Feedback control design for closed-loop oscillations via dominant system theory

Stable oscillations, arising from nonlinear non-equilibrium dynamics, are an important class of behavior in both nature and engineering. For engineering applications like robotic locomotion, it is usually desirable to induce oscillations in a controlled manner. While plenty of feedback control methods have been developed for equilibrium behaviors in control theory, we still lack a systematic design framework to excite non-oscillatory systems into oscillations via feedback control. As a step towards that, this thesis aims to provide theoretically guaranteed feedback controller design protocols for stable oscillations. Inspired by the oscillator models in biology and electrical engineering, we propose two architectures for oscillation control design: the mixed feedback oscillator and the resonant positive feedback oscillator. Our design methods take advantage of the dominant system theory to trigger and modulate stable oscillations in closed loop. For the mixed feedback oscillator, we show how to achieve oscillations by tuning the control parameters: the balance between the positive and negative feedback networks and the global feedback gain. For the resonant positive feedback oscillator, we show how to trigger oscillations by designing its linear and nonlinear control units. Then we show how to implement the two control architectures with basic linear and nonlinear circuit elements. Finally, we extend the control design for the mixed feedback oscillator. Based on linear matrix inequalities (LMIs) for dominance analysis, we develop control algorithms that automatically return state feedback gains that also take robustness and passivity-based interconnection into consideration.