VCL Safe Mode Eclipse Control Law By Steve Rogers
Introduction This is a conceptual check of the safehold lqr control law study performed by the Glory group contained in DN-Glory-ACS-014, section 5.2. The linearized equations of motion and lqr control law are − J −1 K λ 0ω J −1 ω = H
0
u = −[ K ωK B − J −1 K λ A = 0
+ u 0H −I
ω K h ] H 0 0
J −1 B = − I
0 , K λ = − λ 0
λ 0 0 0
0 0
with the C as identity matrix and D = 0 complete the linear model.
The values for the inertial matrix are not those used in the study, but are the more accurate ones used in the dynamic model. The values are: J = [195.36 1.527 1.527 173.38 1.081 -0.580
1.081 -0.580 303.908];
These more accurate values require a state space development instead of the transfer function development used in DN-Glory-ACS-014. The LQR formulation requires the matrices A, B, Q, and R. A and B are given above, whereas, Q and R (the penalty matrices) are
1 / ω max 2 0 0 0 2 0 1 / ω 0 0 max Q= 2 0 0 1 / hmax 0 2 , 0 0 1 / hmax 0 1 / umax 2 0 R= 2 1 / umax 0
ωmax = 0.002 r/s hmax = 0.3 Nms umax = 0.007 Nm In the first comparison the momentum is included, but is not included in the second comparison. A snippet of matlab code showing the design is shown below. Q = diag([tempw tempw tempw temph temph temph]); R = diag([tempu tempu tempu]); K = lqr(A,B,Q,R); % lqr control law gain