Estimation of the regions of attraction for autonomous nonlinear systems

Abstract

A methodology for estimating the region of attraction for autonomous nonlinear systems is developed. The methodology is based on a proof that the region of attraction can be estimated accurately by the zero sublevel set of an implicit function which is the viscosity solution of a time-dependent Hamilton–Jacobi equation. The methodology starts with a given initial domain and yields a sequence of region of attraction estimates by tracking the evolution of the implicit function. The resulting sequence is contained in and converges to the exact region of attraction. While alternative iterative methods for estimating the region of attraction have been proposed, the methodology proposed in this paper can compute the region of attraction to achieve any desired accuracy in a dimensionally independent and efficient way. An implementation of the proposed methodology has been developed in the Matlab environment. The correctness and efficiency of the methodology are verified through a few examples.

Learning To Estimate Regions Of Attraction Of Autonomous Dynamical Systems Using Physics-Informed Neural Networks

“…Yuan and Li show that the viscosity solution to a particular Hamilton-Jacobi PDE yields a special function u xe : X ⊂ R n → R whose zero level set ∂L 0,ux e = {x ∈ X : u xe (x) = 0} is the boundary of the complete ROA (i.e., ∂L 0,ux e = ∂R xe ). While the function generated by solving the Yuan-Li PDE as t → ∞ is not a Lyapunov function, it is related to the backward reachable set of the dynamical system, and therefore can serve as an implicit representation of the ROA just like a Lyapunov function (see [1] for more details). Note that we use the notation ∂L 0,ux e = {x ∈ X : u xe (x) = 0} to denote the zero level set of the function u xe : X ⊂ R n → R and the notation L 0,ux e = {x ∈ X : u xe (x) ≤ 0} to denote the zero sublevel set of the same function.…”

“…Instead, we mention numerical methods here because: (1) we use them to compute high fidelity numerical ROA estimates for the sake of comparison with our neural approach; and (2) they demonstrate a key difficulty in solving the Yuan-Li PDE, namely that the PDE solution tends to be highly discontinuous near the ROA boundary. To overcome the challenge that such discontinuity poses to many numerical methods, we use the same techniques that Yuan and Li propose for solving Hamilton-Jacobi PDEs of the kind used in their theorem [1]. Specifically, we use a fifth order weighted essentially non-oscillatory (WENO5) scheme to discretize the spatial variables of the PDE [32] and then apply a third order total variation decreasing Runge-Kutta (TVDRK3) method to integrate the system through time [33].…”

“…This work focuses on the development and testing of this safety network, while future work will apply this method to motor learning problems controlled by reinforcement learning agents. By estimating the ROA of controlled autonomous dynamical systems, this safety network provides two key benefits: (1) it allows for a general evaluation of the stability of the given control scheme, whether this be a neural or traditional controller; and (2) it provides a stability estimate for individual states within the state space of the dynamical system, thereby allowing us to assess the risk of performing exploratory actions near that state. The safety network accomplishes this task by learning to represent the viscosity solution to a particular partial differential equation (PDE) (see theorem 2.2 or [1] for more details), which we refer to as the Yuan-Li PDE, whose solution as t → ∞ is a special function, related to a Lyapunov function, whose zero level set is the complete ROA.…”

Application of Sum-of-Squares Method in Estimation of Region of Attraction for Nonlinear Polynomial Systems

“…The classification and analysis methods of nonlinear systems are different. In the reference [26], estimation of the regions of attraction for autonomous nonlinear systems is studied. In the reference [27], attraction domain estimation of the model with parametric uncertainties is studied.…”

On the synthesis of a polynomial controller for enlarging the stability domain for nonlinear discrete system

“…At the terms of the problem of synthesis, several numerical approaches allowing the determination of control laws ensuring global or local stability of the controlled system were developed during the latest years (Boukas, 2007; Ellis et al, 2016; El-loumi et al, 2016; Gao et al, 2016; Haghighatnia and Moghaddam, 2012; Yuan and Li, 2018). The focus of this paper is on the estimation and the maximization of the attraction domain of discrete polynomial systems.…”