minimum_states_observability.m

%{
octave:4> help obsv
'obsv' is a function from the file /usr/share/octave/packages/control-3.4.0/obsv.m

 -- Function File: OB = obsv (SYS)
 -- Function File: OB = obsv (A, C)
     Return observability matrix.

     *Inputs*
     SYS
          LTI model.
     A
          State matrix (n-by-n).
     C
          Measurement matrix (p-by-n).

     *Outputs*
     OB
          Observability matrix.

     *Equation*
               | C        |
               | CA       |
          Ob = | CA^2     |
               | ...      |
               | CA^(n-1) |

Additional help for built-in functions and operators is
available in the online version of the manual.  Use the command
'doc <topic>' to search the manual index.

Help and information about Octave is also available on the WWW
at https://www.octave.org and via the [email protected]
mailing list.
%}

%x_vector = [x_1 x_2 x_3 x_4 x_5];
%x_dot_vector = [x_1_dot x_2_dot x_3_dot x_4_dot x_5_dot];
%
%u_vector = [V_s V_d];
%B_matrix = [
%     0 0 ;
%     0 K_1 ;
%     0 0 ;
%     K_2 0 ;
%     0 0 ;
%];

pkg load control;
pkg load symbolic;

% syms K_3
K_3 = 1;

A_matrix = [
     [0 1 0 0 0];
     [0 0 0 0 0];
     [0 0 0 1 0];
     [0 0 0 0 0];
     [K_3 0 0 0 0] ;
     ];

% all observability matrixcies with two observable states
C_matrixies = unique(perms([1 1 0 0 0]), "rows")
%C_matrixies = eye(5)

% for each C matrix
for i = 1:size(C_matrixies,1)
     % calculate observability matrix
     ob = obsv(A_matrix, C_matrixies(i, :));

     disp(["rank of the observability matrix is ", num2str(rank(ob))])
end

% problem 3 angle measurement