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					                                                        자동제어




                                  Lecture Note
                                      on

                 Control System Theory


Text      : G. F. Franklin et al.,
            Feedback Control of Dynamic Systems.
            Addison - Wesley Pub. Co. : 3rd ed.,1994.
Reference : Dorf, Modern Control Systems.
            Kuo, Automatic Control System
            MATLAB Toolbox Manuals.
                                                               자동제어


 ( Ch. 1 & Ch. 2 of Flanklin & Powell : Reading Assignment )

1. An overview of Feedback Control
  • Brief history
  • System classification
         dynamic             linear        time - invariant
         instantaneous       nonlinear     time - varying
         causal
         noncausal (anticipatory)
         linear time - invariant causal dynamic system
                                              자동제어




• Basic concepts concerning model
   • why modeling?
   • how modeling?

   • what about LTI system?
      I/O model
                            what relation ?
      state - space model



• why feedback control?
                                                        자동제어




• How to design feedback control systems?
      analysis (modeling, response)
      synthesis (control spec.)

      analysis : time response, Bode, Nyquist, RL,...
      synthesis : classical
                  modern
                                                         자동제어




Ch. 2 Dynamic Models : Reading Assignment
Ch. 3 Dynamic Response
  • Review of Laplace transform

  • model diagram     Block diagram
                      Signal flow graph & Mason’s rule
  • response vs. pole location & time – domain spec.
  • effects of zeros & additional poles
                                                                                            자동제어




1. Review of Laplace Transform
                                                                     
    two – sided Laplace transform                       F ( s)         f (t ) e  st dt
                                                                   
                                                                     
    one – sided Laplace transform                       F ( s )    f (t ) e  st dt
                                                                     0




     Example : Laplace transform of unit step ftn.
           f (t )  1(t )
                                              st   
                                           e                     1 1
          F ( s )   1(t ) e  st dt                  0       
                      0                      s      0
                                                                  s s
              e  st
          why                      0 ?
                s           s
                                                                 자동제어

〈 cf 〉Fourier transform
                               
                   F (s)         f (t ) e  jwt dt
                              

  only interest in steady – state response
  usually, the frequency response means the steady – state
  response of systems to a sinusoid!

          Sinusoid                   H(s)                y(t)

      y (t )  H ( jw ) e jwt ,    y (t )  H ( jw ) e  jwt

                             cos wt  e  e  jwt 
                                       1 jwt
  In case of a sinusoid
                                       2
   y (t )  H ( jw )e jwt  H (  jw )e  jwt   M ( w) cos(wt   )
           1
           2
  where H  Me j
                                                                                                              자동제어


• Properties of Laplace transform (skip)
          Will only have a look at convolution!
                                                  t
                 f1 (t )  f 2 (t )          0
                                                      f1 ( ) f 2 (t   ) d
                           
L f1 (t )  f 2 (t ) 
                                 t
                            
                           0   0
                                     f1 ( ) f 2 (t   ) d e  st dt

                                       
                               
                               0
                                          f1 ( ) f 2 (t   ) e  st dt d

                                                             
                                                         
                                                    s
                                      f1 ( ) e               f 2 (t   ) e  s (t  ) dt d
                               0
                                                             
                                                         
                                                    s
                                      f1 ( ) e                  f 2 (t ) e  st dt d  F1 ( s ) F2 ( s )
                               0                           0

          Laplace transform is a useful tool of
           algebric analysis !!
                                                                    자동제어


• How to find f (t ) from F (t ) ??
      ① Inverse Laplace transform
                                                 skip.
      ② partial – fraction expansion


• Laplace transform theorems
       Final value theorem : If all poles of s F (s) is stable,
                              then       lim f (t )  lim s F ( s)
                                         t             s 0




       Initial value theorem : For any Laplace transform
                              pair,       lim s F ( s)  f (0  )
                                          s 
                                                                                            자동제어

〈 pf 〉① Final value theorem
         
       
     L f(t) 
                   

                   0
                       
                       f(t) e  st dt  sF ( s )  f (0)
    lim s F ( s )  f (0)  lim  f(t) e  st dt
                                          
                                    
     s 0                            s 0 0
                        f ()  f (0)
     lim f (t )  lim s F ( s )
            t           s 0


        ② Initial value theorem

     s 
                           
     lim s F ( s )  f (0 )  lim    s  0
                                             
                                                  
                                                      
                                                      f(t) e  st dt
                                     0 f(t)dt                                f(t)e  st dt 
                                                      
