Diskretni Sistemi Na Upravuvanje-predavanje1

( )

-

-

2

1.

1.1

.

:

1.

2. ( ) ,

3. ,

4. ,

5. , .

:

.

, ,

1.1 , .

.

1.1

-

3

1.1.1

1.1.

. .

Rttn :

.

1.1.2

0, , 2,... () . 1.2

1.1 0, , 2, ...,

() .

,...,...2,,0 kTTTd

d

d

t

ttxtx

/,0

),()(*

x *x

. .

1.2 ()

1.1.3

() .

-

4

, 1.3.

1.3

. .

. .

1.1.4

1.4.

1.4

.

1.1.5

, . 1.5.

-

5

1.5

)(tx ,

. .

)(tx

() .

)(tx

)( tx ,

, .

. )( tx )(tx

.

)( tx

)( tx

. , , ,

)( tx .

x ().

)(* tx

.. , 0 1.

)(* tx

. 0 1, 0 00, 01, b 10 c 11.

, )(* tx

.

. .

-

6

. .

1.1.6

. () 1.6.

1. 1.7 .

1.6 ()

1.7

2. .

3. ( ).

4. :

- ,

- ,

- .

- ,

- ,

-

7

- ,

- ,

- .

1.2

, . .

, , . , , , . . , , . 1.8, 1.9, 1.10 1.11 , , .

1.8

1.9

1.10

-

8

1.11

, . . .

1.3 . , . . . /. , , . /. / / , , , . . 1.8, 1.9, 1.10 1.11 / / , . , / / , .

1.3

1.2

1.2 () .

-

9

1.12

. . . . . . . . . .

1.13 .

1.13 .

)(tx zi

)(* tx zi .

. )(txi

)(* txi .

. .

)(* tx zi )(* txi ,

. /.

-

10

)()()) kTxkTxkTe izi

.

)(kTm

)(tm

. 1.13 /. .

1.14 .

1.14

/ / 0 .

1.3

1.15 .

, . .

-

11

1.15 .

z , ,

0 . z ,

v . z ,

. / - . / - . () .

1.4

1.16 , 1.17 .

1.16

-

12

(). . . , . . . .

1.17 -

1.5

1.18 - .

1.18 - .

z , . z ,

. zv .

.

-

13

1.6

1.19 .

1.19 .

.

/ / . . .

-

14

2. ,

2.1 ()

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

, , , . , 2.1.

2.1 T0

,...3,2,1,0,, iiTiT .

2.1

2.1 , , . , , , (). ,

-

15

, . .

2.11

. 2.2 .

2.2

, .

, constT . ,

.,...,2,1,1 constTittT iiii

2.2 .

2.3

,...3,2,1,0,,,)(

)()(

*

iiTiTttx

txtx .

2.4

,...3,2,1,0,,,)( iiTiTttx ,

)(tx

)(txf .

2.5 1, 1 , :

,,,0

,,),()(*

iTiTt

iTiTttxtx ,...,3,2,1,0 i

2.6 )(tx :

.0,,0)( ttx

-

16

)(* tx

)(* th 2.3

2.3 )(* th

)()(

)()()(

)()()()(

*

0

0

*

thtx

iTthiTthtx

iTthiTthtxtx

i

i

2.1.2

, , , 2.4.

2.4 ()

. , .

)(* th

)(* t ,

-

17

0

* )()(i

iTtt ,

2.5.

2.5 )(* t

)(* tx :

)()()( ** txttx

)(* tx , 2.4,

2.6.

2.6

, 2.4 2.6,

)(tx ().

().

2.7

)()()( ** txttx .

2.7

2.7

, :

1. ()

T .

-

18

2.

(), )()()( txtxtx

,0:, 00 ttTTt

3. () 0 , T

20 ,

. ( ).

4. ()

(), ,sTT 2.8.

2.8 sT .

, , . , .

, ().

().

2.8 ().

, () (), / . 2.9 .

2.9

2.10 .

-

19

2.10

2.1.3

2.9. , 2.11, , .

2.10 )(tx

.

2.11 I

, 2.12

2.11

2.12 I- (), 2.12b.

2.13 ( )

I- T (), 2.12c.

2.12 , I-

2.12, .

2.14 I- .

-

20

2.15 , .

2.16

2.17

T , NTT iii , .

.

2.1.4

2.13 () , .

2.13 ()

uV iV

(). , , . MOSFET ,. , . . , , , . , , . 2.14 () ().

2.14 ()

-

21

. () 2.15.

2.15 ()

,

, dT

, . ,

, oT ,

. . ,

sT ,

oT . ,

, . () . , . 2.16 () .

2.16 -()

2.2

() . . 2.17

-

22

, / , .

2.17 /

2.17

. 2/q

. , . , . . 3/4,

2/4/3 qPOA . 2/4/3 qPOA

, .

2.3

, 2.17 , 2.1.

IAI < 0,5 q

0,5q < IAI < 1,5q

1,5q < IAI < 2,5q

2,5q < IAI <

0

1/4PO = q

1/2PO = 2q

3/4PO = 3q

00

01

10

11

2.1 /

2.4

/ , 2.18 / .

2.18 /

-

23

, . 2.19 / .

2.19 /

() , () / , , .

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

, , , .

-

24

3.

3.1

3.1

)(* sX :

0

*** )()()( dtetxtxLsX st

3.2 1,,, jjs

3.3 ?

3.4

)(* jX

0

**** )()()()( dtetxdtetxtxFjX tjtj

3.5. 0,0(*0)(* ttxtx .

3.1.1

3.1.

3.1

3.1 :

0

* )()()(k

kTttxtx .

:

00

** )()()()()()(kk

kTtkTxLkTttxLtxLsX

-

25

:

00

* )()()()(k

kTs

k

ekTxkTtLkTxsX

, :

0

* )()(k

kTsekTxsX

js

:

0

* )()(k

kTjekTxjX

3.1.2

:

)()()( ** ttxtx .

:

0

**** )()()()()()( dtettxttxLtxLsX st

3.6 ) )(* t )

)(tx p )(pX .

) )(* t :

0

2

0 00 0

0 0 0

**

1

1......1

)()(

)()()(

kTs

kTsTsTsTskTs

k

st

k

st

st

k

st

eeeeee

dtekTtdtekTt

dtekTtdtettL

Tseq :

111

111 )( TT

TTjTTjTs ee

eeeee

T 1Te

0 , 0Re s . 3.2 Te .

