Chapter 4 Frequency Response

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Chapter 4 Frequency Response

• 4.1 Introduction
• 4.2 Fourier Transfer
• 4.3 Frequency Response Function
• 4.4 Diagram of Frequency Response Function
• 4.5 Frequency Response Function of Elementary
  Unit
• 4.6 Diagram of Complex Frequency Response
  Function

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Chapter 4 Frequency Response

• 4.7 Closed-loop Frequency Response Diagram
• 4.8 Nyquist Stability Criterion
• 4.9 Stability Margin of System
• 4.10 Important Characteristic of Frequency
  Response Function
• 4.11 Researching Dynamics Performance of
  Closed-loop from Open-Loop Frequency
  Characteristic
• 4.12 Summary

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4.1 Introduction

• In previous chapter we examined the use of test
  signals such as a step and a ramp signal. In this
  chapter we consider the steady state response of
  a system to a sinusoidal input test signal. We
  will see that the response of a linear constant
  coefficient system to a sinusoidal input signal
  is an output sinusoidal signal at the same
  frequency as input. However, the magnitude and
  phase of the output signal differ from those of
  the input sinusoidal signal, and the amount of
  difference is a function of the input frequency.

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GH(s) Polar plot(Nyquist diagram) and
logarithmic magnitude and phase diagram(Bode
diagram) Determine the stability of closed-loop
system from Nyquist diagram and Bode diagram
Design compensator form to meet the performance
requirement
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4.2 Fourier transfer

• P160-167.

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• The relationship between Fourier and Laplace
  transforms
• Laplace
• Fourier

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When f(t) is defined only for tgt0, as is often
the case, the two equations differ only in the
complex variables. Thus the Fourier transform can
easily be obtained from Laplace transform just by
setting sjw. As the Fourier and Laplace
transforms are so closely related, the question
is why not always use the Laplace transform? Why
use the Fourier transform at all?
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• The Laplace transform permits us to
  investigate the s-plane location of the poles and
  zeros of a transfer G(s). However, the frequency
  response method allows us to consider the
  transfer function G(jw) and to concern ourselves
  with the amplitude and phase characteristics of
  the system. This ability to investigate and
  represent the character of a system by amplitude
  and phase equations and curves is an advantage
  for the analysis and design of control systems.

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• If we consider the frequency response of the
  closed-loop system, we might have an input r(t)
  that has a Fourier transform in the domain. Then
  the output frequency response of a single-loop
  system can be obtained by substituting sjw in
  the closed-loop system relationship, so we have
  the Fourier transform y(jw) of the output y(t).
  So we can use several measures of the transient
  response which can be related to the frequency
  characteristics and utilized for design purposes.

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The cut-off frequency, wB, at which the
frequency response has declined to of its
zero-frequency value.
In general, the Bandwidth is 510 times of wB .
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4.3 Frequency Response Function

• Definition The frequency response of a system
  is defined as the steady-state response of the
  system to a sinusoidal input signal.
• The sinusoid is a unique input signal, and the
  resulting output signal for a linear system, as
  well as signals throughout the system, is
  sinusoidal in the steady state it differs from
  the input waveform only in amplitude and the
  phase angle.

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The Fourier transform of the input-output
relation of steady-state response of the system .
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• Interpretations
• Suppose the input u(t) Usin(wt), then for
  large t,
• the steady-state output is also a sinusoidal
  of the same frequency with amplitude
  
• and phase .

U
U
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• the magnitude provides
  amplification (gain)
• the phase provides phase shift
• magnitude and phase are functions of frequency

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• The magnitude and phase of the harmonic
  response of a system can be obtained from its
  transfer function by substituting sjw.
• The frequency response of the system is simply

There are two major representations of the
frequency response Bode plot and Nyquist diagram.
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• Advantage of the frequency response method
• The ready availability of sinusoid test signals
  for various ranges of frequencies and amplitudes.
  Thus the experimental determination of the
  frequency response of a system is easily
  accomplished and is the most reliable and
  uncomplicated method for the experimental
  analysis of a system.

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• The design of a system in the frequency domain
  provides the designer with
• Control of the bandwidth of a system
• Some measure of the response of the system to
  undesired noise and disturbance.

