Kshitij Singh answered . 2024-05-18 12:20:49

>> sys1 = minreal(ss(zpk([tf(1,[1 1]);tf(1,conv([1 1],[1 2]));tf(1,conv([1 1],[1 3]))])))

sys1 =
 
  A = 
            x1       x2       x3
   x1   -2.067  -0.2775  -0.1388
   x2   0.2401   -1.212    0.894
   x3  0.03861   0.5588   -2.721
 
  B = 
           u1
   x1  0.9679
   x2  0.6052
   x3   1.303
 
  C = 
            x1       x2       x3
   y1   0.1629   0.6703   0.3352
   y2  -0.6095   0.4695   0.2348
   y3  0.06517   0.4681  -0.2659
 
  D = 
       u1
   y1   0
   y2   0
   y3   0
 
Continuous-time state-space model.

Let's look at its transfer function matrix in zpk form (note the cancellations in the individual elements):

>> zpk(sys1)

ans =
 
  From input to output...
          (s+2) (s+3)
   1:  -----------------
       (s+2) (s+3) (s+1)
 
             (s+3)
   2:  -----------------
       (s+2) (s+3) (s+1)
 
             (s+2)
   3:  -----------------
       (s+2) (s+3) (s+1)
 
Continuous-time zero/pole/gain model.

Now defne a new state space realization using the augmented state vector and check its transfer function in zpk form:

>> sys2=ss([[sys1.a ; sys1.c(1:2,:)*sys1.a] zeros(5,2)],[sys1.b;sys1.c(1:2,:)*sys1.b],[zeros(2,3) eye(2);sys1.c(3,:) zeros(1,2)],[zeros(2,1);sys1.d(3,:)]);
>> zpk(sys2)

ans =
 
  From input to output...
          s (s+2) (s+3)
   1:  -------------------
       s (s+3) (s+2) (s+1)
 
             s (s+3)
   2:  -------------------
       s (s+3) (s+2) (s+1)
 
             (s+2)
   3:  -----------------
       (s+3) (s+2) (s+1)
 
Continuous-time zero/pole/gain model.

The transfer function matrices are the same (after cancellation), but additional poles at the origin show up because of the non-minimal realization. As must be the case, the first two outputs are state variables.

>> sys2.c

ans =

            0            0            0   1.0000e+00            0
            0            0            0            0   1.0000e+00
   6.5167e-02   4.6814e-01  -2.6593e-01            0            0

>> T=[sys1.c(1:2,:);[0 0 1]]

T =

   1.6292e-01   6.7034e-01   3.3517e-01
  -6.0951e-01   4.6951e-01   2.3476e-01
            0            0   1.0000e+00

            
>> rank(T)

ans =

     3

This tansformation T is called a similarity transformation and can used to define a new realization in terms of z (instead of x)

>> sys3=ss2ss(sys1,T)

sys3 =
 
  A = 
               x1          x2          x3
   x1          -1  -4.857e-17  -4.441e-16
   x2           1          -2  -4.441e-16
   x3      0.7396      0.1343          -3
 
  B = 
              u1
   x1          1
   x2  2.116e-16
   x3      1.303
 
  C = 
               x1          x2          x3
   y1           1           0  -5.551e-17
   y2           0           1           0
   y3      0.6513     0.06717        -0.5
 
  D = 
       u1
   y1   0
   y2   0
   y3   0
 
Continuous-time state-space model.

>> zpk(sys3)

ans =
 
  From input to output...
          (s+2) (s+3)
   1:  -----------------
       (s+1) (s+2) (s+3)
 
             (s+3)
   2:  -----------------
       (s+1) (s+2) (s+3)
 
             (s+2)
   3:  -----------------
       (s+1) (s+2) (s+3)
 
Continuous-time zero/pole/gain model.

As expected, the first two outputs of sys3 are now state variables and the transfer function matrix of sys3 is the same as that of sys1.

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