                                                                           
                           lim                                           
                                                                                             
                               s   0                                0
                                                                                             
                           f (0  )  f (0  )
     lim s F ( s )  f (0  )
            s 
                                                             자동제어


Remark : Final value theorem can be only
           applied to stable system!
                  s2                           s2
   Y1 ( s )                 ,   Y2 ( s ) 
              s  1s  3                s  1s  3
                 0. 5      0.5 
  y1 (t )  L 
             -1
                                        
                                   0.5 e t  e 3t  
                s 1      s  3
             -1  0.75     0.25 
  y2 (t )  L                   0.75 e t  0.25 e 3t
                 s 1     s  3

  lim sY1 ( s )  0  lim y1 (t )
   s 0                 t 
  lim sY2 ( s )  0  lim y1 (t )
   s 0                  t 
                                                                      자동제어


• Usage of Laplace transform
   • analyzing the frequency response algebraically

   • solve problems, especially, differential equations

• Laplace transform using MATLAB
   x(t )  A x(t )  B u (t )
   
   y (t )  C x(t )  D u (t )
                or
   y ( n )  a1 y ( n 1)    an 1 y  an  bm u ( m )    b1 u  b0
                                                                  

                           Y ( s)  G( s) u ( s)

    “ ss2tf ”
                                              자동제어



2. System Modeling Diagrams
  (1) The Block Diagram


     Algebraic operator :     +       ,   ·




          symbol          :       ∑   ,
                                                                         자동제어


    Example : Find transform from R to                   y1 !

                              y1       y1  G1 u1
         +    u1
R                   G1                 u1  R  y 2                      G1
          -                                                     y1           R
                                       y 2  G2 u 2                  1  G1G2
              y2
                    G2
                              u2       u 2  y1


R   +          G2        G1        y
     -                                                     G1 G2
                                                      y          R
                                                         1  G1G2


     common method
     but tricky to obtain transform!
                                                             자동제어


(2) Signal Flow Graph

           Block Diagram               SFG

                signal                 node
                block                 branch


용어 : node, branch, forward path, loop gain, path gain


 R    1    u1    G1      y1       R     1       G2        G1 y



                -G2
                                                     -1
                                                        자동제어


Mason’s rule
                       y ( s)    Pi  i
               G( s)         
                       u ( s)      

    System determinant
      1   ( all loop gain )   ( gain products of
         all possible two loops which do not touch)
          ( ~ three ~)   

  Pi  i th forward path gain
   i  i th forward path determinant
      i th forward path를 제거한 SFG의 Δ
                                                    자동제어



    Example
                                             node

R     1       G2        G1 y

                   L1

                   -1

                    L1   G1 G2
                    P  G1 G2 ,
                     1              1  1
                      1  G1 G2
                                G1 G2
                     G (s) 
                              1  G1 G2
                                               자동제어


    Mason’s rule is particularly useful
                                           P.117 예 3.22
    for more complex system!

      H4


                      H6



u     H1              H2              H3           y




      H5                              H7



    Summing junction 의 출력
                               node
             block 의 출력

    SFG 를 그리지 않고 Block Diagram 에 직접
    Mason’s rule 적용가능!
                                                                          자동제어

3. Response vs. Pole Locations
Def : (finite) pole/zero
                                   n( s )
                           G( s) 
                                   d (s)
                           pole   p | d ( p)  0 
                           zero    z   | n( z )  0 

Remark : infinite pole/zero
Consider the Following 2nd order system :
               n2

G( s)  2
        s  2  n s   n
                         2

                                                   underdamped : 0    1
                : damping ratio                  critically damped :   1
                                                  overdamped :   1
                                                      자동제어


① overdamped case (               )1
  have real poles , e.g. –a & -b
  have a look at the step response

                            A           
     y( t )  L1                       
                   s ( s  a )( s  b ) 

                        
           L                e   e ,t  0
               1                     at   bt

              s  s  a s  b

      stability depends on the pole location.

  (NOTE)             n  0  unstable
                                                                자동제어


② critically damped case (   1 )
  real double poles : ( s   n )2  0
  step response

                  2
                         1    1                    
y( t )                                    
                   n                                     n

         s( s   n ) 2
                         s  s          n       ( s   n )2

                 nt            nt
y(t )  1  e           ne
                                                                       자동제어

③ underdamped case (       0  1
                                )
  complex poles :         n  j n       1       2


  step response
                       n2

    y(s) 
           s ( s 2  2  n s   n )
                                  2


                            n
                             2

          
              s [(s    n ) 2   n (1   2 )]
                                    2


            1            s  2  n
             
            s   ( s    n ) 2   n (1  
                                    2             2
                                                      )
            1             s  n
             
            s   ( s    n ) 2   n (1  
                                    2             2
                                                      )
                        n                   n 1  2
                
                     n 1     2   ( s    n ) 2   n (1   2 )
                                                        2
                                                                                                      자동제어



