Why do I get different outputs with bilinear and c2d(sysc,Ts,'Tustin') MATLAB functions?

I have continuous state space model.

Ac = [ 1.3905    -20.8518      -0.7666
       5.1879    -78.8942       6.0869
      -2.8936    130.2226    -857.3009]

Bc = [  90.20      94.7
        16.00      14.0
      -1307.8     953.0]
  
Cc = [1     0      0
      0     1      0
      0     0      1]
  
Dc = [0     0
      0     0
      0     0]
  
dt = 0.05

When I discretize it with Tustin method (bilinear transformation) using c2d function

sysc = ss(Ac,Bc,Cc,Dc);
sysd = c2d(sysc,dt,'Tustin');
[Ad,Bd,Cd,Dd,K] = ssdata(sysd);

output is

Ad = [ 1.0240    -0.3595    -0.0042
       0.0886    -0.3378     0.0044
       0.0063     0.0973    -0.9102]

Bd = [ 4.5570    4.5668
       0.3202    0.5461
      -2.8832    2.1887]
  
Cd = [1.0120     -0.1798      -0.0021
      0.0443      0.3311       0.0022
      0.0032      0.0486       0.0449]
  
Dd = [ 2.2785     2.2834
       0.1601     0.2730
      -1.4416     1.0943]

But when I use bilinear function

[Ad,Bd,Cd,Dd] = bilinear(Ac,Bc,Cc,Dc,1/dt);

output is

Ad = [ 1.0240    -0.3595    -0.0042
       0.0886    -0.3378     0.0044
       0.0063     0.0973    -0.9102]

Bd = [ 20.3796   20.4236
        1.4319    2.4421
      -12.8940    9.7880]
  
Cd = [0.2263     -0.0402      -0.0005
      0.0099      0.0740       0.0005
      0.0007      0.0109       0.0100]
  
Dd = [ 2.2785     2.2834
       0.1601     0.2730
      -1.4416     1.0943]

Can someone explain me why I get different Bd and Cd matrices in these two cases, since these functions should do the same thing, bilinear transformation?

Or these two functions, bilinear and c2d with Tustin method option, are not implemented in the same way?

Answer 1

[Paul]

The state space realization of a linear system is not unique. However, you can compare the results from both methods and show that they have the same input/output relationship. Among other wasy, you can see this with

bode(c2d(sysc,dt,'Tustin'),ss(Ad1,Bd1,Cd1,Dd1,dt));  % Ad1, etc, from bilinear

So apparently c2d and blinear just use different methods to compute the realization of the transformed system. Check the algorithms sections of the doc pages for those functions to see how they work.