Chapter 5 Frequency Response Method

1
Chapter 5 Frequency Response Method

1. Introduction
2. Frequency Response of the typical elements of the
   linear systems
3. Bode diagram of the open loop system
4. Nyquist-criterion
5. System analysis based on the frequency response
6. Frequency response of the closed loop systems

2
5.1 Introduction
Three advantages Frequency
response(mathematical modeling) can be obtained
directly by experimental approaches. easy to
analyze the effects of the system with sinusoidal
voices. easy to analyze the stability of the
systems with a delay element
5.1.1 frequency response For a RC circuit
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5.1 Introduction
Frequency Response(or frequency characteristic)
of the electric circuit.
4
5.1 Introduction
Generalize above discussion, we have

And we name
(phase difference between steady-state output and
sinusoid input )
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5.1.2 approaches to get the frequency
characteristics
1. Experimental discrimination
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5.1.2 approaches to get the frequency
characteristics
2. Deductive approach
Theorem If the transfer function is G(s), we
have
Proof
Where pi is assumed to be distinct pole
(i1,2,3n).
7

• In partial fraction form

Here
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5.1.2 approaches to get the frequency
characteristics

• Taking the inverse Laplace transform

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5.1.2 approaches to get the frequency
characteristics
the steady-state output
The amplitude ratio of the steady-state output
cs(t) versus sinusoid input r(t)
The phase difference between the steady-state
output and sinusoid input
Then we have
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5.1 Introduction

• Examples 5.1.1

Solution
1) Determine the steady-state response c(t) of
the system.
The closed-loop transfer function is
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5.1 Introduction

• The frequency characteristic

The magnitude and phase response
The output response
So we have the steady-state response c(t)
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5.1 Introduction
2) Determine the steady-state error e(t) of the
system.

The error transfer function is
The error frequency response
The steady state error e(t) is
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5.1 Introduction

• 5.1.3 Graphic expression of the frequency response

Graphic expression for intuition
1. Rectangular coordinates plot
Example 5.1.2
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5.1.3 Graphic expression of the frequency response

• 2. Polar plot

The polar plot is easily useful for investigating
system stability.
Example 5.1.3
The magnitude and phase response
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5.1.3 Graphic expression of the frequency response

• The shortage of the polar plot and the
  rectangular coordinates plot to synchronously
  investigate the cases of the lower and higher
  frequency band is difficult.

How to enlarge the lower frequency band and
shrink (shorten) the higher frequency band?
Idea
3. Bode diagram(logarithmic plots)
Plot the frequency characteristic in a semilog
coordinate
First we discuss the Bode diagram in detail
with the frequency response of the typical
elements.
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5.2 Frequency Response of The Typical Elements

• The typical elements of the linear control
  systems refer to Chapter 2.

1. Proportional element
Transfer function
Frequency response
Polar plot
Bode diagram
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5.2 Frequency response of the typical elements

• 2. Integrating element

Transfer function
Frequency response
Polar plot
Bode diagram
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5.2 Frequency response of the typical elements

• 3. Inertial element

Transfer function
1/T
break frequency
Polar plot
Bode diagram
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5.2 Frequency response of the typical elements

• 4. Oscillating element

Transfer function
Make
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5.2 Frequency response of the typical elements

• The polar plot and the Bode diagram

Polar plot
Bode diagram
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5.2 Frequency response of the typical elements

• 5. Differentiating element

Transfer function
1
1
differential
1th-order differential
2th-order differential
Polar plot
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5.2 Frequency response of the typical elements

• Because of the transfer functions of the
  differentiating elements are the reciprocal of
  the transfer functions of Integrating element,
  Inertial element and Oscillating element
  respectively,

that is
the Bode curves of the differentiating elements
are symmetrical to the log?-axis with the Bode
curves of the Integrating element, Inertial
element and Oscillating element respectively.
Then we have the Bode diagram of the
differentiating elements
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5.2 Frequency response of the typical elements

differential
2th-order differential
1th-order differential
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5.2 Frequency response of the typical elements

• 6. Delay element

Transfer function
R1
Polar plot
Bode diagram
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5.3 Bode diagram of the open loop systems

• 5.3.1 Plotting methods of the Bode diagram of
  the open loop systems
• Assume

We have
That is, Bode diagram of a open loop system is
the superposition of the Bode diagrams of the
typical elements.
Example 5.3.1
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5.3 Bode diagram of the open loop systems

