An algorithm for computing robust forward invariant sets of two dimensional nonlinear systems

Abstract

Robustness of nonlinear systems can be analyzed by computing robust forward invariant sets (RFISs). Knowledge of the smallest RFIS of a system, can help analyze system performance under perturbations. A novel algorithm is developed to compute an approximation of the smallest RFIS for two-dimensional nonlinear systems subjected to a bounded additive disturbance. The problem of computing an RFIS is formulated as a path planning problem, and the algorithm developed plans a path which iteratively converges to the boundary of an RFIS. Rigorous mathematical analysis shows that the proposed algorithm terminates in a finite number of iterations, and that the output of the proposed algorithm is an RFIS.Simulations are presented to illustrate the proposed algorithm, and to support the mathematical results. This work may aid future development, for use with higher dimensional systems.

Lyapunov-Net: A Deep Neural Network Architecture for Lyapunov Function Approximation

“…To demonstrate the effectiveness of the proposed method, we apply Lyapunov-Net (1) to three test problems: a twodimensional (2d) nonlinear system from the curve-tracking application [17], a 30d synthetic dynamical system (DS) from [7], and a 2d inverted-pendulum system from [3]. In the first two problems, we aim at finding the Lyapunov function for the given dynamical systems, whereas in the last problem we find both the Lyapunov function and the control u as discussed in Section III-D.…”

“…a) 2d DS in curve tracking: We apply our method to find the Lyapunov function for a 2-dimensional (2d) nonlinear dynamical system (DS) in a curve-tracking application [17]. The dynamical system of x = (ρ, ϕ) is given by ρ = − sin(ϕ), (6a) φ = (ρ − ρ 0 ) cos(ϕ) − µ sin(ϕ) + e.…”

“…We use the following constants in our experiments: e = 0.15, ρ 0 = 1, and µ = 6.42 from [17] as well as ReLU activation.…”

Computing Robust Forward Invariant Sets of Multidimensional Non-linear Systems via Geometric Deformation of Polytopes

“…For comparison, the noise function and system parameters are chosen to be the same as [57]- [59], where the minimal RFIS is computed using a different method. The agreement between the results confirms the validity of Algorithm 2, which converges in 6 iterations as evident from Table I.…”

“…The most significant advantage of geometric approaches is their ability to generalize to wide classes of systems due to their non-reliance on the explicit algebraic representation of system dynamics. This is evident from some recent works such as [57]- [59], where closed polytopes are constructed using path planning algorithms on radial graphs for arbitrary nonlinear systems in continuous time, but limited to systems in R 2 . Another illustration of this can be found in [60], [61], where simplex-based approaches are used for arbitrary nonlinear systems.…”