Link: https://en.wikipedia.org/wiki/Linear%E2%80%93quadratic_regulator
Description: WEBOne of the main results in the theory is that the solution is provided by the linear–quadratic regulator (LQR), a feedback controller whose equations are given below. LQR controllers possess inherent robustness with guaranteed gain and phase margin, and they also are part of the solution to the LQG (linear–quadratic–Gaussian) problem.
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Link: https://underactuated.mit.edu/lqr.html
Description: WEBThe 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.
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Link: https://www.mathworks.com/help/control/ref/lti.lqr.html
Description: WEBControl Systems; Control System Toolbox; Control System Design and Tuning; State-Space Control Design and Estimation; State-Space Control Design; lqr; On this page; Syntax; Description; Examples. LQR Control for Inverted Pendulum Model; LQR Control Using State-Space Matrices; Input Arguments. sys; A; B; Q; R; N; Output Arguments. K; …
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Link: https://www.cds.caltech.edu/~murray/courses/cds110/wi06/lqr.pdf
Description: WEBR. M. Murray Lecture 2 – LQR Control 11 January 2006 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. Two design examples are given: lateral
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Link: https://pages.github.berkeley.edu/ee290-005/sp21-site/assets/lec/LQR_Tomlin.pdf
Description: WEBFebruary 3, 2021. These notes represent an introduction to the theory of optimal control and the linear quadratic regulator (LQR). via Dynamic Programming (making use of the Principle of Optimality). Both approaches involve converting an optimization over a function space to a pointwise optimization.
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Link: https://courses.cs.washington.edu/courses/cse478/20wi/site/resources/lec15_lqr.pdf
Description: WEBDifferent control laws 3 1. PID control 2. Pure-pursuit control 3. Lyapunov control 4. LQR 5. MPC. Recap of controllers 4 PID / Pure pursuit: Worked well, no provable guarantees Lyapunov: Provable stability, convergence rate depends on gains. 5 Control Law Uses model Stability Guarantee
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Link: https://www.youtube.com/watch?v=wEevt2a4SKI
Description: WEBDec 3, 2018 · In this video we introduce the linear quadratic regulator (LQR) controller. We show that an LQR controller is a full state feedback controller where the gai...
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Link: https://www.sciencedirect.com/science/article/pii/S2666720724000171
Description: WEBMar 1, 2024 · Abstract. Linear Quadratic Regulator is one of the most common ways to control a linear system. Despite Linear Quadratic Regulator’s (LQR) strong performance and solid resilience, developing these controllers have been challenging, largely because there is no reliable way to choose the Q and R weighing matrices.
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Link: https://web.stanford.edu/class/ee363/lectures/allslides.pdf
Description: WEBSummary of LQR solution via DP 1. set PN:= Qf 2. for t = N,...,1, Pt−1:= Q+A TP tA−ATPtB(R +BTPtB)−1BTPtA 3. for t = 0,...,N −1, define Kt:= −(R +BTPt+1B)−1BTPt+1A 4. for t = 0,...,N −1, optimal u is given by ulqr(t) = Ktx(t) • optimal u is a linear function of the state (called linear state feedback)
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Link: https://web.stanford.edu/class/ee363/lectures/clqr.pdf
Description: WEBwe’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 ...
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