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.

Linear Quadratic Regulator, a.k.a, LQR, 是线性二次型调节器的缩写。 要说这个LQR，就先得从LQ Optimal Control开始说起。 我们研究一个线性时变系统： \dot {x}=A (t)x+B (t)u\\ y=C (t)x+D (t)u\tag {1} 定义一个时域连续性能泛函: J=\frac {1} {2}\int_ {t_0}^ {t_f} [x (t)^TQx (t)+u (t)^TRu (t)]dt+\frac {1} {2}x^T (t_f)Q_1x (t_f)\tag {2}

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

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 ...

[K,S,P] = lqr(A,B,Q,R,N) calculates the optimal gain matrix K, the solution S of the associated algebraic Riccati equation and the closed-loop poles P using the continuous-time state-space matrices A and B. This syntax is only valid for continuous-time …

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 • …