  We present a new balancing-based structure-preserving model reduction
technique for linear port-Hamiltonian descriptor systems. The proposed method
relies on a modification of a set of two dual generalized algebraic Riccati
equations that arise in the context of linear quadratic Gaussian balanced
truncation for differential algebraic systems. We derive an a priori error
bound with respect to a right coprime factorization of the underlying transfer
function thereby allowing for an estimate with respect to the gap metric. We
further theoretically and numerically analyze the influence of the Hamiltonian
and a change thereof, respectively. With regard to this change of the
Hamiltonian, we provide a novel procedure that is based on a recently
introduced Kalman-Yakubovich-Popov inequality for descriptor systems. Numerical
examples demonstrate how the quality of reduced-order models can significantly
be improved by first computing an extremal solution to this inequality.
