## Electrical, Computer & Energy Engineering Graduate Theses & Dissertations

Spring 4-1-2012

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Electrical, Computer & Energy Engineering

John Hauser

Youjian (Eugene) Liu

Jason Marden

#### Abstract

The pendulum provides a seemingly inexhaustible source of practical applications and interesting problems which have motivated research in a variety of disciplines. In this thesis, we study equations that described a driven pendulum with odd-periodic driving. The equations also describe the under-actuated, double pendulum system called the pendubot. Techniques for trajectory exploration are developed.

For the inverted pendulum, we first wrote the problem as a two point boundary value problem with Dirichlet boundary conditions. Then, we develop an equivalent linear operator that combines a Nemitski operator (or superposition operator) with the linear operator for the unstable harmonic oscillator. By exploring the properties of the Green’s function for the unstable harmonic oscillator with Dirichlet boundary conditions, we developed bounds on various norms that prove useful for determining which parameter values will satisfy invariance and contraction conditions. With a direct application of the Schauder fixed point theorem, we showed that our family of equations representing an inverted pendulum always possessed an odd-periodic solution. Using the Banach fixed point theorem we showed that there is a unique solution within an invariant region of the space of possible solution curves. When there is a unique solution, successive approximations can be used to compute the solution trajectory. To illustrate the power and application of these ideas, we apply them to a pendubot with the inner arm moving at a constant velocity.

For non-inverted trajectories of the pendubot, we presented a necessary condition for trajectories to exist with general periodic forcing. For odd-periodic periodic driving functions this condition is always satisfied. For a driving function of A sin(wt), we found multiple solutions for the outer link. With the trajectories in hand, we demonstrated through simulation and/or physical implementation, the usefulness of maneuver regulation for providing orbital stabilization.

COinS