tau dlya chainikov
TRANSCRIPT
-
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. , -. , -.
. , . , - , .
. , - , , . , - . . - , . - - .
, , - , . - , ( ) - .
. :
1) ; 2) ; 3) , ; 4) .
..-.. .. , ... .. ... .. , , - .
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1. .......................................................................................................................................4 1.1. .................................................................................................................................................4 1.2. .............................................................................................................................4 1.3. ? .................................................................................................7
2. ..........................................................................................................................10 2.1. ? ...................................................................................................10 2.2. ...........................................................................................................................10 2.3. ? .......................................................................................................................11 2.4. ..............................................................................................................12 2.5. .....................................................................................................................13 2.6. ...........................................................................................................................................17
3. .....................................................................................................................20 3.1. ...........................................................................................................20 3.2. ..................................................................................................21 3.3. ............................................................................................................................22 3.4. ( ) .............................................................................24 3.5. .......................................................................................................................25 3.6. ....................................................................................................................26 3.7. .......................................................................29 3.8. ............................................................................................................31 3.9. ..............................................................................32
4. ................................................................................................................34 4.1. ............................................................................................................................................34 4.2. ..........................................................................................................................34 4.3. ...........................................................................................................................36 4.4. .........................................................................................................................38 4.5. ................................................................................................................39 4.6. .......................................................................................................................................40 4.7. .............................................................................................................................41 4.8. ...................................................................................................................42
5. ....................................................................................................................................43 5.1. ........................................................................................................................43 5.2. .....................................................................................................................44 5.3. ......................................................................................................45
6. ......................................................................................................................47 6.1. ....................................................................................................................47 6.2. ...............................................................................................................................47 6.3. ..............................................................................................................................................48 6.4. .....................................................................................................................................50 6.5. ....................................................................................................................57 6.6. ............................................................................................................................62 6.7. .............................................................................................................63 6.8. .................................................................................................................65 6.9. .......................................................................................................................................66
7. ....................................................................................................................................69 7.1. .............................................................................................................................69 7.2. - .................................................................................................................................70 7.3. ..............................................................................................................71 7.4. ..............................................................................................................................72 7.5. .............................................................................................................75 7.6. .................................................................................................................................75 7.7. ...................................................................................76
..........................................................................................................................................................79 ..........................................................................................................80
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1.
1.1. ,
. (, - ), (), ( ). -, , , . , , , , -. , .
?. XIX , (, ). . , - , , (-, ). -.
, , , , . . , - , . , - , ().
- . , control theory.
1.2. 1.2.1. ?
. - , - . , (, ); , - ; -.
. ( ). , - . , .
, ( ), ( ) .
, -. -
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. , .
, - (), , , , , .
, , . , . , , .
1.2.2. , , , . -
, . , . - , ( ), ( ), .
, ( ) , .
, , . - , .
, , , -. , - .
, :
, , ; -
, ; ; ,
; , ; ,
; .
: , ; . , , - . , .
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1.2.3. ? (, , )
( ) . , - . , , - . - :
( ). , , , , - . , , . - ( ).
, ? . , - , . , , , - . ( -) , ( ) . , 3530 , .
, , . , -. ? , . -, , , . , , - .
1.2.4. , ? , .
, , . , ( ). -, . , - , .
. .
. , . ,
()
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( ) , .
, . , . , . , .
1.3. ? , . -
, ( ) - , -, -. .
1.3.1. :
, , - ( , );
( - , );
. , , ( - ), . - , , . , , , .
1.3.2. , ( - );
, ./ ( ).
, , . , , ( ) ().
, . , . , . , - . , , , .., . , -.
- . - , .
1.3.3.
, , ;
, ( -), ;
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-, , . ( ) -
. , - (, ).
, - . - , . - ( ) (- -). .
- , , - (, , .). , -. , - - - .
1.3.4. ,
. , , -, . .
. , . , - . , , . ( -), .
1.3.5. , () ,
, . , .
, . , . , , . - . ? . - , .
-, , , , . , .
, - , (). - . , -, 2 , ( 99% -, 99 100).
1.3.6. . -
, - . , , . , - , .
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1.3.7. -
( ), -, . ( ) . - .
, , - , ( -, ).
(, ) - , . - : -, .
- . ( ) , -.
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2.
2.1. ? (
). : - , , ? - , -, : , , , , -, .
, (). - , , -, ( ) . - , . , (), , -.
2.2. .
, , . -, , , .
, . ( ) , , :
, , x - y. . ][xUy = , y U x.
, . - .
. - ( ), ( ). , - 1 1 /, 2 2 /, - 1. ,
xxU =][ . ,
( 0=t ). tx t , txty =)( ( )(ty - t ). , U ? , , . - )(tx ( !), -
1 , .
U x y
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= t dttxxU0
)(][ .
, , -. , , . S ( 2) , h q ( 3/), S:
= t dttqSth 0 )(1)( ,
:
dttdxtxtxU )()()]([ == & .
, . p. )()( txpty =
p )(tx , , :
dt
tdxtxp )()( = . (1) ? . , -
, i ( ), , - u ( ) :
)()()( tupCdt
tduCti ==
C ( ). , u i :
)()()( tipLdt
tdiLtu ==
L ( ). ( ) ,
. , - ( ) , .
2.3. ? -, -
( , , ). , , .
RLC-, - R ( ), L C. :
dttduCti
tiRdt
tdiLtutu
c
c
)()(
)()()()(
=
++=
, RLC- . )(tiR -
)(ti )(tuc
R L
C )(tu
i
u
i
u
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, , -. . )(tu ,
)(tuc . -
( ). , , . , , , - .
: ( , ) , . , , , ( , - ..), ( ) . - . - .
, - ( ) . - , . , - . , .
, , , - . - , . , ( -).
. , , . - () ( ), . . ( ), , . .
2.4. , . -
(, .) , ().
. 2: : ][][ xUxU = , ( ,
); : , -
: ].[][][ 2121 xUxUxxU +=+
, , . , - .
2 .
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, . , ( ). - , . , , - .
? (), . - , . , - - .