                                                                   n e   t
                                                                             n


                         cos              1                                    sin  n 1   2 t
                  nt
y( t )  1  e                   n
                                                       2
                                                           t 
                                                                  n 1 2

      1
                 1
                    1
                          2
                              e     n   t
                                                       
                                               sin  n 1   2 t                
                                     1           2

     where   tan 1
                                               
                                                   자동제어


• Pole pattern of 2nd order system
                                            î
                                            ¥= 0


                                     jwn
                 ¥
               0<î <1




                        -¥wn
                         î           0
             î
             ¥= 1




                                     -jwn
                                                             자동제어


 • 0    1경우
                                    wd  wn 1   2
                               wn
                                                  cosq  
                               q
                        wn




             : damping ratio
             n : natural frequency
             d : damped natural frequency

stability depends on            n  o          stable
                                     o          unstable
late we will see more complex systems!
                                                                      자동제어



4. Time – domain Specifications
Def :
          • rise time   :   tr           ㆍ setting time :        ts
          • overshoot   :   M Pt         ㆍ peak time :           tp
                 tp


                            Mpt
          1                                                  ¾
                                                             ¡ 10%
         0.9




         0.1

          0                                 ts               t
                tr
typical unit step response of underdamped 2nd order system
                                                                                               자동제어


* relation between time – domain spec. and                                           , n
   1. t r  1.8             ( Rmk :   0.5 인경우)                           zero가 없는 2차원
                 n                                                        시스템 경우만
                                                                           비교적 정확
                                                                 4.6
   1. t s : e
                    t
                      n s
                             0.01                      ts 
                                                                 n

   2. M P & t p :
         t




                   n e         n   t
                                                             n 1   2
      y( t ) 
                                           sin(       )                  e   t cos(
                                                                                n
                                                                                           )  0
                      1       2
                                                                1    2



      sin (     )  1   2 cos (                      )  0
     sin  n 1   2 t  0 ,                        n 1   2 t  0 ,  , 2 , 
                            
      tp 
                  n 1   2
                                                                                                 자동제어

                                                                 
                                                      n                t
                                        1
                                                                                  sin(    )
                                                             n 1   2
M P  y( t p )  1                            e
                                      1
  t
                                            2


                                                              
                                                        
           1
                                      sin    e                                    0    1
                            1                                1  2
      
                                  2

                    e                                                         ,
          1   2
                                                                     자동제어


* How to use time – domain spec. when designing?
 ( time - domain spec.)                ( pole location)

        Given                           1.8
                                 n   
     tr , M P , ts                       tr
                                                            ln M P
                                      (MP ) 
             t
                                                                 t



                                                       2  ln 2 M P
                                              t



                                        4.6                            t


                                  n 
                                         ts
    In other words ,  n det er min es rise time
                          det er min es overshoot
                          n det er min es settling time

                 (see Fig. 3.18 in p. 129)
                                                                       자동제어



Example 3.25 : (Transformation of Spec. to s – plane)
   Find the allowable region of poles if requirements
 and t r  0.6 sec , M P  10% , t s  3 sec
                          t
                                          .
 (sol)
                                                        Im

            1 .8
     n          2.8 rad
                           sec
                                                        j 2.8
             tr                                    wn

         0.6                                     q
                                            wn                  Re
            4.6
      n       1 .5
             ts                                          j 2.8
                                                                               자동제어



5. Effects of Zeros and Additional Poles
* effect of zeros → transient response 의 영향


   1.    real pole 경우 : zero 를 더해서 계수변화
                                2                               2 ( s  1.1 )
   (예)    G1 ( s )                            G2 ( s ) 
                       ( s  1 )( s  2 )                    1.1( s  1 )( s  2 )
                                                                   자동제어

1.   complex pole 경우 : overshoot 증가

                           2
     G1 ( s ) 
                  s 2  2 n s   n2
                          
                    ( s         1 )
                      n
     G2 ( s )  2
                s  2 n s   n2
                       n2             1           s  n2
               2                  
                s  2 n s   n2
                                       n s 2  2 n s   n2

               pole s      j 1   2
              
              
                              n    n

               zero s      n
              
                                                                 자동제어

   1.     4 근방부터 overshoot 에 영향 大
   (cf) How about           0 ? (NMP zero)




                                    0




* effect of extra poles               rise time 증가
                                          n2
              G1 ( s ) 
                         ( s  n  1 )( s 2  2 n   n2 )
                               A          Bs  C
                      
                        s  n  1 s 2  2 n   n2
            n t r에 영향
               4                    ( See Fig. 3.25 - 3.26 )
                                                                                       자동제어



6. Numerical Solutions
 x( t )  f ( x( t ) , u( t ) , t )
                                                    Runge – Kutta Method.

