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Nonlinear dynamics of the inverted pendulum under linear-feedback stabilization

University of British Columbia

Analysis is conducted on a linear control system used for the stabilization of an inverted pendulum, examining the behaviour of the system throughout its global state space. The dynamic equations of the inverted pendulum are obtained, and a linear state-feedback controller is designed, based on this system model, for its stabilization. The behaviour of the resulting closed-loop system is examined through simulation, in two stages. In the first stage, the feedback controller is modified slightly to decouple the pendulum dynamics from those of its supporting cart, permitting examination of the pendulum dynamics in a two-dimensional state space, using phase plane analysis. In this way, a globally valid description of the pendulum dynamics is obtained, revealing the existence of two state attractors in this modified system. The feedback controller is seen to create a stable point attractor in the phase plane, as intended, but the basin of attraction of this point is of unusual shape and limited extent, due to the nonlinearity of the pendulum dynamics. All initial system states outside this basin are drawn to the second state attractor, which is a stable limit cycle. As the second stage of analysis, the system is examined with its original controller; characterization of the overall dynamics then requires a four-dimensional state space. The continued existence of a point attractor at the state space origin is demonstrated, and the basin of attraction of this point is mapped numerically, using a state space sampling algorithm, which might readily be applied to the analysis of other dynamic systems. This basin is presented as a series of three-dimensional sections of the full four-dimensional region; taken together, these sections outline the entire basin map. The boundaries of the origin-bound basin are seen to be highly irregular, and potentially fractal in nature. The existence of an infinite set of such four-dimensional stability regions is demonstrated, and complex, potentially chaotic, behaviour is observed outside the basins of attraction; it is not clear whether or not a four-dimensional counterpart of the limit cycle (from the first stage analysis) exists.

Text

2010-11-16

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http://hdl.handle.net/2429/29971

Electrical and Computer Engineering

University of British Columbia

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