Description of cgi_alloc

cgi_alloc

PURPOSE

CGI_ALLOC - Control allocation based on cascading generalized inverses.

SYNOPSIS

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

DESCRIPTION

 CGI_ALLOC - Control allocation based on cascading generalized inverses.

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

 Approximately solves the bounded sequential least-squares problem

   min ||Wu(u-ud)||   subj. to   u in M

 where M is the set of control signals solving

   min ||Wv(Bu-v)||   subj. to   umin <= u <= umax

 using the method of cascading generalized inverses.

  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]
 imax  max no. of iterations [100]

  Outputs:
  --------
 u     (approximately) optimal solution
 iter  no. of iterations (= no. of pseudoinverse solutions computed)

 See also: SLS_ALLOC, MLS_ALLOC, QP_SIM.

CROSS-REFERENCE INFORMATION

This function calls:

• pinv_sol PINV_SOL - Compute pseudoinverse solution.

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] = cgi_alloc(B,v,umin,umax,Wv,Wu,ud,imax)
0002 
0003 % CGI_ALLOC - Control allocation based on cascading generalized inverses.
0004 %
0005 %  [u,iter] = cgi_alloc(B,v,umin,umax,[Wv,Wu,ud,imax])
0006 %
0007 % Approximately solves the bounded sequential least-squares problem
0008 %
0009 %   min ||Wu(u-ud)||   subj. to   u in M
0010 %
0011 % where M is the set of control signals solving
0012 %
0013 %   min ||Wv(Bu-v)||   subj. to   umin <= u <= umax
0014 %
0015 % using the method of cascading generalized inverses.
0016 %
0017 %  Inputs:
0018 %  -------
0019 % B     control effectiveness matrix (k x m)
0020 % v     commanded virtual control (k x 1)
0021 % umin  lower position limits (m x 1)
0022 % umax  upper position limits (m x 1)
0023 % Wv    virtual control weighting matrix (k x k) [I]
0024 % Wu    control weighting matrix (m x m) [I]
0025 % ud    desired control (m x 1) [0]
0026 % imax  max no. of iterations [100]
0027 %
0028 %  Outputs:
0029 %  --------
0030 % u     (approximately) optimal solution
0031 % iter  no. of iterations (= no. of pseudoinverse solutions computed)
0032 %
0033 % See also: SLS_ALLOC, MLS_ALLOC, QP_SIM.
0034   
0035 % Set default values of optional arguments
0036   if nargin < 8
0037     imax = 100; % Heuristic value
0038     [k,m] = size(B);
0039     if nargin < 7, ud = zeros(m,1); end
0040     if nargin < 6, Wu = eye(m);     end
0041     if nargin < 5, Wv = eye(k);     end
0042   end
0043 
0044   iter = 1;
0045   % Compute the optimal solution disregarding the inequality
0046   % constraints.
0047   A = Wv*B/Wu;
0048   b = Wv*(v-B*ud);
0049   u = ud + Wu\pinv_sol(A,b);
0050 
0051   % Find indeces of infeasible variables.
0052   i_min = u < umin;
0053   i_max = u > umax;
0054   % Set these variables to their limits.
0055   u(i_min) = umin(i_min);
0056   u(i_max) = umax(i_max);
0057   % Let the remaining variables be free.
0058   i_free = ~(i_min | i_max);
0059   
0060   % If the preceeding pseudoinverse solution yielded some variables
0061   % infeasible, redistribute the control effect to the remaining free
0062   % variables, if there are any.
0063   while any([i_min ; i_max]) & any(i_free) & (iter<imax);
0064     iter = iter + 1;
0065     % Now solve for optimal values of the remaining free variables.
0066     % See 2002-02-07.
0067     Wu1 = Wu(i_free,:);
0068     Wu11 = Wu1(:,i_free);
0069     B1 = B(:,i_free);
0070     
0071     A = Wv*B1/Wu11;
0072     b = Wv*(v-B*u-B1*(Wu11\(Wu1*(ud-u))));
0073     % Solve for optimal perturbation.
0074     p1 = Wu11\(pinv_sol(A,b)+Wu1*(ud-u));
0075     % Update solution
0076     u(i_free) = u(i_free) + p1;
0077     
0078     % Find indeces of _new_ infeasible components.
0079     i_min = u < umin;
0080     i_max = u > umax;
0081     % Set these variables to their limits.
0082     u(i_min) = umin(i_min);
0083     u(i_max) = umax(i_max);
0084     % Remove these from the free variables.
0085     i_free = i_free & ~(i_min | i_max);
0086   end

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