State-Space Representation of Control Systems: Interactive Learning Quizzes

Question 1

In the state-space representation of a control system, what condition must be met for the system to be controllable?

Accepted Answer

The rank of the controllability matrix [B, AB, A²B, ..., A^(n-1)B] equals the number of states.

Answer

The eigenvalues of the state transition matrix A are all distinct.

Answer

The system is observable.

Question 2

The controllability matrix of a system is:

Accepted Answer

A rectangular matrix containing the input-to-state transitions.

Answer

A square matrix containing all possible state transitions.

Answer

A rectangular matrix containing the state-to-output transitions.

Answer

A square matrix containing the eigenvalues of the system.

Question 3

A system is said to be observable if:

Accepted Answer

Its state can be uniquely determined from its output.

Answer

Its output can be uniquely determined from its input.

Answer

Its poles can be uniquely determined from its zeros.

Answer

Its zeros can be uniquely determined from its poles.

Question 4

Which of the following is a state-feedback controller?

Accepted Answer

u(t) = Kx(t)

Answer

u(t) = Ky(t)

Answer

u(t) = G(s)e(t)

Answer

u(t) = H(s)y(t)

Question 5

Consider a closed-loop system with the state-space representation: x' = (A-BK)x + Bu. The closed-loop system is asymptotically stable if:

Accepted Answer

All eigenvalues of (A-BK) have negative real parts.

Answer

All eigenvalues of (A-BK) have positive real parts.

Answer

All eigenvalues of (A-BK) are real.

Answer

All eigenvalues of (A-BK) are complex.

Question 6

The optimal state-feedback controller minimizes:

Accepted Answer

The quadratic cost function J = ∫[x'Qx + u'Ru]dt

Answer

The integral absolute error e(t) = |y(t) - r(t)|dt

Answer

The peak overshoot of the system

Answer

The settling time of the system

Question 7

For a linear time-invariant (LTI) system represented in state-space form as x' = Ax + Bu, if all eigenvalues of matrix A are positive, what can you conclude about the system's stability?

Accepted Answer

Unstable

Answer

Asymptotically stable

Answer

Stability cannot be determined

Answer

Marginally stable

Question 8

Which of the following is NOT a state variable in the state-space representation of a control system?

Accepted Answer

Output

Answer

Error

Answer

Velocity

Answer

Acceleration

Question 9

The eigenvalues of the system matrix of a control system determine the:

Accepted Answer

Stability and performance

Answer

Observability

Answer

Transfer function

Answer

Controllability

Question 10

The transfer function of a control system can be obtained from its state-space representation by:

Accepted Answer

Taking the Laplace transform of the system equations

Answer

Evaluating the determinant of the system matrix

Answer

Solving for the output in terms of the input

Question 11

The dimension of the state-space representation of a system is:

Accepted Answer

Equal to the number of state variables

Answer

Equal to the number of inputs

Answer

Equal to the number of outputs

Answer

Equal to the sum of the number of inputs and outputs

Question 12

Of the following, which is a state-space representation of a system?

Accepted Answer

A system of differential equations describing the system

Answer

A system of linear equations describing the system

Answer

A block diagram representing the system

Answer

A transfer function relating the input and output of the system

Question 13

The dimension of the state vector for a system with n states is:

Accepted Answer

n

Answer

n-1

Answer

n+1

Answer

2n

Question 14

The order of a state-space system refers to:

Accepted Answer

The number of states in the system

Answer

The number of inputs to the system

Answer

The number of outputs from the system

Answer

The degree of the characteristic equation

Question 15

A characteristic equation of a state-space system is a polynomial equation that:

Accepted Answer

Has the eigenvalues of the system as its roots

Answer

Describes the system's response to an input

Answer

Relates the input and output of the system

Answer

Represents the system in the frequency domain

Question 16

Which of the following is true about the relationship between the eigenvalues of a state-space system and its stability?

Accepted Answer

If all eigenvalues have negative real parts, the system is stable

Answer

If any eigenvalue has a positive real part, the system is stable

Answer

If all eigenvalues have zero real parts, the system is stable

Answer

The stability of the system is determined by the number of eigenvalues

Question 17

What is the order of a state-space representation of a control system?

