Error in pole placement

I am working on a feedback sytem and I am finding the k:

A=[0 0 0 217.28; 0 0 127.33 0; 0 -1 -3.73 0.97; -1 0 0.97 -3.73]; %4*4
B=[0 0 -.38 -1.39; 0 0 -1.18 0.29].'; %4*2
C=[1 0 0 0]; %1st row of C
D=zeros(1, 2);

% Desired pole locations
desired_poles = -1 * ones(1, size(A, rank(B))); % All poles at -1

% Calculate the feedback gain matrix K using pole placement
k = place(A, B, desired_poles);

Error using place
The "place" command cannot place poles with multiplicity greater than rank(B).

% Create the closed-loop system
Aclosed = A - B * k;
Bclosed = B;
Cclosed = C;
Dclosed = D;

% Check the poles of the closed-loop system
closed_loop_poles = eig(Aclosed);
disp('Poles of the closed-loop system:');
disp(closed_loop_poles);

I'm shown this error

Error using place

The "place" command cannot place poles with multiplicity greater than rank(B).

Answer 1

[Sam Chak]

Hi @Ho Pui Sum

That is the limitation of the place() command, and it is documentated here. In your case, the rank of the input matrix B is 2. Therefore, you are only allowed to place 2 repeated poles. See the workaround below.

A=[0 0 0 217.28; 0 0 127.33 0; 0 -1 -3.73 0.97; -1 0 0.97 -3.73]; %4*4
B=[0 0 -.38 -1.39; 0 0 -1.18 0.29].'; %4*2
rankB = rank(B)     % number of repeated poles cannot be greater than this value

rankB = 2

C=[1 0 0 0]; %1st row of C
D=zeros(1, 2);
sys = ss(A, B, C, D)

sys =
 
  A = 
          x1     x2     x3     x4
   x1      0      0      0  217.3
   x2      0      0  127.3      0
   x3      0     -1  -3.73   0.97
   x4     -1      0   0.97  -3.73
 
  B = 
          u1     u2
   x1      0      0
   x2      0      0
   x3  -0.38  -1.18
   x4  -1.39   0.29
 
  C = 
       x1  x2  x3  x4
   y1   1   0   0   0
 
  D = 
       u1  u2
   y1   0   0
 
Continuous-time state-space model.

% Desired pole locations
% desired_poles = -1 * ones(1, size(A, rank(B))) % All poles at -1
desired_poles = [-1.0001, -1, -1, -0.9999]

desired_poles = 1×4

-1.0001 -1.0000 -1.0000 -0.9999

% Calculate the feedback gain matrix K using pole placement
k = place(A, B, desired_poles)

k = 2×4

0.6710 0.1644 -0.3673 1.0056 -0.2161 0.7879 1.5843 -1.1458

% Create the closed-loop system
Aclosed = A - B * k;
Bclosed = B;
Cclosed = C;
Dclosed = D;

% Check the poles of the closed-loop system
closed_loop_poles = eig(Aclosed);
disp('Poles of the closed-loop system:');

Poles of the closed-loop system:

disp(closed_loop_poles);

   -0.9999
   -1.0001
   -1.0000
   -1.0000