function [ys,us,xs,xu,AA] = ppp_ystar (A,B,C,D,x_0,A_u,U,tau) ## usage: [ys,us,xs,xu,AA] = ppp_ystar (A,B,C,D,x_0,A_u,U,tau) ## ## Computes open-loop moving horizon variables at time tau ## Inputs: ## A,B,C,D System matrices ## x_0 Initial state ## A_u composite system matrix for U* generation ## one square matrix (A_ui) row for each system input ## each A_ui generates U*' for ith system input. ## OR ## A_u square system matrix for U* generation ## same square matrix for each system input ## U Column vector of optimisation coefficients ## tau Row vector of times at which outputs are computed ## Outputs: ## ys y*, one column for each time tau ## us u*, one column for each time tau ## xs x*, one column for each time tau ## xu x_u, one column for each time tau ## AA The composite system matrix ## Copyright (C) 1999 by Peter J. Gawthrop ## $Id$ if (size(A)>0) [n_x,n_u,n_y] = abcddim(A,B,C,D); # System dimensions else n_x = 0; n_y = 0; n_u = 0; endif no_system = n_x==0; [n,m] = size(A_u); # Size of composite A_u matrix square = (n==m); # Is A_u square? n_U = m; # functions per input [n,m] = size(U); if (m != 1) error("U must be a column vector"); endif if n_u>0 if n_u!=length(U)/n_U error("U must be a column vector with n_u*n_U components"); endif else n_u = length(U)/n_U; # Deduce n_u from U if no system endif [n_x0,m_x0] = size(x_0); if n_x0!=n_x error(sprintf("x_0 must be a column with length %i", n_x)); endif [n,m]=size(tau); if (n != 1 ) error("tau must be a row vector of times"); endif if square # Then same A_u for each input ## Reorganise vector U into matrix Utilde Utilde = []; for i=1:n_u j = (i-1)*n_U; range = j+1:j+n_U; Utilde = [Utilde; U(range,1)']; endfor ## Composite A matrix if no_system AA = A_u; else Z = zeros(n_U,n_x); AA = [A B*Utilde Z A_u]; endif xx_0 = [x_0;ones(n_U,1)]; # Composite initial condition else # Different A_u on each input ## Reorganise vector U into matrix Utilde Utilde = []; for i=1:n_u j = (i-1)*n_U; k = (n_u-i)*n_U; range = j+1:j+n_U; Utilde = [Utilde; zeros(1,j), U(range,1)', zeros(1,k)]; endfor ## Create the full A_u matrix (AA_u) with the A_i s on the diagonal # AA_u = []; # for i = 1:n_u # AA_u = ppp_aug(AA_u,ppp_extract(A_u,i)); # endfor AA_u = ppp_inflate(A_u); ## Composite A matrix if no_system AA = AA_u; else Z = zeros(n_U*n_u,n_x); AA = [A B*Utilde Z AA_u]; endif xx_0 = [x_0;ones(n_U*n_u,1)]; # Composite initial condition endif ## Initialise xs = []; # x star xu = []; # x star ys = []; # y star us = []; # u star n_xx = length(xx_0); # Length of composite state ## Compute the star variables for t=tau xxt = expm(AA*t)*xx_0; # Composite state xst = xxt(1:n_x); # x star xut = xxt(n_x+1:n_xx); # x star if length(C)>0 yst = C*xst; # y star else yst = []; endif ust = Utilde*xut; # u star xs = [xs xst]; # x star xu = [xu xut]; # x star ys = [ys yst]; # y star us = [us ust]; # u star endfor endfunction