선형미방의 경우              Runge – Kutta
                      또는 analytic sol. 부터

         x  A x  Bu
         

                                                     
                                                         t
         x( t )  e     A ( t  t0 )
                                        x( t 0 )            e A ( t  t ) Bu( t )d
                                                                    0

                                                      t0

            x( t  1 )  


7. Obtaining Models from Experimental Data (skip)
                                              자동제어



Ch. 4 Basic Properties of feedback
   • adventages of Feedback control
   • PID controller : P. I. D. tuning.
   • Tracking vs. System Type
   • Stablility : Routh – Hurwitz criterion

1. Advantages of FB Control
       disturbance rejection
       sensitivity reduction
       tracking (steady – state error)
       transient response
                                                                       자동제어



         open - loop                             closed - loop
                              d                                    d



                u                          +           u
r         K            G(s)        y   r         K          G(s)        y
                                           -




    ① disturbance rejection
       - open _ loop :        y  d  K Gr
                                    KG         d
       - closed _ loop :      y         r
                                  1 K G    1 K G
                                   K G  1  y d  0
                                                                       자동제어


② sensitivity reduction
                       G  G  G
   - open _ loop :     y  K G r  K G r
                               K ( G  G )
   - closed _ loop :    y                     r
                             1  K ( G  G )
                                        K ( G  G )         KG
                           y                       r           r
                                     1  K ( G  G )      1 K G
                                                  K G
                                                                  r
                                     ( 1  K G  K G )( 1  K G )
                          if K G  K G
                                         K G
                              y                 r
                                      (1 K G ) 2


                           K G  1  y  0
                                                                        자동제어


(cf) Definition of sensitivity

                   T T             G T    ln T
        S    T
                        0
                          S G 
                           
                                 T
                                         
                   G G             T G    ln G
            G



                                GK
         ( 예) T 
                               1 GK
                                                        G       G
                  S    T
                            S    GK
                                       S   1 G K
                                                         K        K
                                                              1 GK
                      G          G          G
                                                       GK
                                 1
                           
                               1 GK
                                                         N
                                                       ln
                               N                         D  SNSD
         ( cf ) T                      S GT 
                                                       G
                                                              G  G
                               D
                                                                            자동제어


  ③ tracking (steady – state error)
     - open _ loop :        e o ( s )  r( s )  y( s )  ( 1  K G ) r
                                            1
     - closed _ loop :       ec ( s )           r
                                        1 KG

     step input :
                                            1
                                                1  K ( 0 ) G( 0 )
          lim e o ( t )  lim s ( 1  KG ( s ) )
          t              s0              s
                                    1   1           1
          lim  e c ( t )  lim s          
          t              s  0 1  KG s   1  K ( 0 ) G( 0 )

(NOTE) 일반적으로              G( 0 )  1             e c (  )  e o (  )
                                                               1
                    if G( 0 ) K ( 0 )  1         ec (  )     , eo (  )  0
                                                               2
                                            But sensitivity problem.
                                                            자동제어


Ramp input :
                                          1         1 KG
       lim e o ( t )  lim s ( 1  KG ) 2  lim
       t              s0              s      s0    s
                                 1     1            1
       lim  e c ( t )  lim s                lim
       t              s  0 1  KG  s2      s0 sK G



Parabola input :
                             1  KG
      lim e o ( t )  lim
      t              s0      s2
                               1
      lim  e c ( t )  lim 2
      t              s  0 s KG




 Remark : System Type vs.             e ss
                                                               자동제어


④ transient response
                                              nc       np
  ←determined by pole location ,          K     , G 
                                              dc       dp
   - open _ loop :     dc d p ( s )  0
   - closed _ loop :   1  KG  0          d c d p  nc n p  0
                                                                 자동제어

In Summary
         KG




                             Stability




                                         measurement noise
                                            model error


                Command                                      w
               disturbance




(NOTE) Feedback 의 단점
       - complexity
       - cost
       - gain 이 줄어든다
       - unstable 가능성.

				
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posted:10/4/2011
language:Korean
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Lingjuan Ma Lingjuan Ma
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