LQR regulator doesn't regulate the system

Hello,

I used the function lqr to get a regulator that helps me to regulate my system.

My function is:

r = lqr(A,B,Q,R)

With this function i get a regulator that gives me two conjugate complex poles. Thus the system is not regulated, but continues to oscillate. I don't get any errors or warnings.

How can I fix this problem? .

[Sam Chak]

Hi, in order to investigate, could you show the matrices for A, B, Q, R?

Also provide the desired settling time if you do not have any specific constraint for the control input u.

If (A, B) is controllable, then the desired settling time is sufficient for the eigenvalue design.

[DoctorCrow]

Hi,

i got the followingen matrices:

A = [0 1 0 0 ; 0 0 14.72 0 ; 0 0 0 1 ; 0 0 -3.066 0]

B = [0 ; 0.001 ; 0 ; -0.000125]

Q = [1 0 0 0 ; 0 1 0 0 ; 0 0 1 0 ; 0 0 0 1]

R = [1]

I checked, that (A,B) is controllable.

Answer 1

[Sam Chak]

Hi @DoctorCrow

Your selection of and results in the closed-loop eigenvalues having negative real parts very close to zero. Consequently, although the system states converge to steady-state, they also oscillate for a much longer period of time.

A = [0 1 0 0; 0 0 14.72 0; 0 0 0 1; 0 0 -3.066 0];
B = [0; 0.001; 0; -0.000125];
Q = [1 0 0 0; 0 1 0 0; 0 0 1 0; 0 0 0 1];
R = 1;
[K1, P, e] = lqr(A, B, Q, R)

K1 = 1×4

1.0000 70.7226 4.6434 333.8955

P = 4×4

1.0e+06 * 0.0001 0.0025 0.0003 0.0120 0.0025 0.1769 0.0116 0.8491 0.0003 0.0116 0.1401 0.0557 0.0120 0.8491 0.0557 4.1219

e = 
  -0.0004 + 1.7510i
  -0.0004 - 1.7510i
  -0.0141 + 0.0141i
  -0.0141 - 0.0141i

sys1 = ss(A-B*K1, B, eye(4), 0);
step(sys1)

Increasing Q should improve the system response, as shown in the following plot.

Q = 1e7*eye(4);
[K2, P, e] = lqr(A, B, Q, R)

K2 = 1×4

1.0e+03 * 3.1623 4.9643 1.8176 7.6882

P = 4×4

1.0e+08 * 0.1570 0.0732 0.2431 0.3328 0.0732 0.1397 0.0489 0.7207 0.2431 0.0489 5.3231 0.2455 0.3328 0.7207 0.2455 5.1509

e = 
  -0.2486 + 1.1213i
  -0.2486 - 1.1213i
  -1.3865 + 0.0000i
  -2.1197 + 0.0000i

sys2 = ss(A-B*K2, B, eye(4), 0);
step(sys2)

[DoctorCrow]

Hi @Sam Chak,

thank u very much for your help.

[Sam Chak]

You are welcome, @DoctorCrow. If you find the solution helpful, please consider accepting ✔ and voting 👍 on the answer. Thanks a bunch! 🙏