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Dynamics of an inverted pendulum with delayed feedback control. (English) Zbl 1170.93366

Summary: We consider an experimental system consisting of a pendulum, which is free to rotate 360 degrees, attached to a cart. The cart can move in one dimension. We describe a model for this system and use it to design a feedback control law that stabilizes the pendulum in the upright position. We then introduce a time delay into the feedback and prove that for values of the delay below a critical delay, the system remains stable. Using a center manifold reduction, we show that the system undergoes a supercritical Hopf bifurcation at the critical delay. Both the critical value of the delay and the stability of the limit cycle are verified experimentally. Our experimental data is illustrated with plots and videos.

MSC:

93D15 Stabilization of systems by feedback
34K35 Control problems for functional-differential equations
34K20 Stability theory of functional-differential equations
70Q05 Control of mechanical systems
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