Previous Year Questions- Time Response Analysis - 1 - Control Systems - Electrical

Q1: Consider the standard second-order system of the form  with the poles p and p^⋆ having negative real parts. The pole locations are also shown in the figure. Now consider two such second-order systems as defined below :
System 1 : ω_n = 3rad/sec and θ = 60°
System 2 : ω_n = 1rad/sec and θ = 70°
Which one of the following statements is correct?  (2024)
(a) Settling time of System 1 is more than that of System 2.
(b) Settling time of System 2 is more than that of System 1.
(c) Settling times of both the systems are the same.
(d) Settling time cannot be computed from the given information.
Ans: (b)
Sol: In system -1
In system -2

Q2: The damping ratio and undamped natural frequency of a closed loop system as shown in the figure, are denoted as ζ and  ω_n, respectively. The values of ζ and ω_n are  (2022)
(a) $\zeta =0.5and\omega n=10rad/s$ζ = 0.5 and ω_n = 10rad/s
(b) ζ = 0.1 and ω_n = 10rad/s
(c) ζ = 0.707 and ω_n = 10rad/s
(d) ζ = 0.707 and ω_n = 100rad/s
Ans: (a)
Sol: Reduced the block diagram:
Transfer function,
Standard form,

Q3: In the given figure, plant  and compensator The external disturbance input is D(s). It is desired that when the disturbance is a unit step, the steady-state error should not exceed 0.1 unit. The minimum value of K is ______.
(Round off to 2 decimal places.)  (2021)
 (a) 12.25
(b) 14.12
(c) 9.54
(d) 6.22
Ans: (c)
Sol: ∴
∴ K_min = 9.54

Q4: Consider a closed-loop system as shown.  is the plant transfer function and G_c(S) = 1 is the compensator. For a unit-step input, the output response has damped oscillations. The damped natural frequency is _____ rad/srad/s. (Round off to 2 decimal places.)  (2021)
(a) 10.9
(b) 4.62
(c) 12.02
(d) 8.05
Ans: (a)
Sol: 
Q5: Consider a negative unity feedback system with the forward path transfer function  where K is a positive real number. The value of K for which the system will have some of its poles on the imaginary axis is ________ .      (2020)
(a) 9
(b) 8
(c) 7
(d) 6
Ans: (b)
Sol: CE is
1 + G(s)H(s) = 0
⇒ s³+3s²+3s+(1+K) = 0  
R.H. criteria:
9 − (1 + K) = 0
⇒ K = 8

Q6: Which of the following option is correct for the system shown below?  (2020)
(a) 4^th order and stable
(b) $3rd$3^rd order and stable
(c) $4th$4^th order and unstable
(d) 3^rd order and unstable
Ans: (c)
Sol: Given system is fourth order system and unstable.
Stablity Status: Since it has one missing term of 's' thus undoubtedly given transfer function is unstable.

Q7: Consider a linear time-invariant system whose input r(t) and output y(t) are related by the following differential equation.
 
The poles of this system are at   (2020)
(a) +2j, -2j
(b) +2, -2
(c) +4, -4
(d) +4j, -4j
Ans: (a)
Sol: Poles: s² + 4 = 0
s = ±j2

Q8: The unit step response y(t) of a unity feedback system with open loop transfer function
 is shown in the figure. The value of K is _______ (up to 2 decimal places).   (2018)
(a) 4
(b) 8
(c) 10
(d) 12
Ans: (b)
Sol: Closed loop transfer function,
Given R(s) = 1/s
 ⇒ K = 8  

Q9: Consider a unity feedback system with forward transfer function given by  The steady-state error in the output of the system for a unit-step input is _________(up to 2 decimal places).   (2018)
(a) 0.25
(b) 0.45
(c) 0.66
(d) 0.85
Ans: (c)
Sol: Steady state error for type-0 and step input, 
= 0.66 uinits 

Q10: Match the transfer functions of the second-order systems with the nature of the systems given below.  (2018)
(a) P-I, Q-II, R-III
(b) P-II, Q-I, R-III
(c) P-III, Q-II, R-I
(d) P-III, Q-I, R-II
Ans: (c)
Sol: ω_n = √15 = 3.872 rad/sec
2ξ × 3.872 = 5
ω_n = √25 = 5 rad/sec  
2ξ × 5 = 10
ξ = 1(Critically damped)
Observing all the options, option (C) is correct.

Q11: Which of the following systems has maximum peak overshoot due to a unit step input?  (SET-2 (2017))
(a)
(b)
(c)
(d)
Ans: (c)
Sol: For maximum peak over shoot M_P ∝ (1/ξ)
ξ = 0.25 for option (C) which is least among all options. Therefore correct option is C.

Q12: When a unit ramp input is applied to the unity feedback system having closed loop transfer function  the steady state error will be  (SET-2(2017))
(a) 0
(b) a/b
(c) (a+K)/b
(d) (a-K)/b
Ans: (d)
Sol: Closed loop transfer function
Open loop transfer function Steady state error for ramp input given to type-1 system = 1/K_V 
where, velocity error coefficient,
 Steady state error,
e_ss = (a-K)/b

Q13: A second-order real system has the following properties:
a) the damping ratio ξ = 0.5 and undamped natural frequency ω_n = 10 rad/s,
b) the steady state value of the output, to a unit step input, is 1.02.
The transfer function of the system is  (SET-2  (2016))
(a)
(b)
(c)
(d)
Ans: (b)
Sol: Damping ratio ξ = 0.5
Undamped natural frequency ω_n = 10  rad/sec
Steady state output toa unit step input C_ss = 1.02
Hence steady state error e_ss = 1.02 − 1.00 = 0.02
∵ Characteristic equation is,
s² + 2ξω_ns + ω²_n = 0
s² + 2 × 0.5 × 10s + 100 = 0
s² + 10s + 100 = 0
From options, if we take option (B) then
C_ss = 1.02
Hence, Option (B) is correct answer.

Q14: The unit step response of a system with the transfer function  is given by which one of the following waveforms?  (SET-2 (2015))
(a) (b) (c) (d) Ans: (a)
Sol: For A,
For B,

Q15:  An open loop control system results in a response of e^−2t (sin5t + cos5t) for a unit impulse input. The DC gain of the control system is ______.  (SET-2 (2015))
(a) 0.82
(b) 0.24
(c) 0.55
(d) 1.47
Ans: (b)
Sol: Putting s = 0,