Optimal Control of Nonlinear Parametrically Excited Beam Vibrations by Spatially Distributed Sensors and Actors

This contribution is focused on a straight composite beam with multiple piezoelectric layers under the action of an axial support motion. In the sense of v. Karman a nonlinear formulation for the axial strain is used and the equations of motion are derived by means of the Hamilton formalism. This system turns out to be special class of infinte dimensional systems, the so called Hamilton AI-systems with exertnal inputs. In order to suppress the excited vibrations two infinite control laws are proposed. The first one is an infinite PD-feedback law and the second one is based on the nonlinear Hinf-design, where an exact solution of the corresponding Hamilton Jacobi Isaacs equation is presented. The necessary quantities for the control laws can be measured by appropriate space-wise shaped sensors and the asymptotic stability of the equilibrium point can be proved.