%% Discrete-time LQ-optimal control on a fixed horizong with fixed final state

%%
% Based on Example 2.2.2 from the second edition of Lewis's book. The latest
% (3rd) edition
% Lewis, Frank L., Draguna Vrabie, and Vassilis L. Syrmo. Optimal Control. 3rd ed. John Wiley & Sons, 2012, freely downloadable at the author's web page https://lewisgroup.uta.edu/FL%20books/Lewis%20optimal%20control%203rd%20edition%202012.pdf
% no longer contains the example.

%% Model of the system and parameterization of the cost function

rN = 3000000;               % The reference value at the end.
N = 30;                     % The length of the time horizon.
x0= 0;                      % The initial state.

A = 1.07;                   % The state-space model "matrix".
B = 1;                      % --//--
R = 1;                      % The penalty on control. Not needed in the scalar case.

%% Computing the weighted finite-time Gramian and computing the optimal control sequence

k = 0:N-1;

G = sum(A.^(2*(N-1-k)));    % The finite-time reachability gramian. Simplified for the scalar case.

u = A.^(N-1-k)/G*(rN-A^N);

%% Plotting the responses

figure(1)
stem(u)
set(gca,'FontSize',12)
xlabel('Year','FontSize',12)
ylabel('Anual investment [CZK]','FontSize',12)

% print -depsc2 ../lectures/figures/saving_strategy.eps
% !epstopdf ../lectures/figures/saving_strategy.eps