Modern Control - Lec 05 - Analysis and Design of Control Systems using Frequency Response

1. ‫ر‬َ‫ـد‬ْ‫ق‬‫ِـ‬‫ن‬،،،‫لما‬‫اننا‬ ‫نصدق‬ْْ‫ق‬ِ‫ن‬‫ر‬َ‫د‬
LECTURE (5)
Analysis and Design of Control
Systems using Frequency Response
Assist. Prof. Amr E. Mohamed

2. Agenda
 Introduction
 Frequency Response
 Bode Plots
 Gain and Phase Margins
 Stability Analysis
 Bandwidth and Cutoff Frequency
 Compensation and Controller Design in the Frequency Domain
2

3. Frequency Response
 For a stable, linear, time-invariant (LTI) system, the steady state
response to a sinusoidal input is a sinusoid of the same frequency but
possibly different magnitude and different phase.
 Sinusoids are the Eigenfunctions of convolution.
 If input is 𝐴 𝑐𝑜𝑠(𝜔 𝑜 𝑡 + 𝜃) and steady-state output is 𝐵 cos(𝜔 𝑜 𝑡 + 𝜑),
then the complex number
𝐵
𝐴
𝑒 𝑗(𝜑−𝜃) is called the frequency response of
the system at frequency 𝜔 𝑜.
3
)(
)(
jT
sT 𝐶(𝑠)
𝐶(𝑗𝜔)
𝑅(𝑠)
𝑅(𝑗𝜔)

4. 4
L.T.I system
tAtr sin)(  )sin()(   tBty
Magnitude: Phase:
A
B 
G(s)
H(s)
＋
-
)(ty)(tr
)()(1
)(
)(
)(
sHsG
sG
sR
sY

  jsjs 
Magnitude: Phase:
)()(1
)(


jHjG
jG
 )]()(1[
)(


jHjG
jG


Steady state response

5. Bode Plots
 The Bode form is a method for the frequency domain analysis.
 The Bode plot of the function G(jω) is composed Bode of two plots:
 One with the magnitude of G(jω) plotted in decibels (dB) versus log10(jω).
 The other with the phase of G(jω) plotted in degree versus log10(ω).
 Feature of the Bode Plots
 Since the magnitude of G(jω) in the Bode plot is expressed in dB, products and
division factors in G() become additions and subtraction, respectively
 The phase relations are also added and subtracted from each other algebraically.
 The magnitude plot of Bode of G(jω) can be approximated by straight-lines
segments which allow the simple sketching of the bode plot without detailed
computation.
5

6. 6
1
2
10log


decDecade :
1
2
2log


octOctave :
1 10 100
Logarithmic coordinate
2 3 4 20
dB


7. Mag
(dB)
Phase
(deg)
1 1 1 1 1 1
This is a sheet of 5 cycle, semi-log paper.
This is the type of paper usually used for
preparing Bode plots.
 (rad/sec)
 (rad/sec)
7

8. A number to decibel conversion
)number(log20 10
decade
8

9. Bode Plots
 In order to simplify the Bode plot, it is convenient to use the so-called Bode form
 Given: the open loop T.F 𝐺 𝑠 𝐻(𝑠) =
𝐴 𝑠+𝑧1 𝑠+𝑧2 … 𝑠+𝑧 𝑚
𝑠 𝑞 𝑠+𝑝1 𝑠+𝑝2 …(𝑠+𝑝 𝑛)(𝑠2+𝑎𝑠+𝜔 𝑜
2)
 Where A, m, q and n are real constants
 The bode form can be expressed as:
𝐺 𝑠 𝐻(𝑠) =
𝐾 1 +
𝑠
𝑧1
1 +
𝑠
𝑧2
… 1 +
𝑠
𝑧 𝑚
𝑠 𝑞 1 +
𝑠
𝑝1
1 +
𝑠
𝑝2
… 1 +
𝑠
𝑝 𝑛
1 +
𝑎
𝜔 𝑜
𝑠 +
𝑠
𝜔 𝑜
2
𝐺 𝑗𝜔 𝐻 𝑗𝜔 =
𝐾 1 +
𝑗𝜔
𝑧1
1 +
𝑗𝜔
𝑧2
… 1 +
𝑗𝜔
𝑧 𝑚
𝑗𝜔 𝑞 1 +
𝑗𝜔
𝑝1
1 +
𝑗𝜔
𝑝2
… 1 +
𝑗𝜔
𝑝 𝑛
1 +
𝑎
𝜔 𝑜
𝑗𝜔 +
𝑗𝜔
𝜔 𝑜
2
9

