Infinite-Dimensional Control Of Nonlinear Beam Vibrations

The beam structure is a kind of very common structure in reality. Researching beam system not only have significant theory value but also have vital project background and applicable value. As the most basic unit of the complicated structure, beam is always a research object of the modern control theory because of its relatively simple character. A lot of relevant achievements have been applied widely in many aspects such as Aero-Space, robotics, bridge construction.The theme of this paper is the infinite-dimensional approach for the active vibration control of composite piezoelectric beam. At first, we derive the mechanical model of composite piezoelectric beam by Hamilton theory without considering hysteresis phenomena or the influence of thermal effects. Secondly three different control strategies are proposed. The first one is an extension of the nonlinear H_∞design to the infinite-dimensional case. The mathematical model turns out to be a member of a special class of nonlinear infinite-dimensional systems, namely the so-called Hamiltonian Affine Input systems. By taking advantage of the special structure and properties of this class of systems, we are able to extend the nonlinear H_∞ design for finite-dimensional affine input systems to the infinite-dimensionalcase. The H_∞ design is traced back to an optimization problem. We find a solutionof the Hamilton-Jacobi-Isaacs equation for the considered mechanical plant. Under certain observability assumptions of the free system, the stability of the closed loop in the sense of Lyapunov can be proved by use of operator hemigroup theory and La Salle invariable set theory. In addition, we expand PD control and fuzzy-PID control to infinite-dimensional, and apply them to the tested piezoelectric beam. Finally, simulation results demonstrate the feasibility and the efficiency of the proposed design methods.