State-Space Matrices

7 views (last 30 days)
Aysel Alimirzayeva
Aysel Alimirzayeva on 3 Nov 2022
Hello.Why do we write A-B*K1 in the example of LQR Control using State-Space Matrices in Matlab?Why is B negative?If anyone knows,can you please explain?

Accepted Answer

Sam Chak
Sam Chak on 3 Nov 2022
It's because of negative feedback.
The LQR function only computes .
Making a substitution
  2 Comments
Aysel Alimirzayeva
Aysel Alimirzayeva on 4 Nov 2022
@Sam Chak Than you very much.
Sam Chak
Sam Chak on 5 Nov 2022
Edited: Sam Chak on 5 Nov 2022
You're welcome, @Aysel Alimirzayeva.
Just for a note, this "state-feedback"
sys1 = ss(A-B*K1, B, C, D)
only works if your reference state is 0.
If your system is tracking a non-zero reference state, then a pre-compensator N is required to be placed at the reference input. N is just a constant gain to rescale input so that the output converges to 1 in the step response
sys2 = ss(A-B*K1, N*B, C, D)

Sign in to comment.

More Answers (1)

Sam Chak
Sam Chak on 5 Nov 2022
I added a simple example to show you. If you like this example, consider voting 👍 the Answer. Thanks!
Say, the reference state is 1.
A = [0 1; 2 3];
B = [0; 1];
K1 = lqr(A, B, eye(2), 1)
K1 = 1×2
4.2361 7.2979
sys1 = ss(A-B*K1, B, [1 0], 0)
sys1 = A = x1 x2 x1 0 1 x2 -2.236 -4.298 B = u1 x1 0 x2 1 C = x1 x2 y1 1 0 D = u1 y1 0 Continuous-time state-space model.
step(sys1, 20)
The step response shows that the output won't reach 1. Thus, a pre-compensator is needed:
N = 1/dcgain(sys1) % pre-compensator
N = 2.2361
sys2 = ss(A-B*K1, N*B, [1 0], 0)
sys2 = A = x1 x2 x1 0 1 x2 -2.236 -4.298 B = u1 x1 0 x2 2.236 C = x1 x2 y1 1 0 D = u1 y1 0 Continuous-time state-space model.
step(sys2, 20)
  1 Comment
Aysel Alimirzayeva
Aysel Alimirzayeva on 5 Nov 2022
Thank you very much for the detailed further explanation.

Sign in to comment.

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!