• G(s)H(s) could be regarded as

Then we have
20dB/dec
-40dB/dec
-20dB/dec
-20dB/dec
-40dB/dec
-40dB/dec
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5.3.2 Facility method to plot the magnitude
response of the Bode diagram
Summarizing example 5.3.1, we have the
facility method to plot the magnitude response of
the Bode diagram
1) Mark all break frequencies in the?-axis of
the Bode diagram.
2) Determine the slope of the L(?) of the lowest
frequency band (before the first break frequency)
according to the number of the integrating
elements -20dB/dec for 1 integrating
element -40dB/dec for 2 integrating
elements
3) Continue the L(?) of the lowest frequency
band until to the first break frequency,
afterwards change the the slope of the L(?)
which should be increased 20dB/dec for the break
frequency of the 1th-order differentiating
element .
The slope of the L(?) should be decreased
20dB/dec for the break frequency of the Inertial
element
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5.3.2 Facility method to plot the magnitude
response of the Bode diagram
Plot the L(?) of the rest break frequencies by
analogy .
Example 5.3.2
The Bode diagram is shown in following figure
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5.3.2 Facility method to plot the magnitude
response of the Bode diagram
-20dB/dec
-60dB/dec
There is a resonant peak Mr at
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5.3.3 Determine the transfer function in terms
of the Bode diagram

1. The minimum phase system(or transfer function)

Compare following transfer functions
The magnitude responses are the same.
But the net phase shifts are different when ?
vary from zero to infinite. It can be illustrated
as following
Sketch the polar plot
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5.3.3 Determine the transfer function in terms
of the Bode diagram

• The polar plot

Im
Re
phase shift -p
phase shift 00
phase shift -p
phase shift p
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5.3.3 Determine the transfer function of the
minimum phase systems in terms of the
magnitude response

• Definition

A transfer function is called a minimum phase
transfer func- tion if its zeros and poles all
lie in the left-hand s-plane. A transfer
function is called a non-minimum phase transfer
function if it has any zero or pole lie in the
right-hand s-plane. Only for the minimum phase
systems we can affirmatively deter- mine the
relevant transfer function from the magnitude
response of the Bode diagram .
2. Determine the transfer function from the
magnitude response of the Bode diagram .
Example 5.3.3
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5.3.3 Determine the transfer function in terms
of the Bode diagram
2 20 200
Example 5.3.4
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5.3.3 Determine the transfer function in terms
of the Bode diagram
Example 5.3.5
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5.3.3 Determine the transfer function in terms
of the Bode diagram
For the non-minimum phase system we must
combine the magnitude response and phase response
together to determine the transfer function.
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5.3.3 Determine the transfer function in terms
of the Bode diagram
Example 5.3.6
All satisfy the magnitude response
But
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5.4 The Nyquist-criterion

• A method to investigate the stability of a
  system in terms of the open-loop frequency
  response.

5.4.1 The argument principle(Cauchys theorem)
Assume
Make
Note si? the zeros of the F(s), also the roots
of the 1G(s)H(s)0
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5.4.1 The argument principle
S-plane
Similarly we have
Fig. 5.4.1
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5.4.1 The argument principle

• If Z zeros and P poles are enclosed by G , then

It is obvious that path G can not pass through
any zeros si or poles pj .
Then we have the argument principle
N P - Z
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5.4.1 The argument principle

• here N number of the F(s) locus encircling
  the origin of the F(s)-plane in the
  counterclockwise direction.

P number of the zeros of the F(s) encircled
by the path G in the s-plane. Z
number of the poles of the F(s) encircled by the
path G in the s-plane.
5.4.2 Nyquist criterion
If we choose the closed path G so that the G
encircles the entire right hand of the s-plane
but not pass through any zeros or poles of F(s)
shown in Fig.5.4.2 .
The path G is called the Nyquist-path.
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5.4.2 Nyquist criterion
S-plane

• When s travels along the the Nyquist-path

Because the origin of the F(s)-plane is
Fig. 5.4.2
equivalent to the point (-1, j0) of the
G(j?)H(j?)-plane, we have another statement of
the argument principle
When ? vary from -? (or 0) ? ? , G(j?)H(j?)
Locus mapped in the G(j?)H(j?)-plane will
encircle the point (-1, j0) in the
counterclockwise direction
here P the number of the poles of G(s)H(s) in
the right hand of the
s-plane. Z the number of the zeros of
F(s) in the right hand of the
s-plane.
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5.4.2 Nyquist-criterion
If the systems are stable, should be Z 0, then
we have

The sufficient and necessary condition of the
stability of the linear systems is When ? vary
from -? (or 0) ? ? , the G(j?)H(j?) Locus
mapped in the G(j?)H(j?)-plane will encircle the
point (-1, j0) as P (or P/2) times in the
counterclockwise direction.
Nyquist criterion
Here P the number of the poles of G(s)H(s)
in the right hand of the s-plane.

