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 [1] belong to

. In particular, both structures are optimal when

and

.