Frequency-Domain Analysis: Master Control System Analysis with Bode, Nyquist, and Root Locus

Question 1

The Nyquist criterion determines the stability of a system based on the:

Accepted Answer

Number of encirclements of the origin in the Nyquist plot

Answer

Slope of the Nyquist plot at the origin

Answer

Area enclosed by the Nyquist plot

Answer

Distance between the Nyquist plot and the origin

Question 2

The gain margin of a system is defined as the:

Accepted Answer

Amount by which the gain can be increased before the system becomes unstable

Answer

Amount by which the gain can be decreased before the system becomes stable

Answer

Difference between the phase margin and 180 degrees

Answer

Ratio of the gain at the crossover frequency to the gain margin

Question 3

The root locus of a system with a pole at -2 and a zero at -1 is a:

Accepted Answer

Straight line from -2 to -1

Answer

Circle centered at the origin with radius 1

Answer

Parabola with vertex at -1.5

Answer

Hyperbola with asymptotes at +2 and -2

Question 4

The phase margin of a system is defined as the:

Accepted Answer

Difference between the phase of the open-loop transfer function at the crossover frequency and 180 degrees

Answer

Sum of the phase of the open-loop transfer function at the crossover frequency and 90 degrees

Answer

Difference between the phase of the closed-loop transfer function at the crossover frequency and 0 degrees

Answer

Negative of the phase of the open-loop transfer function at the crossover frequency

Question 5

Given a Bode plot with a gain margin of 10 dB and a phase margin of 45 degrees, what is the stability status of the control system?

Accepted Answer

Stable

Answer

Unstable

Answer

Marginally stable

Question 6

Which of the following Bode plots represents a system with a phase margin of 45 degrees and a gain margin of 6 dB?

Accepted Answer

A plot with a gain margin of 6 dB and a phase margin of 45 degrees

Answer

A plot with a gain margin of 6 dB and a phase margin of 60 degrees

Answer

A plot with a gain margin of 4 dB and a phase margin of 45 degrees

Question 7

Which of the following frequency-domain techniques can be used to estimate the stability of a system with nonlinear elements?

Accepted Answer

Describing function

Answer

Nyquist plot

Answer

Root locus

Answer

Bode plot

Question 8

When using Nyquist plots to analyze a system, which of the following can be used to estimate the system's stability margin?

Accepted Answer

The distance between the Nyquist plot and the -1+0j point

Answer

The frequency at which the Nyquist plot crosses the imaginary axis

Answer

The frequency at which the Nyquist plot crosses the real axis

Question 9

In a Bode plot, what is the slope of the magnitude response at low frequencies for a zero-pole pair?

Accepted Answer

20 dB/decade

Answer

60 dB/decade

Answer

80 dB/decade

Answer

40 dB/decade

Question 10

What is the phase response of a system at high frequencies with a single pole at s = -2?

Accepted Answer

-90 degrees

Answer

45 degrees

Answer

-90 degrees at low frequencies

Answer

90 degrees

Question 11

According to the Nyquist stability criterion, a control system is stable if:

Accepted Answer

the Nyquist plot does not encircle the point -1+0i

Answer

the Nyquist plot intersects the real axis at a point greater than 1

Answer

the Nyquist plot encircles the point -1+0i clockwise

Answer

the Nyquist plot crosses the negative real axis

Question 12

In root locus analysis, what occurs at a breakaway point?

Accepted Answer

Multiple branches of the root locus merge

Answer

The root locus asymptotes to a vertical line

Answer

The root locus intersects the real axis

Answer

The root locus crosses the imaginary axis

Question 13

What is the definition of the gain margin of a control system?

Accepted Answer

The ratio of the magnitude of the open-loop transfer function at the crossover frequency to that at the unity gain frequency

Answer

The difference between the phase angle of the open-loop transfer function at the crossover frequency and -180 degrees

Question 14

Open-loop control systems are suitable for processes that are:

Accepted Answer

Stable and linear

Answer

Stochastic and time-varying

Answer

Unstable and non-linear

Answer

Non-minimum phase

Question 15

What primary advantage does frequency-domain analysis offer for control system design?

