Positive-real systems under lossless transformations: Invariants and reduced order models. Issue 11 (July 2017) - Buscarino, A.; Fortuna, L.; Frasca, M.; Xibilia, M.G.
Abstract: In this paper, positive-real systems under lossless positive-real transformations are investigated. Let G( s ) be the transfer function matrix of a continuous-time positive-real system of order n and F ( s ) a lossless transfer function of order nF . We prove here that the lossless positive-real transformed system, i.e. G( F ( s )), is also positive-real. Furthermore, the stochastic balanced representation of positive-real systems under lossless positive-real transformations is considered. In particular, it is proved that the positive-real characteristic values πj of G( F ( s )) are the same of G( s ) each with multiplicity nF, independently from the choice of F ( s ). This property is exploited in the design of reduced order models based on stochastic balancing. Finally, the proposed technique is a passivity preserving model order reduction method, since it is proven that the reduced order model of G( F ( s )) is still positive-real. An error bound for truncation related to the invariants πj is also derived.