Upload
ahmed-javaid
View
224
Download
0
Embed Size (px)
Citation preview
7/27/2019 ist and 2nd order system analysis
1/43
First & Second Order
System Analysis
7/27/2019 ist and 2nd order system analysis
2/43
CONTENTS
Introduction
Influence of Poles on Time Response
Transient Response of First-Order System Transient Response of Second-Order
System
7/27/2019 ist and 2nd order system analysis
3/43
INTRODUCTION
The concept of poles andzeros, fundamental to the analysisof and design of control system, simplifies the evaluation of
system response.
The polesof a transfer function are:
i. Values of the Laplace Transform variables s, that cause thetransfer function to become infinite.
The zerosof a transfer function are:
i. The values of the Laplace Transform variable s, that cause
the transfer function to become zero.
7/27/2019 ist and 2nd order system analysis
4/43
INFLUENCE OF POLES ON TIME RESPONSE
The output response of a system is a sum ofi. Forcedresponse
ii. Naturalresponse
a) System showing an input and an output
b) Pole-zero plot of the system
7/27/2019 ist and 2nd order system analysis
5/43
INFLUENCE OF POLES ON TIME RESPONSE
c) Evolution of a system response. Follow the
blue arrows to see the evolution of system
component generated by the pole or zero
7/27/2019 ist and 2nd order system analysis
6/43
System transfer function :
Step response :
0 100 200 300 400 500 6000
10
20
30
40
50
60
70
80
90
100Step Response
Time (sec)
Amplitude
INFLUENCE OF POLES ON TIME RESPONSE
7/27/2019 ist and 2nd order system analysis
7/43
INFLUENCE OF POLES ON TIME RESPONSE
Re(s)
Im(s)
7/27/2019 ist and 2nd order system analysis
8/43
INFLUENCE OF POLES ON TIME RESPONSE
Unstable
Re(s)
Im(s)
7/27/2019 ist and 2nd order system analysis
9/43
INFLUENCE OF POLES ON TIME RESPONSE
Unstable
Re(s)
Im(s)
-1
7/27/2019 ist and 2nd order system analysis
10/43
INFLUENCE OF POLES ON TIME RESPONSE
Unstable
Re(s)
Im(s)
-2
7/27/2019 ist and 2nd order system analysis
11/43
INFLUENCE OF POLES ON TIME RESPONSE
Unstable
Re(s)
Im(s)
faster response slower response
constant
7/27/2019 ist and 2nd order system analysis
12/43
FIRST-ORDER SYSTEM
General form:
Block diagram representation:
By definition itself, the input to the system should be a step
function which is given by the following:
C(s)R(s)1s
K
ssR 1)(
Where,K : Gain
: Time constant
1)()()(
sK
sRsCsG
7/27/2019 ist and 2nd order system analysis
13/43
FIRST-ORDER SYSTEM
General form:
Output response:
1)(
)()(
s
K
sR
sCsG
1
1
1)(
s
B
s
A
s
K
ssC
te
BAtc
)(
)()()( sRsGsC
7/27/2019 ist and 2nd order system analysis
14/43
c(t) = 1 -
At point when t =
c() = 1 -
= 1 0.37
= 0.63
Time constant is the time it takes for thestep-response to increase to 63% of its
final value. In short time constant
measures how fast system can respond.
FIRST-ORDER SYSTEM
7/27/2019 ist and 2nd order system analysis
15/43
FIRST-ORDER SYSTEM
First-order system response to a unit step
7/27/2019 ist and 2nd order system analysis
16/43
FIRST-ORDER SYSTEM
Problem: Find the forced and natural responses for thefollowing systems
7/27/2019 ist and 2nd order system analysis
17/43
TRANSIENT RESPONSE SPECIFICATIONS
Time constant, The time for e-atto decay 37% of its
initial value.
Rise time, tr The time for the waveform to go
from 0.1 to 0.9 of its final value.
Settling time, ts The time for the response to reach,
and stay within 2% of its final value.
a
1
atr
2.2
ats
4
7/27/2019 ist and 2nd order system analysis
18/43
TRANSIENT RESPONSE SPECIFICATIONS
Problem: For a system with the transfer function shownbelow, find the relevant response specifications
i. Time constant,
ii. Settling time, ts
iii. Rise time, tr
50
50)(
s
sG
7/27/2019 ist and 2nd order system analysis
19/43
FIRST ORDER SYSTEMSIMPLE BEHAVIOR.
No overshoot
No oscillations
7/27/2019 ist and 2nd order system analysis
20/43
SECOND ORDER SYSTEM
(MASS-SPRING-DAMPER SYSTEM)
k b
y(t)
F(t)=ku(t)
m
ODE
Transfer Function
7/27/2019 ist and 2nd order system analysis
21/43
SECOND-ORDER SYSTEM
Natural frequency,
n
Frequency of oscillation of the system without damping.
