Analysis and Application of PID (Proportional-Integral-Derivative) Control Loops on the Inverted Pen

by Coyt Barringer

Submitted : Fall 2013

The inverted pendulum problem is a dynamic physics problem involving a pendulum with a center of mass above its pivot point. This pendulum is inherently unstable so an active feedback mechanism is needed to maintain the pendulum in a balanced state. To keep the pendulum balanced, calculus-based algorithms combine sensor data to continuously move the pivot point back under the pendulum’s center of mass. In this implementation, a robot was constructed with the body, and thus, center of mass, fixed above a motorized wheel base acting as the pivot point.  This robot design simulates the inverted pendulum with one degree of freedom, and can be programmed with various algorithms to improve its balancing precision. Unfortunately, the test robot has not reached a functional state due to electrical problems along with programming difficulty. Analysis of the planned balancing algorithm will still be described and analyzed in mathematical terms. Based on this analysis, a properly implemented and tuned PID controller will keep the robot balanced to a very high precision when compared to less complex control algorithms such as a proportional only controller.

The core algorithm studied herein is the PID (Proportional-Integral-Derivative) controller. This algorithm is a standard feedback based control loop that is widely used in industry due to its precision.  It uses the integral and derivative of input variables to predict the best action to take in the present to minimize error in the system. In this case, the inputs are sensor data from the robot providing the vertical angle of the robot in relation to the horizontal floor as well as this angle’s rate of change. Error would be any destabilization of the pendulum, or change from 90 degrees upright.

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