diskretni sistemi na upravuvanje-predavanje1
DESCRIPTION
predavanjaTRANSCRIPT
-
( )
-
-
-
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
TkthkTthkTtL )1()()(
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 :
TkthkTthkTtL )1()()(
TsTs
kTsT
sTskTs
es
Tes
edee
s
e0
0
0
1
Ts
TskTsTsTs
kTs
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e
s
e
se
sTe
s
e 111
Ts
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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
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.
)(* sX .
)(txe t .
:
)()( sXtxeL t . ,
( ).
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)(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 :
)(
)(
...
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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 .
:
- :
)(
)(
...
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01
1
1
1
1
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zQ
zP
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bzbzbzXzzX
n
n
n
m
m
m
m
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)( zX :
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121
1 1*
01
1
1
1
1
1
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i
v
rr
i
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n
n
n
m
m
m
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bzbzbzX
:
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73
2,
*
2,'
*
1
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)(
)()(
)!(
1
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)(
ii
i
i
ii
vzz
v
irz
rv
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d
rv
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:
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i
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:
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*
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
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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
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M
Z - )(zX u )(zX i *
ux
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)()()( 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
*
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zT
zsWWzWWzW
ln1
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Z - .
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2. 7.2 .
)(sX i )(txi .
:
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)()()(*
11
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22
sXsWsX
sXsWsX
ui
ui
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-
77
)()()(
)()()(**
1
*
1
*
2
*
2
*
sXsWsX
sXsWsX
ui
ui
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1
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2 sXsX iu
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1
* sWsWsW
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7.2
3. 7.3 .
7.3
)(sX i )(txi
. :
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)()()()( **1*
1
* sXsWsXWsX uui
(Z- ) .
7.2
7.4 .
)(sX i )(txi .
:
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1
sXsWsX u
n
k
ki
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-
78
)()()( **
1
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n
k
ki
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1
* sXsW u
n
k
k
n
k
k sWsW1
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:
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 :
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*
21
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1* sXsWW
sWsX ui
)(1
)()(
*
21
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1*
sWW
sWsW
:
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)()(
21
1
zWW
zWzW
2. 7.6 .
7.6: .
)(* sX i )(* txi
z - )(txi :
)(1
)()(
*
21
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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
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1
*
1*
sWsW
sWsW
)()(1
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21
1
zWzW
zWzW
5. 7.9 .
Z - :
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)()()(
*
32
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1
*
2
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1*
sWWsW
sWsWsW
)()(1
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321
21
zWWzW
zWzWzW
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81
7.9 , ,
6. 7.10 .
7.10
.
Z - :
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*
2
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21
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.
-
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82
8 M
8.1 -
8.1 .
8.1
:
1-
2-
2 :
mntxtxtx
txtxtxtx
m
umuu
i
n
in
n
in
);(...)()(
)()(...)()(
)(
10
011
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1
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(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
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0
t :
t
txttx
t
tx
dt
dx
)(_()(
-
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83
k
kk
k
kk
t
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t
tx
t
t
tx
t
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k
k
k
k
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n
n
ni
n
n
n
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m
m
muu
(8.2)
:
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t ,...2,1,0, kkTt
T k - k
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k - . 8.2 :
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011
1
1 kxT
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n
n
ni
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n
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m
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muu
(8.3)
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:
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:
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-
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84
)()( jkxkxE j
:
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,...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
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10(
8.4
0d
( ).
8.2 Z
1
0
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j
ji zjxzXzikxZ
8.4 :
m
r
ur
n
r
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n
r
r
j
j
uu
rm
r
r
r
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j
ii
r
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1
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n
r
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r
r
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m
r
m
r
r
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jr
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r
r
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r
r zjxzXzzjxzXz0 0
1
0 0 0
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)(
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)(... .