-

26

3.2 Te

3.7

1

)1(... 12

q

qaaqaqaqa

nn

q

aaqaqaqa k

1...... 12 .

q ,

.

) )(tx p )(pX

0

)()()( dtetxtxLpX pt

jc

jc

ptdpepXj

pXLtx )(2

1)()( 1

.

!

0

* )()()( dtettxsX st

dtetdpepXj

st

jc

jc

pt )()(2

1 *

0

dpdtetpXj

jc

jc

tps

0

)(* )()(2

1

dptLpXj

jc

jc

)()(2

1 *

jc

jc

TpssXdp

epX

j)(

1

1)(

2

1 *)(

3.6,

)(* t Tpse )(1

1

:

-

27

pspsps ReRe0ReRe0)(Re !

3.9.

3.8

k

v zz

zfsidzzfv

1

)(Re2)(

3.9 )(pX

)(tx p :

1. 2

2. )(pX .

3.10 3.9 :

)(tx 0)( tx t ,

)(pX Tpse )(1

1

p ,,0 cc , - , 3.3.

3.3.

3.11. Tpse )(1

1

101 )()( TpsTps ee

3.3 Tpse )(1

1

:

11 jpsw

1)sin(cos1 11111 TjTeeeTTjT

-

28

1sincos 1111 TjeTeTT

0,0sin 11 1 TT eTe , 6.4

3.4 T

e 1

,...3,2,1,1,0sin 11 kkTT

1)1(1cos 11 kTT eke (3.1)

001 111 Te T (3.2)

3.1 :

,...3,2,1,0,2 nnk

,...3,2,1,0,21 nnT

,...3,2,1,0,21 kkT

T

k

21 (3.3)

3.2 3.3 :

,...3,2,1,0,2

11

0** k

T

kjjpsw

,...3,2,1,0,2* kT

kjspk

,...3,2,1,0,2* kT

kjspk

!

jc

jc

Tpsdp

epX

jsX

)(

*

1

1)(

2

1)(

dte

pXj Tps )(1

1)(

2

1

dtepX

j Tps )(1

1)(

2

1

-

29

TpsepXsj

j )(1

1)(Re2

2

1

TpsespXj

j )(1

1Re)(2

2

1

3.12

Tpse )(1

1

*

*

)()(

1

1

1

1Re

k

k

pp

TpsppTps

edt

des

kj

T

kjsp

Tps TeTe 22)(

11

TkjkT

1

)2sin2cos(

1

01

!

T

kjspp

k k

pXT

sX 2*

*)(

1)

)()()2(1 **

0

txLtxLT

jksXT k

3.9 ( 1 2 ) :

)()0(2

1)( ** txLxtxL

k TjksX

Tx

0

21)0(

2

1

.

s j :

k

kjXT

jX )(1

)( 0* .

3.1.

-

30

dpe

pXj

sXTps

jc

jc

)(

*

1

1)(

2

1)(

, , 3.5.

1

)(

*

1

1)(

2

1)( dp

epX

jsX

Tps

1)(1

1)(Re2

2

1Tpse

pXsjj

1)(1

1Re)(2

2

1Tpse

spXjj

3.5 )(pX 1

3.13 )(pX

**

)(

)()(Re

)(

)()(

'ii pppp pf

pqpXs

pf

pqpX

3.14 ipi *

- )(pX

!

n

isTTp

i

i

eepf

pqsX

i1

*'

**

*

1

1

)(

)()(

n

iTjTp

i

i

eepf

pqjX

i1

*'

**

*

1

1

)(

)()(

-

31

3.2 )(* sX

3.15 )(* sX

s

0 :

ZvjvsXsX ),()( 0**

3.16 Z .

: s , v

k

jkjvsXT

jvsX )(1

)( 000*

k

vkjsXT

0)(1

mkmkvkm ,,

m

sXjmsXT

jvsX )()(1

)( *00*

3.17 )(* sX

)( *0 ss () )(* sX

)( 000 jkssjkss () )(* sX ,

)(* sX () ()

.

3.17 3.6.

:

22

,,: 000

jssP

:

22

,,: 000

jssP

)(* sX 0P

0P Zvjv ,0 .

3.3 )(* jX

.

-

32

3.3.1 )(* jX

)( jX

3.7.

, x 3.8.

3.6 )(* jX

3.7

3.8

.:

-

33

0)(* jX 0

. )(* tx

.

3.3.2 ()

3.18 ()

p

p

.

() 3.9.

3.9 ()

3.10.

a) )(tx

3.7, )(txp )(tx ,

)(txp

)(tx .

2

0 p

(

2,

2

00

3.10

3.11 2

0*

-

34

b) )(tx 3.11.

)(* tx )(txp 3.12

3.13, .

3.12 2

0*

3.13 )(txp

,2

0*

)(tx )(txp

)(txp )(tx

. :

*0* ,2

*

*

0

0)(

jX

3.3.3. ( )

3.19 )( jX )(tx

.

2,0)( 0

jX

.

, .

-

35

3.3.4

22

1)(,0),()(

jXthetx t .

)(tx 3.14.

3.14 )()( thetx t

3.15.

,

, * .

3.15 3.14

.

0 :

2,0)( 0

jX

)

3.16

-

36

3.16

3.17 , 3.18 .

3.17

3.18

)

3.19

3.20 .

3.19

-

37

3.20

3.21

3.21

3.22 .

3.23 .

3.20

3.24 .

3.22

-

38

3.23

3.24

3.4

3.21 S . 3.25.

3.25 S

3.22 S 2. 3.26.

3.26 S

-

39

uM

u

u

u

x

x

x

x

.

.

2

1

iN

i

i

i

x

x

x

x

.

.

2

1

3.27.

3.27 S

3.28.

3.28 S

*

*

2

*

*1

*

.

.

uM

u

u

u

x

x

x

x

-

40

4.

4.1

4.1

L .

4.1

L

(

) L , )(ti .

)(ti )(txi L

)(* txu , :

t

o

t

uui dtxidtixtx0

** )()()()()(

0 0

** )()()()( dtxidtix uu

4.1 t :

ttettxttx uu .0)(,0)(0,0)(**

ttettitti .0)(,0)(0,0)(

L , 4.1 4.2.

-

41

4.2 L )(ti

)(txi )(* txu , )(sX i )(

* sX u. ,

L :

)(

)()(

* sX

sXsW

u

i

)(sX i )(* sX u , )(sX i

, )(* sX u

, .

