Computation of the (J, J´)-spectral and (J, J´)-lossless factorizations of a general system

We consider rich extensions of the J-lossless conjugation, (J, J´)-spectral, and (J, J´)- lossless factorizations to become applicable to a descriptor system whose transfer matrix is a completely general rational matrix function. The main novelty is that we allow for arbitrary rank, poles and zeros on the imaginary axis, or at infinity. We use descriptor state-space realizations to obtain constructive formulas for the solutions which bear the same nice expressions and striking simplicity of the regular case. Besides, we remedy numerical drawbacks of alternative approaches by employing a preliminary unitary projection which delineates a subsystem fulfilling all regularity assumptions and for which we actually solve the factorizations.