EEL-3657 Linear Control Systems Final Exam (30 Points)
Name:
➢ Please write your name and read these instructions completely.
➢ This exam consists of 7 problems and you have up to Friday (April 24) midnight to submit your
solution online.
➢ Write legibly, show mathematical details of your solution and simplify your solutions to receive full
credit.
➢ Do not communicate or collaborate with others, an online software will be used to check for
similarities.
Q1. For the system shown in the figure below:
a. Sketch the root locus. (2 points)
b. Construct Routh stable and find the range of positive K that will make the system stable. (2
points)
Q2. Given the control system below:
a. Find K such that there is a 10 percent error in steady state. (2 Points)
b. Use Matlab to plot the output response for the uncompensated and compensated system. (2 Points)
Q.3 A tank system with inflow F(t) and depth (h) is modeled below. Determine the depth h(t) for a
sinusoidal inflow. Assume zero initial conditions. (3 points)
Q4. The feedback system shown below has a plant, a controller, and sensor transfer functions as 𝐺(𝑠),
(𝑠) and 𝐻(𝑠), respectively. Find the output Y(s) and the input 𝑈(𝑠) as a function of the inputs and
transfer functions. (2 Points)
Q5. A cascaded multi-stage industrial process consists of three stages where each stage can be represented
by a first-order system with time constants of 1, 3, and 5 seconds, respectively.
a. Using Matlab, for a unit step input, design a PI controller that will result in an output with an overshoot that is less than 5%, rise time that is less than 7 seconds, settling time that is less than 35
seconds and less than 1% steady state error. Report the system you construct, a plot of the reference
signal and the output, and a plot of the input to the system. Also, report the equation of the controller
you designed with the design values. (5 Points).
b. Use Matlab to plot the root-locus before and after adding the controller. (3 points)
c. Comment on the effect of the controller on the root-locus. (1 point)
d. Discuss how the addition of the integral part of the controller affects the transient, steady state, and
stability of the system. (1 Point).
Q6. Using the bisection method, design a lead compensator Gc for the system shown below such that the
dominant closed loop poles become 𝑠𝑑 = −2 ± 𝑗2√3. (3 points)
Q7. Sketch the Bode plot for the following transfer functions. From your sketch, explain how the transfer
functions affect the different input frequencies in terms of the magnitude and phase. (4 points)