4.1.1

L , 0, , 2, ...,

L , 4.3

4.3 L

0

** )()()()()(j

iii jTtjTxttxtx

4.2 )( jTxi :

0

* )()()( dtixtx ui

:

0

* )()()(k

uu kTkTxx

-

42

:

dtikTkTxtxk

ui )()()()(0 0

:

0 0

)()()()(k

ui dtikTkTxtx

:

0

)()()( kTtidtikT

0)( kT kT

0

)()()(k

ui kTtikTxtx

kTt

0 0

)()()()()(k k

uui TkjikTxkTjTikTxjTx .

!

0 0

* )()()()(j k

ui jTtTkjikTxtx

0 0

* )()()()(k j

ui jTtTkjikTxtx

mkj

0

* )()()()(k km

ui TmktmTikTxtx

0,0)( mmTi

0 0

* )()()()(k m

ui TmktmTikTxtx (4.1)

4.3

*ux *

ix

)(* sW :

)(

)()(

*

**

sX

sXsW

u

i

.

-

43

4.4

*ux *

ix

, :)(* jF

)(

)()(

*

**

jX

jXjF

u

i .

, .

4.1.2

0 0 0

** )()()()( dteTmktmTikTxxLsX st

k m

uii

:

0 00

* )()()()(k

st

m

ui dteTmktmTikTxsX

0

)(

0

* )()()(k

sTmk

m

ui emTikTxsX

0 0

)()(k m

mTskTs

u emTiekTx

0

* )()(m

mTs

u emTisX .

)()()( *

0

* sXemTisX um

mTs

u

)()()( *** sXsWsX ui

0

* )()(m

mTsemTisW . (4.2)

4.2

L .

0

* )()(m

mYs sIemTi

0

*** )()()()(k

kTs tiLekTisIsW .

-

44

)(* ti

. .

4.1.3

)()( ** sIsW

, :

k k

tiLjksWT

jksIT

sIsW )()(1

)(1

)()( *00** .

s. () .

4.1.4

)()( ** sIsW

, :

)(1

1

)(

)()()( *

1*'

***

*tiL

eepf

pqsIsW

n

kTsTp

k

k

k

)(

)()(

pf

pqpI

*kp )(pI :

nkpf k ,...,2,1,0)(*

.

:

kpWppTps k

pWe

ssW)()(

**)(

1

1Re)(

*kp )(pW 1 .

4.1.5

s j ,

:

00

* )()()(m

Tjm

k

Tjk

ui emTiekTxjX

-

45

0

* )()(m

Tjm

u emTijX

0

* )()(k

Tjk

uu ekTxjX .

:

)()()( *** jXjFjX ui

0

* )()(k

TjkekTijF .

s j .

4.1.6

j :

kk

kjFT

kjIT

jF )(1

)(1

)( 00*

4.1.7

j :

n

kTpj

k

k

kepf

pqjF

1)(*'

*

*

1

1

)(

)()(

)(pI )(pW :

)(

)()(

pf

pqpW .

)(pW

:

kpWppTpj k

pWe

sjF)()(

**)(

1

1Re)(

4.2

4.4 .

-

46

4.4

:

)(* sX uk - - )(* txuk

)(* jX uk - - )(* txuk

)(* sX ij - - )(* txij

)(* jX ij - - )(* txij

4.5 )(* sW jk 4.4

-, =1,2,...N, -, =1,2,...,, .

4.6 )(* jFjk

4.4 -, =1,2,...N, -, =1,2,...,, .

4.7 *W N x M, (,)-

)(*, sW kj 4.4

:

)(...)(

.

.

.

)(...)()(

)(...)()(

)(

**

2

*

1

*

2

*

22

*

21

*

1

*

12

*

11

*

sWWsW

sWsWsW

sWsWsW

sW

NMNN

M

M

4.8 *F MN , (,)-

)(*, jF kj 4.4

.

)(...)()(

.

.

.

)(...)()(

)(...)()(

)(

**

2

*

1

*

2

*

22

*

21

*

1

*

12

*

11

*

jFjFjG

jFjFjF

jFjFjF

sF

NMNN

M

M

-

47

5

5.1

)(* txi

.

,

T , .

- () .

5.1 .

5.1

5.2 , 5.3

)(txT .

-

48

5.2

5.3

)(txT

:

TkkTtkTxtxT )1(,),()(

5.3 )(txT

0

)1()()()(k

T TkthkTthkTxtx

5.1.1

:

TkthkTthkTxLtxLsXk

TT )1()()()()(0

0

)1()()(k

TkthLkTthLkTx

s

eekTx

s

e

s

ekTx

TskTs

k

TskkTs

k

1)()(

0

)1(

0

kTsk

Ts

ekTxs

e

0

)(1

-

49

)(1

)( * sXs

esX

Ts

T

)(sWT . :

)(

)()(

* sX

sXsW TT

)()()(1

)( ** sXsWsXs

esX T

Ts

T

s

esW

Ts

T

1

)(

5.1.2

s j

:

1cossin1)(

Tj

T

j

ejF

jT

T

221cossin

)(

TTAT

TTT cos21cossin1 22

2

2sin

)cos1(21

T

T

TT

2cos

2sin2

2sin2

arctansin

1cos

arctan)(

2

TT

T

T

T

T

022tanarctan

TT

, :

0sin

2)( TA (5.1)

-

50

0

)(

T (5.2)

:

00

sin

2)(

j

T ejF

5.1 5.4./

))(,( TA

.

)(txT

)(* tx

T.

5.2 5.5.

5.4

5.5

-

51

5.2

5.6 , 5.7

)(txT .

5.6

5.7 )(txT

5.7

)(txT

)()( * kTxkTxT

*x :

TkTkTtkTtT

TkxkTxkTxtxT )1(,),(

)1()()()(

:

TkthkTthkTtT

TkxkTxkTxtx

k

T )1()()()1()(

)()(0

-

52

5.2.1

:

0

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

)()()(k

TT TkthkTthkTtT

TkxkTxkTxLtxLsX

:

TkthkTthkTtLT

TkxkTx

TkthkTthLkTxtxL

k

k

T

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

)1()()()(

0

0

:

TkthLkTthLTkthkTthL )1()()1()(

kTsTs

TskkTs ese

es

es

111 )1(

dteTkthkTthkTtTkthkTthkTtL st

0

)1()()()1()()( .

