:2012 142

M:""

2012

1

1:

(),,().

() , .

1

():

1 2

4 3

R1Is v1 C1

C2

R2

R3

R4 R5gv1

11 22

44 33

R1Is v1 C1

C2

R2

R3

R4 R5gv1

:v1,():gv1

()4,:

R1R2R3R4R5C1C2v1gv1Is

1

2

3

1 1 0 0 0 1 1 1 0 1

0 1 0 1 1 0 1 0 1 0

0 0 1 1 1 0 0 0 1 0

=

A

:n=4b=10,:(n1)xb=3x10.

/ & /

gv1C1R1

R4+

+

v1 v2

R2

C2

R5s

R3

gv1C1R1

R4+

+

v1 v2

R2

C2

R5s

R3

:2012 242

()(,t=0):

1 2 1 2 2 2 1

2 2 2 4 5 2 4 5 2

4 5 3 4 5 3

1

2

3

0

0

0

s

n

G G sC sC G sC e I

G sC g G G G sC G G e

g G G G G G e

+ + + + + + + = + +

Y

(Gi:Gi=1/Ri).

()()():{C1,C2,R3}.

(...):[vc1,vc2,vR3].

1 2

4 3

R1Is v1

C1

C2

R2

R3

R4 R5gv1 C1

C2

R3

11 22

44 33

R1Is v1

C1

C2

R2

R3

R4 R5gv1 C1

C2

R3

,,(...)...,(,):

1 1 1 1 4 5

2 2 2 1 4 5

3 3 1 4 5

: 0: 0: 0

c R R R S

c R R R

R R R

C

C

R

i i gv i i Ii i gv i ii gv i i

+ + + + = + = =

( )

( )1 1 1 1 1 4 4 5 5

2 2 2 2 1 4 4 5 5

3 3 1 4 4 5 5

00

c R R R S

c R R R

R R R

sC v Gv gv G v G v IsC v G v gv G v G v

G v gv G v G v

+ + + + = + = =

...,:

( ) ( )( )

( ) ( )( )

( )( )

1 1 1 4 5 2 1 3

1 2 2 2 4 5 2 1 3

1 3 3 4 5 2 1 3

00

c c c R S

c c c c R

c R c c R

sC G g v G G v v v Igv sC G v G G v v vgv G v G G v v v

+ + + + + = + + + + = + + + =

:

( ) ( )

( )

( )

1 1 4 5 4 5 4 5 1

4 5 2 2 4 5 4 5 2

4 5 4 5 3 4 5 3

0

0

c s

c

R

sC G G G g G G G G v I

g G G sC G G G G G v

g G G G G G G G v

+ + + + + + + + + + + = + + + +

____________________

:2012 342

2

2:

(),,.

() , .

2

():

1 2

0

3

R1

i1 L1

L2

R2

vs

R4

R5

4 5

r.i1

6 7R3

i1 i3i2

11 22

00

33

R1

i1 L1

L2

R2

vs

R4

R5

44 55

r.i1

66 77R3

i1 i3i2

:i1,():ri1:

R1R2R3R4R5L1L2i1ri1vs

1

2

3

4

5

6

7

1 0 0 0 0 0 0 1 0 0

0 0 0 0 0 1 0 1 0 0

0 0 1 0 0 1 1 0 1 0

0 0 0 1 0 0 0 0 1 0

0 0 0 1 1 0 0 0 0 0

1 0 0 0 0 0 0 0 0 1

0 1 0 0 0 0 1 0 0 0

=

A

(t=0):

ri1L1 L2

R3

+

i1

v2R2

R5

R4

+vs

R1+ ri1

L1 L2R3

+

i1

v2R2

R5

R4

+vs

R1+

:2012 442

( ) ( )( )

( ) ( )( )

( ) ( )

1 1 1 2 2 1 2

3 2 3 2 2 1 2

1 4 5 3 3 2 3

000

sL s R i L s R i i vR i i L s R i ir i R R i R i i

+ + + = + = + + =

:

( )

( )

1 2 1 2 2 2 1

2 2 2 2 3 3 2

3 3 4 5 3

0

0

0

sL s L s R R L s R i v

L s R L s R R R i

r R R R R i

+ + + + + + + = + +

()()():

{R1,i1,vs,R2,R3,ri1,R4},.

1 2

0

3

R1

i1 L1

L2

R2vs

R4

R5

4 5

r.i1

6 7R3

-L1 -R5

-L2

11 22

00

33

R1

i1 L1

L2

R2vs

R4

R5

44 55

r.i1

66 77R3

-L1 -R5

-L2

(..):[iL1,iL2,iR5].,,(...)..,(,):

1 1 3 1

2 2 2 3

5 5 3 1 4

: 0: 0: 0

L R R S

L R R

R R R

L

L

R

v v v vv v vv v ri v

+ = + = + =

( )

( )1 1 3 3 1 1

2 2 2 2 3 3

5 5 3 3 1 4 4

00

L R R S

L R R

R R R

sL i R i Ri vsL i R i R i

R i R i ri R i

+ = + = + =

...,:

( ) ( ) ( )

( ) ( )

( )

1 1 3 1 2 5 1 1

2 2 2 2 3 1 2 5

5 5 3 1 2 5 1 4 5

00

L L L R L S

L L L L R

R L L R L R

sL i R i i i R i vsL i R i R i i i

R i R i i i ri R i

+ = + = + + =

:

1 1 3 3 3 1

3 2 2 3 3 2

3 3 3 4 5 5

0

0

L s

L

R

L s R R R R i v

R L s R R R i

r R R R R R i

+ + + + = + +

_____________________

:2012 542

1

1:

1

().

