Geometric-Topological Limitations and Capabilities for Stability and Safety
Abstract
Achieving stability and safety by feedback control are fundamental and somewhat-dual aims of control theory. They enable complex systems to autonomously regulate themselves to desired equilibrium states or more general behaviors, and to avoid dangerous states. Stability and safety are often easiest to achieve through time-varying or discontinuous control schemes. However, when possible, achieving stability and safety via smooth static feedback control has several practical and theoretical advantages. Thus, it is important to have the capability to determine whether it is possible to achieve this for a given system and desired behavior, or specified set of dangerous states. For a broad class of systems, a necessary and sufficient condition for achievability is the existence of a suitable Lyapunov function. However, using this condition to certify achievability of stability or safety typically requires a numerical or ad hoc analytical search for such a function. Moreover, this condition cannot be used to certify non-achievability directly, since it is impossible to check each of the uncountably infinitely many candidate Lyapunov functions to confirm their unsuitability. Thus, other techniques are needed to certify the (non-)achievability of stability and safety. Most Lyapunov-free techniques in the literature to certify achievability have only been developed for local stabilization of equilibrium states. On the other hand, Lyapunov-free techniques in the literature to certify non-achievability are based on topological obstructions that are fairly coarse, and hence lead to many false positives . In this project, we propose to fill these gaps by exploiting the fundamentally geometric-topological nature of the problem to generate practicable techniques to provide (non-)achievability certificates for stability and safety, with applications and robustness guarantees out of the scope of existing techniques.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Feb 06, 2025
- Source ID
- FA95502410299
Entities
People
- Matthew D. Kvalheim
Organizations
- Air Force Office of Scientific Research
- United States Air Force
- University of Maryland, Baltimore County