State Feedback Control: Master Control Systems with Confidence

Question 1

Which control technique allows for the direct placement of closed-loop poles in desired locations?

Accepted Answer

State feedback

Answer

Output feedback

Answer

Observer design

Answer

Kalman filtering

Question 2

Which performance index is minimized in the design of a linear quadratic regulator (LQR)?

Accepted Answer

Quadratic function of state and control variables

Answer

Integral of squared error

Answer

Sum of absolute values of state variables

Answer

Maximum eigenvalue of the closed-loop system

Question 3

Observability of a system refers to its ability to:

Accepted Answer

Be uniquely determined from its output measurements

Answer

Be effectively controlled using state feedback

Answer

Reject disturbances effectively

Answer

Be stabilized using output feedback

Question 4

In a system with k states, what are the dimensions of the state feedback gain matrix K?

Accepted Answer

k x k matrix

Answer

k x m matrix

Answer

m x k matrix

Answer

m x m matrix

Question 5

In a state feedback control system, the feedback signal is a linear combination of which system variables?

Accepted Answer

State variables

Answer

Input variables

Answer

Output variables

Answer

Error variables

Question 6

In a state feedback control system, which of the following can be placed arbitrarily to achieve desired system behavior?

Accepted Answer

Closed-loop poles

Answer

State variables

Answer

Input variables

Answer

Output variables

Question 7

The Linear Quadratic Regulator (LQR) design method minimizes a cost function that includes which elements?

Accepted Answer

Both state error and control effort

Answer

State error only

Answer

Control effort only

Answer

Neither state error nor control effort

Question 8

In a system with n states, the state feedback gain matrix has dimensions of:

Accepted Answer

n x n

Answer

n x m

Answer

m x n

Answer

m x m

Question 9

In an LQR design, the weighting matrices Q and R primarily determine:

Accepted Answer

The relative significance of state error and control effort

Answer

The closed-loop system's poles

Answer

The system's stability

Answer

The system's sensitivity to disturbances

Question 10

In pole placement design, the desired closed-loop system poles are:

Accepted Answer

Determined based on the system dynamics and desired performance

Answer

Arbitrarily chosen

Answer

Always negative and real-valued

Answer

Always complex and conjugate

Question 11

State feedback control is particularly well-suited for systems with:

Accepted Answer

Multiple inputs and outputs

Answer

Nonlinearities

Answer

Time-varying parameters

Answer

Unknown disturbances

Question 12

What is the purpose of pole placement in state feedback control?

Accepted Answer

To design a controller that moves the closed-loop poles to desired locations

Answer

To design a controller that minimizes the error between the desired and actual state

Answer

To design a controller that stabilizes the system

Answer

To design a controller that optimizes the system performance

Question 13

What is the quadratic cost function used in LQR design?

Accepted Answer

J = ∫(x'Qx + u'Ru)dt

Answer

J = ∫(x'Rx + u'Qu)dt

Answer

J = ∫(x'Qu + u'Rx)dt

Answer

J = ∫(u'Qx + x'Ru)dt

Question 14

What is the advantage of using LQR over pole placement for control system design?

Accepted Answer

It allows for optimization of the system performance

Answer

It is less sensitive to plant uncertainties

Answer

It is easier to implement

Answer

It can be used for nonlinear systems

Question 15

What is the role of the Kalman filter in state feedback control?

Accepted Answer

To estimate the system state from noisy measurements

Answer

To design the state feedback controller

Answer

To stabilize the system

Answer

To optimize the system performance

Question 16

Which of the following is a key assumption in the design of linear quadratic Gaussian (LQG) controllers?

Accepted Answer

The system is linear and the disturbance is Gaussian

Answer

The system is nonlinear and the disturbance is Gaussian

Answer

The system is linear and the disturbance is non-Gaussian

Answer

The system is nonlinear and the disturbance is non-Gaussian

Question 17

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

Accepted Answer

To estimate the system state from output measurements

Answer

To design the state feedback controller

Answer

To stabilize the system

Answer

To optimize the system performance

Question 18

How do controllability and observability impact the design of state feedback controllers?

Accepted Answer

They determine if the system's state can be manipulated and measured.

Answer

They solely affect the closed-loop system's stability.

Answer

Controllability is only crucial for pole placement design, while observability is only needed for observer design.

Question 19

For stability in a state feedback control system, where should the closed-loop poles be positioned?

Accepted Answer

Left-half of the complex plane

Answer

Right-half of the complex plane

Answer

On the imaginary axis

Answer

At the origin

Question 20

In state feedback control, which variable is used to calculate the control action?

Accepted Answer

System state

Answer

System output

Answer

Both system output and state

Question 21

In pole placement design, the desired closed-loop poles are placed via:

Accepted Answer

Solving an eigenvalue problem

Answer

Applying state feedback gains

Answer

Using a state observer

Answer

Adjusting system parameters

Question 22

The performance index used in LQR design is typically expressed as:

Accepted Answer

J = ∫(x'Qx + u'Ru)dt

Answer

J = max(x)

Answer

J = (x-x_d)'(x-x_d)

Answer

J = ∫(e'e)dt

Question 23

The observer gain in a state observer design is chosen to ensure that:

Accepted Answer

The observer poles are placed farther to the left than the system poles

Answer

The observer poles are placed at the same location as the system poles

Answer

The observer poles are placed closer to the system poles

Answer

The observer gain is minimized

Question 24

In the context of implementing state feedback control systems, what is the primary function of state observers?

