Dynamics and stability state-space of a controlled inverted pendulum

A study is conducted into the dynamics of an inverted pendulum under the regulation of a linear-feedback control system. Analysis of the resulting dynamic equations indicates the presence of stable regions within the four-dimensional state space of the system, which are confirmed through simulation. System trajectories are,examined in this state space, and complex, apparently chaotic behaviour is observed outside the stable regions of the state space. An algorithm is developed to map the boundaries of the stable state-space hyper-volume; this algorithm is broadly applicable to the study of other systems. Results are presented, generated by executing the algorithm on the inverted pendulum dynamics. These comprise a series of three-dimensional plots, outlining the four-dimensional stability volume. The boundaries of this volume are seen to exhibit a jagged, possibly fractal, morphology.