Necessary conditions for feedback stabilization and safety

TL;DR

This paper extends Brockett's necessary condition to stabilize general compact sets and ensure safety in control systems, providing new theoretical tools for complex and constrained systems.

Contribution

It generalizes Brockett's condition to stabilizing sets with nonzero Euler characteristic and formulates necessary conditions for safe control laws.

Findings

Generalized stabilizability conditions for compact sets on manifolds.

Derived necessary conditions for safe control law existence.

Showed the new conditions are stronger than Brockett's for point stabilization.

Abstract

Brockett's necessary condition yields a test to determine whether a system can be made to stabilize about some operating point via continuous, purely state-dependent feedback. For many real-world systems, however, one wants to stabilize sets which are more general than a single point. One also wants to control such systems to operate safely by making obstacles and other "dangerous" sets repelling. We generalize Brockett's necessary condition to the case of stabilizing general compact subsets having a nonzero Euler characteristic in general ambient state spaces (smooth manifolds). Using this generalization, we also formulate a necessary condition for the existence of "safe" control laws. We illustrate the theory in concrete examples and for some general classes of systems including a broad class of nonholonomically constrained Lagrangian systems. We also show that, for the special case of stabilizing a point, the specialization of our general stabilizability test is stronger than Brockett's.