AI 中文总结
针对方形非线性控制仿射系统在解耦奇点处反馈线性化失效问题，提出将跟踪转化为实时弧长同伦延拓的奇点鲁棒轨迹跟踪控制器，通过特定矩阵和条件实现有界控制跟踪，经仿真验证了其有效性。
AI 中文摘要
输入-输出反馈线性化在解耦奇点处失效，此时解耦矩阵失去秩，相对度丧失，线性化控制变得无界。本文为方形非线性控制仿射系统开发了一种奇点鲁棒轨迹跟踪控制器，能通过孤立解耦奇点进行有界控制跟踪。该方法将跟踪重铸为实时弧长同伦延拓，用增广矩阵\(A=[\Lambda\mid b]\)的最小范数摩尔-彭罗斯解替代逆解耦矩阵，其中\(b\)是同伦方向。通过横向条件\(w^T b \ne 0\)保持增广矩阵满行秩。所得流在远离奇异集时与反馈线性化一致，跟踪误差为\(O(1/k)\)，每次穿越后重新锁定。理论还刻画了惠特尼折叠处的反射与分支穿越二分法，并将反射情况与菲利波夫滑模相关联。扩展涵盖动态相对度为一的最小相位系统和通过滤波误差减少的任意相对度。仿真包括冗余二自由度机械手、相对度为一和二的装置以及双有源桥串联谐振DC/DC变换器，该方法在保持零电压软开关的同时，能在降压/升压和共振奇点处进行有界反转。
英文摘要
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. 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.
CommentsPython code to reproduce all numerical results is included as ancillary files together with Lean formalization
