Backstepping control of ODE/wave PDE cascaded systems with dynamic boundary conditions

We consider backstepping control for ODE/wave PDE cascaded systems with dynamic boundary conditions. The ODE state is driven by the uncontrolled boundary of the wave PDE. A backstepping compensator is designed such that the closed-loop system is globally asymptotically stable. The controller design and stability analysis are based on coordinate and backstepping transformations. Changes of coordinate recast the original system into a coupled 2×2 first-order hyperbolic system with spatially-varying coefficients cascading into an ODE. Backstepping transformations are exploited to design a compensator for the resulting coupled system. The validity of the proposed design is illustrated by controlling a bit of the oil well drilling system to operate at a constant uniform rotary speed.