Obstructions to the Existence of Universal Stabilizers for Smooth Control Systems

Abstract

This paper states necessary conditions for the existence of “universal stabilizers” for smooth control systems. Roughly speaking, given a control system and a set {\cal U} of reference input functions, by “universal stabilizer” we mean a continuous feedback law that stabilizes the state of the system asymptotically to any of the reference trajectories produced by (arbitrary) inputs in {\cal U}. For an example, consider Brockett’s nonholonomic integrator, with {\cal U} representing a set of uniformly bounded, piecewise continuous functions of time. This system’s state can be asymptotically stabilized to any reference trajectory provided the latter is persistently exciting (PE). By contrast, for constant trajectories (i.e., equilibria), which are not PE, asymptotic stabilization is impossible by means of continuous pure-state feedback, in view of Brockett’s obstruction. However, since this obstruction can be circumvented by the use of time-varying state feedback, one might reasonably expect to be able to design a (time-varying) continuous control law capable of asymptotically stabilizing the state to arbitrary reference trajectories, be they PE or not. Surprisingly, a consequence of the results in this paper is that, for systems with nonholonomic constraints frequently found in control applications, if {\cal U} contains reference functions that are not PE, then the “universal stabilization” problem cannot be solved, even if time-varying feedback is used.