Boundary Stabilization of a Bending and Twisting Beam by Linear Quadratic Regulation

We use Linear Quadratic Regulation (LQR) to stabilize a bending and twisting beam by boundary control . The thin beam is rectangular and of moderate to high aspect ratio. This is the first step in using boundary actuation to stabilize a fluttering wing. The next step will be to add a model of the aerodynamic forces and moments that result from the motions of the wing. We leave this to the future.We find a full state feedback that stabilizes the bending and torsion of the beam using two point actuators located at the root of the beam. One actuator delivers a bending moment to the beam and the other delivers a twisting torque to the beam. The model is the Euler Bernouli beam equation inertially coupled to the wave equation.We express the optimal cost as a Fredholm quadratic form. Using integration by parts and completing the square we derive the Riccati partial differential equation (RPDE) that the kernel of the optimal cost weakly satisfies. This RPDE is elliptic with a quadratic nonlinearity. We seek a continuous, weak solution. The continuity of the solution is important because it confirms that the optimal linear feedback has moved all the open loop eigenvalues off the imaginary axis and into the open left half plane.We use Fourier methods involving two families of eigenfunctions, the family of eigenfunctions of the second order spatial partial differential operator subject to Neumann boundary conditions and the family of eigenfunctions of the fourth order spatial partial differential operator subject to modified cantilever boundary conditions. In this way we convert the RPDE into an infinite dimensional Algebraic Riccati Equation (ARE). The ARE has four uncoupled blocks corresponding to the two families of eigenfunctions in the first or second position.We close with an example where the infinite dimensional ARE is approximated by an 16 dimensional ARE.