Answer: (b) Linear Time invariant systems

Explanation: A linear time invariant (LTI) system provides the same output for the same input irrespective of when input is given. LTI systems are also used to predict the system’s long term behavior.

Hence the correct answer is an option (b).

Answer: (a) Open-loop system

Explanation: The traffic lamp will glow according to the set timing and sequence and is time-dependent. The sequence and time are controlled by relays that work on the pre-programmed time. It does not depend upon the rush of the road.

Hence the correct answer is an option (a).

Answer: (d) 1/s

Explanation: The abrupt change in input of a step function takes place at t=0, and the step’s size is1 unit.

Let the size of the step function be A, where A=1.

The Laplace transform of a function is given by:

L[f(t)] = F(s) = ∫₀^∞f(t) e^-st dt

F(s) = ∫₀^∞Af(t) e^-st dt

F(s) = -A/s (e^-∞-e^-0)

F(s) = A/s

We know, A =1

F(s) = 1/s

Hence the correct answer is an option (d).

Answer: (c) 15V, 7.5V

Explanation:

Given:

G(s) = 2

H(s) = 0.5 and R(s) = 10V

Output voltage:

= (2/1+2x 0.5) x 15 = 15V

Error voltage:

= (1/1+2x 0.5) x 15 = 7.5V

Hence the correct answer is option (c).

Answer: (d) Output of the system depends only on the present input

Explanation: Static systems do not have any feedback system. Hence, the output depends only on the present input.

Hence, the correct answer is an option (d).

Answer: (b) -0.2-0.25e^-t+0.05e^-5t

Explanation: The given function can be written as:

= A/s + B/ (s+1) + C/(s+5)

1 = A(s+1)(s+5) +Bs(s+5) +Cs(s+1)

To calculate the value of A, put s= 0, we get:

1 = A(1)(5)

A = 1/5 = 0.2

Now, to calculate the value of B, put s=-1, we get:

1 = B (-1)(4)

B = -1/4 = -0.25

Similarly, put s=-5, we get:

1 = C (-5) (-4)

C = 1/20 = 0.05

Substituting the value of A, B, and C in F(s), we get:

F(s) = A/s + B/ (s+1) + C/(s+5)

F(s) = 0.2/s – 0.25/ (s+1) + 0.05/(s+5)

We know, Laplace transform of 1/(s + a) = e^-atand 1/s = 1.

f(t) = -0.2-0.25e^-t+0.05e^-5t

Hence the correct answer is option (b).

Answer:

Explanation:

Given: x is the displacement in the above diagram. The Laplace transform of x is X(s). The differential equations governing the system are the balanced force equation at these nodes.

Here,

F is the opposite force due to mass.

F =

f is the opposite force due to friction

F =

k is the ideal elastic spring element that has negligible mass and friction. The force generated by the spring is directly proportional to the displacement of the body.

F = Kx

So, the force equation can be represented as:

Hence, the correct answer is an option (c).

Answer: (a)

Explanation: We will first shift the H1 block after G3

The shifting of a take-off point will make the block as: H1/ G3

Block G2 and G3 are in cascade. The equivalent block will be the product of these two (G2G3).

The gain for that block will be:

G2G3/ (1 + H2G2G3)

As shown, G1 is in cascade, the transfer function of the above system will be:

C(s)/R(s) = [G1G2G3/ (1 + H2G2G3)]/ [1 + G1G2G3/ (1 + H2G2G3) x H1/G3

C(s)/R(s) = G1G2G3/ (1 + H2G2G3 + G2G1H1)

Hence, the correct answer is an option (a).

Answer: (b) Product of all branch gains while traversing the forward path

Explanation: According to Mason’s Gain formula, the transfer function can be calculated as:

T(s) = C(s)/R(s)

Where,

Pk is the forward path gain

∆istheloopgain, which is calculated as:

∆=1-∑(Allloopgain) + ∑(Gainproductoftwonon-touchingloops)-∑(Gainproductofthreenon-touchingloops)

∆_k iscalculatedbyeliminatingallloopstouchingPk

Here, the loop gain is defined as the product of the branch gain that is traversing a forward path.

Hence, the correct answer is an option (b).

The Mason’s gain formula is used to find the overall transfer function of a signal graph.

