Contraction Analysis of Nonlinear DAE Systems

Region of Attraction for Power Systems using Gaussian Process and Converse Lyapunov Function -- Part I: Theoretical Framework and Off-line Study

“…Here we explicitly assume that the system dynamics are available, although our framework can be generally extended to incorporate uncertainties with a predefined complexity level, i.e., the smoothness of the unknown components, in the system's dynamical model. Property (1) implies that the function V (x) decays over time. Equation (2) defines a stable trajectory φ (x,t) with the initial state x.…”

“…where y (i) refers to the observed function value for the input x (i) at the i-th sampling step, and the measurement noise ε is zero-mean, independent and bounded by σ . With the GP approach, we can obtain the posterior distribution over h(x) by using sampling data in the training set {(x (1) , y (1) ), (x (2) , y (2) ), ..., (x (i) , y (i) )}.…”

“…where k N (x) = [k(x (1) , x), ..., k(x (N) , x)] T and K N is the positive definite kernel matrix [k(x, x )] x,x ∈A N .…”

“…Then the value of Lyapunov function at x (i) is estimated by V (x (i) ) with (3). By choosing {(x (1) , V (x (1) )), ..., (x (i) , V (x (i) ))} as the training set, µ i (x) and σ i (x) for the unknown Lyapunov function V (x) can be updated according to (4). On the other hand, if the state trajectory φ (x (i) ,t), t ≥ 0 fails to converge to the origin, the sampling point x (i) is removed from the sampling region X.…”

“…In terms of power systems, the ROA refers to a subspace of operating states that can converge to a steady-state equilibrium. There are various approaches for estimating the ROA of a general nonlinear system such as contraction analysis [1], level sets of Lyapunov function [8,17], sum of square technique [18,19], sampling-based method [20], and so on. Nevertheless, these approaches largely rely on deterministic models and may not be applicable to deal with uncertainties in more realistic systems.…”