Help with state representation of my 3-mass 2-spring system

Starting from this information, I have managed to obtain this state representation. I wanted to ask for your confirmation to know if it is correct, or if it is not, where I made a mistake. Thank you for your attention.

this are my matrix :

A   = [    0 1      0 0      0 0;  
      -k0/m1 0  k0/m1 0      0 0; 
           0 0      0 1      0 0;
           0 0 -k0/m2 0  k0/m2 0; 
           0 0      0 0      0 1; 
           0 0      0 0 -k0/m3 0];
B   = [1/m1; 
          0; 
          0; 
          0; 
          0; 
          0];
C   = [1 0 0 0 0 0]; 
D   =  0;

[Sam Chak]

Hi @Cristian Rohozneanu

It appears to be a nonlinear spring-mass system due to the cubic term . Consequently, the system cannot be mathematically expressed in the conventional linear state-space model , , unless you construct a Takagi–Sugeno fuzzy model for the nonlinear system. Nevertheless, you can still model the system in the control-affine nonlinear state-space model .

[Cristian Rohozneanu]

Hi @Sam Chak

I apologize; I forgot to mention that the cubic term should not be considered. Therefore, k(\delta_i) = k_0 * \delta_i

[Cristian Rohozneanu]

Furthermore, the matrices I have derived are as follows. I apologize for any confusion.

A = [0 0 0 1 0 0;
     0 0 0 0 1 0;
     0 0 0 0 0 1;
    -(k0/m1) (k0/m1) 0 0 0 0;
    (k0/m2) -(2*k0/m2) (k0/m2) 0 0 0;
     0 (k0/m3) -(k0/m3) 0 0 0];

B = [1/m1; 0; 0; 0; 0; 0];
C = [0 1 0 0 0 0]; 

Answer 1

[James Tursa]

The k terms in the equations above appear to be non-linear. Your A matrix doesn't account for the cubic parts.

[Cristian Rohozneanu]

Hi @James Tursa

I forgot to mention that the cubic term should not be considered. Therefore, k(\delta_i) = k_0 * \delta_i

Furthermore, the matrices I have derived are as follows. I apologize for any confusion.

A = [0 0 0 1 0 0;

0 0 0 0 1 0;

0 0 0 0 0 1;

-(k0/m1) (k0/m1) 0 0 0 0;

(k0/m2) -(2*k0/m2) (k0/m2) 0 0 0;

0 (k0/m3) -(k0/m3) 0 0 0];

B = [1/m1; 0; 0; 0; 0; 0];

C = [0 1 0 0 0 0];