Answer: (a) G1G2G3/ (1 + G1H1 + G3H2 + G3G1H1H2)

Explanation: Here, we will calculate the transfer function using mason’s gain formula:

Where,

Pk is the forward path gain

∆istheloopgain, which is calculated as:

∆=1-∑(Allloopgain) + ∑(Gainproductoftwonon-touchingloops)-∑(Gainproductofthreenon-touchingloops)

∆_k iscalculatedbyeliminatingallloopstouchingPk

Now,

Step 1: From the figure, let’s identify the forward path,

P1 = G1G2G3

There is only one forward path in the above figure.

Step 2: Now, let’s find the loop gain. There are two loops:

L1 = -G1H1

L2 = -G3H2

Step 3: Two non-touching loops. There is only one combination of two non-touching loops, as shown below:

L1 L2 = G3G1H1H2

Step 4: Since there are no three non-touching loops, we will directly find the system’s determinant.

∆= 1 – (L1+ L2) + (L1L2)

= 1 + G1H1 + G3H2 + G3G1H1H2

Step 5: Here, we will find ∆₁ because there is only one forward path.

For ∆₁, we will eliminate L1 and L2 because both loop gains are touching the forward path.

L1 = L2 = 0

∆₁=1-0=1

Now, the transfer function is obtained as: C/R = G1G2G3/ (1 + G1H1 + G3H2 + G3G1H1H2)

Hence the correct answer is an option (a).

Answer: (c) 1-e^-t⁄10

Explanation:

The transfer function of a loop is G/ (1+GH)

We will first simplify the above block diagram into the simple system. Let’s calculate the transfer function of the first loop.

TF = [1/ (40s+2)] / [1 + 2/ (40s+2)] = 1/ (40s +4)

Now, two blocks are left in cascade. The equivalent block is the product of these two blocks, as given below:

C(s)/R(s) = 4. (1/ (40s +4)) = 1/ (10s + 1)

The Laplace transform of the step input is 1/s. It means R(s) = 1/s.

C(s) = [1/ (10s + 1)]. R(s)

C(s) = [1/ s (10s + 1)

Taking the inverse Laplace of the above equation, we get:

1-e^-t⁄10

Answer: (d) Underdamped

Explanation: The given equation is: s^2 + 2s + 2 = 0

It is a second-order differential equation. The Laplace transform of a standard form of a second-order differential equation is:

Thus, the system is underdamped.

Hence, the correct answer is an option (d).

Answer: (a) 9, 0.889, 0.762

Explanation: The standard transfer function can be written as:

The given equation is: 81/ (s^2 + 16s + 81)

Comparing the values, we get:

Thus, the undamped natural frequency is 9, and the damping ratio is 0.889.

The Peak time can be calculated as:

Hence, the correct answer is an option (a).

Answer: (c) 2.0, 1.5

Explanation: The standard transfer function can be written as:

The given equation is: 4/ (s^2 + 4s + 4)

Comparing the values, we get:

K= 1

The settling time for a 2 percent band is calculated as:

Settling time = 2 seconds

The settling time for a 5 percent band is calculated as:

Settling time = 1.5 seconds

Hence, the correct answer is an option (c).

Answer: (b) ¼

Explanation: The given transfer function is:

G(s) = (s + 4)/ (ks^2 + s + 4)

The characteristic equation ks^2 + s + 4 = 0

Dividing the equation by k, we get:

S^2 +s/k + 4/k = 0

Hence, the correct answer is an option (b).

Answer: (d) 1/6

Explanation: The step error can be calculated as:

ess = sR(s)/ (1 + G(s))

R(s) = 1/s (in case of unity feedback)

G(s) = 1/ (s+2)(s+3)

ess = (s x 1/s )/ (1 + (1/ (s+2)(s+3)))

ess = 1/(1 + kp)

Where, kp is the step error coefficient

Kp can be calculated as:

Hence, the correct answer is an option (d).

Answer: (c) π⁄4

Explanation: The given transfer function is: G(s) = 25/ (s^2 + 6s + 25)

Comparing the value of the given transfer function with the standard equation

Hence, the correct answer is option (c).

Answer: (d) Decaying exponential function

Explanation: The RL circuit comprises of the resistor and inductor connected in series.

The equation can be written as:

1 = RI(s) + sLI(s)

1 = I(s) [R + sL]

I(s) = 1 / (R +sL)

Taking the inverse Laplace, we get:

The equation clearly depicts that the impulse response is a decaying exponential function.

Hence, the correct answer is option (d).

Answer: (a) -2, (-1, -2)

Explanation: The zeroes can be calculated by equating the numerator to zero:

5 (s + 2) = 0

5s + 10 = 0

5s = -10

s = -2

The poles can be calculated by equating the denominator to zero:

s^2 + 3s + 2 = 0

s^2 + 2s + s + 2 = 0

s (s + 2) + 1 (s + 2) = 0

(s + 1) (s + 2) = 0

s = -1, -2

Hence, the correct answer is an option (a).

