Control lab · inverted pendulum · force-commanded cart

Force Cart-Pole PID Lab

The same balancing problem, but now the PID output is a force in newtons pushed into a cart with mass. The gains take back their textbook roles. Change them while it runs, push the pole, and compare with the velocity-commanded version.

Balancing

Upright for0.0 s

RMS θ · last 5 s0.00°

Peak |F| · last 5 s0.0 N

Cart x · v0.00 m

Keys: ← → push, Space pause, R reset. Click the stage left or right of the pole to push it that way.

Bode and Nyquist

Loop broken at the controller output. Includes actuator lag, the pure delay, the 20 ms D filter and the half-sample delay of the sampled controller. Updates as you change anything.

Closed loop (Nyquist)–

CCW encirclements of −1–one per unstable open-loop pole

Phase margin–

Gain margin ↑ · ↓–raise · lower all gains

Peak sensitivity Ms–aim below 6 dB

Bode magnitude |L(jω)| (dB)

ωc gain crossoverp unstable pole

Bode phase ∠L(jω)

−180° linesx axis: ω in rad/s

Nyquist L(jω)

Log radius

ω > 0ω < 0 (mirror)× critical point −1

Log radius squeezes large gains so the area around −1 stays readable. It keeps the unit circle and the encirclement count unchanged.

Gains

F = Kp·e + Ki·∫e + Kd·ė

Kp · N per rad

Ki · N per rad·s

Kd · N per rad/s

Presets

Linearized loop–

–

Poles–

Damping ratio ζ · ωₙ–

Largest stable Ki–

Ignores motor lag, saturation, sampling and noise.

Plant, sensor and extras

Angle sensor noise (σ)

Angle sensor offset

Motor force lag τ

Controller sample period Ts

Pure delay (sensing + compute + bus)

Force limit ±

Start angle on reset

Push strength (Δω)

Track ends at ±3 m (camera stops following) Centering loop: lean the setpoint to bring the cart home

kx · rad per m

kv · rad per m/s

Control loop, live

Values update in real time. e = θ − θ_ref, so a pole leaning right (θ > 0) pushes the cart right.

Scope

Last 10 seconds of simulated time

Angle θ (degrees)

θθ_ref

Force on the cart (N)

F appliedF command

PID terms (N)

PID

Cart position (m) and velocity (m/s)

How to tune it

A force goes straight into acceleration, so the pole feels the controller output directly. Gravity makes upright an unstable equilibrium, and the controller has to supply a restoring "spring" plus damping. Linearized around upright, with the cart mass M, pole mass m and length L:

plant M·L·θ̈ = (M + m)·g·θ − F (small angles) PD controller F = Kp·θ + Kd·θ̇ closed loop M·L·θ̈ + Kd·θ̇ + (Kp − (M+m)·g)·θ = 0 mass damper spring with Ki M·L·s³ + Kd·s² + (Kp − (M+m)·g)·s + Ki = 0

PKp is the spring

It must beat gravity: Kp > (M+m)·g = 11.8 N/rad here. Below that the pole falls no matter what else you do. Higher Kp gives a stiffer, faster response and bigger forces.

DKd is the damper

Without it, the pole rocks around upright and motor lag makes the rocking grow. Kd is what kills it. It also passes sensor noise straight into the force.

IKi is mostly a liability

It adds a third pole and eats phase. Stability needs Kd·(Kp − 11.8) > M·L·Ki, shown live as "Largest stable Ki". It cannot fix the problems you would expect it to fix (see below).

How this compares with the velocity version

Gain	Commanding velocity	Commanding force
Kp	Damper	Spring, must exceed (M+m)·g
Ki	Spring, must exceed g·L	Optional, limited by Routh
Kd	Extra mass, keep small	Damper, required
Units	m/s per rad	N per rad

The velocity servo in the first lab was itself a controller that hid the cart's mass. That inner loop integrated your command once, which shifted every gain's job up a derivative. With raw force there is no hidden loop, so PD does the work.

