Lecture4 Root Locus Method
Transcript of Lecture4 Root Locus Method
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The Root Locus Analysis
Eng R. L. Nkumbwa
MSc, MBA, BEng, REng.
Copperbelt University
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Stability of Control Systems
Its all about Stability
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Auto-Pilot or Fly-by-Wire Systems
Let us consider the short period approximate
model of the Fly Zambezi 727 aircraft landingat Lusaka International Airport.
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Auto-Pilot or Fly-by-Wire Systems
Where e is the elevator input,
Take the output as , input is e, then formthe transfer function is of the form;
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Auto-Pilot or Fly-by-Wire Systems
For the Zambezi 727 (40Kft, M = 0.8) the
Transfer Function reduces to:
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Auto-Pilot or Fly-by-Wire Systems
Such that, the dominant roots have a frequency
of approximately 1 rad/sec and damping of about0.4 as shown on the pole-zero map below:
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Auto-Pilot or Fly-by-Wire Systems
As the plane continue navigating the sky, we
need to know and analyze where the polesare going as a function of the input command
constant in the above pole-zero map.
How do we know where the poles moves as
the Zambezi 727 system gain changes? This is where Root Locus comes to address
the problem and provide the solutions.
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Root Locus Analysis Intro
In Control Systems I and other previous chapter,we have demonstrated the importance of the polesand zeros of the closed loop transfer function ofthe linear control system on the dynamicperformance of the system.
The roots of the characteristic equation which arethe poles of the closed loop transfer function,determine the absolute and relative stability oflinear SISO Systems.
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Root Locus Analysis Intro
Another important study of the Control systemsis the investigation of the trajectories of theroots of the characteristic equation or simplythe Root Locus When certain systemparameters vary.
The first basic properties of the root loci andthe systematic construction are due to
Wade R. Evans in 1948
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Root Locus Analysis Intro
In general, root locus may be sketched by
following some simple rules and properties.
For plotting the root locus accurately the
MATLAB root locus tool in the Control System
Toolbox (control) or in the Time ResponseAnalysis Tool (time tool) of ACSYS can be used.
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Root Locus Analysis Intro
The root locus technique is not confined only to
the study of control systems.
In general, the method can be applied to study
the behavior of roots of any algebraic equation
with one or more variable parameters.
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Root Locus Example
Consider an illustrative example for the
Radio Volume control in the Course TextBook by Nkumbwa on page 75.
It illustrates how root locus is applied in
volume control of radio systems.
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Root Locus Example: three poles
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Root Locus Analysis Intro
General root locus is hard to determine by hand
and requires Matlab tools such as:rlocus (num,den)
To obtain full result, we can get some important
insights by developing a short set of plotting
rules.
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Defining Root Locus
To start with, lets make sure were clear onexactly what we mean by the words RootLocus plot.
So, what is a Root? A number that reduces an equation to an
identity when it is substituted for onevariable.Roots of this equation are the closed-loop
poles of the feedback system.
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Defining Root Locus
Then, what is a Locus?
The set of all points whose location isdetermined by stated conditions.
The stated conditions here are that 1 + kL (s) =
0for some value ofk, and the points whose 0
locations matter to us are points in the s-plane.
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Defining Root Locus
Now, what is a Root Locus?
The set of all points in the s-plane that satisfy theequation 1 + kL (s) = 0for some 0value ofk.
Root locus is a graphical presentation of the
closed- loop poles as a system parameter is
varied.Root locus is a powerful method of analysis and
design for stability and transient response.
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Defining Root Locus
The root- locus technique is a graphical
method for sketching the locus of the roots in thes-plane as a parameter is varied.
In fact, the root- locus method provides the
engineer with a measure of the sensitivity of the
roots of the system to a variation in theparameter being considered.
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Some Root Locus Basic Questions
What points are on the root locus?
Where does the root locus start?Where does the root locus end?When/where is the locus on the real line? Etc
Answering these and many more questionswill help us understand Root Locustechnique.
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Pole and Zero Locations by R-Locus
Let's say we have a closed-loop transfer
function for a particular system:
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Pole and Zero Locations by R-Locus
Where N is the numerator polynomial and D
is the denominator polynomial of the transferfunctions, respectively.
Now, we know that to find the poles of the
equation, we must set the denominator to 0,and solve the characteristic equation.
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Pole and Zero Locations by R-Locus
In other words, the locations of the poles of a
specific equation must satisfy the followingrelationship:
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Pole and Zero Locations by R-Locus
And from the above equation we can
manipulate an equation such as:
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Pole and Zero Locations by R-Locus
And finally by converting to polar coordinates,
we get:
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Equation for all Gain Values
Now we have 2 equations that govern the
locations of the poles of a system forall gainvalues:
The Magnitude Equation
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Equation for all Gain Values
The Angle Equation
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Root-Locus Design Procedure
In laplace transform domain, when the gain is
small the poles start at the poles of the open loop
transfer function.
