Master Optimal Control: Quizzes and Practice Tests

Question 1

In LQR, the optimal control law is given by:

Accepted Answer

$u = -K x$

Answer

$u = -K y$

Answer

$u = -L x$

Answer

$u = -L y$

Question 2

The stability of an LQR-controlled system is determined by:

Accepted Answer

Both observability/controllability and weight matrices

Answer

Observability and controllability of the system

Answer

Positive definiteness of the weight matrices

Answer

Neither observability/controllability nor weight matrices

Question 3

The Kalman filter covariance matrix represents the:

Accepted Answer

Uncertainty in the state estimate

Answer

Precision of the state estimate

Answer

Accuracy of the state estimate

Answer

Robustness of the state estimate

Question 4

What is a key assumption made in the design of a linear quadratic regulator (LQR)?

Accepted Answer

The system is characterized by linearity and time-invariance.

Answer

The state and control variables are unconstrained.

Answer

The system is observable but not controllable.

Answer

The performance index is a cubic function of the state and control inputs.

Question 5

What is the primary purpose of a Kalman filter?

Accepted Answer

Estimation of a system's state using noisy measurements.

Answer

Control of a system based on complete state information.

Answer

Prediction of a system's future output.

Answer

Optimization of a system's performance.

Question 6

What is the typical objective function used in the design of a linear quadratic regulator?

Accepted Answer

Minimization of the integral of the squared error between the desired and actual state.

Answer

Minimization of the maximum error between the desired and actual state.

Answer

Maximization of the rate of change of the state.

Answer

Minimization of the control effort without considering state error.

Question 7

What is a fundamental difference between linear and non-linear Kalman filters?

Accepted Answer

Linear Kalman filters assume a linear system model, while non-linear Kalman filters assume a non-linear system model.

Answer

Linear Kalman filters use the Kalman gain for state estimation, while non-linear Kalman filters use the extended Kalman gain.

Answer

Linear Kalman filters are more accurate than non-linear Kalman filters.

Answer

Non-linear Kalman filters are less computationally expensive than linear Kalman filters.

Question 8

What is a key advantage of using a Kalman filter in optimal control?

Accepted Answer

It can accurately estimate the system state even when measurements are noisy.

Answer

It simplifies the design of the optimal controller.

Answer

It reduces the computational cost of the optimization problem.

Answer

It improves the stability of the closed-loop system under certain conditions.

Question 9

What is the primary goal of model predictive control (MPC)?

Accepted Answer

To optimize the control input over a specified future horizon.

Answer

To estimate the system state accurately.

Answer

To design a closed-loop controller that guarantees stability.

Answer

To analyze the stability of the system under various conditions.

Question 10

What is the purpose of the weighting matrices in an LQR controller?

Accepted Answer

They allow adjustment of the trade-off between system performance and control effort.

Answer

They represent the uncertainty in the system model.

Answer

They are used for tuning the controller parameters.

Answer

They are not used in LQR controllers.

Question 11

Which of the following is NOT a crucial component of a linear quadratic regulator (LQR)?

Accepted Answer

Optimal control input trajectory

Answer

State feedback gain matrix

Answer

System state matrix

Answer

Control input matrix

Question 12

The primary purpose of a Kalman filter is to:

Accepted Answer

Estimate the state of a system using noisy measurements

Answer

Design a control law for a dynamic system

Answer

Optimize the performance of a system

Answer

Filter unwanted high-frequency noise from a signal

Question 13

In a state feedback control system, the control input is determined based on:

Accepted Answer

Current state of the system

Answer

Reference state of the system

Answer

Past states of the system

Answer

Measurement noise

Question 14

The Hamiltonian in optimal control theory represents:

Accepted Answer

Instantaneous cost of the system

Answer

Total cost of the system over time

Answer

Gradient of the objective function

Answer

State of the system

Question 15

Controllability in optimal control refers to:

Accepted Answer

Ability to steer the system to any desired state

Answer

Ability to observe the system's state

Answer

Ability to minimize the system's cost function

Answer

Stability of the system

Question 16

Which of the following is NOT a common type of cost function used in optimal control?

Accepted Answer

Gaussian cost function

Answer

Quadratic cost function

Answer

Linear cost function

Answer

Exponential cost function

Question 17

The objective function in linear quadratic control (LQR) typically consists of:

Accepted Answer

A weighted sum of squared tracking errors and control effort terms

Answer

Only control effort terms

Answer

Non-quadratic terms

Answer

Only squared tracking errors

Question 18

The Hamilton-Jacobi-Bellman (HJB) equation is primarily used for solving:

Accepted Answer

Optimal control problems with nonlinear system dynamics and potentially non-quadratic objective functions

Answer

System identification problems involving the estimation of unknown system parameters from input-output measurements

Answer

Parameter estimation problems involving system identification and model calibration

Answer

Optimal control problems with linear system dynamics and quadratic objective functions, for which analytical solutions exist

Question 19

In optimal control, which optimization method is commonly employed due to its ability to handle complex system dynamics and constraints?

Accepted Answer

Dynamic programming, which systematically decomposes the problem into smaller, tractable subproblems

Answer

Linear programming, which is primarily suited for solving linear optimization problems with linear constraints

Answer

Genetic algorithms, which employ evolutionary principles to search for optimal solutions

Answer

Gradient descent, which iteratively updates the control inputs based on the gradient of the cost function

Question 20

In nonlinear optimal control, the control law is typically expressed as a:

Accepted Answer

Nonlinear function of the estimated system state, providing continuous feedback control actions

Answer

Proportional-integral-derivative (PID) controller, which combines proportional, integral, and derivative actions

Answer

Fuzzy logic controller, which employs linguistic rules and fuzzy sets to make control decisions

Answer

Fixed gain controller with constant control parameters, regardless of the system state

Question 21

In an optimal control problem, the terminal cost function represents:

Accepted Answer

The desired system state or cost at the end of the optimization horizon

Answer

The initial system state or cost at the start of the optimization horizon

Answer

The running cost accumulated over the entire optimization horizon

Question 22

In model predictive control (MPC), the predictions of future system behavior rely on:

Accepted Answer

A mathematical model of the system that captures its dynamics and behavior

Answer

Artificial neural networks trained on historical data to approximate system behavior

Answer

Recorded historical data of system behavior to predict future trends

Answer

Expert knowledge and heuristics to infer future system states