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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?

I/O model
what relation ?
state - space model

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

analysis : time response, Bode, Nyquist, RL,...
synthesis : classical
modern
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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 ”
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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
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
• PID controller : P. I. D. tuning.
• Tracking vs. System Type
• Stablility : Routh – Hurwitz criterion

disturbance rejection
sensitivity reduction
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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