():

1 2 5R R R 1500= = =

2 7C C 106nF= =

6R 1.1k=

4R 11.3k=

3R 17.6k=

8R 1000=

9R 10908=

41

V (s)H (s)E(s)

=

62

V (s)H (s)E(s)

=

() Bode.

() 1kHz,3kHz.

R6

R1

+

R4

R5

+

C7

C2

R3

+E

0

1

65

43

2

R6

R1

+

R4

R5

+

C7C7

C2

R3

+E

+E

00

11

6655

4433

22

:2012 642

1

(),,V

15 5 2 2

21 3 3 1

37 4 6 7 4

4

eG G C s 0 C s 0e0 G G G G 0eC s G G C s G 0 0e1 0 0 0 E

+ + = + +

():

0 0 1

R5

C2

R3

I-

2 3

R1

1 2 3 5

C7

R4

4 6

R6

0 0 10 1

R5

C2

R3

I-

2 32 3

R1

1 2 3 5

C7

R4

4 64 6

R6

0 0

4 6C1

V-

1 1

3 4R6

C2

R1

E

R5

C7

2 2 5 3

R3

R4

0 00 0

4 64 6C1

V-

1 11 1

3 43 4R6

C2

R1

E

R5

C7

2 2 5 32 2 2 5 5 3 3

R3

R4

:2012 742

5 5 2 2

1 3 1

7 4 6 7

31

5 5 2 2

1 3 3 1

7 4 6 7 4

5 1 4 6 7 7 1 5 2 1 3 2

4 6 7 3 2 1 4 5 2 4 1 3 2

3 2

G G C s 0 C s0 G G 0 GC s G G C s 0 01 0 1 0e (s)H (s)

G G C s 0 C sE(s)0 G G G GC s G G C s G 01 0 0 0

G G (G G C s) C s[ G (G C s) (G G )C s](G G C s)G C s G G (G C s) G (G G )C s

G C C

+ +

+ +

= = = +

+ + +

+ + + + + += =

+ + + + +

=2 2 2 2

7 1 5 4 6 z2 2

p2 23 2 7 6 3 2 1 4 5p

p

s G G (G G ) s s 694480609G C C s G G C s G G G s 8576.4s 61608000s s

Q

+ + + += =

+ + + ++ +

2 3p

1 4 5 2 7

RR R R C C

=

p p 6 7Q R C=

4z p

6

R1R

= +

H2(s)

5 5 2

1 3 3

7 4 6 7 4

42

5 5 2 2

1 3 3 1

7 4 6 7 4

2 2 2 23 2 7 1 5 4 3 5 6 z

2 2p2 23 2 7 6 3 2 1 4 5

pp

G G C s 0 00 G G G 0C s G G C s G 01 0 0 1e (s)H (s)

G G C s 0 C sE(s)0 G G G GC s G G C s G 01 0 0 0

G C C s G G G G G G ) s s 7669000G C C s G G C s G G G s 8576.4s 616080s s

Q

++

+ +

= = = +

+ + +

+ + += = =

+ + + ++ + 00

p,Qp,

1 4z p

3 6

R R1R R

=

()BodeH1(s)()H2(s)().

:2012 842

-400

-300

-200

-100

0

100

Magnitude (dB)

102

103

104

105

106

180

270

360

450

540

Phase (deg)

Bode Diagram

Frequency (rad/sec)

(),H1(s).

________________________

:2012 942

2

2(2()2()):

2() 2()

()Ai=, i=1,2

1 3 5in

2 4

Z (s)Z (s)Z (s)Z (s)Z (s)Z (s)

=

() .

()

0i ii

A aA (s)s

=

1 3 4 5Z (s) Z (s) Z (s) Z (s) R= = = =

21Z (s)

Cs=

Zin(s) .() QL

LLQ

R

=

L R ,().

() ,.

+

-A1

A2

1

2

5

3

4

0

Z1

Zin

Z5

Z4

Z3

Z2

+

-

+

-A1

+

-A1A1

A2

1

2

5

3

4

0

Z1Z1

Zin

Z5Z5

Z4Z4

Z3Z3

Z2Z2

+

-

+

-A2

+

-A1

Z1

1

25

3

4

0

Z3

Zin Z2 Z4

Z5

+

-A2

+

-A1

Z1

1

25

3

4

0

Z3

Zin Z2 Z4

Z5

:2012 1042

2

()()J.,V.

ii

1Y i 1,2,3,4,5.Z

= =

i,i=1,2ei,i=1,2,3,4,5V,:

