6 Root Locus

Click here to load reader

  • date post

    13-Dec-2015
  • Category

    Documents

  • view

    234
  • download

    5

Embed Size (px)

description

6 Root Locus

Transcript of 6 Root Locus

  • Introduction to Control Systems

    VI. (Root Locus)

  • r y+

    - K ( )G s

    ( ) ( )( ) 1 ( )

    Y s KG sR s KG s

    =+

    : 1 ( ) 0KG s+ =

    , K 0 .

    1 ( ) 0( )1 0( )

    ( ) ( ) 01( )

    KG sn sKd s

    d s Kn s

    G sK

    + = + = + = =

  • ( )1( )

    1G s

    s s=

    +

    ( ) .

    ( )2

    1 2

    1 ( ) 0

    1 0 01

    1 1 4: ,

    2

    KG sK s s K

    s s

    Kr r

    + =

    + = + + =+

    =

    -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2-0.8

    -0.6

    -0.4

    -0.2

    0

    0.2

    0.4

    0.6

    0.8Root Locus

    Real Axis

    Imag

    inar

    y A

    xis =sin-1(J)

  • ()

    1 ( ) 0( ) ( ) 1 0

    ( ) 1

    ( ) 180 360

    KG sKG s KG s j

    KG s

    KG s

    + =

    = +

    =

    =

    s 1+KG(s)=0.

    G(s) .

    G(s) G(s) 180 360

  • ( )

    ( )1

    1

    0 1 2 3 4

    ( )

    ( )

    m

    iin

    ii

    s zG s

    s p

    G s

    =

    =

    +=

    +

    =

  • 1: 1+G(s)=0 ( ) : 1+P(s)=0.

    ( )

    ( )1

    1

    1 0

    m

    iin

    ii

    s z

    s p

    =

    =

    ++ =

    +

    3: X .

    2: P(s) :

  • 4: . ) =0 P(s). , P(s). . . ) , . ..

    Re(s)

    Im(s)

    s0

  • r y+

    - K ( )G s

    3 2

    1( )8 32

    G ss s s

    =+ +

    1: 3 2

    3 2

    : 1 08 32

    1( )8 32

    s s s

    P ss s s

    + =

    + +

    =+ +

    2: ( ) ( )( )23 2 28 32 8 32 4 16: 0,-4 4

    : , ,

    s s s s s s s s

    j

    + + = + + = + +

  • 3&4: s=0.

    Re(s)

    Im(s)

    -4

    -4j

    6: .

    5: , o n ( m m > n ).

  • 7: .

    n= m=

    n-m

    ( ) ( )

    ( )

    1 1

    2 1 180 , 0,1,... 1

    n m

    i ii i

    p z

    n mq q n m

    n m

    = =

    =

    +

    = =

  • ( )( )

    ( ) ( ) ( ) ( )

    2

    1 1

    1( )4 16

    3

    0 4 4 4 4 0 83 3

    2 1 2 1180 180 , 0,1,23

    60 ,180 ,300

    n m

    i ii i

    P ss s

    p z j jn m

    q q qn m

    = =

    =+ +

    +

    = = =

    + + = = =

    =

    Re(s)

    Im(s)

    -4

    -4j

    -8/3

  • 8: ( Routh-Hurwitz)

    ( )( )2

    3 2

    3

    2

    1

    0

    1( )4 16

    1 ( ) 0 8 32 0

    Routh

    1 328

    P ss s

    KP s s s s K

    sKs

    bsKs

    =+ +

    + = + + + =

    =256 s=j0 0.

    ( ) ( ) ( )3 20 0 0

    20

    030 0

    8 32 256 0 :

    8 256 032 5.66

    sec32 0

    j j j

    rad

    + + + =

    + = = = + =

    8 32 0 2568

    Kb K = >

  • 9: ( )

    ( )

    ( ) 2

    1( )1

    1 ( ) 01

    10 2 1 02

    P ss s

    KP sK s s s s

    dK s sds

    =+

    + =

    = + =

    = = =

    Re(s)

    Im(s) -4j

    -1/2 -1

    : 0 ( )dKds

    =

    2

    ( ) ( )1 0 ( ) ( ) 0( ) ( )

    ( ) ( ) ( ) ( ) 0( )

    n s d sK d s Kn s Kd s n s

    dK n s d s d s n sds n s

    + = + = =

    += =

  • ( )

    3 2

    3 2

    2

    1( )8 32

    1 ( ) 0 8 32

    3 16 32 0 2.67 1.89

    0

    .

