Time-Domain Analysis Quizzes: Test Your Understanding

Question 1

Which condition must be satisfied for a system to be asymptotically stable?

Accepted Answer

All poles must lie in the left half of the s-plane.

Answer

All poles must lie in the right half of the s-plane.

Answer

All poles must lie on the imaginary axis.

Answer

All poles must lie at the origin.

Question 2

Given a unit impulse response h(t) = e^(-t) * u(t), find the corresponding transfer function, G(s).

Accepted Answer

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

Answer

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

Answer

G(s) = 1

Answer

G(s) = s

Question 3

Consider a system with the following state-space representation: x' = [0 1; -1 0]x + [0; 1]u, y = [1 0]x. Determine if the system is controllable.

Accepted Answer

Controllable

Answer

Uncontrollable

Answer

Marginally controllable

Answer

Cannot be determined from the given information.

Question 4

Which condition is sufficient for a system to be observable?

Accepted Answer

The rank of the observability matrix is equal to the number of states.

Answer

The rank of the observability matrix is greater than the number of states.

Answer

The rank of the observability matrix is less than the number of states.

Answer

The observability matrix is invertible.

Question 5

Consider a system with the following state-space representation: x' = [1 2; -1 -1]x + [0; 1]u, y = [1 0]x. Determine if the system is observable.

Accepted Answer

Observable

Answer

Unobservable

Answer

Marginally observable

Answer

Cannot be determined from the given information.

Question 6

For a linear control system to be stable, which of the following conditions must be met?

Accepted Answer

All poles of the system lie in the left half of the s-plane.

Answer

All poles of the system lie in the right half of the s-plane.

Answer

All zeros of the system lie in the right half of the s-plane.

Question 7

In constructing a state-space representation of a control system, what is the first and most critical step?

Accepted Answer

Deriving the system's governing differential equations

Answer

Defining the system's input and output variables

Answer

Determining the system's eigenvalues

Question 8

In a unity feedback control system, the open-loop transfer function is G(s) = K(s + a) / (s(s + b)). Determine the closed-loop transfer function.

Accepted Answer

K(s + a) / (s(s + b) + K(s + a))

Answer

K(s + a) / (s + b)

Answer

K(s + b) / (s(s + a))

Answer

K(s + b) / (s(s + a) + K(s + b))

Question 9

Which of the following is NOT an advantage of using state-space representation for control system analysis?

Accepted Answer

Is inherently less complex than other methods.

Answer

Provides a systematic approach to modeling dynamic systems.

Answer

Enables direct calculation of system responses.

Answer

Can be easily converted to other representations.

Question 10

Which of the following is the state-space representation of the system defined by the differential equation:

d²y/dt² + 3dy/dt + 2y = u

Accepted Answer

x₁ = y, x₂ = dy/dt

Answer

x₁ = u, x₂ = y

Answer

x₁ = u, x₂ = dy/dt

Answer

x₁ = dy/dt, x₂ = u

Question 11

Which condition must be satisfied for a system to be considered stable?

Accepted Answer

All poles of the system must have negative real parts

Answer

At least one pole of the system must have a positive real part

Answer

All poles of the system must have positive real parts

Question 12

Given a system's unit step response: 
y(t) = 1 - e^(-t)

Determine its steady-state value.

Accepted Answer

1

Answer

-1

Answer

0

Answer

e

Question 13

Define zero-input response of a system.

Accepted Answer

The system's response when the input is zero

Answer

The response of the system when the output is zero

Answer

The response of the system when the initial conditions are zero

Answer

The response of the system when the poles are all at the origin

Question 14

What does the damping ratio of a second-order system represent?

Accepted Answer

The rate of decay of the system's response

Answer

The frequency of the system's oscillations

Answer

The stability of the system

Answer

The magnitude of the system's response

Question 15

Provide the definition of time constant for a first-order system.

Accepted Answer

The time it takes for the system's response to reach 63.2% of its final value

Answer

The time it takes for the system's response to reach 50% of its final value

Answer

The time it takes for the system's response to reach 99% of its final value

Question 16

Which of the following is a state variable in a state-space representation of a system?

Accepted Answer

State

Answer

Parameter

Answer

Output

Answer

Input

Question 17

The transfer function of a system is defined as the Laplace transform of which of the following?

Accepted Answer

System impulse response

Answer

System step response

Answer

System Bode plot

Answer

System frequency response

Question 18

Which of the following methods can be used to check the stability of a control system in the time domain?

Accepted Answer

Routh-Hurwitz criterion

Answer

Nyquist criterion

Answer

Nichols chart

Answer

Bode plot

Question 19

What is the definition of the time constant of a first-order system?

Accepted Answer

Time taken for the output to reach 63.2% of its final value

Answer

Time taken for the output to reach 50% of its final value

Answer

Time taken for the output to reach 99% of its final value

Question 20

Which of the following is a characteristic of a stable system?