Answer: (b) Three

Explanation: The given characteristic equation is: s^3 + 3s^2 + 4s + 1 = 0.

To find the number of roots, we need to create a Routh table, as shown below:

There are no roots and no significant changes in the RHS plane, as shown in the above table. Hence, all three roots lie in the LHS plane.

Hence, the correct answer is an option (b).

Answer: (a) Unstable

Explanation: The given characteristic equation is: s^4 + 5s^3 + 3s^2 + 6s + 5 = 0. We first need to find the roots by creating the Routh’s array table.

Routh’s array table is shown below:

In the first column of the above table, we have two sign changes. It means that two roots are in the RHS plane. Hence, the system is unstable.

Hence the correct answer is an option (a).

Answer: (b) 2

Explanation: For the above equation, we need to find the roots by creating the Routh’s array table. The given equation is: s^3 + Ks^2 + 5s + 10

The table is given below:

For the system to be critically stable, we will put (5K -10)/K = 0

5K – 10 = 0

5K = 10

K = 2

The value of K for which the system is said to be critically stable is 2.

Hence, the correct answer is an option (b).

Answer: (a) 0 < A < 12

Explanation: For the above equation, we need to find the roots by creating the Routh’s array table. The given equation is: s^3 + 3s^2 + 4s + A = 0

The table is given below:

There is no change in sign in the first column of the Routh table. It means that all roots lies in the left half of the s-plane.

Putting A and (12 – A)/3 > 0, we get:

A > 0 (or 0 < A)

(12 – A)/3 > 0

12 – A > 0

12 > A (or A < 12)

From the above equations, we get two values of A, i.e., A > 0 and A < 12. It means that A lies between 0 and 12, as shown below:

0 < A < 12

Hence, the correct answer is an option (a).

Answer: (c) -19.5 < k < 8

Explanation: The two blocks in the above diagram are in cascade. So, the equivalent block will be the product of these two blocks.

G(s) = k (s – 2) /(s +1) (s^2 + 9s + 16)

Now, H(s) = 1, as shown in the above block diagram. The characteristic equation will be: 1 + G(s) H(s)

C(s) = 1 + [k (s – 2) / (s + 1) (s^2 + 9s + 16)] = 0

= s^3 + 10s^2 + 25s + 16 + ks – 2k

= s^3 + 10s^2 + s (25 + k) + 16 – 2k

For the above equation, we need to find the roots by creating the Routh’s array table.

The table is given below:

For stability,

16 – 2k > 0

16 > 2k

8 > k or k < 8

10 (25 + k) – (16 -2k) /10 > 0

250 + 10k -16 + 2k > 0

12k + 234 > 0

12k > -234

k > -19.5

From the two values of k, we can say that it lies between -19.5 < k < 8

Hence, the correct answer is an option (c).

Answer: (b) 0

Explanation: The number of asymptotes in a given system is equal to the number of branches approaching infinity. So, the formula to calculate the number of asymptotes is P -Z. Here, P and Z represent the poles and zeroes.

We know that poles and zeroes are calculated by equating the denominator and numerator to zero. So, for the given open-loop transfer function, we get:

P = 3

Z = 3

So, the number of zeroes at infinity = 3 – 3 = 0

Hence, the correct answer is an option (b).

Answer: (c) Two zeroes at infinity

Explanation: The formula to calculate a number of zeroes at infinity is P – Z. Here, P and Z are the number of poles and zeroes in a given transfer function.

We know that poles are calculated by equating the denominator to zero, and zeroes are calculated by equating the numerator to zero. So, for the above given transfer function, we get:

P = 3

Z = 1

So, the number of zeroes at infinity = 3 – 1 = 2

Hence, the correct answer is an option (c).

Answer: (c) The asymptotes cross the real axis.

Explanation: Centroid is defined as a common point where all the asymptotes intersect on the real axis. The value of centroid is always real. But, it can be located either on the positive or negative real axis.

Answer: (d) -3

Explanation:

We will first calculate the number of branches approaching infinity and then the asymptotes. With the help of asymptotes, we will calculate the value of centroid.

The given transfer function of the system is G(s) = K / [(s + 1) (s + 4 + 4j) (s + 4 – 4j)].

The number of asymptotes is equal to the number of branches approaching infinity. There are no zeroes but three poles.

