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being approximated by a stable G

(s) with the actual error (as opposed to 

just the error bound) satisfying:

 is optimal, that is, there is no other G

 achieving a lower bound.

The first steps of the algorithm are to obtain the Hankel singular values of 
G(s) (by using 

hankelsv( )

) and identify their multiplicities. (Stability of 

G(s) is checked in this process.) If the user has specified 

nsr

 and this does 

not coincide with one of 0,n

1

,n

2

, ... an error message is obtained; generally, 

all the 

σ

 are different, so the occurrence of error messages will be rare. 

The next step of the algorithm is to calculate the sum G(s) = G

(s) + G

(s), 

following [SCL90]. (A separate function 

ophred( )

 is called for this 

purpose.) The controllability and observability grammians P and Q are 
found in the usual way.

AP + PA

′ = –BB′

QA + A

′Q = –C′C

and then a singular value decomposition is obtained of the 
matrix

:

There are precisely n

– n

i – 1

 zero singular values, this being the multiplicity 

of 

σ

. Next, the following definitions are made:

( ) G

( )

–

∞

σ

=

σ

2

–

1

2

0

0 0

1

′

2

′

σ

2

–

=

11

12

21

22

1

′

2

′

=

σ

2

′ QAP

+

(

) V

1

2

(

)

1

2

1

′

2

′

=

1

2

[

]

1

V

2

[

]

=