Getting a symmetric residue matrix from the poly-reference least square complex frequency domain technique

Using Poly-Reference Least Square Complex Frequency Domain (Poly-LSCF) technique, poles and modal participation vectors are simultaneously identified for increasing model orders, to build a stabilization diagram. For a given model order, mode shapes are identified by minimizing the fitting error. Therefore, modal participation vectors and mode shapes result from two different processes. The residue matrix is the product of the mode shape matrix by the modal participation factor matrix: since modal participation vectors and mode shapes are obtained independently, the residue matrix may be non symmetric. On the contrary, modal analysis theory shows that the residue matrix is symmetric for a self-adjoint system, i.e. a given mode shape and the corresponding modal participation vector are proportional to each other. In this paper, to get a symmetric residue matrix, the curve fitting step (LSCF part of the procedure) is iteratively repeated with updated modal participation vectors that are proportional to the mode shapes at the previous step.