Fundamentals of Control Systems - Chapter 5: Analysis of Control System Performance

()( /TteKty 
6 November 2012 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 15

First-order system – Remarks 
 First order system has only one real pole at (1/T), its
transient response doesn’t have overshoot.
Ti t t T i th ti i d f th t f me cons an : s e me requ re or e s ep response o
the system to reach 63% its steady-state value.
 The further the pole ( 1/T) of the system is from the
imaginary axis, the smaller the time constant and the faster
the time response of the system.
 Settling time of the first order system is:
 1lTt  ns
where  = 0 02 (2% criterion) or  = 0 05 (5% criterion)
6 November 2012 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 16
. .
First-order system
The relationship between the pole and the time response 
 The further the pole of the system is from the imaginary axis,
the smaller the time constant and the faster the time
Im s y(t)
response of the system.
Re s
K
0
Pole zero plot Transient response
t0
6 November 2012 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 17
 – 
of a first order system
of the first order
Second-order oscillating system
1222  TssT
K

Y(s)R(s)
 The transfer function of the second-order oscillating system:
2
)( nKKsG  )101(  
 The system has two complex conjugate poles: 
2222 212 nnssTssT  
 ,
Tn
 Transient response: 22
2
.1)()()( nKsGsRsY 
2
2,1 1   nn jp
2 nnsss  
      teKty tn )1(sin1)( 2 )(cos  
6 November 2012 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 18
  n1 2
Second-order oscillating system (cont’)
y(t)
Im s
(1+).K
(1).K
K
R
21  njn
cos = 
e s
0n
21   j

ts
t
0
n
Transient response of a second 
order oscillating system
Pole – zero plot of a second 
order oscillating system
6 November 2012 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 19
Second-order oscillating system – Remark
 A second order oscillation system has two conjugated 
complex poles, its transient response is a oscillation signal.
If  0 t i t = , rans en response 
is a stable oscillation signal 
at the frequency n n
 = 0
 = 0.2
is called natural oscillation 
frequency.
 If 0<<1, transient 
 = 0.4
response is a decaying 
oscillation signal   is 
called damping constant,  = 0.6
the larger the value  , (the 
closer the poles are to the 
real axis) the faster the
6 November 2012 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 20
response decays.
Second-order oscillating system – Overshoot
 Transient response of the second order oscillating system
 
has overshoot.
%100.
1
exp
2  
 POTThe percentage of overshoot:
 The larger the value , 
(the closer the poles are 
to the real axis) the 
(
%
)
smaller the POT.
 The smaller the value , 
(the closer the poles are
P
O
T
(
to the imaginary axis) 
the larger the POT
The relationship
6 November 2012 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 21
between POT and 
Second-order oscillating system – Settling time
 Settling time:
3
n
t s5% criterion:
t 
4s2% criterion:
n
6 November 2012 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 22
Second-order oscillating system
Relationship between pole location and transient response 
 The 2nd order systems that have the poles located in the same
rays starting from the origin have the same damping constant,
th th t f h t th Th f th
Im s y(t)
en e percen age o overs oo s are e same. e ur er
the poles from the origin, the shorter the settling time.
R
K
cos =  e s
0
 
Pole zero plot of a second Transient response of a second
t
0
6 November 2012 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 23
 – 
order oscillating system
order oscillating system
Second-order oscillating system
Relationship between pole location and transient response (cont’) 
 The 2nd order systems that have the poles located in the same 
distance from the origin have the same natural oscillation 
f Th l th l t th i i i th ll
Im s y(t)
requency. e c oser e po es o e mag nary ax s, e sma er 
the damping constant, then the higher the POT.
R
K e s
0
n
t
0
Pole – zero plot of a second Transient response of a second
6 November 2012 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 24
order oscillating system
order oscillating system
Second-order oscillating system
Relationship between pole location and transient response (cont’) 
 The 2nd order systems that have the poles located in the same 
distance from the imaginary axis have the same n, then the 
ttli ti th Th f th th l f th l
Im s y(t)
se ng me are e same. e ur er e po es rom e rea 
axis, the smaller the damping constant, then the higher the POT
R Ke s
0n
t0
Pole – zero plot of a second Transient response of a second
6 November 2012 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 25
order oscillating system
order oscillating system
Transient response of high order system
 High-order systems are the system that have more than 2 poles 
 If a high order system have a pair of poles located closer to the 
imaginary axis than the others then the high order system can be 
i t d t d d t Th i f lapprox ma e o a secon or er sys em. e pa r o po es 
nearest to the imaginary axis are called the dominant poles. 
Im s y(t) R f hi h
Re s
esponse o g
order system
0 Response of second 
order system with 
the dominant poles
High order systems A high order system can be
0
t
6 November 2012 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 26
have more than 2 poles
approximated by a 2nd order system
Performance indices
6 November 2012 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 27
Integral performance indices
 IAE criterion 
(Integral of the Absolute Magnitude of the Error )



