How do i use LQR control in state space form?

I am trying to simulate the step response of the quarter car. for this i am using LQR controller. I have built the state space matrix in the form:

    Xdot=AX+BU+GW;
    Y = CX+DU;

Here, 'W' is the controller force and 'U' is the excitation displacement from the ground. I gave Matrices A, B, C and D matrices, where D is null matrix and C as unity. The problem is where do I put the matrix 'G' which is GAIN matrix from the controller. until now i have the following:

sys = ss(A,B,C,D);

[K,P] = lqr(A,B,Q,R,N); %where N=0

%we will obtain K and P values

sys = ss(A-B*K,B,C,D);

step(sys)

Theoretically, my G matrix is:

G = [U/m1 0; -U/m2 0];

Where must be the 'G' matrix be inputted?

Answer 1

[Azzi Abdelmalek]

You have first to tell us what are the inputs and outputs of your system. Post the parameters representing your system, and post how did you use the lqr function.

[aniket jakati]

The following is the code what i have done until now. I am getting the step response but i am sure whether 'G' matrix must be added to 'B' matrix or not. If yes then why?

m1 = 250;
k = 16812;
c = 1000;
m2 = 50;
kt = 190000;

A = [-c/m1 -k/m1 c/m1 k/m1; 1 0 0 0; c/m2 k/m2 -c/m2 -(k+kt)/m2; 0 0 1 0];
G = [0; 0; kt/m2; 0];
B = [1/m1; 0; -1/m2; 0];
C = [0 1 0 0];
D = 0;

sys = ss(A,B+G,C,D);
step(0.1*sys) % Passive suspension step response
hold on

x = 1000;
Q = diag([x 0 0 0]);
R = 0.001;

[K,P] = lqr(A,B+G,Q,R);

sys = ss(A-B*K,B+G,C,D);

t=0:0.05:5;
step(0.1*sys,t) % Active suspension step response
grid on

Also is it necessary here that the 'Q' matrix must be only Q = diag([x 0 0 0]) and not Q = diag([x x x x]) ?

Sorry for the trouble.

Answer 2

[Kwin]

Both 'W' and 'U' are inputs to your system, however I assume that you will not be able to influence 'U', so it can be considered a disturbance. So 'G' is actually your 'B' matrix. LQR does not consider this disturbance, so you might want to use LQG instead.

Answer 3

[duc nguyentrong]

I am researching about active suspension system using LQR in matlab-simulink software to simulate. I have the code of a quarter-car model but I have a problem running. Hope everyone can point out its faults. thanks