Metriplectic torque for rotation control of a rigid body

Abstract: Metriplectic dynamics couple a Poisson bracket of the Hamiltonian description with a kind of metric bracket, for describing systems with both Hamiltonian and dissipative components. The construction builds in asymptotic convergence to a preselected equilibrium state. Phenomena such as friction, electric resistivity, thermal conductivity and collisions in kinetic theories all fit within this framework. In this paper an application of metriplectic dynamics is presented that is of interest for the theory of control: a suitably chosen torque, expressed through a metriplectic extension of its "natural" Poisson algebra, an algebra obtained by reduction of a canonical Hamiltonian system, is applied to a free rigid body. On a practical ground, the effect is to drive the body to align its angular velocity to rotation about a stable principal axis of inertia, while conserving its kinetic energy in the process. On theoretical grounds, this example provides a class of non-Hamiltonian torques that can be added to the canonical Hamiltonian description of the free rigid body and reduce to metriplectic dissipation. In the canonical description these torques provide convergence to a higher dimensional attractor. The method of construction of such torques can be extended to other dynamical systems describing "machines" with non-Hamiltonian motion having attractors.