    P ss s s

    KP s K s s sdK s s s jds

    dKds

    =+ +

    + = =

    = + + = =

    =

    2

    2

    2

    ( )1

    11 ( ) 0

    0 1 0 1

    sP ss

    sKP s Ks

    dK s sds

    =+

    + = =

    = = = Re(s)

    Im(s) j

    -1

  • 10: [angle of departure] [angle of arrival] .

    ( ) 180 360 s=pi iP s z =

    ( )( )21 2 3

    1

    1

    1

    1( )4 16

    180 360

    90 135 180 360

    405 360 45 360

    45

    P ss s

    =+ +

    =

    =

    = =

    =

    2

    1 2 3

    2

    2

    ( )1

    180 360

    90 90 180 360

    180

    sP ss

    =+

    =

    =

    =

    Re(s)

    Im(s)

    2

    1

    3

    Re(s)

    Im(s)

    3

    2

    1

  • 11: .

    ( )( )21( )4 16

    P ss s

    =+ +

    =256

    =256

  • r y+

    - K

    ( )G s

    2

    11 0sKs+

    + =

    2

    1ss+

    1&2: :

    2

    1( ) sP ss+

    =: 0,0

    : -1,

    3&4:

    Re(s)

    Im(s)

    -1

  • 5,6,7:

    ( ) ( ) ( )1 1

    2

    0 0 1 1

    12 1 180 180 , 0

    n m

    i ii i

    p z

    n mq q

    n m

    = =

    +

    = = =

    += = =

    8: .

    2

    2

    1

    0

    1 ( ) 0 0

    Routh

    1 >0 .

    KP s s Ks K

    s Ks Ks K

    + = + + =

  • 9: ( )

    ( )( )

    22

    22

    2

    2 2

    11 0 ( 1) 0

    2 10

    1 1

    2 2 0( 2) 00, 2

    sK s K ss

    s s ss dKKs ds s

    s s ss s

    s

    ++ = + + =

    + += = =

    + +

    + =

    + =

    =

    10:

    2 1

    1 1

    1

    2 180

    0 2 180 90

    90

    =

    = =

    =

    Re(s)

    Im(s)

    -1 1

    2

  • 11:

    Re(s)

    Im(s)

    -1 -2

    s=-2 ( s=-2). : ( ) ( )22

    2

    s 1 2

    4 44

    K s s

    s sK

    + + = +

    = + +

    =

  • ( )2

    1( )4

    sP ss s

    +=

    +

    1,2,3,4:

    Re(s)

    Im(s)

    -1 -4

    5,6,7:

    ( ) ( ) ( ) ( )1 1

    3 2

    0 0 4 1 32 2

    2 1 180 90 ,270 0,1

    n m

    i ii i

    p z

    n mq q

    n m

    = =

    + +

    = = =

    += = =

    Re(s)

    Im(s)

    -1 -4 -3/2

  • 8: ( )2

    3 2

    3

    2

    1

    0

    11 ( ) 0 1 04

    4 0

    Routh

    14

    >0 .3 4

    sKP s Ks s

    s s Ks K

    KsKs

    KsKs

    ++ = + =

    +

    + + + =

    9:

    ( )

    ( )( ) ( )( )

    ( )

    2

    2 2

    2

    3 2

    2

    41

    3 8 1 40

    1

    2 7 8 0

    2 7 8 0

    0, 1.75 0.97

    s sK

    ss s s s sdK

    ds s

    s s s

    s s s

    s j

    +=

    +

    + + += =

    +

    =

    + + =

    =

  • 10:

    1

    1

    1

    1

    2 180

    0 0 2 180

    90

    90

    z p

    =

    =

    =

    =

    Re(s)

    Im(s)

    -1 1

    z

    -4 p

    11:

    Re(s)

    Im(s)

    -1 -4

  • Matlab

    -25 -20 -15 -10 -5 0 5 10-20

    -15

    -10

    -5

    0

    5

    10

    15

    20Root Locus

    Real Axis

    Imag

    inar

    y A

    xis

    num=1; den=[1,8,32,0]; rlocus(tf(num,den))

    ( )( )21( )4 16

    P ss s

    =+ +

  • -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5-1.5

    -1

    -0.5

    0

    0.5

    1

    1.5Root Locus

    Real Axis

    Imag

    inar

    y A

    xis

    num=[1 1]; den=[1,0,0]; rlocus(tf(num,den))

