On the Minimum Number of Control Laws for Nonlinear Systems with Input-Output Linearisation Singularities

TL;DR

This paper establishes the exact number of control laws needed for global controllability in nonlinear systems with input-output linearisation singularities, using a rigorous mathematical framework and validating with a mechanical example.

Contribution

It introduces and proves the (k+1)-Controller Lemma, determining the minimal control laws for systems with singularities, a novel theoretical result in nonlinear control.

Findings

Exactly k+1 control laws are necessary and sufficient for systems with a k-parameter singularity.

The proof combines approximate linearisation, transversality, and the Implicit Function Theorem.

Validation on the ball-and-beam system confirms the theoretical results.

Abstract

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.