Control Systems MCQ Quiz - Objective Question with Answer for Control Systems - Download Free PDF

Concept:

For a unity feedback system with open-loop transfer function , the steady-state error depends on the input type and the system type.

The system type is determined by the number of integrators (poles at origin) in the open-loop transfer function.

The steady-state error is given by:

• For unit-step input: , where
• For unit-ramp input: , where

Given:

This system has one pole at origin → Type 1 system

Calculation:

Step input:

Ramp input:

Correct Answer: 1) 1, 1

Concept:

The time constant (τ) of an RC circuit is given by the formula:

τ = R_eqC

where R_eq is the equivalent resistance seen by the capacitor, and C is the capacitance.

While finding out the time constant the voltage source will be shorted.

Calculation:

Given:

Resistors: 3Ω and 2Ω will be in parallel
Capacitance: C = 10 F

Solution:

If the resistors are actually in parallel (common in RC circuits):

R_eq = (3Ω × 2Ω)/(3Ω + 2Ω) = 6/5Ω = 1.2Ω

τ = R_eqC = 1.2Ω × 10 F = 12 sec

Final Answer:

The correct time constant is 1) 12 sec when considering parallel resistors.

Explanation:

To determine the centroid of the asymptotes in the root locus of the given loop transfer function, we need to first understand the parameters involved. The given loop transfer function of the closed-loop control system is:

G(s) H(s) = k(s + 1) / [s(s + 2)(s + 3)]

The root locus method involves plotting the roots of the characteristic equation of the closed-loop system as the gain "k" varies from 0 to ∞. The centroid of the asymptotes is a point on the real axis where the asymptotes of the root locus intersect the real axis. This point is important as it provides insights into the system's stability and behavior.

The formula for finding the centroid (σ) of the asymptotes is given by:

σ = (sum of poles - sum of zeros) / (number of poles - number of zeros)

From the given transfer function:

Poles: s = 0, s = -2, s = -3

Zeros: s = -1

Number of poles (P) = 3

Number of zeros (Z) = 1

Sum of poles = 0 + (-2) + (-3) = -5

Sum of zeros = -1

Substituting these values into the formula for the centroid:

σ = (-5 - (-1)) / (3 - 1)

σ = (-5 + 1) / 2

σ = -4 / 2

σ = -2

Therefore, the centroid of the asymptotes in the root locus is at (-2, 0).

Correct Option Analysis:

The correct option is:

Option 3: (-2, 0)

This option correctly identifies the centroid of the asymptotes for the given loop transfer function

Explanation:

To determine the slope in the magnitude plot for the high frequency asymptote of a control system, we need to understand the relationship between the number of poles and zeros in the system.

In control systems, the Bode plot is a graphical representation of a system's frequency response. The slope of the magnitude plot in the high frequency region is influenced by the number of poles and zeros. Specifically, each pole contributes a slope of -20 dB/decade, and each zero contributes a slope of +20 dB/decade.

Given:

• Number of poles (P" id="MathJax-Element-893-Frame" role="presentation" style="position: relative;" tabindex="0">PP $P$ ) = 14
• Number of zeros (Z" id="MathJax-Element-894-Frame" role="presentation" style="position: relative;" tabindex="0">ZZ $Z$ ) = 2

The formula to determine the overall slope in the high frequency asymptote is:

Slope (dB/decade) = -20 × (Number of poles - Number of zeros)

Substituting the given values:

Slope (dB/decade) = -20 × (14 - 2)

Slope (dB/decade) = -20 × 12

Slope (dB/decade) = -240 dB/decade

Therefore, the correct option is:

Option 2: -240 dB/decade

Let's analyze the given Routh array and determine how to replace the row of zeros.

Understanding the Routh Array and Row of Zeros:

• The Routh array is used to determine the stability of a linear time-invariant system.
• A row of zeros in the Routh array indicates the presence of roots on the imaginary axis (jω-axis) or roots that are symmetrical about the origin.
• To continue the Routh array and determine stability, we need to form an auxiliary polynomial from the row above the row of zeros.

Forming the Auxiliary Polynomial:

• The row above the row of zeros is s⁴.
• The coefficients in this row are used to form the auxiliary polynomial.
• The auxiliary polynomial is formed using only even powers of 's'.

In this case, the row s⁴ has coefficients 2, 12, and 16.

Therefore, the auxiliary polynomial is:

A(s) = 2s⁴ + 12s² + 16

Finding the Derivative of the Auxiliary Polynomial:

To replace the row of zeros, we need to find the derivative of the auxiliary polynomial with respect to 's'.

dA(s)/ds = 8s³ + 24s

Replacing the Row of Zeros:

The coefficients of the derivative will replace the row of zeros in the Routh array.

The coefficients of the derivative are 8 and 24.

We can simplify these coefficients by dividing by 8:

8/8 = 1 24/8 = 3

So, the simplified coefficients are 1 and 3.

The row of zeros will be replaced by the coefficients of s³ + 3s.

Therefore, the correct answer is option 2.

Concept:

Time constant 

Calculation:

Taking Laplace transform, we get

40 s X(s) + 2X(s) = 12(s)

Pole will be at -1/20. 

