The Root Locus Analysis
Eng R. L. Nkumbwa
MSc, MBA, BEng, REng.
Stability of Control Systems
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Eng R. L. Nkumbwa@CBU-2010
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Eng R. L. Nkumbwa@CBU-2010
Auto-Pilot or Fly-by-Wire Systems
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Let us consider the short period approximate
model of the Fly Zambezi 727 aircraft landing
at Lusaka International Airport.
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Eng R. L. Nkumbwa@CBU-2010
Auto-Pilot or Fly-by-Wire Systems
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Where δe is the elevator input,
Take the output as θ, input is δe, then form
the transfer function is of the form;
4/13/2015
Eng R. L. Nkumbwa@CBU-2010
Auto-Pilot or Fly-by-Wire Systems
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For the Zambezi 727 (40Kft, M = 0.8) the
Transfer Function reduces to:
4/13/2015
Eng R. L. Nkumbwa@CBU-2010
Auto-Pilot or Fly-by-Wire Systems
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Such that, the dominant roots have a frequency
of approximately 1 rad/sec and damping of about
0.4 as shown on the pole-zero map below:
4/13/2015
Eng R. L. Nkumbwa@CBU-2010
Auto-Pilot or Fly-by-Wire Systems
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As the plane continue navigating the sky, we
need to know and analyze where the poles
are 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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Eng R. L. Nkumbwa@CBU-2010
Root Locus Analysis Intro
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In Control Systems I and other previous chapter,
we have demonstrated the importance of the poles
and zeros of the closed loop transfer function of
the linear control system on the dynamic
performance of the system.
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The roots of the characteristic equation which are
the poles of the closed loop transfer function,
determine the absolute and relative stability of
linear SISO Systems.
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Eng R. L. Nkumbwa@CBU-2010
Root Locus Analysis Intro
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Another important study of the Control systems
is the investigation of the trajectories of the
roots of the characteristic equation or simply
the Root Locus – When certain system
parameters vary.
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The first basic properties of the root loci and
the systematic construction are due to
Wade R. Evans in 1948
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Eng R. L. Nkumbwa@CBU-2010
Root Locus Analysis Intro
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In general, root locus may be sketched by
following some simple rules and properties.
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For plotting the root locus accurately the
MATLAB root locus tool in the Control System
Toolbox (control) or in the Time Response
Analysis Tool (time tool) of ACSYS can be used.
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Eng R. L. Nkumbwa@CBU-2010
Root Locus Analysis Intro
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The root locus technique is not confined only to
the study of control systems.
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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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Eng R. L. Nkumbwa@CBU-2010
Root Locus Example
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Consider an illustrative example for the
Radio Volume control in the Course Text
Book by Nkumbwa on page 75.
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It illustrates how root locus is applied in
volume control of radio systems.
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Eng R. L. Nkumbwa@CBU-2010
Root Locus Example: three poles
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Eng R. L. Nkumbwa@CBU-2010
Root Locus Analysis Intro
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General root locus is hard to determine by hand
and requires Matlab tools such as:
rlocus (num,den)
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To obtain full result, we can get some important
insights by developing a short set of plotting
rules.
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Eng R. L. Nkumbwa@CBU-2010
Defining Root Locus
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To start with, let’s make sure we’re clear on
exactly what we mean by the words “Root
Locus plot.”
So, what is a Root?
“A number that reduces an equation to an
identity when it is substituted for one
variable.”
Roots of this equation are the closed-loop
poles of the feedback system.
4/13/2015
Eng R. L. Nkumbwa@CBU-2010
Defining Root Locus
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Then, what is a Locus?
“The set of all points whose location is
determined by stated conditions.”
The “stated conditions” here are that 1 + kL (s) =
0 for some value of k, and the “points” whose 0
locations matter to us are points in the s-plane.
4/13/2015
Eng R. L. Nkumbwa@CBU-2010
Defining Root Locus
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Now, what is a Root Locus?
The set of all points in the s-plane that satisfy the
equation 1 + kL (s) = 0 for some 0 value of k.
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.
4/13/2015
Eng R. L. Nkumbwa@CBU-2010
Defining Root Locus
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The root- locus technique is a graphical
method for sketching the locus of the roots in the
s-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 the
parameter being considered.
4/13/2015
Eng R. L. Nkumbwa@CBU-2010
Some Root Locus Basic Questions
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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 questions
will help us understand Root Locus
technique.
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Eng R. L. Nkumbwa@CBU-2010
Pole and Zero Locations by R-Locus
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Let's say we have a closed-loop transfer
function for a particular system:
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Eng R. L. Nkumbwa@CBU-2010
Pole and Zero Locations by R-Locus
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Where N is the numerator polynomial and D
is the denominator polynomial of the transfer
functions, respectively.
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Now, we know that to find the poles of the
equation, we must set the denominator to 0,
and solve the characteristic equation.
