Inverted Pendulum — LQR & Pole Placement

CONTROLLER

Q · POSITION (x)

Q · VELOCITY (ẋ)

Q · ANGLE (θ)

Q · ANGULAR VELOCITY (θ̇)

R · CONTROL PENALTY

SYSTEM PARAMETERS

CART MASS (M) 1.0kg

PENDULUM MASS (m) 0.20kg

PENDULUM LENGTH (l) 0.50m

REFERENCE & INITIAL STATE

TARGET POSITION (x*) 0.0m

INITIAL ANGLE (θ₀) 8°

PHYSICAL SYSTEM

TIME RESPONSE θ [°] x [m] u [N]

PENDULUM ANGLE θ

—

θ̇ = — °/s

CART POSITION x

—

ẋ = — m/s

CONTROL FORCE u

—

|u| ≤ 200 N

CLOSED-LOOP

—

mode: LQR

FEEDBACK GAIN u = −K·(x − x*)

K = [ — — — — ]

state: x = [ x , ẋ , θ , θ̇ ]ᵀ

CLOSED-LOOP EIGENVALUES eig(A − B·K)

—

ⓘ Nonlinear model (integrated with RK4): ẍ = [u + m·sinθ·(l·θ̇² − g·cosθ)] / (M + m·sin²θ), θ̈ = [(M+m)·g·sinθ − cosθ·(u + m·l·θ̇²·sinθ)] / [l·(M + m·sin²θ)]. Controllers are designed using linearized A, B around the upright equilibrium (θ=0): LQR solves the continuous Algebraic Riccati Equation (CARE) using the Matrix Sign Function; pole placement assigns closed-loop poles based on damping ratio (ζ) and natural frequency (ωn) targets. The open-loop system is unstable — the pendulum falls in OFF mode. Eigenvalues are computed via Faddeev–LeVerrier + Durand–Kerner. All parameters can be modified live.