This paper addresses the fundamental question of determining the minimum number of distinct control laws required for global controllability of nonlinear systems that exhibit singularities in their feedback linearising controllers. We introduce and rigorously prove the (k+1)-Controller Lemma, which establishes that for an nth order single-input single-output nonlinear system with a singularity manifold parameterised by k algebraically independent conditions, exactly k+1 distinct control laws are necessary and sufficient for complete state-space coverage. The sufficiency proof is constructive, employing the approximate linearisation methodology together with transversality arguments from differential topology. The necessity proof proceeds by contradiction, using the Implicit Function Theorem, a dimension-counting argument and structural constraints inherent to the approximate linearisation framework. The result is validated through exhaustive analysis of the ball-and-beam system, a fourth-order mechanical system that exhibits a two-parameter singularity at the third output derivative.

翻译：本文研究了在存在反馈线性化控制器奇异性的非线性系统中，实现全局可控性所需的不同控制律的最小数量这一基本问题。我们引入并严格证明了(k+1)控制器引理，该引理指出：对于由k个代数独立条件参数化奇异流形的n阶单输入单输出非线性系统，恰好需要k+1个不同的控制律才能实现状态空间的完全覆盖。充分性证明是构造性的，采用近似线性化方法结合微分拓扑中的横截性论证。必要性证明通过反证法进行，利用隐函数定理、维数计数论证以及近似线性化框架固有的结构约束条件。该结果通过球杆系统的详尽分析得到验证，这是一个四阶机械系统，其第三输出导数处存在双参数奇异性。