Eigenstructure Assignment of Output-Nulling Friends by Sylvester Mapping

This paper studies eigenstructure assignment of output-nulling friends for linear time-invariant systems. The internal and external assignment equations are reformulated as generalized Sylvester equations associated with the Rosenbrock polynomial matrix and the state-input polynomial matrix. The prescribed target matrices need not be in Jordan form. This permits the direct use of companion or block-companion, real Schur, or other convenient target coordinates. A complete parameterization is established based on the Sylvester mapping and the Smith form, separating all solutions into a polynomial right-nullspace part and a local torsion part. The latter is crucial for eigenstructure assignment at invariant zeros and for defective invariant-zero chains. Within the split-realization class, the framework parameterizes all output-nulling friends admitting a closed-loop invariant complement that carries the prescribed external structure. Fixed left-kernel bases convert the torsion constraints into unconstrained Smith-coordinate parameters. The contribution is exact, structural, and target-coordinate-covariant. When the target spectrum avoids the relevant rank-drop set, state-direction-space dimension formulas are extended to arbitrary target matrices.