0 0

3 3

4 4

5 52 2 Y3

Y5

Y1T12 T22

Y2

T11 Y4

T21

V-

0 00 0

3 33 3

4 44 4

5 55 52 22 2 Y3

Y5

Y1T12 T22

Y2

T11 Y4

T21

V-

0 0 3 5

1 1

2 3 2 4

J

4

2 1

3

I-

5

00 0 3 5

1 1 1

2 2 3 3 2 4

J

4

2 1

3

I-

5

:2012 1142

15 5

21 2 2

33 3 4 4

41 1

52 2

eY 0 0 0 Y Je0 Y Y Y 0 0 0e0 0 Y Y Y Y 0eA A 1 0 0 0eA 0 0 A 1 0

+ = +

5

1 2 2

3 3 4 4

1

21in

5 5

1 2 2

3 3 4 4

1 1

2 2

1 2 3 4 2 4 1 2 3 4 2 4

5 1 5 2 5 1 2 3 4 2 4

1 0 0 0 Y0 Y Y Y 0 00 0 Y Y Y Y0 A 1 0 00 0 0 A 1eZ

J Y 0 0 0 Y0 Y Y Y 0 00 0 Y Y Y YA A 1 0 0A 0 0 A 1

(Y Y )(Y Y A Y ) A Y (Y Y A Y )( Y Y Y Y Y A Y )(Y Y A Y )

+ + = = =

+ +

+ + + + + + + 1 2 3 5 1 2 2 5 3 4 1 2 1 2A A Y Y (Y Y ) A Y (Y Y )(Y Y A Y ) + + + + +

1 2A A= = 121,2,

1 2

1 3 52 4 2 4in in 1 2A ,A

5 2 4 3 5 1 2 2 5 3 4 3 5 1 2 4

Z (s)Z (s)Z (s)Y Y Y YZ lim Z (A , A )Y Y Y Y Y (Y Y ) Y Y (Y Y ) Y Y Y Z (s)Z (s)

= = = =

+ + +

()V.

0 0 2 4

1 1

2 3 3 5

J5

2 3

4

I-

1

00 0 2 4

1 1 1

2 2 3 3 3 5

J5

2 3

4

I-

1

:2012 1242

11 1

22 2 3 3

34 4 5

41 1

52 2

eY Y 0 0 0 Je0 Y Y Y Y 0 0e0 0 0 Y Y Y 0eA 0 A 1 0 0e0 1 A 0 A 0

+ = +

1

2 2 3 3

4 4 5

1

2 21in

1 1

2 2 3 3

4 4 5

1 1

2 2

1 2 2 4 2 2 4 2 2 5 2 4 2 5 3 4 1 3 5 1

2 1 2 4 2

1 Y 0 0 00 Y Y Y Y 00 0 0 Y Y Y0 0 A 1 00 1 A 0 AeZ

J Y Y 0 0 00 Y Y Y Y 00 0 0 Y Y YA 0 A 1 00 1 A 0 A

A A Y Y A Y Y A Y Y Y Y Y Y Y Y (1 A ) Y Y (1 A )A Y Y Y A Y

+ + = = =

+ +

+ + + + + + + ++ 1 2 5 1 2 4 1 2 5 1 3 4 1 1 3 5 1 1 2 1 3 5Y Y Y Y Y Y Y Y Y Y Y (1 A ) Y Y Y (1 A ) A A Y Y Y+ + + + + + +

1 2A A= = 121,2,

1 2

1 3 52 4in in 1 2A ,A

3 5 1 2 4

Z (s)Z (s)Z (s)Y YZ lim Z (A , A )Y Y Y Z (s)Z (s)

= = =

1 1

0 0

3 34 4 5 52 2 Y3

21

12T22 5

Y2

1 Y4

T11

V-

1 1

0 0

3 34 4 5 52 2 Y3

21

12T22 5

Y2

1 Y4

T11

V-

:2012 1342

()

1 3 5Z (s) Z (s) Z (s) R= = =

22

44

1Z (s) R Z (s)Cs1Z (s) Z (s) RCs

= = = =

2L R C=

()

1 3 4 5Z (s) Z (s) Z (s) Z (s) R= = = =

21Z (s)

Cs=

ii

AA (s)s

=

()2 2

2 1 2 1 2

in 32 3 2 2 32

1 2 1 2 2

3 2 2 22 1 2 1 2

2 3 3 2 2 2 3 31 2 2 1 2

12GCs (2G A GC 2A GC) (A G A A GC)sZ (s)

A G 12G Cs (2G 2A G C A G C) A A Gs s

2GCs (2G A GC 2A GC)s (A G A A GC)s2G Cs (2G 2A G C A G C)s A G s A A G

+ + + + += =

+ + +

+ + + + +=

+ + +

Stodola.

1

:2012 1442

1G R AC L

= >

.

22G 2R AC L

= >

.LR.()

2 22 1 1 2

in 32 3 2 31

2 1 2 2

3 2 2 22 1 1 2

2 3 3 2 2 3 32 1 1 2

12GCs (2G 2A GC) (2A G A A GC)sZ (s)

2A G 12G Cs (2G 2A G C) A A Gs s

2GCs (2G 2A GC)s (2A G A A GC)s2G Cs (2G 2A G C)s 2A G s A A G

+ + + += =

+ + + +

+ + + +=

+ + + +

Stodola.

1>02>0

+..Routh

s3 22G C 312A G s2 3 2

22G 2A G C+ 3

1 2A A G s 6 5

1 1 23 2

2

4A G 2A A G C2G 2A G C

++

0

1 31 2A A G

1,2,.