So, P – Z = 3

Let’s calculate the value of the poles by equating the denominator equal to zero. We get:

Poles located at: -1,

The angle of asymptotes is calculated by:

Θ = (2q + 1) 180 / (P – Z)

Here, q = 0, 1, 2…

The number of asymptotes is equal to the number of branches approaching infinity.

So, we will calculate the asymptotes at value 0, 1, and 2.

For q = 0,

Θ=180⁄3=60degrees

For q = 1,

Θ=(2+1) 180⁄3=180degrees

For q = 2,

Θ=(4+1) 180⁄3=300degrees

Centroid is defined as a common point where all the asymptotes intersect on the real axis.

σ= / (P – Z)

σ= (- 1 – 4 – 4 – 0) / 3

σ=(-9)⁄3

σ=-3

Hence, the correct answer is an option (d).

Answer: (b) (-2, 0)

Explanation:

The given equation for the feedback control system is s^3 + 5s^2 + (K + 6)s + K = 0.

The above equation can also be written as:

s^3 + 5s^2 + Ks + 6s + K = 0

s^3 + 5s^2 + 6s + K(s + 1) = 0

s (s^2 + 5s + 6) + K(s + 1) = 0

1 + K(s + 1) / [s (s^2 + 5s + 6)] = 0

1 + K(s + 1) / s (s^2 + 2s + 3s + 6) = 0

1 + K(s + 1) / s (s + 2)(s + 3) = 0

Now, we will calculate the value of centroid, which is equal to:

σ= / (P – Z)

Here, the number of poles and zeroes are 3 and 1.

σ= [(0 – 2 – 3) + 1] / (3 – 1)

σ=(-4)⁄2

σ=-2

Hence, the correct answer is option (b).

Answer: (d) 65 degrees

Explanation: The phase margin can be calculated as 180 +

Where,

∅ is the phase of G(jw) at the gain cross over frequency.

So, phase margin = 180 + (-115)

= 180 – 115

= 65 degrees

Hence, the correct answer is an option (d).

Answer: (b) Infinite

Explanation: The gain margin indicates the additional gain provided to the system without affecting its stability.

The total phase shift of a second-order system is approximately equal to 180 degrees, which leads to the infinite frequency. Thus, the gain margin is also infinite.

Hence, the correct answer is an option (b).

Answer: (c) 0. 707 Radians / second

Explanation: The given open-loop transfer function is: G(s) = 1 / s(s + 1) (2s + 1)

The imaginary part of the system at phase cross over frequency is zero. Hence, we will equate the imaginary part to zero, as shown below:

It means that the phase cross-over frequency to the system is 0.707 Radians/s.

Hence, the correct answer is an option (c).

Answer: (c) 8

Explanation: The given open-loop feedback control system is: G(s)H(s) = 1 / [(s + 1) ^3]

The phase angle for the cross-over frequency can be calculated by:

-3 tan^-1 ω=-180°

We get the value of ω=√3

Now, the gain margin is the magnitude of the transfer function, as shown below:

= 1 / |G(s)H(s)| = |(j√3+1³|

= 8

Hence, the correct answer is an option (c).

Answer: (a) Π

Explanation: The phase margin can be calculated as: 180 + ∅

Where,

∅ is the phase of G(jω) at the gain cross over frequency.

We will first find the magnitude of the given system, as shown below:

Hence, the correct answer is an option (a).

Answer: (d) The frequency at which the two asymptotes meet.

Explanation: The frequency at which the two asymptotes in the Bode plot meet is termed as corner frequency.

Hence, the correct answer is an option (d).

Answer: (b) 1 and 3

Explanation: The Bode plot is defined as the frequency response plot of the sinusoidal transfer function of a system. The two graphs of the Bode plot are the plot of magnitude and phase angle. Thus, it is in the frequency domain.

The Nyquist plot is considered as the extension of the polar plot. The variation of frequency from infinity to -infinity results in the plot, known as the Nyquist plot. Hence, the Nyquist criterion is in the frequency domain.

Hence, the correct answer is an option (b).

Answer: (d) 0, 2

Explanation: The given plot is of the type 1/ (ST1 + 1)(ST2 + 1)

The above transfer function has order 2 and type 0.

Hence, the correct answer is an option (d).

Answer: (b) 0.42

Explanation: The formula to calculate the damping ratio using the phase margin is:

Damping Ratio = tan∅√cos ∅⁄2

= ((tan45√cos45))⁄2

= 0. 42

Hence, the correct answer is an option (b).