Accepted Answer

Insights into the steady-state response and stability

Answer

Accurate predictions of non-linear effects

Answer

Detailed information about the transient response

Question 16

What is the phase margin of a system whose Nyquist plot encircles the -1 point three times in the clockwise direction?

Accepted Answer

540 degrees

Answer

180 degrees

Answer

0 degrees

Answer

360 degrees

Question 17

Which of the following root locus plots corresponds to a system that is unstable for all values of 'K'?

Accepted Answer

A plot with all roots in the right-half of the complex plane.

Answer

A plot with all roots on the imaginary axis.

Answer

A plot with some roots in the left-half and some in the right-half plane.

Question 18

Which of the following transfer functions represents a high-pass filter?

Accepted Answer

G(s) = s/(s+a)

Answer

G(s) = s^2/(s+a)

Answer

G(s) = 1/(s+a)

Answer

G(s) = a/(s+a)

Question 19

What is the cut-off frequency of a system with a transfer function of G(s) = K/(s^2+2ζωns+ωn^2)?

Accepted Answer

ωn

Answer

ωn^2

Answer

K/ωn

Answer

2ζωn

Question 20

Which of the following frequency-domain analysis techniques is used to determine the stability of a nonlinear system in the time domain?

Accepted Answer

describing function analysis

Answer

Bode plot

Answer

Root locus analysis

Answer

Nyquist plot

Question 21

What is NOT a benefit of employing frequency-domain analysis in control system design?

Accepted Answer

It lacks information regarding the transient behavior of the system

Answer

It allows for the analysis of systems with nonlinearities

Answer

It requires less computational effort than time-domain analysis

Question 22

A Bode plot visually depicts the:

Accepted Answer

Frequency response of a system

Answer

Time response of a system

Answer

Phase margin of a system

Question 23

The root locus plot demonstrates the:

Accepted Answer

Variation of the poles and zeros of a system as a parameter changes

Answer

Time response of a system

Answer

Frequency response of a system

Question 24

A system with a bandwidth of 100 Hz will likely have a rise time of approximately:

Accepted Answer

10 ms

Answer

100 ms

Answer

1 ms

Question 25

The gain margin of a system represents the:

Accepted Answer

Amount by which the loop gain can be increased before instability occurs

Answer

Amount by which the loop gain can be decreased before instability occurs

Answer

Phase margin of the system

Question 26

Which of the following is a limitation of frequency-domain analysis?

Accepted Answer

It cannot analyze systems with time-varying parameters

Answer

It cannot be used to design controllers

Answer

It is computationally demanding

Question 27

What does the phase margin of a control system represent?

Accepted Answer

The difference between the phase angle of the system at the crossover frequency and 180 degrees

Answer

The difference between the phase angle of the system at the zero frequency and -180 degrees

Answer

The difference between the phase angle of the system at the gain crossover frequency and 90 degrees

Question 28

What is the purpose of root locus analysis?

Accepted Answer

To determine the stability and performance of a control system as a parameter varies

Answer

To determine the frequency response of a control system

Answer

To identify the eigenvalues and poles of a control system

Question 29

What is the significance of the crossover frequency in a Bode plot?

Accepted Answer

It is the frequency at which the gain of the system is 0 dB

Answer

It is the frequency at which the system is marginally stable

Answer

It is the frequency at which the phase of the system is -90 degrees

Question 30

What is the advantage of using a Nyquist plot over a Bode plot?

Accepted Answer

It provides a graphical representation of both the magnitude and phase response of the system in a single plot

Answer

It is more accurate at high frequencies

Answer

It is easier to construct

Question 31

Which of the following is a key assumption in frequency-domain analysis?

Accepted Answer

The system is linear and time-invariant

Answer

The system is single-input single-output

Answer

The system is causal

Answer

The system is open-loop

Question 32

Which of the following is a key advantage of using frequency-domain analysis for control systems?

Accepted Answer

It allows us to analyze system stability and performance for a range of input frequencies

Answer

It eliminates the need for time-domain analysis

Answer

It simplifies the process of solving partial differential equations

Question 33

A Bode plot displays the frequency response of a system. What two quantities are typically plotted against frequency?