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• The transfer function describing the sinusoidal
  steady-state behavior of a system can be obtained
  by replacing s with jw in the system transfer
  function G(s). The transfer function representing
  the sinusoidal steady-state behavior of a system
  is then a function of the complex variable jw
  and is itself a complex function G(jw) that
  possesses a magnitude and phase angle. The
  magnitude and phase angle of G(jw) are readily
  represented by graphical plots that provide
  significant insight into the analysis and design
  of control systems.

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• Basic disadvantage of the frequency response
  method
• There is no direct link between the frequency and
  the time domain.
• Direct correlation between the frequency
  response and the corresponding transient response
  characteristics are somewhat tenuous.
• In practice, the frequency response
  characteristic is adjusted by using various
  design criteria that usually result in a
  satisfactory transient response.

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• The transfer function of a system G(s) can be
  described in the frequency domain by the relation

(4.1)
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• Alternatively the transfer function can be
  represented by a magnitude and a
  phase as

(4.2)
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• The graphical representation of the frequency
  response of the system G(jw) can utilize either
  Eq. (4.1) or Eq. (4.2). The polar plot
  representation of the frequency response is
  obtained by using Eq. (4.1) . The coordinates of
  the polar plot are the real and imaginary parts
  of G(jw).

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4.4 Diagram of Frequency Response Function

• 4.4.1 Polar plot (Nyquist diagram)
• 4.4.2 Logarithmic frequency characteristic
  plot(Bode)
• 4.4.3 Logarithmic Magnitude-Phase
  Diagram(Nichols)

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4.4.1 Polar plot (Nyquist diagram)

• Polar plot The plot of frequency response G(j?)
  in the complex plane as ? increases from -8 to
  8, often called Nyquist plot(diagram) of G(s).
• The polar plot is a closed oriented curve in
  the complex plane.

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• Step
• 1.start point magnitude and
  phase(angle).
• 2.end point
• 3.circumrotate direction
• 4.intersection on negative real axis

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• Notes
• 1.

2. The polar plot is useful for investigating
system stability. 3.The limitations of polar
plots The addition of poles or zeros to an
existing system requires the recalculation of the
frequency response, and the calculation in this
manner is tedious and does not indicate the
effect of the individual poles or zeros.
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• Example Polar plot of a transfer function

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• Example Polar plot of

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• Intersection on the real axis

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4.4.2 Logarithmic frequency characteristic plot
(Bode)
Logarithmic frequency characteristic plots
often called Bode plots, (in honor of H.W.Bode,
who used them extensively in his studies of
feedback amplifiers), simplifies the
determination of the graphical portrayal of the
frequency response G(j?), consists of two plots
magnitude plot and phase plot. The magnitude
and frequency are in logarithm scale.

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• 1. Logarithmic magnitude-frequency plot
• An interval of two frequencies with a ratio
  equal to 10 is called a decade.
• A convenient unit for the magnitude is dB or
  B. 1B20dB

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• 2. Logarithmic phase-frequency plot

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• ExampleBode plot for first-order system

Magnitude plot
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The slope of the asymptotic line for this
first-order transfer function is 20dB/decade.
Plot Seek point Q The frequency w1/T is often
called the break frequency or corner frequency.
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• Phase plot

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break frequency
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• 3. Characteristic of Bode plot
• 1)Expand frequency range.
• 2)Easy to draw (asymptote)
• 3)Multiply(divide) can be transformed to
  plus(subtract).

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4) the curve move up or down.(gain) 5)
the curve move left or right.(time constant)
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• 6)
• response inverse frequency plot is the plot
  mirror about the horizontal axis.

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In comparison with the Nyiquist diagrams, the
Bode plots contain additional information about
the system in the frequency data. Knowing the
Bode plot one can construct the corresponding
Nyquist diagram, but the reverse is not possible.
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4.4.3 Logarithmic Magnitude-Phase
Diagram(Nichols)

• Nichols chart
• x-coordinate phase in degree,
• y-coordinate amplitude (in logarithmic
  scale).
• Nichols chart is a graphical tool for
  obtaining closed-loop frequency response from
  open-loop response. It is a popular tool before
  computers are widely available.

