Stabilizability of first-order dynamics in second-order systems

TL;DR

This paper investigates the conditions under which second-order control systems can be designed to replicate prescribed first-order dynamics on manifolds, providing both positive and negative results based on actuation and topology.

Contribution

It characterizes when second-order systems can globally or locally stabilize prescribed vector fields, highlighting the role of actuation and manifold topology.

Findings

Fully actuated systems can stabilize any smooth vector field exponentially.

Underactuated systems on manifolds with nonzero Euler characteristic cannot generally be stabilized.

The Euler characteristic condition is crucial for the negative stabilization result.

Abstract

We study whether second-order systems can be made to behave like prescribed first-order dynamical systems through feedback control. More precisely, we study whether prescribed vector fields on compact smooth manifolds, viewed geometrically as sections of the tangent bundle, can be asymptotically stabilized in a strong sense by second-order control systems on the base manifold. Our class of second-order systems includes most Lagrangian systems, and we obtain both positive and negative results. The positive result asserts that, for fully actuated systems, the section corresponding to any smooth vector field can be made globally exponentially stable, normally hyperbolic, and more. In particular, not only does each closed-loop solution asymptotically have the prescribed velocities, but it also converges to a trajectory of the first-order dynamics generated by the prescribed vector field at an exponential rate. Thus, the closed-loop second-order system asymptotically reproduces the prescribed first-order dynamics. In contrast, the negative result asserts that, for underactuated systems on manifolds with nonzero Euler characteristic, sections corresponding to "almost all" smooth vector fields cannot even be locally asymptotically stabilized. This includes, in particular, all vector fields with only isolated zeros. An example shows that the Euler characteristic assumption is necessary for the negative result.