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Controller reduction proceeds by implementing the same connection rule 
but on reduced versions of the two transfer function matrices. 

When K

 has been defined through Kalman filtering considerations, the 

spectrum of the signal driving K

 in Figure 4-5 is white, with intensity Q

. 

It follows that to reflect in the multiple input case the different intensities 
on the different scalar inputs, it is advisable to introduce at some stage a 
weight 

 into the reduction process.

After preliminary checks, the algorithm steps are:

1.

Form the observability and weighted (through Q

) controllability 

grammians of E(s) in Equation 4-7 by

(4-8)

(4-9)

2.

Compute the square roots of the eigenvalues of PQ (Hankel singular 
values of the fractional representation of Equation 4-5). The maximum 
order permitted is the number of nonzero eigenvalues of PQ that are 
larger than 

ε. 

3.

Introduce the order of the reduced-order controller, possibly by 
displaying the Hankel singular values (HSVs) to the user. Broadly 
speaking, one can throw away small HSVs but not large ones.

4.

Using 

redschur( )

-type calculations, find a state-variable 

description of E

(s). This means that E

(s) is the transfer function 

matrix of a truncation of a balanced realization of E(s), but the 

redschur( )

 type calculations avoid the possibly numerically 

difficult step of balancing the initially known realization of E(s). 
Suppose that:

5.

Define the reduced order controller C

(s) by

(4-10)

so that

1 2

⁄

–

(

)′

–

(

)P

+

–

′

=

–

(

)

–

(

)′Q

+

′

′C

–

–

=

Aˆ

′

–

(

)S

,

′

=

=

′

–

–

(

)S

=

( )

–

(

)

1

–

=