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. 3
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>
> ! (t) = ; Jy ; Jx ! ! + Mz (t)
: _z
Jz x y Jz
Ji Mi !i (i = x y z) { -
,
(x y z):
Jy = Jz = J -
Mi (t) 0 (i = x y z).
.
66
2. -

, ( -
- ) -
,
" "{ -
. -
. -
, .
, .
-
.
2.1.
A (1.3)
si i = 1 2 : : : n : .
2.1.1.
, . . si 6= sj i 6= j i j = 1 2 : : : n
si {
, , : Imsi = 0: 1

nn -
A = diagfs1 s2 : : : sn g 53, 115] , ,
2s 03
0 0 ::: 0
1
60 07
s2 0 : : : 0
6
07
60 7
0 s3 : : : 0
A = 6 .. ... 7 : (2.1)
...
6. 7
6 7
40 05
0 0 : : : sn;1
0 0 0 ::: 0 sn
fsig
, -
(2.1). -
,
1
, -
(2.1) (2.4).
-
.
. 2.1.3. ( ., , 53, 66, 115]).

67
sIn ; A, -
, sIn ; A = diagfs; sig: -
,
-
Qn
i=1 (s ; si)
53]. , A(s) =
.
, -
n -
u(t) 2R (m =
. ,
1) " -
", . . -
x:
8
> x1(t) = s1x1 (t) + b1u(t)
>_
> x (t) = s x (t) + b u(t)
< _2 22 2
(2.2)
...
>
>
>
: xn (t) = sn xn (t) + bn u(t):
_
i 6= j):
, xi (t) xj (t) ( -
, { -
(n- ) n
( ) .
.
, -
A .
l = m = 1 T{ ,
B = b1 b2 : : : bn C = c1 c2 : : : cn : (2.2) -
, xi -
b
Wi (s) = s ;i s i = 1 2 : : : n :
i Pn
y(t) = Cx(t) i=1 cixi (t) ,

n
X Ki
W(s) = Ki = ci bi:
i=1 s ; si
, A -
, -
(
).

68
2.1.2.
A-
. , -
2

, A -
(2.1), -
. .
-
( ) 53, 115].
-
si i+1 = i i | -
( )
Ai = ; i i : (2.3)
i i
Ai(s) = (s ;
+ i = s ;2 i s+ i + i :
i) si i+1 = i
2 2 2 2 2

|i . A
(
):
2 3
s1 0 0 0 ::: 0
6 0 s2 0 0 7
::: 0
6. 7
...
...
6 .. 7
:::
6 7
6 0 ::: 0 s 7
0 ::: 0
6 7
q
60 7:
::: 0 ::: 0
A=6 (2.4)
7
1 1
60 7
::: 0 ; 1 ::: 0
6 7
1
6 .. 7
...
... ...
6. 7
6 7
4 5
0 ::: 0 r r
0 ;r
0 ::: r

s1 : : : sq -
11
sq+2i;1 q+2i = |i i = 1 2 ::: r
i
22 (2.3).
-
2
,
, si i+1 = i | i (|2 = ;1)
- i = Resi i+1 i=
jImsi i+1j:


69
(2.4),
. 2.1.1. . 67,
q r
Y Y
det(sIn ; A) = (s ; si) (s2 ; 2 j s + + j2 ):
2
j
i=1 j=1
, A -
si .
x , " -
" q+r
. m=l=1

q+r
X
W(s) = Wi (s) (2.5)
i=1
8
Ki
>
> i = 1 ::: q
< s ; si
Wi (s) = > d0j s + dj
s2 ; 2 j s + 2 + j2 j = i ; q i = q + 1 : : : q + r:
>
:
j
, -
-
, .
2.1. 3

A (2.1)
(2.4)
. -
, A
. -
A
(. 1 . 67).
( ) -
3
( ) -
. ,
76, 95, 66],
-
76]. ,
-
.



70
. 2.1. ,
(2.4).

2.1.3. .
A n -
: s1 { l1 P 2 {
s l2 : : : sp {
p l = n:
lp : -
i=1 i
-
- -
(2.1), (2.4). -
A,
53, 66, 115]. A
: 4

2J 03
0 :::
1
60 J2 : : : 07
A = 6 .. 7 (2.6)
... 5
: : : ...
4.
0 ::: 0 Jr
Ji i = 1 2 : : : r { , :
( )
A
,
4 -
.




