4.1 Introduction

In Chapter 3, we introduced the truncated balanced realization (TBR) method for compact modeling of interconnect circuits. In this chapter, we introduce a new passive TBR method, which only requires the solving of generalized Lyapunov equations and is able to preserve the structure of the reduced models just like the structure-preserving Krylov subspace method of [37].

In the TBR method, two steps are involved in the reduction process: the balancing step aligns the states such that states can be controlled and observed equally. The truncating step then throws away the weak states, which usually leads to much smaller model. The major advantage of TBR methods is that TBR methods can give deterministic global bound for the approximate error and can give nearly optimal models in terms of errors and model sizes [60].

Standard TBR algorithms, which solve the Lyapunov equations (linear matrix equations) do not necessarily preserve passivity. To ensure passivity, positive-real TBR (PR-TBR) has to be carried out [88, 131] by solving more difficult Lur e or Riccati equations, which can be computationally prohibitive as they are non-linear (quadratic) matrix equations.

Given a state-space model in descriptor form

where E, A ? R ^{n n} , B ? R ^{n p} , C ? R ^{p n} , D ? R ^{p p} , and y(t), u(t) ? R ^p. When E= I, (4.1) is in standard state-space form. Note that the descriptor form is the natural form of the circuit MNA matrices for interconnect circuits,...