Global Stabilization of Compressible Flow Between Two Moving Pistons

TL;DR

This paper develops boundary feedback control laws to globally stabilize the flow of a viscous compressible gas between two pistons, ensuring exponential convergence without internal state measurements.

Contribution

It introduces a novel control Lyapunov functional approach for stabilizing a coupled PDE-ODE system modeling piston-gas dynamics, requiring only boundary measurements.

Findings

Achieves global asymptotic stability with exponential convergence.

Designs boundary feedback laws that do not need internal state measurements.

Validates the stabilization method through theoretical analysis.

Abstract

This paper studies the global feedback stabilization problem of a system with two pistons and the area between them containing a viscous compressible fluid (gas) modeled by the Navier-Stokes equations. The control input is the force applied on the left piston (boundary input) and the overall system consists of two nonlinear Partial Differential Equations and four nonlinear Ordinary Differential Equations. Global feedback stabilizers are designed for the overall system by means of the Control Lyapunov Functional methodology. The closed-loop system exhibits global asymptotic stability with an exponential convergence rate. The proposed stabilizing boundary feedback laws do not require measurement of the density and velocity profiles inside the area between the pistons and simply require measurements of the gas density and velocity at the position of the actuated piston.