4/13/2015
Eng R. L. Nkumbwa@CBU-2010
Pole and Zero Locations by R-Locus
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In other words, the locations of the poles of a
specific equation must satisfy the following
relationship:
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Eng R. L. Nkumbwa@CBU-2010
Pole and Zero Locations by R-Locus
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And from the above equation we can
manipulate an equation such as:
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Eng R. L. Nkumbwa@CBU-2010
Pole and Zero Locations by R-Locus
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And finally by converting to polar
coordinates, we get:
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Eng R. L. Nkumbwa@CBU-2010
Equation for all Gain Values
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Now we have 2 equations that govern the
locations of the poles of a system for all gain
values:
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The Magnitude Equation
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Eng R. L. Nkumbwa@CBU-2010
Equation for all Gain Values
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The Angle Equation
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Eng R. L. Nkumbwa@CBU-2010
Root-Locus Design Procedure
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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 the
poles move towards a zero.
4/13/2015
Eng R. L. Nkumbwa@CBU-2010
Root-Locus Design Procedure
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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 poles
will approach.
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Eng R. L. Nkumbwa@CBU-2010
Note
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Remember that, Poles are marked on the
graph with an 'X' and zeros are marked with
an 'O‘ by common convention.
4/13/2015
Eng R. L. Nkumbwa@CBU-2010
Root-Locus Design Procedure
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We can start drawing the root-locus by first
placing 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 and
denominator of the system transfer function.
4/13/2015
Eng R. L. Nkumbwa@CBU-2010
Root-Locus Design Procedure
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Next, we examine the real-axis.
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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.
4/13/2015
Eng R. L. Nkumbwa@CBU-2010
Root-Locus Design Procedure
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Now, a root-locus line starts at every pole.
Therefore, any place that two poles appear to be
connected 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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Eng R. L. Nkumbwa@CBU-2010
Note
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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.
4/13/2015
Eng R. L. Nkumbwa@CBU-2010
Root-Locus Design Procedure
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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.
4/13/2015
Eng R. L. Nkumbwa@CBU-2010
Root-Locus Design Procedure
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If a pole is traveling towards infinity, it always
follows an asymptote.
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The number of asymptotes is equal to the
number of implicit zeros at infinity.
4/13/2015
Eng R. L. Nkumbwa@CBU-2010
Root-Locus Construction Rules
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Rule 1: Starting Point (K=0)
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Rule 2: Terminating Point (K=infinity)
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The root locus terminates at open loop zeros which
include those at infinity.
Rule 3: Number of Distinct Root Loci
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The root locus starts at open loop poles. Or there is
one branch of the root-locus for every root of b(s).
There will be as many root loci as the highest
number of finite open loop poles or zeros.
4/13/2015
Eng R. L. Nkumbwa@CBU-2010
Root-Locus Construction Rules
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Rule 4: Symmetry of the Root Loci
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Rule 5: Angle of Asymptotes
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The root loci are symmetrical with respect to the
real axis and all complex roots are conjugate.
The root loci are asymptotic to straight lines at
large values and the angle of asymptotes is given
by
4/13/2015
Eng R. L. Nkumbwa@CBU-2010
Root-Locus Construction Rules
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Rule 6: Asymptotic Intersection
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The asymptotes intersects the real axis at the
point given by
4/13/2015
Eng R. L. Nkumbwa@CBU-2010
Root-Locus Construction Rules
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Rule 7: Root Locus Location on the Real
Axis
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Rule 8: Locus Breakaway Point
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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.
The points at which the root locus break away can
be calculated by the following:
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Eng R. L. Nkumbwa@CBU-2010
Root-Locus Construction Rules
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Rule 9: Angle of Departure and Arrival
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Rule 10: Point of Intersection with the
Imaginary Axis
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Find the formula
Rule 11: Determination of K
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Find the formula
Find the formula
And many more rules and equations
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Eng R. L. Nkumbwa@CBU-2010
Root Locus Example
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A single- loop feedback system has a
characteristic equation as follows:
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Eng R. L. Nkumbwa@CBU-2010
Root Locus Example
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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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Eng R. L. Nkumbwa@CBU-2010
Root Locus Example
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Eng R. L. Nkumbwa@CBU-2010
Root Locus Example
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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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Eng R. L. Nkumbwa@CBU-2010
Root Locus Example
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The intersection of the asymptotes is:
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Eng R. L. Nkumbwa@CBU-2010
Root Locus Example
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The angles of the asymptotes are:
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Eng R. L. Nkumbwa@CBU-2010
Root Locus Example
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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 of
the root locus:
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Eng R. L. Nkumbwa@CBU-2010
Root Locus Example
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Eng R. L. Nkumbwa@CBU-2010
Compensator design using the root locus
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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 found
accurately to give quantitative design
information.
4/13/2015
Eng R. L. Nkumbwa@CBU-2010
Compensator design using the root locus
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The root locus typically allows us to choose the
proper 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 that
point on the root locus.
Thus, we are limited to those responses that
exist along the root locus.
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Eng R. L. Nkumbwa@CBU-2010
Possible Root Locus
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Eng R. L. Nkumbwa@CBU-2010
Possible Response Options
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Eng R. L. Nkumbwa@CBU-2010
Wrap Up…
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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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Eng R. L. Nkumbwa@CBU-2010