Control by interconnection of the timoshenko beam

In this paper,the dynamical control of a mixed finite and infinite dimen-sional mechanical system is approached within the framework of port Hamiltonian systems. As an applicative example of the presented methodology, a flexible beam with a mass under gravity field connected to a free end. modeled according to the Timoshenko theory and in distributed port Hamiltonian form, is considered. The control problem is approached by generalization of the concept of structural invariant (Casimir function) to the infinite dimensional case and the so-called control by interconnection control technique is extended to the infinite dimensional case. In this way finite dimensional passive controllers can stabilize distributed parameter systems by shaping their total energy. i.e. by assigning a new minimum in the desired equilibrium configuration that can be reached if a dissipation effect is introduced.