Steady state error –P, PD, PID Controllers,

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IFETCE R-2023 Academic Year – 2024-2025
IFET COLLEGE OF ENGINEERING
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DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING

Subject Title : Control Systems Engineering Title : Question Bank
Subject Code : 23EC4404 Unit : II
Dept/Year/Semester : ECE/II /IV Type :100% Numerical

UNIT II TIME DOMAIN SPECIFICATIONS
Measures of performance of the standard second order system – Steady state error –P, PD, PID
Controllers.
Activity: To learn to use LabVIEW/Matlab to generate and manipulate polynomials and transfer
functions.

PART A (2 MARKS)
Steady State error
1. Define steady state error. (or) What is a steady state error? K2
2. 10 K1
A system has open loop transfer function G( s)  . What is the
s( s  1)( s  2)
3
steady state error when it is subjected to input r (t )  1  2t  t 2
2
3. Give the steady state errors due to various standard inputs for type 2 systems K2
4. 10 K2
For servo mechanisms with open loop transfer function, G( s)  what
( s  2)( s  3)
type of input signal gives constant steady state error and calculate its value.
5. What is velocity error constant, positional error constant and acceleration error K1
constant? Write their units
6. Determine the error coefficients for the system having K2
(s+2)
G(s)H(s) =
s(1+0.5s)(1+0.2s)
7. Define static error constant. K2
Performance of second order systems
8. Define damping ratio. Classify second order systems depending on the value of the K1
damping ratio.
9. Define unit impulse and unit step signals. K1
10. Define unit ramp and parabolic signals. K1
11. 5 K2
Find the open loop steady state gain and time constants for the system G ( s) 
s2
using time constant form of the given pole zero form.
12. Find the unit impulse response of a system G(s) = 5s/(s + 2) with zero initial K2
conditions using Laplace transform.
13. Define rise time. How it is calculated for various types of systems? K1
14. With reference to time response of a control system, define peak time. K2
15. Write the expression for Peak overshoot, settling time, delay time and rise time. K1
16. The damping ratio of the system is 0.6 and the natural frequency of oscillation is 8 K1
rad/sec. Determine the rise time.
17. The damping ratio of a system is 0.75 and the natural frequency of oscillation is 12 K1
rad/sec. Determine the peak overshoot and the peak time.

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18. The damping ratio and natural frequency of oscillation of a second order system is K1
0.5 and 8 rad/sec respectively. Calculate the resonant peak and resonant frequency.
19. The closed loop transfer function of a second order system is given by K1
2
400/s +2s+400. Determine the damping ratio and natural frequency of oscillation.
20. Classify the system based on damping.(Or)How a control system is classified K2
depending on the value of damping?
21. Using standard form of transfer function of a second order system, find the impulse K2
9
response of the system G ( s )  2
s  4s  9
22. What is damped frequency of oscillation? K1
nd
23. Write the expression of unit step response of 2 order system for under damped K2
system.
Introduction and design of PD, PI, PID controllers
24. Why derivative controller is not used in control systems? K1
25. What is meant by PI controller? Discuss the effect and expression of PI controller K2
on system performance. (or) What is the effect of PI controller on the system
performance?
26. What is Proportional Control (P Controller)? Write its effect on system K1
performance
27. What is meant by PD controller? K2
Electrical system design using op-amps
28. Construct a circuit diagram of PID controller using op-amp. K1
29. Illustrate PI controller using op-amp. K1
30. Draw the Equivalent circuit of differentiator and sign changer of PD controller K1
using op-amp.
PART:B (16 MARKS)
1. (i). Derive and plot the response of a first order system for step input. 4 K2
(ii). Derive an expression for the response of under damped second order system 12 K2
for unit step input.
2. (i). Derive the expression for the time domain response of a second order systems 8 K2
for a step input when the system is (a) undamped and (b) critically damped.
(ii). Derive time domain specifications for a second order system subjected to a
step input. 8 K2
3. (i) A positional control system with velocity feedback is shown in fig. find the 8 K3
response of the system with unit step input.




8 K3
(ii). The unity feedback system is characterized by an open loop transfer function
K
G(s)= .Determine the gain K, so the system will have a damping ratio of
S ( S  10)
0.5 for this value of K. Determine settling time, peak overshoot and time at peak
overshoot for a unit step input.

