Infinite-dimensional control of nonlinear beam vibrations by piezoelectricactuator and sensor layers

An infinite-dimensional approach for the active vibration control of a mult ilayered straight composite piezoelectric beam is presented. In order to co ntrol the excited beam vibrations, distributed piezoelectric actuator and s ensor layers are spatially shaped to achieve a sensor/actuator collocation which fits the control problem. In the sense of von Karman a nonlinear form ulation for the axial strain is used and a nonlinear initial boundary-value problem for the deflection is derived by means of the Hamilton formalism. Three different control strategies are proposed. The first one is an extens ion of the nonlinear H-infinity-design to the infinite-dimensional case. It will be shown that an exact solution of the corresponding Hamilton-Jacobi- Isaacs equation can be found for the beam under investigation and this lead s to a control law with optimal damping properties. The second approach is a PD-controller for infinite-dimensional systems and the third strategy mak es use of the disturbance compensation idea. Under certain observability as sumptions of the free system, the closed loop is asymptotically stable in t he sense of Lyapunov. In this way, flexural vibrations which are excited by an axial support motion or by different time varying lateral loadings, can be suppressed in an optimal manner. A numerical example serves both to ill ustrate the design process and to demonstrate the feasibility of the propos ed methods.