Real Axis

Imaginary Axis

Nyquist Diagrams

-2 0 2 4 6 8 100

0.5

1

1.5

2

2.5

3

3.5

From: U(1)

To: Y(1)

ReZ in

ImZ in

Z ina

Real Axis

Imaginary Axis

Nyquist Diagrams

0 5 10 15 200

50

100

150

200

250

300

350

From: U(1)

To: Y(1)

ReZ in

ImZ in

Z inb

().

().

:2012 1542

()L=1mH,Gainbandwidth106(),

G=1031C=109F

12 3 6 2

ina 15 3 9 2 3 3

2*10 s 10 sZ2*10 s 10 s 10 s 10

=

+ + +

12 3 6 2

inb 15 3 9 2 3 3

2*10 s 4*10 s 3sZ2*10 s 4*10 s 2*10 s 10

+ +=

+ + +

{ }{ }

6 2 3 15 3 3 9 2 15 3ina

La 6 2 3 9 2 3 15 3 15 3ina

Im Z 10 (10 2*10 ) (10 10 )( 2*10 )QRe Z 10 (10 10 ) (10 2*10 )( 2*10 )

+ = =

+

{ }{ }

6 2 3 15 3 3 9 2 15 3inb

Lb 6 2 3 9 2 3 15 3 15 3inb

Im Z 4*10 (2*10 2*10 ) (10 4*10 )(3 2*10 )QRe Z 4*10 (10 4*10 ) (2*10 2*10 )(3 2*10 )

+ = =

+

0 2 4 6 8 10

x 107

-2

-1.5

-1

-0.5

0

0.5

1

Q La

0 2 4 6 8 10

x 104

0

500

1000

1500

2000

2500

3000

Q Lb

QL() QL()

G=1041C=1011F

(),

:2012 1642

0 2 4 6 8 10

x 107

-5

-4

-3

-2

-1

0

1

2Q

La

w

() ,().

.()(0,107),

0 2 4 6 8 10

x 107

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Q Lb

().

___________________________

:2012 1742

1

1:

1

Vo, () N.()

oV (s)G(s)=I(s)

() Lyapunov.I(t)

0I(t)=I (t) . t.

(),

oV 3L

1L

2L

3C1C

4C2C

I

+

_

oV 3L

1L

2L

3C1C

4C2C

II

+

_

oV 3L

1L

2L

3C1C

4C2C

I

+

_

1

2

3

4

0

oV 3L

1L

2L

3C1C

4C2C

II

+

_

11

22

33

44

00

:2012 1842

3L1L

2L

3C1C

4C2C

I

0

4

3

2

1

3L1L

2L

3C1C

4C2C

I

00

44

33

22

11

,

1 2 3 4 1 2 3

T

C C C C L L Lx= v v v v i i i

Kirchhoff1

1

1

C1 L

dvC =I-i

dt

15

1 1

dx 1 1= I- xdt C C

KirchhoffI,C2,L1,L2

2

1 2

C2 L L

dvC =I-i -i

dt

25 6

2 2 2

dx 1 1 1= I- x - xdt C C C

Kirchhoff1

3

1 3

C3 L L

dvC =i -i

dt

35 7

3 3

dx 1 1= x - xdt C C

KirchhoffC4,L1,L2,L3

4

1 1 3

C4 L L L

dvC =i i -i

dt+

:2012 1942

45 6 7

4 4 4

dx 1 1 1= x + x - xdt C C C

KirchhoffL1,C1,C2,C3,C4

1

1 2 3 4

L1 C C C C

diL =v v -v v

dt+

51 2 3 4

1 1 1 1

dx 1 1 1 1= x + x - x xdt L L L L

KirchhoffL2,C2,C4

2

2 4

L2 C C

diL =v -v

dt

62 4

2 2

dx 1 1= x - xdt L L

KirchhoffL3,C3,C4

3

3 4

L3 C C

diL =v +v

dt

73 4

3 3

dx 1 1= x + xdt L L

1 2o C C 1 2V =v +v =x +x

1

2 21 1

2 23 3

3 3

4 44 4 4

5 5

6 61 1 1 1

7 7

2 2

3 3

-10 0 0 0 0 0C-1 -10 0 0 0 0C C

x x1 -10 0 0 0 0x x

C Cx x

1 1 -1d 0 0 0 0x = xC C Cdt

x x1 1 -1 -1 0 0 0x xL L L L

x x1 -10 0 0 0 0

L L1 10 0 0 0 0L L

1

2

1C1

C0+ I0000

:2012 2042

[ ]oV 1 1 0 0 0 0 0 x= ()

1

2

3

4

1 -1s+ -s 0s s

e (s) 11 -1-s 2s+ 0 e (s) 0s s + I(s)e (s)-1 2 00 s+ -s

s s e (s) 0-1 10 -s 2s+s s

1,Cramer,

o 1

-1I(s) -s 0s

1 -10 2s+ 0s s

20 0 s+ -ss

-1 10 -s 2s+V (s) e (s) 1 s sG(s)= = =1 -1I(s) I(s) I(s) s+ -s 0s s

1 -1-s 2s+ 0s s

-1 20 s+ -ss s

-1 10 -s 2s+s s

1 -12s+ 0 -ss s

20 s+ -ss

-1 1-s 2s+s s

1 -1 1 -1s+ -s s+ 0s s s s

1 1 1 -12s+ 2s+ 0 - -s 0s s s s

-1 2 -1 20 s+ s+s s s

s

=

=

1s+ -s 0s

1 -1s -s 2s+s s

-1-s 0 -ss s

+

4 2

6 4 2

s(2s +11s +8)s +8s +11s +4

=

:2012 2142

().,G(s).,

n

o i i i i ii=1

V (t)= A G(j ) [ t+Arg(G(j ))+ ]

iG(j ) 0=

iG(j ) 0=

s=ji.

