Backstepping control of an aeroelastic system with large Mach numbers

We design a controller for flow-induced vibrations of an infinite-band membrane, with flow running across the band and only above it, and with actuation only on the trailing edge of the membrane. Due to the infinite length of the membrane, the dynamics of the membrane in the spanwise direction are neglected, namely, we employ a 1-D model that focuses on streamwise vibrations. This framework is inspired by a flow along an airplane wing with actuation on the trailing edge. The model of the flow-induced vibration is given by a wave PDE with an anti-damping term throughout the 1-D domain. Such a model is based on linear aeroelastic theory for Mach numbers above 0.8. To design a controller we introduce a 3-stage backstepping transformation. The first stage gets the system to a critically anti-damped wave equation, changing the stiffness coefficient´s value but not its sign. The second stage changes the system from a critically anti-damped to a critically damped equation with an arbitrary damping coefficient. The third stage adjusts stiffness arbitrarily. The controller and backstepping transformation map the original system into a target system given by a wave equation with arbitrary positive damping and stiffness.