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PRESENTATION ON
PERFORMANCE OF A SECOND-
ORDER SYSTEM
Main content
 Test input signals
 Performance of a second-order system
 Effects of a third pole and a zero on system response
 Steady-state error analysis
Introduction
 Transient response
 Steady-state response
 Design specifications
 How to get compromise?
A distinct advantage of feedback control
system is the ability to adjust the transient
and steady-state response
Test input signals
 Step input
 Ramp input
 Parabolic input
 Sinusoidal input
 Unit impulse input
The standard test input signals commonly used
are:
Representation of test signals
 Step:
 Ramp:
 Parabolic:
 sinusoidal:
22
3
2
2
sin
1
0
2
1
1
0,
1
0),(1







s
A
tA
s
tt
s
tt
s
tt
Input time domain frequency domain
Unit impulse response






otherwise
t
t
,0
22
,
1
)(

Unit impulse:
)]([)( 1
sGLtg 
System impulse response:
 


t
sRsGLdrtgty )]()([)()()( 1

System response is the convolution integral of g(t) and r(t):
Standard test signal
The standard test signals are of the general
form:
n
ttr )(
And its Laplace transform is:
1
!
)( 
 n
s
n
sR
Performance indices
 Time delay t d
 Rise time t r
 Peak time t p
 Settling time t s
 Percent overshoot
%
Transient Performance:
Steady-state Performance: Steady-state error
Response and performance of a
second-order system
 Model of 2nd-order system
 Roots of characteristic equation (Poles)
22
2
2)(
)(
)(




sssR
sY
sT
12
2,1   nns
The response depends on and n
Unit step response of 2nd-
order system
 If ,2 positive real-part roots, unstable
 If ,2 negative real-part roots, underdamped
 If ,2 equal negative real roots, critically damped
 If ,2 distinct negative real roots, overdamped
 If ,2 complex conjugate roots, undamped
0
10  
1
1
0
Case 1: underdamped
 Oscillatory response
 No steady-state error
Case 2: critically damped
 Mono-incremental response
 No Oscillation
 No steady-state error
Case 3: overdamped
 Mono-incremental response
 slower than critically damped
 No Oscillation
 No steady-state error
Performance evaluation
( underdamped condition)
 Performance indices evaluation
1 Time delay
2 Rise time
3 Peak time
4 Percent overshoot
5 Settling time
Effects of a third pole and a zero
on 2nd-order system response
 Effect of a third pole
 Effect of a third zero
 Dominant poles
References
1. Modern Control System by Richarc C. Drof and Robert H.
Bishop,11th Edition Person Int.
2. Modern Control Engineering by Katsuhiko Ogata, 4th Edition,
Prentice Hall of India.
3. Automatic Control Systems by Benjamin C.Kuo, 8th Edition, Farid
Golnaraghi, John Wiley & Sons.
4. Control Systems Engineering by Nagrath and Gopal New Age
Publication
5. Feedback and Control Systems by Joseph J Distefano 2nd Edition
TMH
Performance of second order system

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Performance of second order system

  • 1. PRESENTATION ON PERFORMANCE OF A SECOND- ORDER SYSTEM
  • 2. Main content  Test input signals  Performance of a second-order system  Effects of a third pole and a zero on system response  Steady-state error analysis
  • 3. Introduction  Transient response  Steady-state response  Design specifications  How to get compromise? A distinct advantage of feedback control system is the ability to adjust the transient and steady-state response
  • 4. Test input signals  Step input  Ramp input  Parabolic input  Sinusoidal input  Unit impulse input The standard test input signals commonly used are:
  • 5. Representation of test signals  Step:  Ramp:  Parabolic:  sinusoidal: 22 3 2 2 sin 1 0 2 1 1 0, 1 0),(1        s A tA s tt s tt s tt Input time domain frequency domain
  • 6. Unit impulse response       otherwise t t ,0 22 , 1 )(  Unit impulse: )]([)( 1 sGLtg  System impulse response:     t sRsGLdrtgty )]()([)()()( 1  System response is the convolution integral of g(t) and r(t):
  • 7. Standard test signal The standard test signals are of the general form: n ttr )( And its Laplace transform is: 1 ! )(   n s n sR
  • 8. Performance indices  Time delay t d  Rise time t r  Peak time t p  Settling time t s  Percent overshoot % Transient Performance: Steady-state Performance: Steady-state error
  • 9. Response and performance of a second-order system  Model of 2nd-order system  Roots of characteristic equation (Poles) 22 2 2)( )( )(     sssR sY sT 12 2,1   nns The response depends on and n
  • 10. Unit step response of 2nd- order system  If ,2 positive real-part roots, unstable  If ,2 negative real-part roots, underdamped  If ,2 equal negative real roots, critically damped  If ,2 distinct negative real roots, overdamped  If ,2 complex conjugate roots, undamped 0 10   1 1 0
  • 11. Case 1: underdamped  Oscillatory response  No steady-state error
  • 12. Case 2: critically damped  Mono-incremental response  No Oscillation  No steady-state error
  • 13. Case 3: overdamped  Mono-incremental response  slower than critically damped  No Oscillation  No steady-state error
  • 14. Performance evaluation ( underdamped condition)  Performance indices evaluation 1 Time delay 2 Rise time 3 Peak time 4 Percent overshoot 5 Settling time
  • 15. Effects of a third pole and a zero on 2nd-order system response  Effect of a third pole  Effect of a third zero  Dominant poles
  • 16. References 1. Modern Control System by Richarc C. Drof and Robert H. Bishop,11th Edition Person Int. 2. Modern Control Engineering by Katsuhiko Ogata, 4th Edition, Prentice Hall of India. 3. Automatic Control Systems by Benjamin C.Kuo, 8th Edition, Farid Golnaraghi, John Wiley & Sons. 4. Control Systems Engineering by Nagrath and Gopal New Age Publication 5. Feedback and Control Systems by Joseph J Distefano 2nd Edition TMH