Damping ratio,
It is measure of the degree of resistance to change in thesystem output.
7/27/2019 ist and 2nd order system analysis
22/43
SECOND-ORDER SYSTEM
General form:
Roots of denominator:
22
2
2 nn
n
ss
KsG
Where,
K : Gain : Damping ratio
n : Undamped natural frequency
02 22 nnss
122,1 nns Generically
2
2,1 1 nn js
7/27/2019 ist and 2nd order system analysis
23/43
POLAR VS. CARTESIAN REPRESENTATIONS.Cartesian representation :
Imaginary part (frequency)
Real part (rate of decay)
7/27/2019 ist and 2nd order system analysis
24/43
POLAR VS. CARTESIAN REPRESENTATIONS.Cartesian representation :
Imaginary part (frequency)
Real part (rate of decay)
Polar representation :
damping ratio
natural frequency
7/27/2019 ist and 2nd order system analysis
25/43
POLAR VS. CARTESIAN REPRESENTATIONS.Cartesian representation :
Imaginary part (frequency)
Real part (rate of decay)
Polar representation :
damping ratio
natural frequency
7/27/2019 ist and 2nd order system analysis
26/43
SECOND-ORDER SYSTEM
7/27/2019 ist and 2nd order system analysis
27/43
SECOND-ORDER SYSTEM
7/27/2019 ist and 2nd order system analysis
28/43
SECOND-ORDER SYSTEM
Step responses for second-order system damping cases
7/27/2019 ist and 2nd order system analysis
29/43
SECOND-ORDER SYSTEM
Second-order response as a function of damping ratio
7/27/2019 ist and 2nd order system analysis
30/43
SECOND-ORDER SYSTEM
Second-order response as a function of damping ratio
7/27/2019 ist and 2nd order system analysis
31/43
SECOND ORDER SYSTEM RESPONSE.
Unstable
Re(s)
Im(s)
Undamp
ed
Overdamped or Critically damped
Underdamped
Underdamped
7/27/2019 ist and 2nd order system analysis
32/43
SECOND-ORDER SYSTEM
Second-order underdamped responses for damping ratiovalue
7/27/2019 ist and 2nd order system analysis
33/43
TRANSIENT RESPONSE SPECIFICATIONS
Second-order underdamped response specifications
7/27/2019 ist and 2nd order system analysis
34/43
TRANSIENT RESPONSE SPECIFICATIONS
Rise time, Tr The time for the waveform to go from 0.1 to 0.9 of its final
value.
Peak time, Tp The time required to reach the first
or maximum peak.
Settling time, Ts The time required for the transients
damped oscillation to reach and stay
within 2% of the steady-state value.
2
1
n
pT
n
sT
4
7/27/2019 ist and 2nd order system analysis
35/43
TRANSIENT RESPONSE SPECIFICATIONS
Percent overshoot, %OS The amount that the waveform overshoots the steady-state, or
final value at peak time, expressed as a percentage of the
steady-state value.
%100% )1/(2
eOS
)100/(%ln
)100/ln(%
22OS
OS
7/27/2019 ist and 2nd order system analysis
36/43
SYSTEM PERFORMANCE
Percent overshoot versus damping ratio
7/27/2019 ist and 2nd order system analysis
37/43
SYSTEM PERFORMANCE
Lines of constant peak time Tp, settling time Tsand percentovershoot %OS
Ts2< Ts1Tp2< Tp1
%OS1< %OS2
7/27/2019 ist and 2nd order system analysis
38/43
SYSTEM PERFORMANCE
Step responses of second-order underdamped systems aspoles move
a) With constant
real part
b) With constant
imaginary part
7/27/2019 ist and 2nd order system analysis
39/43
SYSTEM PERFORMANCE
Step responses of second-order underdamped systems aspoles move
c) With constant dampingratio
Th ti f t l t i t
7/27/2019 ist and 2nd order system analysis
40/43
40
The time response of a control system consists
of two parts:
1. Transient response
- from initial state to the final
statepurpose of control
systems is to provide a desired
response.
2. Steady-state response
- the manner in which the
system output behaves as t
approaches infinitythe error
after the transient response has
decayed, leaving only the
continuous response.
7/27/2019 ist and 2nd order system analysis
41/43
THEOREMS
Initial Value Theorem
Final Value Theorem (Steady state) If all poles of sX(s) are in the left half plane
(LHP), then
7/27/2019 ist and 2nd order system analysis
42/43
FURTHER READING
Chapter 4
i. Nise N.S. (2004). Control System Engineering (4th Ed), John
Wiley & Sons.
Chapter 5
i. Dorf R.C., Bishop R.H. (2001). Modern Control Systems (9thEd), Prentice Hall.
7/27/2019 ist and 2nd order system analysis
43/43
THE END