LFRT version 2.0: order reduction



This page describes two functions for order reduction after realization. Note that after realization, parameter commutativity is ignored (i.e., a*b-b*a connot be reduced to zero).


Function minlfr1

A system is defined, it will be used to illustrate the differences between both functions.

lfrs a b c d
sys1 = [a*c+a*d+b*c+b*d      1/(a*b*c);...
        c+d                  1/(a*b*c)];

size(sys1)

LFR-object with 2 output(s), 2 input(s) and 0 state(s).
Dimension of constant block in uncertainty matrix: 2
Uncertainty blocks (globally (16 x 16)):
 Name  Dims  Type   Real/Cplx   Full/Scal      Bounds
 a     4x4   LTI       r           s           [-1,1]
 b     4x4   LTI       r           s           [-1,1]
 c     5x5   LTI       r           s           [-1,1]
 d     3x3   LTI       r           s           [-1,1]
The function minlfr1 considers separately the parameters for order reduction, so, the expected order reduction corresponds to the following factorized from.
   = [a+b  1/(a*b*c) ; 1  1/(a*b*c)] * ...
     [c+d  0 ; 0 1];
i.e., order 10:

sys2 = minlfr1(sys1);
size(sys2)

LFR-object with 2 output(s), 2 input(s) and 0 state(s).
Dimension of constant block in uncertainty matrix: 2
Uncertainty blocks (globally (10 x 10)):
 Name  Dims  Type   Real/Cplx   Full/Scal      Bounds
 a     3x3   LTI       r           s           [-1,1]
 b     3x3   LTI       r           s           [-1,1]
 c     3x3   LTI       r           s           [-1,1]
 d     1x1   LTI       r           s           [-1,1]

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Function minlfr

The function minlfr considers all parameters simultaneously for factorization, the expected order reduction corresponds to the following factorized form (considering the same system sys1 as above):

    = [a+b  1 ; 1  1] * ...
      [c+d  0 ; 0 1/(a*b*c)];
i.e., order 7:

sys3 = minlfr(sys1);
size(sys3)

LFR-object with 2 output(s), 2 input(s) and 0 state(s).
Dimension of constant block in uncertainty matrix: 1
Uncertainty blocks (globally (7 x 7)):
 Name  Dims  Type   Real/Cplx   Full/Scal      Bounds
 a     2x2   LTI       r           s           [-1,1]
 b     2x2   LTI       r           s           [-1,1]
 c     2x2   LTI       r           s           [-1,1]
 d     1x1   LTI       r           s           [-1,1]

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