Description of fxp_alloc

fxp_alloc

PURPOSE

FXP_ALLOC - Control allocation using a fixed-point iterations.

SYNOPSIS

function [u,iter] = fxp_alloc(B,v,umin,umax,Wv,Wu,ud,gam,u,imax)

DESCRIPTION

 FXP_ALLOC - Control allocation using a fixed-point iterations.

  [u,iter] = fxp_alloc(B,v,umin,umax,[Wv,Wu,ud,gamma,u0,imax])

 Solves the weighted, bounded least-squares problem
 
   min ||Wu(u-ud)||^2 + gamma ||Wv(Bu-v)||^2

   subj. to  umin <= u <= umax
 
 using a fixed-point iteration algorithm.

  Inputs:
  -------
 B     control effectiveness matrix (k x m)
 v     commanded virtual control (k x 1)
 umin  lower position limits (m x 1)
 umax  upper position limits (m x 1)
 Wv    virtual control weighting matrix (k x k) [I]
 Wu    control weighting matrix (m x m) [I]
 ud    desired control (m x 1) [0]
 gamma weight (scalar) [1e3]
 u0    initial point (m x 1)
 imax  no. of iterations [100]
 
  Outputs:
  -------
 u     (approximately) optimal solution
 iter  no. of iterations

 See also: WLS_ALLOC, WLSC_ALLOC, IP_ALLOC, QP_SIM.

CROSS-REFERENCE INFORMATION

This function calls:

This function is called by:

• alloc_sim ALLOC_SIM - Control allocation simulation.
• qp_ca_sl Wrapper used in the QP control allocation Simulink block.

SOURCE CODE

0001 function [u,iter] = fxp_alloc(B,v,umin,umax,Wv,Wu,ud,gam,u,imax)
0002 
0003 % FXP_ALLOC - Control allocation using a fixed-point iterations.
0004 %
0005 %  [u,iter] = fxp_alloc(B,v,umin,umax,[Wv,Wu,ud,gamma,u0,imax])
0006 %
0007 % Solves the weighted, bounded least-squares problem
0008 %
0009 %   min ||Wu(u-ud)||^2 + gamma ||Wv(Bu-v)||^2
0010 %
0011 %   subj. to  umin <= u <= umax
0012 %
0013 % using a fixed-point iteration algorithm.
0014 %
0015 %  Inputs:
0016 %  -------
0017 % B     control effectiveness matrix (k x m)
0018 % v     commanded virtual control (k x 1)
0019 % umin  lower position limits (m x 1)
0020 % umax  upper position limits (m x 1)
0021 % Wv    virtual control weighting matrix (k x k) [I]
0022 % Wu    control weighting matrix (m x m) [I]
0023 % ud    desired control (m x 1) [0]
0024 % gamma weight (scalar) [1e3]
0025 % u0    initial point (m x 1)
0026 % imax  no. of iterations [100]
0027 %
0028 %  Outputs:
0029 %  -------
0030 % u     (approximately) optimal solution
0031 % iter  no. of iterations
0032 %
0033 % See also: WLS_ALLOC, WLSC_ALLOC, IP_ALLOC, QP_SIM.
0034   
0035 % Number of variables
0036   m = length(umin);
0037   
0038   % Set default values of optional arguments
0039   if nargin < 10
0040     imax = 100; % Heuristic value
0041     [k,m] = size(B);
0042     if nargin < 9, u = (umin+umax)/2; end
0043     if nargin < 8, gam = 1e3;         end
0044     if nargin < 7, ud = zeros(m,1);  end
0045     if nargin < 6, Wu = eye(m);      end
0046     if nargin < 5, Wv = eye(k);      end
0047   end
0048 
0049   % Change of variables.
0050   x = u - ud;
0051   xmin = umin - ud;
0052   xmax = umax - ud;
0053   
0054   % Compute the weight epsilon used by Burken et al.
0055   e = 1/(gam+1);
0056   
0057   % m = number of variables (actuators)
0058   m = size(B,2);
0059   
0060   % Notational differences: Q1 = Wu'Wu
0061   %                Q2 = Wv'Wv
0062   %                CzBu*-dz = v
0063   
0064   % Common factor.
0065   BzT_Q1 = B'*Wv'*Wv;
0066   % Just above eq. (25).
0067   H = (1-e) * (BzT_Q1 * B) + e*Wu'*Wu;
0068   % Eq. (25).
0069   w = 1/norm(H,'fro');
0070   % Matrices used for updating solution:
0071   % u[k] = sat(Fv-Gu[k-1])
0072   Fv = (1-e) * w * BzT_Q1 * v;
0073   G  = w*H - eye(m);
0074     
0075   for iter = 1:imax
0076     % Compute next point without considering the constraints.
0077     x = Fv - G*x;
0078     % Saturate to box constraints.
0079     x = max(xmin,min(xmax,x));
0080   end
0081     
0082   % Back to original variables.
0083   u = x + ud;

---

Generated on Wed 25-Aug-2004 14:38:35 by m2html © 2003