Performance
Specifications
3
제어응용연구실
⊙ synopsis
▣ Tracking Systems
▣ Forced Response
▣ Power-of-Time Error Performance
▣ Performance Indices and Optimal System
▣ System Sensitivity
▣ Time Domain Design
제어응용연구실
⊙ Analyzing Tracking Systems
○ Tracking Systems
: Control system that creates an output which tracks the input to
some level of tolerance.
제어응용연구실
Ex)
- Step input
T (s) 
50
s 2  4 s  50
50
1
s4
1
Y ( s)  T ( s)  


2
s s 2  4 s  50
 s  s( s  4s  50)
y(t )  1  1.04e2t cos(6.78  163.6 )
제어응용연구실
Forced
component
Natural
component
Forced
component
Natural
component
※ The analysis and design of tracking systems can be separated into two parts
1. The characteristic roots (poles)
Determine the character of the system’s natural response component
2. Tracking of the reference input
the forced response of the system
제어응용연구실
○ Natural Response, Relative Stability, and Damping
- The relative stability : the distance into the left half of the complex plane
제어응용연구실
- A pair of complex conjugafe characteristic roots
yi (t )  Aeat cos(bt   )
  cos
제어응용연구실
s1 , s2  a  jb
⊙ Forced Response
○ Steady State Error
error E ( s)  R( s)  Y ( s)  R( s)  T ( s) R( s)
 [1  T (s)]R( s)  TE ( s) R( s)
 TE ( s )  1  T ( s ) 
E ( s)
R( s )
Zero initial conditions
※ The poles of TE (s )= The poles of T(s)
The forced part of the error signal
e forced (t )  r (t )  y forced (t )
y forced (t )  r (t )  e forced (t )  0
제어응용연구실
=> Perfect tracking
○ Initial and Find Values
- Initial value
y (0)  lim[ sY ( s )]
s 
 4s 4  3s 3  s 2  s  1
Y ( s) 
3s 5  2s 4  s 3  s  10
Ex)
4
y (0)  lim[ sY ( s )]  
s 
3
※ If there is m impulse in y(t) at t=0, then y(0) will be infinite
제어응용연구실
- The final value (steady state value)
lim y (t )  lim[ sY ( s )]
t 
Ex)
s 
 4s 4  2s 2  7 s  3
Y ( s) 
s 3  9s 2  2s
 lim y (t )  lim[ sY ( s )] 
t 
s 
3
2
Multiple poles
 4s 4  2s 2  7 s  3
Y ( s) 
s( s 3  9s 2  2s)
 lim y (t )  lim[ sY ( s )]  
t 
제어응용연구실
s 
6s  7
※ Y ( s) 
s ( s 3  s 2  2 s  3)
RHP
제어응용연구실
 s3
s2
s1
s0
1
2
-1
3
2
3
 lim[ sY ( s )] 
s 
7
3
※ Initial and Final Value Theorems to Representative Laplace Transform terms.
제어응용연구실
○ Steady State Errors to Power-of-time Inputs
R0 ( s ) 
1
Ri ( s )  i 1
s

1
 r (t )  u (t )
s
1
R1 ( s )  2  r (t )  tu (t )
s
1
1 2
R2 ( s )  3  r (t )  t u (t )
s
2
Error signal is
1
Ei ( s )  TE ( s ) Ri ( s )  i 1 TE ( s )
s
제어응용연구실
Ex)
T (s) 

6
s 2  3s  2
s 2  3s  4
TE ( s)  1  T ( s)  2
s  3s  2
Its error to the standard ramp input
s 2  3s  4
E ( s)  TE ( s) R( s)  2 2
s ( s  3s  2)

