Using dynamic extensions for the backstepping control of hyperbolic systems

TL;DR

This paper introduces a novel dynamic extension method for boundary control of hyperbolic PDE systems, enabling advanced control design such as input-output decoupling and improved stability, extending finite-dimensional control concepts to PDEs.

Contribution

It develops a systematic approach to dynamic boundary control of hyperbolic PDEs using dynamic extensions, enhancing control design flexibility and stability analysis.

Findings

Dynamic extensions homogenize transport velocities on the spatial domain.

Backstepping transformation simplifies static feedback design.

Achieves input-output decoupling with guaranteed internal stability.

Abstract

This paper systematically introduces dynamic extensions for the boundary control of general heterodirectional hyperbolic PDE systems. These extensions, which are well known in the finite-dimensional setting, constitute the dynamics of state feedback controllers. They make it possible to achieve design goals beyond what can be accomplished by a static state feedback. The design of dynamic state feedback controllers is divided into first introducing an appropriate dynamic extension and then determining a static feedback of the extended state, which includes the system and controller state, to meet some design objective. In the paper, the dynamic extensions are chosen such that all transport velocities are homogenized on the unit spatial interval. Based on the dynamically extended system, a backstepping transformation allows to easily find a static state feedback that assigns a general dynamics to the closed-loop system, with arbitrary in-domain couplings. This new design flexibility is also used to determine a feedback that achieves complete input-output decoupling in the closed loop with ensured internal stability. It is shown that the modularity of this dynamic feedback design allows for a straightforward transfer of all results to hyperbolic PDE-ODE systems. An example demonstrates the new input-output decoupling approach by dynamic extension.