, -, . - , . , - , - .
, 1.3 , , . , - .
2.5. ,
. . - , , - - .
2.5.1. .
, . - S, S0.
, h ( ) q ( 3/). ,
2
2vhg = . ( /3), 81,9g /2 , v - ( /). ghv 2= . , - vSq = 0 ,
hq = , (2) gS 20= . , , . - ( ), .
q
h
S
0S
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, (2) , h . - (2) hkq = , k -. ? .
, 0 1 . - ,
hq = . 1= , 1=k .
, , - 0,1 0,6. , k (-, 1,2), - , - , .
, 5,0=h . . ,
hq = )22;5,0( , -
707,022
21
5,05,0
===== hh hdh
dqk .
k, )22;5,0( ,
bkhq += . b 354,0
425,0
22
22 =+=+= bbbkh ,
42
22 += hq . (3)
, (3) , , , . , ]2[ hU ][2 hU :
422]2[ += hhU , ]2[
222][2 hUhhU += .
. , (3) , -
);( 00 qh , . (3) -,
0 1
1 707,0=k
h
q
0.50 1
1
1=k
h
q
2,1=k
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42)(
22
00 ++=+ hhqq . (4) (3) );( 00 qh ,
42
22
00 += hq . (4)
hq =22 . (5)
, - () );( 00 qh . (5) , - .
- . , ( ).
2.5.2. , -
(2) , - , ( ). 2.3, -. , , , . , .
, ( - ) - , . .
, , , , . , , , , Q . Q, h.
, t Q q -. , , tQ , - tq . , S, :
tS
qQh = )( . 0t ,
[ ])()(1)( tqtQSdt
tdh = . , . , )(tq )(th - )()( thtq = .
)()(1)( thS
tQSdt
tdh = . (6)
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: )(tQ ( ) - )(th (). .
() , , -
. , 0)( =dt
tdh (6),
22
0 QhhQ == . (7)
Q h - . Q h.
, , , 0QQ = 0hh = (7), - .
QQQ += 0 hhh += 0 , Q h .
. - ),( yxf ),( 00 yx :
),(),(),(),(),( 000000 yxFyyyxfx
xyxfyxfyxf +
++= ,
x
yxf
),( 00 y
yxf
),( 00 ),( yxf x y ),( 00 yx , ),( yxF (, ..). -
x y , ),( yxF , -,
yy
yxfxx
yxfyxfyxf +
+ ),(),(),(),( 000000 . (8) (8) (6), x -
Q, y h. ,
hSh
SQ
ShSh
SQ
SQ 21,11 =
=
.
(8)
hhS
QS
hS
QS
hS
QS
+0
00 2111 .
QQQ += 0 hhh += 0 (6) , dthd
dthhd =+ )( 0 .
hhS
QS
hS
QSdt
hd +0
00 211 .
, 0Q 0h , 01
00 = hSQS , -
:
Qkhkdt
hdQh + , (9)
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02 hS
kh=
SkQ
1= . , hk 0h , - . .
( ) . ,
)()()( tQkthkdt
tdhQh =+ . (10)
, , - ),( 00 hQ . hk - .
2.6. , . -
, q - ( ).
, , -, . , , . , -
h0 ( ).
, , -- . h - Q ( 3/). , h - , Q . , h -.
, -. q ( 3/) , - 1 .
h qQ S . t , t
StqtQth = )()()( .
:
= t dttqtQSth 0 ))()((1)( .
0=t , - ( 0)0()0( qqQ == ), . ( ). , ,
)()(),()( 00 tqqtqtQqtQ +=+= , )(tQ )(tq . , ,
= t dttqtQSth 0 ))()((1)( .
Q
q h
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)(th , )(tQ )(tq . -, ( - ):
[ ])()(1)( tqtQSdt
tdh = . 1=S 2.
. :
)()()( 0 ththte = . K ( -
, -), [ ])()()()( 0 ththKteKtq == . . , . - . - , - ( ). , , )(tm , .
K.
, , . , q ( ). (. ) 1=K , 5=K .
:
( , q ) - ( 0=h );
, h ; , K ;
, . , K, -.
, , ( ).
t
h
0 5=K
1=K
+
0h hQ
e
q
+
m
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, - (, ), K -. q (- ).
K ( ) - , . - , . - K .
: . , , K, , K, .
, - (-). - :
-? K ( )? ? , ? ( -
q ) ? , ?
, - , - .
t
Q
0
5=K1=K t
h
05=K
1=K
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3.
3.1. -
, .
, , -. )(tu ( ), )(t ( ).
. , . , )(t ( ) , )(t .
, . ( - , , ), , - (). , - .
. -
)(t )(t , dt
tdt )()( = . -, )(t . -
)()()( tMtMdt
tdJ H= , M (t) ( H), MH (t) (, H). J ( 2). , ( , ).
. M (t) ,
)()( tiCtM M = , MC , , (- ); )(ti ( ),
)()()( tiRtetu += , )(te () ( ) R ( ). , :
)()( tCte = , C . = MCk1 = Ck2 , -
)()()( 1 tMtikdttdJ H= , = 2)( kte , dt
tdt )()( = , )()()( tiRtetu += . (11) (11) , .
, (). , -
e u
i
R
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. (11) , )(ti . )(t , :
)()()()( 2122
tMdt
tdktuRk
dttdJ H
=
, , )(t , )()()()( 2212
2
tMtukdt
tdRkk
dttdJ H=+ . (12)
, )(tu )(tM H )(t . (11), ( )(te
)(ti ) . (12) -.
. .
, . , . , , - (, -).
3.2. , , -
, . - , .
, , 0)( =tM H ( ). -, )()( tt &= , (12)
)()()(
)()(
121 tuRJ
ktRJkkt
tt
+==
&
&
:
)(0
)()(
010
)()(
121 tuRJ
ktt
RJkk
tt
+
=
&&
(13)
)(t )(t t . , 0t )(tu 0tt . - )(t , )(t )(tu ( 0tt < ) . )(t )(t - ,
)()(
tt
.
