Output-feedback stabilization of an underactuated network of N interconnected n + m hyperbolic PDE systems

TL;DR

This paper presents a novel output-feedback control method for stabilizing a chain of interconnected hyperbolic PDE systems with only boundary measurements and actuation at one end, using integral transformations and recursive design.

Contribution

It introduces a new integral transformation approach to handle in-domain couplings and designs a recursive stabilizing controller for underactuated hyperbolic PDE networks.

Findings

Successfully stabilizes the PDE network with boundary control.

Ensures output trajectory tracking and input-to-state stability.

Provides a state observer for delayed state reconstruction.

Abstract

In this article, we detail the design of an output feedback stabilizing control law for an underactuated network of N subsystems of n + m heterodirectional linear first-order hyperbolic Partial Differential Equations interconnected through their boundaries. The network has a chain structure, as only one of the subsystems is actuated. The available measurements are located at the opposite extremity of the chain. The proposed approach introduces a new type of integral transformation to tackle in-domain couplings in the different subsystems while guaranteeing a ''clear actuation path'' between the control input and the different subsystems. Then, it is possible to state several essential properties of each subsystem: output trajectory tracking, input-to-state stability, and predictability (the possibility of designing a state prediction). We recursively design a stabilizing state-feedback controller by combining these properties. We then design a state-observer that reconstructs delayed values of the states. This observer is combined with the state-feedback control law to obtain an output-feedback controller. Simulations complete the presentation.