Equivalence between SYLTDMOR1 and moment matching

Consider an LTI system with piecewise constant input and zero initial state. Choose Laguerre polynomials; Legendre polynomials with odd reduced order; Chebychev polynomials of the first kind with odd reduced order; or Chebychev polynomials of the second kind with odd reduced order. Let SYLTDMOR1 denote the time-domain model-order-reduction method presented in the source, and let $\tilde{A}$ and $\tilde{E}$ be the matrices in its Sylvester equation. Equivalence conjecture. SYLTDMOR1 is equivalent to the moment matching method, with expansion points chosen as the eigenvalues of the matrix pencil $(-\tilde{A},\tilde{E})$ arising in the Sylvester equation; these points depend on the chosen orthogonal polynomials and the reduced order.

The claim concerns the connection between Laguerre-based time-domain model reduction and moment matching. The paper reports numerical evidence for the required distinctness of the generalized eigenvalues, while the analogous equivalence for SYLTDMOR2 is stated as a theorem. The conjecture is resolved: the equivalence between Laguerre-based time-domain model reduction and moment matching was proved by Eid.