem sa PI regulatorom

Polinomijalni

regulatorRipl momenta 0.06 NmPI regulatorRipl momenta 5 Nm

6.51.PI .

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

80

100

120

140

160

180

Vreme (s)

Brzina(ob/min)

Uporedni prikaz odziva brzine motora za sistem sa polinomijalnim regulatoromi sistem sa PI regulatorom pri prelasku konfiguracije 2 u konfiguraciju 1

Polinomijalni

regulator

Ripl brzine 0.05 ob/min

PI regulator

Ripl brzine 5 ob/min

6.52.

PI 21.


108/145

108

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-6

-4

-2

0

2

4

6

8

10

12

Vreme (s)

Momenat(Nm)

Uporedni prikaz odziva elektromagnetnog momenta motora za sistem sa polinomijalnimregulatorom i sistem sa PI regulatorom pri prelasku konfiguracije 2 u konfiguraciju 1

Polinomijalni

regulator

Ripl momenta 0.06 Nm

PI regulator

Ripl momenta 5 Nm

6.53.PI 2

1.

PI (6.50.), PI . , 30 ms PI. PI 100 , PI10 /0,1 / . ; PI5 Nm, , 0,06 Nm.

2 1. PI . , PI 100 , .

PI.


109/145

109

7

, . , setup-. , .

7.1

, .. , , . ,

. ..

. on-line . , . , off-line , , ,

(batch processing). off-line() , .

, , , ( ARX ARMAX ), , , .

, . ,

. ,


110/145

110

. (, ) . , , .

.

, , . . , off-line. on-line , .

. , ,

.

. , , . , () () . , . . ,

. (resiuals), .,

. : , () (). , , , , , .

, ..

, , . .

7.1.1

, . .

:


111/145

111

o ,

o () , o ,

().

, . :

1 - .

2 - . . .

. .

3 - ( ) .

4 - .

5- .

, , , ( 3) ( 4). (1 2).

7.1.2

.

7.1.2.1

, :

( ) ( ) ( )

( ) ( )mTdTkTubTdTkTub

mTkTyaTkTyakTy

m

umuu

++=

=+++

...

...

1

1 , (7.1)

:


112/145

112

( ) ( )

( ) ( ) ,

,

0

0

YkTYkTy

UkTUkTu

=

= (7.2)

U(kT) Y(kT) U0Y0; d

( ). Z (7.1),

:

( ) ( )

( )( )( )

d

m

m

m

mdup z

zaza

zbzbz

zA

zB

zu

zyzG

+++

++===

K

K1

1

1

1

1

1

1. (7.3)

7.1

n(kT), 7.1.

n(kT)(autoregressivemoving average) , (kT) (), :

( ){ }

( )[ ] ( ) ( ){ } ( ) .,cov

,0

2 =+=

==

vkTvkTvEkTv

kTvEv (7.4)

2v , ( ) -:

( )[ ]

( ) .00

01

,1lim1

22

=

=

= =

za

za

vkTvN

N

kN

v

(7.5)

:

( ) ( )

( )( )( ) pp

p

p

vzczc

zdzd

zD

zC

zv

znzG

+++

+++===

...1

...1

1

1

1

1

1

1

. (7.6)

(7.3) (7.6) 7.1:


113/145

113

( ) ( )

( ) ( )

( )( )

( )zvzC

zDzuz

zA

zBzy d

1

1

1

1

+= . (7.7)

( ) ( ) ( )111 ,, zCzBzA ( )1zD , . mnd.

n(kT)y(kT), .

( )1zC z. (7.7) , RMA BJ

(Box-Jenkins) . :

RMAX ML :

( ) ( )

( )

( ) ( )

( )

( )zv

zA

zDzuz

zA

zBzy d

1

1

1

1

+= . (7.8)

RX LS :

( ) ( )

( ) ( )

( ) ( )zv

zAzuz

zA

zBzy d

11

1 1

+= . (7.9)

OE (Output-Error) :

( ) ( )

( ) ( ) ( )zvzuz

zA

zBzy d +=

1

1

. (7.10)

7.1.2.2

)(),...,2(),1( mkykyky )(),...,2(),1( mdkudkudku k,

k-1.

