control theory

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CAREER POINT UNIVERSITY
Major Assignment
of
Root locus Technique
Submitted To:- Submitted By:-
Mr.Somesh Sir Nilesh Prajapati
B.tech 6th sem
Branch Mechanical

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INTRODUCTION
Root locus is a graphical presentation of the closed loop poles as a
system parameter is varied. The root locus also gives a graphic
presentation of a system’s stability. Before presenting root locus,
let us review two consepts that we need for the ensuing
discussion
(1) The Control System Problem,
(2) Complex Numbers

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General steps for drawing the Root Locus of the
given system:
Step 1: Determine the open loop poles, zeros and number of
branches from given G(s)H(s).
Step 2: Draw the pole zero plot and determine the region of real axis
for which the root locus exists. Also determine the number of
breakaway points (This will be explained, while solving the problems).
Step 3: Calculate angle of asymptotes.
Step 4: Determine the centroid.
Step 5: Calculate the breakaway points (if any).

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Step 6: Calculate the intersection point of root
locus with imaginary axis.
Step 7: Calculate the angle of departure or angle of
arrivals if any.
Step 8: from above steps draw the overall sketch of
the root locus.
Step 9: Predict the stability and performance of the
given system by the root locus.

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Root Locus Summary
Root locus shows evolution of closed-loop poles as one
parameter changes
• A simple set of rules allow the loci to be sketched
• If details are needed, good computer packages are available to
plot root locus, Matlab: rlocus, rlocfind, rltool
• To develop insight into control design, a control engineer will
be able to determine the main features of root locus without a
computer

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Evaluation of a Transfer Function
Consider a Transfer Function to be
• Evaluate the function at a test point s = -3+j4, i.e. find out the magnitude
and phase of the transfer function at the test point.
• Following the geometric technique or using simple complex algebra you may
find out that :
• Zero length - √20, φz = 116.60
• Pole at Origin: Pole length 5, φp1 = 126.90
• Pole at -2: Pole length √17, φp2 = 104.00
• Magnitude of the transfer function:

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Minimum Phase Systems
It should be emphasized that from the root locus method it follows
that
the systems having unstable open-loop zeros become unstable for
large
values of the static gain . Such kinds of systems are called non
minimum
phase systems, in contrast to minimum phase systems whose
definition is
given below.

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Conclusion
In this handout we showed how to determine the location of the
poles of a
control system through a root locus plot. These plots are
important because
they indicate whether a system is stable or not and can also be
used to estimate
the transient response.

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