Preservation of properties in discrete-time systems under substitutions

For discrete-time systems, using substitutions of the variable z by M₁ functions, we give results on preservation of stability, stabilization, positive real (PR), strictly positive real (SPR), bounded real (BR), and strictly bounded real (SBR) functions, along with the ℋ∞-norm. These results can be interpreted in the sense of robust control or uniform systems. Based on these properties, we present results about preservation of the SPR Lemma and absolute stability for discrete-time descriptor systems using linear matrix inequalities (LMIs) and substitutions.