Tracking Through Decoupling Singularities: A Singularity-Robust Homotopy-Continuation Extension of Feedback Linearization
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| b], where b is the homotopy direction. A transversality condition wT b ≠ 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. The core results are formalized and machine-checked in the Lean proof assistant, including the multi-parameter (higher-corank) repair criterion, the uniqueness of the sliding motion at the fold, and the machine-checked treatment of Filippov solutions. 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.
Related
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BibTeX
@misc{borisevich2026tracking,
title = {{Tracking Through Decoupling Singularities: A Singularity-Robust Homotopy-Continuation Extension of Feedback Linearization}},
author = {Alex Borisevich},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.10436},
eprint = {2607.10436},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.10436},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Tracking Through Decoupling Singularities: A Singularity-Robust Homotopy-Continuation Extension of Feedback Linearization
AU - Alex Borisevich
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2607.10436
UR - https://arxiv.org/abs/2607.10436
AB - 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| b], where b is the homotopy direction. A transversality condition w^T b ≠ 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. The core results are formalized and machine-checked in the Lean proof assistant, including the multi-parameter (higher-corank) repair criterion, the uniqueness of the sliding motion at the fold, and the machine-checked treatment of Filippov solutions. 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.
ER -