Practical stabilization of driftless systems on Lie groups: the transverse function approach

A general control design approach for the stabilization of controllable driftless nonlinear systems on finite dimensional Lie groups is presented. The approach is based on the concept of bounded transverse functions, the existence of which is equivalent to the systemʹs controllability. Its outcome is the practical stabilization of any trajectory, i.e., not necessarily a solution of the control system, in the state-space. The possibility of applying the approach to an arbitrary controllable smooth driftless system follows in turn from the fact that any controllable homogeneous approximation of this system can be lifted (via a dynamic extension) to a system on a Lie group. Illustrative examples are given.