Nonlinear Bilateral Output-Feedback Control for a Class of Viscous Hamilton-Jacobi PDEs

TL;DR

This paper develops explicit nonlinear boundary control and estimation strategies for viscous Hamilton-Jacobi PDEs with bilateral actuation and sensing, enabling trajectory tracking and stability analysis within a regional context, demonstrated through traffic flow applications.

Contribution

It introduces a novel feedback linearizing transformation combined with backstepping for bilateral boundary control of viscous Hamilton-Jacobi PDEs, providing explicit control and observer designs with regional stability guarantees.

Findings

Explicit boundary control laws for viscous Hamilton-Jacobi PDEs.

Constructed nonlinear observer using bilateral boundary measurements.

Validated methods through traffic flow control simulations.

Abstract

We tackle the boundary control and estimation problems for a class of viscous Hamilton-Jacobi PDEs, considering bilateral actuation and sensing, i.e., at the two boundaries of a 1-D spatial domain. First, we solve the nonlinear trajectory generation problem for this type of PDEs, providing the necessary feedforward actions at both boundaries. Second, in order to guarantee trajectory tracking with an arbitrary decay rate, we construct nonlinear, full-state feedback laws employed at the two boundary ends. Third, a nonlinear observer is constructed, using measurements from both boundaries, which is combined with the full-state feedback designs into an observer-based output-feedback law. All of our designs are explicit since they are constructed interlacing a feedback linearizing transformation (which we introduce) with backstepping. Due to the fact that the linearizing transformation is locally invertible, only regional stability results are established, which are, nevertheless, accompanied with region of attraction estimates. Our stability proofs are based on the utilization of the linearizing transformation together with the employment of backstepping transformations, suitably formulated to handle the case of bilateral actuation and sensing. We illustrate the developed methodologies via application to traffic flow control and we present consistent simulation results.