Nonlinear Dynamic Systems Parameterization Using Interval-Based Global Optimization: Computing Lipschitz Constants and Beyond

Abstract

Numerous state-feedback and observer designs for nonlinear dynamic systems (NDS) have been developed in the past three decades. These designs assume that NDS nonlinearities satisfy one of the following <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">function set classifications:</i> bounded Jacobian, Lipschitz continuity, one-sided Lipschitz, quadratic inner-boundedness, and quadratic boundedness. These function sets are characterized by <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">constant</i> scalars or matrices bounding the NDS’ nonlinearities. These constants depend on the NDS’ operating region, topology, and parameters and are utilized to synthesize observer/controller gains. Unfortunately, there is a near-complete absence of algorithms to compute such bounding constants. In this article, we develop analytical and computational methods to compute such constants. First, for every function set classification, we derive analytical expressions for these bounding constants through global maximization formulations. Second, we utilize a derivative-free interval-based global maximization algorithm based on the branch-and-bound framework to numerically obtain the bounding constants. Third, we showcase the effectiveness of our approaches to compute the corresponding parameters on some NDS such as highway traffic networks and synchronous generator models.