An LMI Framework for Contraction-based Nonlinear Control Design by Derivatives of Gaussian Process Regression

TL;DR

This paper develops an LMI-based control design framework for nonlinear systems using Gaussian process regression to address the integrability problem and handle unknown dynamics with stochastic error modeling.

Contribution

It introduces a novel use of Gaussian process regression to solve the integrability problem in contraction-based control design and incorporates stochastic learning errors into the LMI framework.

Findings

LMI framework effectively designs controllers for nonlinear systems.

GPR-based approach handles unknown dynamics with probabilistic error bounds.

Framework is suitable for discrete-time nonlinear control applications.

Abstract

Contraction theory formulates the analysis of nonlinear systems in terms of Jacobian matrices. Although this provides the potential to develop a linear matrix inequality (LMI) framework for nonlinear control design, conditions are imposed not on controllers but on their partial derivatives, which makes control design challenging. In this paper, we illustrate this so-called integrability problem can be solved by a non-standard use of Gaussian process regression (GPR) for parameterizing controllers and then establish an LMI framework of contraction-based control design for nonlinear discrete-time systems, as an easy-to-implement tool. Later on, we consider the case where the drift vector fields are unknown and employ GPR for functional fitting as its standard use. GPR describes learning errors in terms of probability, and thus we further discuss how to incorporate stochastic learning errors into the proposed LMI framework.