)(tx , (- ) )(tu . (13)
)()()( tuBtxAtx +=& (14)
=)()(
)(tt
tx
,
=RJkkA 210
10
=RJ
kB 10
. (14) )(tu -
)(tx , -.
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- , , )(ty :
)()()()()()(tuDtxCtytuBtxAtx
+=+=&
(15)
--. :
[ ] [ ] )(01)()(
01)()( txtt
tty =
== ,
[ ]01=C 0=D . , [ ]10=C . (15), C D , -
. , .
J , R 1k 2k - , A , B , C D (15) . , , .
(15) - , - .
, (15) . , , )0(x
0=t . , )0(x )(tu 0>t - .
(15) , , - )(tx . , tt 0 , t - , .
tt = [ ] tuBxAxtxxtx ++=+ )0()0()0()0()0()( & , , . )( tx )( tu ,
)()()( tuDtxCty + . , [ ] ttuBtxAtxttxtxtx ++=+ )()()()()()2( & ,
)2()2()2( tuDtxCty + . , () 0>t . , - , t , . -. A , B , C D , ( , ) (15).
3.3. -
. ( ), 0 1 0=t . :
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h(t):
, , , . , .
, -. , , -.
:
)()()( txktydt
tdyT =+ , (16) k , T , ( ). . - (16) 1)( =tx ( 0>t ),
+=TtCkty exp)( 1 ,
1C . - , , 0)0( =y ,
kC =1
==
Ttktyth exp1)()( . (17)
(17) T, :
, T y , - k , (16). - , , .
, , - .
U 1(t) h(t)
0 1 2 3 4 5 6
k
t
y
cT 1=
cT 5,0=
0 t
h(t)
0 t
1
1(t)
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3.4. ( ) , , . ,
. , . )-) , - . , .
, , ? -
, ( ) -. , - )(t . ( ) , , 0=t , , - ( ) :
==
0,00,
)(tt
t ,
= 1)( dtt . , -, (. ).
- - )(t1 . , t , , .
(-) - w(t):
, , , - , , .
- - , .
, /1 .
)]()([1)( = tttx 11 , )( t1 , =t , , (. ).
0 t 0 t 0 t
) ) ) )
0 t
)(t 1
U (t) w(t)
0 t
w(t)
0 t
(t)
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, )(t1 )( t1 , /1 . -, )(t1 )(th ,
)]()([1)( = ththty . 0 , ,
dttdhththtw )()()(lim)(
0==
,
, . , 0 t:
= t dwth0
)()( . (17) , -
:
=
=Tt
Tk
Ttk
dtdtw expexp1)( .
. , )(tx )(ty
==0
)()()()()( dwtxdtwxtyt
.
)(tw )(tx . , , .
, - , , , .
3.5. , -
. - .
, )(tx )(ty :
)()()()()( 010122
2 txadttdxatyb
dttdyb
dttydb +=++ (18)
)1,0( =iai )2,1,0( =ibi .
0 t0 t
)(t1 )( t11
11
)(tx
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dtdp = , )(tx -
dt
tdxtxp )()( = . , )(txp p )(tx , , )(tx .
)(tx )(ty
)()()(),()()(),()()( 222
tpxdt
tdxtxtypdt
tydtytpydt
tdyty ====== &&&& . (18), )()()()()( 0101
22 txatpxatybtpybtypb +=++ . (19)
)(ty (19) )(tx : )()()()( 0101
22 txapatybpbpb +=++ . (20)
(20) , 012
2 bpbpb ++ )(ty , - 01 apa + )(tx . (, ) (20) 01
22 bpbpb ++ ,
)()()()(01
22
01 txpWtxbpbpb
apaty =+++= , (21)
)()( txpW ,
01
22
01)(bpbpb
apapW +++= . (22)
)(tx . , )()()( txpWty = , (18), .
)( pW , (18). - , .
)(W , (22) - p . - () .
)(W , , ; , ; , , . -
, 1
1+ ; 1+
,
( ), 1
12
+++
-
. ,
. , 23
1)( 2 ++=
W 1= 1= 2= .
3.6. 3.6.1. ?
, - . , -
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. , , - -, , () .
)(tf , )}({ tfL :
==0
)()}({)( dtetftfsF stL . (23)
)(sF )(tf (). s - , , (23) 3.
)}({ sF1-L )(tf )(sF :
+
==
j
j
st dsesFj
sFtf
)(
21)}({)( 1-L , (24)
1=j , , 4. (24) ,
. , - -, ate ,
1)}({ =tL , s
t 1)}({ =1L , as
e at += 1}{L . (25)
3.6.2. . -,
(23) (24), , , : )}({)}({)}()({ 2121 tftftftf LLL +=+ , (26) )}({)}({)}()({ 2121 sFsFsFsF
1-1-1- LLL +=+ . (27) -, )(tf
)0()()( fsFsdt
tdf =
L ,
)(sF )(tf , )0(f 5 0=t . , - s . i - - is ( ).
, - - ( 0=t t ), : )(lim)0( sFsf
s= , )(lim)( 0 sFsf s = . (28)
3 , tMetf sRe ) stetf )( t (23) .
4 , - )(tf . -, (24) .
5 0=t , , - t .
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3.6.3. (18):
)()()()()( 010122
2 txadttdxatyb
dttdyb
dttydb +=++ (29)
, , . ,
)(sX )(sY : )()()()()( 0101
22 sXassXasYbssYbsYsb +=++
)(sY )(sX : )()()()( 0101
22 sXasasYbsbsb +=++ .
012
2 bsbsb ++ , )()()()(
012
2
01 sXsWsXbsbsb
asasY =+++= ,
012
2
01)(bsbsb
asasW +++= . (30)
(22) (30) , )(sW , s , p , (22).
, - -.
(30) : - .
3.6.4. -
. - (16):
)()()( txktydt
tdyT =+ (31) )()( ttx 1= . - )(ty , .
-. (30), - )(sX )(sW . (. (25)), (31), -:
ssX 1)( = ,
1)( += Ts
ksW .
11
1)( +=+= TskT
sk
Tsk
ssY .