0
)( dtteJIAE
 ISE criterion 
(Integral of the Square of the Error)


2
 ITAE criterion

0
)( dtteJISE
(Integral of Time multiplied by the Absolute Value of the Error)


)( dttetJ
6 November 2012 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 28

0
ITAE
Optimal systems
 A control system is optimal when the selected performance 
index is minimized
minIAEJ when 707.0
iJ 50
 Second order system:
m nISE when .
minITAEJ when 707.0
y(t) =0.3
=0.5
=0.707
0
t
=0.9
6 November 2012 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 29
Transient response of second order systems
ITAE optimal control
 ITAE is usually used in design of control system 
 An n-order system is optimal according to ITAE criterion if the
denominator of its transfer function has the form:
6 November 2012 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 30
ITAE optimal control (cont’)
 Optimal response according to ITAE criterion 
y(t)
1st d tor er sys em
2nd order system
3rd order system
4th order system
0
t
6 November 2012 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 31
Relationship between frequency domain 
performances & time domain performances
6 November 2012 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 32
Relationship between frequency response & steady state error
R(s) Y(s)
G(s)+
)()(lim)()(lim
00
 jHjGsHsGK sp  
)()(lim)()(lim
00
 jHjGjsHsGsK sv  
)()()(lim)()(lim 2
0
2
0
 jHjGjsHsGsK sa  
6 November 2012 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 33
Relationship between frequency response & steady state error
R( ) Y( )s
G(s)+
s
 Steady state error of the closed-loop system depends on the
magnitude response of the open-loop system at low
frequencies but not at high frequencies.
 The higher the magnitude response of the open-loop system
at low frequencies, the smaller the steady-state error of the
closed-loop system.
 In particular, if the magnitude response of the open-loop system 
is infinity as frequency approaching zero, then the steady-state 
6 November 2012 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 34
error of the closed-loop system to step input is zero.
Relationship between frequency response & transient response
R(s) Y(s)
G(s)+
 In the frequency range  <c , because then:1)( jG
1
)(
)(
)(1
)(
)(  



jG
jG
jG
jG
jGcl
 In the frequency range  >c , because then:1)( jG
)(
)()(
)(

jG
jGjG
jG
 Bandwidth of the closed-loop system is approximate the gain
1)(1
 jGcl 
6 November 2012 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 35
crossover frequency of the open-loop system.
Relationship between frequency response & transient response
Bode plot of a open-loop system Bode plot of the corresponding 
closed-loop system
6 November 2012 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 36
Relationship between frequency response & transient response
R(s) Y(s)
G(s)+
 The higher the gain crossover frequency of open-loop system,
the wider the bandwidth of closed-loop system  the faster the
response of close-loop system, the shorter the settling time.
 4
f
c
qd
c
t  
 The higher the phase margin o the open-loop system, the 
smaller the POT of closed-loop system. In most of the cases, 
if the phase margin of the open-loop system is larger than 600
6 November 2012 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 37
then the POT of the closed-loop system is smaller than 10%.
Ex: relationship between gain crossover frequency & settling time
R(s) Y(s)
)1080)(110(
10)( sG
G(s)+
..  sss
6 November 2012 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 38
Ex: relationship between gain crossover frequency and settling time
R(s) Y(s)
)11.0(
50)(  sssG
G(s)+
6 November 2012 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 39
Example of relationship between phase margin and POT
R(s) Y(s)
)1080)(110(
6)(  ssssG
G(s)+
..
6 November 2012 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 40
Ex of relationship between phase margin and POT (cont’)
R(s) Y(s)
)11.0(
6)(  sssG
G(s)+
6 November 2012 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 41