Bài giảng Fundamentals of Control Systems - Chapter 3: System dynamics - Huỳnh Thái Hoàng

me response:
 Impulse response: )(11
1
1)( 1 te
TTs
tg T
t 



L
 Step response: )(1)1(
)1(
1)( 1 te
T
th T
t 


 L
ss 
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 26
Time response of first-order lag factor
(a) Weighting function (b) Transient function
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 27
Frequency response of first-order lag factor
F 1 requency response:
1
)(   TjjG
1Magnitude response:
221
)(  TM 
)()( 1  Ttg Phase response:
 221lg20)(  TL 
 Approximation of the Bode diagram by asymptotes:
 : the asymptote lies on the horizontal axis
 : the asymptote has the slope of 20dB/dec
T/1
T/1
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 28
Frequency response of first-order lag factor (cont’)
corner frequency 
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 29
Bode diagram Nyquist plot
First-order lead factor
T f f ti )( rans er unc on: 1TssG
 Time response:
 Step response: )(1)()1()( 1 ttT
s
Tsth 


   L
)()()()( ttTthtg    Impulse response:
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 30
Time response of first-order lead factor
Transient function
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 31
Frequency response of first-order lead factor
F requency response:
Magnitude response:
1)(   TjjG
221)(  TM 
 221lg20)(  TL 
)()( 1  Ttg Phase response:
 Approximation of the Bode diagram by asymptotes:
 : the asymptote lies on the horizontal axisT/1
 : the asymptote has the slope of +20dB/decT/1
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 32
Frequency response of first-order lead factor (cont’)
corner 
frequency
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 33
Bode diagram Nyquist plot
Second-order oscillating factor
1 Transfer function:
 Time-domain dynamics:
12
)( 22  TssTsG  )10(  
 Impulse response:  tetg ntn n )1(sin
1
)( 2
2

  

 Step response:

    teth tn )1(sin1)( 2 n1 2
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 34
Time response of second-order oscillating factor
(a) Weighting function (b) Transient function
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 35
Frequency response of second-order oscillating factor
1 Frequency response:
M it d
12
)( 22   TjTjG
1)(M agn u e response:

222222 4)1(  TT 
222222 4)1(l20)(  TTL 
 Phase response:
g  


  221 1
2)( 
T
Ttg
 Approximation of the Bode diagram by asymptotes:
 
 : the asymptote lies on the horizontal axis
 : the asymptote has the slope of 40dB/dec
T/1
T/1
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 36
Frequency response of second-order oscillating factor
Corner 
frequency
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 37
Bode diagram Nyquist plot
Time delay factor
T Transfer function:
 Time-domain dynamics:
sesG )(
 Impulse response:   )()( 1 Ttetg Ts   L
 Step response: )(1)( 1 Tt
s
eth
Ts






L
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 38
Time response of time delay factor
(a) Weighting function (b) Transient function
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 39
Frequency response of time delay factor
 Frequency response:
M it d
 TjejG )(
1)(M 0)(Lagn u e response:
 T)( Phase response:
 
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 40
Frequency response of time delay factor (cont’)
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 41
Bode diagram Nyquist plot
Dynamics of control systems
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 42
Time response of control systems
 Consider a control system which has the TF G(s):
nn
mm
mm bsbsbsbsG   

1
1
1
10)( 
nn asasasa  110 
 Laplace transform of the transient function:
sasasasas
bsbsbsb
s
sGsH nn
mm
mm
)(
)()(
1
1
10
1
1
10

  



nn
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 43
Remarks on the time response of control systems
 If G(s) does not contain a ideal integral or derivative factor then:
 Weighting function decays to 0
1   mm bbbb 0lim)(lim)(
1
1
10
110
00


 





nn
nn
mm
ss asasasa
ssssssGg 

 Transient function approaches to non-zero value at steady
state:
0.1lim)(lim)(
1
1
10
1
1
10
00











n
m
nn
nn
mm
mm
ss a
b
asasasa
bsbsbsb
s
sssHh 

6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 44
Remarks on the time response of control systems (cont’)
 If G(s) contain a ideal integral factor (a = 0) then:n
 Weighting function has non-zero steady-state:
1    bbbb mm 0lim)(lim)(
1
1
10
110
00


 




 sasasa
ssssssGg
n
nn
mm
ss 

 Transient function approaches infinity at steady-state
 
 bsbsbsbh
mm
1
1
101li)(li)(   




 sasasas
sssH
n
nn
mm
ss
1
1
10
00
.mm 
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 45
Remarks on the time response of control systems (cont’)
 If G(s) contain a ideal derivative factor (b = 0) then:m
 Weighting function approaches zero at steady-state.
1   mm bbb 0lim)(lim)(
1
1
10
110
00


