Feedback Linearization of Hyperbolic PDEs with Volterra Nonlinearities

TL;DR

This paper extends feedback linearization techniques from parabolic to hyperbolic PDEs, simplifying controller design for certain nonlinear PDEs without solving complex kernel PDEs.

Contribution

It generalizes the feedback linearization framework to hyperbolic PDEs and constructs controllers for a specific class of nonlinear PDEs without kernel PDEs.

Findings

Successfully extended feedback linearization to hyperbolic PDEs.

Constructed controllers for a subclass of nonlinear PDEs without kernel PDEs.

Provided a new approach inspired by geometric nonlinear control theory.

Abstract

Alberto Isidori's framework of geometric nonlinear control, and particularly of feedback linearization, is the inspiration behind PDE backstepping: apply a transfromation of the state to cast the plant into a canonical form, bring all the non-canonical effects within the "span" of (boundary) control, and close the design with a feedback that makes the closed loop evolve in accordance with well-studied stable dynamics. The specificity of this approach is that, for PDEs, there is not one canonical form (like Brunovsky for ODEs) but the canonical forms are PDE-class-specific. When conducting this process for nonlinear PDEs, where the "transformation of the state" is performed using a nonlinear Volterra series indexed by the spatial variable, enormous technical challenges arise. One has to deal with kernels governed by PDEs on simplex domains growing in dimension to infinity, capture the growth rates of these kernels of the "direct transformation," and conduct the same for the "inverse transformation" without directly studying its Volterra kernels. So far, this agenda has been executed only once, two decades ago: for parabolic PDEs by Vazquez and Krstic [Automatica, 2008]. Generalization attempts have not followed because of the immense complexity involved in feedback-linearizing nonlinear PDEs. In this paper, dedicated to Professor Isidori, we convert the PDE feedback-linearizing methodology of 2008 from the parabolic to a hyperbolic class and, for a transport-adapted subclass of Chen-Fliess series, construct controllers without kernel PDEs.