Solution_A1_NPTEL_Control_Engg__Jan_April_2020

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NPTEL
Control Engineering
Jan-April 2020
Assignment 1 Solution
1. Consider a system with input x(t) and output y(t) related by y(t) = x2(t) +x(t−3). The system is
(a) Causal [Correct]
(b) Non causal
(c) Time varying
(d) None of these
Solution:
Let us put the value of t in the above equation
t = 0, y(0) = x2(0) + x(−3)
t = 1, y(1) = x2(1) + x(−2)
t = 2, y(2) = x2(2) + x(−1)
t = −1, y(−1) = x2(−1) + x(−4)
We can observe from the above value of output that the output only depends on the present and
past value of input. Hence, the system is causal. It can be easily observed from the above relation
that the system is time-invariant.
2. Consider the following statements for a linear system given by y = f(x) with f : Rn → Rn
1. f(x1 + x2) = f(x1) + f(x2)
2. f [x(t+ T )] = f [x(t)] + f [x(T )]
3. f(Kx) = Kf(x)
(a) 1, 2 and 3 are correct
(b) 1 and 2 are correct
(c) 1 and 3 are correct [Correct]
(d) Only 2 is correct
Solution:
Any linear system satisfies the laws of superposition and homogeneity.
If an input consists of weighted sum of several signals, then the output is the superposition, that
is, the weighted sum of the responses of the system to each of those signals.
If y = f(x) and say, y1 = f(x1) and y2 = f(x2), then
f [(x1) + (x2)] = f(x1) + f(x2) = y1 + y2
means that the system satisfies the superposition theorem.
A system is said to be homogeneous if, any input signal say y = f(x) is multiplied by an arbitrary
scalar K i.e., f(Kx), then output f(Kx) = Kf(x), or scaling any input signal scales the output
signal by the same factor.
1
3. Consider a system with input x(n) and output y(n) related by y(n) = cosx(n) is the system linear?
(a) Yes
(b) No [Correct]
Solution:
A direct consequence of the Superposition property is that, for linear systems, an input which is
zero for all time results in an output which is zero for all time. When input x(n) = 0 then the
output y(n) = cos 0 = 1. Hence, for zero input, output is 1 which is not equal to zero and hence
the given system is nonlinear.
4. Which of the following is/are true about a dynamic system (Note: mutiple correct answer possible)
(a) memory elements are present [Correct]
(b) no memory elements
(c) output depends only on present input
(d) output depends on past and present inputs [Correct]
Solution:
A dynamic system is defined as the system in which the system retains or stores information about
input values at times other than the current time. The concept of memory in the system corresponds
to the presence of mechanism in the system related to memory.
5. In translational mechanical system, the damping is generally provided by
(a) Static friction
(b) Coulomb friction
(c) Viscous friction [Correct]
(d) Spring friction
Solution:
Viscous friction can be safely used to provide damping in mechanical system. Both static friction as
well as coulomb friction is not suitable to be used for the purpose of damping. The reason for static
friction is that it is only in existence as long as body is stationary, but as soon as the body starts
moving, the static friction vanishes. Coulomb friction has constant magnitude and does not vary
with the velocity of the body; therefore, it is not suitable to be used for the purpose or damping.
Further, spring friction is not a correct term at all.
6. In order to double the time period of a simple pendulum, the length of the string should be
(Hint: The time period of a simple pendulum: It is defined as the time taken by the pendulum to
finish one full oscillation)
(a) halved
(b) doubled
(c) quadrupled [Correct]
(d) none of the above
Solution:
Time period of a simple pendulum is given by
Tp =
√
L
g
2
If the length of the string is made four times then the time period will become double, as shown
below
Tpnew =
√
4L
g
= 2Tp
7. Consider a simple mass-spring-friction system as shown in the figure, where K1, K2 are spring
stiffness‘s, f is friction, M is mass, F is force, and x is displacement.
The force acting on the system is given by
(a) F = M d
2x
dt2 + f
dx
dt + (K1 +K2)x [Correct]
(b) F = M d
2x
dt2 + f
dx
dt + (K1 −K2)x
(c) F = M d
2x
dt2 + f
dx
dt + (K1K2)x
(d) F = M d
2x
dt2 + f
dx
dt
Solution:
The mechanical equivalent of the system is shown in the figure below
MF 1K 2K
x
f
From the above figure, mechanical equation of given system is
F = M
d2x
dt2
+ f
dx
dt
+K1x+K2x
F = M
d2x
dt2
+ f
dx
dt
+ (K1 +K2)x
8. For the given rotational system consisting of motor coupled to an inertial load through a shaft with
spring constant K, the motor torque acting on the load is given by
3
(a) T (t) = J d
2θ(t)
dt2 +B
dθ(t)
dt
(b) T (t) = J d
2θ(t)
dt2 +B
dθ(t)
dt +Kθ(t) [Correct]
(c) T (t) = J d
2θ(t)
dt2 +B
dθ(t)
dt + θ(t)
(d) T (t) = J dθ(t)dt +B
dθ(t)
dt + θ(t)
Solution:
The mechanical equivalent of the system is shown in the figure below
JT K
θ
B
From the above figure, mechanical equation of given system is
T (t) = J
d2θ(t)
dt2
+B
dθ(t)
dt
+Kθ(t)
9. The input is vi(t) and output is v0(t). Which of the following equation correctly describe the
dynamics of the electrical circuit shown below is
4
(a) vi(t) = R
di(t)
dt + L
d2i(t)
dt2 +
1
C i(t); vo(t) =
1
C
∫
i(t)dt
(b) vi = Ri(t) +
1
C
∫
i(t)dt; vo(t) =
1
C
∫
i(t)dt+ Ldi(t)dt
(c) v0(t) = Ri(t) + L
di(t)
dt +
1
C
∫
i(t)dt; vo(t) =
1
C
∫
i(t)dt
(d) vi(t) = Ri(t) + L
di(t)
dt +
1
C
∫
i(t)dt; vo(t) =
1
C
∫
i(t)dt [Correct]
Solution:
Applying KVL to loop, we get
vi(t) = Ri(t) + L
di(t)
dt
+
1
C
∫
i(t)dt
vo(t) =
1
C
∫
i(t)dt
5