A tuning recipe

Set Ki = 0 and Kd = 0. Raise Kp until the pole stops falling over and starts rocking around upright. That happens just above 11.8; go to about 3× that, say 40.
Raise Kd until a push settles in about one overshoot. For a target ζ, use Kd ≈ 2·ζ·√(M·L·(Kp − 11.8)). With Kp = 40 and ζ = 0.7 that gives Kd ≈ 7.4.
Turn sensor noise up to 0.5° and watch the force trace. The jitter scales with Kd. If it bothers you, lower Kd or filter harder; a real motor heats up on that buzz.
Try Ki last, and watch "Largest stable Ki". Push it near the limit and the pole starts a slow, growing wobble. The "Too much I" preset sits past it.
Stress it. Push harder, raise the motor lag τ, lower the force limit. If F sits flat at the limit in the scope, the gains are asking for more than the motor has.

Reading the scope

What you see	Likely cause	Try
Pole falls in the first second, cart chases it	Kp ≤ 11.8: spring weaker than gravity	Raise Kp
Steady rocking that grows over a few seconds	Too little Kd, or motor lag eating phase	Raise Kd, lower τ
Slow wobble that builds even with good Kp and Kd	Ki past the Routh limit	Lower Ki
Pole creeps back slowly, cart moves a lot	Overdamped, Kd too large for Kp	Lower Kd or raise Kp
Fuzzy force trace	D term amplifying sensor noise	Lower Kd
Force flat at the limit after a push	Motor saturation	Softer gains or a stronger motor
Cart cruises or accelerates away while the pole looks fine	No position feedback, or a sensor offset	Turn on the centering loop

Why the I term can't save you from an offset

Set the sensor offset to 1°. The controller now holds the measured angle at zero, so the real pole leans 1° the other way. A pole can only stay leaned if the cart keeps accelerating under it, so the cart speeds off and never stops. Adding Ki doesn't help, because from the angle alone the controller cannot tell an offset from a real lean.

The fix is information, not a gain. Turn on the Centering loop: the outer loop sees the cart running away and leans θ_ref until the cart stops, which finds the true upright on its own. The cart settles a little off center, and adding an integrator to the outer loop would remove that last bit. Keep the outer loop much slower than the inner one.

Reading the Bode and Nyquist plots

The plant has an unstable pole p = √((M+m)g/(M·L)) ≈ 3.4 rad/s. That changes the rules you may know from stable plants: the Nyquist curve must circle −1 counter-clockwise once, and a positive phase margin alone proves nothing. Load "Kp below (M+m)g" to see a 60°+ phase margin on a loop that falls over.

Low frequencies. With Ki = 0 the curve starts on the negative real axis at −Kp/((M+m)g). It must start left of −1, which is the Kp > 11.8 rule again. How far left is the downward gain margin: how much you could lower every gain before the pole falls.
Crossover. The PD zero at ω = Kp/Kd adds phase lead and swings the curve around −1. Read the phase margin where |L| = 0 dB; 35–60° is the usual target. Keep ωc above about 2·p: you can't stabilize an unstable pole with a loop slower than the pole.
Phase eaters. Motor lag, the D filter and the sample delay all bend the phase down near crossover. Drag τ up and watch the margin shrink and the Nyquist curve slide toward −1 until the encirclement flips.
Ki. It adds −90° at low frequency, so the curve now starts at infinity. Near the Routh limit, the extra lag reaches crossover and the margin collapses.
Ms. The peak of |1/(1+L)| is the inverse of the closest distance from the curve to −1. Below 6 dB means no frequency gets amplified more than 2× by feedback.

What the simulation includes

Full nonlinear cart-pole: cart M = 1 kg, point-mass pole m = 0.2 kg at L = 1 m, cart friction 0.1 N·s/m, light pivot friction. Physics at 1 kHz.
Controller sampled every Ts (10 ms, 100 Hz, by default). An optional pure delay holds each command back before the motor sees it, the way sensing, computation and a slow bus would. Both appear in the Bode and Nyquist panel as phase lost: ω·(Ts/2 + delay).
Derivative taken on the measured angle with a 20 ms low-pass, so changes to θ_ref cause no derivative kick.
Motor force follows the command through a first-order lag τ and is clipped to the force limit. The integrator stops growing when its share reaches that limit (anti-windup); the I block outlines in red when it does.