Reference Tracking AND Observer Design for Space-Fractional Partial Differential Equation Modeling Gas Pressures in Fractured Media

TL;DR

This paper develops a backstepping-based control and observer design for space-fractional PDEs modeling gas pressures in fractured media, addressing stability, tracking, and disturbance estimation with numerical validation.

Contribution

It introduces a novel backstepping approach for space-fractional PDEs and designs an adaptive observer for joint state and disturbance estimation.

Findings

Proves well-posedness, uniqueness, and stability of the FPDEs.

Develops a backstepping control for pressure tracking in fractured media.

Validates the approach through numerical simulations.

Abstract

This paper considers a class of space fractional partial differential equations (FPDEs) that describe gas pressures in fractured media. First, the well-posedness, uniqueness, and the stability in $L_(\infty{R})$of the considered FPDEs are investigated. Then, the reference tracking problem is studied to track the pressure gradient at a downstream location of a channel. This requires manipulation of gas pressure at the downstream location and the use of pressure measurements at an upstream location. To achiever this, the backstepping approach is adapted to the space FPDEs. The key challenge in this adaptation is the non-applicability of the Lyapunov theory which is typically used to prove the stability of the target system as, the obtained target system is fractional in space. In addition, a backstepping adaptive observer is designed to jointly estimate both the system's state and the disturbance. The stability of the closed loop (reference tracking controller/observer) is also investigated. Finally, numerical simulations are given to evaluate the efficiency of the proposed method.