liseis thematon sae 3_09

... : . . . : : 5

E: . . ..

. :

: 10/3/2009

1.1 1 1.2 0.5 2.1 0.5 2.2 0.5 2.3 0.5 2.4 0.5 3.1 0.4 3.2 0.4 3.3 0.5 3.4 0.5 3.5 0.7 4.1 0.8 4.2 0.5 4.3 0.7 5.1 0.5 5.2 1 5.3 0.5 10

. , . . , .

1

1

_+K G(s)

_+K G(s)

22

s bG(s) b 0s (s 5)(s 1 N )

+= + + + 2 .

0 K 20< < . 1. b ( )

0.05 0.12 0.23 0.28 0.67 1.02 1.56 1.87 1.98 2.30 2.56 2.65 2.78 2.84 2.96 3.02 3.11 3.16 3.24

2. c

. ( )

1.91 1.83 1.75 1.69 1.62 1.58 1.55 1.53 1.49 1.45 1.41 1.37 1.35 1.29 1.24 1.22 1.20 1.17 1.15

a=N2+1

2 4 3

22p(s) s (s 5)(s 1 N ) K(s b) s (5 a)s 5as Ks Kb= + + + + + = + + + + +

Routh

s4 1 5a Kb s3 5+a K 0 s2 (5 a)5a K

(5 a)+

+ Kb

s 2[(5 a)5a K]K Kb(5 a)(5 a)

(5 a)5a K(5 a)

+ ++

+ +

s0 Kb

2

, 1>0 5+a>0

2(5 a)5a K 5a 25a K 0+ = + > 2[(5 a)5a K]K Kb(5 a) 0+ + >

Kb>0

=0 =20 . =0 s=0. b0, 0 G(s) . =20 . s Routh ,

2 2[(5 a)5a K]K Kb(5 a) [(5 a)5a 20]20 20b(5 a) 0+ + = + + =

2

2

5a 25a 20b(5 a)+ = +

, Routh

s4 1 5a 20b s3 5+a 20 0 s2 (5 a)5a 20

(5 a)+

+ 20b

s (5 a)5a 202(5 a)+

+ 0

s0 20b b>0 Routh .

2(5 a)5a 20B(s) s 20b(5 a)+ = ++

2c 2

20b(5 a) 205a 25a 20 5 a

+ = =+ + 2

N2 0 1 2 3 4 5 6 7 8 9 b 0.28 1.02 1.56 1.98 2.30 2.56 2.78 2.96 3.11 3.24 c 1.83 1.69 1.58 1.49 1.41 1.35 1.29 1.24 1.20 1.15

3

2 Nyquist. () , (),

-5 0 5 10 15-10

-8

-6

-4

-2

0

2

4

6

8

10

-0.2 -0.1 0 0.1 0.2 0.3

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

. .

Nyquist G(s) () 180 -180 +, >0. () .

Real Axis

Imag

inar

y A

xis

Nyquist Diagrams

-1000 -900 -800 -700 -600 -500 -400 -300 -200 -100 0-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4From: U(1)

To:

Y(1

)

Nyquist Diagram

Real Axis

Imag

inar

y Ax

is

-0.1 -0.08 -0.06 -0.04 -0.02 0

-5

-4

-3

-2

-1

0

1

2

3

4

5

x 10-3

System: gReal: -0.0499Imag: -8e-006Frequency (rad/sec): -1.42

A

-1/K

. . D, -1/ Nyquist . 0

=0

=0 , 2 =5a-5b-ab

2 ,

[ ][ ]

2 2 2

2 2 2 2 2 2 2 2

-5ba b a 5 (5a-5b-ab)-5ba-5 b a ( 25)( a ) (5a-5b-ab) 5a-5b-ab+25 5a-5b-ab+a ( 25)( a )

1 1= -(5 a)(5b 5a ab) 20

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

2

2

5a 25a 20b(5 a)+ = +

b 2 2

2c 2

5a 25a 20 205a b(5 a) 5a (5 a)(5 a) 5 a+ = + = + =+ +

1

:

Routh s

2[(5 a)5a K]K Kb(5 a) 0+ + =

2

[(5 a)5a K]b(5 a)+ = +

2

[(5 a)5a 20] 5ab(5 a) 5 a+ <

Kmax b Kmax (0,20) (Kmax=20).

2

Kharitonov . .

1. .

