81 lines
2.0 KiB
Matlab
81 lines
2.0 KiB
Matlab
% === Problem 3b: Effect of Sampling Period Ts on Estimation Accuracy ===
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clear; close all;
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% True parameters
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m = 0.75;
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L = 1.25;
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c = 0.15;
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g = 9.81;
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mL2_true = m * L^2;
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mgL_true = m * g * L;
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theta_true = [mL2_true; c; mgL_true];
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% Parameters of input
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A0 = 4;
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omega = 2;
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% Time settings
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T_final = 20;
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dt = 1e-4;
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t_full = 0:dt:T_final;
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% Simulate system with very fine resolution
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odefun = @(t, x) [
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x(2);
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(1/(m*L^2)) * (A0*sin(omega*t) - c*x(2) - m*g*L*x(1))
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];
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x0 = [0; 0];
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[t_sim, x_sim] = ode45(odefun, t_full, x0);
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% Sampling periods to test
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Ts_list = [0.01, 0.05, 0.1, 0.2, 0.5];
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n_cases = length(Ts_list);
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rel_errors_all = zeros(3, n_cases);
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for i = 1:n_cases
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Ts = Ts_list(i);
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% Resample at this Ts
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t_sampled = t_sim(1):Ts:t_sim(end);
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q = interp1(t_sim, x_sim(:,1), t_sampled);
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u = A0 * sin(omega * t_sampled);
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N = length(t_sampled);
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% Compute dq and ddq with central differences
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dq = zeros(N,1);
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ddq = zeros(N,1);
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for k = 2:N-1
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dq(k) = (q(k+1) - q(k-1)) / (2*Ts);
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ddq(k) = (q(k+1) - 2*q(k) + q(k-1)) / Ts^2;
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end
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% LS Estimation
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idx = 2:N-1; % Truncate to valid range (2:N-1)
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X = [ddq(idx), dq(idx), q(idx)'];
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y = u(idx).';
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theta_hat = (X' * X) \ (X' * y);
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rel_error = abs((theta_hat - theta_true) ./ theta_true) * 100;
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rel_errors_all(:, i) = rel_error;
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% Print
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fprintf('Ts = %.3f s → mL^2=%.4f (%.2f%%), c=%.4f (%.2f%%), mgL=%.4f (%.2f%%)\n', ...
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Ts, theta_hat(1), rel_error(1), ...
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theta_hat(2), rel_error(2), ...
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theta_hat(3), rel_error(3));
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end
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% === Plot ===
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figure('Name', 'Problem 3b - Effect of Sampling Period', 'Position', [100, 100, 1000, 600]);
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plot(Ts_list, rel_errors_all', '-o', 'LineWidth', 2, 'MarkerSize', 4);
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legend({'mL^2', 'c', 'mgL'}, 'Location', 'northwest');
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xlabel('Sampling Period Ts [sec]');
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ylabel('Relative Error [%]');
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title('Effect of Ts on Parameter Estimation');
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grid on;
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saveas(gcf, 'output/Prob3b_SamplingPeriodEffect.png');
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