    2

    1( ) sP ss+

    =

  • ( )21( )

    4sP s

    s s+

    =+

    num=poly([-1]); den=poly([0,0,-4]); rlocus(tf(num,den))

    -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5-10

    -8

    -6

    -4

    -2

    0

    2

    4

    6

    8

    10Root Locus

    Real Axis

    Imag

    inar

    y A

    xis

  • ( )2

    1( )12

    sP ss s

    +=

    +

    ( ) ( ) ( ) ( )1 1

    : 0,0,12 : 1, ,3 2

    0 0 12 1 112 2

    2 1 180 90 ,270 , 0,1

    n m

    i ii i

    p z

    n mq q

    n m

    = =

    + +

    = = =

    += = =

    Re(s)

    Im(s)

    -1 -12 -11/2

  • 3 2

    3

    2

    1

    0

    12

    Routh

    112

    >0 .11 12

    s s Ks K

    KsKs

    KsKs

    + + +

    ( )

    ( )( ) ( )( )

    ( )( )

    2

    2 2

    2

    2 2

    2

    121

    3 24 1 120

    1

    12 3 27 24 0

    2 15 24 0

    0, 5.18, 2.31

    s sK

    ss s s s sdK

    ds s

    s s s s s

    s s s

    s

    +=

    + + + +

    = =+

    + =

    + + =

    =

  • :

    Re(s)

    Im(s)

    -1 -12

  • ( )21( )12

    sP ss s

    +=

    +

    num=poly([-1]); den=poly([0,0,-12]); rlocus(tf(num,den))

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

    -8

    -6

    -4

    -2

    0

    2

    4

    6

    8

    10Root Locus

    Real Axis

    Imag

    inar

    y A

    xis

  • ( ) ( )1 1

    : 0,-2,-1 2j : , , ,4 4

    0 2 1 1 14

    2 1 180 45 ,135 , 45 , 135 0,1,2,3

    n m

    i ii i

    p z

    n mq q

    n m

    = =

    = = =

    += = =

    ( ) ( )( )21( )

    2 1 4P s

    s s s=

    + + +

    ( )

    4 3 2

    4

    3

    2

    1

    0

    4 9 10 0 Routh

    1 94 10

    13 22 65 4 13

    s s s s K

    Kss

    KsKs

    Ks

    + + + + =

    06565 4 0 16.254

    K

    K K

    >

    > < =

    0 16.25K <

  • 0 s=j0 =16.25

    ( ) ( ) ( ) ( )

    ( )

    4 3 20 0 0 0

    4 3 20 0 0 0

    4 20 0

    30 0

    2 20 0 0 0

    4 9 10 16.25 0

    4 9 10 16.25 0 :

    9 16.25 0

    4 10 052 2 5 0 1.582 sec

    j j j j

    j j

    rad

    + + + + =

    + + =

    + = + =

    = = =

    -1+2j

    1 2 3 180 360

    116.6 63.4 90 180 360

    270 180 360 90

    90

    dep

    dep

    dep

    dep

    =

    =

    = + =

    =

  • ( )( )

    4 3 2

    3 2

    2

    4 9 10

    4 12 18 10 0

    1 4 8 10 0

    1, 1 1.22

    K s s s sdK s s sds

    s s s

    s j

    =

    = =

    + + + =

    =

  • ( ) ( )( )21( )

    2 1 4P s

    s s s=

    + + +

    num=1; den=poly([0,-2,-1+2i,-1-2i]); rlocus(tf(num,den))

    -8 -6 -4 -2 0 2 4 6-8

    -6

    -4

    -2

    0

    2

    4

    6

    8Root Locus

    Real Axis

    Imag

    inar

    y A

    xis

    Introduction to Control SystemsSlide Number 2Slide Number 3Slide Number 4Slide Number 5Slide Number 6Slide Number 7Slide Number 8Slide Number 9Slide Number 10Slide Number 11Slide Number 12Slide Number 13Slide Number 14Slide Number 15Slide Number 16Slide Number 17Slide Number 18Slide Number 19Slide Number 20Slide Number 21Slide Number 22Slide Number 23Slide Number 24Slide Number 25Slide Number 26Slide Number 27Slide Number 28Slide Number 29Slide Number 30Slide Number 31Slide Number 32Slide Number 33Slide Number 34