# Phase-amplitude reduction

This study combines phase-amplitude reduction with nonlinear machine-learning sparse-sensor reconstruction to estimate phase and amplitude sensitivity fields from only three airfoil-surface sensors and analytically derive optimal actuation waveforms for fast synchronization of post-stall airfoil wake shedding; the identified optimal forcing alters wake frequency eight times faster than standard sinusoidal actuation, and the amplitude-penalized waveform suppresses lift-coefficient fluctuation relative to the optimal forcing without amplitude penalty.

This study combines phase-amplitude reduction with nonlinear machine-learning sparse-sensor reconstruction to estimate phase and amplitude sensitivity fields from only three airfoil-surface sensors and analytically derive optimal actuation waveforms for fast synchronization of post-stall airfoil wake shedding; the identified optimal forcing alters wake frequency eight times faster than standard sinusoidal actuation, and the amplitude-penalized waveform suppresses lift-coefficient fluctuation relative to the optimal forcing without amplitude penalty.