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Given an th-order minimal asymptotically stable system , we consider the class of minimal state-space realizations which minimize the trace of , where is the controllability and the observability Grammian. We show that the optimal structures are defined by with optimal cost where is the sum of singular values. For , the internally balanced structure belongs to while if , the optimal noise structures of Mullis and Roberts  belong to . In particular, both structures are optimal when and .