Balancing an inverted pendulum with delayed feedback

A homogeneous rod on a cart, stabilised by delayed PD control. Past the critical delay τ_c the upright position loses stability in a Hopf bifurcation and the cart starts to rock. For larger derivative gain K_d the oscillation then breaks its left–right symmetry before the pendulum falls.

angle x(t) [rad] mean tilt x̄ over one Hopf period cart position y(t) [m] scene: delayed state seen by the controller

Statestable

critical delay τ_c–

Hopf period 2π/ω_c–

amplitude of x (last 3 s)–

mean tilt x̄–

cart speed ẏ–

time0.0 s

Scenarios

Delay τ–

speed

Feedback gains

K_p on x [N/rad]

K_d on ẋ [N·s/rad]

k_p on y [N/m]

k_d on ẏ [N·s/m]

M = 1 kg, m = 0.3 kg, L = 1 m, g = 9.81 m/s². τ_c is recomputed from the linearisation whenever the gains change.

Equations of motion

Rod of length L and mass m (centre of mass at l = L/2, moment of inertia mL²/12 about it), cart of mass M driven by a horizontal force F. x is the tilt from upright, y the cart position.

(M + m) \ddot{y} + m l (\ddot{x} \cos x - {\dot{x}}^{2} \sin x) = F (t)

\frac{4}{3} l \ddot{x} + \ddot{y} \cos x - g \sin x = 0, l = \frac{L}{2}

Delayed PD feedback

All four states are measured τ seconds late. The cart gains are much smaller than the angle gains. For an inverted pendulum they carry the same sign as the angle gains: the cart briefly moves away from the origin to lean the rod back towards it.

F (t) = K_{p} x (t - τ) + K_{d} \dot{x} (t - τ) + k_{p} y (t - τ) + k_{d} \dot{y} (t - τ)

Hopf bifurcation

Linearising about x = y = 0 gives the characteristic equation

λ^{4} - a λ^{2} + \frac{e^{- λ τ}}{D} [(K (λ) - \frac{4}{3} l k (λ)) λ^{2} + g k (λ)] = 0

with K(λ) = K_p + K_dλ, k(λ) = k_p + k_dλ, D = l(4M+m)/3, a = (M+m)g/D.

A pair of roots crosses the imaginary axis at λ = ±iω_c when τ = τ_c. With K_d = 7 the bifurcation is supercritical: the amplitude grows like √(τ − τ_c) until, about 10 % above τ_c, the pendulum falls.

Symmetry breaking

The equations are invariant under (x, y) ↦ (−x, −y), and the Hopf orbit inherits that symmetry: its mean tilt x̄ is zero. The fall corresponds to a connecting orbit. For larger K_d the orbit first loses its symmetry. In simulations without cart feedback, the pendulum falls from the symmetric orbit for K_d = 4. Symmetry breaking comes first at about 1.18 τ_c for K_d = 5.5, 1.13 τ_c for K_d = 7 and 1.03 τ_c for K_d = 13. In each case the fall follows only at 1.26–1.28 τ_c. The pendulum then oscillates about a tilted mean x̄ ≠ 0. Averaging the cart equation over one period gives the balance

K_{p} \overline{x} + k_{p} \overline{y} + k_{d} \overline{\dot{y}} = 0

Without cart feedback (k_p = k_d = 0) the leaning oscillation is a stable periodic orbit of the angle equation, and the cart accelerates steadily towards the lean. With the small cart gains the balance cannot hold for long. The cart runs off, its own feedback pushes the tilt over to the other side, and the cart runs back. You see a fast oscillation with a slowly flipping mean tilt, and large, slow cart sweeps.