Answer: (c) 60 dB

Explanation: The Phase margin can be calculated as:

The gain margin (GM) = 20 log (1 / |G(s)H(s)|)

At, s = jω

We get: (GM) = 60dB

Hence, the correct answer is an option (c).

Answer: (c) PID Controller

Explanation: The PID is the combination of proportional, integral, and derivate control modes. Such a combination makes it the most powerful controller.

Hence, the correct answer is an option (c).

Answer: (a) 61. 52%

Explanation: The error the rate of 1.2% per second c neb calculated as:

e = 1.2% x 2 = 1.2/100 x 2 = 2.4%

Now, let’s calculate the Pout (controller output) = 4 (2.4%+ KD x 1.2%) + Po

= 4 (2.4 / 100 + 0.4 x 1.2 /100) + 50/100

= 4 x 2.88/100 + 50/100

= 11.52/100 + 50/100

= 61.52/100

= 61.52%

Hence, the correct answer is an option (a).

Answer: (a) PD controller

Explanation: The Proportional Derivative controller is preferred to handle fast process load changes.

Hence, the correct answer is an option (a).

Answer: (c)

Explanation: Lag compensator does not suppress steady state performance. Instead, it improves steady state performance.

Hence, the correct answer is an option (c).

Answer: (a) Between 0 and 1

Explanation: The given transfer function is:

(1 +aTs) / (1 + Ts)

We will first calculate the poles and zeroes of the given transfer function.

Here,

Zero = -1/aT

Pole = -1/T

The pole in the given system is nearer to the jω axis (origin). The 0 will be far from the axis, such that the value of a < 1. It means that the value lies between 0 and 1.

Hence, the correct answer is an option (a).

Answer: (d)

Explanation: Phase lead is responsible for improving the damping and reducing the system’s steady-state error.

Hence, the correct answer is an option (d).

Answer: (b) 30 degrees

Explanation: The phase shift provided by the lead compensator can be calculated using the formula:

The standard transfer function of a lead compensator is represented as:

(s +1/aT) / (s + 1/bT)

Comparing the value of the given transfer function to the standard transfer function, we get:

a = 1/6

b = 1/2

Putting the values of a and b in the phase shift formula:

Hence, the correct answer is an option (b).

Answer: (b) Controllable but unstable

Explanation:

We know,

Here,

The product of matrix A and B is,

Thus, the rank of the system is 2.

The determinant of the system is non-zero. Hence, the system is controllable.

Now, we will check for stability.

SI – A = 0

The above matrix can be written in the form of an equation:

s^2 + 3s -6 = 0

Now, we will find the roots using Routh’s array table, as shown below:

There are two sign changes in the first column. Thus, the system is unstable.

Hence, the correct answer is an option (b).

Answer: (d) Sum of the principal diagonal elements

Explanation: The sum of Eigenvalue of a given matrix is generally called the trace of that matrix. The trace of a matrix is the sum of the diagonal elements of a matrix.

Hence, the correct answer is an option (d).

Answer: (d)

Explanation: Let x1(t) = a(t) and x2(t) = ẋ1

Thus, the given characteristic equation can be written as:

ẋ2 +3×2 + 2×1 = r(t)

The matrix representation will be:

Where, A is the matrix:

We know the characteristic equation to calculate Eigenvalue is |SI – A| = 0

The equation thus formed is:

s (s + 3) + 2 = 0

s^2 + 3s + 2 = 0

s^2 + 2s + s + 2 = 0 (factorization to find Eigenvalues)

(s + 2)(s + 1) = 0

S = -2, -1

Thus, the required Eigenvalues are -2 and -1.

Hence, the correct answer is an option (d).

Answer: (c) forward path = 3, loops = 3, and non-touching loops = 0

Explanation: Forward path is the path from the input to the output node in the given signal flow graph. Here, there are three forward paths.

The number of loops in the given signal flow graph is three, as shown in the below image:

The non-touching loops are considered as non-touching when there are no common nodes between them. There are no non-touching loops in the given signal flow graph. It is because all the loops touch each other.

Hence, the correct answer is an option (c).

Answer: (b) 2, 1, 5, 3, 4

Explanation: The correct order to find the transfer function of the free body diagram is given below:

The free-body diagram is created by drawing each mass separately and marking all the forces acting on the node. Write one differential equation for each of the free body diagram and take the Laplace transform and convert the differential equations into the algebraic equations. Rearrange the equations in the s-domain to find the output and input ratio by eliminating the unwanted variables. In this way, we can easily calculate the transfer function of the free body diagram.

Hence, the correct answer is an option (b).