Work 1: Add problem 2 matlab + report

Work 1/scripts/Problem1.m

@@ -1,3 +1,5 @@
+% === Problem 1: Simulation of the responce ===
+
 % Parameters
 m = 0.75;
 L = 1.25;
@@ -6,9 +8,9 @@ g = 9.81;
 A0 = 4;
 omega = 2;
 
-% Time span
+% Time span for 20s simulation
 tspan = [0 20];
-dt = 1e-3;
+dt = 1e-4;
 t_eval = 0:dt:20;
 
 % ODE Function
@@ -20,28 +22,29 @@ odefun = @(t, x) [
 x0 = [0; 0];                        % Initial conditions
 [t, x] = ode45(odefun, t_eval, x0); % Solve
 
-% Plots
+% Plots for 20 sec
 figure('Name', 'System responce', 'Position', [100, 100, 1280, 860]);
 subplot(2,1,1);
 plot(t, x(:,1)); ylabel('q(t) [rad]'); grid on; title('Γωνία');
 subplot(2,1,2);
 plot(t, x(:,2), 'r'); ylabel('dq(t) [rad/s]'); xlabel('t [sec]'); grid on; title('Γωνιακή Ταχύτητα');
-
 saveas(gcf, 'Prob1_responce_20s.png');
 
+% === Sampling for Problem 2 ===
+Ts = 0.1;                          % Sampling period
+sample_data(t, x, Ts, A0, omega, 'problem1_data.csv');
 
-% Time span
+% --- Extended simulation to 90 sec ---
 tspan = [0 90];
 t_eval = 0:dt:90;
 
-x0 = [0; 0];                        % Initial conditions
-[t, x] = ode45(odefun, t_eval, x0); % Solve
+x0 = [0; 0];
+[t, x] = ode45(odefun, t_eval, x0);
 
-% Plots
+% Plots for 90 sec
 figure('Name', 'System responce', 'Position', [100, 100, 1280, 860]);
 subplot(2,1,1);
 plot(t, x(:,1)); ylabel('q(t) [rad]'); grid on; title('Γωνία');
 subplot(2,1,2);
 plot(t, x(:,2), 'r'); ylabel('dq(t) [rad/s]'); xlabel('t [sec]'); grid on; title('Γωνιακή Ταχύτητα');
-
-saveas(gcf, 'Prob1_responce_90s.png');
+saveas(gcf, 'Prob1_responce_90s.png');

Work 1/scripts/Problem2a.m

@@ -0,0 +1,79 @@
+% === Problem 2: Parameter Estimation using Least Squares ===
+
+% True data for comparison
+m = 0.75;
+L = 1.25;
+c_true = 0.15;
+g = 9.81;
+
+mL2_true = m * L^2;
+mgL_true = m * g * L;
+
+theta_true = [mL2_true; c_true; mgL_true];
+
+% Load sampled data from Problem 1
+data = readtable('problem1_data.csv');
+t = data.t;
+q = data.q;
+dq = data.dq;
+ddq = data.ddq;
+u = data.u;
+
+% Build regression matrix X and target vector y = u
+X = [ddq, dq, q];  % columns correspond to coefficients of [mL^2, c, mgL]
+y = u;
+
+% Least Squares estimation: theta_hat = [mL^2; c; mgL]
+theta_hat = (X' * X) \ (X' * y);
+
+% Extract individual parameters (optional interpretation)
+mL2_est = theta_hat(1);
+c_est   = theta_hat(2);
+mgL_est = theta_hat(3);
+
+% Reconstruct ddq_hat using the estimated parameters
+ddq_hat = (1 / mL2_est) * (u - c_est * dq - mgL_est * q);
+
+% Numerical integration to recover dq̂(t) and q̂(t)
+Ts = t(2) - t(1);
+N = length(t);
+dq_hat = zeros(N,1);
+q_hat = zeros(N,1);
+
+% Initial conditions
+dq_hat(1) = dq(1);
+q_hat(1) = q(1);
+
+for k = 2:N
+    dq_hat(k) = dq_hat(k-1) + Ts * ddq_hat(k-1);
+    q_hat(k) = q_hat(k-1) + Ts * dq_hat(k-1);
+end
+
+% Estimation error
+e_q = q - q_hat;
+
+% === Plots ===
+figure('Name', 'Problem 2 - LS Estimation', 'Position', [100, 100, 1280, 800]);
+
+subplot(3,1,1);
+plot(t, q, 'b', t, q_hat, 'r--');
+legend('q(t)', 'q̂(t)');
+title('Actual vs Estimated Angle');
+ylabel('Angle [rad]');
+grid on;
+
+subplot(3,1,2);
+plot(t, e_q, 'k');
+title('Estimation Error e_q(t) = q(t) - q̂(t)');
+ylabel('Error [rad]');
+grid on;
+
+subplot(3,1,3);
+bar(["mL^2", "c", "mgL"], theta_hat);
+title('Estimated Parameters');
+ylabel('Value');
+grid on;
+saveas(gcf, 'Prob2_20s_Ts0.1.png');
+
+fprintf('   Actual Parameters:  mL^2=%f, c=%f, mgL=%f\n', theta_true(1), theta_true(2), theta_true(3));
+fprintf('Estimated Parameters:  mL^2=%f, c=%f, mgL=%f\n', theta_hat(1), theta_hat(2), theta_hat(3));