Accepted Answer

Providing estimates of system states that cannot be directly measured

Answer

Enhancing the stability of the controlled system

Answer

Eliminating the need for state feedback control

Question 25

In a linear quadratic regulator (LQR) control problem, what is the optimal control law?

Accepted Answer

u = -K*x

Answer

u = -x*K

Answer

u = K*x

Question 26

Which of the following is NOT a crucial step in LQR design?

Accepted Answer

Calculating the Kalman filter gain

Answer

Selecting appropriate weighting matrices (Q and R) to balance control effort and system performance

Answer

Solving the algebraic Riccati equation to obtain the optimal gain matrix

Answer

Determining the system's controllability and observability to ensure the existence of a stabilizing controller

Question 27

In state feedback control, what is the primary purpose of an observer?

Accepted Answer

To estimate the system's state variables, which may not be directly measurable, using available sensor measurements.

Answer

To amplify the control signal, increasing its effectiveness in driving the system.

Answer

To provide a reference signal for the controller, guiding the system towards the desired state.

Answer

To filter out noise and disturbances from the control signal, improving system performance.

Question 28

What is the fundamental difference between full-state feedback and output feedback in control systems?

Accepted Answer

Full-state feedback utilizes all system state variables for control, while output feedback relies solely on measurable system outputs.

Answer

Full-state feedback requires more complex control algorithms than output feedback, making it less practical.

Answer

Output feedback provides better disturbance rejection capabilities compared to full-state feedback.

Question 29

Which of the following is NOT an advantage of using LQR control design?

Accepted Answer

Guaranteed system stability, as LQR design does not inherently guarantee stability.

Answer

Ease of implementation, as LQR controllers can be implemented using simple algebraic equations.

Answer

Optimal performance, as LQR aims to minimize a quadratic cost function based on desired system behavior.

Answer

Robustness to disturbances, as LQR design incorporates a weighting matrix on the control effort to balance performance and robustness.

Question 30

A system's state-space representation is given by: 
ẋ = Ax + Bu, y = Cx. The matrix A has a pair of complex conjugate eigenvalues. To stabilize the system using state feedback, where should the closed-loop poles be located?

Accepted Answer

In the left half of the complex plane.

Answer

On the imaginary axis.

Answer

At the origin.

Answer

In the right half of the complex plane.

Question 31

In a linear quadratic regulator (LQR) design, which matrix is used to compute the optimal control input?

Accepted Answer

Control gain matrix (K).

Answer

Controllability matrix (B).

Answer

State transition matrix (Φ).

Answer

Observability matrix (C).

Question 32

In a closed-loop control system with state feedback, how does the selection of feedback gains influence the system's behavior?

Accepted Answer

It affects both the transient response and steady-state error.

Answer

It does not affect either the transient response or steady-state error.

Answer

It only affects the steady-state error.

Answer

It only affects the transient response.

Question 33

In the context of state feedback control, what does the state vector 'x' represent?

Accepted Answer

The system's internal variables.

Answer

The system's outputs.

Answer

The system's disturbances.

Answer

The system's inputs.

Question 34

Given a system with the following state-space representation: 
ẋ = [0 1; -1 -2]x + [1 0; 0 1]u, y = [1 0]x. What condition guarantees the system's controllability?

Accepted Answer

The rank of the controllability matrix (B) is equal to the number of states.

Answer

The system is stable.

Answer

The eigenvalues of matrix A are all distinct.

Answer

The system is observable.

Question 35

In state feedback control, the controller's output is directly determined by which of the following?

Accepted Answer

The values of the system's state variables

Answer

The system's input variables

Answer

The system's output variables

Question 36

Which of the following techniques is specifically designed for creating a state feedback controller?

Accepted Answer

Pole placement

Answer

Nyquist plot

Answer

Root locus

Answer

Bode plot

Question 37

The Linear Quadratic Regulator (LQR) method aims to solve which optimization problem?

Accepted Answer

Minimizing a quadratic cost function that represents system performance

Answer

Minimizing a nonlinear cost function related to system performance

Answer

Maximizing a linear cost function related to system performance

Question 38

What are the key parameters that influence the behavior of an LQR controller?

Accepted Answer

Weighting matrices that define the relative importance of different system states and control effort

Answer

The gain matrix of the state feedback controller

Answer

The system's transfer function

Question 39

What is the primary advantage of state feedback control compared to other control methods?

Accepted Answer

It allows direct manipulation of the system's internal dynamics by using state variables

Answer

It provides improved robustness to disturbances and uncertainties

Answer

It simplifies the overall system design and complexity

Question 40

What is a major limitation of state feedback control?

Accepted Answer

It requires knowledge of all system states, which may not always be available for measurement

Answer

It can increase the overall cost of the system

Answer

It can reduce the overall reliability of the controlled system

Answer

It can lead to complex controller design processes

Question 41

In a state feedback system, how is the state feedback gain matrix typically determined?

Accepted Answer

By solving an algebraic Riccati equation, which ensures optimal performance based on the cost function

Answer

By applying the Nyquist stability criterion to ensure system stability

Answer

Through trial and error, adjusting the gain matrix until desired performance is achieved

Answer

By using the root locus method to achieve desired closed-loop pole locations

Question 42

What is the primary function of an observer in a state feedback control system?

Accepted Answer

To estimate the values of unmeasured states based on available measurements

Answer

To generate the control signal for the system

Answer

To measure the system's outputs

Answer

To provide disturbance rejection for the system