 





nn
nn
m
ss asasasa
ssssssGg 

 Transient function approaches zero at steady-state.
01lim)(lim)( 1
1
10 


  

m
mm sbsbsbsssHh .
1
1
10
00

 



nn
nnss asasasas 
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 46
Remarks on the time response of control systems (cont’)
 If G(s) is proper (m  n) then h(0) = 0.
0.1lim)(lim)0( 1
1
1
10 


  

m
mm sbsbsbsHh 
110    nnnnss asasasas 
 If G(s) is strictly proper (m < n) then g(0) = 0.
0lim)(lim)0(
1
1
10
1
1
10 




  

 nn
m
mm
ss asasasa
sbsbsbsGg 

 nn
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 47
Frequency response of control system
 Consider a control system which has the transfer function
G(s). Suppose that G(s) consists of basis factors in series:
l GG )()(


i
i ss
1
 Frequency response: l jGjG )()(


i
i
1

l MM )()( l
Ph l
Magnitude response:


i
i
1



i
iLL
1
)()( 
 The Bode diagram of a system consisting of basic factors in series
 ase response:


i
i
1
)()( 
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 48
equals to the sum of the Bode diagram of the basic factors.
Approximation of Bode diagram
 Suppose that the TF of the system is of the form:
)()()()( 321 sGsGsGKssG 
(>0: the system has ideal derivative factor(s)
 Step 1: Determine all the corner frequencies  =1/T and sort
<0: the system has ideal integral factor(s))
i i ,
them in ascending order 1 <2 < 3
 Step 2: The approximated Bode diagram passes through the
point A having the coordinates:
  0  0lg20lg20)(  KL
where 0 is a frequency satisfying 0 1 then it is
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 49
possible to chose 0 =1.
Approximation of Bode diagram (cont’)
 Step 3: Through point A draw an asymptote with the slope: , 
 ( 20 dB/dec  ) if G(s) has  ideal integral factors
 (+ 20 dB/dec  ) if G(s) has  ideal derivative factors
The asymptote extends to the next corner frequency.
 Step 4: At the corner frequency i =1/Ti , the slope of the 
asymptote is added with: 
 (20dB/dec  i) if Gi(s) is a first-order lag factor (multiple i)
 (+20dB/dec  ) if G (s) is a first order lead factor (multiple  )  i i - i
 (40dB/dec  i) if Gi(s) is a 2nd order oscillating factor (multiple i)
 (+40dB/dec  i) if Gi(s) is a 2nd order lead factor (multiple i)
The asymptote extends to the next corner frequency.
 Step 5: Repeat the step 4 until the asymptote at the last
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 50
corner frequency is plotted.
Approximation of Bode diagram – Example 1
 Plot the Bode diagram using asymptotes:
)101,0(
)11,0(100)( 

ss
ssG
Based on the Bode diagram, determine the gain cross
frequency of the system.
 Solution:
 Corner frequencies:
(rad/sec) 100
01,0
11
2
2  T(rad/sec) 101,0
11
1
1  T
 The Bode diagram pass the point A at the coordinate:
 1
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 51
  40100lg20lg20)( KL 
Approximation of Bode diagram – Example 1 (cont’)
L(), dB
A
20dB/dec
40

20dB/dec
0dB/dec
20

c
0

lg
100 10110-1
10-1 2
102
3
 In the Bode diagram, the gain crossover frequency is 103
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 52
rad/sec.
Example 2 – Bode diagram to transfer function 
D t i th t f f ti f th t hi h h th e erm ne e rans er unc on o e sys em w c as e
approximation Bode diagram as below:
L(), dB
60 0dB/dec
20dB/d40
54
A
D E
 ec
20 0dB/dec
26
B C
0
lg10-1 21.301
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 53
g1 g2 g3
Example 2 – Bode diagram to transfer function (cont’)
2654 (dB/dec) 40
301.12

 The slope of segment CD:
 The corner frequencies:
(rad/sec) 510 7.01 g 7.020
26400lg 1 g 
 301.1lg 2 g  (rad/sec) 2010 301.12 g
 2lg 3 g  (rad/sec) 1001023 g
 The transfer function has the form: 2
3
2
21
)1(
)1)(1()( 

sTs
sTsTKsG
100 40lg20  KK
0.2111 T 0.05112 T 0.01113 T
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 54
51g 202g 1003g