2. . s . Kharitonov s s0 ABCD . , b , AC . , Kharitonov

2 3 4 2 30 1 2 3 4a a s a s a s a s 20b 5as (5 a)s s+ + + + = + + + + 4

, Routh.

s4 1 5a 20b s3 5+a 0 0 s2 5a 20b s 20b(5 a)

5a +

0

s0 20b

0 5 10 15 20 25 30 35 40 45 500

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

A

B C

D

6

2

_+K G(s)

_+K G(s)

2

s 3G(s)s s a

+= + + >0

a = 0+10

0 . 1.

-3 . 2.

G(s) (1) .

3. 20% .

4. 2sec .

2

s bG(s)s s a

+= + +

2 2p(s) s s a K(s b) s (1 K)s a Kb= + + + + = + + + + x+jy p(s).

2 2 2(x jy) (1 K)(x jy) a Kb x y (1 K)x a Kb j[2xy (1 K)y] 0+ + + + + + = + + + + + + + =

2 2x y (1 K)x a Kb2xy (1 K)y 0

0 + + + + =+ + =

y=0

K 2x 1=

7

, 2 2 2 2x y ( 2x)x a ( 2x 1)b x y 2xb a b 0 + + + = + =

2 2 2(x b) y b a b+ + = +

b, 3

2R b a b 6 a= + = + 2

3

1

2

1

2

2

1

1

3

1

2

1

2

2

1

1

1 .

(11)- (111)- (221) = -180

(311)=180 - (11)

(311)=180 - (111)

(312)=180 - (221)

(311)= (311)+ (312)

(311)=(112)

22 12 2 11. 1122

8

(221)+(112)=90 - (112) +90 + (311)=180

12 221 . 1, 2 12. 1 12 12 .

R=Z11

11 j 4a 1

2 + = 1=-3+j0

22

1 11 4a 1 4a 1R Z 3 0 6.25 a 6

2 2 4 = = + = + = +

( 1 2). 2. . (-,-3] 0=0.

9

-14 -12 -10 -8 -6 -4 -2 0 2-6

-4

-2

0

2

4

6Root Locus

Real Axis

Imag

inar

y Ax

is=0.456

d=/2A

B

3.

p 2M exp 0.2

1

= = =+

2 2 2n np(s) s 2 s s (1 K)s a 3K= + + = + + + +

1 0.4562 a 3

+ = >+ ,

2(1 ) 4*0.208*(a 3 )+ > +

10

2 0.496K 1 0.832a 0 + >

2

1,20.496 0.496 4(1 0.832a)

2 =

20.496 0.496 4(1 0.832a)2

+ > ( =0.456). a . 0 0 1 2 3 4 5 6 7 8 9 > 2.96 3.11 3.26 3.39 3.52 3.65 3.77 3.88 3.99 4.10

4.

p 2 2n

t 21 1a 3K 1

2 a 3

= = + + +

<

2 24(a 3 ) (1 )2.467

2 4 + =

3o

_+K G(s)

_+K G(s)

1

50(s 1)G(s)(s 1)(s N 3)

+= + + 1 . 1) G(j) 2) G(j) 3) Nyquist G(s). 4) , Nyquist

5) =2

. .

a=N1+3

H G(s) s=j 2

2 2 2 2 2 2

2 2 2

2 2 2 2 2 2

50( j 1) 50( j 1)( j 1)( j a) 50(1 2j )(a j )G(j )( j 1)( j a) (1 )(a ) (1 )(a )

50[a(1 ) 2 ] 50[2a (1 )]j jY(1 )(a ) (1 )(a )

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

2 2

2 2 2 2 2 2

50[2a (1 )] 50 [2a 1 ]Y 0(1 )(a ) (1 )(a )

+= =+ + + + =

=0

= -50/a

12

=

= 0

2.

2 2 2

2 2 2 2 2 2

50[a(1 ) 2 ] 50[a (2 a) ] 0(1 )(a ) (1 )(a )

+ + = = =+ + + +

aa 2

=

2

22

a 2(a 1)a a 5050 2a 1a 2 a 2a 2 a 2 50 a 2Y

a a a 1 aa 1 (a 1)1 a 2 aa 2 a 2 a 2 a 2

= = + + =

=

= 0

Re{G(j)}=0 ( Im{G(j)}=0) Im{G(j)} ( Re{G(j)}) (( ). .

3. Nyquist G(s) N1=5

13

-12 -10 -8 -6 -4 -2 0 2 4 6 8-8

-6

-4

-2

0

2

4

6

8Nyquist Diagram

Real Axis

Imag

inar

y Ax

is

4. s=1. Nyquist .

-1/

, .

