Linearization of Nonlinear Systems

Linearization simplifies complex nonlinear systems by approximating them with linear models around a specific operating point. This allows e

%% Performed By Rushikesh.Bhavsar Roll no 08
%% Linearization of Nonlinear Systems
% Define time vector
t = 0:0.01:10;

% Input waveform: sinusoidal signal
u = sin(t);

% Nonlinear system: y = u^2 + sin(u)
y_nonlinear = u.^2 + sin(u);

% Linearization around u = 0 (Taylor expansion)
% dy/du = 2u + cos(u) → at u = 0: dy/du = 0 + 1 = 1
% So linearized system: y ≈ u
y_linear = u;

% Plotting the waveforms
figure;
plot(t, u, 'b', 'LineWidth', 1.5); hold on;
plot(t, y_nonlinear, 'r--', 'LineWidth', 1.5);
plot(t, y_linear, 'g:', 'LineWidth', 1.5);
xlabel('Time (s)');
ylabel('Amplitude');
title('Waveform Comparison: Input, Nonlinear Output, Linearized Output');
legend('Input u(t)', 'Nonlinear Output y(t)', 'Linearized Output y_{lin}(t)');
grid on;