Discussion
i) If the open loop systems are stable, that is
P 0, then
for the stable open-loop systems, The
sufficient and necessary condition of the
stability of the closed-loop systems is
When ? vary from -? (or 0) ? ? , the G(j?)H(j?)
locus mapped in the G(j?)H(j?)-plane will not
encircle the point (-1, j0).
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5.4.2 Nyquist-criterion

• ii) Because that the G(j?)H(j?) locus encircles
  the point (-1, j0) means that the G(j?)H(j?)
  locus traverse the left real axis of the point
  (-1, j0) , we make

G(j?)H(j?) Locus traverses the left real axis
of the point (-1, j0) in the counterclockwise
direction positive traversing.
G(j?)H(j?) Locus traverses the left real axis
of the point (-1, j0) in the clockwise direction
negative traversing.
Then we have another statement of the Nyquist
criterion
The sufficient and necessary condition of the
stability of the linear systems is When ? vary
from -? (or 0) ? ? , the number of the net
positive traversing is P (or P/2).
Here the net positive traversing the
difference between the number of the positive
traversing and the number of the negative
traversing .
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5.4.2 Nyquist-criterion

• Example 5.4.1

The polar plots of the open loop systems are
shown in Fig.5.4.3, determine whether the systems
are stable.
stable
stable
unstable
unstable
Fig.5.4.3
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5.4.2 Nyquist-criterion

• Note

the system with the poles (or zeros) at the
imaginary axis
Example 5.4.2
There is a pole s 0 at the origin in this
system, but the Nyquist path can not pass
through any poles of G(s)H(s).
Idea
Im
at the s 0 point we have
Re
Fig. 5.4.4
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5.4.2 Nyquist-criterion

• It is obvious that there is a phase saltation
  of the G(j?)H(j?) at ?0, and the magnitude of
  the G(j?)H(j?) is infinite at ?0.

Fig.5.4.5
In terms of above discussion , we can plot the
systems polar plot shown as Fig.5.4.5.
The closed loop system is unstable.
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Example 5.4.3
Determine the stability of the system applying
Nyquist criterion.
Solution
Similar to the Example 5.4.2, the systems
polar plot is shown as Fig.5.4.6 .
The closed loop system is unstable.
5.4.3 Application of the Nyquist criterion in
the Bode diagram
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5.4.3 Application of the Nyquist criterion in
the Bode diagram
G(j?)H(j?) locus traverses the left real axis
of the point (-1, j0) in G(j?)H(j?)-plane ?
L(?)0dB and f(?) -180o in Bode diagram (as
that mentioned in 5.4.2).
We have the Nyquist criterion in the Bode diagram

The sufficient and necessary condition of the
stability of the linear closed loop systems is
When ? vary from 0? ? , the number of the net
positive traversing is P/2.
Here the net positive traversing the
difference between the number of the positive
traversing and the number of the negative
traversing in all L(?)0dB ranges of the
open-loop systems Bode diagram.
positive traversing f(?) traverses the
-180o line from below to above in the open-loop
systems Bode diagram negative
traversing f(?) traverses the -180o line
from above to below.
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5.4.3 Application of the Nyquist criterion in
the Bode diagram

• Example 5.4.4

The Bode diagram of a open-loop stable system
is shown in Fig.5.4.7, determine whether the
closed loop system is stable.
Solution
Because the open-loop system is stable, P 0
.
In terms of the Nyquist criterion in the Bode
diagram
The number of the net positive traversing is
0 ( P/2 0 ).
The closed loop system is stable .
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5.4.4 Nyquist criterion and the relative
stability (Relative stability of the
control systems)

• In frequency domain, the relative stability
  could be described by the gain margin and the
  phase margin.
• 1. Gain margin Kg

2. Phase margin ?c
3. Geometrical and physical meanings of the Kg
and ?c
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5.4.4 Nyquist criterion and the relative
stability
The geometrical meanings is shown in Fig. 5.4.8.
The physical signification
Kg amount of the open-loop gain in decibels
that can be allowed to increase before the
closed-loop system reaches to be unstable. For
the minimum phase system
Kggt1 the closed loop system is stable .
?c amount of the phase shift of G(j?)H(j?)
to be allowed before the closed-loop system
reaches to be unstable.
For the minimum phase system ?cgt0 the closed
loop system is stable .
52
5.4.4 Nyquist criterion and the relative
stability
Attention
For the linear systems
The changes of the open-loop gain only alter
the magnitude of G(j?)H(j?).
The changes of the time constants of G(s)H(s)
only alter the phase angle of G(j?)H(j?).
Example 5.4.5
The open loop transfer function of a control
system is
(1) Determine Kg and ?c when K 1 and t 1.
(2) Determine the maximum K and t based on K 1
and t 1.
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5.4.4 Nyquist criterion and the relative
stability
Solution
(1) Determine Kg and ?c ( K 1 and t 1)
(2)
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5.4.4 Nyquist criterion and the relative stability

