One Equation to Rule Them All -- Part I: Direct Data-Driven Cascade Stabilisation

TL;DR

This paper introduces a data-driven control framework that uses solutions to Sylvester equations derived from data, enabling robust cascade stabilization, model reduction, and output regulation with noise impact analysis.

Contribution

It develops a method to solve Sylvester equations from data and analyzes noise effects, providing a unified approach for cascade stabilization and related problems.

Findings

Data-efficient cascade stabilization method demonstrated

Noise propagation bounds established

Numerical example validates approach

Abstract

In this article we present a framework for direct data-driven control for general problems involving interconnections of dynamical systems. We first develop a method to determine the solution of a Sylvester equation from data. Such solution is used to describe a subspace that plays a role in a large variety of problems. We then provide an error analysis of the impact that noise has on this solution. This is a crucial contribution because, thanks to the interconnection approach developed throughout the article, we are able to track how the noise propagates at each stage, and thereby provide bounds on the final designs. Among the many potential problems that can be solved with this framework, we focus on three representatives: cascade stabilisation, model order reduction, and output regulation. This manuscript studies the first problem, while the companion Part II addresses the other two. For each of these settings we show how the problems can be recast in our framework. In the context of cascade stabilisation, we consider the 2-cascade problem, the effect of noise through the cascade, as well as N-cascade case, and we demonstrate that our proposed method is data efficient. The proposed designs are illustrated by means of a numerical example.