State Space with Disturbance

Hello, I am working on a project which involves disturbance term w in the state equation as below.

I want to convert this standard state space form which is Ax+Bu to use lqr function and then using R,Q to get K,S,e terms. Is there a way to obtain K,S,e in this case or how can I convert to standard form? A, B1, and B2 matrices are as below.

if true
  % code
M1f= 1; %kg
M1r= 1; %kg
M2=2.9; %kg
J=2.18; %kgm²
l=3; %m
lf=1.5; %m
K1f=2717; %N/m
K1r=2717; %N/m
C1f=5; %N/m
C1r=5; %N/m
K2f=970; %N/m
K2r=970; %N/m
C2f=7.5; %Ns/m
C2r=7.5; %Ns/m
m61= -K2f/M1f;
m62= 0;
m63= K1f/M1f;
m64= 0;
m65= (-K2f*lf)/M1f;
m66= (-C1f-C2f)/M1f;
m67= 0;
m68= C2f/M1f;
m69= (-C2f*lf)/M1f;
m71= 0;
m72= -K2r/M1r;
m73= 0;
m74= K1r/M1r;
m75= K2r*(l-lf)/M1r;
m76= 0;
m77= (-C1r-C2r)/M1r;
m78= C2r/M1r;
m79= (C2r*(l-lf))/M1r;
m81= K2f/M2;
m82= K2r/M2;
m83= 0;
m84= 0;
m85= ((K2f*lf)-K2r*(l-lf))/M2;
m86= C2f/M2;
m87= C2r/M2;
m88= (-C2f-C2r)/M2;
m89= ((C2f*lf)-C2r*(l-lf))/M2;
m91= (-K2f*lf)/J;
m92= (K2r*(l-lf))/J;
m93= 0;
m94= 0;
m95= (-K2f*(lf^2)-K2r*((l-lf)^2))/J;
m96= (-C2f*lf)/J;
m97= (C2r*(l-lf))/J;
m98= ((C2f*lf)-(C2r*(l-lf)))/J;
m99= ((-C2f*(lf^2))-(C2r*((l-lf)^2)))/J;
A=[0 0 0 0 0 1 0 -1 0;0 0 0 0 0 0 1 -1 0;0 0 0 0 0 -1 0 0 0;0 0 0 0 0 0 -1 0 0;...
  0 0 0 0 0 0 0 0 1;m61 m62 m63 m64 m65 m66 m67 m68 m69;...
  m71 m72 m73 m74 m75 m76 m77 m78 m79;m81 m82 m83 m84 m85 m86 m87 m88 m89;...
  m91 m92 m93 m94 m95 m96 m97 m98 m99];
B=[0 0 0 0;0 0 0 0;1 0 0 0;0 1 0 0;0 0 0 0;C1f/M1f 0 -1/M1f 0;0 C1r/M1r 0 -1/M1r;...
  0 0 1/M2 1/M2;0 0 -lf/J (l-lf)/J];
B1=B(:,1:2);
B2=B(:,3:4);
C=[0 0 0 0 0 0 0 1 0;0 0 0 0 0 0 0 0 1];
D=[0 0];
end
Thank you in advance

Answer 1

[Tim Grunert]

Hello, just ignore B1 to use lqr(...) there is no impact from the controller to w anyway.

Maybe you need some feed forward for w and/or the setpoint to be statically exact. Or you use different approaches like lqi(...) instead.