Enforcing contraction via data

TL;DR

This paper develops data-driven conditions for designing feedback controllers that ensure system contractivity and stability, even in the presence of certain disturbances, by solving semidefinite programs based on system data.

Contribution

It introduces a novel data-dependent approach for enforcing contractivity in unknown nonlinear systems using semidefinite programming, including disturbance robustness and integral control design.

Findings

Data-based semidefinite programs guarantee contractivity.

Conditions are robust to sinusoidal disturbances with unknown amplitudes and phases.

Designed controllers achieve reference tracking and disturbance rejection.

Abstract

We present data-based conditions for enforcing contractivity via feedback control and obtain desired asymptotic properties of the closed-loop system. We focus on unknown nonlinear control systems whose vector fields are expressible via a dictionary of functions and derive data-dependent semidefinite programs whose solution returns the controller that guarantees contractivity. When data are perturbed by disturbances that are linear combinations of sinusoids of known frequencies (but unknown amplitude and phase) and constants, we remarkably obtain conditions for contractivity that do not depend on the magnitude of the disturbances, with imaginable positive consequences for the synthesis of the controller. Finally, we show how to design from data an integral controller for nonlinear systems that achieves constant reference tracking and constant disturbance rejection.