Seeking Help to Find Kalman Decomposition Matrices

I need to find the matrix for a Kalman decomposition of the following problem. It would be even better if I could obtain the symbolic matrix as well.

Specifically, I am looking for the matrices At, Bt, and Ct.

Please help me as I'm having trouble finding the correct matrices, and my thesis is incomplete without solving this issue.

Thanks!!

[Sam Chak]

@Edoardo Moroni, I'm not entirely clear on your approach to solving the problem and determining the Kalman Decomposition Matrices. Could you please provide a description of the system and explain why you introduce τ at ?

[Edoardo Moroni]

hi thanks to try to help me.

so the system is in the form

X_dot = A(x) + B

y = C(x)

just dont consider the [tau], it is there only because i screen the general problem

i need to find the At Bt Ct matrix of the decomposition of Kalman.

like :

At = T * A * T ^-1

Bt = T^-1 * B

Ct = C * T

better if i can find it in symbolic value

Answer 1

[Sam Chak]

Hi @Edoardo Moroni

A Kalman decomposition of the system can be obtained by using the minreal() function as follows:

% Parameters
m1   = 1;
m2   = 1;
m3   = 1;
k0   = 1;

% State-space system
A21  = [-k0/m1    k0/m1      0;
         k0/m2 -2*k0/m2  k0/m2;
             0    k0/m3 -k0/m3];
A    = [zeros(3) eye(3);
        A21    zeros(3)];
B    = [1/m1 zeros(1,5)]';
C    = [1 zeros(1,5)];
D    =  0;
sys  = ss(A, B, C, D)

sys =
 
  A = 
       x1  x2  x3  x4  x5  x6
   x1   0   0   0   1   0   0
   x2   0   0   0   0   1   0
   x3   0   0   0   0   0   1
   x4  -1   1   0   0   0   0
   x5   1  -2   1   0   0   0
   x6   0   1  -1   0   0   0
 
  B = 
       u1
   x1   1
   x2   0
   x3   0
   x4   0
   x5   0
   x6   0
 
  C = 
       x1  x2  x3  x4  x5  x6
   y1   1   0   0   0   0   0
 
  D = 
       u1
   y1   0
 
Continuous-time state-space model.

% Kalman decomposition, where T is the transformation matrix
[sysr, T] = minreal(sys)

1 state removed.

sysr =
 
  A = 
              x1         x2         x3         x4         x5
   x1     0.1925         -1     0.7698  5.551e-17    -0.7698
   x2     0.3333  2.996e-65    -0.6667  1.245e-64    -0.3333
   x3    -0.9623          2    -0.3849         -1     0.3849
   x4     0.3333  2.996e-65     0.3333  1.245e-64     0.6667
   x5     0.3849          1    -0.1925         -1     0.1925
 
  B = 
              u1
   x1     0.5774
   x2   5.19e-65
   x3     0.5774
   x4  2.157e-64
   x5    -0.5774
 
  C = 
              x1         x2         x3         x4         x5
   y1     0.5774   5.19e-65     0.5774  2.157e-64    -0.5774
 
  D = 
       u1
   y1   0
 
Continuous-time state-space model.

T = 6×6

0.5774 -0.0000 -0.0000 -0.6667 0.3333 0.3333 0.0000 1.0000 -0.0000 0.0000 0.0000 0.0000 0.5774 -0.0000 -0.0000 0.3333 -0.6667 0.3333 0.0000 -0.0000 1.0000 0.0000 0.0000 0.0000 -0.5774 0.0000 0.0000 -0.3333 -0.3333 0.6667 0 0.0000 0.0000 -0.5774 -0.5774 -0.5774

% Check by manually transforming the original matrices
At   = T*A*T';
Bt   = T*B;
Ct   = C*T';
Dt   = D;
Ksys = ss(At, Bt, Ct, Dt)

Ksys =
 
  A = 
               x1          x2          x3          x4          x5          x6
   x1      0.1925          -1      0.7698   5.551e-17     -0.7698     -0.3333
   x2      0.3333   2.996e-65     -0.6667   1.245e-64     -0.3333     -0.5774
   x3     -0.9623           2     -0.3849          -1      0.3849     -0.3333
   x4      0.3333   2.996e-65      0.3333   1.245e-64      0.6667     -0.5774
   x5      0.3849           1     -0.1925          -1      0.1925      0.3333
   x6   8.919e-65    1.11e-16   3.729e-65   -1.11e-16   1.265e-64  -1.545e-64
 
  B = 
              u1
   x1     0.5774
   x2   5.19e-65
   x3     0.5774
   x4  2.157e-64
   x5    -0.5774
   x6          0
 
  C = 
              x1         x2         x3         x4         x5         x6
   y1     0.5774   5.19e-65     0.5774  2.157e-64    -0.5774          0
 
  D = 
       u1
   y1   0
 
Continuous-time state-space model.

[Edoardo Moroni]

Hello , thank you for your help

can u try to calculate this for C = [ 0 1 0 0 0 0]

because i cant do it and i dont know why

thanks

[Sam Chak]

Hi @Edoardo Moroni

I don't know what went wrong. Can you show you MATLAB code so that we can investigate the issue? This way, you will also learn so that you won't the same mistake in the future.

[Edoardo Moroni]

Hi @Sam Chak!

thanks for helping me!

i only changed the C in the Matlab code that u sent me.

but i'm thinking that maybe my system is wrong, can u check if my matrix (A,B,C,D) are correct?

m1 = 1

m2 = 1

m3 = 1

k0 = 1

i have to find the matrix of the space rappresentation of this sistem

[Sam Chak]

Hi @Edoardo Moroni,

Can you include matrices in MATLAB code? This will allow me to compare it line by line with the code in my Answer and identify any mistakes you may have made.