9 2 2
6
3 2
 2 

s
s
s 1 s  2
Forced
component
e(t ) 
제어응용연구실
9
3
 2t  6e t  e  2t
2
2
Forced
component
Natural
component
Natural
component
(after t=0)
※ The forced component of the error can do only one of three things
1. The forced error can be zero
Output equals the power-of-time reference input
2. The forced error can be a constant
Output and reference input differ by a constant
3. The forced error can involve a nonzero term proportional to t or a higher
power of t
The error grows without bound
제어응용연구실
※ These three situaticns are easily distinguished, without calculating e(t),
by applying the final-value theorem to E(s)
* The final value of e(t) is zero
situation 1
* The final value of e(t) is a finite nonzero constant
situation 2
* E(s) has more than a single pole at s=0,
the fimal value of e(t) approaches infinity
제어응용연구실
situation 3
⊙ Power-of-time Error Performance
○ System type number
: the order of the pole of T(s) at s=0
- the type number is 0


i) Step input  R( s ) 
A

s 
lim estep (t )  lim sTE ( s ) R ( s )
t 
s 
 lim sTE ( s )
s 
A
 ATE (0)
s
Constant
제어응용연구실
ii) Ramp input
A

 R( s)  2 
s 

lim eramp (t )  lim sTE ( s ) R ( s )
t 
s 
A
 lim sTE ( s ) 2  
s 
s
※ Higher power-of-t input give infinite steady state error
제어응용연구실
- the type number is 1
TE (s ) has one factor of s in the numerator
i) Step input
A
lim estep (t )  lim sTE ( s )
 ATE (0)  0
t 
s 
s
ii) Ramp input
lim eramp (t )  lim sTE ( s )
t 
s 
A
TE ( s )

A
lim
s 
s2
s
※ Higher power-of-t input, the error of such a system is infinite, since
has a repeated s=0 denominator root
A
E
(
s
)

T
(
s
)
E
제어응용연구실
sn
- the type number is 2
TE (s ) has one factor of s 2 in the numerator
i) Step input , ramp input
zero steady state error
ii) Parabolic input
A
TE ( s )
lim e parabolic (t )  lim sTE ( s ) 3  A lim
t 
s 
s 
s
s2
제어응용연구실
○ Unity Feedback Systems
TE ( s )  1  T ( s )  1 

제어응용연구실
G( s)
1  G( s)
1
1  G ( s)
Y ( s)
K ( s  2)
K
T ( s) 


R( s) 1  K ( s  2) s  2  K
A
i) step input 
 R( s ) 

s 

Y ( s)  T ( s) R( s)  T ( s)
E ( s )  R( s )  Y ( s ) 
제어응용연구실
A
s
A
A s2 
[1  T ( s)] 


s
s  s2 K 
2A
 s2 
lim e(t )  lim sE ( s )  lim A


t 
t 
s 
 s2 K  2 K
Can be made aibitraily small in this case by choosing a sufficiently
large amplifier gain K
ii) Ramp input
A

 R( s)  2 
s 

K3
K2
 A  s  2  K1
E ( s)   2 




s
s2
s2 K
 s  s  2  K 
K2t  withoutbormd
※ ramp, paratolic , or higher power-of-t input
제어응용연구실

T ( s) 
i) step input
K s( s  2)
K
 2
1  K s ( s  2)
s  2s  K
A

 R( s )  
s

A
A  s 2  2s

E ( s )  R( s )  Y ( s )  [1  T ( s )] 
2
s
s 
s
 2s  K

lim e(t )  lim sE ( s )  0
t 
ii) ramp input
s 
lim e(t )  lim
t 
제어응용연구실
s 
A( s  2)
2A

s 2  2s  K
K




○ Unity Feedback Error Coefficients
Steady state error coefficients of unity feedback system
K i  lim s i G ( s )
s 0
TE ( s)  1  T ( s)  1 
E ( s )  TE ( s ) R( s ) 
G( s)
1

