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Uniform global stability of switched nonlinear systems in the Koopman operator framework

Christian Mugisho Zagabe, Alexandre Mauroy

Abstract

In this paper, we provide a novel solution to an open problem on the global uniform stability of switched nonlinear systems. Our results are based on the Koopman operator approach and, to our knowledge, this is the first theoretical contribution to an open problem within that framework. By focusing on the adjoint of the Koopman generator in the Hardy space on the polydisk (or on the real hypercube), we define equivalent linear (but infinite-dimensional) switched systems and we construct a common Lyapunov functional for those systems, under a solvability condition of the Lie algebra generated by the linearized vector fields. A common Lyapunov function for the original switched nonlinear systems is derived from the Lyapunov functional by exploiting the reproducing kernel property of the Hardy space. The Lyapunov function is shown to converge in a bounded region of the state space, which proves global uniform stability of specific switched nonlinear systems on bounded invariant sets.

Uniform global stability of switched nonlinear systems in the Koopman operator framework

Abstract

In this paper, we provide a novel solution to an open problem on the global uniform stability of switched nonlinear systems. Our results are based on the Koopman operator approach and, to our knowledge, this is the first theoretical contribution to an open problem within that framework. By focusing on the adjoint of the Koopman generator in the Hardy space on the polydisk (or on the real hypercube), we define equivalent linear (but infinite-dimensional) switched systems and we construct a common Lyapunov functional for those systems, under a solvability condition of the Lie algebra generated by the linearized vector fields. A common Lyapunov function for the original switched nonlinear systems is derived from the Lyapunov functional by exploiting the reproducing kernel property of the Hardy space. The Lyapunov function is shown to converge in a bounded region of the state space, which proves global uniform stability of specific switched nonlinear systems on bounded invariant sets.
Paper Structure (21 sections, 15 theorems, 115 equations, 3 figures)

This paper contains 21 sections, 15 theorems, 115 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Stability of switched systems
  4. Uniform stability
  5. Lie-algebraic conditions in the linear case
  6. Lie-algebraic condition in the nonlinear case
  7. Koopman operator approach to dynamical systems
  8. Koopman operator
  9. Koopman operator on reproducing kernel Hilbert spaces
  10. Switched Koopman systems and Lie-algebraic conditions
  11. Main result
  12. A first remark on the existence of the common invariant flag
  13. A common Lyapunov function for switched nonlinear systems
  14. Corollaries of the main result in the Hardy space context
  15. Global uniform asymptotic stability for polynomial vector fields
  16. ...and 6 more sections

Key Result

Theorem 2.4

Suppose that $X \subseteq \mathbb{R}^n$ is compact and forward-invariant with respect to the flow induced by the subsystems (eq:switch). The switched system (eq:switch) is GUAS on $X$ if and only if all subsystems share a CLF on $X$.

Figures (3)

  • Figure 1: For the switched system \ref{['eq:illustration_1']}, the region of attraction is estimated with the quadratic CLF $V(x_1,x_2)=x_1^2+x_2^2$ (magenta curve), with the CLF \ref{['clf_example_2']} truncated to the total degree 4 (i.e. $k_{max}=5$) (red curve), and with \ref{['clf_example_2']} truncated to the total degree 8 (i.e. $k_{max}=14$) (green curve).
  • Figure 2: The switched system is shown to be GUAS on a hypercube of edge length $2\rho$ that depends on the parameter $\eta$.
  • Figure 3: For the switched system \ref{['eq:illust2']} with $\eta=1$, the region of attraction is estimated with the quadratic CLF $V(x_1,x_2)=x_1^2+x_2^2$ (magenta curve), with the CLF \ref{['clf_example_3']} truncated to the total degree 4 ($k_{max}=5$) (red curve), and with \ref{['clf_example_3']} truncated to the total degree 8 ($k_{max}=14$) (green curve).

Theorems & Definitions (43)

  • Definition 2.1: Switched system
  • Definition 2.2: Uniform stability
  • Definition 2.3: Common Lyapunov function liberzon2003switching
  • Theorem 2.4: mancilla2000condition
  • Definition 2.5: Solvable Lie algebra
  • Theorem 2.6: liberzon1999stability
  • Definition 2.7: Invariant flag
  • Proposition 2.8: zagabe2021switched
  • Definition 2.9: Koopman semigroup lasota1998chaos
  • Definition 2.10: Koopman generator lasota1998chaos
  • ...and 33 more