• Example 5.4.6

The G(j?)H(j?) polar plot of a system is shown in
Fig.5.4.9.
(1) Determine Kg
(2) Determine the stable range of the open loop
gain.
Solution
(1) Determine Kg
(2) Determine the stable range of the open
loop gain.
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5.5 System analysis based on the frequency
response

• 5.5.1 Performance specifications in the
  frequency domain

1. For the closed loop systems
The general frequency response of a closed loop
systems is shown in Fig. 5.5.1
(1) Resonance frequency ?r
(2) Resonance peak Mr
(3) Bandwidth ?b
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2. For the open loop systems
(1) Gain crossover frequency ?c
For the unity feedback systems, ?c ?b , because
(2) Gain margin Kg
(3) Phase margin ?c
58
5.5 System analysis based on the frequency
response
Generally Kg and ?c could be concerned with
the resonance peak Mr Kg and ?c ? Mr ?.

?c could be concerned with the resonance
frequency ?r and bandwidth ?b ?c? ?r and
?b?.
5.5.2 Relationship of the performance
specifications between the frequency and
time domain
The relationship between the frequency
response and the time response of a system can be
expressed by following formula
But it is difficult to apply the formula .
59
5.5.2 Relationship of the performance
specifications between the frequency and
time domain

• (1) Bandwidth ?b(or Resonance frequency ?r) ?
  Rise time tr
• Generally ?b(or ?r )? tr ? because of the
  time scale theorem

alike ?c? tr ? because of ?c ?b .
For the large ?b , there are more
high-frequency portions in c(t), which make the
time response to be faster.
60
5.5.2 Relationship of the performance
specifications between the frequency and
time domain
(2) Resonance peak Mr ? overshoot sp
Normally Mr ? sp ? because of the large
unbalance of the frequency signals passing to
c(t) .
Kg and ?c ? sp ?is alike because of Kg and ?c
?Mr ?.
Some experiential formulas
61
5.5.2 Relationship of the performance
specifications between the frequency and
time domain
(3) A(0) ? Steady state error ess
So for the unity feedback systems
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5.5.2 Relationship of the performance
specifications between the frequency and time
domain

• (4) Reproductive bandwidth ?M ? accuracy of
  Reproducing r(t)

Reproductive bandwidth ?M
for a given ?M , ??higher accuracy of
reproducing r(t) . for a given ?, ?M ? higher
accuracy of reproducing r(t) .
Demonstration
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5.5.2 Relationship of the performance
specifications between the frequency and time
domain
For the frequency spectrum of r(t) shown in
Fig.5.5.3 .
5.5.3 Relationship of the performance
specifications between the frequency and the
time domain for the typical 2th-order system
For the typical 2th-order system
64
5.5.3 Relationship of the performance
specifications between the frequency and the
time domain for the typical 2th-order system
65
5.5 System analysis based on the frequency
response

• 5.5.4 three frequency band theorem
• The performance analysis of the closed loop
  systems according to the open loop frequency
  response.

1. For the low frequency band
the low frequency band is mainly concerned
with the control accuracy of the systems.
The more negative the slope of L(?) is , the
higher the control accuracy of the systems. The
bigger the magnitude of L(?) is, the smaller the
steady state error ess is.
2. For the middle frequency band
The middle frequency band is mainly concerned
with the transient performance of the systems.
?c?tr ? Kg and ?c ?sp ?
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5.5.4 three frequency band theorem
The slope of L(?) in the middle frequency band
should be the 20dB/dec and with a certain width
.
3. For the high frequency band
The high frequency band is mainly concerned
with the ability of the systems restraining the
high frequency noise. The smaller the magnitude
of L(?) is, the stronger the ability of the
systems restraining the high frequency noise is.
Example 5.5.1
Compare the performances between the system
?and system ?
67
5.5 System analysis based on the frequency
response
Solution
The ability of the system ? restraining the
high frequency noise is stronger than system ?
?
Example 5.5.2
For the minimum phase system, the open loop
magnitude response shown as the Fig. 5.5.5.
Determine the systems parameter to make the
system being the optimal second-order system and
the steady-state error esslt 0.1.
Solution
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5.6 Frequency response of the closed loop systems
5.6 Frequency response of the closed loop systems
How to obtain the closed loop frequency
response in terms of the open loop frequency
response.
5.6.1 The constant M circles How to obtain the
magnitude frequency response of the closed loop
systems in terms of the open loop frequency
response (refer to P495)
5.6.2 The constant N circles How to obtain the
phase frequency characteristic of the closed loop
systems in terms of the open loop frequency
response (refer to P496)
5.6.3 The Nichols chart How to obtain the closed
loop frequency response in terms of the open loop
frequency response
(refer to P496)
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Chapter 5 Frequency Response Methods