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OCP_NumericApproach_1_MintimeandEnergy.m
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%%
% NMPC -- Theory and Applications
% Course at Karlsruhe Institute of Technology
%
% Optimal loop Optimal control
%
%
% Autor: Hesham Hendy
% Email: [email protected]
% Date: 08-10-2019
% Status: final
clear;
clc;
addpath('C:\Program Files\MATLAB/casadi-windows-matlabR2016a-v3.5.0');
import casadi.*;
opti = casadi.Opti();
const = constants();
%% System limits
% Notice (for me) compare to direct_single_shooting.m codes, also in CasADi’s examples collection
t_0 = 0;
N = 50; % % number of control intervals % Time horizon %horzions (optimizied)
%% optimized Variables
% The control trajectory is parameterized using some piecewise smooth approximation, typically piecewise constant.
x = opti.variable(4, N + 1); % Optimization of 4 states (amount of parameters) on N control intervals + 1 predicted state
u = opti.variable(2, N); % Optimization of 2 inputs controls on N control intervals
tendopt = opti.variable(); % Optimization of the value of the time step ( the optimized final time)
timeSteps = tendopt / N;
%% objective function
% S = 1e-4; % weight matrix to act on time
% R = 1e-02 * eye(2); % weight matrix to act on input torque
%
% % Notice: Infeasible Problem Detected sometimes !
%
% J = S * tendopt; % responsible for time optimality
J = 0;
t = t_0;
%
R = [1 0;
0 1];
R_ = [1 0;
0 1];
x_0 = const.x_0;
x_end = const.x_end;
for i = 1:N
% continuity constraint
x_next = explicitEuler(x(:, i), u(:, i), timeSteps, tendopt);
opti.subject_to(x(:,i+1) == x_next);
J = J + u(:, i)' * R * u(:, i) + (u(:, i) - u(:, i-1))' * R_ * (u(:, i) - u(:, i-1)) ; %through that ,a compromise bet minimal time and minimal control energy
end
opti.minimize(J + tendopt);
%% constraints formulation
opti.subject_to(x(:, 1) == x_0);
opti.subject_to(x(:, end) == x_end);
opti.subject_to(const.omega_lb <= x(3, :) <= const.omega_ub);
opti.subject_to(const.omega_lb <= x(4, :) <= const.omega_ub);
opti.subject_to(const.u_lb / 1000 <= u(1, :) <= const.u_ub / 1000);
opti.subject_to(const.u_lb / 1000 <= u(2, :) <= const.u_ub / 1000);
opti.subject_to(0 <= tendopt <= 3 );
%% solve
opti.solver('ipopt'); % Ipopt, a library for large-scale nonlinear optimization.
sol = opti.solve();
% Remeber I became an error by running and after placing diffrent break
% points I discovered that the input is tiny so I decided to magnifiy (scale) it by
% solving and unscale it again !
% Run on command window
% opti.debug.value(x_next)
% opti.debug.value(x,opti.initial())
% opti.debug.show_infeasibilities()
x_sol = full(sol.value(x));
u_sol = full(sol.value(u)) .* [1000; 1000];
t_opt_sol = full(sol.value(tendopt));
timeSteps = t_opt_sol/ N; % length of a control interval
t = 0:timeSteps:t_opt_sol;
%% plot in q1-q2-plane
figure;
subplot(2, 2, 1);
hold on;
grid on;
plot(t, x_sol(1, :),'g', 'LineWidth', 2);
xlabel('sec');
ylabel('rad');
title('Angles on q1 plane');
subplot(2, 2, 2);
hold on;
grid on;
plot(t, x_sol(2, :),'g', 'LineWidth', 2);
xlabel('sec');
ylabel('rad');
title('Angles on q2 plane');
subplot(2, 2, 3);
hold on;
grid on;
plot(t, x_sol(3, :),'g', 'LineWidth', 2);
xlabel('sec');
ylabel('rad/s');
title('Angualr Velocity on q1 plane');
subplot(2, 2, 4);
hold on;
grid on;
plot(t, x_sol(4, :),'g', 'LineWidth', 2);
xlabel('sec');
ylabel('rad/s');
title('Angualr Velocity on q2 plane');
%% plot in x-y-plane
figure;
x_ = const.l1 .* cos(x_sol(1, :)) + const.l2 .* cos(x_sol(1, :) + x_sol(2, :));
y_ = const.l1 .* sin(x_sol(1, :)) + const.l2 .* sin(x_sol(1, :) + x_sol(2, :));
subplot(2, 1, 1);
hold on;
grid on;
plot(t, x_, 'LineWidth', 2);
xlabel('sec');
ylabel('m/sec');
title('Motion on x-plane');
subplot(2, 1, 2);
hold on;
grid on;
plot(t, y_, 'LineWidth', 2);
xlabel('sec');
ylabel('m/sec');
title('Motion on y-plane');
%% plot inputs
figure;
fig3 = gcf;
fig3.PaperUnits='inches';
fig3.PaperPosition = [0 0 8 8];
subplot(2, 1, 1);
axis([0, 3, -1500, 1500]);
hold on;
grid on;
stairs(t(2:length(t)), u_sol(1, :), 'LineWidth', 2);
xlabel('sec');
ylabel('N.m');
title('Applied Torque by U1');
subplot(2, 1, 2);
axis([0, 3, -1500, 1500]);
hold on;
grid on;
stairs(t(2:length(t)), u_sol(2, :), 'LineWidth', 2);
xlabel('sec');
ylabel('N.m');
title('Applied Torque by U2');
%% ========================================================================
% subfunctions
% =========================================================================
function x_next = explicitEuler(x, u, h ,t_opt)
%Solving an ODR using one-Step Method
% Computation is performed without iteration
x_next = x + h .* stateSpace_ODE(t_opt, x, u); % revise Buchter Diagram
end