:
Tsk
sksY
/1)( += .
(27), :
+
=
Tsk
skty
/111)( 11 LL .
(25):
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=Ttkkty exp)( 0>t ,
(17). - .
(28) - )(ty :
)(lim)0( sYsys
= , )(lim)( 0 sYsy s = .
ssX 1)( =
)(lim)0( sWys = , )0()( Wy = .
,
01
lim)0( =+= Tsky
s, kWy == )0()( .
)0(W , -, .
3.7. ,
)()()()()()(tuDtxCtytuBtxAtx
+=+=&
, )(tu , )(ty )(tx , - . ( - ),
)()()(
)()()(sUDsXCsY
sUBsXAsXs+=+=
(32)
, )(sX , : )()()( sUBsXAIs = ,
I , , - . 1)( AIs , )(sX :
)()()( 1 sUBAIssX = (32) [ ] )()()()()()( 11 sUDBAIsCsUDsUBAIsCsY +=+= . , :
DBAIsCsUsYsW +== 1)()()()( . (33)
, , . , - . .
012
23
012
2)(bsbsbs
asasadsW ++++++= ,
d, )2,1,0( =iai )2,1,0( =ibi ,
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[ ] .,,100
,100010
210
210
dDaaaCBbbb
A ==
=
= (34)
( ), -. A , , . B , . , - .
, , ( ) .
,
[ ] 0,25,01,01
,04
5,03 ==
=
= DCBA . (33),
[ ]23
101
045,03
00
25,01)()( 2
1
+++=
=+=
sss
ss
DBAsICsW .
. (34)
[ ] 0~,11~,10~,
3210~ ==
=
= DCBA . , , (33), . , . A, B, C D, -
DDCPCPBBPAPA ==== ~,~,~,~ 11 , P ( ). ( !). )(' tx , )()(' txPtx = . ,
=0125.00
P .
)(sW , , - 1+s , . ,
21)( += ssW .
: 0,1,1,2 ==== DCBA . (35) ( , ) - . ? , - , - . - .
( ) , . - , - . (. . 6.4).
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3.8. (, ), :
ttx sin)( = , (36) ( ). , - ( t ) 6, A :
))(sin()()( += tAty . - . , t (, t -). t , ( ).
0> ( ), - , , .
)(sW ,
)()( jWA = , )(Re)(Imarctg)(arg)(
jWjWjW == .
)( jW , )(sW js = , 1=j . jQPjW +=)( -
, 22)( QPjW += PQjW arctg)(arg = .
)( jW , - . )(P )(Q ( )( jW ) -.
)(A )( ( ) - ( ). . - 1)( >A , ,
1)(
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, - , 2/1 .
. - , (36) )(ty . , - .
,
. . - , . )(tr - )(tx )(ty , )(A ( )(tx )(ty ) )( .
3.9. . 60- , -
, , - , - . .
)(A (): , - ( lg ), )(lg20)( ALm = , (). () lg .
, 10 ( ). - () .
:
yx
r
)(A
0
)(A
0
)(A
0
)(A
0
1 2 1 2
yx
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1) )()( 21 sWsW :
)(lg20)(lg20)(lg20 21 AAA += ; (37) )()()( 211 += ; (38)
2) , 20 / ( ), 40 / ..
, - . C , - .
10-1
100
101-40
-20
0
L m( )
10-1
100
101-90
-45
0
()
( ) ( )
11)( += TssW 1=T .
, , -, , - ,
01)0( =W . 0)0( =W , ks ( 0>k ), -
k . 20k /. =)0(W , ,
ks . 20 k /. -
. m , n , )(20 nm /. 110 == nm . , ,
20 /, , 20 ( !).
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4. , -
( ). , . ,
)()...()()( 21 sWsWsWsW N= , , .
4.1. , ,
0)0( = kW , . , - ( ).
() . - ksW =)( . , , - , . 1(t) ( - )(t ) , k ,
)0()( >= tkth )()( tktw = . , k
, :
kA =)( , 0)( = . 4.2. ,
)()()( txktydt
tdyT =+ (39)
1
)( += TsksW . k , 0>T
, . , , .
. 3.3 3.4
=Ttkth exp1)( ,
=
Tt
Tktw exp)( .
:
-
.. , 2008
35
, k , - 0=t Tt = . ( 5%) T3 . -, .
111)1(
1)( 222222 ++=+
=+=
TjkT
Tk
TTjk
TjkjW .
)( jW . 0 , ( ). , - )0;5,0( k k5,0 . ( ) )0;(k ( ).
,
Tc1= . (
), kLm lg20 . 20 /, -
. 0 90 , c 45 .
, - , .
,
0
Re
Im
k
0=
L m( )
-90
-45
0
( )
Tc1=
20 /klg20
0 t
)(twk
T0 t
)(th
k
T
-
.. , 2008
36
)()()( txktydt
tdyT = (40) , (39) ( ). - :
= 1exp)(Ttkth ,
=
Tt
Tktw exp)( .
, 0>T , - t . : , .
.
, - - , - . - , , - . , -.
- - , - -, , -
. - ( ). , 1T , 2T 3T
)1)(1(1)(32
1
+++=
sTsTsTsW
-,
)1)(1(1)(32
11 +
+=sTsT
sTsW , )1)(1(
1)(32
11 ++
=sTsT
sTsW , )1)(1(
1)(32
13
+=sTsT
sTsW
-.
4.3.
1)(
12
2 ++=
sbsbksW ,
- ( , 04 22
1
-
.. , 2008
37
12
)( 22 ++= sTsTksW (41)
k , T ( ), ( 10
-
.. , 2008
38
4.4. , .
, . , () .
)()( txkdt
tdy = , (42)
sksW =)( . (42)
+= t dxkyty0
)()0()( . ( 1)( =tx 0t ) - ( 0)0( =y ), :
tkth =)( . , , , - , 0=t , 1. ktw =)( ( 0t ).
kj
jkjW ==)( . , -
20 /. , 0= . , , - .
1= klg20 , k= , - 1)( =jW . = 90)( .
