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function [y,x,u,t,U,U_c,U_l] = ppp_nlin_sim (system_name,A_u,tau,t_ol,N_ol,w,extras)
## usage: [y,x,u,t,U,U_c,U_l] = ppp_nlin_sim (system_name,A_u,tau,t_ol,N,w)
##
##
## Simulate nonlinear PPP
## Copyright (C) 2000 by Peter J. Gawthrop
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function [y,x,u,t,UU,UU_c,UU_l] = ppp_nlin_sim (system_name,i_ppp,i_par,A_u,w_s,N_ol,extras)
## usage: [y,x,u,t,U,U_c,U_l] = ppp_nlin_sim (system_name,A_u,tau,t_ol,N,w)
##
##
## Simulate nonlinear PPP
## Copyright (C) 2000 by Peter J. Gawthrop
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endif
## Names
s_system_name = sprintf("s%s", system_name);
## System details
par = eval(sprintf("%s_numpar;", system_name));
x_0 = eval(sprintf("%s_state;", system_name))
[n_x,n_y,n_u] = eval(sprintf("%s_def;", system_name));
## Sensitivity system details
sympars = eval(sprintf("%s_sympar;", s_system_name));
pars = eval(sprintf("%s_numpar;", s_system_name));
## Times
n_tau = length(tau);
dtau = tau(2)-tau(1); # Optimisation sample time
Tau = [0:dtau:tau(n_tau)+eps]; # Time in the moving axes
n_Tau = length(Tau);
dt = t_ol(2)-t_ol(1);
n_t = length(t_ol);
T_ol = t_ol(n_t)+dt
## Weight and moving-horizon setpoint
weight = [zeros(n_y,n_Tau-n_tau), ones(n_y,n_tau)];
ws = w*ones(1,n_tau);
## Create input basis functions
[n_U,junk] = size(A_u);
## For moving horizon
eA = expm(A_u*dtau);
u_star_i = ones(n_U,1);
u_star_tau = [];
for i = 1:n_Tau
u_star_tau = [u_star_tau, u_star_i];
u_star_i = eA*u_star_i;
endfor
## and for actual implementation
eA = expm(A_u*dt);
u_star_i = ones(n_U,1);
u_star_t = [];
for i = 1:n_t
u_star_t = [u_star_t, u_star_i];
u_star_i = eA*u_star_i;
endfor
if extras.verbose
title("U*(tau)")
xlabel("tau");
plot(Tau,u_star_tau)
endif
## Check number of inputs adjust if necessary
if n_u>n_U
disp(sprintf("Augmenting inputs with %i zeros", n_u-n_U));
u_star_tau = [u_star_tau; zeros(n_u-n_U, n_Tau)];
u_star_t = [u_star_t; zeros(n_u-n_U, n_t)];
endif
if n_u<n_U
error(sprintf("n_U (%i) must be less than or equal to n_u (%i)", \
n_U, n_u));
endif
## Indices of U and sensitivities
FREE = "[";
free_name = "ppp";
for i=1:n_U
FREE = sprintf("%ssympars.%s_%i sympars.%s_%is", FREE, free_name, \
i, free_name, i);
if i<n_U
symbol = "; ";
else
symbol = "];";
endif
FREE = sprintf("%s%s", FREE, symbol);
endfor
FREE
free = eval(FREE);
[k_x,k_w,K_x,K_w] = ppp_lin(A,B,C,D,A_u,0,tau);
## Main simulation loop
y = [];
x = [];
u = [];
t = [];
t_last = 0;
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endif
## Names
s_system_name = sprintf("s%s", system_name);
## System details -- defines simulation within ol interval
par = eval(sprintf("%s_numpar;", system_name));
simpar = eval(sprintf("%s_simpar;", system_name))
x_0 = eval(sprintf("%s_state(par);", system_name))
[n_x,n_y,n_u] = eval(sprintf("%s_def;", system_name));
## Sensitivity system details -- defines moving horizon simulation
simpars = eval(sprintf("%s_simpar;", s_system_name))
sympars = eval(sprintf("%s_sympar;", s_system_name));
pars = eval(sprintf("%s_numpar;", s_system_name));
## Times
## -- within opt horizon
n_Tau = round(simpars.last/simpars.dt)
dtau = simpars.dt
Tau = [0:n_Tau-1]'*dtau;
[n_tau,n_w] = size(w_s);
tau = Tau(n_Tau-n_tau+1:n_Tau);
w = w_s(length(w_s)); # Final value of setpoint
## -- within ol interval
n_t = round(simpar.last/simpar.dt)
dt = simpar.dt;
t_ol = [0:n_t-1]'*dt;
T_ol = n_t*dt;
## Weight and moving-horizon setpoint
# weight = [zeros(n_y,n_Tau-n_tau), ones(n_y,n_tau)];
# w_s = w*ones(n_tau,1);
## Create input basis functions
[n_U,junk] = size(A_u);
## For moving horizon
eA = expm(A_u*dtau);
u_star_i = ones(n_U,1);
u_star_tau = [];
for i = 1:n_Tau
u_star_tau = [u_star_tau; u_star_i'];
u_star_i = eA*u_star_i;
endfor
## and for actual implementation
eA = expm(A_u*dt);
u_star_i = ones(n_U,1);
u_star_t = [];
for i = 1:n_t
u_star_t = [u_star_t; u_star_i'];
u_star_i = eA*u_star_i;
endfor
if extras.verbose
title("U*(tau)")
xlabel("tau");
plot(Tau,u_star_tau)
title("U*(t)")
xlabel("t_ol");
plot(t_ol,u_star_t)
endif
## Check number of inputs adjust if necessary
if n_u>n_U
disp(sprintf("Augmenting inputs with %i zeros", n_u-n_U));
