What is an LQ problem?

An optimal-control problem in which the system dynamics are linear and the performance cost is quadratic in the state and control.

What do $Q$ and $R$ represent in a typical LQ cost?

$Q$ weights state deviations, while $R$ weights control effort. Their relative values express the designer's performance-versus-effort priorities.

What is state feedback in LQR?

A control rule that computes the input from the current state, typically $u=-Kx$ in continuous time or $u_k=-F_kx_k$ in discrete time.

What is the purpose of the Riccati equation in LQR?

It determines a matrix $P$ used to calculate the optimal state-feedback gain for the specified model and cost.

Why must an engineer still review an LQR controller?

The engineer must choose appropriate weights and verify that the resulting behavior meets design goals, often through simulation and iteration.

Control systemsLesson 2 of 2 in the course PID control

Linear–Quadratic Regulation: Riccati Methods, Horizons, and MPC

5 flashcards · 5 quiz questions · 5 match pairs · 5 fill-in sentences · English

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Summary

The linear–quadratic regulator (LQR) is an optimal-control method for choosing control actions that balance system performance against control effort. It applies when a system has linear dynamics and performance is measured with a quadratic cost.

A continuous-time linear system can be written as

where $x$ is the system state, $u$ is the control input, and $A$ and $B$ describe how the system evolves and how the input affects it. In discrete time, the corresponding model is $x_{k+1}=Ax_k+Bu_k$ .

All flashcards

What is an LQ problem?: An optimal-control problem in which the system dynamics are linear and the performance cost is quadratic in the state and control.
What do $Q$ and $R$ represent in a typical LQ cost?: $Q$ weights state deviations, while $R$ weights control effort. Their relative values express the designer's performance-versus-effort priorities.
What is state feedback in LQR?: A control rule that computes the input from the current state, typically $u=-Kx$ in continuous time or $u_k=-F_kx_k$ in discrete time.
What is the purpose of the Riccati equation in LQR?: It determines a matrix $P$ used to calculate the optimal state-feedback gain for the specified model and cost.
Why must an engineer still review an LQR controller?: The engineer must choose appropriate weights and verify that the resulting behavior meets design goals, often through simulation and iteration.

Quiz questions

In infinite-horizon discrete-time LQR, which equation determines the steady-state matrix $P$ ?Answer: The discrete-time algebraic Riccati equation (DARE).
What is the defining sequence in receding-horizon control?Answer: Optimize over a horizon, apply the first input, update the state, and solve again.
With suitable quadratic cost weights, what kind of optimization results from adding linear state and input constraints to finite-horizon LQR?Answer: A constrained convex optimization problem.
In regulator terminology, what defines a quadratic–quadratic regulator (QQR)?Answer: Quadratic state dynamics with a quadratic cost.
What setting is associated with the linear–quadratic–Gaussian (LQG) problem?Answer: Linear-quadratic control in the presence of Gaussian noise.

Based on the Wikipedia article Linear–quadratic regulator, licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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