1  G( s) 1  G( s)
A
s i [1  G ( s )]
 lim sE ( s)  lim
s 0
제어응용연구실
s 0
A

 R( s)  i 
s 

A
s i 1[1  G ( s)]
i) Step input ( i = 1 )
Steady state error to input
A
A
 lim

s 0 1  G ( s )
1  K0
ii) Ramp input ( i = 2 )
Steady state error to input
 lim
s 0
A
A

sG ( s )
K1
iii) Higher power-of-t input ( i = 2, 3, 4 ... )
Steady state error to input
제어응용연구실
A
A
 lim i 1
 lim i 1
s 0 s
[1  G ( s )] s 0 s G ( s )
A

K i 1
※ Steady State Error Coefficients of Unity Feedback System
제어응용연구실
Ex)
Type 0 system
G (s) 
K
s2
K
K 0  lim G ( s ) 
s 0
2
Steady state error to input

※ When an integrator is added
A
2A

1  K0
2 K
type 1
K

s 0
s 0 s ( s  2)
A
Steady state error to input 
0
1  K0
K 0  lim G ( s )  lim
K
K
K1  lim

s 0 s  2
2
Steady state error to input 
제어응용연구실
A
2A

K1
K
⊙ Performance Indices and Optimal Systems
A commonly used performance index

2
I s   estep
(t ) dt
0
제어응용연구실
Ex)
A conmonly used performance index is the integral of the square of the error
to a step input
제어응용연구실
Desired to choose the parameter K to give minimum
Integral square error to a step input
T ( s) 
2 s( s  3)  K ( s  3)
2  Ks
 2
1  2 s( s  3)
s  3s  2
s 2  (3  k ) s
TE ( s )  1  T ( s ) 
s 2  3s  2
The error to a step input
E ( s) 
A
s 3 K
 K  2  K 1 
TE ( s)  A 2
 A


s
s  3s  2
s

1
s

2


e(t )  L1[ E(s)]  A[(K  2)et  ( K  1)e2t ]u(t )
e2 (t )  A2 [(K  2)e2t  2( K  1)(K  2)e3t
 ( K  1) 2 e4t ]u(t )
제어응용연구실
 I s (k ) 


0
 e 2t 
e (t )dt  A [(K  4 K  4)
 2 



3t
e 
 e 4t
2
2
 2( K  3K  2)
 3 
  ( K  2 K  1)
 4



2
2
2
 

]0


1
1
 1 
 A2 ( K 2  4K  4)   2( K 2  3K  2)   ( K 2  2K  1 
 2
 3
 4 

A2

[ K 2  6 K  11]
12
dIs
A2


[ 2 K  6]  0
dK
12
제어응용연구실
 K  3
n 2
T ( s)  2
2
s  2 n s  n
For unit step input

s 2  2 n s
1
1
E ( s )  [1  T ( s )]   2

s
s  s  2 n s  n 2 
s  2 n
s  2 n
 2

2
s  2 n s  n
( s   n ) 2  (n 1   2 ) 2

s   n
( s   n ) 2  (n 1   2 ) 2




제어응용연구실

1 
2
2


1


n

 ( s   ) 2  ( 1   2 ) 2
n
n

e(t )  L1[ E ( s)]


 e  nt cos(n 1   2 t ) 

 I ( s) 
1 

sin(n
2


1   2 t )


0
e 2 (t )dt


0


e 2nt cos(n 1   2 t ) 



1 

2
sin(n


1   2 t ) dt

2
 let t   nt
dt   n dt



cos( 1   2 t ) 
Is 
e
1  2t  
n
0
1 

2


sin( 1   2 t ) dt 

2
제어응용연구실
Figure : Integral square error performance measure
for a certain second order system with adjustable damping ratio
※ Minimal mean square error to a step input for the system occurs for
제어응용연구실
  0.5
- Other useful performance indices
IM 
I TS 
I TM 
Figure : other performance measure
제어응용연구실


0


0


0
e(t ) dt    0.67
te 2 (t )dt    0.6
t e(t ) dt    0.7
- Hurwitz determinant method