u_star_tau = [u_star_tau; zeros(n_u-n_U, n_Tau)];
u_star_t = [u_star_t; zeros(n_u-n_U, n_t)];
endif
if n_u<n_U
error(sprintf("n_U (%i) must be less than or equal to n_u (%i)", \
n_U, n_u));
endif
## Compute linear gains
[A,B,C,D] = eval(sprintf("%s_sm(par);", system_name));
B = B(:,1); D = D(:,1);
[k_x,k_w,K_x,K_w] = ppp_lin(A,B,C,D,A_u,0,tau');
## Main simulation loop
y = [];
x = [];
u = [];
t = [];
t_last = 0;
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elseif strcmp(extras.U_initial,"zero")
U = zeros(n_U,1);
else
error(sprintf("extras.U_initial = \"%s\" is invalid", extras.U_initial));
endif
## Reverse time to get "previous" U
U = expm(-A_u*T_ol)*U;
for i = 1:N_ol
## Compute initial U from linear case
U_l = K_w*w - K_x*x_0;
## Compute initial U for next time from continuation trajectory
U_c = expm(A_u*T_ol)*U;
## Create sensitivity system state
x_0s(1:2:2*n_x) = x_0;
## Set next U (initial value for optimisation)
if strcmp(extras.U_next,"linear")
U = U_l;
elseif strcmp(extras.U_next,"continuation")
U = U_c;
elseif strcmp(extras.U_next,"zero")
U = zeros(n_U,1);
else
error(sprintf("extras.U_next = \"%s\" is invalid", extras.U_next));
endif
## Put initial value of U into the parameter vector
pars(free(:,1)) = U;
## Compute U
tick = time;
if extras.max_iterations>0
[U, U_all, Error, Y] = ppp_nlin(s_system_name,x_0s,u_star_tau,tau,pars,free,ws,extras);
else
Error = [];
endif
opt_time = time-tick;
printf("Optimisation %i took %i iterations and %2.2f sec\n", i, \
length(Error), opt_time);
# title(sprintf("Moving horizon trajectories: Interval %i",i));
# grid;
# plot(tau,Y)
u_ol = U'*u_star_t(1:n_U,:);
## Simulate system
## (Assumes that first gain parameter is one)
U_ol = [u_ol; zeros(n_u-1,n_t)]; # Generate the vector input
[y_ol,x_ol] = eval(sprintf("%s_sim(x_0, U_ol, t_ol, par);", system_name));
## Extract state for next time
x_0 = x_ol(:,n_t);
y = [y y_ol(:,1:n_t)];
x = [x x_ol(:,1:n_t)];
u = [u u_ol(:,1:n_t)];
UU = [UU, U];
UU_l = [UU_l, U_l];
UU_c = [UU_c, U_c];
t = [t, t_ol(:,1:n_t)+t_last*ones(1,n_t) ];
t_last = t_last + t_ol(n_t);
endfor
## Rename returned matrices
U = UU;
U_l = UU_l;
U_c = UU_c;
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elseif strcmp(extras.U_initial,"zero")
U = zeros(n_U,1);
else
error(sprintf("extras.U_initial = \"%s\" is invalid", extras.U_initial));
endif
## Reverse time to get "previous" U
U = expm(-A_u*T_ol)*U;
for i = 1:N_ol # Main loop
## Compute initial U from linear case
U_l = K_w*w - K_x*x_0;
## Compute initial U for next time from continuation trajectory
U_c = expm(A_u*T_ol)*U;
## Create sensitivity system state
x_0s(1:2:2*n_x) = x_0;
## Set next U (initial value for optimisation)
if strcmp(extras.U_next,"linear")
U = U_l;
elseif strcmp(extras.U_next,"continuation")
U = U_c;
elseif strcmp(extras.U_next,"zero")
U = zeros(n_U,1);
else
error(sprintf("extras.U_next = \"%s\" is invalid", extras.U_next));
endif
pars(i_ppp(:,1)) = U; # Put initial value of U into the parameter vector
## Compute U by optimisation
tick = time;
if extras.max_iterations>0
[U, U_all, Error, Y] = ppp_nlin(system_name,x_0s,pars,simpars,u_star_tau,w_s,i_ppp,extras);
pars(i_ppp(:,1)) = U; # Put final value of U into the parameter vector
else
Error = [];
endif
opt_time = time-tick;
printf("Optimisation %i took %i iterations and %2.2f sec\n", i, \
length(Error), opt_time);
##title(sprintf("Moving horizon trajectories: Interval %i",i));
##grid;
##plot(tau,Y)
## Generate control
u_ol = u_star_t*U; # Not used - just for show
## Simulate system over one ol interval
## (Assumes that first gain parameter is one)
##U = [u_ol zeros(n_t,n_u-1)]; # Generate the vector input
[y_ol,y_s,x_ol] = eval(sprintf("%s_ssim(x_0s, pars, simpar, u_star_t);", s_system_name));
x_0 = x_ol(n_t+1,:)'; # Extract state for next time
y_ol = y_ol(1:n_t,:); # Avoid extra points due to rounding error
x_ol = x_ol(1:n_t,:); # Avoid extra points due to rounding error
y = [y; y_ol];
x = [x; x_ol];
u = [u; u_ol];
UU = [UU; U'];
UU_l = [UU_l; U_l'];
UU_c = [UU_c; U_c'];
t = [t; t_ol+t_last*ones(n_t,1) ];
t_last = t_last + T_ol;
endfor
endfunction
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