5.8

kT Tk )1( .

5.8 kT Tk )1(

:

,)1(,0

)1(,,1

,0,0

)1()(

Tkt

TkkTt

kTt

TkthkTth

TkkT )1(, , :

TkthkTthkTtL )1()()(

-

53

dteTkthkTthkTt stTk

kT

)1(1

)1()()( .

kTt

0 KTt

TTkt )1(

ddtkTt


T T T T

skTs

skTsskTskTss eds

eed

sedeedee

0 0 0 0

)()(

. (5.3)

.

5.1 :

T T

T vduuvudv0 0

0

5.3 u ve s :


TsTs

kTsT

sTskTs

es

Tes

edee

s

e0

0

0

1

Ts

TskTsTsTs

kTs

Tes

e

s

e

se

sTe

s

e 111

Ts

TskTs

Tes

e

s

e 1

:

0

1)(

)(k

TskTs

T s

eekTx

sX

0

1)1()(

k

TsTskTs

Tes

e

s

e

T

TkxkTx

00

1)1(

11)1()(

k

TsTskTs

k

TsTskTs

Tes

e

Ts

eTkxe

Tse

s

ekTx

mk 1

-

54

0

1)1(

k

TsTskTs

Tes

e

Ts

eTkx

1

1

m

TsTs

TsmTs

Tes

ee

Ts

exmT (5.4)

5.4 :

0

2)1(1

1)1()()(k

TsTsTs

TsTskTs

T eeTs

ee

Tse

s

ekTxsX

0

11)1()(

k

TsTsTs

kTs

Ts

ee

Tse

s

ekTx

02

2 )(11

)1(k

kTsTs ekTxTss

e

)(11

)1()( *2

2 sXTss

esX TsT

(5.5)

:

)(

)()(

* sX

sXsW TT

)()()( * sXsWsX TT

5.5 :

2

2 11)1()(Tss

esW TsT

5.2.2

s j

:

2

2

2

2 1)1()

11)1()(

T

jTe

TjejF jTjTT

5.9 .

-

55

5.9

5.9 () () .

6 Z-

6.1 z

Z . Z .

z:

sTez .

z:

zjezz arg .

z zarg

)sin(cos jez j

cos

zz jjz 21

-

56

z 21 z . 6.1

z z .

6.1 z z -

s :

21 nn js

:

cos , sarg , sn

221 1sin1cos2 TjTeeez nnTTjT nnn Tne

, 21 Tn

s :

js

sRe sIm

)sin(cos TjTeeez TTjT

Te , T .

6.2 s z

s - z - :

sTez

.

z - s - :

0

1jkljz

Ts (6.1)

.

6.1 6.1:

-

57

zesT

zjTjT ezee arg

js . zje arg

,...3,2,1,0,)2(argarg kee kzjzj

)2(arg kzjTjT ezee

zTze T ln

,...3,2,1,0,2arg kkzT

:

kjzjzkzjzTjT 2argln)2(argln .

:

kjzTjTzjzz 2lnarglnln

0ln12

ln1

jkzTT

kjz

Tjs

.

!

sTe :

,...3,21,,0,)(2 0 keezeez jksTsTkjsTsT

s -

Zmm ,0 , Z ,

z -, s - z -.

s - z -

__

0P :

2,

2,,: 00

__

0

jssP

s - z -, __

kP :

,...3,2,1,0,2

)12(,

2

)12(,,: 00

__

k

kkjssPk

s - z - __

0P s -

z -. 6.2 s -.

-

58

6.2 s -

s - z -.

6.2.1

2,

2

00 jj

2,

2

00 jj s -

z - :

TjTez Tj sincos

1 T .

:

22

00 T

0,0,2

0

00

,02

,0 0

2

0 .

6.3 s z -, 6.4 .

-

59

6.3 s - z

,...3,2.1,2

)12(,

2

)12( 00

k

kjkj

s - z -.

6.4 z -

6.2.2

,2

0js

__

0P s -

2

0j 2

0j .

1z , . z - z - .:

TjT

Tj

TsT eeeeeez

2

0

(6.2)

:

TjT

Tj

TsT eeeeeez

20

(6.3)

-

60

6.5 Te

6.2 6.3 z -.

6.5 Te

:

0 z

1,00, z

10 z

,1,0 z

z .

6.6 ___

0P

z -, 6.7 z -, z -.

6.6 ___

0P

-

61

6.7

,2

0js s - z -

0,1 , 1, .

,...3,2,1,2

)12( 0

kk

js

6.2.3 s

z - .:

TsT eez (6.4)

6.8 Te ,

6.4 s - z .

6.8 Te

:

1,00, z

10 z

,1,0 z

z .

s -

1,0 z -, s - ,1 z -

-

62

. 6.9 s - 6.10 z -, . z -.

,...3,2,1,,0 kjks

6.2.4 s - z -

:

s - z -,

s - z -,

s z -.

6.9 s -

6.10 s - z -

6.11 , 6.12 .

-

63

6.11 s -

6.12 z -

6.2.5

constjs ,0,,

. z - .:

TTjTsT eeeez

Te .

6.13 , 6.14 z -.

13.

-

64

6.14 s - z -

6.2.6

s 6.15.

:

21 nn js

z :

21 TjT nn eez

21, Tje nTn .

6.16 s - z -.

6.3 Z

:x

6.2 x :

)(kTx Zk 1,2...,,0)( kkTx , I k

k

)()( kTxktx

)(lim)( txkTxkTt

.

6.3 Z (Z -) x , ,

)()( zXkTxZ

-

65

0

)()()(k

kzkTxzXkTxZ .

6.15

6.16 s - z -.

Z - T . T x Z -.

. T Z - x . Z - x . . 6.17 .

x I )(* sX :

0

* )()(k

kTsekTxsX

-

66

Z - x :

0

)()(k

kzkTxzX

6.17 z

.

ZkjkzT

ssXzX

,ln1

*

0

)()(

0k .

6.3.1 Z -

6.4 Z :

)()()()( 22112211 txZtxZtxtxZ .

: Z - :

0

22112211 )()()()(k

kzkTxkTxtxtxZ

0 0

1

2211 )()(k k

k zkTxzkTx

)()( 2211 txZtxZ

.

5 x .

6.5 Z - x nT :

1. :

1

0

)()()(n

k

kn zkTxzXznTtxZ

6.18 )(tx

nT .