71
{ (Imsj = 0)
2s 1 0 0 03
0 :::
j
6 0 sj 1 0 ::: 0 07
60 0 s 0 07
1 :::
6 7
6 0 0 0j 0 07
sj :::
Ji = 6 . 7 (2.7)
6 . . . ... 7
...
6 .. : : : 7
6 7
40 sj 1 5
::: 0
0 ::: 0 0 sj
{ sj = j | j
2 03
100 :::
j j
6; j j 0 1 0 ::: 07
60 0 0 ::: 0 7
10
6 7
6 0 0 ;j j
0 ::: 0 7:
j01
Ji = 6 . 7 (2.8)
6 ... 7
j
... ... ...
6 .. 7
6 7
40 5
::: 0 j j
0 ;j j
0 :::
- A (2.6) -
( ) .
( . 53, 115]) .
. -
,
.2
Ji (2.7) 11
Ji
lj lj (2.8) { 2 2 2lj 2lj ( j {
sj ). , -
,
: Ji = si ,
: Ji = ; i i
{ :
i i
, . 2.1.2. . 69, (2.4)
(2.6) .
, -
.
si -
. , , . 1.8. . 60,
A1 = 0 1 A2 = 0 0 -
00 00
s1 2 = 0:
72
, A1
22 A2
J1 = J2 = 0 1 1: ,
-
, .. -
. ,
-
53, 115].
-
, -
(2.6), .
W(s) ,
(2.5),
8
Bi(s)
> ;
>
>
> (s ; si)li
>
>
>
>
<
Wi (s) = > (2.9)
> D (s)
>
> (s ; 2 s i+ 2 + 2 )li ;
>2
>
>
> i i i
:

li ;1 2lj ;1
Bi (s) Dj (s)
.
-
-
53, 115]:
sIn ; A
{
sIn ;A
{
{ -
.
-
. -
53, 66, 115].




73
2.2.
{ -
( ) 3], -
-
" ",
47, 102]
1, 174]. 5
A
2 3
0 1 0 ::: 0
6 7
0 0 1 ::: 0
6 7
... ...
A=6 7 (2.10)
::: :::
6 7
4 5
0 0 0 ::: 0 1
;an ;an;1 ;an;2 : : : ;a2 ;a1
a1 a2 : : : an { .6
. ,
A(s) = sn + a1sn;1 + a2sn;2 + + an;1 s + an: ,-
-
A: -
, . 7

( . 53, 115] . 3.2.1. .
84). , -
.
B
.
u(t)2R . . m = 1: 8
B n1
- .
T
B= 0 0 1] : (2.11)
-
5
.
, . . 3.2. . 84.
6

A = 0 In;1 :
7
;aT
8
A
.
, (2.10), . 1, 3, 174].
.

74
, -
8
> x1 (t)
>_ = x2 (t)
> x (t)
> _2 = x3 (t)
>
<
... (2.12)
>
>
> xn;1 (t) = xn (t)
_
>
>
: xn (t) = ;an x1 (t) ; an;1 x2 (t) ; ; a1 xn (t) + u(t)
_
8
> y1 (t) = c1 1x1 (t) + c1 2x2 (t) + : : : c1 nxn (t)
<
...
>
: yl (t) = cl 1 x1 (t) + cl 2 x2 (t) + : : : cl n xn (t)
ci j l n- C -
. ,
(2.12)
.9
n- , ..
66]. -
,
(2.12), . 2.2.




. 2.2. (2.12) ( ).
(2.12),
, l = 1 C = c1 c2 : : : cn : -
xj
,
9
yi (t) ,
i- C ci 2 .

