Bilateral Boundary Control of One-Dimensional First- and Second-Order PDEs using Infinite-Dimensional Backstepping

TL;DR

This paper extends the infinite-dimensional backstepping method to design bilateral boundary controllers for 1-D PDE systems, enabling improved control effort and robustness with two actuators at both ends.

Contribution

It introduces a novel bilateral control design framework for 1-D PDEs using backstepping, expanding beyond traditional unilateral approaches.

Findings

Bilateral controllers reduce control effort compared to unilateral ones for large coefficients.

Explicit solutions demonstrate effectiveness of the bilateral control laws.

The approach facilitates future sensor/actuator feedback and fault-tolerant control designs.

Abstract

This paper develops an extension of infinite-dimensional backstepping method for parabolic and hyperbolic systems in one spatial dimension with two actuators. Typically, PDE backstepping is applied in 1-D domains with an actuator at one end. Here, we consider the use of two actuators, one at each end of the domain, which we refer to as bilateral control (as opposed to unilateral control). Bilateral control laws are derived for linear reaction-diffusion, wave and 2X2 hyperbolic 1-D systems (with same speed of transport in both directions). The extension is nontrivial but straightforward if the backstepping transformation is adequately posed. The resulting bilateral controllers are compared with their unilateral counterparts in the reaction-diffusion case for constant coefficients, by making use of explicit solutions, showing a reduction in control effort as a tradeoff for the presence of two actuators when the system coefficients are large. These results open the door for more sophisticated designs such as bilateral sensor/actuator output feedback and fault-tolerant designs.