-

67

6.18 )(tx

nT

2. :

)()( zXznTtxZ n

6.19 )(tx

nT

6.19 )(tx

nT

. 1. 2.

1. Z - x nT :

kk

zTnkxnTtxZ0

)()(

)(0

)( nk

k

n zTnkxz

mnk

nmk 0

mk

:

kk

n zmTxznTtxZ

0

)()( (6.5)

6.5 :

-

68

0

1

0

)()()(m

n

k

mmn zmTxzmTxznTtxZ (6.6)

6.6

Z - x m k :

1

0

)()()(n

k

kn zkTzXznTtxZ

2. Z - x nT :

kk

zTnkxnTtxZ0

)()(

knk

n

k

k zTnkxzTnkx

)()(1

0

mnk

nmk 0

11 mnk

0 mmk

mk

:

1

0

)()( )()()(nm m

nmmn zmTxzmTxnTtxZ (6.7)

6.7 :

.0,0)( iiTx

)()()(0

zXzzmTxznTtxZm

nmn

.

)(sX ( ) )(zX Z -

x ;

)()(,)()( txZzXtxLsX .

Z - x

)(sX :

)()()(__

zXtxZsXZ

-

69

)(__

sXZ )(* sX

s zT

ln1

:

z

Ts

sXsXZln

1

*__

)()(

.

)(* sX .

)(txe t .

:

)()( sXtxeL t . ,

( ).

6.6 )(zX Z - x Z -

)(txe T :

)()( zeXtxeZ Tt . .

00

))(()()(k

kT

k

kTt zekTxzkTxetxeZ

zez T1

)()()()( 10

1 zeXzXzkTxtxeZT

k

kt

.

.

Z - n - )(txn )(tx

6.7 Z - n - )(txn

)(tx :

1

0

1 )0()1()()1()(n

i

iinnn xzzzXztxZ .

. ! :

)()()( txTtxtx

:

)()()()(2 txTtxtxtx

)()()()2( txTtxTtxTtx

).()(2)2( txTtxTtx

n - :

-

70

)(...)()( 1 txtxtx nn . .

1n . 1n . :

)()()( txZTtxZtxZ

)0()()1()()0()( zxzXzzXxzXz

11

0

111 )0()1()()1(i

ii xzzzXz

)0()0(0 xx . 1n .

1n 1n . :

)()()()(1 txZTtxZtxZtxZ nnnn

)0()()1( xztxZz nn

)0()0()1()()1()1()(1

0

11 xzxzzzXzztxZ nn

i

iinnn

1

0

11 )0()0()1()1()()1(n

i

niinn xzxzzzzXz

)0()1()0()1()()1( 1101 nnn zzzzXz )0()1()0()1(...)0()1( 01122 nnn zzz

11

0

111 )0()1()()1(n

i

iinn xzzzXz .

1n 1n . n Nn .

6.3.2 Z

Z - )(zX )(1 zXZ

G

k dzzzXj

zXZ 11 )(2

1)(

G 1)( kzzX .

6.3.3 Z

6.8 )0(x x 0t :

)(lim)0( zXxx

.

-

71

: Z - :

0 1

)()0()()(k k

kk zkTxxzkTxzX

z :

)0()(lim xzXx

.

6.9 )()1( zXz

1* z

)()1(lim1

zXzz

:

)()1(lim)(lim1

zXzkTxxk

: Z - )(tx x

:

)0()()1()( zxzXztxZ (6.8)

:

)()()( txZTtxZtxZ

00

)()1(k

kk

k

zkTxzTkx

0 0

11)()1(lim)(lim

k k

kk

zzzkTxzTkxtxZ

M

k

M

k

kk

MzzkTxzTkx

0 01

)()1(limlim

...))(2()1)(()0(limlim 211

zzTxzTxxMz

MMM zTMxzzMTx ))1(()(... )1(

...))(2()1)(()0(limlim 2011

0

1

zzTxzTxx

zM

MMM zTMxzzMTx ))1(()(...

0)1(

)0()1(lim xTMxM

.

kM 1 kM . 6.8 :

-

72

)0()()1(lim)(lim11

zxzXztxZzz

)0()(lim)0()()1(lim1

xkTxxzXzkz

)()1(lim)(lim1

zXzkTxzk

.

.

6.4 Z

6.4.1 Z

Z Z -.

Z - )(zX

Z - :

)(...)()( 1 zXzXzX m

m

i

i zXZkTx1

1 )()( .

)(zX

)(zX :

)(

)(

...

...)(

01

1

1

1

1

1

zQ

zP

azazaz

zbzbzbzX

n

n

n

m

m

m

m

(6.9)

6.9 nm 00 b 1na .

:

- :

)(

)(

...

...)()(

01

1

1

1

1

11

zQ

zP

azazaz

bzbzbzXzzX

n

n

n

m

m

m

m

- )( zX

)( zX :

vvv

zzzzzzzQ )...()()()( **2*

121

1 1*

01

1

1

1

1

1

)(...

...)(

i

v

rr

i

ir

n

n

n

m

m

m

mi

zz

c

azazaz

bzbzbzX

:

-

73

2,

*

2,'

*

1

*

)(

)()(

)!(

1

)(

)(

ii

i

i

ii

vzz

v

irz

rv

i

vzzi

ir

zQ

zPzz

d

d

rv

zQ

zPc

c

:

1 1*

1

)()(

i

v

rr

i

irzz

zczX

:

1 1

*

1*)(

)!1()!1(

!

)(

1)(

i

v

r

k

ir

i

ir

i

zkrk

k

zckTx .

6.4.2

Z - :

G

k dzzzXj

kTx 1)(2

1)(

G 0z 1)( kzzX .

G izXzz

kk

i

zzXsjdzzzX)(

11*)(Re2)(

)(

1*)(Re)( sXzz

i

k

i

zzXskTx

.

*ix )(zX )(

)()(

zQ

zPzX :

** 1*1 ))(()!(

1)(Re

i

i

i

i

i zz

kv

irv

rv

izz

k zzzzXdz

d

rvzzXs

.

6.4.3 1z

Z - x :

...)(...)2()()0()( 21 kzkTxzTxzTxxzX

:

)(lim)0( zXxz

)(lim)( 2 zXz

zTxz

)(lim!2

1)2( 22 zX

zz

zzTx

z

-

74

)(...lim!