75
(1.25) -

cn sn;1 + cn;1sn;2 + + c2s + c1 = B(s) : (2.13)
W(s) = sn + a sn;1 + a s + + an;1s + an A(s)
2 n;2
1

, -
A(s) B(s)
. -
-
A i- :
-
, . 3, 1, 174].
,
(2.10), (2.11). -
, . . 3.2.
7.2.
2.3.
-
( ), " -
-
".
, y(t) 2 R l = 1 ( . . SISO- MISO- ).
A , -
(2.10), B ,
1 n-
C = 1 0 ::: 0 0 : (2.14)

8
>_
> x1 (t) = x2 (t) + b1 1u1 (t) + + b1 mum (t)
> x (t)
> _2 = x3 (t) + b2 1u1 (t) + + b2 mum (t)
>
>
>
< ...
(2.15)
> xn;1 (t)
>_ = xn (t) + bn;1 1u1 (t) + + bn;1 mum (t)
>
> xn (t) ;anx1(t) ; an;1x2(t) : : : ; a1xn(t)+
>_ =
>
>
: + bn 1 u1(t) + + bn m um (t)
y(t) = x1(t)
bi j n m- B: -
,
(2.15), . 2.3. -
76
. 2.3. (2.15) ( ).

A(s) (2.15) -
A:
B(s) .
,
MIMO- . ,
(. . . 3.2. 7.3.)
-
-
. , -
( ), -
1, 3, 174], A
, C=
0 ::: 0 1 : , 7. . 166, -
3, 47].
2.4.
1. ,
2 3 2 3
240 abc
32 51 40 3 25 40 d 05:
;1 1 05 001 00e



77
2 3
1 0 ;2
A=41 2 1 5
2
03
1
2
.
3. , A
T
A 3].
4. , n n-
A T , -
˜
A = TAT ;1 . -
s1 s2 : : : sn , -
, 3].
5. , A
">0
A , , " ,
(2.1) 174].
6. : -
. , -
, -
(
k+1 k). -
- -
. -
-
. -
.
> 0: ( < :) 0< 1
.
,
174].
k r k] u k] . -
:
;
r k] ; ;r k] ; (r k] + u k])
r k + 1] = (2.16)
u k] + r k] ; (r k] + u k]) :
u k + 1] =


78
. (2.16) (1.5) -
x = colfr ug:
. -
A (2.16).
. ,
0 min 1 ;
,
,
. , 0
2:
7. . 6, -
s k] sk+
1] = s k] + u k]: , u k]
174].
, -
-
, .




79
3.

-
-
. -
n n- T ,
˜ ˜
A = TAT ;1 B =
ABC
˜
TB C = CT ;1 .1
T ;1
, -
Rn -
3, 53, 66, 115]. ,
1 e2 : : : en g
2 feg = fe

ffg = ff1 f2 : : : fng fi ffg -
feg . .
Pn
fi = j=1 pjiej i = 1 2 : : : n , -
, f1 f2 : : : fn ] = e1 e2 : : : en]P e1 e2 : : : en] =
f1 f2 : : : fn]P ;1 T = P ;1:
-
T ,-
.
˜ ˜
A = TAT ;1
n n- AA
T -

˜ det T 6= 0:
TA = AT (3.1)
(3.1) n2 -
. -
, , 53]. ,
. -
.
.
,
1
˜ ˜˜
A (A B)),
( ,
T
n-
, -
2
n
( )
3, 53].


80
3.1. -
-
( { -
)
˜ ˜ ˜
A. B C
" " -
˜ ˜
B = TB C = CT ;1 :
T -
T ,
˜ ˜
A = TAT ;1 A , . 2.1.
. 67. -
˜
A A: -
˜
, A ,
2.1.. T -
(3.1) -
. -
A: ,
si A .
3.1.1.
-
. ,
n n- A -
xi 6= 0
si -
0

53, 115]
Ax0 = six0 : (3.2)
i i
, { ,
A -
. ,
-
6= 0
, .. x0 {
i
x0 A: -
i

,
Rn. 3
A
,
3
,

81
˜
A -
˜
A = diagfs1 s2 : : : sn g Imsi = 0 i = 1 2 : : : n:
(3.2) ,
˜
A , ,
x0 = ei = 0 : : : |{z} : : : 0] T
˜i 1
i
x0
˜i .
, , -
. ,
˜
A
. , T -
(2.4)

T = x0 x0 : : : x0 ];1 (3.3)
n
1 2

x0 (i = 1 2 : : : n) { A:
i
, si { , xi { - 0

A, . . Axi = six0 x0 6= 0
n n- 0
ii
(i = 1 2 : : : n): ,
,
˜ ˜
;1 detT 6= 0 A = diagfs1 s2 : : : sng:
.. A = TAT
P = x0 x0 : : : x0 ]:
n
12
-
˜
, AP = P A:
˜
A = P ;1AP
xi -
0
;1
,T =P (3.3).
. -
fxi g , ,
0

T (3.3). -
53, 115], si
A , -
.
3.1.2.
-
- (2.4).
. -
.


