Robust control design of underactuated 2× 2 PDE-ODE-PDE systems

In this letter, we design a robust stabilizing controller for a system composed of two sets of linear heterodirectional hyperbolic PDEs, with actuation at one boundary of one of the PDEs, and couplings at the middle boundary with ODEs in a PDE-ODE-PDE configuration. The system is underactuated since only one of the PDE systems is actuated. The design approach employs a backstepping transformation to move the undesired system couplings to the proximal boundary (where the actuation is located). We can then express this target system as a time-delay neutral system for which we can design an appropriate control law to obtain an exponentially stable target system.