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% Example 3.31 
% 
% Pole placement -- using transformation matrix 
% 
disp('Pole placement -- using transformation matrix') 
a=[0 1 0; 0 0 1; -6 -11 -6]; 
b=[0; 0; 10]; 
% Step 1 
cam=ctrb(a,b); 
disp('The rank of controllability matrix') 
rc=rank(cam) 
% Step 2 
beta=poly(a) 
% Step 3 
a1=beta(2); a2=beta(3); a3=beta(4); 
w=[a2 a1 1; a1 1 0; 1 0 0]; 
t=cam*w; 
% Step 4 
j=[-2+2*sqrt(3)*i   0    0 
    0     -2-2*sqrt(3)*i 0 
    0               0  -10]; 
alph=poly(j) 
aa1= alph(2); aa2=alph(3); aa3=alph(4); 
% Step 5 
k=[aa3-a3 aa2-a2 aa1-a1]*(inv(t)) 
% 
% 
% 
% Pole placement -- using Ackermann's formula 
% 
disp('Pole placement -- using Ackermanns formula') 
a=[0 1 0; 0 0 1; -6 -11 -6]; 
b=[0; 0; 10]; 
% Step 1 
cam=ctrb(a,b); 
disp('The rank of controllability matrix') 
rc=rank(cam) 
% Step 2 
j=[-2+2*sqrt(3)*i   0    0 
    0     -2-2*sqrt(3)*i 0 
    0               0  -10]; 
alph=poly(j); 
% Step 3 
phi=polyvalm(alph,a); 
% Step 4 
k=[0 0 1]*(inv(cam))*phi 
% 
% Pole placement -- using place function in MATLAB 
% 
disp('Pole placement -- using place function in MATLAB') 
p=[-2+2*sqrt(3)*i -2-2*sqrt(3)*i -10]; 
k=place(a,b,p)