82
, -
A , -
(| = ;1)
si i+1 = | = Resi i+1 i =
2
i i i
jImsi i+1j: A n -
.
3.1.1. . 81, . -
˜
A=
diagfs1 s2 : : : sn g T
,
. -
, - -
(2.4) 47, 79].
si i+1 = | -
i i
xi x0 : -
0
i+1
2R 6= 0
79, 115], ,
xi , x0 { - . -
0
i+1
, xi+1 = conj(x0 ) conj( ) { -
0
i
.
hi hi+1
hi = 1 (x0 + x0 ) 1
hi+1 = 2| (x0 ; x0 ): (3.4)
2 i i+1 i i+1

hi hi+1 , -
,
. -
Rn { -
A . 4




T = x0 x0 : : : hj hj+1 : : : hq+r;1 hq+r ];1
1 2

- x0 , hj hj+1 {
i
sj j+1 = | j: -
j
˜ TAT ;1
A= T
, -
4
A A( -
XA X x2 XA
), ,
Ax 2 X A 115]. -
X A = f0g X:
-
.

83
- -
(2.4),
x0 hj -
i
P = T ;1:
-
. , ,
(2.6) , T -
(3.1). -
T , , 47].
, A -
(2.10), -
, -
.
, -
xi = 1 si s2 : : : sn;1]T (i = 1 2 : : : n): -
0
i i
,
T -
, - , .
3.2. -
3.2.1. -

( . . . 2.2. 2.3.) -
A (2.10).
, B,
{ C:
, -
(2.10).
53, 115], (2.10) -
A(s) = det(sIn ; A) -
.5 : , -
f(s)
53].
5
A,
f(A) = 0: , A(s) -
{ , .
, -
.

84
, -
. ,
, -
(2.10).
(2.10)
(2.6).
-
-
( - )
, -
(2.10) 115]. -
.
. A
- ( -
53, 115]), A
2L 0 ::: 03
1
6 0 L2 : : : 07
A = 6 .. 7 (3.5)
... 5
4 . : : : ...
0 : : : 0 Lr
Li i = 1 2 : : : r { (2.10). -
,
A:
A -
(2.10), -
.
A
(s)
A . ,
n: deg (s) = n
deg (s) -
.
A
, -
. (s)
53]: A(s) = (s)d(s) A(s) { -
(A(s) = det(sIn ;A) ) d(s) {
sIn ; A adj(sIn ; A) :
T




85
3.2.2.
. 2.2. . 74,
{ u(t) 2 R:
, -
.
˜
n n- A A n- -
˜
b b: T ,

˜
˜ b = Tb
A = TAT ;1 (3.6)
˜˜
(A b) (A b)
.. .
1.8. . 60, -
. 6
˜ ˜
b (3.6) A: -
˜˜ ˜
Ab = ATb: (3.6) ( . .
˜˜
Ab = TAb:
3.1 . 86), , , -
˜
˜ ˜2 b = TA2b:
A (3.1), A
, :
˜
b = Tb
˜˜
Ab = TAb
::: (3.7)
˜˜
An;1 b = TAn;1b:
n n-
˜ ˜ ˜˜ ˜˜
Q = b Ab : : : An;1 b] Q = b Ab : : : An;1 b]: (3.8)
(3.7)
˜
Q = QT:
˜
: det(sIn ; A) det(sIn ; A) detQ 6=
˜
0 detQ 6= 0 -
T
˜
T = QQ;1 (3.9)
, .
6
˜
AA
, -
.

86
˜˜
Ab Ab (3.6):
˜
˜
A = TAT ;1 b = Tb: 7
, -
2.2. . 74, SISO SIMO-
. A -
(2.10), B= 0
˜˜
T
0 : : : 0 1] : AB
. -
ai -
n +a1 sn;1 +a2 sn;2 + +an;1s+an
A(s) = s -
˜ = B = 0 0 : : : 0 1]T ,
˜
b = B, b
A:
˜ detQ 6=
QQ (3.8).
˜ 6= 0
0 detQ -
(3.9). -
˜ ˜ = CT ;1 :
C C
. -
˜B ˜
, A
˜
detQ 6= 0 ( -
ai ): -
Q . , -
˜B ˜
A -
˜
Q 3, 47].
3.2.3.

