Generalize feedback
Work through composition operators
The expanded setting retains the classical problem while making its structural ingredients easier to isolate.
Paper project · Nonlinear control
Continuous Feedback Stabilization of Nonlinear Control Systems by Composition Operators
Overview
Build feedback from a section of the vector field.
The paper enlarges the usual feedback framework using composition operators and obtains a characterization of continuous feedback stabilizability without starting from a control Lyapunov function.
The central bridge is a local section of the system vector field. Under the appropriate local invertibility, the section determines both an associated autonomous field and a feedback law. Continuous stabilizability of the original control system is then equivalent to stability of that associated system, and the construction yields a universal formula for the stabilizing feedbacks.
Choose a local section
Invert the vector field locally
A section α selects a state-control pair for each nearby velocity and satisfies f∘α=id.
Reduce to stability
Recover feedback from the section
When the state component of the section is locally invertible, its inverse defines an associated field and its control component supplies the feedback law.
From a local section to a feedback law
Choose stable dynamics, then recover the control.
The paper’s polynomial example makes the factorization concrete. Compare the uncontrolled plant, the associated stable field, and the original plant under the feedback obtained from the section; the last two trajectory families agree exactly.
Citation
Use or reference this work.
The article is published open access; the arXiv version is also linked above.
BibTeX citation
@article{christopherson2022composition,
title = {Continuous Feedback Stabilization of Nonlinear Control Systems by Composition Operators},
author = {Christopherson, Bryce A. and Mordukhovich, Boris S. and Jafari, Farhad},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
volume = {28},
eid = {30},
pages = {1--22},
year = {2022},
doi = {10.1051/cocv/2022022}
}