L m( )
-180
-90
0
( )
20 /klg20
1=0
k=
0 t
)(tw
k
0 t
)(th
k=tg
-
.. , 2008
39
4.5. .
dt
tdxkty )()( = , )()( txpkty = , sksW =)( .
, )(t1 0=t - )(t .
)()( tkth = , dt
tdktw )()( = . , - , , . - .
20 /, 0)( =mL k1= .
1= kLm lg20)1( = . (- ) , .
,
o90 . , ttx sin)( = )90sin(cos)( +== ttty .
, - , . , - , . .
. - , . -
dttdxkty
dttdyT )()()( =+
1
)( += TskssW .
. :
, .
L m( )
0
90
180
()
klg20
1=0
k/1=
20 /
-
.. , 2008
40
, ( - Tc /1= ) , . , - 90 .
4.6. , .
, .
, - , vL /= , L ( ), v ( /). , - ( ).
- . , .
, . -
)()( = txty . - - :
)()()()}({)(00
sXedtetxedtetxtysY sstsst
==== L ,
sesW =)( .
L
L m()
0
45
90
()
T/1=
20 /
t
x
t
y
-
.. , 2008
41
, , . jejW =)( . :
1)()( == jWjA , == )(arg)( jWj . , - , , .
4.7.
)(
1)(~sW
sW = )(sW ( ). , . .
)()()( jQPjW += , )(P )(Q - .
)()()( 22 QPA += , )()(arctg)(
PQ= .
)()()()(
)()(1
)(1)(~ 22
QP
jQPjQPjW
jW +=+== ,
)(1
)()(1)(~
22 AQPA =+= , )()()(arctg)(~
==PQ .
,
)(lg20)(
1lg20)(~lg20 AAA == , )()(~ = .
, - .
, , 1)( += TssW . - , , .
sesW =)(~ , 1 , -
=)( . , -
L m( )
0
45
90
( )
Tc1=
20 /
0
-
.. , 2008
42
, . . , , . , . , , ( !), - .
4.8.
. - , . .
)1)(1()1()(
31
2
012
2
01
++=++
+=sTsT
sTkbsbsb
asasW .
)3,...1( =iTi . 321 TTT >> .
1
1)1(1
1)(3
21 +
+= sTsTsTksW . (43) , , (
12 sT ). (37)
(38), - )(sW (43) - -.
, -
1
11Tc
= , , klg20 .
1
11Tc
= 20 /,
22
1Tc
= 12 sT . 3
31Tc
= () ,
20 / . -
. , : , .
L m( )
11
1Tc
=
20 /klg20
22
1Tc
= 3
31Tc
=
20 /
-
.. , 2008
43
5.
5.1. , (, ,
, ). , , - . , . - .
, :
, p , ;
, - s, - .
. , - , -, (t s), s, .
, . , , - . , .
. x , y . e , u - , ( ). g ( ), m .
( ) ( )(0 sR ).
+
xC(s) P(s)
H(s)
y u
R0(s)
e
g
m
321 xxx ++ 1x
3x
2x
21 xx 1x
2x
xx
x
)(sW )(sY )(sX
)( pW )(ty )(tx
-
.. , 2008
44
5.2. ()
. - , - . .
, - :
, [ ] )()()()()()()()()()( 212121 sXsWsWsXsWsXsWsYsYsY +=+=+= ,
)()()()()()( 2112 sXsWsWsYsWsY == .
, )()()( 1 sEsWsY = ,
)()()()()()( 2 sYsWsXsFsXsE == . [ ])()()()()( 21 sYsWsXsWsY = . )(sX ,
[ ] )()()()(1)( 121 sXsWsWsWsY =+ )()()(1)()(
21
1 sXsWsW
sWsY += . ( x f ), - :
)()(1)()(
21
1
sWsWsWsW = .
, . - , :
, -. :
x W(s)
f y x
W(s)
f
y W(s)
x W1(s)
W2(s)
e y
)()(1)(
21
1
sWsWsW
+y x
f
x W1(s)
W2(s)
1y
2y
yW1(s)+W2(s)
x W1(s)
1y W1(s)
y
y x
W1(s)W2(s) y x
-
.. , 2008
45
, :
:
5.3. -
. (x, g m), y, u e. , 9 - , -.
( -
), :
)(1)()(
0
0
sRsRsR += .
:
x . -
. , )(sC , )(sR )(sP :
+
xC(s) P(s)
H(s)
y uR (s)
e g
m
+
xC(s) P(s)
H(s)
y u R0(s)
e
g
m
x W(s)
f y x
1/W(s)
f
y W(s)
xW(s)
1y xW(s)
2y
1y
1/W(s)2y
x W(s)
1y x W(s)
2y
1y
W(s) 2y
-
.. , 2008
46
- :
)()()()(1)()()()(
sPsRsCsHsPsRsCsW += .
u e, :
)(sWu , - )(sWe ( , , ). , :
)()()()(1)()(
sPsRsCsHsCsWu += , )()()()(1
1)(sPsRsCsH
sWe += . , . - .
+
x1
H(s)C(s)R(s)P(s)
ee+
x C(s)
H(s)R(s)P(s)
ue
+
x C(s)R(s)P(s)
H(s)
ye
-
.. , 2008
47
6.
6.1. ? , , .
. - - ().
:
, ( ) ;
, - ( );
- ;
, - , .
6.2. ,
)(sW , )(sX . )()()( sXsWsY = . -
, )(sW )(sX , -
)()()(
ssnsW W= , )(
)()(sdsnsX
X
X= . , )(s )(sd X ,
))...()(()( 21 Nssss = , ))...()(()( 21 MX ssssd = , 7. ),...1( Nii = ),...1( Mjj = )(sW )(sX .
)()()( sXsWsY =
M
M
N
N
sb
sb
sb
sa
sa
sasY +++++++= ......)( 2
2
1
1
2
2
1
1 .
),...1( Niai = ),...1( Mjbj = , :
.))(()(,))(()(ji s
jjsiissXsWbssXsWa == ==
, )()( sXsW . ia jb .
)(ty , )(sY . (., , (25))
7 (- ) .
