function [name,T,y,u,ys,us,J] = ppp_ex11 (ReturnName)
## usage: [name,T,y,u,ys,us,T1,du,dus] = ppp_ex11 (ReturnName)
##
## PPP example
## $Id$
## Example name
name = "Input constraints +-1.5 on u* at tau=0,0.5,1,1.5,2";
if nargin>0
return
endif
## System
A = [-3 -3 -1
1 0 0
0 1 0];
B = [1
0
0];
C = [0 -0.5 1];
D = 0;
[n_x,n_u,n_y] = abcddim(A,B,C,D);
## Controller
t = [6:0.02:7]; # Time horizon
A_w = 0; # Setpoint
A_u = ppp_aug(laguerre_matrix(3,2.0), A_w); # Input functions
Q = ones(n_y,1);;
## Constraints
Gamma = [];
gamma = [];
## Constraints - u
Tau_u = [0:0.5:2];
one = ones(size(Tau_u));
limit = 1.5;
Min_u = -limit*one;
Max_u = limit*one;
Order_u = 0*one;
## Constraints - y
Tau_y = []; # No output constraints
one = ones(size(Tau_y));
limit = 1.5;
Min_y = -limit*one;
Max_y = limit*one;
Order_y = 0*one;
## Simulation
W=1;
x_0 = zeros(3,1);
## Constrained - open-loop
disp("Designing controller");
[k_x,k_w,K_x,K_w,Us0,J_uu,J_ux,J_uw] = ppp_lin (A,B,C,D,A_u,A_w,t,Q); # Unconstrained design
[Gamma_u,gamma_u] = ppp_input_constraint (A_u,Tau_u,Min_u,Max_u);
Gamma = Gamma_u;
gamma = gamma_u;
## Constrained OL simulation
disp("Computing constrained ol response");
[u,U] = ppp_qp (x_0,W,J_uu,J_ux,J_uw,Us0,Gamma,gamma);
T = [0:t(2)-t(1):t(length(t))];
[ys,us] = ppp_ystar (A,B,C,D,x_0,A_u,U,T);
## Unconstrained OL simulation
disp("Computing unconstrained ol response");
[uu,Uu] = ppp_qp (x_0,W,J_uu,J_ux,J_uw,Us0,[],[]);
[ysu,usu] = ppp_ystar (A,B,C,D,x_0,A_u,Uu,T);
title("Constrained and unconstrained y*");
xlabel("t");
grid;
figure(1);
plot(T,ys,"-;y*: constrained;", T,ysu, "--;y*: unconstrained;")
## Non-linear - closed-loop
disp("Computing constrained closed-loop response");
[T,y,u,J] = ppp_qp_sim (A,B,C,D,A_u,A_w,t,Q, \
Tau_u,Min_u,Max_u,Order_u, \
Tau_y,Min_y,Max_y,Order_y,W,x_0);
title("Constrained closed-loop response");
xlabel("t");
grid;
figure(2);
plot(T,y,"-;y;", T,u,"--;u;");
# ## Compute derivatives.
# dt = t(2)-t(1);
# du = diff(u)/dt;
# dus = diff(us)/dt;
# T1 = T(1:length(T)-1);
##plot(T1,du,T1,dus);
endfunction