Accepted Answer

The number of state variables

Answer

The sum of the number of input and output variables

Answer

The number of input variables

Answer

The number of output variables

Question 18

Which method is commonly used to design state feedback controllers?

Accepted Answer

Pole placement

Answer

Root locus method

Answer

Bode plot method

Answer

Nyquist stability criterion

Question 19

Which statement does not describe a limitation of state-space representation?

Accepted Answer

Provides a complete and accurate system representation.

Answer

Can be computationally expensive for large-scale systems.

Answer

May be unsuitable for systems with nonlinearities.

Answer

Requires knowledge of the system's internal dynamics.

Question 20

Determine the time response of a system with state-space representation ẋ = [[-1, -2], [0, 1]]x and input u(t) = 1.

Accepted Answer

x(t) = [e^(-t) - e^(-2t), (1/2)e^(-t) - (1/2)e^(-2t)]

Answer

x(t) = [e^(-t) + e^(-2t), (1/2)e^(-t) + (1/2)e^(-2t)]

Answer

x(t) = [(1/2)e^(-t) - (1/2)e^(-2t), e^(-t) - e^(-2t)]

Answer

x(t) = [(1/2)e^(-t) + (1/2)e^(-2t), e^(-t) + e^(-2t)]

Question 21

Identify the controllability matrix used to determine the controllability of a system with state-space representation ẋ = [[0, 1], [-2, -3]]x + [0, 1]u.

Accepted Answer

[B, AB, A^2B]

Answer

[B, A^2B, AB]

Answer

[A^2B, AB, B]

Answer

[AB, B, A^2B]

Question 22

What is a key distinction between state-space representation and transfer function representation of control systems?

Accepted Answer

State-space representation captures the system's internal states, while transfer function representation only shows the input-output relationship.

Answer

State-space representation is unsuitable for multi-input-multi-output systems, unlike transfer function representation.

Answer

Transfer function representation is more user-friendly than state-space representation.

Question 23

Which of the following is the state-space representation of a system with transfer function G(s) = 1/(s^2 + 2s + 1)?

Accepted Answer

ẋ = [-2 -1]x + [1]u, y = [1 0]x

Answer

ẋ = [1 -2]x + [1]u, y = [1 0]x

Answer

ẋ = [2 -1]x + [0]u, y = [0 1]x

Answer

ẋ = [-1 2]x + [0]u, y = [0 1]x

Question 24

A state-space system is given by ẋ = Ax + Bu, y = Cx + Du. Which of the following is the controllability matrix of the system?

Accepted Answer

R = [B; AB; A^2B; ...]

Answer

R = [C; CA; A^2C; ...]

Answer

R = [B; CB; CA^2; ...]

Answer

R = [C; AB; CA^2; ...]

Question 25

Which of the following conditions ensures that a state-space system is controllable?

Accepted Answer

The controllability matrix has full rank.

Answer

The observability matrix has full rank.

Answer

The system matrix A is invertible.

Answer

The input matrix B is invertible.

Question 26

Which of the following conditions ensures that a state-space system is observable?

Accepted Answer

The observability matrix has full rank.

Answer

The system matrix A is invertible.

Answer

The controllability matrix has full rank.

Answer

The output matrix C is invertible.

Question 27

A state-feedback controller is given by u = Kx. Which of the following is the closed-loop system matrix?

Accepted Answer

A_cl = A - BK

Answer

A_cl = K - BA

Answer

A_cl = A + BK

Answer

A_cl = KA - B

Question 28

Which of the following is the characteristic equation of a state-space system?

Accepted Answer

det(sI - A) = 0

Answer

det(A - sI) = 0

Answer

det(sI + A) = 0

Answer

det(I - sA) = 0

Question 29

According to the Lyapunov stability theorem, which of the following ensures asymptotic stability of a state-space system?

Accepted Answer

A positive definite Lyapunov function with a negative definite derivative.