Accepted Answer

All poles are in the left-half plane

Answer

Some poles are in the left-half plane and some are in the right-half plane

Answer

All poles are in the right-half plane

Answer

All poles are on the imaginary axis

Question 21

The unit impulse function is a function that is:

Accepted Answer

Zero everywhere except at t=0, where it is equal to 1

Answer

Proportional to time

Answer

Constant over time, equal to 1

Question 22

Which of the following is an advantage of using a state-space representation over a transfer function representation?

Accepted Answer

Can handle systems with multiple inputs and multiple outputs

Answer

Easier to analyze system stability

Answer

More intuitive to understand

Answer

Requires less computation

Question 23

The step response of a system is the output of the system when the input is a:

Accepted Answer

Unit step function

Answer

Unit parabolic function

Answer

Unit impulse function

Answer

Unit ramp function

Question 24

Consider the differential equation that describes a system: $\frac{d^2y}{dt^2} + 3\frac{dy}{dt} + 2y = x(t)$. Determine the transfer function of this system.

Accepted Answer

$\frac{1}{s^2 + 3s + 2}$

Answer

$\frac{s^2}{s^2 + 3s + 2}$

Answer

$\frac{s}{s^2 + 3s + 2}$

Answer

$\frac{1}{s^2 - 3s + 2}$

Question 25

A system has poles at -2 and -4. Determine whether this system is stable.

Accepted Answer

Yes, the system is stable.

Answer

Stability cannot be determined from pole locations.

Answer

The system is marginally stable.

Answer

No, the system is unstable.

Question 26

In state-space representation, what does matrix 'A' represent?

Accepted Answer

System dynamics matrix

Answer

Feedforward matrix

Answer

Output matrix

Answer

Input matrix

Question 27

According to the final value theorem, what information can be obtained about a system's response?

Accepted Answer

Steady-state value of the output for a given input.

Answer

Initial value of the output.

Answer

Peak value of the output.

Question 28

Which technique is NOT used for time-domain analysis?

Accepted Answer

Bode plot

Answer

Laplace transform

Answer

State-space analysis

Answer

Convolution

Question 29

Consider the state-space equations of a system: ẋ = Ax + Bu, y = Cx + Du. What does matrix 'A' represent?

Accepted Answer

The system's internal dynamics, describing how state variables change over time.

Answer

The system's output matrix, which maps state variables to outputs.

Answer

The system's input matrix, which maps inputs to state changes.

Question 30

A system's step response shows a significant overshoot and a prolonged settling time. What can be concluded about the system's stability?

Accepted Answer

The system is underdamped, indicating potential instability issues.

Answer

The system is critically damped, exhibiting optimal performance.

Answer

The system is overdamped, resulting in a slow response but stability.

Question 31

Which of the following methods is NOT a common time-domain analysis technique?

Accepted Answer

Bode Plot Analysis

Answer

Ramp Response

Answer

Step Response

Answer

Impulse Response

Question 32

A system has a pole at s = -2 and a zero at s = -4. What does this imply about its stability and response?

Accepted Answer

The system is stable due to the negative pole, and its response will have a faster rise time due to the zero.

Answer

The system is unstable due to the negative pole, and its response will be slow.

Question 33

The Routh-Hurwitz criterion is a tool used to:

Accepted Answer

Determine the stability of a system by analyzing the coefficients of its characteristic polynomial.

Answer

Analyze the system's response to different inputs in the time domain.

Answer

Calculate the system's transfer function from its state-space equations.

Question 34

A system is considered asymptotically stable if:

Accepted Answer

Its output eventually settles to a constant value after an initial transient response.

Answer

Its output oscillates indefinitely.

Answer

Its output grows unbounded over time.

Question 35

The time constant of a first-order system determines:

Accepted Answer

The rate at which the system reaches its steady-state value after a step input.

Answer

The amplitude of the system's steady-state response.

Answer

The frequency of any oscillations in the system's response.

Question 36

Which of the following statements about a system's impulse response is TRUE?

Accepted Answer

The impulse response represents the system's output when subjected to a brief, high-magnitude input.

Answer

The impulse response is always a sinusoidal function.

Answer

The impulse response is always equal to the system's transfer function.

Question 37

A system is described by the state-space equations: 
ẋ = Ax + Bu
 y = Cx + Du
What does the matrix 'A' represent in this context?

Accepted Answer

The system's internal dynamics, determining how the state variables evolve over time.

Answer

The input matrix, mapping control inputs to state changes.

Answer

The output matrix, relating state variables to the system's output.

Question 38

A system's step response exhibits a decaying oscillatory behavior. Which of the following characteristics is most likely true about the system's poles?

Accepted Answer

The poles are complex conjugate pairs with negative real parts.

Answer

The poles are all real and negative.

