National Instruments ni matrixx xmath 用户手册

© National Instruments Corporation

The treatment of j

ω-axis or right half plane poles in the above schemes is 

crude: they are simply copied into the reduced order controller. A different 
approach comes when one uses a so-called matrix fraction description 
(MFD) to represent the controller, and controller reduction procedures 
based on these representations (only for continuous-time) are found in 

fracred( )

. Consider first a scalar controller 

. One 

can take a stable polynomial 

 of the same degree as d, and then 

represent the controller as a ratio of two stable transfer functions, namely

Now 

 is the numerator, and

 the denominator. Write 

 as 

. Then we have the equivalence shown in Figure 4-1.

Figure 4-1.  Controller Representation Through Stable Fractions

Evidently, C(s) can be formed by completing the following steps:

1.

Construction of the one-input, two-output stable transfer function 
matrix

(which has order equal to that of   or  ).

2.

Interconnection through negative feedback of the second output to the 
single input.

These observations motivate the reduction procedure:

•

Reduce G to G

; notice that G is stable. Let G

 be

( )

( ) d s

( )

⁄

=

( )

( )

( )

----------

( )

( )

----------

1

–

⁄

⁄

⁄

1 e d

⁄

+

⁄

⁄

=

⁄

⁄

=