Comparison of Model Reduction Methods with Applications to Circuit Simulation

Abstract

Summary. We compare different model reduction methods applied to the dynamical system of a coupled transmission line: balanced truncation (BT), truncation by balancing one gramian (or PMTBR - poor man’s truncated balanced reduction), positive real balanced truncation (PRBT) and its Hamiltonian implementation (PRBT-Ham), PRIMA, spectral zero method (SZM) and its Hamiltonian implementation (SZM-Ham), and finally, optimal H2. Their performance is analyzed in terms of several criteria such as: preservation of controllability, observability, stability and passivity, relative H2 and H∞ norms, and the computational cost involved. This paper presents different reduction methods together with results obtained by applying each method on a dynamical system given by a coupled transmission line. In Sect. 2, a modified nodal analysis (MNA)-similar representation of the system is derived. The model reduction methods are grouped in two main categories, gramian based and Krylov based, discussed in sections 3 and 4 respectively. Sect. 3 outlines the theory behind gramian based reduction methods: BT, PMTBR and PRBT. Krylov based reduction methods PRIMA, SZM and optimal H2 are described in Sect. 4. In Sect. 5 we compare all methods in terms of: preservation of some important properties like controllability, observability, stability and passivity, the relative H2 and H∞ norms and in terms of the computational cost. In Sect. 6, error systems resulting from different methods are compared. This allows us to identify frequency ranges where one particular method approximates the original system more accurately. Sect. 7 presents additional results obtained with the optimal H2 method. Finally, Sect. 8 summarizes our analysis and motivates further research.