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Comment:Added R and P arguments for LQ ppp
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SHA3-256: 13958eaa9b1296ba6b4208f3f2d71abfb86aafbde60c9c858823135b6dda13a2
User & Date: gawthrop@users.sourceforge.net on 2002-12-11 12:38:25
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Context
2002-12-11
12:41:01
Solves ARE for PPP check-in: 09c187373a user: gawthrop@users.sourceforge.net tags: origin/master, trunk
12:38:25
Added R and P arguments for LQ ppp check-in: 13958eaa9b user: gawthrop@users.sourceforge.net tags: origin/master, trunk
12:36:12
Added R and P arguments so that ppp_quad_lin can be used here check-in: cf62983f7f user: gawthrop@users.sourceforge.net tags: origin/master, trunk
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Modified mttroot/mtt/lib/control/PPP/ppp_lin.m from [342b9867d7] to [db890617e0]. 

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function [k_x,k_w,K_x,K_w,Us0,J_uu,J_ux,J_uw,J_xx,J_xw,J_ww,y_u,cond_uu] = ppp_lin(A,B,C,D,A_u,A_w,tau,Q,max_cond);
  ## usage:   [k_x,k_w,K_x,K_w,Us0,J_uu,J_ux,J_uw,J_xx,J_xw,J_ww,y_u,cond_uu] = ppp_lin(A,B,C,D,A_u,A_w,tau,Q,max_cond)
function [k_x,k_w,K_x,K_w,Us0,J_uu,J_ux,J_uw,J_xx,J_xw,J_ww,y_u,cond_uu] = ppp_lin(A,B,C,D,A_u,A_w,tau,Q,R,P,max_cond);
  ## usage:   [k_x,k_w,K_x,K_w,Us0,J_uu,J_ux,J_uw,J_xx,J_xw,J_ww,y_u,cond_uu] = ppp_lin(A,B,C,D,A_u,A_w,tau,Q,R,P,max_cond)
  ##
  ## Linear PPP (Predictive pole-placement) computation 
  ## INPUTS:
  ## A,B,C,D: system matrices
  ## 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.
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  ## Check some dimensions
  [n_x,n_u,n_y] = abcddim(A,B,C,D);
  if (n_x==-1)
    return
  endif

  ## Default Q
  ## Default Q (output weight)
  if nargin<8
    Q = ones(n_y,1);
  endif

  ## Default order
  ## Default R (input weight)
  if nargin<9
    R = zeros(n_u,1);
  endif

  ## Default P (terminal weight)
  if nargin<10
    P = zeros(n_x,n_x);
  endif

  ## Default condittion number
  if nargin<11
    max_cond = 1e20;
  endif
  
  [n_U,m_U] = size(A_u);
  if (n_U != n_u*m_U)&&(n_U != m_U)
    error("A_u must be square or have N_u rows and N_u/n_u columns");
  endif
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  endif
  

  [n_t,m_t] = size(tau);
  if n_t != 1
    error("tau must be a row vector");
  endif
  dt = tau(2) - tau(1);		# Sample interval

  [n_Q,m_Q] = size(Q);
  if ((m_Q != 1)&&(m_Q != m_t)) || (n_Q != n_y)
    error("Q must be a column vector with one row per system output");
  endif

  if (m_Q == 1)			# Convert to vector Q(i)
    Q = Q*ones(1,m_t);		# Extend to cover full range of tau
  endif
  
  ##Set up initial states
  u_0 = ones(m_U,1);
  w_0 = ones(m_W,1);

  ## Find y_U - derivative of y* wrt U.
  i_U = 0;
  x_0 = zeros(n_x,1);		# This is for x=0
  y_u = [];			# Initialise
  x_u_t = [];			# Initialise
  Us = [];			# Initialise
  for i=1:n_U			# Do for each input function U*_i
    dU = zeros(n_U,1);
    dU(++i_U) = 1;		# Create dU/dU_i 
    [ys,us] = ppp_ystar (A,B,C,D,x_0,A_u,dU,tau); # Find ystar and ustar
    [ys,us,xs] = ppp_ystar (A,B,C,D,x_0,A_u,dU,tau); # Find ystar and ustar
    y_u = [y_u ys'];		# Save y_u (y for input u)  with one row for each t.
    Us = [Us us'];		# Save u (input)  with one row for each t.
    Us = [Us us'];		# Save u (input)  with one row for each
				# t.
    x_u_t = [x_u_t xs(:,m_t)];	# x_u at terminal time
  endfor

  Ws = [];			# Initialise
  if n_W>0
    ## Find w*
    i_W = 0;
    x_0 = zeros(n_x,1);		# This is for x=0
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  ## Add up cost derivatives for each output but weighted by Q.
  ## Scaled by time interval
  ## y_u,y_x and Ws should really be 3D matrices, but instead are stored
  ## with one row for each time and one vector (size n_y) column for
  ## each element of U

  ## Scale Q
  Q = Q/m_t;			# Scale by T/dt = number of points
  Q = Q*dt;			# Scale to give correct units
  ## Non-w bits
  for i = 1:n_y			# For each output
    QQ = ones(n_U,1)*Q(i,:);	# Resize Q
    J_uu = J_uu + (QQ .* y_u(:,i:n_y:n_yu)') * y_u(:,i:n_y:n_yu);
    J_ux = J_ux + (QQ .* y_u(:,i:n_y:n_yu)') * y_x(:,i:n_y:n_yx);
    QQ = ones(n_x,1)*Q(i,:);	# Resize Q
    J_xx = J_xx + (QQ .* y_x(:,i:n_y:n_yx)') * y_x(:,i:n_y:n_yx);
  endfor

  ## Input weighting (scalar for the moment)
  if (n_u>1)
    warning("Sorry, cant do n_u>1 just now");
  endif

  ## Scale R
  R = R*dt;			# Scale to give correct units
  for i = 1:m_t
    Ust = Us(i,:);
    J_uu = J_uu + Ust'*R*Ust;
  endfor
  
  ## Exit if badly conditioned
  cond_uu = cond(J_uu);
  if cond_uu>max_cond
    error(sprintf("J_uu is badly conditioned. Condition number = 10^%i",log10(cond_uu)));
  endif

  ## Terminal constraint
  tau_last = tau(m_t);
  x_x_t = expm(A*tau_last);	# deriv. of x*(tau_2) wrt x(0)
  J_ux = J_ux + x_u_t'*P*x_x_t;
  J_uu = J_uu + x_u_t'*P*x_u_t;

  ## w bits
  if n_W>0
    for i = 1:n_y			# For each output
      QQ = ones(n_U,1)*Q(i,:);	# Resize Q
      J_uw = J_uw + (QQ .* y_u(:,i:n_y:n_yu)') * Ws (:,i:n_y:n_yw);
      QQ = ones(n_x,1)*Q(i,:);	# Resize Q
      J_xw = J_xw + (QQ .* y_x(:,i:n_y:n_yx)') * Ws (:,i:n_y:n_yw);
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