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4.5 Frequency Response Function of Elementary
Unit

• We will first develop rules for plotting the
  individual simple (elementary) unit, and later
  combine those in a single plot.
• 1. Proportion(constant gain) Unit

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• The logarithmic gain curve is simply a
  horizontal line on the Bode diagram.
• If the gain is a negative value, -K, the
  logarithmic gain remains 20lgK,the negative sign
  is accounted for by the phase angle, -1800.

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• 2. Integrator(poles at the origin)

The slope of the magnitude curve is -20dB/dec.
The phase angle is -900.
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• In this case the slope due to the multiple
  poles is 20ndB/dec. The phase angle is -900n.

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• 3. Differentiator (a zero at the origin)

The slope of the magnitude curve is 20dB/dec.
The phase angle is 900.
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• 4. First-order Unit (a pole on the real axis)

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Approximately, the magnitude response
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• Definition
• Break frequency the intersection of the
  magnitude response curve and the horizontal axis.

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Magnitude response is composed of two segments
one is 0dB,another branchs slope is
-20dB/dec.The intersection of the two asymptotes
occurs w1/T. Error curve symmetry on w1/T,
and the maximum error when w1/T is 3dB. Phase
response range from 00 to -900,center symmetry
about the point(1/T,-450).
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• 5. First-order Differential Unit (a zero on
  the real axis)
• The frequency response can be obtained by the
  reciprocal relationship (inverse frequency
  response)

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• 6. Oscillations Unit (complex conjugate poles)

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The slope is -40dB/dec.
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? 0.1
? 0.3
? 0.5
? 0.7
? 1
? 0.5
? 0.3
? 1
? 0.1
? 0.7
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Magnitude response is composed of two segments
one is 0dB,another curve with a slope of
-40dB/dec.The intersection of the two asymptotes
occurs w1/T. Phase response range from 00 to
-1800,center symmetry about the point(1/T,-900)
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• Error curve symmetry on w1/T.
• The difference between the actual magnitude
  curve and the asymptotic approximation is a
  function of the damping ratio ?and must be
  accounted for when ?lt0.707.

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The maximum value of the frequency response Mr
occurs at the resonant frequency wr. When the
damping ratio approaches zero, then wr approaches
wn, the natural frequency. The resonant frequency
is determined by taking the derivative of the
magnitude of M with respect to the normalized
frequency and setting it equal to zero. The
resonant frequency is represented by the relation
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• And the maximum value of the magnitude is

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• 7. Second-order Differential Unit (complex
  conjugate zeros)
• symmetry with respect to the frequency axis.

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• 8. Non-minimum phase system
• A transfer function is called a minimum phase
  transfer function if all its zeros lie in the
  left-hand s-plane. It is called a non-minimum
  phase transfer function if it has zeros in the
  right-hand s-plane.
• The meaning of the term minimum phase the
  range of phase of a minimum phase transfer
  function is the least possible or minimum
  corresponding to a given amplitude curve, whereas
  the range of the non-minimum phase curve is
  greater than the minimum possible for the given
  amplitude curve.

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• (1) Transfer function of a delay of Td seconds
  and the frequency response is

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• A time delay added into the loop does not
  change the magnitude but increases the phase lag
  of the frequency response.
• The main effects of time delay are to reduce
  phase margin reduce the closed-loop bandwidth.

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• (2) Unstable unit

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is the same as the oscillation unit.
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• Conclusion
• Asymptote
• Corner frequency
• Slope
• Symmetry
• Phase angle range.

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4.6 Diagram of Complex Frequency Response
Function

• 1. Decompose TF to the elementary unit.

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• 2. Calculate the break frequencies.
• 3. Calculate the slope and gain of the low
  frequency(only consider the proportion and the
  integrators)

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• 4. Obtain the asymptotic magnitude plot for the
  complete transfer function.
• 5. Furthermore, obtain the actual curves
  only at specific important frequencies.

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• Phase plot
• 1. Write the phase angle equation.
• 2. Calculate the range and estimate the change
  direction.
• 3. List the phase angle of some points (the
  every corner frequency point and wc)
• 4. Draw the phase angle curve.

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• Example The Bode diagram of the TF
• 1.
• 2.
• 3.

break frequency
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• 4. asymptotic magnitude plot
• slope ?lt0.4,-20dB/dec,?gt0.4,-40dB/dec,?gt2,-20dB/
  dec,?gt10,-40dB/dec,
• ?gt20,-60dB/dec,
• 5.revise.
• 6.
• Definition

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• 7. Phase plot
• Range from 900 to 2700.