( ) ( ) ( ) ( ) ( ) ( )1,...,1,1,1,...,1,1 2221 kbkbkbkakaka mm .

:( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ,1...111...11

1

1

kemdkukbdkukb

mkykakykaky

m

m

=

+++ (7.11)

e(k) y(k) . k.

y(k) ( )1 kky , :

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ,11...11

1...111

1

1

=+++

=

kkmdkukbdkukb

mkykakykakky

Tm

m

(7.12)


114/145

114

( )kT m

k-1,

( ) ( ) ( ) ( ) ( )[ ]mdkudkumkykykT = ...1...1 , (7.13)

( )1 k :

( ) ( ) ( ) ( ) ( ) ( ) ( )kbkbkbkakakak mm ...... 2121= . (3.14)

k,:

( ) ( ) ( ) ( ) ( ) ( )11 == kkkykkykyke T . (7.15)

m+d , k. k=1,2,,m+d+N , N+1

( ) ( ) ( ) ( )kekkky T += 1 , (7.16)

( ) ( ) ( ) ( )NdmeNdmNdmNdmy +++++++=++ 1 , (7.17)

:

( ) ( ) ( ) ( ) ( )[ ]NdmydmydmydmyNdmyT +++++++=++ ...21 , (7.18)

( ) ( ) ( ) ( ) ( )[ ]NdmedmedmedmeNdmeT +++++++=++ ...21 , (7.19)

( )

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

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

+++++++

+++++

=

=++

NuNmuNmuNdyNdmyNdmy

umumudydmydmy

umumudydmydmy

Ndm

...21...21

...............................................................................................................................

1...11...1

0...21...21

. (7.20)

. ,

= , (7.21), ,

( ) ( ) ( ) ( )++

+=

=++++=Ndm

dmk

TkeNdmeNdmeJ 2 , (7.21)

(7.17), :

( ) ( ) ( ) ( )1 ++++++=++ NdmNdmNdmyNdme . (7.22)

mN 2 :

yTT

= . (7.23)


115/145

115

, (7.22) (7.21), (7.24):

( ) TTTTTT yyyyJ += . (7.24)

T ,

(7.23) (7.22) . (7.23)

( ) ( ) ( ) ( )NdmyNdmNdmPNdm T ++++++=++ 1 , (7.25)

( )NdmP ++ :

( ) ( ) ( )[ ] 1++++=++ NdmNdmNdmP T . (7.26),

, . (7.20). , :

. md.. ( ) ( ) 0UkTUkTu = 0U

.. ( )kTu m

..

( ) ( ) 0YkTYkTy = , 0Y 00 UKY = .

. T ( ){ } 0=keE .

[45],[57].

7.1.3 MATLAB 6.0

MATLAB 6.0 , toolbox-[58]. , , . System IdentificationToolbox [59]. System Identification Toolbox , 7.1.1.. 2 - ,

. ,


116/145

116

. 3 , , .

4 . . , (maximal

likelihood) , . 5, , . toolbox- toolbox-ControlSystem Toolbox [60]Signal Processing Toolbox [61].

7.2 -

, setup . setup :

(DC brushless ) Vickers- ( MOOG), , 7.2. - , ;

DBM03 Vickers-(MOOG) ,

PC RS232/RS485 DBM03 , .

7.2 -

DBM03 Vickers- (MOOG) 7.3.


117/145

117

. DBM03. DC brushless . (DSP) , .

(IFOC), 0.0003s(300 s). , , , ,.

7.3DBM03setup-

DBM03:

; ; 10 kHz; IGBT ; ; , ,

, , ,

RS232/RS485 PC .

DBM03 RS232/RS485 PC DBM Windows , :

communication setup;

terminal screen; ,

(. text box-) command screen;


118/145

118

, (download), (upload) drive test;

digital scope, save or load.

7.4. main menu .

7.4 main menuWindowsDBM

7.5 drive test WindowsDBM


119/145

119

setup- 4096 ( 122 ), stimulus . 7.5.

7.3

, , . , 7.1. .

7.3.1 (1)

5, ( D/ ), -(/D ).

, -, . ( )kTu , 7.1.2.:

. ( ) ( ) 0UkTUkTu = 0U .

