Development of passivity preserving model order reduction techniques

Abstract

This research examines the passivity preserving frequency weighted model order reduction(MOR) techniques for linear time invariant (LTI) systems. In this research different singleand double sided passivity preserving techniques for MOR are presented. Different combinationsof Lyapunov and arithmetic Riccati equations (ARE’s) are used to deduce the controllabilityand observability Gramians from frequency weighted and un-weighted systems.First of all, augmented system (a system with both input and output frequency weights) istransformed into a new system using two different transformations. Then by using the sametransformations, weighted controllability and observability Gramians (which are obtainedfrom weighted ARE’s) are transformed into new weighted controllability and observabilityGramians.For double sided passivity preserving, three schemes are presented in which both controllabilityand observability Gramians are weighted. In first scheme, an ARE based transformedweighted controllability Gramian and a Lyapunov based weighted observability Gramian and vice versa are used for balancing the system. In second scheme, an ARE based transformedweighted controllability Gramian and an ARE based weighted observability Gramianand vice versa are used to balance the system. In third scheme, an ARE based transformedweighted controllability Gramian and an ARE based transformed weighted observabilityGramian are used for balancing purpose.For single sided passivity preserving, five schemes are presented in which either a controllability Gramian is weighted and an observability Gramian is un-weighted or a controllability Gramian is un-weighted and an observability Gramian is weighted. In first scheme, an ARE based un-weighted controllability Gramian and a Lyapunov based weighted observabilityGramian and vice versa are used to balance the system. In second scheme, anARE based weighted controllability Gramian and an ARE based un-weighted observabilityGramian and vice versa are used for balancing. In third scheme, an ARE based transformedweighted controllability Gramian and an ARE based un-weighted observability Gramianand vice versa are used for balancing purpose. In fourth scheme, an ARE based transformedweighted controllability Gramian and a Lyapunov based un-weighted observabilityGramian and vice versa are used for balancing the system. In fifth scheme, an ARE basedtransformed weighted controllability Gramian and an ARE based un-weighted observabilityGramian and vice versa are used for balancing. Several practical examples using differentweighting functions are given to show the effectiveness of the proposed schemes.