21

The remainder of the proof
is achieved by using the order reduction by one-step results and by
noting that

obtained
by the kth order partitioning is internally balanced with balanced Gramian given
by *S**1 **=*diag *(**s**1 **I**s**1**, **s**2 **I**s**2**,
, **s**r **I**s**r**) *

Let *E**k**(s)=G**k+1**(s)-G**k**(s)* for *k*=1,2,
,*N*-1 and let *G**N**(s)=G(s).* Then

Since *G**k**(s) *is a reduced-order model obtained from the
internally balanced realization of *G**k+1**(s) *and the bound for one-step
order reduction holds.

Noting that

by the definition of Ek(s), we have

This is the desired upper
bound.

Continued