Matrix Fraction Description in Large Scale MIMO Descriptor Systems: Matrix Polynomials Approaches

The matrix transfer function (MTF) is fundamental to the analysis and control of multivariable descriptor systems, especially under zero initial conditions. Its importance lies in its direct relation to input–output behavior and its natural use in frequency-domain methods. Unlike classical approaches that obtain MTF through companion linearizations or indirect Weierstrass–Kronecker reductions, our method derives a closed-form MFD directly from the descriptor pencil λE−A, avoiding linearizations and preserving descriptor structure. This yields (i) an explicit parameterization of state feedback gains via finite/infinite Jordan pairs, (ii) a normalization law that removes impulsive modes by design, and (iii) improved reproducibility through block-polynomial operations suited to large-scale MIMO systems. The framework further extends eigenstructure assignment to descriptor models, combining clarity of analysis with practical control design. These results establish a systematic basis for scalable methods in MIMO descriptor systems.