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

Nikolaos D. Tantaroudas

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.

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

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.
Paper Structure (40 sections, 8 theorems, 28 equations, 4 figures, 1 table)

This paper contains 40 sections, 8 theorems, 28 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Literature review and motivation
  3. Contributions
  4. Paper organisation
  5. Mathematical Preliminaries
  6. Nonlinear control systems
  7. Lie derivatives and Lie brackets
  8. Relative degree and input--output linearisation
  9. Singularities and the singularity manifold
  10. Elements of differential topology
  11. The Ball-and-Beam System: Lagrangian Derivation
  12. Physical description and coordinate system
  13. Kinetic and potential energy
  14. Euler--Lagrange equations
  15. State-space representation
  16. ...and 25 more sections

Key Result

Lemma 1

If $M \pitchfork N$ in $\mathbb{R}^n$, then $M \cap N$ is a smooth submanifold with $\mathrm{codim}(M \cap N) = \mathrm{codim}(M) + \mathrm{codim}(N)$. In particular, if $\mathrm{codim}(M) + \mathrm{codim}(N) > n$, then generically $M \cap N = \emptyset$.

Figures (4)

  • Figure 1: Schematic of the ball-and-beam system. The beam rotates about a fixed pivot with angle $\theta$ from horizontal. The ball position $r$ is the signed distance from pivot to ball centre along the beam. The torque $\tau$ is the control input.
  • Figure 2: Hybrid controller results ($A = 0.4$ m, $|\theta| \leq 30^\circ$). Top row (left to right): position tracking ($r(t)$ vs $y_d(t)$); tracking error convergence; beam angle $\theta$ with $\pm30^\circ$ constraint boundaries. Bottom row: control input $u$; control coefficient $a(x) = 2Bx_1x_4$ showing periodic zero-crossings at singularity encounters; active controller selection indicating Law 1 (away from singularities), Law 2 (near one singularity) and Law 3 (near intersection $\mathcal{S}_1 \cap \mathcal{S}_2$).
  • Figure 3: Singularity manifold analysis. Top row: distance to $r = 0$ singularity ($\mathcal{S}_1$) and distance to $\dot{\theta} = 0$ singularity ($\mathcal{S}_2$) for trajectories starting near different components. Bottom row: control coefficient magnitude $|a(x)|$ and distance to Control Law 2 singularity $|\cos(\theta)|$, confirming the necessity of multiple controllers.
  • Figure 4: State-space partition projected onto the $(x_1, x_4)$ plane. $\mathcal{S} = \mathcal{S}_1 \cup \mathcal{S}_2$ (red, blue) separates the state space. Law 1 operates in the green regions. The pure parts $X_1$ and $X_2$ require Laws 2 and 3.

Theorems & Definitions (26)

  • Definition 1: Lie Derivative
  • Definition 2: Lie Bracket
  • Definition 3: Relative Degree
  • Definition 4: Singularity Manifold
  • Definition 5: $k$-Parameter Singularity
  • Remark 1
  • Definition 6: Transversality
  • Lemma 1: Codimension of Transverse Intersections Hirsch1976
  • Theorem 2: Implicit Function Theorem KrantzParks2002
  • Proposition 3: Ball-and-Beam has a 2-Parameter Singularity
  • ...and 16 more