13 - Kontrolabilnost Sistema i Povratna Sprega Po Stanju Sistema

., , 2013/14.

/

9.1

.

, , .

.

- - -

),(

),(

uxgy

uxfdtdx

==

nRtx )(pRtu )(mRty )(

DuCxy

BuAxdtdx

+=+=

(9.2)

/

buyadtdya

dtyda

dtyd

n

n

nn

n

=++++

121

1

L

=

=

11

2

1

/

/

nnn dtyd

dtdyy

x

xx

x MM

1

21 1

00

01000010

xy

bux

aaadtdx

n

=

+

= M

LL

LL

/ (1)

/ , ,

duxbxbxby nnn ++++= 1211 L

[ ] duxbbby

ux

aaadtdx

nn

n

+=

+

=

11

21 1

00

01000010

L

ML

LLL

9.1

.9.1

sincos)(sincos)(

2

2

mglpmlmlJFmlpbmlpmM

=++++=++

&&&&&&&&&&

, m J, , l m , b , g , p .9.1.

9.1..

u=F, . 9.1,

=0 sin0, cos1 (..)

Tppx ],,,[ &&=Tpy ][ =

++

++

=

sinsin

coscos 21

2 mglumlpb

mlmlJmlmM

p

p

p

dtd

&&&&

&&

9.1..

u

MmlmMJml

MmlmMJmlJ

p

p

MmlmMJmlb

MmlmMJmMmgl

MmlmMJbmlJ

MmlmMJglm

p

p

dtd

++

++++

++

+++

+++

++=

2

2

2

22

2

2

2

22

)(

)(

00

0)()(

)(0

0)(

)()(

0

10000100

&&

&&

xy

=00100001

() , =0.

,

( ) , .

, =0.

)( 4321 && kpkkpkKxu +++==[ ]Tppx &&= [ ]4321 kkkkK =

u(t) y(t) , (9.2) . A, B C (9.2) , D .

(9.7)

- . (9.2) Txz =

uBzATBuzTATBuAxTdtdz ~~)( 1 +=+=+=

DuzCDuzCTDUCxy +=+=+= ~1DDCTCTBBTATA ==== ~,~,~,~ 11

(1)(1)

z(t) x(t),

(9.9) , (9.7) (9.9) (9.2).

+== t tAtA duBeTTxeTtzTtx0

)(~1~11 )(~)0()()( A~

x(t)

.

uxdtdx

nn

+

=

MO2

1

2

1

0

0

uxx iiii +=&

9.2

=

n

A

0

0

2

1

O

=

kkn

kk

kk

k

t

tt

At

0

0

)( 21

O

=

tn

t

t

At

e

ee

e

0

02

1

O

(Jordan)

i ixi Ji.

=

kJ

JJ

J

LO

L

0

0

2

1

=

i

i

i

iJ

LO

L

01

101

i

=

tJ

tJ

tJ

Jt

ke

ee

e

LOMLL

00

0000

2

1

=

t

titt

tittt

tJ

i

iii

iiii

i

e

ietete

ietettee

e

LM

L

000

)!2(...0

)!1(!22

12

.

( s-) .

s-.

x(0)- (9.14) u(t)

x(t) t>0 x(0)?

BuAxdtdx +=

(1)(1)

: Za stawa modela kontinualnog sistema ija je jednaina staa (9.14) aemo da su potpuno kontrolabilna ako je za svako poetno stae x(0) za bilo koje eeno terminalno stae x* mogue nai upravae u(t) u nekom konanom vremenskom intervalu [0,] tako da se obezbedi uslov x()=x*.

:

u(t) (0,t) x(t), , B, AB,..., An-1B .

n.

.

dtBuedBuetxt

At

tA )()()(00

)( == 0t)(...)()( 1

110 +++= nnA AAIe

+++= t t t nn dtuBAdtuABdtuBtx0 0 0

11

10 )()(...)()()()()(

[ ]BAABBW nC 1= L

, x(0)=0, u(t)0,

9.5

(9.16)

. ( )

(9.16) ( ).

uBzAuz

aaaa

dtdz

nn

~~

0

001

0100

00100001

121

+=

+

=

MM

L

=

1000

101

~ 21121

1

LM

LL

n

n

C

aaaaa

W

. 9.3.

S

S

u S , u S.

.9.3

uxdtdx

uxdtdx

+=

+=

22

11

=

1111

CW

z=Tx .[ ]BAABBW nC 1= L

TABTBTATBA == 1~~BTATBTATTATTBTATBA 211212 )(~~ ===

M.~~ BTABA nn =[ ] [ ] CnnC TWBAABBTBABABW === 11 ~~~~~~ LL

1~ = CCWWT

.

-

,

K AZ(s)

Cxy

BuAxdtdx

=+=

))(()( txftu =

)()()( trKtKxtu r+=

nnnn

Z pspspsBKAsIsA ++++=+= 111)det()( L)det( BKAsI +

rBKxBKAdtdx

r+= )(

9.8

(): :

r(t)=r0 ess=0

[ ]xyuxdtdx 01;

10

0010 =

+

=

rKxkxku r+= 2211

[ ]xyrKx

kkdtdx

r

011010

21

=

+

= 122

21

1det ksks

ksks ++=

+

200

2 2)( ++= sssAZ 202201 2, == kk2

0=rK

.

9.9

!

... 0 .

[ ]xyuxdtdx 01;

01

0010 =

+

=

rKxkxku r+= 2211

)(0

1det 11

221 ksssksskks +=+=

++

x2 je .

, () .

9.10 ,

(9.24)

[ ] .~0

001

0100

00100001

~~

21

121

zbbbzCy

uz

aaaa

uBzAdtdz

n

nn

L

MM

L

==

+

=+=

(1)

(9.24)

.

100

0010001

det)(

121

+

=

s

saaaas

sD

nn

n

M

L

nnnn

n asasassD +++= 111)( L

rKzkzkzkrKzKu rnnr +=+= ~~~~ 2211 L

[ ] .0

00

0100

00100001

~~~~

21

112211

zbbby

r

K

z

kakakaka

dtdz

n

rnnnn

L

MM

L

=

+

=

(2)

. . :

ess=0

:

x: -

, .

nnnnnn

Z kaskaskassA~)~()~()( 11

111 +++++++= L

nnn apk

apk

apk

=

==

~

~

~

222

111

n

n

n

nnr b

pbkaK =+=~

Txz =

CW

1~ = CCWWT rKKxrKTxKu rr +=+= ~

;~~~ 1== CCWWKTKK CW~rKzKu r+= ~