-
.. , 2008
48
tMttt
Ntt ebebebeaeaeaty N M2121 ......)( 2121
+++++++= . (44) , te t , 0 . (44) :
, ;
, ( )(ty ), ),...1( Nii = ),...1( Mii = ( );
)(sW )(sX , )(ty (-) ;
)(sW )(sX ( - ), .
, )()( sXsW , - ia / jb (44). , , , - )(sW )(sX ( )()( sXsW ).
(44), ( ),...1( Nii = ) - )(s . - , -, )()( sXsW . , -, . )(s - , - ( ) ( . . 6.4).
6.3. .
, ,
-
.. , 2008
49
)(sC )(sP , )(sX -
)()()(
sdsnsC
C
C= , )()()(
sdsnsP = ,
)()()(
sdsnsX
X
X= .
)()()(
)()(11)(
ssdsd
sPsCsW Ce =+= ,
)()()()()( snsnsdsds CC += . ,
ssX /1)( = . . 6.2, - )(sWe ( )(s ) )(sX . )(sWe , ( . . 6.4). - )(sWe .
sbsY
ssPsCsXsWe +=+= )(
1)()(1
1)()( 0 .
)(0 sY , - b :
)0()0()0(
)0()0(11
=+=dd
PCb C .
. 6.2, - bte = )(limt .
, , , - 0)0( =Cd ( ) 0)0( =d ( ).
, - ( 0=s js = ). , - t , -
)()(
)()()()()(
sdsn
ssdsdsXsW
X
XCe = .
, , , )()( sdsdC , , .
. , , -
( s/1 ). )()( sdsdC s , )(sX 0=s )()( sXsWe . , )()( sPsC -
s , ( ). .
+
x C(s) P(s)
y u
e
-
.. , 2008
50
- ( )(sX 0=s ). - . ,
)(1)()( sGs
sPsC = , 0> )(sG 0=s , - - .
11
2210 ...)(
++++= txtxtxxtx
)1,...0( = ixi . , , -
. , - , ( ) . , - , . , - ( -). , - , )(te
=0
0)( dtte .
? : , , . ( - , ).
, , - . .
6.4. 6.4.1. ?
. , , , . , - . . - , , - 1986 .
, , , -, . , , - . - , - .
-
.. , 2008
51
, , - , .
, -, .
- . , , - .
, - . , , ; , , -
; ( -
) ( ).
6.4.2. , -
. , -.
, , ( ). , - ( ) . , - , . ( ), ( ). , -, , ( ) - .
, , . , .
6.4.3. - ,
, , . , , - ( ). , , , .
, . - . ( !) ( ), -.
6.4.4. -,
, . , ,
, )(ty , . .
( t ) , . ( ), , -.
-
.. , 2008
52
, , -, , . , , - ( ).
6.4.5. , , ,
. - )(tx ,
),()( txfdt
tdx = (45) , - t )(tx , . )(tx , )(1 tx )(2 tx , (45)
=
=
),()(
),()(
22
11
txfdt
tdx
txfdt
tdx
),(1 txf ),(2 txf . . -
, -, . ,
0),( * =txf , *x 8. , )0(0 xx = ( -
), . ( )(tx ) (45) .
, , )(tx , *x , t *x .
, , , , , )(tx *x t . - .
) , - . - . , , - , , , , .
, , , ( ), . , - .
) , : .
8 , 0* =x . , )(tx *)()(~ xtxtx = , )(~ tx .
) )
-
.. , 2008
53
.. 9, -.
, )(tx . *x , *x , , *x , , )( :
, )(tx -
-
.. , 2008
54
6.4.6.
, :
( ) ( ) - ( );
, : ( ), ;
- , , ;
-, ( , ) . , , -
, . )(sW . , ( ) ),...,1( Nii = ( -):
))...()(()(
)()()(
21 N
WW
ssssn
ssnsW == ,
)(snw )(s . , : tN
tt Neaeaeaty +++= ...)( 21 21 , (47) ),...,1( Niai = , . , )(ty ,
),...,1( Nii = . - .
- )(s )(sW , - .
( 10) , ( ), . ,
10 )(s , , , j+ j . - . 1=j .
1x
2x
1x
2x
1x
2x
) ) )
Im
Re
-
.. , 2008
55
. , )(s (, 01 = ), -
, , . , - . , 101 == ee t t ,
tN
t Neaeaaty +++= ...)( 221 . , , , 1a . , , - .
, : j=1 j=2 . , -
. (47) tjea 1 tjea 2 , -
( ) )sin(cos11 tjtaea
tj += , )sin(cos22 tjtaea tj = . ( , ), ( ). -, 1a 2a -, , jcba +=1 ,
jcba =2 . tctbeaea tjtj sin2cos221 =+
.
6.4.7. ,
, . , ,
, . , -:
)()()( tButAxtx +=& , )(tx , )(tu , A B . ( ), )()( tAxtx =& . (48) , A .
, A
=2
1
00
A .
(48) ( ):
)()()()(
22
11
txtxtxtx
==
&&
1 2 . , . , ( ), .
A , , 0)det( = AI , I , det . )det( AI . , A
))((0
0det
00
1001
det)det( 212
1
2
1
=
=
= AI
-
.. , 2008
56
, 1 2 . ( -
, ), . ( ), -. - , 11.
, , - , . , ,
[ ] )(10)()(
01
)(10
01)(
txty
tutxtx
=
+
=
= 1001
A 1 1 , -, .
(. 3.7):
[ ]1
110
1001
1001
10)(1
+=
=
sssW .
( ) 1)( += ss , - 1 , ! , - 2, 1. , , .
, . , - , , .
6.4.8. -
. , )(s :
1) , ;
2) , ;
3) , , .
, -.
11 ( ) , .
-
.. , 2008
57
6.5. , -
. (- , ), , . , ( ) -.
- . , - , ( ) - , , .