)1()( 222 zX

zz

zz

zz

kkTx

k

z

k

6.20

)(ti

6.5 Z Z

)(ti

6.20.

6.10. Z - Z - )(zW :

)(

)()(

zX

zXzW

u

i .

Z - . z - s - .

6.21.

6.21

:

ZtxZzX ukuk )()( * - k - )(* txuk .

ZtxZzX ijij )()( * - j - )(* txij .

6.11 )(zW jk Z -

6.21 j -, Nj ,...,2,1 k -,

Mk ,...,2,1 , .

-

75

6.12. W MN , ),( kj -

)(, zW kj 6.21 Z -

.

)(...)()(

.

.

.

)(...)()(

)(...)()(

)(

21

22221

11211

zWzWzW

zWzWzW

zWzWzW

zW

NMNN

M

M

Z - )(zX u )(zX i *

ux

*ix :

)()()( zXzWzX ui

)()()( 1 zXzWZkTx ui

.

-

76

7

7.1 s- .

7.1

1. 7.1 .

7.1

)(sX i )(txi .

:

)()()()()()( **21 sXsWsXsWsWsX uui

)()()()()( ***** sXsWsXsWsX uui

)()()()()( **21**

21 sXsWWsXsWsW uu

7.2 )()()( *2*

1

*

21 sWsWsWW )()(*

22

*

21 sWWsWW

zT

zsWWzWWzW

ln1

*

2121 )()()(

Z - .

7.3 )()()( 2121 zWzWzWW

2. 7.2 .

)(sX i )(txi .

:

)()()(

)()()(*

11

*

22

sXsWsX

sXsWsX

ui

ui

-

77

)()()(

)()()(**

1

*

1

*

2

*

2

*

sXsWsX

sXsWsX

ui

ui

)()()()( **2*

1

* sXsWsWsX ui

)()( *1*

2 sXsX iu

)()()( *2*

1

* sWsWsW

)()()( 21 zWzWzW

7.2

3. 7.3 .

7.3

)(sX i )(txi

. :

)()()( 1 sXsWsX ui

)()()()( **1*

1

* sXsWsXWsX uui

(Z- ) .

7.2

7.4 .

)(sX i )(txi .

:

)()()( *

1

sXsWsX u

n

k

ki

-

78

)()()( **

1

* sXsWsX u

n

k

ki

)()( *

1

* sXsW u

n

k

k

n

k

k sWsW1

** )()(

:

n

k

k zWzW1

)()(

7.4

7.3

1. 7.5 .

7.5

)(sX i )(txi .

:

)()()( *1 sEsWsX i

-

79

)()()( **1

* sEsWsX i (7.1)

, :

)()()()()( *12 sEsWsWsXsE u

)()()()( **12** sEsWWsXsE u

)()(1

1)( *

*

21

* sXsWW

sE u

(7.2)

7.1 7.2 :

)()(1

)()( *

*

21

*

1* sXsWW

sWsX ui

)(1

)()(

*

21

*

1*

sWW

sWsW

:

)(1

)()(

21

1

zWW

zWzW

2. 7.6 .

7.6: .

)(* sX i )(* txi

z - )(txi :

)(1

)()(

*

21

*

1*

sWW

sXWsX ui

)(1

)()(

21

1

zWW

zXWzX ui

.

3. 7.7 ..

)(* sX i )(* txi

z - )(txi :

-

80

)(1

)()(

*

21

*

1*

sWW

sXWsX ui

,

)(1

)()(

21

1

zWW

zXWzX ui

.

7.7 .

4. 7.8 .

7.8

Z - :

)()(1

)()(

*

2

*

1

*

1*

sWsW

sWsW

)()(1

)()(

21

1

zWzW

zWzW

5. 7.9 .

Z - :

)()(1

)()()(

*

32

*

1

*

2

*

1*

sWWsW

sWsWsW

)()(1

)()()(

321

21

zWWzW

zWzWzW

-

81

7.9 , ,

6. 7.10 .

7.10

.

Z - :

)()(1

)()(

*

2

*

1

*

1*

sWsW

sWsW

,

)()(1

)()(

21

1

zWzW

zWzW

.

-

82

8 M

8.1 -

8.1 .

8.1

:

1-

2-

2 :

mntxtxtx

txtxtxtx

m

umuu

i

n

in

n

in

);(...)()(

)()(...)()(

)(

10

011

)1(

1

)(

(8.1)

(

) *ux

8.1 . , 8.1. .

)(tx t :

t

txttx

dt

dx

t

)()(lim

0

. , k -

)(tx t :

t

txttx

dt

xd kk

tk

k

)()(lim

)1()1(

0

t :

t

txttx

t

tx

dt

dx

)(_()(

-

83

k

kk

k

kk

t

tx

t

tx

t

t

tx

t

tx

dt

txd

)(...

)(

)(

)()(2

_2(2

)2(

)1(

Tt :

T

tx

dt

dx )(

k

k

k

k

T

tx

dt

txd )()( .

:

)()(..)()(0

011

1

1 txT

txT

txT

txT

iii

n

n

ni

n

n

n

)(...)()( 10

0 txT

txT

txT

u

m

m

muu

(8.2)

:

,0d

t ,...2,1,0, kkTt

T k - k

)(kxu )(kxi

k - . 8.2 :

)()(..)()(0

011

1

1 kxT

kxT

kxT

kxT

iii

n

n

ni

n

n

n

)(...)()( 10

0 kxT

kxT

kxT

u

m

m

muu

(8.3)

8.3 .

:

)1()( kxkEx

)()1()()()()1()( kxEkxkExkxkxkx

)()1()()1)(1()()1()()( 22 kxEkxEEkxEkxkx

:

)()1()( kxEkx jj

-

84

)()( jkxkxE j

:

)()1(...)1()( 011 kxkxnkxnkx iiinin

,...2,1,0),(...)1()( 10 kmkxkxkx umuu

n

r

m

r

urir rkxrkx0 0

)()( (8.4)

:

i

j

ji

jm

jm

im

i

j

ji

jn

jn

in

ji

jm

T

ji

jn

T

0

0

)1(

10(

8.4

0d

( ).