s=0

1,2,3,4

j2.153511 121 4*2*8s

16j0.9287

= =

s=js=j

( )

1 1 2 2

1 1 1 1 2 2 2 2

1 1 2 2 1 1 2 2

2 2 21 2 1 2 1

A (t+ )+A (-t+ )==A (t)( )+A (t)( )-A (t)( ) A (t)( )==(t)[A ( )-A ( )]+(t)[A ( ) A ( )]=

=(t) +(t) (t)+(t)

++

= + = + 22 *(t+)

12 21 2

22 21 2

K= +K=

+

n=3

1=0

2=2.1535

3=0.9287

:2012 2242

2

2:

(), vs(t),vout(t) : x(t) = [vc, iL1, iL2]T.

() R1=R2=1, C=2F, L1=L2=1H, : G(s)=Vout(s)/Vs(s). () ,=0,

, vc(0)=2V, iL1(0)=iL2(0)=0.

2

(),:

1 2R1

-C

3L1 4

0

vs

5

L2

C

ica.i c

R2

11 22R1

-C

33L1 44

00

vs

55

L2

C

ica.i c

R2

......C.....L1L2:

1 2

11 1 1

22 2 2

0

0

0

cL L

Lc s R

LR c

dvC i idtdiL v v RidtdiL R i vdt

+ = + + = + =

iR1 iR2 (R1R2):

R1

vs

L1iL1

y = voutvc R2C

L2iL2+

-

+

.ic

ic+-

R1

vs

L1iL1

y = voutvc R2C

L2iL2+

-

+

.ic

ic+-

:2012 2342

( )

( )( )

( )( )

1 2

11 1

1

22 2 1 2

2

1

1

1

cL L

Lc L s

Lc L L L

dv i idt Cdi v R i vdt L

di v R i ai aidt L

= = +

= +

,:

( )

11 1

1 1 1

2 22 2 ( )( )

2 2 2

1 10 0

1 10

01 1

c c

L L s

L L

tt

C Cv vRd i i v

dt L L Li iR Ra a

L L L

= + +

xxB

A

:

( ) ( )( )2 2 2 2 2 2 2 1 2out R R L c L L Ly v v R i R i ai R i a i i= = = = = :

( )[ ] [ ]2 2 1

2

0 1 0c

L s

L

v

y aR a R i v

i

= + +

DC

():

( )

0 0.5 0.5 0

1 1 0 1

1 1 0sv

a a

= + +

BA

x x

( )[ ] [ ]1

2

0 1 0c

L s

L

v

y a a i v

i

= + +

DC

,:

( ) 1( )H s s = +C I A B D

( )[ ]

( )

10.5 0.5 0

( ) 0 1 1 1 0 1

1 1 0

s

H s a a s

a s a

= + + + +

( )[ ] ( )11 21 31

12 22 32

13 23 33

01( ) 0 1 1

0

H s a as

= +

I A

:2012 2442

( ) ( )[ ] ( ) ( )( )21

22 22 23

23

1 1( ) 0 1 1H s a a a as s

= + = + +

I A I A

( )

( )( ) ( ) ( )

0.5 0.5

1 1 0 1 1 0.5 1 0.5 1

1 1

s

s s s s s a s a s a

a s a

= + = + + + + + + + +

+ +

I A

( ) ( )( ) ( ) ( )2 3 22 1 1 2 2 1s s s a s a s s a s a s = + + + + + + = + + + + +I A

( )( ) ( ) ( )2 2 2220.5

1 1 0.5 1 0.51 1

ss s a s a s

s a+ = = + + + = + + +

+ +

( )( )2 3230.5

1 0.51

sas

a+

= = +

:

( ) ( )( ) ( )( )

( ) ( )

222 23

3 2

1 0.5 1 0.51( )2 2 1

a s a s a asa aH ss s a s a s

+ + + + + + + + = = + + + + +I A

( ) ( )

2

3 20.5( )

2 2 1asH s

s a s a s +=

+ + + + +

()(),=0:

( ) [ ] [ ] ( )AME11 21 31

112 22 32

13 23 33

21( ) (0) 0 0 1 0

0

y t ss

= =

C I A xI A

L

( )

( )

( )( ) ( ) ( )( )

( )( ) ( )AME

1 3

132 2

1 12 1

1 02 2 1 2( )1 1 0.5 1 0.5 1 1 1 1

s

sy ts s s s s s s s s s s

++

+= = = =

+ + + + + + + + + + +I AL

:

( ) ( ) ( )

AME

1 1 22 22

1 1 4 3 3( ) 2 2 sin ( )3 21 1 3

2 2

t

y t e t u ts s

s

= = = + + + +

L L

____________________________________

:2012 2542

1

1:

1

() .()

1 2 31 C 4 C 8 Ce v , e v , e v= = = ,

1 2 3

TC C Cx v v v = .