10. Basic Terms
 Constant gain G s H s = K or G jω 𝐻 jω = K
 Integral factors G s H s =
1
𝑠
or G jω 𝐻 jω =
1
jω
 Derivative factors G s H s = 𝑠 or G jω 𝐻 jω = jω
 First order factors G s H s = 1 +
𝑠
𝑎
±1
or G jω 𝐻 jω = 1 +
𝑗𝜔
𝑎
±1
 Quadratic factors G s H(𝑠) = 1 +
2z
𝑤 𝑛
𝑠 +
𝑠
𝑤 𝑛
2 ±1
or
G jω 𝐻 jω = 1 +
2z
𝑤 𝑛
𝑗𝜔 +
𝑗𝜔
𝑤 𝑛
2 ±1
10

11. The Bode Plot for a constant gain K
)( jT
4K
4/1K
4K
4/1K
|G(jω)H(jω)| = 20𝑙𝑜𝑔10 𝐾
zerojj  ))H(G( 
11

12. The Bode Plot for a derivative factor
 3
j
 3
j
 2
j
 2
j
j
j
dB/decade20 nslope
 90n
1
12

13. The Bode Plot for an integral factor
  3
j
1
  2
j
  1
j
  3
j
  2
j
  1
j
dB/decade20 nslope
 90n
13

14. The Bode Plot for a first order factor 𝟏 +
𝑠
𝑎
Magnitude:
Phase:
])(1log[10
)(1log20)1(
2
2
a
aa
j
dB




aa
j
 1
tan)1( 

01log100  dB
a
a

 
adB
a
dB
aa
ja
log20log20
log201




 
o
GH
a
a 00tan0 1
 
 
o
GH
a
a 90tan 1
 
 
01.32log1011  dBja
14

15. The Bode Plot for a Second order factor















2
2
1)()(
nn
j
jjHjG




z











1,)log(40
1,)2log(20
1,0
)()(


nn
n
n
jHjG




















1,180
1,90
1,0
)()( 0
0


n
o
n
n
jHjG







2
12
22
2
10
2
)(1
2
tan)()(2))(1()()(
2)(1log20)()(2))(1()()(
n
n
nn
nn
dB
nn
jHjGjjHjG
jHjGjjHjG























































2
2
1)()(
nn
s
ssHsG

z
15

16. The Bode Plot for a Second order factor
16















2
2
1)()(
nn
j
jjHjG




z


17. Bode plots for
 𝐺 𝑠 𝐻(𝑠) = 𝑠;
 𝐺(𝑠)𝐻(𝑠) =
1
𝑠
;
 𝐺(𝑠)𝐻(𝑠) = 1 +
𝑠
𝑎
 𝐺(𝑠)𝐻(𝑠) =
1
1 +
𝑠
𝑎
17

18. Example 5.1: Draw the Bode plot for a system transfer function T(s)
 
  
 
  
































1
2
1
1
3
5.1
)()(
1
2
12
1
3
3
)()(
21
3
)()(
s
ss
s
sHsG
s
ss
s
sHsG
sss
s
sHsG
Bode Mag. plot for Example 5.1:
a. components;
b. composite 18

19. Example 5.1: Draw the Bode plot for a system transfer function T(s)
Bode phase plot for Example 5.1:
a. components;
b. composite 19
 
  
 
  
































