Learning controllers for performance through LMI regions

TL;DR

This paper develops convex data-driven control methods using LMI regions to ensure desired transient performance for unknown linear systems, extending classical model-based eigenvalue placement conditions to noisy data scenarios.

Contribution

It introduces convex programs for data-driven control that enforce eigenvalue placement within LMI regions, generalizing classical conditions to unknown systems with noise.

Findings

Convex programs successfully enforce eigenvalue placement within LMI regions.

Methods extend classical model-based eigenvalue conditions to data-driven settings.

Numerical examples compare different data-based conditions for control design.

Abstract

In an open-loop experiment, an input sequence is applied to an unknown linear time-invariant system (in continuous or discrete time) affected also by an unknown-but-bounded disturbance sequence (with an energy or instantaneous bound); the corresponding state sequence is measured. The goal is to design directly from the input and state sequences a controller that enforces a certain performance specification on the transient behaviour of the unknown system. The performance specification is expressed through a subset of the complex plane where closed-loop eigenvalues need to belong, a so called LMI region. For this control design problem, we provide here convex programs to enforce the performance specification from data in the form of linear matrix inequalities (LMI). For generic LMI regions, these are sufficient conditions to assign the eigenvalues within the LMI region for all possible dynamics consistent with data, and become necessary and sufficient conditions for special LMI regions. In this way, we extend classical model-based conditions from a seminal work in the literature to the setting of data-driven control from noisy data. Through two numerical examples, we investigate how these data-based conditions compare with each other.