Class 34 - 35 Modeling of a Inverted Pendulum

System Modeling Coursework

P.R. VENKATESWARANFaculty, Instrumentation and Control Engineering,

Manipal Institute of Technology, ManipalKarnataka 576 104 INDIAPh: 0820 2925154, 2925152

Fax: 0820 2571071Email: [email protected], [email protected]

Web address: http://www.esnips.com/web/SystemModelingClassNotes

Class 34-35: Modeling of Inverted Pendulum

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http://www.esnips.com/web/SystemModelingClassNotes

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.


Contents

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Dynamics of Inverted Pendulum•

Transfer function of Inverted Pendulum


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.


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


Force analysis


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:


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:


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:


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):


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),


Transfer function

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then substituting into the second equation:

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Re-arranging, the transfer function is:


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:


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…

Class 34 - 35 Modeling of a Inverted Pendulum

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