Abstract Many control applications are built around systems that operate near a repeating cycle. Engineers often talk about phase, timing, and synchronization, but in high dimensional state spaces the notion of phase is a derived and fragile object. This paper reverses the usual viewpoint. Instead of extracting a phase function from the dynamics and then building controllers on top of it, we begin from a prescribed phase field on a neighborhood of a periodic orbit, treat that field as a design variable, and design feedback so that it behaves as a clock-like phase coordinate. The closed loop system is a control affine nonlinear plant with state feedback. A smooth phase field is given around a hyperbolic periodic orbit, and the feedback is required to make the phase advance at a constant rate along all trajectories in a tubular neighborhood, while directions transverse to phase are contracting. The main relations are directional derivative conditions on the closed loop vector field and simple inequalities in transverse coordinates. We derive pointwise conditions on the instantaneous control authority that guarantee the existence of such feedback laws and provide local constructions in simple settings. The formulation is time domain and geometric and it treats “phase” as a structural closed loop property that can be designed, rather than extracted, around nonlinear oscillations.
Shen Zeng (Mon,) studied this question.