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IMA Journal of Mathematical Control and Information 1986 3(1):43-58; doi:10.1093/imamci/3.1.43
© 1986 by Institute of Mathematics and its Applications
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Balanced, Normal, and Intermediate Realizations of Nonrational Transfer Functions

N. J. YOUNG

Department of Mathematics, University of Glasgow Scotland

For a very general transfer-function matrix G(z) we construct a one-parameter family (A{alpha}, B{alpha}, C{alpha}) of state-space realizations of G(z), continuously indexed by {alpha} {varepsilon} (0, 1), such that the realizations corresponding to {alpha} equal to 0, 1/2, and 1 are output-normal, balanced, and input-normal respectively. The realization with parameter {alpha} has the property that M1–{alpha} = W{alpha}, where M and W are the observability and controllability gramians of the system respectively. (A{alpha}, B{alpha}, C{alpha}) has bounded reachability and observability operators and ||Aalpha;|| ≤ 1. The class of admissible G(z) contains all matrix functions bounded and analytic in the complement of the closed unit disc and vanishing at infinity, and also contains some unbounded functions.


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