Frequency-Domain
Control Systems
Eng R. L. Nkumbwa
Copperbelt University
School of Technology
2010
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Eng R. L. Nkumbwa @ CBU 2010
Its all Stability of Control Systems
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Frequency Response Roadmap
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We will cover the following:
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General frequency analysis in Control
Engineering
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Eng R. L. Nkumbwa @ CBU 2010
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 @ CBU 2010
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:
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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,
Such that the time domain properties of the
system can be predicted based on the
frequency-domain characteristics.
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Example: Gun Positional Control
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Eng R. L. Nkumbwa @ CBU 2010
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
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design of control
systems. Eng R. L. Nkumbwa @ CBU 2010
Characteristics of Frequency Response
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Frequency response methods are a good complement
to the root locus techniques:–
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Can infer performance and stability from the same plot
Can use measured data rather than a transfer function model
Design process can be independent of the system order
Time delays are handled correctly
Graphical techniques (analysis and synthesis) are quite simple.
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Eng R. L. Nkumbwa @ CBU 2010
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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Time & Frequency-Domain Specs.
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So, what are time-domain specifications by
now? Am sure u all know what they are?
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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 @ CBU 2010
Time-Domain Specifications
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Frequency-Domain Specifications
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Eng R. L. Nkumbwa @ CBU 2010
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 @ CBU 2010
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 @ CBU 2010
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 @ CBU 2010
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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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 @ CBU 2010
Further Frequency Response
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What is the Importance of Frequency methods?
They are a powerful technique to design a
single-loop 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 @ CBU 2010
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 @ CBU 2010
Variety of Frequency domain Analysis
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Bode Plot – Log |G(jω)| and Phase of G(jω) vs.
Log frequency.
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Polar (Nyquist) plot – Re vs.Im of G(jω) in
complex plane.
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Simplest tool for visualization and synthesis
Typically plot 20log|G| which is given the symbol dB
Hard to visualize, not useful for synthesis, but gives
definitive tests for stability and is the basis of the
robustness analysis.
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Eng R. L. Nkumbwa @ CBU 2010
Variety of Frequency domain Analysis
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Nichols Plot – |G(jω)| vs. Phase of G(jω), which
is very handy for systems with lightly damped
poles.
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Eng R. L. Nkumbwa @ CBU 2010
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.
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 @ CBU 2010
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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Eng R. L. Nkumbwa @ CBU 2010
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 @ CBU 2010
Characteristics of Frequency Domain
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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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Obtaining Frequency Response
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Concept of Frequency Response
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Obtaining Magnitude M and Phase Ø
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Eng R. L. Nkumbwa @ CBU 2010
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 @ CBU 2010
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 @ CBU 2010
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 @ CBU 2010
Plotting Bode Plots
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Eng R. L. Nkumbwa @ CBU 2010
Amplitude Ratio (AR) on log-log plot
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Start from steady-state gain at ω=0. If GOL includes either
integrator or differentiator it starts at infinity or 0.
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Each first-order lag (lead) adds to the slope –1 (+1) starting
at the corner frequency.
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Each integrator (differentiator) adds to the slope –1 (+1)
starting at zero frequency.
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A delays does not contribute to the AR plot.
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Eng R. L. Nkumbwa @ CBU 2010
Phase angle on semi-log plot
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Start from 0°or -180°at ω =0 depending on the sign of steadystate gain.
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Each first-order lag (lead) adds 0°to phase angle at ω =0, adds
-90°(+90°) to phase angle at ω = ∞ , and adds -45°(+45°)to
phase angle at corner frequency.
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Each integrator (differentiator) adds -90°(+90°)to the phase
angle for all frequency.
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A delay adds -ωθ to phase angle depending on the frequency.
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Try Solving the Following Using Bode
Technique
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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 @ CBU 2010
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 @ CBU 2010
Nyquist is an alternative representation
of frequency response
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Compact (one plot)
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Wider applicability of stability analysis than Bode
plot
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High frequency characteristics will be shrunk near
the origin.
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Inverse Nyquist diagram: polar plot of G(jw)
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Combination of different transfer function
components is not easy as with Nyquist diagram as
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Eng R. L. Nkumbwa @ CBU 2010
with Bode plot.
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 @ CBU 2010
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 @ CBU 2010
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 @ CBU 2010
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) which is a plot of L(jw) in the polar
coordinates of M [L(jw)] versus Re[L(jw)] as
ω varies from 0 to ∞.
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Eng R. L. Nkumbwa @ CBU 2010
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 @ CBU 2010
The Argument Principle
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If we have a contour, Γ (capital gamma), 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 @ CBU 2010
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 @ CBU 2010
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 @ CBU 2010
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 @ CBU 2010
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 @ CBU 2010
Nyquist Manke’s View Cont…
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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 @ CBU 2010
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 @ CBU 2010
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 @ CBU 2010
Benefits of Frequency Response
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Frequency responses are the informative
representations of dynamic systems
Example of an Audio Speaker
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Eng R. L. Nkumbwa @ CBU 2010
Benefits of Frequency Response
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Lets now look at a Mechanical or Civil
Engineering example of frequency domain,
say a structure like a bridge.
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Eng R. L. Nkumbwa @ CBU 2010
Frequency Stability Tests
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Want tests on the loop transfer function
L(s)=Gc(s)G(s) that can be performed to
establish stability of the closed-loop system
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Eng R. L. Nkumbwa @ CBU 2010
Frequency Stability Tests
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Any more worries about freqtool…
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