() .()x(0)0 E(t)=0, x(t) ,

.() R5 ,()()

.

()V.

+

4

R3

0

R1

R2

R4

Vo

_

-

+

C3

-

+

-

+

C1

-

+

C2

E+

R5

R6

2

15

3

87

6

9+

44

R3

00

R1

R2

R4

Vo

_

-

+

C3

-

+

-

+

-

+

C1

-

+

C2

E+

R5

R6

22

1155

33

8877

66

99

I-

R1 R3

2 31 2

4 7

0 0 1

C1

R2R4

3 5

6 4 8 9

R6R5

C3

C2

I-

R1 R3

2 32 32 31 21 2

4 74 74 7

0 00 00 0 1 1

C1

R2R4

3 53 53 5

6 6 4 4 8 8 9 9

R6R5

C3

C2

:2012 2642

.

11 1

22 2

33 5 4

46 3

5

eG C s 0 0 0 0 0eG C s 0 0 0 0e0 G G 0 G 0e0 0 G C s 0 0e0 0 0 0 1 E(s)

=

()

dfL sF(s) f (0)dt

=

,.

1 11

1

de G edt C

= (1)

2 21

2

de G edt C

= (2)

643

3

Gde edt C

= (3)

5e E(t)= (4) 3 2 5 3 4 5G e G e G e 0+ + = (5)

11 1 Cx e v= = (6)

22 4 Cx e v= = (7)

33 8 Cx e v= = (8)

(4)(5)(3)

0 0

C1

V-

1 1 3 6

R4R1

2 4

R2

4 85 9

7 5 3 2

R6R5

R3

E C3

C2

0 00 0

C1

V-

1 11 1 3 63 6

R4R1

2 42 4

R2

4 84 85 95 9

7 7 5 5 3 3 2 2

R6R5

R3

E C3

C2

:2012 2742

[ ]

1 11 1

2 22 2

3 3 55

4 6 33 6 3

1

2

3

1 0 0R Cx x 0

d 1x 0 0 x 0 E(t)dt R C

x x RR R R C0 0

R R C

xy 0 0 1 x

x

= +

=

() =0,

e

1 11e

x x 2e2 2

3e5

3 6 3

1 0 0R C x 0

dx 1 0 0 x 0dt R C

x 0R0 0

R R C

=

= =

1e 2e

3e

x x 0x

= ==

x3.

1 1

2

2 2 1 1

5

3 6 3

1s 0 0R C1 1(s) det(sI A) det s 0 s s

R C R CR0 s

R R C

+

= = = +

() =0

11 1 1

1 1

dx 1 x x (t 0) x (0)dt R C

= = =

1 1

1 tR C

1 1x (t) e x (0)

=

x2

1 1

1 tR C2

1 12 2 2 2

dx 1 1x e x (0)dt R C R C

= =

:2012 2842

1 1 1 1

t1 1tR C R C1 1 1 1

2 22 2 2 20 0

x (0) x (0)R Cx (t) x (0) e d eR C R C

= =

1 1

1 tR C1 1

2 2 12 2

R Cx (t) x (0) x (0) 1 eR C

=

x3

3 52

3 6 3

dx R x (t)dt R R C

=

1 1

1t tR C5 5 1 1 1

3 3 2 2 1 1 1 13 6 3 3 6 3 2 20

R R x (0)R Cx (t) x (0) x ( )d tx (0) t R C e R CR R C R R C R C

= = +

1 1

1 tR C5 1 1 1

3 3 2 1 1 1 13 6 3 2 2

R x (0)R Cx (t) x (0) tx (0) t R C e R CR R C R C

= + +

x3(t),.() R5=0,

[ ]

1 11 1

2 22 2

3 3 5

4 6 3

1

2

3

1 0 0R Cx x 0

d 1x 0 0 x 0 E(t)dt R C

x x R0 0 0 R R C

xy 0 0 1 x

x

= +

=

1 1

2

2 2 1 1

1s 0 0R C1 1 (s) det(sI A) det s 0 s s

R C R C0 0 s

+

= = = +

:2012 2942

1 11e

2e2 2

3e

1 0 0R C x 0

1 0 0 x 0R C

x 00 0 0

=

1e

3e 2e

x 0x , x

=

x2,x3.x1(t),x2(t)(4)x3(t)

33 3

dx 0 x (t 0) x (0)dt

= = =

3 3x (t) x (0)=

Lyapunov.

_______________________

:2012 3042

1

1. . 2.

2V (s)G(s)=

F(s)

3. Bode G(s) i=i N/m, Mi=i kg Bi=i Ns/m.

1. . .

M2M1

B3B1K2

F(t)

K1

x2,v2B2

x1,v1

M2M1

B3B1K2

F(t)

K1

x2,v2B2

x1,v1

0

1

M1

2

B1

K1

B3M2

B2

F

K2

00

11

M1

22

B1

K1

B3M2

B2

F

K2

21

0

1 1C =M I=F2 2C =M

22

1L =K

3 3G =B1 1G =B

11

1L =K

2 2G =B

2211

00

1 1C =M I=F2 2C =M

22

1L =K

3 3G =B1 1G =B

11

1L =K

2 2G =B

:2012 3142

2.

