MATLAB Project # 1

Objective:

Write a script to plot time-domain response of overdamped and underdamped control systems based on user given output functions Y(s).
Programming Steps:

1. Write down your name, date, course name, and project number at the beginning of your script as comments.

2. Explain the objective of your program as comments.

3. Ask user to enter coefficients of the numerator of the output function as a vector.

4. Ask user to enter coefficients of the denominator of the output function as a vector.

5. Use residue function to calculate the partial fractions corresponding to the output function.

6. Find the smallest non-zero real part of the root of denominator (characteristic equation) [smallest non-zero absolute value of the real part of P vector from residue funsmction]. This will be the dominant root of the response and will define the shape of the response, i.e. how long it is going to take for the transient response to decay.

7. Calculate the time constant τ = 1/dominant root

8. Calculate the time domain response corresponding to each of the partial fraction term up to 10τ. Keep your interval to be τ/100. Note that time domain response for each of the partial fraction term for overdamped and underdamped systems can be given by R(k)eP(k)t , where R(k) is the k-th numerator of the partial fraction term and P(k) is the corresponding k-th root of the denominator.

9. Add up all of the individual responses from each of the partial fraction term to get the total response y(t).

10. Plot y(t) vs. t

11. From your program, calculate the peak time, peak value, and percent overshoot for the output

Note: Do not use any built-in function to calculate the time domain response directly from the coefficients of the output function.

MATLAB Project # 2

Objective:

Write a script to plot Root-Locus corresponding to the characteristic equation of a given transfer function.

Programming Steps:

1. Write down your name, date, course name, and project number at the beginning of your script as comments.

2. Explain the objective of your program as comments.

3. From the characteristic equation q(s) = 1 + K*J(s), ask user to enter the coefficients of the numerator and denominator of J(s) as vectors.

4. In your script, change the value of the control gain K from zero to a large number.

5. Evaluate roots of the characteristic equation q(s) = 0

⇒ 1+ K.(^mΠ_i=1 (s + z_i))/(ⁿΠ_j=1 (s + p_j)) = 0 ⇒ (ⁿΠ_j=1 (s + p_j)) + K(^mΠ_i=1 (s + z_i)) = 0 each time you change the value of the control gain [you can use MATLAB roots function].

6. Plot the roots of each characteristic equation (with different value of K) on the same graph. Make sure to use some style instead of using simple line plot else your plot will be incorrect.

Note: Do not use any built-in root-locus function.

MATLAB Project # 3

Objective:

Write a script to create magnitude and phase Bode plots of a user-given transfer function for a control system.
Programming Steps:

1. Write down your name, date, course name, and project number at the beginning of your script as comments.

2. Explain the objective of your program as comments.

3. Ask user to enter the range of the angular frequency ω over which frequency response will be plotted.

4. Ask user to enter the transfer function of the control system H(s) by entering coefficients of the numerator of H(s) as a vector and coefficients of the denominator of H(s) as another vector.

5. Calculate the magnitude and phase (in degrees) of the transfer function.

6. Plot magnitude (dB) and angle (degrees) vs. semilog ω axis either in two separate figures or two sub-plots in the same figure.
Based on your interval, your plots can be quite accurate and not as approximate as we usually discuss in the theory.

Attachment:- Simulation in MATLAB.zip