Observer-Based Output-Feedback Backstepping Stabilization of Continua of Hyperbolic PDEs and Application to Large-Scale $n+m$ Coupled Hyperbolic PDEs
Jukka-Pekka Humaloja, Nikolaos Bekiaris-Liberis
TL;DR
The paper addresses stabilizing continua of linear hyperbolic PDEs and transferring that stability to large-scale $n+m$ hyperbolic PDEs. It develops a backstepping-based observer for continua with a Lyapunov functional for the estimation error and proves well-posedness and exponential stability of the closed-loop system, enabling a separation principle. The authors extend the continuum design to large-scale networks by solving continuum kernel equations to obtain observer-based output-feedback laws, and they introduce a virtual continuum with resets to leverage continuum-approximation properties for the finite-dimensional system. A numerical example demonstrates that continuum kernels can be computed in closed form and that the continuum observer-based controller achieves stabilization with complexity largely independent of $n$, highlighting potential computational advantages for very large networks.
Abstract
We develop a non-collocated, observer-based output-feedback law for a class of continua of linear hyperbolic PDE systems, which are viewed as the continuum version of $n+m$, general heterodirectional hyperbolic systems as $n\to\infty$. The design relies on the introduction of a novel, continuum PDE backstepping transformation, which enables the construction of a Lyapunov functional for the estimation error system. Stability under the observer-based output-feedback law is established by using the Lyapunov functional construction for the estimation error system and proving well-posedness of the complete closed-loop system, which allows utilization of the separation principle. Motivated by the fact that the continuum-based designs may provide computationally tractable control laws for large-scale, $n+m$ systems, we then utilize the control/observer kernels and the observer constructed for the continuum system to introduce an output-feedback control design for the original $n+m$ system. We establish exponential stability of the resulting closed-loop system, which consists of a mixed $n+m$-continuum PDE system (comprising the plant-observer dynamics), introducing a virtual continuum system with resets, which enables utilization of the continuum approximation property of the solutions of the $n+m$ system by its continuum counterpart (for large $n$). We illustrate the potential computational complexity/flexibility benefits of our approach via a numerical example of stabilization of a large-scale $n+m$ system, for which we employ the continuum observer-based controller, while the continuum-based stabilizing control/observer kernels can be computed in closed form.