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

Sebastian A. Nugroho, Ahmad F. Taha, and Vu Hoang

Hundreds of state-feedback and observer designs for nonlinear dynamic systems (NDS) have been proposed in the past three decades. The designs assume that NDS nonlinearity satisfies one of the following function set classifications: bounded Jacobian, Lipschitz continuity, one-sided Lipschitz, and quadratic inner-boundedness. These function sets are defined by constant scalars or matrices that bound NDS nonlinearities. These constants (i) depend on the NDS' operating region, topology, and parameters and (ii) are used to design observer/controller gains. Unfortunately, there is a near-complete absence of algorithms to compute such bounding constants. This paper investigates a combination of analytical and computational methods to compute such constants. First, and for each function set classification, we derive analytical expressions for these bounding constants through global maximization formulations. Second, we develop a scalable interval-based global maximization algorithms to numerically obtain the bounding constants. The algorithms come with performance guarantees. Third, we showcase the effectiveness of the proposed approaches to find the corresponding parameters on some NDS models for traffic networks, and synchronous generator models.

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