Adaptive output feedback control of flow-induced vibrations of a membrane at high Mach numbers

We address the problem of adaptive outputfeedback stabilization for flow-induced vibrations of a membrane at high Mach numbers. The aeroelastic instability is compensated by using boundary control and anti-collocated measurement. The membrane is infinite in spanwise and the streamwise vibrations are modeled by a one-dimensional wave Partial Differential Equation (PDE) with an aerodynamic forcing term. Based on Piston theory, the term is represented by an anti-damping term multiplying a constant coefficient and a convective term multiplying an unknown constant coefficient. We then transform the wave PDE to a one-dimensional 2 × 2 first-order hyperbolic PDEs with constant coupling coefficients. Before introducing an adaptive output feedback controller for the hyperbolic PDEs, we present a nonadaptive explicit state feedback controller. To deal with the absence of both full-state measurements and parameter knowledge, the adaptive outputfeedback design is used; the design is based on an observer canonical form which is obtained through a change of variables and a backstepping transformation. The form enables us to design an explicit state observer. We employ gradient-based parameter estimators. For the closed loop system, we achieve convergence of the state of the wave PDE to zero. We validate our result with a simulation.