Optimal control of perturbations for inferring response functions in dynamical systems

Response functions of dynamical systems are measured by analyzing how they respond to perturbations. Recent advancements have allowed for such measurements at minute length and time scales. These techniques have broad applications, from rheological experiments to measuring viscoelasticity, and extending into microfluidic control and micro-robotics. However, for systems with poorly characterized responses, maximizing the signal-to-noise ratio (SNR) across a specified bandwidth, under given constraints — based on time series data of a dynamical system — remains a significant challenge.

Here, we propose an optimal control framework using an ansatz given by multiple superimposed sine waves with control over their frequencies, amplitudes, and phases. We can characterize their response function with limited training data availability. With both simulations and experiments, we demonstrate the efficacy of our optimal control algorithm for a feedback control system that obeys linear response theory. We then extend this method to more complex, non-linear cases, such as networks of dynamical systems. Our method provides a principled approach to time series-driven estimation of response functions through optimal perturbations.