Time-optimal control of harmonic oscillators at resonance

Harmonic oscillators are widespread in industrial applications. One of their key properties is the large amplification of excitation signals at their eigenfrequency - broadly known as resonance. This paper investigates how to excite harmonic oscillators such that resonance is reached in minimum time. We use Pontryagin´s Minimum Principle to derive time-optimal control laws to reach resonance for both the undamped and damped harmonic oscillator. For the former our derivations even lead to a time-optimal feedback law given through a switching curve. For application purposes we compare this feedback law with a much simpler sliding-mode based control law for controlling a simplified model of an industrial vibratory device. In order to smoothly apply both control laws, additional state and parameter estimators are presented to address the time-varying nature of the application.