,
AC { . 2.3. . 76, -
. SISO- MISO-
(y(t) 2 R l = 1):
.
˜
n n- A A n- -
c c:
˜ T ,
7
T . ,
T detT 6=
(3.1)
T
0:
7. . 166, .
30, 83].


87
˜ c
˜ = cT ;1
A = TAT ;1 (3.10)
˜c
(A c) (A ˜)
.. .
1.8. . 60, -
. 8
˜
c
˜ (3.10) A:

˜A = cAT ;1: (3.10),

˜A = cAT ;1:
, -
˜ c˜
˜A2 = cA2 T ;1:
A (3.1),
3.2.2., , -

c = cT ;1
˜

˜A = cAT ;1
::: (3.11)
c˜ = cAn;1T ;1:
˜An;1
n n-
2
c3 2
˜3
c
˜
cA 7 Q = 6 ˜.A 7 :
... 7 ˜ 6 c.. 7
6
Q=6 (3.12)
4 5 4 5
˜˜
cAn;1 cAn;1
(3.11)
˜
Q = QT ;1:
˜
det(sIn ; A) det(sIn ; A) detQ 6=
˜
0 detQ 6= 0 -

˜
T = Q;1 Q (3.13)
˜c
Ac A˜ (3.10).
,
SISO-, MISO- . A -
(2.10), C= 1
T
, , -
8
˜
A A
,
.

88
˜˜
0 : : : 0 0]: AC
. ai -
n + a1 sn;1 + a2 sn;2 +
A(s) = s +
c = C,
an;1s + an A:
c˜ ˜
˜ = C = 1 0 ::: 0 0], Q Q (3.12).
˜
detQ 6= 0 detQ 6= 0 -
(3.13).
˜ ˜
B B = TB:
˜˜ ˜
Q = In :
. AC
˜ 6= 0
detQ ai : -
, -
Q . , -
T (3.13)
˜
T = Q, . . B = QB:
3.2.4.
-
.
1. . -
. 1.4.2. . 41,
(1.33)
ay = ;2:10 c;1 ] amz = 29:4 c;2] a!zz = 2:18 c;1] amz =
m
;2 ]:
60:7 c ,
{ , :
2 3 2 3
;2:1 0 2:1 0
A= 4 29:4 ;2:18 ;29:4 5 B = 4 ;60:7 5 C = 0 0 1]: (3.14)
0 1 0 0

A: , -
si xi (i = 1 2 3), -
0




det(sIn ; A) = s3 + 4:28s2 + 34s s1 = 0 s2 3 = ;2:14 5:42|
2 3 2 3 2 3
0:707 0:0525;0:039| 0:0525+0:039|
x0 = 4 0 5 x0 = 4;0:952+0:247| 5 x0 = 4;0:952;0:247| 5 :
1 2 3
0:707 0:10+0:014| 0:10;0:014|
,
˜
A - (2.4), -
89
. 3.1.2. . 82. -
,T (3.4)
h2 = 0:0525 ;0:952 0:10] h3 = ;0:039 0:247 0:136]T : -
2 3 2 3
0:707 0:0525 ;0:039 ;1 1:22 0:0874 0:190
T = 4 0 ;0:952 0:247 5 = 4 ;1:38 ;0:982 1:38 5
;5:34 0:262
0:707 0:10 0:136 5:34

2 3 2 3
;5:30
00 0
˜ ˜
A = 4 0 ;2:14 5:42 5 B = 4 59:6 5
0 ;5:42 ;2:14 ;15:9
˜
C = 0:707 0:01 0:136 ] :
-
(1.36) . 42 (2.5)
W(s)=W1 (s)+W2 (s) W1 (s)= ;3:75 W2 (s)= s23:75s ; 44:6 :
s +4:28s+34:0
-
: 9

2
;127 3 2 3
0 0 00 1
˜
Q = 4 ;60:7 132 1:50 103 5 Q = 4 0 1 ;4:28 5
0 ;60:7 1 ;4:28 15:6
132
2 3
;7:84 10;3 0 0
;0:0165 5
T = 4 0:0165 0
;0:0346 ;0:0165 0:0346
2 3 23
0 1 0 0
˜ ˜ ˜
C = ;127 ;60:7 0 ] :
A = 40 15 B = 405
0
0 ;34:0 ;4:28 1
, , -
9
,
(2.13), . 76.
T,
" " .