6.5.1. ,
nnnn asasasas ++++= 1110 ...)( ,
. , ),...,0( niai = - , , . - . 2>n , -- . () .
nH - nn , )(s : ,...,, 531 aaa ( ), -
; ,...,, 420 aaa ( ); 1 -
, .. , ( 5=n )
=
531
420
531
420
531
5
0000000000
aaaaaaaaa
aaaaaa
H ( 00 >a )
. )(s - , n nH ( ) -.
, , - . 1n -. , 5=n
011 >= aD , 020
312 >= aa
aaD , 0
0 31420
531
3 >=aaaaaaaa
D , 0
00
00
420
531
420
531
4 >=aaaaaa
aaaaaa
D .
-
.. , 2008
58
5H , 4555 det DaHD == . 05 >a , 04 >D 05 >D .
, . . , 2=n . 3=n - 32
21
30)( asasasas +++= ,
=
31
20
31
3
000
aaaaaa
H ( 00 >a ).
,
0302120
312 >== aaaaaa
aaD . (49)
, :
)1)(1(1)(
21 ++=
sTsTsP ,
sKsC =)( .
, K ( ) .
)()()(1)()()(
sK
sPsCsPsCsW =+= ,
KssTTsTTKssTsTs ++++=+++= 22132121 )()1)(1()( .
0>K . (49),
212121
11TT
KTKTTT ++ .
, 21
110TT
K +
-
.. , 2008
59
6.5.2. ,
. )(sL - , )( jL .
, - , = KL )0( , K .
)( jL , - . 0 - , )0;(K - ( )(sL , ). , - , )( jL )0;1( . ( ), - ( ).
, -
)0;1( . 1)( =A = 180)( . ,
, (), .
c , 1)( =cA , .
, 180 ; )0;1( .
Re
Im
K1
==
180)(1)(
A
Re
Im
K1Re
Im
K1
+
x L(s)
ye
-
.. , 2008
60
)(sL 0=s ( -
), . - , . - , - . )(sL k 0=s , k 90 . , )(sL 1 2 0=s . - )0;1( .
, -
( ). , .
, -
. , - ( )0;1( ). , - . , , -.
)(sL ( ), , )0;1( . , - -.
Re
Im
K1
Re
Im
K1
c
Re
Im
1Re
Im
1
-
.. , 2008
61
, , , - 2/l , l )(sL . )0;1( - .
1=l .
)0;1( . -
( ) )0;1( , 2/2/1 l= - .
( ), ( -). 2/2/1 l= .
6.5.3. .
, - . , - c , 1)( =cA 0)(lg20)( == ccm AL . , 180 . - . 1 , ( ), 2 , 3 c .
,
= 180)( . - , . - = 180)( ( ), . - 2/l , l )(sL .
mL
c
180
0
0
90
1
2
3
Re
Im
1K
21
1+Re
Im
1 K
21+
-
.. , 2008
62
6.6.
, (). - ( ).
, -
( t ). , - 2 y . -, tt > , . , 2% 5%. -, T - Tt 3= ( 5%).
, - maxy
12 y :
%100max =
y
yy . ( , ). , - .
, - )(sW , , - .
22 )1()/1(
)1(1)( +
+=++=
sasa
sassW ,
a , . as /1= . , - ( 0>a ) ( -), (
0
-
.. , 2008
63
, a . 0>a ( ) , , a. a -. , , .
6.7. ( -
), ( ). - . , - , , , -, . , , - , , - .
. - mg , , - . .
gm A
g 1lg20= , 1
-
.. , 2008
64
:
, , - 180 .
, ( ) , . - 0);1( .
M.
:
M ,
,
. 12 = M
MR ,
0;122
MM . -
M.
)(A
0A maxA
0
max
AAM =
Re
Im
K1
mL
180
0
0
90
cmg
m
g
-
.. , 2008
65
1=M ( ) )0;5,0( . M .
6.8. , -
)(s . , )(s , . ( - ).
. ( 1 4) - (2 3). 1, .
,
, 3=t .
2, 3 4 . , , ,
, . , , -
. - j - ( ):
= .
, , , 1 - .
, . - (
Re
Im
1
2
3
4
Re
Im
K1
0,1=M1,1=M
5,1=M
3=M
-
.. , 2008
66
min ) max . .
6.9. 6.9.1. ?
() ( ) . - . , -. (). .
, : -
; -
;
(, ). , , -
(). .
6.9.2. , ,
, :
1)()(
20
10
+++=
sTksP
, 0k 0T , 1 2 .
, - KsC =)( .
)(1)()( 1020 ++++= kKsTs .
1 2 . , , 20 +T )(1 10 ++ kK , ( ).
, 00 >k 00 >T , 1 2 0k 0T -
Re
Im
min
tgmax =
-
.. , 2008
67
. , 020 >+T 2 . ,
0)(1 10 >++ kK 10
1+
>k
K .
- 1 ,
max10min
1+=>
kKK .
, -, minKK > , - , 1 2 .
, -
nn
nn sasasaas ++++= 110)( K ,
naaa ,,, 10 K , ),,1( niua iii Kl =
-
.. , 2008
68
(50) , , - , .
, , )( jm -. , (50), , )(0 jW , -.
-
.. , 2008
69
7.
7.1. , :
, -
g . , . , ( )(sWg , g y ) - ( )(sW , x y )
)()()(1)()(
sPsRsCsPsWg += , )()()(1
)()()()(sPsRsC
sPsRsCsW += . )(sC , - .
, [ ] )()(1)( sPsWsWg = . , , . , -.
, , , . ( )(tx )(te )
)()()(11)(
sPsRsCsWe += .
, , , . , , - . )(sC ( - ), )(sC . , . - .
. -, - . -, , , . - .
. , - )(tx , - )( jW 1 ( )()( txty ). , 0)( jW , - . , , - , 0)( jWg .
+
xC(s) P(s)
y u
R (s)
e
g
-
.. , 2008
70
, : 1) 1)( jW ,
; 0)( jWg , , ;
2) 0)( jW , - ; )()( jPjWg , - , .
7.2. - , -
. -, .
- KsC =)( . )(te , - K . - , - .
, - IK ,
sKKsC I+=)( , += tI dtteKtKetu
0
)()()( .
- -. - , , - . -.
