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On the dual linear periodic time‐delay system: Spectrum and Lyapunov matrices, with application to ℋ 2 analysis and balancing
Author(s) -
Michiels Wim,
Gomez Marco A.
Publication year - 2020
Publication title -
international journal of robust and nonlinear control
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.361
H-Index - 106
eISSN - 1099-1239
pISSN - 1049-8923
DOI - 10.1002/rnc.4970
Subject(s) - lyapunov function , mathematics , lyapunov equation , dual (grammatical number) , matrix norm , lti system theory , affine transformation , linear system , norm (philosophy) , control theory (sociology) , matrix (chemical analysis) , computer science , pure mathematics , mathematical analysis , eigenvalues and eigenvectors , nonlinear system , control (management) , art , physics , materials science , literature , quantum mechanics , artificial intelligence , political science , law , composite material
Summary We present novel theoretical concepts for linear time‐periodic systems with multiple delays, which are closely related to the spectral properties and Lyapunov matrices. At the basis of the main results is the associated dual system, constructed by transposition of the systems matrices and affine transformations of their arguments. We introduce, for the first time, the concepts of theℋ 2norm and the dual Lyapunov matrix of periodic systems with delays. We show that the primal and dual system have the sameℋ 2norm, characterized by primal and dual delay Lyapunov equations, which extend the well‐known results for time‐invariant systems with delays, and periodic systems without delays. Having at hand the pair of primal‐dual Lyapunov matrices, along with some energy interpretations, allow us to generalize the concept of position balancing and explore its potential for model reduction. The obtained results are illustrated by several examples, including the delayed Mathieu equation.