1 2 21 2 1 2

1

22 22 2 3 2

K K KB +B +M s+ + -B - V (s) 0s s s =V (s)K K F(s)-B - B +B +M s+

s s

Cramer,

22

2 22 3 2 2

1

1 2 21 2 1 2

2 22 2 3 2

K0 -B -s

K KF(s) B +B +M s+ B +V (s) 1 s sG(s)=K K KF(s) F(s) D(s)B +B +M s+ + -B -s s s

K K-B - B +B +M s+s s

= =

( ) ( )

( )( ) ( ) ( )( ) ( )

( )

21 2 1 2 3 2 1 2

1 2 2 3 1 2 2 1 2 1 2 2 3 2 1 2

2 21 2 2 2 2 2 22 2

D(s)=M M s + M B +B +M B +B s+

1+ M K + B +B B +B +M K +K + K +K B +B +K B +B +s

1 1 1+ K +K K -B -2B K -Ks s s

3.

2 1 2 1 2 4 3 2

22+ s(s+1)sG(s)=2s +s(6+5)+2+15+6+(6+15)s +6s -4-8s -4s s +5.5s +9.5s +6.5s+1

=

Bode

2

2

s(s+1)G(s)=(s+3.0469)(s+0.2116)(s +2.2416s+1.5515)

s(s+1)s s 2.2416 s1+ 1+ 1+ s+

3.0469 0.2116 1.5515 1.5515

=

=

=1 rad/sec G(s)

=0.2116 rad/sec, =3.0469 rad/sec = 1.5515 1.2456= rad/sec G(s).

Bode G(s) Bode .

:2012 3242

Frequency (rad/sec)

Phase (deg); Magnitude (dB)

Bode Diagrams

-50

-40

-30

-20

-10From: U(1)

10-2 10-1 100 101-200

-150

-100

-50

0

50

100

To: Y(1)

___________________________

2

F1(t) 2. 1. x1, x2. 2. . 3. x2(t)

2

1

X (s)G(s)F (s)

=

4. .

F2

K

F1

x2

B1

B2

M1

M2

x1

F2

K

F1

x2

B1

B2

M1

M2

x1

:2012 3342

( -2) 1. 1 x1

(x2-x1) -1dx1/dt -F1(t) .

,

21 1

1 2 2 1 1 12

d x dxM K (x x ) B F (t)dt dt

= (1)

2 x2

-(x2-x1) -2dx2/dt F2(t)=2g 2.

,

21 2

2 2 2 1 2 22

d x dxM K (x x ) B F (t)dt dt

= + (2)

(1) (2) . 2. 1 2x , x ,

21

0

1 1C =M2 2 2I =F =M g1 1I =F

2 2C =M1L=K

2 2G =B1 1G =B

2211

00

1 1C =M2 2 2I =F =M g1 1I =F

2 2C =M1L=K

2 2G =B1 1G =B

3. Laplace ( ), (1) (2)

( )( )

21 1 1 2 1

21 2 2 2 2

M s B s K X (s) KX (s) F (s)

KX (s) M s B s K X (s) F (s)

+ + =

+ + + = (3)

x2, X2(s)

21 1 1

22 2

1 12

2 2

M s B s K F (s)K F (s)

X (s)M s B s K K

K M s B s K

+ +

= =+ + + +

:2012 3442

( )21 1 1 24 3 2

1 2 2 1 1 2 2 1 1 2 1 2

KF (s) M s B s K F (s)M M s (M B M B )s (M K M K B B )s (B K B K)s

+ + += =

+ + + + + + +

211 1

2

F (s)K M s B s KF (s)D(s) D(s)

+ +=

24 3 2

1 1 2 2 1 1 2 2 1 1 2 1 2

X (s) KG(s)F (s) M M s (M B M B )s (M K M K B B )s (B K B K)s

= =

+ + + + + + +

4.

21 1

22 2

M s B s K K(s)

K M s B s K+ +

= = + +

4 3 21 2 2 1 1 2 2 1 1 2 1 2M M s (M B M B )s (M K M K B B )s (B K B K)s= + + + + + + +

_____________________________

:2012 3542

1

2

Y(s) 1 sG(s)U(s) s s 1

= =

+ +

1. .2.

{ }t 0 t

dylim lim y(t)dt

.3.

G(s).4. .1. Laplace

2 2

1 s 1 1 1 sY(s) G(s)*U(s) *s s 1 s s s s 1

+= = =

+ + + +

1 1t t2 21 3 1 3y(t) u(t) e t ( 3) u(t) u(t) e t u(t)

2 2 33 34 4

= =

u(t).

0 2 4 6 8 10 12-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4Step Response

Time (sec)

Amplitude

:2012 3642

2.

{ } { } 2 2t s 0 s 0 s 01 s 1 1 slim y(t) lim sY(s) lim s* * lim 1

s s 1 s s s 1 = = = = + + + +

Laplacedy/dtsY(s)y(0)=sY(s),

{ } 2t 0 s s sdy dy 1 slim lim s*L lim s*s*Y(s) lim s* 1dt dt s s 1

= = = = + +

3. n

n n 11 nD(s) s a s ... a

= + + +

{ } { }t s 0 s 0 s 0

n

1 s 1 1 s 1lim y(t) lim sY(s) lim s* * limD(s) s D(s) a

= = = =

{ }t 0 s s s

dy dy 1 slim lim s*L lim s*s*Y(s) lim s* 0dt dt D(s)

= = = =

{ }k k

k kk k 2t 0 s s s

d y d y 1 slim lim s*L lim s*s *Y(s) lim s * 0 k n 1dt dt s s 1

= = = = < + +

{ }n 1 n 1

n 1 n 1n 1 n 1t 0 s s s

d y d y 1 slim lim s*L lim s*s *Y(s) lim s * 1dt dt D(s)

= = = =

,an.yTaylory(t).4. .

