Singularities in feedback-linearizing controllers are treated as failure points — regions of state space where the control law breaks down, the denominator hits zero, the system becomes momentarily uncontrollable. The standard response: avoid them, patch around them, design trajectories that never pass through them. The singularity is the problem.
Tantaroudas (arXiv:2603.18947) reframes singularities as a counting tool. For an nth-order single-input single-output nonlinear system with k algebraically independent singularity conditions, exactly k+1 distinct control laws are both necessary and sufficient for complete state-space coverage. Not “at least k+1” — exactly k+1. The singularity structure doesn't just indicate where one controller fails; it precisely determines the minimum architecture of the complete control system.
The sufficiency proof uses approximate linearization with transversality arguments from differential topology: k+1 controllers can always be arranged so their valid regions cover the entire state space, with each controller's singularity manifold falling within another controller's valid region. The necessity proof uses contradiction via the Implicit Function Theorem: k controllers cannot cover a space punctured by k independent singularity conditions, because the codimension count forces gaps.
Validated on a ball-and-beam system with a two-parameter singularity, requiring exactly 3 controllers. Not 2 (insufficient), not 4 (redundant). Three.
The structural insight: the defect tells you the cure. The number of places a controller fails is not a measure of how broken the system is — it's a specification of how many controllers you need. The obstacle is the design parameter. What looks like a catalogue of failure modes is actually a minimal architecture prescription.
Tantaroudas, "On the Minimum Number of Control Laws for Nonlinear Systems with Input-Output Linearisation Singularities," arXiv:2603.18947 (2026).