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36
2
0.4
10
20
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Bode plot can be readily obtained by Matlab
num0 0 10 den1 3 2 0 bode(num,den)
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• The Bode diagram of a transfer function G(s)
  is obtained by adding the plot due to each
  individual pole and zero.

The factors are 1. A constant gain K5 2. A pole
at the origin 3. A pole at ?2 4. A zero at
?10 5. A pair of complex poles at wwn 50
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Plot the magnitude characteristic 1. The
constant gain is 20log514 dB 2. The magnitude
of the pole has a slope of 20 dB/decade and is 0
dB at ?1. 3. Beyond the break frequency at ?
2, the slope is - 20 dB/decade and below the
break frequency 0 dB/decade. 4. Beyond the break
frequency at ? 10, the slope is 20 dB/decade
and below the break frequency 0 dB/decade. 5. At
? ?n 50, the slope of the magnitude for the
pair of complex poles is - 40 dB/decade. Because
the damping ratio is ? 0.3, the approximation
must be corrected to the actual magnitude.
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By adding the asymptotes due to each factor, we
obtain the total asymptotic magnitude as
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Plot the phase characteristic
Range from 900 to 2700.
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• Nyquist diagramm
• Minimum Phase System,Open-loop TF

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• Exp.

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-5/3
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• Nyquist diagram can be readily obtained by Matlab
  
• num0 0 10
• den1 3 2 0
• nyquist(num,den)
• v-5 1 3 3axis(v)

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4.7 Closed-loop frequency response diagram
A typical control system where L(s) is the loop
transfer function.
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The closed-loop frequency response
is bandwidth of the closed-loop system is
defined as the frequency wB such that
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For standard second-order system the bandwidth
is roughly wB. Larger bandwidth means
smaller rise time (more high frequency components
get passed).
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Type ?M??(?????) ?N??(?????) ?????? ????
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4.8 Nyquist stability criterion

• 4.8.1 Mapping Theorem
• 4.8.2 Nyquist Stability Criterion
• 4.8.3 Examples of Nyquist Stability Analysis
• 4.8.4 ???Nyquist??Nyquist????

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• A frequency domain stability criterion was
  developed by H.Nyquist in 1932 and remains a
  fundamental approach to the investigation of the
  stability of linear control systems. The Nyquist
  stability criterion is based on a theorem in the
  theory of the function of a complex variable due
  to Cauchy. Cauchys theorem is concerned with
  mapping contours in the complex s-plane, and
  fortunately the theorem can be understood without
  a formal proof that uses complex variable theory.

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4.8.1 Mapping theorem

• Cauchys theorem If a contour C, in the
  s-plane encircles P poles and Z zeros of W(s)
  and does not pass through any poles or zeros of W
  (s) and the traversal is in the clockwise
  direction along the contour, the corresponding
  contour C in the W (s) -plane encircles the
  origin of the W (s)-plane NZ-P times in the
  clockwise direction.

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• Suppose W(s) is the rational function of s. That
  is

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• As an example of the use of Cauchys theorem,
  consider the pole-zero pattern shown in Fig.(a)
  with the contour C, to be considered. The contour
  encloses and encircles three zeros and one pole.
  Therefore we obtain N3-12, and C completes
  two clockwise encirclements of the origin in the
  W(s)-plane, as shown in Fig.(b).

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contour C
W(s)
contour C
origin
(a)
N3-12
(b)
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Example When N is negative. For the pole and
zero pattern and the contour Gs shown below, one
pole is encircled and no zeros are encircled.
Therefore NZ-P-1. Since the sign of N is
negative, the encirclement moves in the
counterclockwise direction.
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4.8.2 Nyquist Stability Criterion

• Assistant function W(s)
• Open-loop TF
• Construct

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• Summary
• (1)W(s)s poles are open-loop poles.
• (2)W(s)s zeros are close-loop poles.(The
  numerator of W(s) is the CL characteristic
  polynomial P(s)N(s))
• (3)The origin of W(s)-plane 0 is the (-1,j0)
  point of Q(s)-plane, the jw-axis offsets by 1 to
  the right.
• Instead of plotting W(s), we can plot Q(s),
  and count the encirclements of (-1,j0).

