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Generalized Lyapunov conditions for k-contraction: analysis and feedback design

Andreu Cecilia, Samuele Zoboli, Daniele Astolfi, Ulysse Serres, Vincent Andrieu

Abstract

Recently, the concept of k-contraction has been introduced as a promising generalization of contraction for dynamical systems. However, the study of k-contraction properties has faced significant challenges due to the reliance on complex mathematical objects called matrix compounds. As a result, related control design methodologies have yet to appear in the literature. In this paper, we overcome existing limitations and propose new sufficient conditions for k-contraction which do not require matrix compounds computation. Notably, these conditions are also necessary in the linear time-invariant framework. Leveraging on these findings, we propose a feedback design methodology for both the linear and the nonlinear scenarios which can be used to enforce k-contractivity properties on the closed-loop dynamics.

Generalized Lyapunov conditions for k-contraction: analysis and feedback design

Abstract

Recently, the concept of k-contraction has been introduced as a promising generalization of contraction for dynamical systems. However, the study of k-contraction properties has faced significant challenges due to the reliance on complex mathematical objects called matrix compounds. As a result, related control design methodologies have yet to appear in the literature. In this paper, we overcome existing limitations and propose new sufficient conditions for k-contraction which do not require matrix compounds computation. Notably, these conditions are also necessary in the linear time-invariant framework. Leveraging on these findings, we propose a feedback design methodology for both the linear and the nonlinear scenarios which can be used to enforce k-contractivity properties on the closed-loop dynamics.
Paper Structure (38 sections, 24 theorems, 130 equations, 5 figures)

This paper contains 38 sections, 24 theorems, 130 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on $k$-contraction
  3. Definition of $\mathit{k}$-contraction
  4. Infinitesimal $\mathit{k}$-contraction
  5. Sufficient conditions based on additive matrix compounds
  6. Limitations of matrix compound-based conditions
  7. $k$-contraction for linear systems
  8. Generalized Lyapunov necessary and sufficient conditions
  9. Computational burden of Theorem \ref{['thm:k_contraction_LMI_cLTI']}
  10. $\mathit{k}$-order stabilizability
  11. $\mathit{k}$-contractive feedback design
  12. $k$-contraction for nonlinear systems
  13. Sufficient conditions
  14. Relaxing conditions for the planar case
  15. $\mathit{k}$-contractive feedback design
  16. ...and 23 more sections

Key Result

Lemma 1

If system eqn:original_system is $k$-contractive for an integer $1\leqslant k<n$, then it is also $(k+1)$-contractive.

Figures (5)

  • Figure 1: Flow of a $2$-contractive system. The initial submanifold of initial conditions is described by $\Phi$. The volume of this submanifold decreases exponentially along the trajectories of the system.
  • Figure 2: Flow of an infinitesimally $3$-contractive system.
  • Figure 3: Number of variables to be estimated by Theorem \ref{['thm:demidovich']} (dashed) and by Theorem \ref{['thm:k_contraction_LMI_cLTI']} (solid) in function of $k$. Colors refer to different $n$.
  • Figure 4: Evolution of two trajectories of the system \ref{['eqn:Rossler']}. The first (blue) has an initial condition $[0.1,0.1,0]$ and the second (red) $[0.099,0.1,0]$.
  • Figure 5: Evolution of three trajectories of the system \ref{['eqn:Rossler_mod']}. The first (blue) has an initial condition $[0.2,0.5,0]$, the second (red) $[-0.3,-0.3,-0.5]$ and the third (yellow) $[0.2,-0.5,-0.3]$.

Theorems & Definitions (48)

  • Remark 1
  • Remark 2
  • Definition 1: $\boldsymbol k$-contraction
  • Lemma 1
  • Lemma 2
  • Definition 2: Multiplicative Compound bar2023compound
  • Definition 3: Infinitesimal $\boldsymbol k$-contraction
  • Theorem 1
  • Definition 4: Additive Compound bar2023compound
  • Theorem 2
  • ...and 38 more