8.2 Z

1

0

)()()(i

j

ji zjxzXzikxZ

8.4 :

m

r

ur

n

r

ir rkxZrkxZ00

)()(

n

r

r

j

j

uu

rm

r

r

r

oj

j

ii

r

r zjxzXzzjxzXz0

1

00

1

)()()()(

n

r

n

r

r

j

m

r

m

r

r

j

jr

uru

r

r

jr

iri

r

r zjxzXzzjxzXz0 0

1

0 0 0

1

0

)()()()(

)(

0

0 0

1

0

1

0

)(

0

0

)()(

)()(

zY

n

r

r

r

n

r

m

r

r

j

jr

ur

r

j

jr

ir

u

zW

n

r

r

r

m

r

r

r

i

z

zjxzjx

zX

z

z

zX

)()( 1 zXZkx ii

-

85

9

9.1 , ,

9.1 kTt ,

,...,...,)1(, NTTkkT

,...,...,)1(, NTTkkT

.

.

9.2 kTt ,

,

,...,...,)1(, NTTkkT

,...,...,)1(, NTTkkT .

, .

9.3 kT () ,

nxxx ,...,, 21 .

kTt )(),...,(),( 21 kxkxkx n .

9.4 nx

T

nxxxx ),...,,( 21

, x(k) kTt .

x kTt (

) :

);;;()( 1,00 0 kkuxxkkkx (9.1)

:

0k - Tk0

0x - Tk0 .

:

)1(),...,1(),( 001,0 kxkxkxx uuukku

9.1

-

86

01,00 );;;( 0 xxxkk kku

:

)()()1( kBxkAxkx u

.

:

)()( ikxkx uu

9.5 :

1. Tkt 00

:

);;;(;;);( 1,001,00 00 kkuikiku xxkkxxikik

2. :

)();()( kxkxkx uii .

.

00k

);;0;();;();;;( 1,00000 kuuu xxkxxkxxkk

9.6 :

1. :

);;();;();;( 202210112211022011 uuuu xxkaxxkaxaxaxaxak

2. i :

);();();( 22211122112211 uiuiuui xxaxxaxaxaxaxa

.

)(kx .

)(kxM k

d . 9.1

.

-

87

9.1

9.1.1

:

)()1(..._1()( 011 kxakxankxankxa iiinin

nmmkxbkxbkxb umuu ),(...)1()( 10 .

.

:

)()()1( kbxkAxkx u

)()()( kdxkxckx uT

i

:

RdcbA nnnn ,,, .

:

;

0...000

1...000

.

.

.

0...010

0...001

0

1

2

1

a

a

a

a

A

n

n

n

nnn

nnn

bab

bab

bab

b

00

22

11

.

.

.;

0

.

.

.

0

1

c ; nbd

-

88

:

;

...

1...0000

.

.

.

0...0100

0...0010

13210

naaaaa

A

1

.

.

.

0

0

b ;

nnn

nnn

n

n

bab

bab

bab

bab

c

11

22

11

00

.

.; nbd .

.

Mjxuj ,...,2,1, jb

B :

)()()1( kBxkAxkx u

)()()( kxdkxckx uTT

i

:

MbbbB ...21 ; Mj

bab

bab

bab

b

njj

njnnj

njnnj

j ,...,1,

.

.

.

,00,

,22,

,11,

TnMnn bbbd ,,2,1 ... .

:

)()()1( kBxkAxkx u

)()()( kDxkCxkx ui

:

MNnNMnnn RDCBA ,,, .

9.1.2

9.2 .

-

89

9.2

9.2 :

uxBxAdt

dx

ui xDxCx

.

9.3 9.2 .

9.3 , 9.2 .

:

)()()(

txBtxAdt

tdxuT

(9.2)

)()()( txDtxCtx uTi

(9.3)

:

3,2,1,0,)1(,,)1()()()( kTkkTtTkthkTthkTxtx uuT

9.2 9.3

9.2 9.3 :

t

uT

tAtA

TuT dxBeexexxtA

0

0,00 )();;0;(

(9.4)

:

TkkTuTxkTxkTTk )1(,);(;;)1(

TkuTxxTk )1(,00;;0;)1(

(9.5)

:

-

90

TuTTkkTuT xxTxkTxkTTk ,00)1(, ~;~;0;);(;;)1(

(9.6)

:

)(~0 kTxx

)(~

)1(,,0 kTxxx uTkkTuTTuT (9.7)

9.4 9.7 9.6 :

T

uT

ATATA

TuT dxBeexexxT0

0,00 )(~~~;~;0;

)()(0

kTxBdeekTxe u

T

ATATA

(9.8)

9.8 9.6 :

TkkTuTxkTxkTTk )1(,);(;;)1(

)()(0

kTxBdeekTxe u

T

ATATA

(9.9)

9.5 9.9 :

TkxxxTk TkuT )1(;;0;)1( )1(,00

)()(0

kTxBdeekTxe u

T

ATATA

(9.10)

:

BdeABeA

T

ATA

0

;

9.10 :

)()()!( kBxkAxkx u (9.11)

9.11 0d

.

:

)()()( kDxkCvkx ui

:

DDCC

, .

.

-

91

9.1.3

:

)()()1( kBxkAxkx u (9.12)

)()()( kDxkCvkx ui (9.13)

9.7 (, )

1. :

.,,0

)(;;0;;;0);1(

0

1,00,00

M

u

n

ukuku

xxk

kBxxxkAxxk

2. :

.,;;0;0 000M

u

n

u xxxxx

:

M

uukuu kx 0,0,0, .

9.8 i ( ,

) 10.1 10.2

1.

)();;0;();;0;( 1,00,00 kDxxxkCxxk ukukui

2. :

)0())0(;;0;0( 0 iui xxx

10.1 10.2 :

x(k+1)=Ax(k) (9.14)

xi(k)=Cx(k) (9.15)

9.14 n .

9.9 X: nnd n

9.14 .

X(k+1)AX(k)

.

Y(k)=X(k)C

9.14 X(k) . .

(0)=I

-

92

:

(k+1)A(k), (0)=I

.

9.10 9.14 9.15 :

(k)=Ak , k0

9.11 9.14 9.15 :

1.()(k)=(j+k), j,k=1,2,

2. (0)=

3.det(k)0, k=0,1,2,.. TAeA

4. det0 (-k)=-1 ()

x :

0

1

0)(j

jx

9.14 9.15 .

9.12 10.1 10.2 :

)0;;0;( 0 uxk (k)x0 +

k

j 1

(k-j)Bxu(j-1).

9.13 10.1 10.2 :

(k;0;x0;xu[0,k-1])=(k)x0.

