Averaging of zero dynamics for systems controlled by the vertically transverse function approach

Abstract

In this paper, a method is explored to introduce dissipation in the closed-loop, zerodynamics systems that arise in the vertically transverse function approach (VTFA) recently proposed by the authors. The VTFA is an attempt to extend the transverse function approach (TFA) of Morin and Samson to deal with practical point-stabilization of a class of critical simple mechanical systems on Lie groups. This class comprises systems that are not kinematic reductions and hence fall beyond the scope of application of the TFA as originally formulated. The VTFA gives rise to a nontrivial zero dynamics that ultimately determines the qualitative nature of the trajectories and, in order to constrain the velocity (or “fiber”) coordinates to vanish asymptotically, as required in typical applications, dissipation must be injected into the zero dynamics. Reported below is a possible way to reach that goal based on the adjunction of additional auxiliary inputs, via generalized vertically transverse functions, and the use of nonlinear, high-order averaging theory. Also included is an illustrative example as well as numerical simulations suggesting that the feedback laws herein designed yield promising results.