Accepted Answer

Magnitude (gain) and phase

Answer

Rise time and settling time

Answer

Amplitude and period

Answer

Gain margin and phase margin

Question 34

What information does the gain crossover frequency provide about a system's stability?

Accepted Answer

It indicates the frequency at which the system's open-loop gain is 0 dB

Answer

It reveals the system's damping ratio

Answer

It determines the system's steady-state error

Question 35

A Nyquist plot is used to determine the stability of a closed-loop system by analyzing its open-loop frequency response. Which of the following statements about Nyquist plots is TRUE?

Accepted Answer

It can determine stability without needing to explicitly calculate the closed-loop poles.

Answer

It plots the closed-loop transfer function in the frequency domain.

Answer

It only provides information about the system's steady-state error.

Answer

It directly shows the system's time-domain response to a sinusoidal input.

Question 36

In root locus analysis, the plot depicts the movement of closed-loop poles as a system parameter varies. What do the points where the root locus branches cross the imaginary axis signify?

Accepted Answer

The system's transition between stability and instability.

Answer

The system's steady-state error, indicating accuracy.

Answer

The system's natural frequency, representing oscillations.

Answer

The system's damping ratio, a measure of response decay.

Question 37

Phase margin is a key indicator of relative stability in control systems. A system possessing a phase margin of 60 degrees is generally considered:

Accepted Answer

Stable with good relative stability, exhibiting desirable response characteristics.

Answer

Unstable, with unbounded output for bounded inputs.

Answer

Stable, but with poor relative stability, potentially leading to oscillations.

Answer

Marginally stable, at risk of instability with parameter variations.

Question 38

Lead compensators are frequently employed in control system design to enhance performance. What is the primary purpose of using lead compensators?

Accepted Answer

To improve transient response speed and reduce overshoot, enhancing stability.

Answer

To attenuate high-frequency noise, improving signal-to-noise ratio.

Answer

To decrease the system's bandwidth, limiting the range of frequencies effectively controlled.

Answer

To reduce steady-state error, improving accuracy in tracking reference signals.

Question 39

In frequency-domain analysis, the concept of 'bandwidth' is crucial. What does the term 'bandwidth' specifically refer to in this context?

Accepted Answer

The range of frequencies over which the system's gain remains above a specified level, typically -3 dB, indicating effective signal transmission.

Answer

The time it takes for the system's output to settle within a certain tolerance band of the final value.

Answer

The maximum amplitude of the system's response, representing the peak output magnitude.

Answer

The difference between the upper and lower cutoff frequencies where the gain drops to zero.

Question 40

Frequency-domain analysis offers numerous benefits in control system analysis. However, it also has certain limitations. Which of the following is NOT a benefit of using frequency-domain analysis?

Accepted Answer

It always provides a complete and unambiguous representation of nonlinear system behavior.

Answer

It provides insights into the system's response to various frequencies, aiding in controller design.

Answer

It enables the analysis of systems with complex dynamics through transfer function manipulations.

Answer

It offers intuitive graphical tools like Bode and Nyquist plots for stability and performance assessment.

Question 41

The Nyquist stability criterion provides a graphical method for determining the stability of a system. How does a Nyquist plot achieve this?

Accepted Answer

By plotting the open-loop transfer function in the complex plane as a function of frequency, revealing encirclements of critical points related to stability.

Answer

By plotting the root locus of the system, highlighting pole locations as parameters vary.

Answer

By plotting the closed-loop transfer function in the frequency domain, directly showing resonant peaks.

Answer

By plotting the system's step response, illustrating settling time and overshoot characteristics.

Question 42

The root locus method is a powerful technique for analyzing control system behavior as parameters change. What specifically does a root locus plot depict?

Accepted Answer

The trajectories of the closed-loop poles in the complex plane as a specific system parameter varies, providing insights into stability and response changes.

Answer

The frequency response of the system, showing gain and phase variations with frequency but not directly pole locations.

Answer

The poles and zeros of the open-loop transfer function, revealing fundamental system dynamics but not closed-loop behavior.

Answer

The time-domain response of the system to various inputs, illustrating transient and steady-state behavior.

