Feedback Stabilization of Relative Equilibria for Mechanical Systems with Symmetry

This paper is an outgrowth of the work of Bloch, Krishnaprasad, Marsden and Sánchez de Alvarez [1992], where a feedback control that stabilizes intermediate axis rigid body rotation using an internal rotor was found. Stabilization is determined by use of the energy-Casimir (Arnold) method. In the present paper we show that this feedback controlled system can be written as the Euler-Lagrange equations for a modified Lagrangian: a velocity shift associated with a change of connection turns the free (unforced) equations into the feedback controlled equations. We also show how stabilization of the inverted pendulum on a cart can be achieved in an analogous way. We provide a general systematic construction of such controlled Lagrangians. The basic idea is to modify the kinetic energy of the free Lagrangian using a generalization of the Kaluza-Klein construction in such a way that the extra terms obtained in the Euler-Lagrange equations can be identified with control forces. The fact that the controlled system is Lagrangian by construction enables one to make use of energy techniques for a stability analysis. Once stabilization is achieved in a mechanical context, one can establish asymptotic stabilization by the further addition of dissipative controls. The methods here can be combined with symmetry breaking controls obtained by modifying the potential energy and also can be used for tracking.