Time constant 

• The root locus is the strongest tool for determining stability and the transient response of the system as it gives the exact pole-zero location and also their effect on the response
• A Bode plot is a useful tool that shows the gain and phase response of a given LTI system for different frequencies
• The Nyquist plot in addition to providing absolute stability also gives information on the relative stability of stable systems and degree of instability of the unstable system
• Routh-Hurwitz criterion is used to find the range of the gain for stability and gives information regarding the location of poles

Concept:

A transfer function is defined as the ratio of Laplace transform of the output to the Laplace transform of the input by assuming initial conditions are zero.

TF = L[output]/L[input]

For unit impulse input i.e. r(t) = δ(t)

⇒ R(s) = δ(s) = 1

Now transfer function = C(s)

Therefore, the transfer function is also known as the impulse response of the system.

Transfer function = L[IR]

IR = L-1 [TF]

Calculation:

Given the differential equation is,

By applying the Laplace transform,

s² Y(s) + 4 Y(s) = 6 R(s)

Poles are the roots of the denominator in the transfer function.

⇒ s² + 4 = 0

⇒ s = ±2j

Lag compensator:

Transfer function:

If it is in the form of

 , then a

If it is in the form of

 , then a > b

Pole zero plot:

The pole is nearer to the origin.

Filter: It is a low pass filter (LPF)

Maximum phase lag frequency:

Maximum phase lag:

ϕm is negative

Lead compensator:

Transfer function:

If it is in the form of

 , then a > 1

If it is in the form of

 , then a

Pole zero plot:

The zero is nearer to the origin.

Filter: It is a high pass filter (HPF).

Maximum phase lead frequency:

Maximum phase lead:

ϕm is positive

Concept:

KP = position error constant = 

Kv = velocity error constant = 

Ka = acceleration error constant = 

Steady-state error for different inputs is given by

From the above table, it is clear that for type – 1 system, a system shows zero steady-state error for step-input.

Concept:

Closed-loop transfer function = 

For unity negative feedback system Open-loop transfer function (G(s) H(s)) can be found by subtracting the numerator term from the denominator term 

Application:

Open-loop transfer Function

For DC gain s = 0

∴ open-loop gain 

Concept:

Minimum phase system: It is a system in which poles and zeros will not lie on the right side of the s-plane.  In particular, zeros will not lie on the right side of the s-plane.

For a minimum phase system,

Where P & Z are finite no. of poles and zeros of G(s)H(s)

Non-minimum phase system: It is a system in which some of the poles and zeros may lie on the right side of the s-plane. In particular, zeros lie on the right side of the s-plane.

Stable system: A system is said to be stable if all the poles lie on the left side of the s-plane.

Application:

As one zero lies in the right side of the s-plane, it is a non-minimum phase transfer function.

As there no poles on the right side of the s-plane, it is a stable system.

Explanation:

The controller is a device that is used to alter or maintain the transient state & steady-state region performance parameter as per our requirement.

Proportional Controller- 

The standard Proportional Controller as shown:

In space-form - 

In time-domain form - 

p(t) = Kp e(t) + po

Where,

po = controller output with zero error  

Kp = proportional gain constant.

Some effects of the proportional controller are as follows:

• The P-controller can stabilize a first-order system, can give a near-zero error, and improves the settling time by increasing the bandwidth.
• It also helps in reducing the steady-state error which makes the system more stable.
• The slow response of an over-damped system can be made faster with the help of the proportional controller. Hence option (2) is the correct answer.

​Important Points

Effects of Proportional Integral (PI) controllers:

• Increases the type of the system by one
• Rise time and settling time increases and Bandwidth decreases
• The speed of response decreased i.e. transient response becomes slower
• Decreases the steady-state error and steady-state response is improved
• Decreases the stability

Effects of Proportional Derivative (PD) controllers:

• Decreases the type of the system by one
• Reduces the rise time and settling time
• Rise time and settling time decreases and Bandwidth increases
• The speed of response is increased i.e. transient response is improved
• Improves gain margin, phase margin, and resonant peak
• Increases the input noise
• Improves the stability

The transfer function of a control system is defined as the ratio of the Laplace transform of the output variable to Laplace transform of the input variable assuming all initial conditions to be zero.

It is also defined as the Laplace transform of the impulse response.

If the input is represented by R(s) and the output is represented by C(s), then the transfer function will be:

Routh-Hurwitz criterion:

• Using the Routh-Hurwitz method, the stability information can be obtained without the need to solve the closed-loop system poles. This can be achieved by determining the number of poles that are in the left-half or right-half plane and on the imaginary axis.
• This involves checking the roots of the characteristic polynomial of a linear system to determine its stability.
• It is used to determine the absolute stability of a system.

Other methods of determining stability include:

Root locus:

• This method gives the position of the roots of the characteristic equation as the gain K is varied.
• With Root locus (unlike the case with Routh-Hurwitz criterion), we can do both analysis (i.e., for each gain value we know where the closed-loop poles are) and design (i.e., on the curve we can search for a gain value that results in the desired closed-loop poles).

Nyquist plot:

• This method is mainly used for assessing the stability of a system with feedback.
• While Nyquist is a graphical technique, it only provides a limited amount of intuition for why a system is stable or unstable, or how to modify an unstable system to be stable.