%   
% Written by:
% -- 
% John L. Weatherwax                2006-08-28
% 
% email: wax@alum.mit.edu
% 
% Please send comments and especially bug reports to the
% above email address.
% 
%-----

close all; clc; clear; 

% set the constants of this problem: 
t_0     = 0.; 
t_f     = 12.; 
xair    = 45;
xd      = 70; 
x0      = 45; % the initial conditions on x_1(t) 
T       = 4;
k1      = 1.0; 

% first solve for mu:
num = ( xd - ( xair + (x0 - xair)*exp( -(t_f - t_0)/T ) ) );
den = -( (k1^2)*T )*( 1 + exp( - 2*(t_f - t_0)/T ) )/4;
mu = num/den;

% integrate the dynamical equations (we have the explicit expression for x_1(t) so the ODE integration is not necessarily needed): 
sol = ode45( @(t,x) sect_5_prob_3_ode_fn(t,x, t_f,T,xair,k1,mu), [t_0,t_f], [x0] );
t = sol.x;

% plot the solution x(t): 
figure; plot( t, sol.y, '-x' ); xlabel('time'), ylabel('x_1(t)'); title('state')
%saveas( gcf, '../../WriteUp/Graphics/Chapter3/sect_5_prob_3_x_t.eps', 'epsc' ); 

% plot the control u(t): 
u = -mu*(k1/2)*exp( (t-t_f)/T );
figure; plot( t, u, '-o' ); xlabel('time'); ylabel('u'); title('control');
%saveas( gcf, '../../WriteUp/Graphics/Chapter3/sect_5_prob_3_u_t.eps', 'epsc' );