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with controllability and observability grammians given by,

in which the diagonal entries of 

Σ are in decreasing order, that is, 

σ

1

≥ σ

2

≥ ···, and such that the last diagonal entry of Σ

1

 exceeds 

the first diagonal entry of 

Σ

2

. It turns out that Re

λ

(

)<0  and 

λ

(A

11

–A

12

21

)< 0, and a reduced order model G

(s) can be 

defined by:

The attractive feature [LiA89] is that the same error bound holds as for 
balanced truncation. For example,

Although the error bounds are the same, the actual frequency pattern of 
the errors, and the actual maximum modulus, need not be the same for 
reduction to the same order. One crucial difference is that balanced 
truncation provides exact matching at 

ω = ∞, but does not match at DC, 

while singular perturbation is exactly the other way round. Perfect 
matching at DC can be a substantial advantage, especially if input signals 
are known to be band-limited. 

Singular perturbation can be achieved with 

mreduce( )

. Figure 2-1 shows 

the two alternative approaches. For both continuous-time and discrete-time 
reductions, the end result is a balanced realization. 

In Hankel norm approximation, one relies on the fact that if one chooses an 
approximation to exactly minimize one norm (the Hankel norm) then the 
infinity norm will be approximately minimized. The Hankel norm is 
defined in the following way. Let G(s) be a (rational) stable transfer 

Σ

Σ

1

0

0

Σ

2

=

=

=

22

1

–

22

1

–

x·

11

12

22

1

–

21

–

(

)x

1

12

–

22

1

–

2

+

(

)u

+

=

1

2

22

1

–

21

–

(

)x

2

22

1

–

2

–

(

)u

+

=

ω

( ) G

ω

( )

–

∞

2tr

Σ

2

≤