- DK :
sKs
KKsC DI ++=)( , dttdeKdtteKtKetu D
t
I)()()()(
0
++= . - (--). - . , - (-), .
. - . - , , . - , , . , - , - :
1)( +++= sT
sKs
KKsCD
DI ,
DT . DT , - , .
, - . -, , - -.
-
.. , 2008
71
( ), -:
1)(
)()(
01
012
2
+++=+++=
sTsK
sKKsC
bsbsasasasC
D
DI .
7.3. -
, , , - (. . 6.8). , . .
012
01
)()()(
nsdsnsn
sdsnsP ++
+== .
01
01
)()()(
bsbasa
sdsnsC
C
C
++== ,
010 ,, baa 1b , . -
0000011001102
011113
1
01012
0101
)()(
))(())(()()()()()(
bdansbdbdanansbbdansbbsbdsdsasansnsdsdsnsns CC
+++++++++=++++++=+=
.
, , )(s ,
012
23)( +++= ssss ,
)2,...,0( =ii . s ,
000000
1011001101
2011112
13
:
:
:
1:
=+=+++
=++=
bdansbdbdanans
bbdansbs
=
0
1
2
0
1
0
1
00
1010
11
1
00
100100
bbaa
dnddnn
dn.
=
0
1
2
1
00
1010
11
0
1
0
1 1
00
100100
dnddnn
dn
bbaa
.
, ( ) . , , )(sn )(sd ,
-
.. , 2008
72
)(sP . )(s .
, , , , 1N , N :
)}(deg),(max{deg sdsnN = , deg . )(s .
, -, .
7.4.
, (). :
1) - (, ) ;
2) ( - ), ; . )()()( sRsPsG = ,
, ( , -). )(lg20)(0 jGL = . - )(L , - : )()()( 0 LLL C = . (51) , :
1) )( jL , ?
2) )(sC (51)? , :
; ; ( ) ; ; ( ).
. (-
), . - - , - )()( sCsG ( ).
, , . . , - () , - ():
-
.. , 2008
73
, 20 /, Tc /1= . . , : 3T. , . , . - :
(, ) - , 0=mL ; - ;
, 20 /;
t ,
c t3= .
. , - . , . - , .
. , -. . - 20 /, -, - .
20 /. - ( 2,1
-
.. , 2008
74
40 / .
. - , - )(sG .
( ) 20 / , - . c :
2/3 == c t /. , - -
s
c ,
clg20)( =L ( ).
20 -40 / 1 , 15)( =mL .
24,11262,5101015lg20 4/3120/15
11
===== ccc
/.
, , . ( , ), -, .
. ( 0/1 T
-
.. , 2008
75
1). , ,
- .
7.5. ,
)(2 sC :
)()()(1)()()()()( 2
sPsRsCsPsRsCsCsW += .
)(2 sC ( , , ), . , , - )(sC , , )(sW )(2 sC . , ( - ).
, )(tx )(ty , , 1)( sW . ,
)(1
)()()()()()(1)(
12 sWsPsRsC
sPsRsCsC =+= , (52)
)()()(1
)()()()(1 sPsRsCsPsRsCsW += -
. (52) , )(2 sC ()
)(1 sW . )(1 jW -, , )(2 sC -
. , 1
1)(1 += TssW 1)(2 += TssC , - . , (52) . (52) , .
, , -, , .
7.6. g - ,
( ):
+
x C(s) P(s)
yu
R (s)
e
g
C2(s)
-
.. , 2008
76
[ ]
)()()(1)()()(1)( 3
sPsRsCsPsRsCsWg +
= . g ,
)(
1)(3 sRsC = , (53)
0)( =sWg . (), - g . , ( ), (52).
, , )(3 sC , - , .
, )(3 sC (53) , . , , . - )(3 sC , , (53) .
7.7. , .
, . - . , , , -, .
:
)()(1
)()()(sPsC
sPsCsW += . (54) , . -,
)()()( sPsQsW = , )()(1
)()(sPsC
sCsQ += (55)
+
x C(s) P(s)
y u
e
+
x C(s) P(s)
y1u
R (s)
e
g
C2(s)
C3(s)
3uu
-
.. , 2008
77
(55) )(sP )(sQ , )(sQ . : )(sQ , - , (55):
)()(1
)()(sPsQ
sQsC = . (56) , )(sQ , )(sW (55) . , )(sP , , (56), . , (56) - . (56) - , - (D.C. Youla).
(56) )(sQ , -. (56) . , - )(sC ( -). )(sQ .
)(/1)( sPsQ = , 1)( =sW , . ,
( ), )(sQ . ,
. ,
)(sQ - . - (56).
, ,
1
1)( = ssP . 1)( =sQ , (56)
21)(
=sssC .
1
121)()(
=ss
ssPsC -
() . 2)1()1)(2(1)( =+= sssss
, . , (56) .
, .
)()()(
sdsnsP = , )(sn )(sd . )(sf ,
)(sn )(sd . )(sP -
)()()(
sVsUsP = ,
)()()(
sfsnsU =
)()()(
sfsdsV = .
, )(sX )(sY , - 1)()()()( =+ sYsVsXsU . (57)
-
.. , 2008
78
)()()()()()()(
sQsUsYsQsVsXsC
+= , (58) )(sQ . (58) - ( ) -, . (58) (54), , (57), [ ] )()()()()( sUsQsVsXsW += .
)(sQ , - ( , , ) , , (58).
11)( = ssP ,
)()()(
sVsUsP = ,
11)(,
11)( +
=+= sssV
ssU .
(57) , ,
13)(,
14)( +
+=+= sssY
ssX .
1)( =sQ (58)
23)( +
+=sssC .
)()( sPsC , .
-
.. , 2008
79
,
. , , , . , - , , (. ), .
, . , , - , -.
, . - , . , - .
, .
-
.. , 2008
80
( )
1. .. . .: , 1989.
2. .. . . .: , 2005.
3. .. .: , 1986. 4. .., .. 4- . .:
, 2003. 5. ., . .: ,
, 2004. 6. .., .., .. . .: , -
, 2004.