CA2LQ - Incorporate control allocator into LQ controller. R1 = ca2lq(B,R2,Wu) Given a dynamic system . . x = Ax + B1u, B1 = B2*B <=> x = Ax + B2v, v = Bu this function calculates the weighting matrix R1 such that the design criterium min Integral (x'Qx + u'R1u) dt (LQ control) u gives the same linear optimal control law as min Integral (x'Qx + v'R2v) dt (LQ control) v min ||Wu u|| subj. to Bu = v (control allocation) u Controller structures: x ------- u x ------- v ---- u --->| Q, R1 |---> <=> --->| Q, R2 |--->| Wu |---> ------- ------- ---- LQ ctl LQ ctl Ctl alloc See also LQ2CA, LQR, WPINV.
0001 function R1 = ca2lq(B,R2,Wu) 0002 0003 % CA2LQ - Incorporate control allocator into LQ controller. 0004 % 0005 % R1 = ca2lq(B,R2,Wu) 0006 % 0007 % Given a dynamic system 0008 % . . 0009 % x = Ax + B1u, B1 = B2*B <=> x = Ax + B2v, v = Bu 0010 % 0011 % this function calculates the weighting matrix R1 such that the 0012 % design criterium 0013 % 0014 % min Integral (x'Qx + u'R1u) dt (LQ control) 0015 % u 0016 % 0017 % gives the same linear optimal control law as 0018 % 0019 % min Integral (x'Qx + v'R2v) dt (LQ control) 0020 % v 0021 % 0022 % min ||Wu u|| subj. to Bu = v (control allocation) 0023 % u 0024 % 0025 % Controller structures: 0026 % 0027 % x ------- u x ------- v ---- u 0028 % --->| Q, R1 |---> <=> --->| Q, R2 |--->| Wu |---> 0029 % ------- ------- ---- 0030 % LQ ctl LQ ctl Ctl alloc 0031 % 0032 % See also LQ2CA, LQR, WPINV. 0033 0034 % Thesis, Theorem 10.4: 0035 R1 = Wu^2+B'*(R2-inv(B*inv(Wu)^2*B'))*B;