On the driven inverted pendulum

"We explore the solutions of the driven inverted pendulum system l#0308=gsin?å-al(t)cos?ä where al(??) is a bounded lateral acceleration. We show that, for lateral accelerations that are constant before some initial time, an inverted trajectory always exists and remains within a diamond shaped region in the state space. Functional analytic techniques are also developed to provide further insight into the nature of the inverted pendulum trajectories. Associated to the driven inverted pendulum is a time varying linear system. We show that this system always possesses an exponential dichotomy, allowing for the development of a successive approximation algorithm for finding the desired inverted pendulum trajectory. We show that the curve obtained from one iteration of this algorithm is a very good estimate of the required inverted trajectory. As that curve is obtained by filtering the quasi-static angle trajectory by a noncausal time varying low pass filter with weighting function with a shape similar to h(t) = exp?êÆ??0|t|, we find that the current pendulum angle is influenced by the values of the lateral acceleration within only a few seconds of the current time. These results are important as the driven inverted pendulum is a common susbsystem in systems ranging from motorcycles and bicycles to rockets and aircaft."