Previous Year Questions- Mathematical Models of Physical Systems - 2 | Control Systems - Electrical Engineering (EE) PDF Download

Q16: The response h(t) of a linear time invariant system to an impulse δ(t), under initially relaxed condition is h(t) = e^−t + e^−2t. The response of this system for a unit step input u(t) is  (2011)
(a) u(t) + e^−t + e^−2t
(b) $(e-t+e-2t)u\left(t\right)$(e^−t + e^−2t)u(t)  
(c) $(1.5-e-t-0.5e-2t)u\left(t\right)$(1.5 − e^−t − 0.5e^−2t)u(t)
(d) e^−tδ(t) + e^−2tu(t)
Ans: (c)
Sol: Transafer function of system is impulse response of the system with zero initial conditions.
Transfer function H(s) = L(e^−t + e^−2t)

Q17: A function y(t) satisfies the following differential equation :
 where δ(t) is the delta function. Assuming zero initial condition, and denoting the unit step function by u(t), y(t) can be of the form  (2008)
(a) e^t
(b) e^-t
(c) e^tu(t)
(d) $e-t$e^−t u(t)
Ans: (d)
Sol: Taking (L.T.) on both sides
 Y(s)(s + 1) = 1
Taking inverse laplace transform
y(t) = e^−tu(t)

Q18: The system shown in figure below
can be reduced to the form
with        (2007)
(a) X = c₀s + c₁,  Y = 1/(s² + a₀s + a₁), Z = b₀s + b₁ 
(b) X = 1, Y = (c₀s + c₁)/(s² + a₀s + a₁), Z = b₀s + b₁ 
(c) X = c₁s + c₀, Y = (b₁s + b₀)/(s² + a₁s + a₀), Z = 1
(d) X = c₁s + c₀, Y = 1/(s² + a₁s + a), Z = b₁s + b₀ 
Ans: (d)
Sol: The block diagram can be redrawn as
Signal flow graph of the block diagram,
There are two forward paths:
These are four individual loops
All the loop touch forward paths
Using Masson's gain formula
Comparing equation (i) and (ii), we get
Hence, Option (D)is correct.

Q19: When subject to a unit step input, the closed loop control system shown in the figure will have a steady state error of  (2005)(a) -1
(b) -0.5
(c) 0
(d) 0.5
Ans: (c)
Sol: Using signal flow graph,
Forward path gains
Individual loop,
Loop touches forward paths, therefore,
Using Mason's gain formula,
Steady state value of error, using final value theoram,

Q20: The unit impulse response of a second order under-damped system starting from rest is given by
c(t) = 12.5e^−6tsin8t, t ≥ 0
The steady-state value of the unit step response of the system is equal to  (2004)
(a) 0
(b) 0.25
(c) 0.5
(d) 1
Ans: (d)
Sol: Transfer function of a system is the unit impulse response of the system.
when input is unit step, R(s) = 1/s
Steady-state value of response, using final value theorem

Q21: For the block diagram shown in figure, the transfer function C(s)/R(s) is equal to  (2004)
(a)
(b)
(c)
(d)
Ans: (b)
Sol: Method-1: Using block-diagram reduction technique.
So, transfer function
Method-2: Using signal flow graph
Three forward paths,
The number of individual loap = 0
So graph determinant = Δ = 1
and Δ₁ = Δ₂ = Δ₃ = 1
Applying Mason's gain formula

Q22: For a tachometer, if θ(t) is the rotor displacement in radians, e(t) is the output voltage and K_t is the tachometer constant in V/rad/sec, then the transfer function,  will be  (2004)
(a) K_tS²
(b) $Kt/S$K_t/S
(c) K_tS
(d) K_t
Ans: (c)
Sol: θ(t) = rotor displacement in radians
ω(t) = dθ/dt = angular speed in rad/sec
Output voltage;
Taking laplace transform on both sides

Q23: The block diagram of a control system is shown in figure. The transfer function G(s) = Y(s)/U(s) of the system is (2003)
(a)
(b)
(c)
(d)
Ans: (b)
Sol:  Integrator are represented as 1/s in s-domain
As per the block diagram, the corresponding signal flow graph is drawn
One forward path P₁ = 2/s²
The individual loops are,
L₁ and L₂ are non-touching loops
L₁L₂ = 36/s²
The loops touches the forward path Δ₁ = 1
The graph determinant is  
Applying mason's gain formula,

Q24: A control system with certain excitation is governed by the following mathematical equation

The natural time constant of the response of the system are   (2003)
(a) 2 sec and 5 sec
(b) 3 sec and 6 sec
(c) 4 sec and 5 sec
(d) 1/3 sec and 1/6 sec
Ans: (b)
Sol: Natural time constant of the response depends only on poles of the system.

Q25: A control system is defined by the following mathematical relationship

 The response of the system as t → ∞ is  (2003)
(a) x = 6
(b) x = 2
(c) x = 2.4
(d) x = -2
Ans: (c)
Sol: Taking (LT) on both sides
Responce at t → ∞
Using final value theorem

Q26: The transfer function of the system described by  with u as input and y as output is  (2002)
(a)
(b)
(c)
(d)
Ans: (a)
Sol: