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Abstract
Commutative controllers provide a convenient means for the systematic adjustment of the gain/phase characteristic of generalized Nyquist diagrams. Due to the irrational nature of the eigenvectors of transfer function matrices, practical commutative controllers are based on eigenvector approximations. This paper investigates the use of rational approximations, highlights stability difficulties that arise in connection with unstable fixed modes and derives the necessary and sufficient conditions that overcome these difficulties. The results of the paper are