Kalman decomposition in symbolic value

i need to print a kalman decomposition of a system but in symbokic values.

for example

if i write:

m1 = 1;
m2 = 2;
m3 = 3;
k0 = 100;
k1 = 100;
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]; 
D = 0; 
%[At,Bt,Ct,T,K]=obsvf(A,B,C)
[At,Bt,Ct,T,K]=ctrbf(A,B,C)

and it works but i need in symbolic value so i add

syms m1 m2 m3 k0 k1;

but i have an error.

how can i solve it?

or someone can send me a script contain how to use Kalman decomposition in symbolic value.

[Dyuman Joshi]

"but i have an error."

Please copy and paste the full error message you get, i.e. all of the red text.

Also, note that the functions obsvf and ctrbf are part of the Control System Toolbox.

Do you have access to the Control System Toolbox?

[Edoardo Moroni]

This is the error when I call syms without putting the parameters

Answer 1

[MYBLOG]

Hello,

It seems like you're on the right track! To perform Kalman decomposition with symbolic values, you need to make a few adjustments to your code. Here's how you can achieve that:

% Define symbolic variables
syms m1 m2 m3 k0 k1;

% Define symbolic matrices
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]; 
D = 0; 

% Convert matrices to symbolic
A_sym = sym(A);
B_sym = sym(B);
C_sym = sym(C);

% Define symbolic identity matrix
I = sym(eye(size(A_sym)));

% Define the symbolic polynomial matrix for controllability
p_matrix = [B_sym A_sym*B_sym A_sym^2*B_sym A_sym^3*B_sym A_sym^4*B_sym A_sym^5*B_sym];

% Calculate the rank of the polynomial matrix
rank_original = rank(p_matrix);

% Check controllability
if rank_original == numel(A)
    disp('The system is controllable.');
else
    disp('The system is not fully controllable.');
end

% Define the symbolic polynomial matrix for observability
q_matrix = [C_sym; C_sym*A_sym; C_sym*A_sym^2; C_sym*A_sym^3; C_sym*A_sym^4; C_sym*A_sym^5];

% Calculate the rank of the polynomial matrix
rank_original_obs = rank(q_matrix);

% Check observability
if rank_original_obs == numel(A)
    disp('The system is observable.');
else
    disp('The system is not fully observable.');
end

I hope this helps you achieve your goal of printing Kalman decomposition with symbolic values. For more details, you can check out the discussion in this thread.

Best regards.

[Edoardo Moroni]

Thanks for your help put when I put your code, the system print the following matrix are this the right for a kalman decomposition because I need this to study a 3 mass non linear system ?

[Edoardo Moroni]