Late lumping of transformation-based feedback laws for boundary control systems

TL;DR

This paper introduces a late-lumping feedback design method for boundary control systems that approximates unbounded feedback operators by decomposing them into bounded and unbounded parts, ensuring convergence to desired dynamics.

Contribution

It proposes a novel late-lumping scheme suitable for infinite-dimensional systems, enabling approximation of complex feedback laws while maintaining exact realization of unbounded components.

Findings

Convergence of closed-loop dynamics to desired behavior

Applicable to backstepping and flatness-based designs

Demonstrated on a hyperbolic system

Abstract

Late-lumping feedback design for infinite-dimensional linear systems with unbounded input operators is considered. The proposed scheme is suitable for the approximation of backstepping and flatness-based designs and relies on a decomposition of the feedback into a bounded and an unbounded part. Approximation applies to the bounded part only, while the unbounded part is assumed to allow for an exact realization. Based on spectral results, the convergence of the closed-loop dynamics to the desired dynamics is established. By duality, similar results apply to the approximation of the observer output-injection gains for systems with boundary observation. The proposed design and approximation steps are demonstrated and illustrated based on a hyperbolic infinite-dimensional system.