A property of internally balanced and low noise structures

Given an
N
th-order minimal asymptotically stable system
H(z)
, we consider the class
e
of minimal state-space realizations which minimize the trace of
\gamma^{2}K + W
, where
K
is the controllability and
W
the observability Grammian. We show that the optimal structures are defined by
W = \gamma^{2}K
with optimal cost
2\gammaS
where
S
is the sum of singular values. For
\gamma = 1
, the internally balanced structure belongs to
e
while if
\gamma = S/ N
, the optimal noise structures of Mullis and Roberts [1] belong to
e
. In particular, both structures are optimal when
\gamma = 1
and
S = N
.