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Preservation of SPR functions and stabilization by substitutions in SISO plants

In this paper, we present sufficient conditions to preserve stability by rational function substitutions, It is proved that by making substitutions with strictly positive real (SPR) zero relative degree functions, it is possible to preserve the stability and coprime factorization of transfer functions. It is also proved that these functions are closed under function composition. Finally, we prove some results about robust stabilization using constant compensators, SPR robust functions, and robust stability, based on the result over closure of SPR functions.