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Stability analysis of state‐time‐dependent nonlinear hybrid dynamical systems
Author(s) -
Asahara Hiroyuki,
Kousaka Takuji
Publication year - 2019
Publication title -
ieej transactions on electrical and electronic engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.254
H-Index - 30
eISSN - 1931-4981
pISSN - 1931-4973
DOI - 10.1002/tee.22807
Subject(s) - monodromy matrix , nonlinear system , state transition matrix , mathematics , monodromy , perturbation (astronomy) , matrix (chemical analysis) , ordinary differential equation , orbit (dynamics) , stability (learning theory) , dynamical systems theory , mathematical analysis , differential equation , control theory (sociology) , computer science , physics , symmetric matrix , pure mathematics , quantum mechanics , engineering , eigenvalues and eigenvectors , materials science , control (management) , aerospace engineering , machine learning , artificial intelligence , composite material
In this paper, we present an improved method for analyzing the stability of the nonlinear hybrid dynamical systems (NHDSs) using a monodromy matrix. We define an n ‐dimensional NHDS and focus on an orbit whose initial values exist near the periodic orbit. Because the system has nonlinearity, the circuit equation cannot be expressed in the form of a linear ordinary differential equation, meaning that the existing monodromy‐matrix‐based stability analysis method cannot be used because it requires a matrix exponential. Therefore, we propose a general theory for calculating orbital perturbations during Poincaré observation. Perturbation through switching events is expressed by a state‐transition matrix, which we refer to as the saltation matrix. By computing the perturbations, we obtain the monodromy matrix, whose characteristic multipliers denote the stability of the periodic orbit. We apply the proposed algorithm to an interrupted electric circuit with a nonlinear characteristic to confirm its validity. © 2018 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.