A numerical Koopman-based framework to estimate regions of attraction for general vector fields

Abstract

In this paper, we develop a comprehensive framework to estimate regions of attraction of equilibria for dynamics associated with general vector fields. This framework combines Koopman operator-based methods with rigorous validation techniques. A candidate Lyapunov function is constructed with approximated Koopman eigenfunctions and further validated through polynomial approximation, either with SOS-based techniques or with a worst-case approach using an adaptive grid. The framework is general, not only since it is adapted to non-polynomial vector fields, but also since the Koopman operator can be approximated with general bases yielding non-polynomial Lyapunov functions. The performance of the method is illustrated with several numerical examples.

Figures (6)

• Figure 1: The flowchart shows how the Lyapunov candidate is constructed depending on the nature of the vector field and of the basis functions (solid lines). Polynomial Lyapunov functions are obtained only when the basis functions are monomials. Dashed lines represent the polynomial approximations (Taylor or minimax) that are applied to the vector field or to the Lyapunov function.
• Figure 2: Hypothetical example of an adaptive grid obtained through our proposed method. Blue cells contain the level set $\partial\Omega_{\gamma_1}$ or $\partial\Omega_{\gamma_2}$ since \ref{['crossing-cell']} holds. In contrast, the green cell does not satisfy \ref{['crossing-cell']}, although it intersects the level set $\partial\Omega_{\gamma_1}$. The hatched cells satisfy \ref{['hatched']}.
• Figure 3: Inner approximations of the ROA are obtained for the dynamics (\ref{['sys1']}). The boundary of the ROA is in red. (Left) The approximation of the ROA is obtained with monomial basis functions up to degree 3 and 5. The approximation obtained with SOS-based validation is in black while the approximation obtained with the adaptive grid is in yellow (monomials up to degree 3) and blue (monomials up to degree 5). The inset shows the boundary of the estimated ROA near the equilibrium, when it is computed with the adaptive grid. (Right) The approximation of the ROA is obtained with 25 Gaussian radial basis functions with $\mathcal{D}_c = [-1,1]^2$ and $\eta = 0.9$. The approximation obtained with SOS-based validation (performed with a minimax polynomial approximation of degree 12) is in black while the approximation obtained with the adaptive grid is in yellow.
• Figure 4: Inner approximations of the ROA are obtained for the dynamics (\ref{['sys2']}). The boundary of the ROA is shown in red. (Left) The approximation of the ROA is obtained with monomial basis functions of degree 5, for a Taylor approximation of the vector field of order 5 (with $c_1=c_2 = 0.7)$) (dashed line) and of order 15 (with $c_1=c_2=2\times 10^{-4}$) (solid line). (Right) The approximation of the ROA is obtained with 9 Gaussian radial basis functions (with $\mathcal{D}_c = [-1,1]^2$ and $\eta = 0.1$), for a Taylor approximation of the vector field of order 5 and 15. The Lyapunov function is computed with the approximated vector field. The orthogonal projection is computed over the set $[-0.1,0.1]^2 \subset \mathbb{X}$.
• Figure 5: Inner approximations of the ROA are obtained for the dynamics (\ref{['sys3']}). Black curves are the smallest and largest level sets of the Lyapunov function obtained with the minimax approximation of the vector field (polynomials up to degree 12, approximation error set to 0.028). Blue curves are the smallest and largest level sets of the Lyapunov function obtained with a 5th order Taylor expansion of the second component of the vector field (with $c_2=1.6\times10^3$). In both cases, basis monomials are used up to degree 5. By combining both results, the sublevel set $\Omega_{\gamma_{2}^{\mathrm{rem}}}$ provides an inner approximation of the ROA, and of the set of trajectories that remain in $\mathbb{X}$ (dashed red curves).

Theorems & Definitions (5)

• Proposition 1
• proof
• Definition 1
• Remark 1
• Remark 2