Efficient Positive-Real Balanced Truncation of Symmetric Systems Via Cross-Riccati Equations

We present a highly efficient approach for realizing a positive-real balanced truncation (PRBT) of symmetric systems. The solution of a pair of dual algebraic Riccati equations in conventional PRBT, whose cost constrains practical large-scale deployment, is reduced to the solution of one cross-Riccati equation (XRE). The cross-Riccatian nature of the solution then allows a simple construction of PRBT projection matrices, using a Schur decomposition, without actual balancing. An invariant subspace method and a modified quadratic alternating-direction-implicit iteration scheme are proposed to efficiently solve the XRE. A low-rank variant of the latter is shown to offer a remarkably fast PRBT speed over the conventional implementations. The XRE-based framework can be applied to a large class of linear passive networks, and its effectiveness is demonstrated through numerical examples.