1. Invariance of characteristic values and L∞ norm under lossless positive real transformations
- Author
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Luigi Fortuna, Arturo Buscarino, Mattia Frasca, and Maria Gabriella Xibilia
- Subjects
Lossless compression ,0209 industrial biotechnology ,Pure mathematics ,Transfer function matrix ,Computer Networks and Communications ,Applied Mathematics ,020206 networking & telecommunications ,02 engineering and technology ,LTI system theory ,Algebra ,020901 industrial engineering & automation ,Control and Systems Engineering ,Norm (mathematics) ,Signal Processing ,0202 electrical engineering, electronic engineering, information engineering ,Positive-real function ,Algebraic number ,2ND-ORDER MODES ,REAL TRANSFORMATIONS ,DIGITAL-FILTERS ,TIME-SYSTEMS ,Mathematics - Abstract
In this paper the invariance of the characteristic values and of the L ∞ norm of linear time-invariant (LTI) systems under lossless positive real transformations is proven. Given a LTI system with transfer function matrix G ( s ) , the transformation s ← F ( s ) with F ( s ) being an arbitrary lossless positive real function of order nF is considered, and the algebraic Riccati equations (AREs) allowing to assess some properties of the transformed system G ( F ( s ) ) are investigated. It is proven that, under such transformations, the solutions of the AREs associated to system G ( F ( s ) ) are related to those of G ( s ) . From this property, it derives that G ( F ( s ) ) and G ( s ) have the same L ∞ norm and that the characteristic values of G ( F ( s ) ) are those of G ( s ) , each with multiplicity nF.
- Published
- 2016