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Unformatted text preview: MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Mechanical Engineering 2.004 Dynamics and Control II Fall 2007 Problem Set #10 Posted: Friday, Nov. 30, ’07 Due: Friday, Dec. 7, ’07 m θ l T 1. Inverted pendulum Consider the inverted pendulum system shown above. It consists of a point mass m attached to the end of a rigid rod of length l . The rod’s mass is negligible. An input torque T is applied to the rod base in order to control the pendulum rotation angle θ . a) Derive the inverted pendulum’s equation of motion; then linearize the equa- tion that you derived by assuming that the angle θ is very small ( θ 1rad). Don’t forget to include the effect of gravity in your model! b) With the angle θ as output and torque T as input, and using numerical values l = 1 . 09m, m = 0 . 8417kg, derive the transfer function of the inverted pendulum system. c) Deduce from the transfer function that the inverted pendulum system is unstable . Describe an experiment you could do with everyday objects to verify this result. Note: Your transfer function should have two real poles at s = ± 3. If you cannot quite arrive at this result, assume it to be true and move on to the remaining questions. 1 d) We will now perform a sequence of attempts to stabilize and control the inverted pendulum. First we consider proportional (P) control. Draw the block diagram of the proportional control system, indicating clearly...
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- Fall '08
- Mechanical Engineering, Inverted pendulum, verted pendulum