Learning Region of Attraction for Nonlinear Systems

Abstract

Estimating the region of attraction (ROA) of general nonlinear autonomous systems remains a challenging problem and requires a case-by-case analysis. Leveraging the universal approximation property of neural networks, in this paper, we propose a counterexample-guided method to estimate the ROA of general nonlinear dynamical systems provided that they can be approximated by piecewise linear neural networks and that the approximation error can be bounded. Specifically, our method searches for robust Lyapunov functions using counterexamples, i.e., the states at which the Lyapunov conditions fail. We generate the counterexamples using Mixed-Integer Quadratic Programming. Our method is guaranteed to find a robust Lyapunov function in the parameterized function class, if exists, after collecting a finite number of counterexamples. We illustrate our method through numerical examples.

Revisiting PGD Attacks for Stability Analysis of High-Dimensional Nonlinear Systems and Perception-Based Control

“…To see this, we apply the Lagrange multiplier theorem to (10) and obtain −∇L T (ξ k ) + 2λC Cξ = 0 and Cξ = r, where λ is the Lagrange multiplier. This leads to (11). The above PGD update rule (10) can be viewed as a special case of the so-called Frank-Wolfe (or conditional gradient) algorithm [23].…”

“…There is also a connection between (11) and the alignment condition in the controls literature [24], since the initial condition can be viewed as an input applied at t = 0. For p = 2, the implementation of ( 11) is straightforward.…”

“…Third, in general, the control action may depend on the past output measurements, and thus the coupling of system states at different time steps can be complicated. The first two issues will cause trouble for existing Lyapunov-based ROA analysis methods using semidefinite programming [5]- [8] or mixed-integer programs [9]- [11]. Due to the last issue, the methods of Lyapunov neural networks [12]- [14] or other stability certificate learning methods [15]- [17] may also be not applicable since these methods typically require the control action to depend on the current state.…”

Revisiting PGD Attacks for Stability Analysis of Large-Scale Nonlinear Systems and Perception-Based Control

“…Third, in general, the control action may not just be a function of the current state, and the coupling of system states at different time steps can be complicated. The first two issues will cause trouble for existing Lyapunov-based ROA analysis methods using semidefinite programming (Yin et al, 2021;Hu et al, 2020;Jin and Lavaei, 2020;Aydinoglu et al, 2021) or mixed-integer programs (Chen et al, 2020(Chen et al, , 2021Dai et al, 2021). Due to the last issue, the methods of Lyapunov neural networks (Richards et al, 2018;Chang et al, 2019; or other stability certificate learning methods (Kenanian et al, 2019;Giesl et al, 2020;Ravanbakhsh and Sankaranarayanan, 2019) may also be not applicable since these methods typically require the control action to depend on the current state.…”

Neural Lyapunov Control of Unknown Nonlinear Systems with Stability Guarantees

“…The other is to encode the verification problem into an SMT problem. In the first category, [13,14] find a robust Lyapunov function for an uncertain nonlinear system or a hybrid system which can be partly approximated by piecewise linear neural networks with the help of a counterexample-guided method using mixed integer quadratic programming (MIQP), and the corresponding robust ROA is estimated. With a similar technique, [15] synthesizes a neural-network controller and Lyapunov function simultaneously to certify the dynamical system's stability by converting the dynamics, the Lyapunov function, and the controller into piecewise linear functions and solving it with mixed-integer linear programming (MILP).…”