Nonlinear control of networked dynamical systems

TL;DR

This paper introduces a mathematical framework combining dimensionality reduction, bifurcation theory, and model discovery to control high-dimensional nonlinear networked systems by manipulating fixed points.

Contribution

It presents a novel method that uses low-dimensional subspaces and bifurcation analysis to design control signals for complex nonlinear systems.

Findings

Successfully controls biological bistable systems

Applies to high-dimensional networks and memory models

Demonstrates effective fixed point switching

Abstract

We develop a principled mathematical framework for controlling nonlinear, networked dynamical systems. Our method integrates dimensionality reduction, bifurcation theory and emerging model discovery tools to find low-dimensional subspaces where feed-forward control can be used to manipulate a system to a desired outcome. The method leverages the fact that many high-dimensional networked systems have many fixed points, allowing for the computation of control signals that will move the system between any pair of fixed points. The sparse identification of nonlinear dynamics (SINDy) algorithm is used to fit a nonlinear dynamical system to the evolution on the dominant, low-rank subspace. This then allows us to use bifurcation theory to find collections of constant control signals that will produce the desired objective path for a prescribed outcome. Specifically, we can destabilize a given fixed point while making the target fixed point an attractor. The discovered control signals can be easily projected back to the original high-dimensional state and control space. We illustrate our nonlinear control procedure on established bistable, low-dimensional biological systems, showing how control signals are found that generate switches between the fixed points. We then demonstrate our control procedure for high-dimensional systems on random high-dimensional networks and Hopfield memory networks.