Frequency-Domain of Control
Systems
Eng R. L. Nkumbwa
Copperbelt University
2010
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Eng. R. L. Nkumbwa
Its all Stability of Control Systems
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Introduction
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In practice, the performance of a control system is
more realistically measured by its time domain
characteristics.
The reason is that the performance of most
control systems is judged based on the time
response due top certain test signals.
In the previous chapters, we have learnt that the
time response of a control system is usually
more difficult to determine analytically,
especially for higher order systems.
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Eng. R. L. Nkumbwa
Introduction
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In design problems, there are no unified
methods of arriving at a designed system that
meets the time-domain performance
specifications, such as maximum overshoot,
rise time, delay time, settling time and so on.
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Introduction
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On the other hand, in frequency domain, there
is a wealth of graphical methods available that
are not limited to low order systems.
It is important to realize that there are correlating
relations between frequency domain
performance in a linear system,
So the time domain properties of the system
can be predicted based on the frequencydomain characteristics.
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Eng. R. L. Nkumbwa
Example: Gun Positional Control
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Why use Frequency-Domain?
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With the previous concepts in mind, we can
consider the primary motivation for conducting
control systems analysis and design in the
frequency domain to be convenience and the
availability of the existing analytical tools.
Another reason, is that, it presents an alternative
point of view to control system problems, which
often provides valuable or crucial information in
the complex analysis and design of control
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systems.
Frequency-Domain Analysis
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The starting point for frequency-domain
analysis of a linear system is its transfer
system.
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Eng. R. L. Nkumbwa
Time & Frequency-Domain Specs.
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So, what are time-domain specifications by
now?
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Ok, what of frequency domain specifications?
What are they?
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Lets look at the pictorials views…
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Eng. R. L. Nkumbwa
Time-Domain Specifications
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Frequency-Domain Specifications
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Wrap Up…
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The frequency response of a system directly
tells us the relative magnitude and phase of a
system’s output sinusoid if the system input is
a sinusoid.
What about output frequency?
If the plant’s transfer function is G (s), the
open-loop frequency response is G (jw).
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Eng. R. L. Nkumbwa
Further Frequency Response
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In previous sections of this course we have
considered the use of standard test inputs,
such as step functions and ramps.
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However, we will now consider the steadystate response of a system to a sinusoidal
input test signal.
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Eng. R. L. Nkumbwa
Further Frequency Response
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The response of a linear constant-coefficient
linear system to a sinusoidal test input is an
output sinusoidal signal at the same frequency
as the 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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Eng. R. L. Nkumbwa
Further Frequency Response
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We will now examine the transfer function G(s)
where s = jw and graphically display the complex
number G(jw) as w varies.
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The Bode plot is one of the most powerful
graphical tools for analyzing and designing
control systems, and we will also consider polar
plots and log magnitude and phase diagrams.
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Eng. R. L. Nkumbwa
Further Frequency Response
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How is this different from Root Locus?
The information we get from frequency response
methods is different than what we get from the
root locus analysis.
In fact, the two approaches complement each
other.
One advantage of the frequency response
approach is that we can use data derived from
measurements on the physical system without
deriving its mathematical model.
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Eng. R. L. Nkumbwa
Further Frequency Response
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What is the Importance of Frequency methods?
They are a powerful technique to design a singleloop feedback control system.
They provide us with a viewpoint in the frequency
domain.
It is possible to extend the frequency analysis idea
to nonlinear systems (approximate analysis).
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Eng. R. L. Nkumbwa
Who Developed Frequency Methods?
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Bode, Nyquist, Nichols and others, in the
1930s and 1940s.
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Existed before root locus methods.
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Eng. R. L. Nkumbwa
What are the advantages?
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We can study a system from physical data and
determine the transfer function experimentally.
We can design compensators to meet both steady
state and transient response requirements.
We can determine the stability of nonlinear systems
using frequency analysis (out of the scope of this
lecture).
Frequency response methods allow us to settle
ambiguities while drawing a root locus plot.
A system can be designed so that the effects of
undesirable noise are negligible.
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Eng. R. L. Nkumbwa
What are the disadvantages?
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Frequency response techniques are not as
intuitive as root locus.
Find more cons
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Concept of Frequency Response
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The frequency response of a system is the
steady state response of a system to a sinusoidal
input.
Consider the stable, LTI system shown below.
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Eng. R. L. Nkumbwa
Concept of Frequency Response
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The input-output relation is given by:
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Concept of Frequency Response
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Concept of Frequency Response
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Obtaining Magnitude M and Phase Ø
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Concept of Frequency Response
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For linear systems, M and Ø depend only on
the input frequency, w.
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So, what are some of the frequency
response plots and diagrams?
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Eng. R. L. Nkumbwa
Frequency Response Plots and
Diagrams
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There are three frequently used representations
of the frequency response:
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Nyquist diagram: a plot on the complex plane
(G(jw)-plane) where M and Ø are plotted on a
single curve, and w becomes a hidden
parameter.
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Eng. R. L. Nkumbwa
Frequency Response Plots and
Diagrams
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Bode plots: separate plots for M and Ø, with the
horizontal axis being w is log scale.
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The vertical axis for the M-plot is given by M is
decibels (db), that is 20log10(M), and the vertical
axis for the Ø -plot is Ø in degrees.
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Eng. R. L. Nkumbwa
Frequency Response Plots and
Diagrams
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Log-magnitude versus phase plot which
is called the Nichols plot.
Now, let us consider each of the techniques
in more detail in the following chapters.