Work 1/scripts/Problem2b.m

@@ -0,0 +1,88 @@
+% === Problem 2b: Estimation using only q(t) and u(t) ===
+
+% True parameter values
+m = 0.75;
+L = 1.25;
+c_true = 0.15;
+g = 9.81;
+
+mL2_true = m * L^2;
+mgL_true = m * g * L;
+
+theta_true = [mL2_true; c_true; mgL_true];
+
+% Load sampled data
+data = readtable('problem1_data.csv');
+t = data.t;
+q = data.q;
+u = data.u;
+
+Ts = t(2) - t(1);
+N = length(t);
+
+% Compute dq and ddq using central differences
+dq = zeros(N, 1);
+ddq = zeros(N, 1);
+
+for k = 2:N-1
+    dq(k) = (q(k+1) - q(k-1)) / (2 * Ts);
+    ddq(k) = (q(k+1) - 2*q(k) + q(k-1)) / Ts^2;
+end
+
+% Build regression matrix and output vector
+X = [ddq, dq, q];
+y = u;
+
+% Least Squares estimation
+theta_hat = (X' * X) \ (X' * y);
+
+% Parameter extraction
+mL2_est = theta_hat(1);
+c_est   = theta_hat(2);
+mgL_est = theta_hat(3);
+
+% Reconstruct ddq_hat from estimated parameters
+ddq_hat = (1 / mL2_est) * (u - c_est * dq - mgL_est * q);
+
+% Integrate to get dq_hat and q_hat
+dq_hat = zeros(N,1);
+q_hat = zeros(N,1);
+dq_hat(1) = dq(1);
+q_hat(1) = q(1);
+
+for k = 2:N
+    dq_hat(k) = dq_hat(k-1) + Ts * ddq_hat(k-1);
+    q_hat(k) = q_hat(k-1) + Ts * dq_hat(k-1);
+end
+
+% Estimation error
+e_q = q - q_hat;
+
+% === Plots ===
+figure('Name', 'Problem 2b - LS Estimation from q only', 'Position', [100, 100, 1280, 800]);
+
+subplot(3,1,1);
+plot(t, q, 'b', t, q_hat, 'r--');
+legend('q(t)', 'q̂(t)');
+title('Actual vs Estimated Angle from q(t) only');
+ylabel('Angle [rad]');
+grid on;
+
+subplot(3,1,2);
+plot(t, e_q, 'k');
+title('Estimation Error e_q(t) = q(t) - q̂(t)');
+ylabel('Error [rad]');
+grid on;
+
+subplot(3,1,3);
+bar(["mL^2", "c", "mgL"], theta_hat);
+title('Estimated Parameters');
+ylabel('Value');
+grid on;
+
+% Save figure
+saveas(gcf, 'Prob2b_20s_Ts0.1.png');
+
+% Print results
+fprintf('   Actual Parameters:  mL^2=%f, c=%f, mgL=%f\n', theta_true(1), theta_true(2), theta_true(3));
+fprintf('Estimated Parameters:  mL^2=%f, c=%f, mgL=%f\n', theta_hat(1), theta_hat(2), theta_hat(3));

Work 1/scripts/problem1_data.csv

@@ -0,0 +1,202 @@
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Work 1/scripts/sample_data.m

@@ -0,0 +1,39 @@
+function sample_data(t, x, Ts, A0, omega, filename)
+% SAMPLE_DATA - Resamples pendulum simulation data and exports to CSV
+%
+% Usage:
+%   sample_pendulum_data(t, x, Ts, A0, omega, filename)
+%
+% Inputs:
+%   t        - time vector from ODE solver (e.g., from ode45)
+%   x        - state matrix [q(t), dq(t)]
+%   Ts       - sampling period (in seconds)
+%   A0       - input amplitude for u(t) = A0 * sin(omega * t)
+%   omega    - input frequency (in rad/s)
+%   filename - output CSV filename
+
+    % Time vector for resampling
+    t_sampled = t(1):Ts:t(end);
+
+    % Resample q and dq using interpolation
+    q_sampled  = interp1(t, x(:,1), t_sampled);
+    dq_sampled = interp1(t, x(:,2), t_sampled);
+
+    % Estimate second derivative using central finite differences
+    ddq_sampled = zeros(size(q_sampled));
+    N = length(q_sampled);
+
+    for k = 2:N-1
+        ddq_sampled(k) = (q_sampled(k+1) - 2*q_sampled(k) + q_sampled(k-1)) / Ts^2;
+    end
+
+    % Evaluate input u(t)
+    u_sampled = A0 * sin(omega * t_sampled);
+
+    % Create table and export to CSV
+    T = table(t_sampled', q_sampled', dq_sampled', ddq_sampled', u_sampled', ...
+        'VariableNames', {'t', 'q', 'dq', 'ddq', 'u'});
+
+    writetable(T, filename);
+    fprintf('Sampling data saved to file: %s\n', filename);
+end