obtain the family of step responses in matlab

To obtain the family of step responses in MATLAB, we need to first design a control system using the feedback function. Once the system is designed, we can use the step function to simulate the step response of the system. We can then plot the step response using the plot function.

Here is an example code snippet that shows how to obtain the family of step responses in MATLAB for a second-order system:

main.m
% Define the transfer function of the system
num = [1];
den = [1 1 1];
sys = tf(num, den);

% Define a range of values for the system parameter
K = 0.1:0.1:1;

% Loop over the range of parameter values and compute the step response
figure;
hold on;
for i = 1:length(K)
    % Design the control system using feedback
    Kp = K(i);
    Ki = 0;
    Kd = 0;
    C = pid(Kp, Ki, Kd);   
    T = feedback(C*sys, 1);
    
    % Simulate the step response
    [y, t] = step(T);
    
    % Plot the step response
    plot(t, y);
end

% Add labels to the plot
title('Step Response');
xlabel('Time (s)');
ylabel('System Output');
legend(num2str(K'));
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This code will generate a plot of the step response for the second-order system with varying values of the system parameter K. Each curve in the plot corresponds to a different value of K, and the legend shows the corresponding value of K for each curve.

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