90
-
. Q (3.13)
2 3 2 3
0 0 1 0
˜
B = 4 ;60:7 5 :
Q=4 0 05
1
29:4 ;2:18 ;29:4 132
˜
A
˜
, C = 1 0 0]:

MATLAB-
10


v,r]=eig(A)
eig
{ x0 ( -
i
v) si (
r)
h1=1/2*(v(:,2)+v(:,3)) h2=1/2*(v(:,2)-v(:,3))/j
{" " (3.4)
Pd= v(:,1),h1,h2] Td=inv(Pd)
Pd
{ -
Td (3.3)
Ad=Td*A*Pd Bd=Td*B Cd=C*Pd
{ - -
˜ = TAT ;1 B = TB C = CT ;1
˜ ˜
A
Ad1=Ad(1,1) Bd1=Bd(1,1) Cd1=Cd(1,1) Dd1=0
Ad2=Ad(2:3,2:3) Bd2=Bd(2:3,1)
Cd2=Cd(1,2:3) Dd2=0
{
nd1,dd1]=ss2tf(Ad1,Bd1,Cd1,Dd1,1)
nd2,dd2]=ss2tf(Ad2,Bd2,Cd2,Dd2,1)
{ W1 (s) W2 (s) -
(2.5).
a=poly(A) % { -
Af= zeros(2,1),eye(2) -a(4:-1:2)] Bc= 0 0 1]'
, -
10
, 2 . 1.6.4.
. 53.

91
˜˜
{ AB
Qc= B,A*B,Ab2*B]
Qc = Bc,Af*Bc,Afb2*Bc]
˜
{ Qc Qc
Tc=Qc /Qc
{ T (3.9)
Ac=Tc*A/Tc Bc=Tc*B Cc=C/Tc
{ ( -
Cc, Ac, Bc )
˜
Co = 1 0 0] % { C
Qo= C C*A C*Ab2]
Qo = Co Co *Af Co *Afb2]
˜
{ Qo Qo
Po=Qo /Qo To=inv(Po)
{ T (3.13)
Ao=To*A*Po Bo=To*B Co=C*Po
{ .
2. -
. . 1.4.3. . 31
-
(1.18). -
k
47]: M = 1 c;1 ] M = 1 ], L0 = 0:842 ].
h i
11
C = ; L0 0 L0 0 :
. 11

-
.
A : s1 2 = 3:41 s3 =0 s4 =;1
2 3 2 3 2 3 2 3
;0:477
0 0 0:707
607 607 607 6 0:477 7
x0 = 60:281 7 x0 = 6;0:281 7 x0 = 60:707 7 x0 = 6;0:521 7 :
4 5 4 534 544 5
1 2

0:960 0:960 0 0:521

2
;1:78 ;0:40 1:78 0:523 2 3
3:4 0 0 0
6 1:78 ;0:74 ;1:78 0:527 ˜ 6 0 ;3:4 0 07
T = 6 1:41 1:41 7 6 7
0 5 A= 4 0
4 05
0 00
0 ;1
0 2:10 0 0 0 0
, , -
11
(1.18) .

92
˜ ˜
B = ;0:40 ;0:74 1:4 2:1]T C = 0:33 ;0:33 0
;0:053]: , -

W(s)=; s 0:133 + s 0:243 ; 0:111 :
; 3:41 + 3:41 s + 1
,
.
˜
C: -
, s3 = 0
.
-
. -
( 7. . 166.).
, -
, -
(
).
.
det(sIn ; A) = s4 + s3 ; 11:6s2 ; 11:6s :

2 3 2 3
;1
0 1 0 0 0 1 1
;1 ;1 7
˜6 0 7 Q = 61
A = 60 0 1 1
7 6 7
;16:5 5
40 15 40
0 0 0 0
0 11:6 11:6 ;1:0 0 ;16:5 16:5
0
2 3 2 3
0 ;0:086
00 0 1 0 0
;1 7 ;0:086 7
˜6 6
Q = 6 0 0 ;1 12:6 7 T = 6 0
1 0 0 7

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