Invariance of characteristic values and L∞ norm under lossless positive real transformations

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.