0

om 180 = +

a . 1 0 1 2 3 4 5 6 7 8 9 m 90.57o 91.15o 91.72o 92.29o 92.87o 93.44o 94.01o 94.59o 95.16o 95.74o

, 0.5 Nyquist.

4

[ ]0.5 1 0 0x(k+1)= x(k)+ u(k) y(k)= 1 0 x(k) x(0)=-0.25 0 1 0

1. , . 2.

2x(M)=

1

.

M-12 2

1i=1

J(M)=(N +1)u (0)+ u (i) 1 , J.

3. =3 ( ), u(0), u(1) u(2) J(3).

[ ]c 0 1P = B AB = 1 0

c

0 1det(P )=det 1 0

1 0 =

16

.

o

C 1P = =

CA 0.5 10

o

1 0det(P )=det 1 0

0.5 1 =

.

2z-0.5 -1p(z)=det(zI-A)=det =z -0.5z+0.250.25 z

2

1,20.5 0.5 4*0.25 0.5j 0.75z = =

2 2

1,20.5j 0.75 1z =

2 2=

. - Routh . Routh . , . Jury. Routh

1+sz=1-s

2

2 2 21+sz=1-s

1+s 1+s(s)=(1-s) *p(z) =(1-s) -0.5 +0.25 =1.75s +1.5s+0.75

1-s 1-s

2.

17

0.5 1 0 0x(1)= x(0)+ u(0)= u(0)

-0.25 0 1 1

.

[ ]0.5 1 0 u(1) 0 1 u(1) 2x(2)= x(1)+ u(1)= B AB =-0.25 0 1 u(0) 1 0 u(0) 1 =

. u(0)=2u(1)=1

T 2 2

1J(2)=(N +1)u (0)+u (1)=25

1 . 1 0 1 2 3 4 5 6 7 8 9

J(2) 5 9 13 17 21 25 29 33 37 41

3. ,

2

u(2) u(2)0 1 0.5 2

x(3)= B AB A B u(1) = u(1)1 0 -0.25 1

u(0) u(0)

=

.

u(0)=q

u(1)=2-0.5*q

u(2)=1+0.25*q

2 2 2 2 2 2 2

1 1 1J(3)=(N +1)u (0)+u (1)+u (2)=(N +1)q +(2-0.5q) +(1+0.25q) =(N +1.3125)q -1.5q+5

1dJ =(2N +2.625)q-1.5=0dq

1

1.5q=u(0)=2N +2.625

18

( )1

1.5u 1 2 0.5*2N +2.625

=

( )1

1.5u 2 1 0.25*2N +2.625

= +

2 2

11 1

1.5 1.5 1.5J(3)=(N +1) + 2 0.5* + 1 0.25*2N +2.625 2N +2.625 2N +2.625 +

2

1

1 . 1 0 1 2 3 4 5 6 7 8 9

u(0) 0.5714 0.3243 0.2264 0.1739 0.1412 0.1188 0.1026 0.0902 0.0805 0.0727 u(1) 1.7143 1.8378 1.8868 1.9130 1.9294 1.9406 1.9487 1.9549 1.9597 1.9636 u(2) 1.1429 1.0811 1.0566 1.0435 1.0353 1.0297 1.0256 1.0226 1.0201 1.0182 J(3) 4.5714 4.7568 4.8302 4.8696 4.8941 4.9109 4.9231 4.9323 4.9396 4.9455 5

2

2

d 0dt + =

1. Tx = &

. 2.

. ;

3. =0 Lyapunov 22 1V(x) 0.5x 1 (x )= +

;

1.

1 2

2 1

x xx (x )

==

&&

19

21

x 0(x ) 0=

=

1

2

x n , nx 0

= =

&

e1

0x

0 =

e2x 0 =

2. xe1

12

12

1 1

1 21

x 012 2x 0

x 01 2x 0

f fx x 0 1 0 1

(x ) 0f f 1 0x x

==

==

= = =

xe2

12

12

1 1

1 22

x12 2x 0

x1 2x 0

f fx x 0 1 0 1

(x ) 0f f 1 0x x

==

==

= = =

1

21 1

s 1p (s) det(sI ) det s 1

1 s = = = +

. . 2

22 2

s 1p (s) det(sI ) det s 1

1 s = = =

p2(s) . Lyapunov .

20

3. x1e V(x) x1=x2=0.

1 2 2 1 2 11 2

dV V Vx x x (x ) x [ (x )] 0dt x x

= + = + = & & 0

dV/dt V/x , Lyapunov . Lyapunov ( ).

21

liseis thematon sae 3_09

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