I have the following system:
$A= \begin{pmatrix} 0&1&0&0\\0&0&1&0\\0&0&0&1\\-21.6&-22.5&-16.8&-6.9 \end{pmatrix} $ and $B = \begin{pmatrix} 0 & 1 \\ 0 & 0 \\ 0 & 0 \\ 1 & 0 \end{pmatrix} $
and $C = (\begin{matrix} 0.2430 & 1.0800 & 0.9000 & 0 \end{matrix} )$
D = 0.
The Objective is to design an Integral Control Tracking system with state-feedbacks, I have tried to solve it while it has only one output, but I had no luck. Then I tried to make an arbitrary output and added [0 0 1 1] as the second row of the C, but again I had no luck designing an Integral Control system. I'm doing as this PDF says(using [A-$\lambda$I B] nullity vectors as eigen vectors), I'm trying to put the systems poles on -1 -1.25 -1.5 -1.75 -2.
here is my matlab code:
B_2 = [B_new; zeros(2,2)];
C_c_pr = [C_c;0 0 1 1];
C_2 = [C_c_pr zeros(2,2)];
A_2 = [A_c zeros(4,2);C_2];
null1_2 = null([A_2+eye(6) B_2]);
null2_2 = null([A_2+1.25*eye(6) B_2]);
null3_2 = null([A_2+1.5*eye(6) B_2]);
null4_2 = null([A_2+1.75*eye(6) B_2]);
null5_2 = null([A_2+2*eye(6) B_2]);
null6_2 = null([A_2+1.9*eye(6) B_2]);
V_two = [null1_2(1:6,2) null2_2(1:6,2) null3_2(1:6,2) null4_2(1:6,2) null5_2(1:6,2) null6_2(1:6,2)];
Q_two = [null1_2(7:8,2) null2_2(7:8,2) null3_2(7:8,2) null4_2(7:8,2) null5_2(7:8,2) null6_2(7:8,2)];
K_two = -Q_two * V_two^-1;
A_2_new = [A_c - B_new*K_two(:,1:4) B_new*K_two(:,5:6); -C_2];
G_two = C_2*(s*eye(6)-A_2_new)^-1 * [zeros(4,2);eye(2)];