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Chapter8 Root-Locus Technique

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					Root-Locus Technique

         8
•   Root Loci
•   Properties of Root Loci
•   Root Sensitivity
•   Root Contours
•   Root Loci of Discrete-Data Systems
R(s)                               C(s)
                     G(s)

                     H(s)


       C ( s)     G ( s)
                                          (a)
       R( s) 1  G ( s)  H ( s)

  Characteristic equation 1+G(s)H(s)=0 ,   (1)
Where, G(s)H(s) is defined as fallow

                  k ( S  z1)(S  z 2)  ( S  z m)
G ( s) H ( s) 
                  (S    p )(S  p ) ( S  p )
                          1        2               n

                    KQ ( s )
             =                               (2)
                     P(s)
eq (2) into eq (1)
                KQ ( s )
             1          0                   (3)
                 P( s)
eq (3) into eq (a)

c( s )    G( s)     G ( s) P( s)
                                            (4)
k ( s) 1  KQ( s) P( s)  KQ( s)
            p( s)
The charicteristic eq from eq (4)

 1  G(s) H (s)  P(s)  K Q(s)  0

F (s)  P(s)  K Q(s)  0                 (5)

 The parameters of P(s) Q(s) are all flxed
 Root-Loci
 Root-Contours

• RL (Root-loci)   ;   0k 
• CRL (complementary RL) ;    k  0
• RC (Root contours) ; Contours of
  roots when more than one
  parameters variable
• Root-locus     ;
                      k  
            From eq (5)

    k=0  root ;        p(s)
    k=   root ;        Q(s)

    if k is varied zero to  than Root-locus
*   start from open-loop pole finished to
    open-loop zero
Ex) s( s  2)(s  3)  k ( s  1)  0 의 k  0인 점을 확도


                               k ( s  1)
  1  G ( s) H ( s)  1                     0
                          s ( s  2)( s  3)
                     k ( s  1)
   G(s) H (s) 
                s ( s  2)( s  3)                           j


   K=0인점            s  0, s  2, s  3
   K=    인 점 s  1                                              
                                                  x x        x
                                                  -3 –2 -1
Root-Locus Technique :
       Developed by W.R Evans 1948
 If the system has a variable loop gain as follow fig.
    R(s)                                      Y (s)
                     k          G(s)



*   Root-Locus
    [The variable loop gain k    the location of the closed-
    loop poles]
    The characteristic of the transient response
    The system stability
      Design Problem


[the approprite gain selection]
[addition of compensator]
  By using the root-locus method the designer can

   predict the effects on the location of the closed-
   loop poles of varying the gain value or adding
   open-loop poles and/or open-loop zeros.
Definition
Root Loci    ; System parameter; fixed

                    k        ;variable
     
RL ( 0  k   )+ CRL (    k  0 )

RC (Root contours)      the Root-Loci of
   multiple-variable parameter at a timer
  Root-Locus Plots
R(s)                      Y (s) Y ( s)         G(s)
              G(s)               R( s)     1  G ( s) H ( s)

               H (s )           1  G( s) H ( s)  0

  Condition on Magnitude

       G(s) H (s)  1  j 0
       G( s) H ( s)  1
Let’s K is isolated as a multiplying factor

G(s)H (s)  K G1 (s) H 1 (s)  1  j0
                        1
 G1 ( s ) H 1 ( s )    K          1 

   Angle condition
     G ( s ) H ( s )   (2i  1)       ,       i  0,1,2,3,
                                    
                         180  i 360        ,   i  0,1,2,

G(s)H(s) involves a gain parameter K, and the
characteristic equation may be written as
                                                K ( s  z1)(s  z 2)  ( s  z m)
G ( s) H ( s)  K G1 ( s) H 1 ( s ) 
                                                (s        p )(s  p )(s  p )
                                                             1           2           n

                                   m
                                   s  zk              1
      G (s) H
        1           1
                        ( s)     k 1
                                   n
                                                      
                                                        K
                                                                           K  
                                   s        p
                                  j 1            j




for          0  K   ( RL)
                           m                           n

      G (s) H       (s)                (s  z k )            (s    p )  (2i  1)
        1       1                                                            j
                           k 1                       j 1
for       k  0             (CRL)
                      m                 n

  G (s) H
       1     1
                 (s)      (s  z k )       (s    p )  2i
                                                        j
                     k 1               j 1


                                       i  0,1,2,

      The difference between the sums of the angles of
      the vectors drawn from the zeros and those from
                                  s
      the poles of G(s)H(s) to 1 is an odd multiple of 180
                                                                  




