Using dynamic extensions for the backstepping control of hyperbolic systems

TL;DR

The paper addresses boundary control of general heterodirectional hyperbolic PDEs with spatially varying transport velocities by introducing dynamic extensions that homogenize transport delays on the unit interval. It combines a preliminary backstepping transform with a carefully constructed dynamic extension to yield an extended plant in which a static feedback of the extended state can assign arbitrary in-domain couplings and achieve input-output decoupling, while preserving internal stability. The results extend naturally to PDE-ODE systems and are supported by an example that demonstrates complete input-output decoupling in the closed loop. This dynamic-extension framework provides a modular, design-flexible approach that broadens the class of target dynamics accessible with backstepping and offers a path toward parabolic and other distributed-parameter systems applications.

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.

Figures (4)

• Figure 1: Idea of homogenization by dynamic extension.
• Figure 2: Typical characteristics visualize the kernel equations \ref{['eq:kernel2']}, with the at $\zeta=0$ and $\zeta=z$ highlighted in color. Due to the common transport velocities in $\bar{\Lambda}^-(z)$ and $\bar{\Lambda}^+(z)$, respectively, each block of $L(z,\zeta)$ in \ref{['eq:defMK']} is represented by only one element.
• Figure 3: Norms $||\chi(t)||$ and $||\bar{w}(t)||$ of the dynamically extended system state and the controller state in closed loop.
• Figure 4: Evolution of $y(t)$, $\bar{v}(t)$ and $u(t)$.