提出新方法让控制系统在奇异点处仍能稳定跟踪，控制量始终有界。
Tracking Through Decoupling Singularities: A Singularity-Robust Homotopy-Continuation Extension of Feedback Linearization
- 用连续弧长同伦法替代传统反馈线性化，避免奇异点失效
- 通过最小范数解实现控制有界，跟踪误差为O(1/k)
- 适用于机械臂、电源转换等复杂系统，可处理多种奇异情况
输入输出反馈线性化在去耦奇异点处会失效，此时去耦矩阵秩降、相对阶消失，线性化控制量无界。本文针对平方非线性控制仿射系统，提出一种奇异点鲁棒的轨迹跟踪控制器，可在孤立去耦奇异点处持续跟踪且控制量有界。方法将跟踪问题重构为实时弧长同伦延续，等价于连续时间牛顿/Davidenko流，用增广矩阵 $A=[Λackslash b]$ 的最小范数Moore-Penrose解替代逆去耦矩阵，其中 $b$ 为同伦方向。通过左零空间向量 $w$ 满足横截条件 $w^T b \ne 0$，确保增广矩阵在一般一阶秩损下仍满行秩。所得流在奇异集外与反馈线性化一致，跟踪误差为 $O(1/k)$，每次穿越后重新锁定。理论还揭示了惠特尼折叠处的反射与分支穿越二分性，并将反射情形关联至Filippov滑模。扩展涵盖动态相对阶一的最小相位系统及任意相对阶的滤波误差降低方法。仿真包括冗余2-自由度机械臂、相对阶一与相对阶二系统，以及双主动桥串联谐振直流/直流变换器，该方法在降压/升压和谐振奇异点处实现有界逆运算，同时保持零电压软开关特性。
原文摘要 · Abstract (English)
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.
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