Answer

The poles are all real and positive.

Question 39

Which of the following techniques is NOT typically used for time-domain analysis of control systems?

Accepted Answer

Frequency response analysis

Answer

Impulse response analysis

Answer

Step response analysis

Answer

Ramp response analysis

Question 40

A system with a dominant pole located at -5 on the s-plane is subjected to a step input. What can be expected about the system's response?

Accepted Answer

The response will exhibit an exponential decay with a time constant of 0.2 seconds.

Answer

The response will oscillate with a frequency of 5 Hz.

Answer

The response will rise quickly and settle to a steady-state value.

Question 41

A control system is designed with a closed-loop pole at -10. What does this indicate about the system's stability?

Accepted Answer

The system is stable, as the pole is in the left half-plane.

Answer

The system is unstable, as the pole is in the right half-plane.

Answer

The system is marginally stable, as the pole lies on the imaginary axis.

Question 42

Consider a system with a transfer function G(s) = 1/(s^2 + 4). What is the system's natural frequency?

Accepted Answer

2 rad/s

Answer

Cannot be determined from the transfer function.

Answer

4 rad/s

Answer

1 rad/s

Question 43

A system's impulse response is characterized by a rapid rise followed by a slow exponential decay. What can be concluded about the system's poles?

Accepted Answer

The system likely has a dominant real pole and potentially other poles with faster decay rates.

Answer

The system has only complex poles with negative real parts.

Answer

The system has a single pole at the origin.

Question 44

A system's step response exhibits a steady-state error. Which of the following actions is likely to reduce this error?

Accepted Answer

Increasing the system's gain.

Answer

Adding a low-pass filter to the system.

Answer

Decreasing the system's damping ratio.

Answer

Decreasing the system's gain.

Question 45

Given the transfer function G(s) = (s+1)/(s^2+2s+1), what is the order of the system?

Accepted Answer

2

Answer

4

Answer

3

Answer

1

Question 46

Consider a system with the transfer function G(s) = 1/(s+1). What is the impulse response of this system?

Accepted Answer

e^(-t)

Answer

t

Answer

1

Question 47

Given the state-space representation [dx/dt] = [2 -1] [x] + [1] [u], what is the system's characteristic equation?

Accepted Answer

s^2 - 2s + 1 = 0

Answer

s^2 + 2s + 1 = 0

Answer

s^2 - 1 = 0

Answer

s^2 + 1 = 0

Question 48

Which of the following is an advantage of time-domain analysis over frequency-domain analysis?

Accepted Answer

Can provide more detailed information about system dynamics.

Answer

Less computationally expensive for high-order systems.

Answer

Easier to apply to nonlinear systems.

Question 49

A system is described by the state-space representation: 
ẋ = Ax + Bu 
 y = Cx + Du. Which of the following matrices represents the system's transfer function?

Accepted Answer

C(sI - A)^-1B + D

Answer

B(sI - A)^-1C + D

Answer

D(sI - C)^-1A + B

Answer

A(sI - B)^-1C + D

Question 50

A system has the following transfer function: G(s) = 10/(s^2 + 2s + 1). What is the system's natural frequency (ωn)?

Accepted Answer

1 rad/s

Answer

0.5 rad/s

Answer

2 rad/s

Answer

10 rad/s

Question 51

A second-order system has a damping ratio (ζ) of 0.5. What type of response would you expect from this system?

Accepted Answer

Underdamped

Answer

Undamped

Answer

Overdamped

Answer

Critically damped

Question 52

A system's step response is characterized by a steady-state error. Which of the following techniques can be used to reduce this error?

Accepted Answer

Proportional-Integral (PI) control

Answer

Proportional (P) control

Answer

Derivative (D) control

Answer

Feedforward control

Question 53

A control system is said to be stable if:

Accepted Answer

All poles of its transfer function are in the left-half of the s-plane.

Answer

All poles of its transfer function are in the right-half of the s-plane.

Answer

All zeros of its transfer function are in the left-half of the s-plane.

Question 54

The impulse response of a system represents its output when the input is:

Accepted Answer

A unit impulse function.

Answer

A step function.

Answer

A ramp function.

Answer

A sinusoidal function.

Question 55

A system is said to be controllable if:

Accepted Answer

It is possible to transfer the system from any initial state to any desired final state using an appropriate input.

Answer

It is possible to transfer the system from any initial state to a specific final state only.

Question 56

What is the main advantage of using state-space representation compared to transfer functions?

Accepted Answer

It can handle multiple inputs and outputs, and allows for easier analysis of complex systems.

Answer

It provides a more intuitive representation of system dynamics.

Answer

It is simpler to derive and understand than transfer functions.

Question 57

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

Accepted Answer

The system will exhibit an oscillatory response with a decay rate determined by the real part of the pole.

Answer

The system will be unstable due to the imaginary part of the pole.