9.14 i 10.1 10.2 :

i ( k;0;x0;xu[0,k])=C[(k)x0+

k

j 1

(k-j)Bxu(j-1)]+Dxu(k).

9.15 i 10.1 10.2

:

i (k;0;x0;xu[0,k] C(k)x0.

9.14

:

x(k+1)=Ax(k)+Bxu(k) (9.16)

xi(k)=Cx(k)+Dxu(k) (9.17)

9.16 9.17 .

-

93

9.16 9.16 9.17

xu(k)=0u M . .

9.17 x* 9.16 9.17 xr :

x*= xr

:

(k;x*;0u) x*.

9.18 x* x*= xr 9.16 9.17 :

(-) x*=0x.

9.19 :

1. 9.16 9.17 xr=0x. 2. 9.16 9.17

xr=0x nii ,...,2,1,1 .

3. , ni ,...,2,1 1Ai 9.16 9.17

((-) x=0 n

x 0 .

. :

1. xr=0x 9.16 9.17

(-) x*=0x.

nn .

2.

(-) x*=0x

xr=0x

det((-)0

det( 0) A

1 xr=0x.

3. , -

(-) x*=0x

(-) :

det((-)=0

-

94

det( 0) A

1 .

9.2

x :

22

2

2

1

2/1 ...)( nT xxxxxx

K , ,

:

xxK :

9.2.1

9.20 xr=0x 9.16.9.17

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

0)(,0 x

;(k ux 0;0 ,...2,1,0) k

9.4

9.21 xr=0x, 9.16 9.17

0 ,0 ),( LL 0 :

1. 0x

2. ;(k ux 0;0 )

-

95

3. 9.22 xr=0x 9.16 9.17

( ) )(,0 ,,0 ),( LL 0x

;(k ux 0;0 ) , ,...2,1 LLk

( 0x k ).

9.5

9.23 xr=0x 9.16 9.17 .

9.24 xr=0x 9.16 9.17 .

9.25 xr=0x 9.16 9.17 .

9.26 xr=0x 9.16 9.17 .

9.22

9.27 9.16 9.17 :

1. xr=0x ,

2. xr=0x .

3. xr=0x .

9.23

9.28 xr=0x 9.16 9.17 :

1. i

:

nii ,...,2,1,1

-

96

2. i

:

1,1 ii v

9.29 xr=0x 9.16 9.17 :

1. i :

ni ,...,2,1 1i

2. i :

ni ,...,2,1 2,1 ii v

1 2.

9.30 xr=0x 9.16 9.17

i

:

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

9.2.4

Z - 9.16 9.17 :

)()()(

)()()0()(

zCXzCXzX

zBXzAXzxzzX

ui

u

)()()0()()(

)()()0()()(11

11

zXDBAzICxAzIzCzX

zBXAzIxAzIZzX

ui

u (9.18)

)()()0()();;( 1110 zBXAzIxAzIzZxxk uu

)()()0()()( 111 zXDBAzICxAzIzCZkx ui

Z - 9.16 9.17 9.18 0=0:

)()()()( 1 zXzWzXDBAzICzX uui

)()()(

1)()( 1 zDfBAzICadj

zfDBAzICzW

det)( zf (z-A)

, .

-

97

9.31 *z Z - W(z) 9.16

9.17 )(ZWij .

9.32 Z - W(z) .

9.33 9.16 9.17 ( xr=0x )

*iz Z - W(z) :

nizi ,...,2,1,1* .

-

98

10 , ,

10.1

:

x(k+1)+Ax(k)+Bxu(k) (10.1)

xi(k)=Cx(k)+Dxu(k) (10.2)

10.1 10.2 ( 10.1 10.2) :

zzf det()(

n

i

n

i

i azaA0

1,) .

10.1.1 -

- k kk 22 :

01321

2340

12310

01321

23401

12310

00...00...

..

..

..

...000...

...00...00

00...00...

..

..

..

...000...0

...00...00

aaaaaa

aaaa

aaaaaa

aaaaaa

aaaaaa

aaaaaa

nnknknkn

kkk

kkkn

nkkk

knknknn

knknknnn

k

10.1 10.1 10.2

k :

nkkk ,...,2,1,0)1(

- .

-

99

10.1.2 -

- :

1

2

na

0a

1na

1a

2na

2a

2a

2na

1a

1na

0a

na

3

4

nb

1b

1nb

2b

2nb

3b

2b

1nb

1b

nb

5

6

nc

2c

1nc

3c

2nc

4c

2c

nc

.

.

.

2n-3

ns

1ns

2ns

.

n

n

naa

aab

0

0 ;

10

1

1

n

n

naa

aab ; ...

10

1

1aa

aab

nn

n

n

nbb

bbc

1

1 ;

11

2

1

n

n

nbb

bbc ; ...

21

1

2bb

bbc

nn .

.

10.2 10.1 10.2 :

1. 0)1( f 0)1()1( fn

2.

01 aan

1bbn

2ccn

.

.

.

2 nn ss

-

100

10.2

10.2.1 Z -

, z -

)(1

)()( 1

zW

zWzW

ok

)(zWok z - . :

)(1)( zWzT ok

0)( zT

.

10.1 D. :

(

z - )(zWok )

)(zWok , D

,1 R

10.1 D

z D .

P )(zWok D.

N )(zT D

. CAP 0N .

-

101

)(zT z D

2)( PN

)(zT P2 .

Z - .

10.3 P z -

)(zWok , D.

z D , )(zWok

okW (-1,j0) P

.

10.3

:

x(k+1)+Ax(k)+Bu(k) (10.3)

xi(k)=Cx(k)+Du(k) (10.4)

u.

10.4 x0 10.3 10.4 0K u 0, ,2,..., (-1) :

xKuxk 0);;( 1,00

10.5 10.3 10.4

nx 0

u(0), u(1), . . .,u(K-1) :

xKuxk 0);;( 1,00

10.6 10.3 10.4

nBAABBU n )...( 1

10.4

10.7 x0 10.1 10.2 0K xi(0), xi(1),, xi(K) xu(0), xu(1),,xu(K).

y=xi-Dxu

y=Cx(k)

10.1 10.2 .

-

102

10.8 x0n 10.1 10.2

0K xi(0), xi(1),, xi(K) .

10.9 10. 10.2 x0

n )0;;( 0 ui xk , =1,2,...,

.

10.10 10.1 10.2

nCACACS TnTTTT 1)(... .

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