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• Contour D this contour in the s-plane
  encloses the entire right half s-plane.
• The contour consists of three segments

This part of the contour provides the familiar
F(jw).
A semicircular path of radius infinity.
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• The application of Cauchys theorem
• As s travels along the contour D in the
  clockwise direction, the corresponding contour in
  the W(s)-plane encircles the origin of the
  W(s)-plane NZ-P times in the clockwise
  direction.
• Z is the number of close-loop poles in the
  right-half s-plane.
• P is the number of open-loop pole in the
  right-half s-plane.

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Nyquist stability criterion can be stated as
follows A feedback system is stable if and only
if, for the contour in the W(s)-plane, the
number of counterclockwise encirclements of the
origin point is equal to P, i.e. the number of
poles of W(s)(open-loop poles) with positive real
parts. (NZ-P, Z0, so N-P)
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• Notes
• (1) The origin of W(s)-plane 0 is the (-1,j0)
  point of Q(s)-plane.
• (2) If P0,i.e. minimum phase system, then
  N0,the stable condition is the contour in the
  Q(s)-plane does not encircle the (-1,j0) point.
  

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(3)
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(4)
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• (5) The conclusion can be expanded to the
  system including delay unit.
• (6) If the contour Q(jw) overpass the (-1,j0)
  point, that is one close-loop pole on the
  jw-axis, the system is critically stable.

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• (7)System with r poles at the origin (contour
  D pass some poles of Q(s))
• The supplement curve must be drawn.
• The small semicircular detour around the pole at
  the origin can be represented by setting

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• Nyquist stability criterion For a system with P
  open loop poles in the right-half
  plane(unstable poles) to be stable, the open loop
  frequency response must encircle the (-1,j0)
  point P time counter-clockwise in the s-plane.
• If the number of counter-clockwise
  encirclements is N?P, then the closed-loop system
  is unstable with Z unstable poles, where ZNP.

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• If the open-loop transfer function is stable,
  i.e., P0, then the closed-loop, system is stable
  if and only if the Nyquist plot of G(s) does not
  encircle the (-1,j0) point.

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4.8.3 Examples of Nyquist Stability Criterion

• Exp.1
• K20,
• determine the stability of the system.

We need to find the cross-over point and compare
it with -1!
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-1
20
-0.3
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Nyquist plot of G(s) does not encircle the 1
point. Because P0, we can know the system is
stable!
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Exp.2 K100

• The end point and circle direction is
  unchanged. But the cross-over point is changed.

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100
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-1.48
-1
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• ConclusionNyquist diagram encircles the 1
  point two times in the direction of clockwise.
  P0, so the system is unstable. N2,Z2, so there
  are two poles of close-loop on the right half
  s-plane.

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• Exp.3
• K2,determine the systems stability.
• There is a open-loop pole on the jw-axis, so
  the supplement curve must be drawn.

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-0.18
-1
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ConclusionNyquist diagram does not encircle
the 1 point, P0, so the system is stable.
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• Exp.4K20

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-1.8
-1
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ConclusionNyquist diagram encircles the 1
point two times in the direction of clockwise.
P0, so the system is unstable. N2,Z2, so there
are two poles of close-loop on the right half
s-plane.
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• Exp.5non-minimum phase system

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ConclusionNyquist diagram encircles the 1
point one time in the direction of
counter-clockwise. N-1,P1,ZNP0, so the
system is stable. The changing range of K is
Kgt3.
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• Exp.6The open-loop TF is
• Determine the changing range of K.

The -1 point located on A or C, the system is
stable. The 1 point located on B or D, the
system is unstable.
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• We can get
• -1 locus on A, Klt19.2, stable
• -1 locus on B, 19.2ltKlt334, unstable
• -1 locus on C, 334ltKlt13200, stable
• -1 locus on D, Kgt13200, unstable
• So the changing range of K is
• 0lt Klt19.2 and 334ltKlt13200.

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4.8.4 ???Nyquist??Nyquist????

• ????

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4.9 Stability Margin of System

• The closeness of the GH(j?) cure to
• -1 is a measure of the relative stability of
  the system. There are two numbers reflecting this
  measure ----- Gain and Phase Margin.

