Costa asked . 2021-10-04

Cannot compute Riccati solution for LQI controller

u=0.5;
v=0;
omega=0;
m=50;
psi=0;
c_d_Long=0.5;
c_dr=0.6;
c_d_Lat=1;
rho=1000;
L=1.2;
W=0.275;
H=0.416;
I=38;

A =[0, 0, - v*cos(psi) - u*sin(psi),    cos(psi),      -sin(psi),    0;
0, 0,   u*cos(psi) - v*sin(psi),   sin(psi),  cos(psi),   0;
0, 0, 0, 0,    0,  1;
0, 0,  0, -((3*W*c_d_Long*rho*u^2*1)/10 + (3*W*c_d_Long*rho*u*sign(u))/10)/m, omega,      v;
0, 0, 0,  -omega, -((3*L*c_d_Lat*rho*v^2*1)/10 + (3*L*c_d_Lat*rho*v*sign(v))/10)/m, -u;
0, 0,  0,   0,     0, - (300*L*c_dr*omega^2*rho*1)/38347 - (300*L*c_dr*omega*rho*sign(omega))/38347];


B =[ 0,          0,         0,          0,         0,         0,          0,          0;
  0,          0,         0,          0,         0,         0,          0,          0;
   0,          0,         0,          0,         0,         0,          0,          0;
 1/m,        1/m,       1/m,        1/m,         0,         0,          0,          0;
    0,          0,         0,          0,       1/m,       1/m,        1/m,        1/m;
825/76694, -825/76694, 825/76694, -825/76694, 325/38347, 325/38347, -325/38347, -325/38347];

C=[1 0 0 0 0 0;
    0 1 0 0 0 0;
    0 0 1 0 0 0;
    0 0 0 1 0 0;
    0 0 0 0 0 0;
    0 0 0 0 0 1];
    
SYS=ss(A,B,C,0);

Co = ctrb(SYS);
C=rank(Co);

Ob=obsv(SYS);
O=rank(Ob);

Q=eye(12);
R=eye(8);
klqi=lqi(SYS,Q,R)

As you can see I have a 6-dimension state with 8 input and 6 output. When i Run this code i have the following error:

Cannot compute a stabilizing LQR gain (the Riccati solution S and gain matrix K are infinite).
This could be because:
  * A has unstable modes that are not controllable through B,
  * Q,R,N values are too large,
  * [Q N;N' R] is indefinite,
  * The E matrix in the state equation is singular.

I had the same issue in the past but every time the problem was the controllabilty of the state but in this case I haven't this problem (rank of the controllabilty matrix= 6)

lqi , riccati solution