The LQR algorithm is essentially an automated way of finding an appropriate state-feedback controller. As such, it is not uncommon for control engineers to prefer alternative methods, like full state feedback, also known as pole placement, in which there is a clearer relationship between controller parameters and controller behavior. Difficulty ...

The simplest case, called the linear quadratic regulator (LQR), is formulated as stabilizing a time-invariant linear system to the origin. The linear quadratic regulator is likely the most important and influential result in optimal control theory to date.

The linear quadratic regulator (LQR) is a well-known design technique that provides practical feedback gains. For the derivation of the linear quadratic regulator, we assume the plant to be written in state-space form ̇x = Ax + Bu, and that all of the n …

One of these [Kalman and Bertram 1960], presented the vital work of Lyapunov in the time-domain control of nonlinear systems. The next [Kalman 1960a] discussed the optimal control of systems, providing the design equations for the linear quadratic regulator (LQR).

This lecture provides a brief derivation of the linear quadratic regulator (LQR) and describes how to design an LQR-based compensator. The use of integral feedback to eliminate steady state error is also described.

Standard LQR: ! How to incorporate the change in controls into the cost/ reward function? ! Soln. method A: explicitly incorporate into the state by augmenting the state with the past control input vector, and the difference between the last two control input vectors. ! Soln. method B: change of variables to fit into the standard LQR

quadratic regulator (LQR). There exist two main approaches to optimal control: 1. via the Calculus of Variations (making use of the Maximum Principle); 2. via Dynamic Programming (making use of the Principle of Optimality). Both approaches involve converting an optimization over a function space to a pointwise optimization.

we’ll solve LQR problem using dynamic programming for 0 ≤ t ≤ T we define the value function Vt: Rn → R by Vt(z) = min u Z T t x(τ)TQx(τ)+u(τ)TRu(τ) dτ +x(T)TQfx(T) subject to x(t) = z, x˙ = Ax+Bu • minimum is taken over all possible signals u : [t,T] → Rm • Vt(z) gives the minimum LQR cost-to-go, starting from state z at ...

Summary: LQR Control Application #1: trajectory generation • Solve for (xd, yd) that minimize quadratic cost over finite horizon • Use local controller to track trajectory Application #2: trajectory tracking • Solve LQR problem to stabilize the system • …

Goal 1: Outline linear quadratic optimal control (LQR). Goal 2: Introduce Concepts of Model Predictive Control (MPC). Goal 3: A stability proof for linear quadratic MPC.