Cannot compute Riccati solution for LQI controller

Hi to everybody!

This is my code right now:

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)

Hope somebody can help me.

Thank you in advance.

Answer 1

[Paul]

I'm not sure it's relevant, but should the fifth row of SYS.c be all zeros?

I suspect the issue is with stabilizability of the augmented system. Note that the code as posted is checking controllability of the plant, but the LQR solution requires stabilizability of the augmented plant, i.e., the plant combined with the integral control. I think the augmented system (not incuding the reference input) can be formed as

sysaugmented = series(SYS,ss(zeros(6),eye(6),eye(6),0)); % with SYS.c = eye(6)

Then the rank of the controllability matrix is

>> rank(ctrb(sysaugmented)) 

ans =

     9

So the augmented system is not controllable. The eigenvalues are:

>> eig(sysaugmented)

ans =

         0
         0
         0
         0
         0
         0
         0
         0
         0
   -0.6188
         0
         0

So at least two of the uncontrollable modes are on the imaginary axis, which (I think) means the augmented system is not stabilizable, which violates one of the conditions for the solution to exist as stated here:

doc lqi

[Leonardo Costa]

Aaah okok.

The C in my code is an eye(6), the fifth row with all zeros is because I was trying to change something and I Past the wrong C matrix ahaha.

I understood what you mean, do you know if there is a chance to stabilize an augmented state like the mine?

I'm not fully immersed in the theory of the LQI, but I understood that the LQI command creates an augmented state doubling the dimensions of the initial state and then it solves like a LQR. Is it correct?

In my case I don't need an augmented state of 12x12 but only of 7x7. Can I create manually the augmented state use the LQR command?

Hope that I explained well with my poor english 😅.

Thank you in advance

[Paul]

Yes, you can augment the plant manually with the integrator and then use the lqr() command to compute the feedback gains,subject to the standard consraints for LQR solutions.

[Leonardo Costa]

Thank you very much!