Generalization of all stabilizing compensators for finite-dimensional linear systems

For finite-dimensional linear systems, the Youla-Kucera parameterization (YKP) with a Q parameter over RH∞ is assumed to satisfy the Diophantine identity. However, the stability is guaranteed if the Diophantine equation is the “U(RH∞)“ equality, but not if it is the “identity” equality. However, Vidyasagar's structure with an H parameter over U(RH∞) is an observer-controller configuration that satisfies the Diophantine equation. This study discusses the deficiency of the Diophantine identity; expands the YKP using an H parameter over U(RH∞), and expands the Vidyasagar's structure using a Qv parameter over RH∞ so that both of the expanded parameterizations satisfy the Diophantine equation and are equivalent for all stabilizing compensators. Moreover, an equation that relates to Q, Qv, and H will be introduced to establish relationships among the YKP, Vidyasagar's structure and both expanded parameterizations.