State-Space Representation of Control Systems: Interactive Learning Quizzes

Question 1

Define the key characteristic of a state-space representation in control systems.

Accepted Answer

Employs a vector of variables to represent the system's state

Answer

Describes the system's input-output relationship using differential equations

Answer

Assumes a strictly linear relationship between input and output

Answer

Requires knowledge of the system's transfer function

Question 2

Define controllability in the context of a state-space system.

Accepted Answer

The ability to reach any state from any other state

Answer

The ability to control every output independently

Answer

The system's asymptotic stability

Answer

The system's ability to return to an equilibrium point

Question 3

Define observability in relation to a state-space system.

Accepted Answer

The ability to determine every state from the output

Answer

The ability to uniquely identify every input from the output

Answer

The ability to reach the system from any initial state

Answer

The system's inherent stability

Question 4

Consider the state-space model given by x' = Ax + Bu, where A = [[1, 2], [0, 1]] and B = [[1], [0]]. The characteristic equation of the system is:

Accepted Answer

s^2 + 2s - 1 = 0

Answer

s^2 - 2s - 1 = 0

Answer

s^2 - s - 2 = 0

Answer

s^2 + s - 2 = 0

Question 5

A state-space model is said to be in the Jordan canonical form if:

Accepted Answer

The system matrix A is a diagonal matrix with the eigenvalues on the diagonal.

Answer

The system matrix A is a triangular matrix.

Answer

The system matrix A is a symmetric matrix.

Answer

The system matrix A is an identity matrix.

Question 6

The observability grammar of a linear system is:

Accepted Answer

A set of rules that define the observable subspace of the system.

Answer

A set of equations that define the state of the system.

Answer

A set of matrices that represent the system.

Answer

A set of vectors that span the state space of the system.

Question 7

Which of the following is an advantage of using state-space models for control system analysis?

Accepted Answer

They provide a more comprehensive representation of the system than transfer functions.

Answer

They are more intuitive than transfer functions.

Answer

They are easier to analyze than transfer functions.

Answer

They are more accurate than transfer functions.

Question 8

In a state-space model, the state transition matrix captures the relationship between:

Accepted Answer

State and state

Answer

State and input

Answer

State and output

Answer

Input and output

Question 9

Controllability of a state-space system ensures that:

Accepted Answer

Any state can be reached from any initial state using a finite input sequence

Answer

The system is stable and will not diverge

Answer

Any state can be observed using a finite output sequence

Answer

The system has a unique equilibrium point

Question 10

Observability of a state-space system requires:

Accepted Answer

The rank of the observability matrix to be equal to the state space dimension

Answer

The eigenvalues of the state transition matrix to be distinct

Answer

The system to be controllable

Answer

The system to have a unique equilibrium point

Question 11

The state feedback gain matrix in a state-space feedback control system is responsible for:

Accepted Answer

Driving the system state towards desired values

Answer

Minimizing the error between output and reference

Answer

Increasing the system bandwidth

Answer

Reducing sensitivity to disturbances

Question 12

A major application of state-space representation in control engineering is:

Accepted Answer

Control system design and analysis

Answer

Signal processing and filtering

Answer

Power system modeling

Answer

Economic forecasting

Question 13

The derivative of the state vector in a state-space model is defined in terms of:

Accepted Answer

Both state and input vectors

Answer

State vector only

Answer

Output vector only

Answer

Input and output vectors

Question 14

Which of the following determines whether a linear time-invariant system is controllable?

Accepted Answer

The controllability matrix's rank is equal to the number of states.

Answer

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

Answer

All of the system matrix's eigenvalues are different.

Answer

The system is stable.

Question 15

What is the necessary and sufficient condition for observability of a linear time-invariant system?

Accepted Answer

The observability matrix's rank equals the number of states.

Answer

The controllability matrix's rank equals the number of states.

Answer

The eigenvalues of the system matrix are all distinct.

Answer

The system is stable.

Question 16

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

Accepted Answer

A polynomial obtained by equating the transfer function's denominator to zero.

Answer

A polynomial obtained by equating the transfer function's numerator to zero.

Answer

A polynomial obtained by multiplying the transfer function's numerator and denominator.

Answer

A polynomial obtained by dividing the transfer function's numerator by its denominator.

Question 17

How are the state-space model and transfer function of a system related?

Accepted Answer

The transfer function and state-space model are equivalent representations of the same system.

Answer

The transfer function is the state-space model's Laplace transform.

Answer

The state-space model is the transfer function's inverse Laplace transform.

Answer

The transfer function and state-space model are not related.

Question 18

What is the purpose of a state observer?

Accepted Answer

Estimating a system's state from inputs and outputs.

Answer

Controlling a system's state through feedback.

Answer

Designing a controller for a system.

Answer

Analyzing a system's stability.

Question 19

Which of the following is NOT a primary design consideration for control systems?

Accepted Answer

Cost

Answer

Stability

Answer

Performance

Answer

Robustness

Question 20

Which of the following is a linear transformation?

Accepted Answer

The function f(x) = Ax, where A is a matrix

Answer

The function f(x) = sin(x)

Answer

The function f(x) = x^2

Answer

The function f(x) = |x|

Question 21

Which of the following is an eigenvector of the matrix A?

Accepted Answer

A non-zero vector v such that Av = λv

Answer

A non-zero vector v such that A + λI = 0

Answer

A non-zero vector v such that Aλ = λA

Answer

A non-zero vector v such that A - λI = 0

Question 22

The state vector of a linear time-invariant system is defined as the vector of:

Accepted Answer

The minimal set of variables that completely describe the system's future behavior given its present state and inputs.

Answer

The system's state space variables.

Answer

The system's output variables.

Answer

The system's input variables.

Question 23

Which of the following is a necessary condition for a linear time-invariant system to be controllable?

Accepted Answer

The controllability matrix has full rank.

Answer

The system is stable.

Answer

The eigenvalues of the system matrix are distinct.

Answer

The system has more states than inputs.

Question 24

A system is observable if:

Accepted Answer

The output vector can be expressed as a function of the state vector and the input vector.

Answer

The system is controllable.

Answer

The system has more outputs than states.

Answer

The eigenvalues of the system matrix are distinct.

Question 25

Which of the following is NOT an application of state-space representation?

Accepted Answer

Tuning PID controllers.

Answer

Analyzing system stability.

Answer

Predicting system response to inputs.

Answer

Designing optimal controllers.

Question 26

The state feedback control law is given by:
      u = -Kx
    Where K is the state feedback gain matrix. What is the purpose of the state feedback control?

Accepted Answer

To stabilize the system and achieve desired performance.

Answer

To minimize the system's steady-state error.

Answer

To reduce the system's sensitivity to disturbances.

Question 27

The following system is represented in state-space form:
      ẋ = [1 0]x + [0 1]u
      y = [2 1]x
    Is the system observable?

Accepted Answer

Yes, because the observability matrix has full rank.

Answer

Yes, because the system is stable.

Answer

No, because the system is not controllable.