. ( )kTu m , m .

. T ( ){ } 0=keE .

, ( pseudorandom

binary sequence). ( ), , . () (7.4). .

, , . .

:


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1. DBM03 162 . , 92 .

2. DBM03 sTs 0003.0= . , setup-. 3.9.2. :

Hz

T

f

s

800

4

10

. (7.27)

, , 200 Hz. , :

ssTT s 0024,00003,088min === . (7.28)

3.

, MATLAB

-randn [58], , :

( ){ }

( )[ ] ( ) ( ){ } ( ) .1,cov

,1

,0

2

2

==+=

==

==

v

vv

kTvkTvEkTv

kTvEv

(7.29)

testM (9

2 ), testM (

92 ). (7.29) , .

1., 2. 3. MATLAB-m (), :


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121

...randn('state',sum(100*clock)); % Podeava se stanje generatora sluajnih brojeva

for i=1:7:4088,z=randn;

for j=1:1:7,if z>=0x=512;y=512;

elsex=65024;y=-512;

endB(i+j)=x;N(i+j)=y;

end

endfid = fopen('no1-8-9-bin.txt','wt')fprintf(fid,'%5i\n',B);fclose(fid);...

m MATLAB, . PC setup- DBM DBM03 . drive test dwnload stimulus

(from file), execute, upload response (to file) ( 7.5), , . .

7.3.2 (2)

MATLAB- System Identification Toolbox

7.1.1. MATLAB-ident, 7.6. . Preprocess , Operations (7.7). : Select channels... , Selectexperiments... Merge experiments..., , Select range

, Remove means Remove trends


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, Filter... Resample... , . .

7.6 ident MATLAB-SystemIdentification Toolbox-


Data Views , chek box-: Time plot Data spectra . , , , , , . [62]. )(tx

, :


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100

101

102

103

104

105

100

10

5

1010

y

1

Periodogram za eksperiment 1

100

101

102

103

104

105

10-5

100

105

Frequency (rad/s)

u1

7.81

0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75-4

-3

-2

-1

0x 10

5

y1

Ulazni i izlazni signali eksperimenta 2

0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75

-500

0

500

Vreme (s)

u1

7.92


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100

101

102

103

104

105

100

10

5

1010

y

1


100

101

102

103

104

105

10-5

100

105

Frequency (rad/s)

u1

7.92

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-3

-2.5

-2

-1.5

-1

-0.5

0x 10

5

y1

Ulazni i izlazni signali eksperimenta 3

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-600

-400

-200

0

200

400

600

Vreme (s)

u1

7.103


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127

100

101

102

103

104

105

100

10

5

1010

y

1


100

101

102

103

104

105

10-5

100

105

Frequency (rad/s)

u1

7.114

7.3.3 (3 4)

,() . MATLAB-System IdentificationToolbox-,Operations , Estimate ( 7.12) . : Parametric models... , Spectral model... Correlation model... . Parametric models... 7.13 : RX ( 7.9), RMAX ( 7.8), OE(Output-Error) ( 7.10), BJ (Box-Jenkins) ( 7.7), . , , , . RX AIC (Akaike Information Criterion) [59].

ident, Model Views , chek box-, Model output Model resids.Chek box Model output () , ()


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() . . Chek box Model resids . (residuals), a

.chek box-, chek box-: Transient resp ( ), Frequency resp ( ),Zeros and poles Noise spectrum .


7.13 Parametric models... Estimate


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129

, 7.8, 7.9, 7.10 7.11. : RX, RMAX , OE (Output-Error) BJ (Box-Jenkins) .

)(),(),( 111 zCzBzA )( 1zD , d, . ()(Model output) . (residuals), , , , :

1,3)deg(,2)deg(,4)deg(,3)deg( ===== dCDAB . (7.33)

(7.14). , , , . , , () (Model output). 3, , 2 7.15 , :

RX 52,9915%, RX () 51,4162%,

RMAX 49,7871%, OE(Output-Error) 78,1835% BJ(Box-Jenkins) 74,5174%.



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, . (residuals) : OE (Output-Error) BJ (Box-Jenkins) , OE (Output-Error) .

