ca2lq

 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;