Post on 10-Apr-2015
System Modeling Coursework
P.R. VENKATESWARANFaculty, Instrumentation and Control Engineering,
Manipal Institute of Technology, ManipalKarnataka 576 104 INDIAPh: 0820 2925154, 2925152
Fax: 0820 2571071Email: pr.venkat@manipal.edu, prv_i@yahoo.com
Web address: http://www.esnips.com/web/SystemModelingClassNotes
Class 34-35: Modeling of Inverted Pendulum
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 2
WARNING!
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I claim no originality in all these notes. These are the compilation from various sources for the purpose of delivering lectures. I humbly acknowledge the wonderful help provided by the original sources in this compilation.
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For best results, it is always suggested you read the source material.
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Contents
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Dynamics of Inverted Pendulum•
Transfer function of Inverted Pendulum
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Problem statement
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The cart with an inverted pendulum, shown below, is "bumped" with an impulse force, F.
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Determine the dynamic equations of motion for the system, and linearize about the pendulum's angle, theta = Pi (in other words, assume that pendulum does not move more than a few degrees away from the vertical, chosen to be at an angle of Pi).
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Find a controller to satisfy all of the design requirements given below.
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System Diagram
M mass of the cart 0.5 kg
m mass of the pendulum 0.5 kg
b friction of the cart 0.1 N/m/sec
l length to pendulum center of mass 0.3 m
I inertia of the pendulum 0.006 kg*m^2
F force applied to the cart
x cart position coordinate
theta pendulum angle from vertical
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Force analysis
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System equations
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Summing the forces in the Free Body Diagram of the cart in the horizontal direction, you get the following equation of motion:
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Note that you could also sum the forces in the vertical direction, but no useful information would be gained.
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Summing the forces in the Free Body Diagram of the pendulum in the horizontal direction, you can get an equation for N:
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System equations
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If you substitute this equation into the first equation, you get the first equation of motion for this system:
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To get the second equation of motion, sum the forces perpendicular to the pendulum. Solving the system along this axis ends up saving you a lot of algebra. You should get the following equation:
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System equations
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To get rid of the P and N terms in the equation above, sum the moments around the centroid of the pendulum to get the following equation:
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Combining these last two equations, you get the second dynamic equation:
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System equations
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The equations should be linearized about theta = Pi.
Assume that theta = Pi + ø
(ø
represents a small angle from
the vertical upward direction). Therefore, cos(theta) = -1, sin(theta) = -ø, and (d(theta)/dt)^2 = 0.
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After linearization the two equations of motion become (where u represents the input):
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Transfer function
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To obtain the transfer function of the linearized system equations analytically, we must first take the Laplace transform of the system equations. The Laplace transforms are:
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Since we will be looking at the angle Phi as the output of interest, solve the first equation for X(s),
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Transfer function
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then substituting into the second equation:
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Re-arranging, the transfer function is:
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Transfer function
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From the transfer function above it can be seen that there is both a pole and a zero at the origin. These can be canceled and the transfer function becomes:
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And, before we break…
A monk was imprisoned. Within a week, he was to be killed. In the prison, he heard a beautiful verse from a scripture sung by his co-prisoner. He requested him to teach that verse. The co-prisoner asked, “
what is the purpose of
learning if you are going to die within one week?”
The monk answered: “Exactly for the same reason you learn something if you are going to die within forty five years”.
Thanks for listening…