Question 43

Gain margin and phase margin are closely related stability measures in frequency-domain analysis.  Given a control system with a crossover frequency of 10 Hz and a gain margin of 20 dB, what can be inferred about its phase margin?

Accepted Answer

The phase margin cannot be precisely determined from the given information alone; a Bode plot or similar analysis is required.

Answer

The phase margin is approximately 90 degrees, implying a system on the verge of instability.

Answer

The phase margin is approximately 60 degrees, indicating a fixed relationship between gain and phase margins.

Answer

The gain margin provides no information about the phase margin; they are independent measures.

Question 44

The bandwidth of a control system is a fundamental concept related to its frequency response.  Which of the following definitions accurately describes the bandwidth in this context?

Accepted Answer

The range of frequencies over which the system's closed-loop gain remains above -3 dB, signifying the range of effectively controlled frequencies.

Answer

The frequency range over which the system remains stable, which is not necessarily related to signal attenuation.

Answer

The frequency at which the system exhibits a 90-degree phase shift, known as the phase crossover frequency.

Answer

The difference between the break frequencies of the dominant poles in the system, related to but not equivalent to the bandwidth.

Question 45

Which of the following is a common application of frequency-domain analysis in control system design?

Accepted Answer

Analyzing the stability and performance of a closed-loop system.

Answer

Analyzing the transient response of a system.

Answer

Designing a state-space controller.

Question 46

In root locus analysis, the angle of departure of a root from a pole is calculated as:

Accepted Answer

180 degrees minus the sum of angles of all other poles and zeros.

Answer

90 degrees plus the angle of the pole.

Answer

180 degrees plus the angle of the pole.

Answer

90 degrees minus the angle of the pole.

Question 47

A system's Bode plot exhibits a phase margin of -45 degrees. What can you infer about the system's stability?

Accepted Answer

The system is unstable.

Answer

The system is stable.

Answer

The system's stability cannot be determined from the phase margin alone.

Answer

The system is marginally stable.

Question 48

What is the primary purpose of utilizing a Nyquist plot in control system analysis?

Accepted Answer

To determine the stability of a closed-loop system and identify potential instability issues.

Answer

To analyze the frequency response of a system's open-loop transfer function.

Answer

To design a controller for a specific system.

Question 49

Which of the following statements accurately describes the root locus method?

Accepted Answer

The root locus plot illustrates the movement of closed-loop poles as a system parameter is varied.

Answer

The root locus plot is used to determine the system's steady-state error.

Answer

The root locus plot only applies to open-loop systems.

Question 50

A system has a gain margin of 10 dB. What does this indicate about the system's stability?

Accepted Answer

The system has a significant margin of safety before becoming unstable.

Answer

The system is marginally stable.

Answer

The system is unstable.

Question 51

In a Bode plot, what does the slope of the magnitude curve represent?

Accepted Answer

The rate of change in the system's gain with respect to frequency.

Answer

The system's phase shift at a specific frequency.

Answer

The system's bandwidth.

Question 52

A system's Nyquist plot encircles the point (-1, 0) on the complex plane. What can you conclude about the system's stability?

Accepted Answer

The system is unstable.

Answer

The system is marginally stable.

Answer

The system's stability cannot be determined from the Nyquist plot alone.

Answer

The system is stable.

Question 53

Which of the following is NOT typically used to assess control system performance?

Accepted Answer

Open-loop gain.

Answer

Rise time.

Answer

Overshoot.

Answer

Settling time.

Question 54

What is the primary function of a lead compensator in a control system?

Accepted Answer

To increase the system's phase margin and enhance stability.

Answer

To reduce the system's bandwidth.

Answer

To increase the system's steady-state error.

Question 55

How does the location of the dominant closed-loop poles affect a system's transient response?

Accepted Answer

Poles closer to the imaginary axis result in a slower transient response.

Answer

The location of the poles does not affect the transient response.

Answer

Poles closer to the imaginary axis result in a faster transient response.

Question 56

A system has a dominant pole at -2 + 3i. What can you conclude about the system's response?

Accepted Answer

The system will exhibit an oscillatory response with a natural frequency of 3 rad/s.

Answer

The system will exhibit a critically damped response.

Answer

The system will exhibit an overdamped response.