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This rather crude approach to the handling of the unstable part of a 
controller is avoided in 

fracred( )

, which provides an alternative to 

wtbalance( )

 for controller reduction, at least for an important family 

of controllers.

The major steps of the algorithm are as follows:

1.

Check dimension, syntax, stability of 

SysV

, closed-loop stability, and 

decomposition of C(s) into the sum of a stable part (poles in Re[s] < 0) 
and unstable part (poles in Re[s]

≥ 0); 

stable( )

 is used for this 

purpose.

2.

Compute input (right) weight and/or output (left) weight as appropriate 
for the specified type.

Table 4-2.  Error Measure Interpretation for wtbalance

"input stab"

A stability robustness argument, based on breaking the loop at the controller 
output, indicates that if C is stabilizing for P and the error measure is less 
than 1, then Cr is stabilizing for P. The smaller the error measure is, the 
greater the stability robustness.

"output stab"

A similar stability robustness argument, but based on breaking the loop 
at the controller input, indicates that if C is stabilizing for P and the error 
measure is less that 1, then C

 is stabilizing for P. The smaller the error 

measure is, the greater the stability robustness. 

"match"

If T = PC(I + PC)

–1

 and T

 = PC

(I + PC

)

–1

 are the two closed-loop transfer 

function matrices, then T – T

 to first order in C – C

 is given by 

(I + PC)

–1

P[C

– C][I + PC]

–1

, so that the error measure looks at matching 

of the closed-loop transfer function matrix.

"match spec"

It may be important to match closed-loop transfer function matrices more 
at certain frequencies than others; frequency weighting is achieved by 
introducing V(s). Frequencies corresponding to larger values of |V(j

ω)| or 

V(j

ω)V*(jω) will be the frequencies at which T(jω) and T

(j

ω) should have 

smaller error.

"input spec"

This is the one error measure that is not associated with a plant, or 
closed-loop of some kind. It simply allows the user to emphasize certain 
frequencies in the reduction procedures.