Analog Data-Driven Theory and Estimation of the Region of Attraction Using Sampled-Data

TL;DR

The paper addresses estimating the region of attraction of a locally exponentially stable equilibrium for unknown nonlinear autonomous systems using analog trajectory data, without sampling. It introduces a data-driven optimization to fit a linear vector field along a trajectory: minimize $J(\bar{A}, x_0)=\int_0^\infty \| f(s(t,x_0)) - \bar{A} s(t,x_0)\|^2 \mathrm{d}t$, under the persistency of excitation condition $\Gamma_2 \succ 0$, yielding the global minimizer $\hat{A}(x_0)= \Gamma_1 \Gamma_2^{-1}$. The authors prove strict convexity of $J$ when $\Gamma_2 \succ 0$, continuity of $\hat{A}(x_0)$ near the origin, and Lyapunov stability (and Hurwitzness under observability) of the minimizer, providing a solid foundation for a data-driven ROA estimator. A novel geometric-flow algorithm refines an initial ROA guess by evolving the boundary using a residual energy $E(x_0)$ and a tanh mapping, guaranteeing convergence to the exact ROA boundary under suitable growth and unboundedness assumptions. Simulations on Vand der Pol with a bounded ROA, a system with unbounded ROA, and a counterexample not satisfying the growth condition illustrate the method's accuracy and the need for conservative adjustments in challenging cases.

Abstract

The contributions of this technical note are twofold. Firstly, we formulate an optimization problem to obtain a linear representation of a nonlinear vector field based on a system's trajectory. We also prove that its cost function is strictly convex, given the trajectory is persistently exciting. Under certain observability conditions, we provide results that guarantee the Hurwitz stability of the global minimizer. Secondly, we present a novel algorithm based on point-wise geometric flows to estimate the boundary of the region of attraction. We show that the algorithm converges to the exact boundary of the region of attraction under certain assumptions on the system dynamics. Finally, we validate the results using simulations on various nonlinear autonomous systems.

Figures (3)

• Figure 1: The evolution of the boundary of the ROA is shown, where $\partial M_i$ represents the boundary submanifold at the $i^{\mathrm{th}}$ iteration. The algorithm is initialized with the curve $\partial M_0$ and is shown to asymptotically converge to $\partial M_\infty$, which is the true boundary of the ROA.
• Figure 2: The evolution of $\partial M_i$ for a system with an unbounded ROA. The solid black curves represent the actual ROA. Since the algorithm was terminated in finite time, a conservative estimate $\partial M_{450}$ is obtained.
• Figure 3: The system has an unbounded ROA, but assumption \ref{['eqn: condition on growth']} is not satisfied. Hence, only a conservative estimate can be obtained by setting $\gamma=0.7$. The actual boundary of the ROA is shown as solid black curves.

Theorems & Definitions (19)

• Lemma 1
• proof
• Theorem 1
• proof
• Remark 1
• Remark 2: Gradient Descent in $\mathbb{R}^{n\times n}$
• Remark 3: Linear Systems
• Theorem 2
• proof
• Remark 4: Finite Length Trajectories