1
2
1
1
3
5.1
)()(
1
2
12
1
3
3
)()(
21
3
)()(
s
ss
s
sHsG
s
ss
s
sHsG
sss
s
sHsG

20. Example 5.2: Draw the Bode plot for a system transfer function T(s)
Bode Mag. plot for Example 5.2:
a. components;
b. composite
 
  
































108.0
5
1
2
1
3
06.0
)()(
2522
3
)()(
2
2
s
ss
s
sHsG
sss
s
sHsG
20

21. Example 5.2: Draw the Bode plot for a system transfer function T(s)
Bode phase plot for Example 5.2:
a. components;
b. composite
21
 
  
































108.0
5
1
2
1
3
06.0
)()(
2522
3
)()(
2
2
s
ss
s
sHsG
sss
s
sHsG

22. Relative stability (Gain and Phase Margins)
 A transfer function is called minimum phase when all the poles and zeros
are LHP and non-minimum-phase when there are RHP poles or zeros.
 The phase margin (PM): is that amount of additional phase lag at the gain
crossover frequency required to bring the system to the verge of instability.
𝑃𝑀 = 180 + phase of GH measured at the gain crossover frequency 0 𝑑𝐵
 The gain margin (GM): is the reciprocal of the magnitude |G(jw)| at the
frequency at which the phase angle is
𝐺𝑀 = 0 − 𝑑𝐵 𝑜𝑓 𝐺𝐻 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 𝑎𝑡 𝑡ℎ𝑒 𝑝ℎ𝑎𝑠𝑒 𝑐𝑟𝑜𝑠𝑠𝑜𝑣𝑒𝑟 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 −180
Minimum phase system Stable
22

23. 23
Open loop transfer function :
Closed-loop transfer function :
)()( sHsG
)()(1 sHsG
Open loop Stability  poles of in LHP)()( sHsG
)0,0()0,1(
Re
Im
RHP
Closed-loop Stability 
poles of in left side of (-1,0))()( sHsG

24. Gain and Phase Margins (Cont.)
 The gain margin indicates the system gain can be increased by a factor of GM
before the stability boundary is reached.
 The phase margin is the amount of phase shift of the system at unity
magnitude that will result instability
Fig. 5.17 Phase and gain margin definition
(a) Stable system (b) Unstable system
24

25. Example 5.3:
 For the system shown in Fig.5.18, determine the gain margin, phase
margin, phase-crossover frequency, and gain-crossover frequency.
 Solution:
 The Bode plot for this system is shown in Fig. 5.19.
)(sR )(sC
  1025
)1(20
2


ssss
s
25

26. Example 5.3:
4.0131
9.9293 dB
0.4426
103.6573º
 The gain margin = 9.929 and the phase margin = 103.7 degree
26

27. Bandwidth and Cutoff Frequency
-3 dB
cutoff frequency
bandwidth
27

28. Using Matlab For Frequency Response
 Instruction: We can use Matlab to run the frequency response for the
previous example. We place the transfer function in the form:
 The Matlab Program
28
>> num = [5000 50000];
>> den = [1 501 500];
>> Bode (num,den)
]500501[
]500005000[
)500)(1(
)10(5000
2





ss
s
ss
s

29. Relationship between System Type and Log-Magnitude Curve
 Consider the unity-feedback control system. The static position,
velocity, and acceleration error constants describe the low-frequency
behavior of type 0, type 1, and type 2 systems, respectively.
 For a given system, only one of the static error constants is finite and
significant.
 The type of the system determines the slope of the log-magnitude
curve at low frequencies.
 Thus, information concerning the existence and magnitude of the
steady-state error of a control system to a given input can be
determined from the observation of the low-frequency region of the
log-magnitude curve.
29

30. Determination of Static Position Error Constants.
 Consider the type-0
unity-feedback control
system,
30

31. Determination of Static Position Error Constants.
 Consider the type-1
unity-feedback control
system,
31

32. Determination of Static Position Error Constants.
 Consider the type-2
unity-feedback control
system,
32

33. 33