YTU FACULTY OF ELECTRICAL & ELECTRONICS ENGINEERING
DEPARTMENT OF CONTROL & AUTOMATION ENGINEERING
KOM3751 CONTROL SYSTEMS, FINAL EXAM
Name, Surname:
Student number:
Signature:
Date: January 14, 2021
Duration: 75 mins.
Grading:
Problem 1:
Problem 2:
Problem 3:
Problem 4:
40
20
20
20
Problem 1. Consider that a feedback system has the open-loop poles of −4, −1 ± 𝑗 and no zeros (only K).
(a) Sketch the root-locus of the system.
(b) Find the stability margin of the system for 𝐾 > 0.
(c) Find the value of gain for which the closed-loop transfer function will have a real pole at −5.
(d) What would be the maximum percent overshoot and settling time at this gain?
Hint: Show if the 2nd order approximation is valid, then use the formula given at the footer.
(e) What would be the steady-state errors when the inputs are 𝑟(𝑡) = 0.625𝑢(𝑡) and 0.625𝑡𝑢(𝑡)?
(f) Design a PID controller to speed up the step response twice while keeping the same maximum overshoot
and improving the steady-state performance.
Hint: Design PD first then add a PI controller.
(g) What would be the steady-state errors for the same inputs of (d) after the PID controller?
Problem 2. Suppose that a PID controller has the following transfer function,
20(𝑠 + 8)(𝑠 + 0.1)
𝑠
(a) Find the proportional, integral and derivative coefficients of the controller, 𝐾𝑃 , 𝐾𝐼 and 𝐾𝐷 , respectively.
(b) Determine the proportional coefficient and, reset (integral) and rate (derivative) times of the controller,
𝐾𝑃 , 𝑇𝑖 and 𝑇𝑑 , respectively.
𝐺𝑃𝐼𝐷 (𝑠) =
Problem 3. Now suppose that a controller or compensator is defined as follows,
𝑉𝑜 (𝑠) 𝐾(𝑠/𝛼 + 1)
=
𝑉𝑖 (𝑠)
𝑠/𝛽 + 1
(a) Determine the characteristics of controller for the values of 𝐾 = 0.25, 𝛼 = 2 sec, 𝛽 = 8 sec,
(b) What does this controller used for? function. (Hint: For example, “It is a PI controller since it has a pole
near the origin and a zero further to the left. It improves the steady-state performance of the system…”)
(c) Draw the Bode magnitude and phase plots of this controller.
𝐺𝑐 (𝑠) =
Problem 4. Consider the Bode plots given.
(a) What would be the gain, K, in linear unit
at low frequencies?
(b) What would be the break (corner)
frequency?
(c) What are the slopes at low and high
frequencies in magnitude plots?
(d) What are the slopes at low, medium and
high frequencies in the phase plot?
(e) Propose a transfer function that would
give this Bode plot.
𝑇𝑝 =
𝜋
𝜔𝑛√1−𝜁 2
4
2
, 𝑇𝑠 ≅ 𝜁𝜔 , %OS= 100. 𝑒 −𝜁𝜋⁄√1−𝜁 , 𝜁 =
𝑛
−ln(%𝑂𝑆/100)
√𝜋2+ln2 (%𝑂𝑆/100)
Good luck! Şeref Naci Engin
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