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• Phase Margin(PM)
• DefinitionLet ?c be the gain crossover
  frequency, i.e.,
• PMgt0(PMlt0) indicates a stable (an unstable)
  system.

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• Gain Margin(GM) Kg
• Let ?g be the phase crossover frequency (which
  the phase angle reaches 1800), i.e.,

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• GMgt1(gt0 dB) indicates a stable closed-loop
  system and the system will remain stable if the
  loop gain increase is less than GM.
• GMlt1(lt0 dB) indicates an unstable closed-loop
  system and a reduction of loop gain at least GM
  is required for the system to become stable.

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• From the Nyquist plot we can also determine
  the gain margin and phase margin of the system.
• The phase of the point at which the plot cross
  the unit circle centered at the origin (0, 0)
  gives phase margin.
• The magnitude of the point at which the plot
  cross the negative real axis defines the gain
  margin.

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Stability criterion The system is stable when
the Gain Margin is greater than one and the Phase
Margin is positive.
crossover frequency
1/Kg
0
A
phase margin
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• 6?Stability from Bode Plots Gain and Phase
  Margins on Bode plots.

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4.10 Important Characteristic of Frequency
Response Function

• ??????
• ???????????G(j?)????

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4.11 Researching dynamics performance of
closed-loop from open-loop frequency
characteristic

• System type and steady state error from Bode
  plots
• Consider a general Open Loop TF

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• At low frequencies , the above will be
  approximately equal to s j?0

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• The system type can be determined from the low
  frequency portion of L(j?).
• the system is type 0 if slope of the low
  frequency asymptote is 0.
• the system is type 1 if slope of the low
  frequency asymptote is -1 or -20dB/dec.
• the system is type 2 if slope of the low
  frequency asymptote is -2 or -40dB/dec.

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• Error constants
• for a type 0 system, the position error constant
  is the gain at ?0, KpL(j0).
• for a type 1 system, the velocity error constant
  is the value of the low frequency asymptote at
  ?1 .
• for a type 2 system, the acceleration error
  constant is the value of the low frequency
  asymptote at ?1 .

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high-frequency
Low-frequency
middle-frequency
Desired big K
-20db/dec lt-20db/dec Bode plot
-40de/dec , definite range
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• ????II???,K????????
• ?K???????

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• ?????????????????????????
• ?????????
• 1??????????(??)???
• ????????

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• 2????? ??????? ?????
• ???? ??????

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• ??K?,????, ??,?????????????????
• ????????????
• 3???????????????
• ???????????,?????????,?????
• 4?????????(?)?

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• ????????
• 1??????????30dB????-15dB?????
• 2??????????20dB????-20dB?????
• 3? ???????????-1?????

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• Roughly speaking, controller design based on
  Bode plot seeks to achieve
• large low-frequency gain, i.e., large error
  constants, for small steady-state error
• large enough gain crossover frequency for speed
  of response ( we will see that ?c is closely
  related to closed-loop bandwidth)
• large enough stability margins for robustness and
  transient response
• fast enough decrease in L(j?) for ?gt?c to avoid
  noise amplification

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• Basic disadvantage of the frequency response
  method is that there is no direct link between
  the frequency and the time domain.
• But for a simple second-order system, we can
  answer this question by considering the
  time-domain performance in terms of overshoot,
  settling time, and other performance criteria
  such as integral squared error.
• For high-order system, there are just some
  estimate formulae.

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4.12 Summary

• Frequency response of a system was defined as
  the steady-state response of the system to a
  sinusoidal input signal.
• Several alternative forms of frequency response
  plots were considered, including the polar plot
  and Bode plots.
• The asymptotic approximation for drawing the
  Bode diagram simplifies the computation
  considerably.

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• Several performance specifications, such as the
  maximum magnitude Mr, wr and wB, in the frequency
  domain were discussed.
• The relationship between the time-domain
  specifications and the specifications in
  frequency domain was discussed for a second-order
  system.
• Finally, the log magnitude versus phase diagram
  was considered for graphically representing the
  frequency response of a system.

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• In the frequency domain, Nyquists criterion can
  be used to determine the stability of a feedback
  system.
• Nyquists criterion provides two relative
  stability measures gain margin and phase margin.
• They are the indices of the transient
  performance, based on the correlations between
  the frequency domain and the transient response.

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• The End of Chapter 4