Output-Feedback Boundary Control of Thermally and Flow-Induced Vibrations in Slender Timoshenko Beams

This work is motivated by the engineering challenge of suppressing vibrations in turbine blades of aero engines, which often operate under extreme thermal conditions and high-Mach aerodynamic environments that give rise to complex vibration phenomena, commonly referred to as thermally-induced and flow-induced vibrations. Using Hamilton's variational principle, the system is modeled as a rotating slender Timoshenko beam under thermal and aerodynamic loads, described by a mixed hyperbolic-parabolic PDE system where instabilities occur both within the PDE domain and at the uncontrolled boundary, and the two types of PDEs are cascaded in the domain. For such a system, we present the state-feedback control design based on the PDE backstepping method. Recognizing that the distributed temperature gradients and structural vibrations in the Timoshenko beam are typically unmeasurable in practice, we design a state observer for the mixed hyperbolic-parabolic PDE system. Based on this observer, an output-feedback controller is then built to regulate the overall system using only available boundary measurements. In the closed-loop system, the state of the uncontrolled boundary, i.e., the furthest state from the control input, is proved to be exponentially convergent to zero, and all signals are proved as uniformly ultimately bounded. The proposed control design is validated on an aero-engine flexible blade under extreme thermal and aerodynamic conditions.