Proximate time-optimal control of a second-order flexible structure

We develop a Proximate Time-Optimal Ser-vomechanism (PTOS) for a harmonic oscillator plant with no damping. We approximate the signum function inherent to the bang-bang, time-optimal solution with a gain and saturator. We prove that if certain design restrictions are met, this control law will drive any initial state into an invariant region B about the origin of the phase-plane in finite-time. We then show that once the state enters this region, it is asymptotically stable about the origin, by demonstrating that a Lyapunov function exists for the region B. Finally, we present simulation results using this control law on a simplified model of an atomic force microscope.