Inverted pendulum

A cart of mass $M$ rolls along a track. A pole of mass $m$ and half-length $L$ is hinged to the cart by a free pivot, so the moment it tilts, gravity pulls it over. The single thing we get to control is a horizontal force $F$ on the cart. Two things to track:

• $x$— the cart’s position from the middle of the track.
• $\theta$— the pole’s tilt from vertical (zero = upright).

Mental picture: balancing a broom on your palm. You don’t push the broom directly — you move your hand. Same trick.

Newton’s laws say any acceleration must come from a net force or torque. Sum the forces on the cart, sum the torques on the pole, and keep careful track of how the pole’s rotation drags on the cart’s motion (and vice versa).

The result is a coupled pair of nonlinear equations with $\sin \theta$ and $\cos \theta$ in them. But for balancing, $\theta$ stays small, so we use the approximations $\sin \theta \approx \theta$ and $\cos \theta \approx 1$. The equations collapse into:

Read these in plain English. The first one is $F=ma$for the whole system. The second one says the pole’s angular acceleration depends on (a) how fast the cart is jerking (which swings the pole around) and (b) how far the pole has already tilted — because gravity on a tilted pole creates a torque that tilts it more.

That last bit, $mgL\theta$, is positive feedback. More tilt → more torque → more tilt. Left alone, this is a runaway — and the controller’s whole job is to break it.

A transfer function $G(s)$ answers one question: poke the input $F(t)$, what shape does the output $\theta (t)$ take? Picture it as a box that takes force in and gives angle out.

The trick to deriving $G(s)$is the Laplace transform. Don’t worry about the details — just hold this fact: it turns $d/dt$ into multiplication by a variable $s$. Differential equations turn into algebra. Solve the linearized pair from Step 1 for $\Theta (s)/F(s)$:

The most important thing about a transfer function is its poles — the values of $s$that make the denominator zero. Poles are the system’s natural modes: how it moves when nobody is controlling it. Setting the denominator to zero and solving for $s$:

Plugging in the values from the parameters table, that’s about $s\approx \pm 3.97\text{rad/s}$. Two poles, on the real axis, symmetric about the origin:

A pole on the right side of the $s$-plane means the system grows on its own — any tiny error blows up like $e^{+3.97t}$. The time constant $1/3.97\approx 0.25$seconds is how long it takes for a microscopic disturbance to tip the pole noticeably. That’s gravity, written in math: a ball balanced on top of a hill.

The job of any controller is to feed force back based on what we measure, in a way that drags that right-half-plane pole into the left half — where things decay instead of blow up.

Look back at the transfer function: $G(s)=-1/(\dots s^2-\dots )$. There’s a minus sign in front. It says: push the cart right, the pole tips left. Obvious in hindsight — yank a rug right, the broom on it falls left.

Now suppose the cart has drifted left and you want to bring it back to center. The naïve thing is to push right. But the moment you push right, the pole tips left — and now you have a falling pole on top of a cart still on the wrong side of the track. Worse, not better.

The maneuver that actually works is sneakier. To move the cart right, you first push left for a brief moment. That tilts the pole right. Now you chase the rightward-leaning pole by pushing right, and the cart ends up centered with the pole upright.

The system has to go the wrong way before it goes the right way.

This is called non-minimum phase. You’ve felt it:

• A bicycle: to turn left, you steer right first (countersteering).
• A car backing up: turn the wheel one way, the rear swings the other.
• An aircraft pulling up: the elevator on the tail pushes down first, which makes the nose pitch up.

Every one of these needs a controller patient enough to commit to short-term wrongness in exchange for long-term rightness. As we’ll see, this is exactly what PID is bad at.

A controller is a rule: look at the system, decide what force to apply. To use it, we wrap it around the plant in a feedback loop — measure the output, compare against where we want it, feed the difference into the controller.

For us the reference is “pole upright” — i.e. ${\theta }_{\text{ref}}=0$, so the error is just $e=\theta$. Now: what goes inside the controller box?

PID — proportional, integral, derivative — is the most common answer in industry. It is three opinions about the same error, added together.

• P · Proportional — “How wrong is it now?” Push back proportionally to the current error. Like a spring. Term: $K_p\theta$.
• I · Integral — “How long has it been wrong?” Sum the error over time. If something keeps the pole leaning by a hair, the integral grows until it overpowers the bias. Term: $K_i{\int }_0^t\theta d\tau$.
• D · Derivative — “How fast is it getting worse?” Push back against the rate of change. Like a shock absorber. Term: $K_d\dot{\theta }$.

Add the three voices together:

But this only fights tilt. The pole could be perfectly upright while the cart slides off the end of the track. So we add a second, gentler piece of feedback that pulls the cart back toward the middle:

Defaults you can play with above: $K_p=40,K_i=0,K_d=8$, with cart gains fixed at $K_x=1.0,K_{\dot{x}}=1.5$. The output is clipped to $|F|\le 20\text{N}$ and the integrator is clamped to $|\int \theta dt|\le 5$so it can’t run away.

Tuning intuition: turn $K_p$ up → stiffer, but the pole overshoots. Add $K_d$ → damping, oscillations die down. Add $K_i$only if a small steady lean won’t go away on its own.

PID is famous because it works well enough on a huge fraction of real systems. Cart-pole is one of those systems where it just barely works, and the seams show. Each issue below is reproducible in the sim above.

None of this means PID is bad. It is the right tool for thermostats, motor speed loops, drone attitude — single-input, single-output problems where the dynamics are mild. Cart-pole is the canonical “PID can just barely” demo. That’s exactly why it’s the standard teaching system: it shows you everything PID can do, and the cliff where the next chapter of control theory starts.