0 0.2 0.4 0.6 0.8 1 1.2 1.4-4

-3

-2

-1

0

1

2

3

4x 10

5

Vreme (s)

Izmeren i simuliran izlaz nad setom validacionih podataka

OE model fit: 78.1835

ARMAX model fi t: 49 .7871

ARX (IV algo ritam) model fit :51.4162

ARX model f it: 52.9915

Izmeren izlaz

BJ model fit: 74.5174

Pozicijamotora

(dig.

inc)

7.15

OE (Output-Error) . () (Model output) :

1 68,4513%, 2 91,8832%, 3 89,9255% 4 98,1873%.

7.16, 7.17, 7.18

7.19:


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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-6.5

-6

-5.5

-5

-4.5

-4

-3.5

-3x 10

5

Time

Meren i simuliran izlaz

Meren izlaz eksperiment 1

Simuliran izlaz

7.16 4 1

0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0 x 10

5

Vreme (s)

Poloaj(inc)

Izmeren i simuliran izlaz

Izmeren izlaz eksperiment 2

Simulirani izlaz

7.17 4 2


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0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-2.5

-2

-1.5

-1

-0.5

0x 10

5

Time



Simuliran izlaz

7.18 4 3

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4 x 10

5

Time



Simuliran izlaz

7.19 4 4


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:

( ) ( )

( ) ( ) ( ) ( )ze

zzzz

zzzzezu

zF

zBzy +

+++

=+=

4321

321

1

1

9605.0882.388.596.31

9701.097.1, (7.34)

:

9605.0882.388.596.3

9701.097.1

)(

)()(

234

23

+++

==zzzz

zzz

zu

zyzGidentif . (7.35)

7.3.4 (5)

, . . 7.20, 7.21, 7.22 7.22

, : ( ), ( ) .

Od z i v n a s t e p p o b u d u

Vreme (sec)

Pozicija(dig.)

0 0.005 0.01 0.0150

200

400

600

800

1000

1200

7.20 )(zGidentif


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134

"Bodeov" dijagram


Fa

za(deg)

Amplituda(d

B)

-50

0

50

100

150

100

101

102

103

104

-180

-150

-120

7.21 )(zGidentif

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Dijagram polova i nula u z ravni

I

m

z

R e z )

7.22 )(zGidentif


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136

Odziv na jedinicnu step pobudu

Vreme (sec)

Amplituda

0 0.005 0.01 0.015-20

0

20

40

60

80

100Sistem sa elasticnom osovinom (teorij sko razmatranje)

Identifi kovan sistem

7.23

Bodeov dijagram amplitudske i fazne frekventne karakteristike


Faza(stepen)

Amplituda(dB)

-100

-50

0

50

100

102

103

104

-450

-360

-270

-180

-90

0Sistem sa elasticnom osovinom (teorij sko razmatranje)

Identifi kovan sistem

7.24


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7.4 SETUP-A

, setup -:

9605.0882.388.596.3

9701.097.1

)(

)()(

234

23

+++

==zzzz

zzz

zu

zyzGidentif . (7.35)

DBM03 , :

z

zzGs

44)(

= , (7.36)

, :

9605.09205.296.2

8805.388.74)()(

23

2

++

==zzz

zzzGzGG sidentifis . (7.37)

, . )(zR , )(zS )(zT , :

( ) ( )237127.0229128.2069832.21)( 231 ++= zzzzzR , (7.38)

28203.083209.081752.026745.0)( 23 ++= zzzzS , (7.39)

1370928.06283420.0571220.0)( 2 += zzzT . (7.40)

MATLAB-Simulink. - (PI) , [26], [27]. 7.25 i 7.26 setup,

1400 /, . , 7.27 i 7.28 setup, 5 Nm, .

- . , , setup , (one-mass system), ( PI

) .


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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

200

400

600

800

1000

1200

1400

1600Odziv brzine motora na referencu brzine od 1400 ob/min

Vreme (s)

Brzina(ob/min)

PI regulator

RST polinomijalni regulator

7.25

1400 /PI(RST).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-2

0

2

Ž. marković - metod za sintezu digitalnog zakona upravljanja servo sistemom sa elastično...

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