When gain becomes infinity, the poles move to
overlap the zeros of the system.
This means that on a root-locus graph, all thepoles move towards a zero.
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Root-Locus Design Procedure
Only one pole may move towards one zero
and this means that there must be the same
number of poles as zeros.
If there are fewer zeros than poles in the
transfer function, there are a number of
implicit zeros located at infinity that the poleswill approach.
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Note
Remember that, Poles are marked on the
graph with an 'X' and zeros are marked with
an 'O by common convention.
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Root-Locus Design Procedure
We can start drawing the root-locus by firstplacing the roots of b(s) on the graph with an
'X'.Next, we place the roots of a(s) on the graph,
and mark them with an 'O'.
Where b(s) and a(s) are the numerator anddenominator of the system transfer function.
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Root-Locus Design Procedure
Next, we examine the real-axis.
Starting from the right-hand side of the graph
and traveling to the left, we draw a root-locus
line on the real-axis at every point to the left of
an odd number of poles on the real-axis.
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Root-Locus Design Procedure
Now, a root-locus line starts at every pole.
Therefore, any place that two poles appear to beconnected by a root locus line on the real-axis,
the two poles actually move towards each other,
and then they "breakaway", and move off the
axis. The point where the poles break off the axis is
called the breakaway point.
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Note
It is important to note that the s-plane is
symmetrical about the real axis, so whatever is
drawn on the top half of the S-plane, must be
drawn in mirror-image on the bottom-half plane.
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Root-Locus Design Procedure
Once a pole breaks away from the real axis,
they can either travel out towards infinity (to
meet an implicit zero) or they can travel to
meet an explicit zero, or they can re-join the
real-axis to meet a zero that is located on the
real-axis.
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Root-Locus Design Procedure
If a pole is traveling towards infinity, it always
follows an asymptote.
The number of asymptotes is equal to the
number of implicit zeros at infinity.
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Root-Locus Construction Rules
Rule 1: Starting Point (K=0) The root locus starts at open loop poles. Or there is
one branch of the root-locus for every root of b(s).
Rule 2: Terminating Point (K=infinity) The root locus terminates at open loop zeros which
include those at infinity.Rule 3: Number of Distinct Root Loci
There will be as many root loci as the highest number
of finite open loop poles or zeros.
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Root-Locus Construction Rules
Rule 4: Symmetry of the Root Loci The root loci are symmetrical with respect to the
real axis and all complex roots are conjugate.
Rule 5: Angle of Asymptotes The root loci are asymptotic to straight lines at
large values and the angle of asymptotes is givenby
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Root-Locus Construction Rules
Rule 6: Asymptotic Intersection The asymptotes intersects the real axis at the point
given by
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Root-Locus Construction Rules
Rule 7: Root Locus Location on the Real
Axis The root loci may be found on portions of the real axis
to the left of an old number of open loop poles and
zeros.
Rule 8: Locus Breakaway Point The points at which the root locus break away can be
calculated by the following:
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Root-Locus Construction Rules
Rule 9: Angle of Departure and Arrival Find the formula
Rule 10: Point of Intersection with the
Imaginary Axis Find the formula
Rule 11: Determination of K Find the formula
And many more rules and equations
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Root Locus Example
A single- loop feedback system has a
characteristic equation as follows:
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Root Locus Example
We wish to sketch the root locus in order to
determine the effect of the gain K. The poles
and the zeros are located in the s-plane as:
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Root Locus Example
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Root Locus Example
The root loci on the real axis must be located
to the left of an odd number of poles and
zeros and are therefore located as shown on
the figure above in heavy lines.
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Root Locus Example
The intersection of the asymptotes is:
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Root Locus Example
The angles of the asymptotes are:
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Root Locus Example
There are three asymptotes, since the number of
poles minus the number of zeros, n m = 3.
Also, we note that the root loci must begin at
poles, and therefore two loci must leave the
double pole at s = - 4. Then, with the asymptotes
as sketched below, we may sketch the form ofthe root locus:
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Root Locus Example
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Compensator design using the root locus
The root locus graphically displays both transient
response and stability information.
The locus can be sketched quickly to get a
general idea of the changes in transient
response generated by changes in gain.
Specific points on the locus can also be foundaccurately to give quantitative design
information.
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Compensator design using the root locus
The root locus typically allows us to choose theproper loop gain to meet a transient response
specification.As the gain is varied, we move through different
regions of response. Setting the gain at a particular value yields the
transient response dictated by the poles at thatpoint on the root locus.
Thus, we are limited to those responses thatexist along the root locus.
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Possible Root Locus
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Possible Response Options
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Wrap Up
Root Locus is a very important techniques
that can be used for compensation design of
various control systems
Do further research on this topic
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