Output-Feedback Stabilization of an Underactuated Network of $N$ Interconnected $n+ m$ Hyperbolic PDE Systems

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.