Stabilisation of a Rotating Beam Clamped on a Moving Inertia with Strong Dissipation Feedback

Abstract

In this paper we consider the stabilization problem of a beam clamped on a moving inertia actuated by an external torque and force. The beam is modelled as a distributed parameter port-Hamiltonian system (PDEs), while the inertia as a finite dimensional port-Hamiltonian system (ODEs). The control inputs correspond to a torque applied by a rotating motor and a force applied by a linear motor. In this paper we propose the use of a strong dissipation term in the control law, consisting of the time derivative of the restoring force at the clamping point. After a change of variables, the closed loop system shows dissipation at the boundaries of the PDEs. In this preliminary work we show that the closed loop operator is the generator of a contraction C 0 -semigroup in a special weighted space, with norm equivalent to the standard one. Further, we prove the asymptotic stability of the closed loop system and we show the effectiveness of the proposed control law in comparison with a PD controller with the help of numerical simulations.

Stabilization of a class of mixed ODE–PDE port-Hamiltonian systems with strong dissipation feedback

“…Control design and stability analysis for m-ODE-PDE has raised a significant attention from many researchers because of its wide range of applications. In particular, m-PDE-ODE equations enclose models of rotating and/or translating beams [1,2,3], controlled nanotweezer used for DNA manipulation [4] as well as electric transmission lines with load [5].…”

Finite-time stabilization of an overhead crane with a flexible cable submitted to an affine tension

“…The stabilization of coupled PDE-ODE systems has attracted the attention of the control community since several decades. We can mention applications such as the control of a rotating body-beam without natural damping [11] as well as the case of a rotating and translating body-beam [20], a slide-flute [4], a switched power converter with a transmission line [13], turbulent fluid motion and traffic flow [31], and the overhead crane taking into account the flexibility of the cable, as discussed below.…”

Finite-time stabilization of an overhead crane with a flexible cable submitted to an affine tension

“…The stabilization of hybrid PDE-ODE systems has attracted the attention of the control community since several decades. We can mention applications such as the control of a rotating body-beam without natural damping [11] as well as the case of a rotating and translating body-beam [19], a slide-flute [4], a switched power converter with a transmission line [13], turbulent fluid motion and traffic flow [29], and the overhead crane taking into account the flexibility of the cable, as discussed below.…”