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Continuity Conditions for Piecewise Quadratic Functions on Simplicial Conic Partitions are Equivalent

Magne Erlandsen, Tomas Meijer, W. P. M. H. Heemels, Sebastiaan van den Eijnden

TL;DR

The paper addresses the problem of enforcing continuity for piecewise quadratic Lyapunov functions on partitions of the state space used in conwise linear systems. It proves that, on simplicial conic partitions, several existing continuity conditions—ranging from direct boundary equalities to parametrizations and LMIs—are formally equivalent, ensuring no additional conservatism is introduced by the chosen form. A technical lemma related to a non-strict projection result underpins the proof, and the authors illustrate the results with numerical examples that also contrast equality-based and parametrized approaches. The findings enable practitioners to select the most convenient continuity condition for PWQ Lyapunov analysis based on practicality and numerical properties, simplifying stability analysis of PWL systems.

Abstract

Analysis of continuous-time piecewise linear systems based on piecewise quadratic (PWQ) Lyapunov functions typically requires continuity of these functions over a partition of the state space. Several conditions for guaranteeing continuity of PWQ functions over state space partitions can be found in the literature. In this technical note, we show that these continuity conditions are equivalent over so-called simplicial conic partitions. As a consequence, the choice of which condition to impose can be based solely on practical considerations such as specific application or numerical aspects, without introducing additional conservatism in the analysis.

Continuity Conditions for Piecewise Quadratic Functions on Simplicial Conic Partitions are Equivalent

TL;DR

The paper addresses the problem of enforcing continuity for piecewise quadratic Lyapunov functions on partitions of the state space used in conwise linear systems. It proves that, on simplicial conic partitions, several existing continuity conditions—ranging from direct boundary equalities to parametrizations and LMIs—are formally equivalent, ensuring no additional conservatism is introduced by the chosen form. A technical lemma related to a non-strict projection result underpins the proof, and the authors illustrate the results with numerical examples that also contrast equality-based and parametrized approaches. The findings enable practitioners to select the most convenient continuity condition for PWQ Lyapunov analysis based on practicality and numerical properties, simplifying stability analysis of PWL systems.

Abstract

Analysis of continuous-time piecewise linear systems based on piecewise quadratic (PWQ) Lyapunov functions typically requires continuity of these functions over a partition of the state space. Several conditions for guaranteeing continuity of PWQ functions over state space partitions can be found in the literature. In this technical note, we show that these continuity conditions are equivalent over so-called simplicial conic partitions. As a consequence, the choice of which condition to impose can be based solely on practical considerations such as specific application or numerical aspects, without introducing additional conservatism in the analysis.

Paper Structure

This paper contains 9 sections, 4 theorems, 53 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Cones and partitions
  5. Technical Lemma
  6. Continuity Conditions
  7. Proof of Theorem \ref{['thm:1']}
  8. Examples
  9. Conclusion

Key Result

Lemma 1

Let $\boldsymbol{U}\in\mathbb{R}^{m\times n}$, $\boldsymbol{V}\in\mathbb{R}^{p\times n}$ and $\boldsymbol{Q}\in\mathbb{S}^n$. Consider the following statements: Then, item:P1 holds if and only if item:P2 and item:P3 hold.

Figures (3)

  • Figure 1: Example illustrating the difference between the regions considered in the first two items of Theorem \ref{['thm:1']}. The figure includes the distinct extremal rays of two example cones, $\mathcal{S}_1$ and $\mathcal{S}_2$ (in red and green), their shared extremal rays (in blue), their boundary region, $\mathcal{S}_1 \cap \mathcal{S}_2$ (also in blue), and the extended boundary region, $\mathop{\mathrm{im}}\nolimits Z_{ij}$ (in cyan).
  • Figure 2: Contour plot of the PWQ function \ref{['eq:ex1']} in example \ref{['ex:1']}.
  • Figure 3: Position $z$ of the system analysed in example \ref{['ex:ex2']}, with initial condition $(z(0), \dot z(0)) = (-1, 0)$.

Theorems & Definitions (7)

  • Lemma 1
  • proof
  • Corollary 1
  • Corollary 2
  • Theorem 1
  • Example 1
  • Example 2