J   e (t )dt  
0
2
j
 j
E ( s) E (s)ds
N n 1S n 1 ㆍㆍㆍ N1s  N 0
N ( s)
E ( s) 

D( s)
Dn S n  Dn 1S n 1 ㆍㆍㆍ D1s  D0
J 
제어응용연구실

j
 j
N ( s) N ( s)
ds
D( s ) D(  s )
s 3 K
A 2
s  3s  2
Ex)
D2  1, D1  3, D0  2, N1  A, N0  A(3  K )
Using Table
N D  N 0 D2
J2  1 0
2 D0 D1 D2
2
2
 ( A2 / 12)(K 2  6K  11)
dJ 2
 ( A2 / 12)( 2 K  6)  0
dK
 K  3 and J 2  A2 / 6
제어응용연구실
⊙ System Sensitivity
○ Calculating the Effects of changes in Parameters
T ( s) 
T ( s) 
400 ( s  2)
400

1  400 ( s  2)
s  402
400 ( s  2 K1 )
400

1  400 ( s  2 K1 ) s  400 2 K1
Unit step input
제어응용연구실
s  2 K1
2 K1
1
lim s [1  T ( s)]  lim

s 0
s 0 s  400 2 K
400 2 K1
s
1
400K 2 ( s  2)
400K 2
T ( s) 

1  400K 2 ( s  2) s  400K 2  2
Unit step input
s2
2
1
lim s [1  T ( s)]  lim

s 0
s 0 s  400K  2
400K 2  2
s
2
400 ( s  2)
400
T ( s) 

1  400K 3 ( s  2) s  400K3  2
Unit step input
s  400( K 3  1)  2 400( K 3  1)  2
1
lim s [1  T ( s)]  lim

s 0
s

0
s  400K 3  2
400K 3  2
s
제어응용연구실
○ Sensitivity Functions
T / T
a T
a T
S a  lim
 lim

a  0 a / a
a  0 T a
T a
Ex)
G
10
30
T 


1  GH
1  10 3 13
The sensitivity of T to changer in G
SG
G T
1


T G
1  GH

제어응용연구실
1
3

1  10 3 13
1

 G  10, H  
3

The sensitivity of T to changer in H
SH 

H T
 GH

T H
1  GH
1

G

10
,
H



3

 10 3
10

1  10 3
13
The sensitivity of T(s) to changer in a
Sa 
제어응용연구실
a T
1
a
 a( s  a  K )

T a
(s  a  K )2
saK
Ex)
T( s , a )
10
 s  3 


 2

s
s

as

10


 
10
 s  3 

1 
 2

 s  s  as  10 

10 s  30
s 3  as 2  20 s  30
T
 s 2 (10s  30)
 3
a
( s  as2  20s  30) 2
Sa





a T
a
 s 2 (1 0s  3 0)


 

1 0s  3 0
T a
   ( s 3  a s2  2 0s  3 0) 2 

 3

2

s

a
s

2
0
s

3
0




 s2
  3

2
 s  a s  2 0s  3 0 
제어응용연구실
For the nominal value of a=2
 2s 2
Sa  3
s  2 s 2  20s  30
Sa 
a T
T a
T ( s )
a
a
 2s

sa 
a( s )
a
a s 3  2s 2  20s  30
T ( 0)
0
a ( 0 )
T( s ,a )
T (0)  0
10
 s  3 


 2

10s  30
s
s

2
s

10

 
 
