Linear Quadratic Regulator (LQR) controller for time-varying matrix B

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Linee Teh
Linee Teh 2020-4-19
评论： Ameer Hamza 2020-4-20
Hey, I am here to ask for help in the coding for LQR controller. The code is provided as follow.
t = linspace(0, 5640); % default 100 points
i = 83*pi/180; % inclination angle in rad
w0 = 0.0011; % angular velocity in rad/s
b_1 = sin(i)*cos(w0*t);
b_2 = -cos(i)*ones(size(t));
b_3 = 2*sin(i)*sin(w0*t);
J1 = 4;
J2 = 4.2;
J3 = 4;
% choose Q & R
Q = diag([0.001 0.001 0.001 0.001 0.001 0.001]);
R = diag([1e3 1e3 1e3]);
% Define system
dfw = [0 0 w0*(-J1+J2-J3)/J1; 0 0 0; w0*(J1-J2+J3)/J3 0 0];
dfq = [2*(w0^2)*(J3-J2)/J1 0 0; 0 0 0; 0 0 2*(w0^2)*(J1-J2)/J3];
ZERO = zeros(1,1,numel(b_1));
b1 = reshape(b_1, 1,1,[]);
b2 = reshape(b_2, 1,1,[]);
b3 = reshape(b_3, 1,1,[]);
A = [dfw dfq; 0.5*eye(3) zeros(3)];
C = eye(6);
D = zeros(6,3);
for n = 1:length(t)
B = [ ZERO b3/J1 b2/J1;
-b3/J2 ZERO b1/J2;
b2/J3 -b1/J3 ZERO;
zeros(3,3,numel(b_1))]
[K, S, e] = lqr(A,B,Q,R);
end
The state-space model has constant A, C and D matrices but B matrix is varying with time.
When I computed this code, error coming out by saying that "Error using lqr (line 43): The "lqr" command operates on a single model."
Can any expert on LQR controller help me to solve this question?

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Ameer Hamza
Ameer Hamza 2020-4-19
Since your matrix B has 100 slices in the 3rd dimension, you will have 100 K, S, and e matrices. Replace your for loop with
B = [ ZERO b3/J1 b2/J1;
-b3/J2 ZERO b1/J2;
b2/J3 -b1/J3 ZERO;
zeros(3,3,numel(b_1))];
for n = 1:length(t)
[K(:,:,n), S(:,:,n), e(:,:,n)] = lqr(A,B(:,:,n),Q,R);
end
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Linee Teh
Linee Teh 2020-4-20
Noted with thanks. By ignoring the disturbance, the coding that you have provided is very useful and easy to understand!

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