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Eng. R. L. Nkumbwa
Frequency-Domain Systems
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We can plot G(jw) as a function of w in three
ways:
– Bode Plot.
– Nyquist Plot.
– Nichols Plot (we may not cover this).
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Eng. R. L. Nkumbwa
Nyquist Diagram or Analysis
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The polar plot, or Nyquist diagram, of a
sinusoidal transfer function G(jw) is a plot of the
magnitude of G(jw) versus the phase angle of
G(jw) on polar coordinates as w is varied from
zero to infinity.
Thus, the polar plot is the locus of vectors |G(jw)|
LG(jw) as w is varied from zero to infinity.
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Eng. R. L. Nkumbwa
Nyquist Diagram or Analysis
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The projections of G(jw) on the real and
imaginary axis are its real and imaginary
components.
The Nyquist Stability Criteria is a test for
system stability, just like the Routh-Hurwitz test,
or the Root-Locus Methodology.
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Eng. R. L. Nkumbwa
Nyquist Diagram or Analysis
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Note that in polar plots, a positive (negative)
phase angle is measured counterclockwise
(clockwise) from the positive real axis. In the
polar plot, it is important to show the
frequency graduation of the locus.
Routh-Hurwitz and Root-Locus can tell us
where the poles of the system are for
particular values of gain.
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Eng. R. L. Nkumbwa
Nyquist Diagram or Analysis
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By altering the gain of the system, we can
determine if any of the poles move into the
RHsP, and therefore become unstable.
However, the Nyquist Criteria can also give us
additional information about a system.
The Nyquist Criteria, can tell us things about the
frequency characteristics of the system.
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Eng. R. L. Nkumbwa
Nyquist Diagram or Analysis
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For instance, some systems with constant gain
might be stable for low-frequency inputs, but
become unstable for high-frequency inputs.
Also, the Nyquist Criteria can tell us things about
the phase of the input signals, the time-shift of
the system, and other important information.
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Eng. R. L. Nkumbwa
Nyquist Kuo’s View
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Kuo et al (2003) suggests that, the Nyquist
criterion is a semi-graphical method that
determines the stability of a closed loop system
by investigating the properties of the frequency
domain plot, the Nygmst plot of L(s) is a plot of L
(jw) in the polar coordinates of M [L(jw)] versus
Re[L(jw)] as w varies from 0 to ∞.
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Eng. R. L. Nkumbwa
Nyquist Xavier’s View
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While, Xavier et al (2004) narrates that, the
Nyquist criterion is based on “Cauchy’s Residue
Theorem” of complex variables which is referred
to as “Principle of Argument”.
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Eng. R. L. Nkumbwa
The Argument Principle
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If we have a contour, Γ, drawn in one plane (say
the complex laplace plane, for instance), we can
map that contour into another plane, the F(s)
plane, by transforming the contour with the
function F(s).
The resultant contour, Γ F(s) will circle the origin
point of the F(s) plane N times, where N is equal
to the difference between Z and P (the number of
zeros and poles of the function F(s), respectively).
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Eng. R. L. Nkumbwa
Nyquist Criterion
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Let us first introduce the most important
equation when dealing with the Nyquist criterion:
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Where:
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N is the number of encirclements of the (-1, 0) point.
Z is the number of zeros of the characteristic equation.
P is the number of poles of the open-loop
characteristic equation.
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Eng. R. L. Nkumbwa
Nyquist Stability Criterion Defined
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A feedback control system is stable, if and only if
the contour ΓF(s) in the F(s) plane does not
encircle the (-1, 0) point when P is 0.
A feedback control system is stable, if and only if
the contour ΓF(s) in the F(s) plane encircles the
(-1, 0) point a number of times equal to the
number of poles of F(s) enclosed by Γ.
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Eng. R. L. Nkumbwa
Nyquist Stability Criterion Defined
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In other words, if P is zero then N must equal
zero. Otherwise, N must equal P. Essentially, we
are saying that Z must always equal zero,
because Z is the number of zeros of the
characteristic equation (and therefore the
number of poles of the closed-loop transfer
function) that are in the right-half of the s plane.
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Eng. R. L. Nkumbwa
Nyquist Manke’s View
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While Manke (1997) outlines that, the Nyquist
criterion is used to identify the presence of roots
of a characteristic equation of a control system in
a specified region of s-plane.
He further adds that although the purpose of
using Nyquist criterion is similar to RHC, the
approach differs in the following respect:
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Eng. R. L. Nkumbwa
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The open loop transfer G(s) H(s) is considered instead
of the closed loop characteristic equation 1 + G(s)
H(s) = 0
Inspection of graphical plots G(s) H(s) enables to get
more than YES or NO answer of RHC pertaining to
the stability of control systems.
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Eng. R. L. Nkumbwa
Kuo’s Features of Nyquist Criterion
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Kuo also outlines the following as the features
that make the Nyquist criterion an attractive
alternative for the analysis and design of control
systems:
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In addition to providing the absolute stability, like the
RHC, the NC also gives information on the relative of
a stable system and the degree of instability.
The Nyquist plot of G(s) H(s) or of L (s) is very easy to
obtain.
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Eng. R. L. Nkumbwa
Kuo’s Features of Nyquist Criterion
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The Nyquist plot of G(s) H(s) gives information on
the frequency domain characteristics such as Mr,
Wr, BW and others with ease.
The Nyquist plot is useful for systems with pure
time delay that cannot be treated with the RHC
and are difficult to analyze with root locus method.
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Eng. R. L. Nkumbwa
Any more worries about freqtool…
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