10
 s  3 
 s 3  2s 2  (10  10b) s  30b
1  b
 2

 s  s  2 s  10 
제어응용연구실
T
 (10s  30)(10s  30)

b
( s 3  2 s 2  10s  10bs  30b) 2
For the nominal value of b=1
T
 (10s  30)(10s  30)

b
( s 3  2 s 2  20s  30) 2
Sb



   (1 0s  3 0)(1 0s  3 0) 
b T
1


 
1 0s  3 0
T b
( s 3  2 s 2  2 0s  3 0) 2 
 



 3

2

  s  2 s  2 0s  3 0  
 10s  30


  3
2
 s  2 s  20s  30 

T ( s )
b
b
 10s  30

sa 
T ( s)
b
b s 3  2s 2  20s  30
T (0)  b 

(1)
T (0)
 b 
제어응용연구실
○ Sensitivity to Disturbance signals
1
TD ( s ) 
s2
For a unit step disturbance input
1 1
Y ( s) 
s s2
lim y (t )  lim sY ( s ) 
t 
제어응용연구실
t 
1
2
1 ( s  2)
1
TD ( s) 

1  [ K ( s  2)]
s2 K
For a unit step disturbance input
1
1

Y ( s)  

s  s2 K 
lim y (t )  lim sY ( s ) 
제어응용연구실
t 
t 
1
2 K
Ex)
1 0( s  3)

 s ( s 2  2 s  1 0)
Y ( s)
TR ( s ) 
 
1 0( s  3)
R( s)
1 

s ( s 2  2 s  1 0)



1 0( s  3)

 3
2

  s  2 s  2 0s  3 0 



10

Y (s)
s 2  2s  1 0
TD ( s ) 
 
1 0( s  3)
D( s )
1 

s ( s 2  2 s  1 0)



1 0s

 3
2

  s  2 s  2 0s  3 0 


제어응용연구실
The system is stable
s3 1
s2 2
s1 5
s 0 30
20
30
A unit step disturbance
10s
1
lim s TD ( s)  lim 3
0
2
t 
s

0
s  2s  20s  30
s
제어응용연구실
※ Consider the open-loop system
The same transfer function relating Y(s) and R(s) as does the feedback system
however, TD ( s ) 
Y (s)
10
 2
D( s )
s  2 s  10
A unit step disturbance
제어응용연구실
10
1
lim s TD ( s )  lim 2
1
s 0
s

0
s  2s  10
s
⊙ Time Domain Design
○ Ziegler-Nichols compensation
( type = 0 )
K d s 2  K p s  Ki
Ki
Gc ( s)  K p 
 Kd s 
s
s
K p  Pr oportional gain , Ki  integralgain
K d  Derivative gain
제어응용연구실
1. A proportional compensator is applied so that
Gc (s)  K p
K p0  thegain is adjusted until thesystembecomesmarginallystable
T0  theperiodof oscillation is designated
2. The compensator is defined by
1
Gc ( s )  K p (1 
 Td s )
Ti s
 Ki  K p / Ti , Kd  K pTd
제어응용연구실
제어응용연구실
Ex)
64
GTp ( s )  3
s  14 s 2  56 s  64
 i) The fist step
Gc (s)  K p  K p  K p 0
Gc ( s)G p ( s)
64
T ( s) 
 3
1  Gc ( s)G p ( s)
s  14s 2  56s  64(1  K p )
The Routh-Hurwitz table is formed
s3 1
s 2 14
s 1 1/14[784-64(1+
)
s 0 30(1+ K
p
 s
제어응용연구실
Row to zero
20
64(1+
)
Kp
)] K p0
 K p 0  11.25
The complex conjugate roots :
 j 7.483
Normal Ziegler-Nichols
P compensator
K p  5.63
PI compensator
K p  5.06 Ti  0.697
PID compensator
K p  6.75 Ti  0.420
Td  0.105
Quarter-Wave Ziegler Nichols
PID compensator
제어응용연구실
Ti  7.15
Td  0.140
○ Chien-Hrones-Reswick Compensation
1. Tg and Tu
2. R  Tg / Tu
제어응용연구실
( type = 1 )
Ex)
Tu  0.20sec
Tg  1.304 0.2  1.104sec
R  1.104 / 0.2  5.52
PID compensator
Overdamped
K p  3.31
Ti  1.104
Td  0.10
제어응용연구실
K p  5.24
20% overshoot
Ti  1.49
Td  0.094