added nlp problem

sec3_predefined_utilities.m

+clear all; close all; clc;
+import casadi.*
+

sec4_solving_NLP.m

@@ -2,41 +2,40 @@ clear all; close all; clc;
 import casadi.*
 
 %% Hanging chain
-N = 15;
-m = 40/N; % mass [kg]
-D = 70*N;        % spring constant [J/m^2]
-g = 9.81;        % gravitational constant [m/s^2]
-L = 5/N;  % reference length [m]
-
-opti = Opti();
-
-x = opti.variable(N);
-y = opti.variable(N);
-
-yr = opti.parameter();
-
-V = 0.5*D*sum((sqrt(diff(x).^2+diff(y).^2)-L).^2,1);
-V = V + g*sum(m*y,1);
-
-opti.minimize(V);
-opti.subject_to([x(1);y(1)] == [-2;0]);
-opti.subject_to([x(N);y(N)] == [2;yr]);
-
-opti.set_initial(x,linspace(-2,2,N));
-
-% Quiet
-options = struct;
-options.qpsol = 'qrqp';
-options.convexify_strategy = 'regularize';
-options.print_iteration = false;
-options.print_time = false;
-options.print_status = false;
-options.print_header = false;
-options.max_iter = 10000;
-options.qpsol_options.print_iter = false;
-options.qpsol_options.print_header = false;
-
-opti.solver('sqpmethod',options);
-
-par_nlp = opti.to_function('par_nlp',{yr},{x(round(N/2))},{'yr'},{'y'});
+N = 40;     % Number of points (chosen by us)
+m = 40 / N; % Weight of each section
+D = 70 * N; % Stifness coeficient
+g = 9.81;   % gravitation acceleration
+L = 1;      % length of the chain
+
+% Set up optimization environment
+opti_var = casadi.Opti();
+
+% Problem 1: Simple hanging chain
+% Decision variables: positions
+x = opti_var.variable(1, N);
+y = opti_var.variable(1, N);
+
+Vg = g*sum(m*y);
+Vs = 0.5*D*sum((x(1:N-1)-x(2:N)).^2+(y(1:N-1)-y(2:N)).^2);
+
+cost_NLP = Vg + Vs;
+opti_var.minimize(cost_NLP);
+
+opti_var.subject_to(x(1)   == -2);
+opti_var.subject_to(y(1)   == 1);
+opti_var.subject_to(x(end) == 2);
+opti_var.subject_to(y(end) == 1);
+
+opti_var.subject_to(y >= cos(0.1 * x) - 0.5);
+opti_var.solver('ipopt');
+
+sol_NLP = opti_var.solve();
+
+% Plotting the results
+figure; hold on;
+plot(sol_NLP.value(x), sol_NLP.value(y), '-o');
+xs = linspace(-2, 2, 1000);
+plot(xs, cos(0.1 * xs) - 0.5, '--r');
+grid on;