q The remainder of the proof
is achieved by using the order reduction by one-step results and by
noting that
q
q obtained
by the kth order partitioning is internally balanced with balanced Gramian given
by S1 =diag (s1 Is1, s2 Is2,
, sr Isr)
Let Ek(s)=Gk+1(s)-Gk(s) for k=1,2,
,N-1 and let GN(s)=G(s). Then
Since Gk(s) is a reduced-order model obtained from the
internally balanced realization of Gk+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.