4. (i). A unity feedback system is characterized by the open loop transfer function 8 K4
1
𝐺 (𝑠) = . Determine the steady state errors for unit-step, unit-
𝑠(0.5𝑠+1)(0.2𝑠+1)
ramp and unit-acceleration inputs. Also determine the rise time, peak time, peak

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overshoot and settling time of the step response of the input.
(ii). A positional control system with velocity feedback is shown in fig. What is the
response c(t) to the unit step input? Also calculate the rise time peak time 8 K4
maximum overshoot and settling time if the damping ratio is 0.5.




5. (i). The open loop transfer function of unity feedback system is given by 10 K3
K
G( s)  where K and T are positive constants. By what factor, should the
s(sT  1)
amplifier gain K be reduced, so that the peak overshoot of unit step response of the
system is reduced from 75% to 25%.
(ii). A unity feedback control system is characterized by the open loop transfer 6 K3
0.4𝑠+1
function 𝐺 (𝑠) = . Determine its transient response for unit step input.
𝑠(𝑠+0.6)
Evaluate the maximum overshoot and corresponding peak time.
6. (i). A unity feedback system has the forward transfer function 𝐺 (𝑠) = 8 K4
𝐾1 (2𝑠+1)
. When the input r(t)=1+6t, determine the minimum value of K1 so
𝑠(5𝑠+1)(1+𝑠)2
that the steady error is less than 0.1
(ii). The open loop transfer function of a servo system with unity feedback is 8 K4
10
𝐺 (𝑠) = . Evaluate the static error constants of the system.Obtain the
𝑠(0.1𝑠+1)
steady state error of the system, when subjected to an input given by the
𝑎
polynomial, 𝑟(𝑡) = 𝑎0 + 𝑎1 𝑡 + 22 𝑡 2
7. Derive the expressions that shows the effects of P, PI, PD, PID controllers on the 16 K2
system response.
8. Derive an expression to find steady state error of a closed loop control system. 16 K2

9. (i). Calculate the delay time, rise time, peak overshoot and settling time for the 8 K3
system whose natural frequency of oscillation is 10rad/sec and damping factor is
0.707
(ii). For a unity feedback system with open loop transfer function, 𝐺 (𝑠) = 8 K3
10(𝑠+2)
, find the error constants. Also calculate the steady state error for the input
𝑠 2 (𝑠+1)
3 2 1
𝑅 (𝑠) = − + .
𝑠 𝑠2 3𝑠 3

10. (i). Find the unit step response of the system shown below. 8 K4




(ii). Consider the unity feedback closed loop system with forward transfer function
25
8 K4
, 𝐺 (𝑠)= . Obtain the rise time , peak time, maximum overshoot and
𝑠(s+5)
settling time when the system is subjected to unit step input.

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Format Number: 2c Ver: 1.0
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11. (i). A certain unity negative feedback control system has the following forward 8 K3
K  s+2 
path transfer function G  s  = . The input applied is r(t) = 1+3t. Find
s  s+5  4s+1
the minimum value of K so that the steady state error is less than 1.
(ii). The closed loop transfer function of a second order system is given by 8 K3
100
T s = 2
. Determine the damping ratio, natural frequency of
s +10s+100
oscillations, rise time, settling time and peak overshoot.
12. Find out the position, velocity and acceleration error coefficients for the following 16 K4
unity feedback systems having forward loop transfer function as,
100 K
(a). 𝐺 (𝑠) = (b). 𝐺 (𝑠) =
(1+0.5s)(1+2s) 𝑠(1+0.1s)(1+s)
K K(1+s)(1+2s)
(c). 𝐺 (𝑠) = 2 (𝑠2 (d). 𝐺 (𝑠) = 2 (𝑠2
𝑠 +8s+100) 𝑠 +4s+20)
13. (i). The open loop transfer function of a unity feedback system is given by 8 K4
20
G s  = . The input function is r(t)=2+3t+t2. Determine generalized error
s  s+2 
coefficient and steady state error.
𝑑2 𝑐
(ii). A closed loop servo motor is represented by the differential equation + 8 K4
𝑑𝑡 2
𝑑𝑐
8 = 64𝑒. Where c is the displacement of the output shaft, r is the
𝑑𝑡
displacement of the input shaft and e = r – c. Determine the undamped natural
frequency, damping ratio and percentage maximum overshoot for unit step input.
14. Determine the steady state errors for the following inputs 5u(t), 5tu(t), 5t2u(t) to a 16 K4
100  s+2  s+6 
system whose pen-loop transfer function is given by G  s  = .
s  s+3 s+4 
15. Explain the construction and design of P, PD controllers using op-amp. 16 K2

16. Explain the design of PI, PID controllers using op-amp. 16 K2