For difference values of K, any points         s on the CRL
                                                 1
must satisfy the condition:
The difference between the sums of the angles of the
vectors drawn from the zeros and from the poles of
             s                            
G(s)H(s) to 1 is an even multiple of 180 , including
zero degrees.
Once the root loci are constructed , the value of K along the
loci can be determined by                n
                                         s   p
                                        j 1
                                  K 
                                                   j
                                          m
                                         s  zk
                                        k 1
Ex)                        k ( s  z1)                       KA
      G( s) H ( s)                          G( s) H ( s) 
                       s(s    p )(s  p )                  BCD
                                2        3
Case of RL
     (s1  z1)         s
                        1                ( s1    p )  (s  p )
                                                   2      1   3


  z1  p1  p 2  p3  (2i  1) (i  0,1,2)

Case of CRL

      
       z1      p1       p2
                              p3  2i


      K 
             s sp sp
              1     1        2       1        3
                                                  
                                                    BCD
                s z    1        1
                                                     A
    Ex) F ( s )  ( s  1)          에서   s  3  j 4
                  s ( s  2)
          인점의 크기와 각도는?
                 j
s  3  j 4
                               4
                              3

                              2
                              1

    3      2       
                     1

                                                        
                                         (s  1)  116.27
          A   4 2            20 ,
                 2        2
                                                   
B
     2
          3  5,
            2              (s)     104.04
     4
C  17
                                               
                           (s  2)  126.87


M          
               A           s 1         s            s2
              BC

                      20
                
                                                           
                           116.27       104.4 126.87
                    5 17

                             114.34
                                          

                 0.217
                                                      Y (s)
Ex) R(s)                                      1
                           K             s ( s  2)



                                 K
     H ( s )  1, G ( s ) 
                            s ( s  2)

Closed-loop characteristic eq

                      2S  K  0
                 2
             s
             s 1, 2
                       1  1  K
1.K  1, s  s  1
               1        2
                                                         j
2.K  0, s  0, s  2
               1            2
                                           K 
 3.0  K  1           , 음의 실근

 4. 1  K   ,음의 복소 공액근
 5.    K  0                           K  0 K 1K  0
                                K     2      1 0   K  
s 은(+)실근, s
 1                 2   는 (-)실근
Properties and Construction
                       of the Root-Loci
1) K=0 and K=          points

 The k=0 points on the root loci are at the
     poles of G(s)H(s)

 The K=   point on the root loci are the
     zeros of G(s)H(s)
  (근궤적은 극점에서 출발하여 영점에서 끝난다.)
2) Number of Branches
               on the Root-Loci
   The Number of branches of the Root-
  Loci is equal to the order of the
  polynomial
 (근궤적의 수는 영점수와 극점수중 큰것과 일치)


3) Symmetry of the Root-Loci
    The Root loci are Symmetrical with
  respect to the real axis of the s-plane
  (근궤적은 실수측에 대해 대칭)
4) Asymptotes of Root-Loci

  Number of Asymptotes

       (n  m) p  z

  Angles of Asymptotes

             2i  1
        i  p  z *180 , n  m (RL)
                       




              2i
      i 
                     
                 *180 , (CRL)
             pz
5) Intersect of the Asymptotes (Centroid)

            finite poles of G(s)H(s)   finite   zeros ofG(s)H(s)
    1
         
                                    nm


6) Angles of Departure and Angles
                              of Arrival of the Root-Loci
  the angle of departure of arrival of a root-locus at a
pole or zero , respectively, of G(s)H(s) denetes the angle of
the taryent to the locus near the point

 D  180 (2i 1)                 A  180 (2i 1) 
                                                   
여기서  는 특정극(영점) 에서 다른 개루프 즉 (영점) 이 이루는
각의 합이다.

                         m                  n
                                                (s    p)
          G (s) H (s)        (s  zi)                  j
                        i 1            j 1

                                        
                       (2i  1) *180


Ex)      p2
                D      (order 3)

               ( p1  D  p3)  (2i  1) *180
                                                       
          z1



         z1  p1  p3)
                               k ( s  2)
Ex)     G(s) H (s) 
                       ( s  1  j )( s  1  j )

                       for K  0, find             D
                                                        at s  1  j



        
        z1   p1      p2
                           180(2i  1) ,           i0


      45  p1  90  180


       
       D     p1
                   180  90  45  225   135

				
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