Tracking Through Decoupling Singularities: A Singularity-Robust Homotopy-Continuation Extension of Feedback Linearization
2026-07-11 • Robotics
Robotics
AI summaryⓘ
The authors address a problem in controlling certain complex systems where usual methods fail at special points called decoupling singularities, causing control signals to become infinite. They create a new control approach that can safely guide the system through these tricky points using a mathematical technique called homotopy continuation, ensuring the control stays bounded. Their method works like the traditional one when away from singularities, switches smoothly during crossings, and can handle different types of system complexities. They also explore specific behaviors near singularities and demonstrate their approach through simulations on robotic arms and power converters.
Input-output feedback linearizationDecoupling singularitiesControl-affine systemsHomotopy continuationMoore-Penrose pseudoinverseRelative degreeTrajectory trackingFilippov sliding modeWhitney foldsNonlinear control
Authors
Alex Borisevich
Abstract
Input--output feedback linearization fails at decoupling singularities, where the decoupling matrix loses rank, the relative degree is lost, and the linearizing control becomes unbounded. This paper develops a singularity-robust trajectory-tracking controller for square nonlinear control-affine systems that tracks through isolated decoupling singularities with bounded control. The method recasts tracking as real-time arc-length homotopy continuation, equivalently a continuous-time Newton/Davidenko flow, and replaces the inverse decoupling matrix by the least-norm Moore--Penrose solution of an augmented matrix $A=[Λ\mid b]$, where $b$ is the homotopy direction. A transversality condition $w^T b \ne 0$, with $w$ in the left null space of the decoupling matrix, keeps the augmented matrix full row rank through a generic rank-one loss. The resulting flow agrees with feedback linearization away from the singular set, tracks with $O(1/k)$ error, and re-locks after each crossing. The theory also characterizes the reflection-versus-branch-crossing dichotomy at Whitney folds and relates the reflection case to a Filippov sliding mode. Extensions cover dynamic relative-degree-one minimum-phase systems and arbitrary relative degree via filtered-error reduction. Simulations include a redundant 2-DOF manipulator, relative-degree-one and relative-degree-two plants, and a dual-active-bridge series-resonant DC/DC converter, where the method performs bounded inversion across buck/boost and resonance singularities while preserving zero-voltage soft switching.