Linear Quadratic Regulator (LQR)
Optimal state feedback control for an inverted pendulum on a cart
Uncontrolled
LQR Controlled
Cart position
θ =
0.00
rad | x =
0.00
m | Cost J =
0.00
LQR Weights
Q — position penalty:
1.0
Q — angle penalty:
10.0
R — control effort:
1.0
Initial Conditions
Pendulum angle:
0.20
rad
Reset Simulation
Pause
LQR
minimizes J = ∫(xᵀQx + uᵀRu)dt
Higher
Q angle
→ more aggressive stabilization.
Higher
R
→ gentler control effort.
Uses linearized pendulum dynamics:
mẍ = u − ml θ̈ cos θ
(m+M)l θ̈ = (m+M)gθ − u cos θ