_________________________

:2012 3742

2

J1=J2=J3=1,B1=B2=3=1,K1=K2=1.2,3J1,J2J2,J3

1.

1 1 2 2 3 3x=[ ]

3.2. .3. .

1. J1

11

d =dt

1 1 1 21 2 1 1 1

1 1 1 1

d K B B 1=- ( - )- - + Tdt J J 2J J

J2

22

d =dt

32 1 2 21 2 2 3 2 2 2

2 2 2 2 2

Bd K K B 1= ( - )- ( - )- - + Tdt J J 2J 2J J

J3

33

d =dt

3 322 3 3

3 3

d BK= ( - )- dt J 2J

,

3B2

3,3T2,2,2T1,1,1

1 21J2J1 J3

2B2

2B2

3B2

3B2

3,3T2,2,2T1,1,1

1 21J2J1 J3

2B2

2B2

3B2

:2012 3842

[ ]

1

2

3

0 1 0 0 0 0 0 0-1 -1.5 1 0 0 0 1 0

T0 0 0 1 0 0 0 0x= x+

T1 0 -2 -1 1 0 0 10 0 0 0 0 1 0 00 0 1 0 -1 -0.5 0 0

y= = 0 0 0 0 0 1 x

2.

2 3 4 5cP = B AB A B A B A B A B =

0 0 1 0 1.5 0 11 0 1.5 0 1.25 1 2.50 0 0 1 0 1 1

=0 1 0 1 1 1 30 0 0 0 0 0 10 0 0 0 0 1 1.5

1,2,3,4,686.

2

o 3

4

5

C 0 0 0 0 0 1CA 0 0 1 0 1 -0.5CA 0 0 -0.5 1 0.5 -0.75

P = =CA 1 0 -2.75 -1.5 1.75 0.875CA -1.5 1 3.875 -1.25 2.375 1.3125CA -2.25 -3 4.8125 5.125 2.5625 3.0313

0,.3.

[ ]

-1

-1

3 2

6 5 4 3 2 6 5 4 3 2

s -1 0 0 0 0 0 01 s+1.5 -1 0 0 0 1 00 0 s -1 0 0 0 0

G(s)=C[sI-A] B= 0 0 0 0 0 1-1 0 2 s+1 -1 0 0 10 0 0 0 s -1 0 00 0 -1 0 1 s+0.5 0 0

s s +1.5s +ss +3s +6.75s +8.75s +6.5s +3s s +3s +6.75s +8.75s +6.5s +3

=

=s

______________________________

:2012 3942

312S1S2.Qi,i=1,2

ii

xQ =R

xix21. .2. R=1sec/m2,S1=4m

2,S2=10m2Q21m

3/sec,u0,x10x20.

3. x1=x10+x1,x2=x20+x2,u=u0+u,y=x2,.

4. .

1. 1.

11 1

dxu-Q =Sdt

11

xQ =R

11

1 1

dx 1 1=- x + udt RS S

Q2

u

Q1

h1

h2

Q2

u

Q1

h1

h2

:2012 4042

2

21 2

2 2

dx 1 1= x - xdt RS RS

11 11

2 2

2 2

1- 0 1RSx x

S ux x1 1- 0

RS RS

= +

2.

32 1Q =Q =u=1m /s

ii

xQ =R

10 20x =x =1m

3. .

1 21 2 33

1 1 1

1 2 11

2 2 2

x =x =1mx =x =1m1 2 2 2 u=1m /su=1m /s

f f 1 f- 0 1x x RS u S= =f f 1 1 f- 0x x RS RS u

11 11

2 2

2 2

1- 0 1RSx xd S u

x x1 1dt - 0RS RS

= +

[ ] 122

xy x 0 1

x = =

___________________

:2012 4142

4

0 -1 0 0 01 0 0

x= 0 0 -1 x+ 1 0 u y= x0 1 0

0 0 0 0 1

1. .2. G(s).3. .1.

2c

0 0 -1P = B AB A B = 1 0 0

0 1 0

3,,.

o2

1 0 00 1 0

C0 -1 0

P = CA =0 0 -1

CA

3, ,.2.

-1

-1

2 3

2 3

2

2

s 1 0 0 01 0 0

G(s)=C[sI-A] B= 0 s 1 1 00 1 0

0 0 s 0 1

1 -1 1-1 1s s s 0 0

1 0 0 1 -1 s s0 1 00 1 0 1 -1s s

0 11 s s0 0s

=

= =

:2012 4242

2 3

2

-1 1s sdet[G(s)] det 01 -1s s

= =

3. .

3o

s 1 0 (s)=det(sI-A)=det 0 s 1 =s

0 0 s

., s=0 , .

s=0

0 -1 0rank(sI-A) =rank 0 0 -1 =2

0 0 0

0.

__________________________________________________________________________