A relative error model reduction method using balancing

This paper proposes to use Enns´s frequency weighted balanced truncation with as a weighting function the inverse of the outer factor of the system to be reduced. This results in a relative error model reduction method very similar to balanced stochastic truncation (BST), in the sense that in both methods, there is a spectral factorization of G(s)G^T(-s), involving the solution of a Riccati equation, and a controllability gramian to be computed. It is shown that the computation of the observability gramian of the weighted system in Enns´s method can be reduced in this case to the computation of the observability gramian of the weighting function only. Moreover, a Schur decomposition of the Hamiltonian associated with the Riccati equation can be found with little effort, so that the Riccati equation may be solved at a much smaller cost