Answer

A positive semidefinite Lyapunov function with a negative semidefinite derivative.

Question 30

In the state-space equation ẋ = Ax + Bu, the stability of the system can be determined by evaluating the eigenvalues of which matrix?

Accepted Answer

A

Answer

B-A

Answer

B

Answer

A+B

Question 31

Which of the following is a primary method for designing state-space controllers?

Accepted Answer

Pole placement

Answer

Root locus

Answer

Nyquist stability criterion

Answer

Bode plot analysis

Question 32

The state-space representation of a discrete-time control system differs from that of a continuous-time system primarily in terms of:

Accepted Answer

The time derivative operator

Answer

The input matrix

Answer

The state vector

Answer

The output matrix

Question 33

The state transition matrix Φ(t) for a continuous-time system is given by:

Accepted Answer

e^(At)

Answer

A^-1 * e^(At)

Answer

e^(-At)

Answer

A * e^(At)

Question 34

The observability of a state-space system refers to the ability to:

Accepted Answer

Determine the state vector from the output measurements

Answer

Predict the future behavior of the system

Answer

Control the system to a desired state

Question 35

Which of the following is a necessary and sufficient condition for a state-space system to be controllable?

Accepted Answer

The controllability matrix is full rank

Answer

The system is stable

Answer

The observability matrix is full rank

Answer

The eigenvalues of A are distinct

Question 36

In the context of state-space representations, an 'eigenvalue' refers to:

Accepted Answer

A characteristic value of the system matrix

Answer

A value that represents the system's input

Answer

A value that determines the system's performance

Answer

A value that determines the system's stability

Question 37

What is the primary advantage of using state-space representation over transfer function representation for control systems?

Accepted Answer

State-space representation allows modeling systems with multiple inputs and outputs, as well as time-varying parameters.

Answer

Transfer function representation is more suitable for analyzing system stability.

Answer

Transfer functions are generally easier to derive from physical system equations.

Question 38

Given a state-space model, how can you determine the system's transfer function?

Accepted Answer

Apply the Laplace transform to the state-space equations and solve for the output-to-input ratio.

Answer

Calculate the eigenvalues of the state matrix. While eigenvalues influence system behavior, they do not directly yield the transfer function.

Answer

Determine the controllability and observability of the system. These properties do not directly lead to the transfer function.

Question 39

A system is considered controllable if:

Accepted Answer

It is possible to steer the system to any desired state from any initial state using the input.

Answer

While stability is desirable, it is not the defining characteristic of controllability.

Answer

Controllability is not related to the presence of distinct poles.

Question 40

A system is considered observable if:

Accepted Answer

It is possible to determine the initial state of the system by observing its output.

Answer

While stability is desirable, it is not the defining characteristic of observability.

Answer

Observability is not related to the presence of distinct poles.

Question 41

Which of the following techniques can be used to design a state-feedback controller for a system represented in state-space?

Accepted Answer

Pole placement

Answer

The Nyquist stability criterion is used for analyzing stability, not designing controllers.

Answer

Bode plot analysis is used for frequency response analysis, not controller design in state-space.

Answer

The root locus method is used for designing controllers in the frequency domain, not state-space.

Question 42

What is the purpose of the state observer in a control system?

Accepted Answer

To estimate the system's state variables when they are not directly measurable.

Answer

The purpose of the controller is to generate feedback signals.

Answer

Stabilizing the system by canceling its poles is achieved through pole placement or other control techniques.

Question 43

The eigenvalues of the state matrix A determine the:

Accepted Answer

System's stability and response characteristics. Eigenvalues directly influence the system's transient response and stability.

Answer

Observability and controllability are related to system structure, not solely determined by eigenvalues.

Answer

Controllability and observability are related to system structure, not solely determined by eigenvalues.

Question 44

A system is considered asymptotically stable if:

Accepted Answer

All eigenvalues of the state matrix have negative real parts. This ensures that the system's response decays to zero over time.

Answer

Asymptotic stability is not related to the presence of distinct poles.

Answer

While a bounded output is a characteristic of stability, it does not guarantee asymptotic stability.