Question

Write a MATLAB program that w design a PD compensator assuming second-order approximations as follows. . Allow the user to input the desired percent overshoot, peak time and gain required to meet a steady-state error specification Display the gain-compensated Bode plot . Calculate the required phase margin and bandwidth. . Display the pole, zero, and gain of the PD compensator. Display the compensated Bode plot ·Output the step response of the PD-compensated system to test your second-order approximation. [Implement your program on a unity feedback system with open loop re- sponse given as follovw G(s) = s(s +2)(s +6) with percent overshoot-10%, peak time-0.1s. Ke-30.1

Answer 1

Hello,

For the first part of taking input from user you can use the input function to take the input as follows :

input_text = 'Enter desired peak overshoot : ';
peak_overshoot = input(input_text);
input_text = 'Enter Peak Time : ';
peak_time = input(input_text);
input_text = 'Enter Gain Required : ';
gain_req = input(input_text);

For the next parts, take a random range of input for the input e(t) lets name it input_data. You can define this using linspace or start:step:stop methods.

Then take its laplace transform in MATLAB using :

answer = laplace(input_data);

Another way is defining input_data in terms of symbols like

syms a t
input_data = exp(-a*t);
answer = laplace(input_data);

The output of the the PD controller is then defined as :

output = answer * (Kp) * (1 + (Kd/Kp)*1i*w);

This output is then multiplied with G which is as defined in the question as :

G = K / (1i*w * (1i*w + 2) * (1i*w + 6));

So final output is C = output * G;

For displaying the plots use the matlab plot command.

plot (x,y); where x is the values you want on the X-axis and y the corresponding values on the Y-axis.

But for the bode plot you would like to go for a semilogy graph instead between gain and frequency so :

semilogy(frequency,gain);

where frequency is the range of values of "s" and gain will be final_output / input plotted with frequency on X-axis and gain in dB / 20 plotted along the Y-axis.

Required phase margin and bandwidth can be calculated using the cursor in the matlab plot that you get which displays the X and Y values of a point you select on the graph, difference between the X-axis values for gain 0 or the Y-axis plot of 1/20 will be giving you the bandwidth while phase margin can be calculated using the slope of the line, -20dB/decade being 90 degree and -40dB/decade being 180 degrees.

Poles and zeros are plotted when you plot the graph itself, real zeros are the points where the plot starts increasing its slope towards a more positive value and real poles are the points where the graph changes its slope towards the more negative value. All this can be found out using the cursor in matlab figure.