Side branch
Learn feedback by making one prediction, running one simulator state, and explaining one result at a time across several plant models.
Core model
Target -> Error -> Controller -> Output -> Plant -> Measurement -> Error
The loop keeps repeating. A controller compares the target with the measurement, sends an output command to the plant, then corrects again after the next measurement.
Check: follow one dashed pulse a full lap. How many blocks does a target change pass through before the controller sees its own correction come back as a new measurement?
How to work
Find the target line, measured height, output command, plant motion, and disturbance controls. The controller only sees measured error, not perfect truth.
Myth: the controller sees the true height. It only sees the measurement — add noise, lag, or deadband later and the same tuning starts to fail.
Predict: what changes when the target moves but the payload stays the same?
Explain: which loop block creates the remaining error?
A step response shows how the plant moves after the target changes. Good tuning is a tradeoff between speed, overshoot, settling, steady error, effort, and saturation.
Check: change the target once and name two metrics that moved.
Explain: what got faster, and what did that speed cost?
An on/off controller is a relay. Hysteresis gives it a band, so small measurement noise does not switch the output at every instant.
Myth: a narrower band is more precise. A narrow band switches on every tiny measurement wobble; the band is what stops the chatter.
Predict: what happens if the hysteresis band is too narrow while measurement is noisy?
Explain: why did a wider band calm the relay?
Proportional control pushes harder when error is larger. Raising Kp can reduce steady error, but high Kp can overshoot or oscillate.
Myth: more gain is always better. Higher Kp shrinks steady error but buys overshoot, effort, and oscillation.
Check: raise Kp until steady error improves, then stop when the response starts hunting.
Explain: where does higher Kp help, and where does it start to hurt?
Derivative action reacts to measured motion. It can calm overshoot, but noisy or delayed measurement can make output jitter or correct late.
Myth: D makes it accurate. D is damping — it calms motion but adds no missing force, and it amplifies noisy or late measurement.
Predict: what will happen to output effort when sensor noise increases?
Explain: when did D add damping, and when did it amplify bad measurement?
Integral action stores past error. It helps remove persistent offset from load, but it can keep growing while the actuator is stuck at its limit.
Myth: I makes the controller stronger. I is memory, not force — while the actuator is maxed it just stores error it cannot use.
Check: run once with anti-windup off, then turn it on and compare saturation recovery.
Explain: what changed after anti-windup stopped storing unreachable error?
A fixed target with a changing payload is a different problem from a target step. Tune for recovery after the disturbance, not just a pretty first move.
Predict: which metric should get worse when the payload becomes a sine load?
Explain: why is disturbance rejection different from following a target step?
Noise, lag, and deadband change what the controller believes. A controller tuned on clean measurement can behave badly when feedback is late or jumpy.
Check: keep the same PD tuning and switch between constant payload, sensor lag, and deadband.
Explain: what did the sensor make the controller believe?
Transfer
The hover lesson gives the vocabulary. Now keep the same words and change the plant: the controller still sees target, measurement, error, and output, but the behavior that proves a good tuning changes.
Predict first: if a P-only loop fails, which signature will prove why — steady error, overshoot, no settling, saturation, or falling away from the target?
The beam behaves like a double integrator: angle becomes acceleration, acceleration becomes motion, and P-only control can keep feeding the hunt. D is the structural damping that lets the ball settle.
Myth: D only polishes a loop that already works. On this plant, without D the proof metric is often settling time: the loop never earns a clean settled value.
Predict: with P-only, will the ball park calmly, overshoot once, or keep rolling through the target?
The beam needs D because position error alone does not know how fast the ball is already moving. P tilts the beam toward the target, but the ball carries velocity through it and the loop keeps correcting late. D reacts to motion, so it damps the crossing and turns a hunting response into a settling one. The proof is the response plot: P-only keeps crossing; PD gives a visible settling time.
The tank integrates net flow. The valve can add water but cannot pull it out, and a constant drain needs a constant valve opening. A P controller creates that opening only by keeping a nonzero error.
Myth: an integrating tank naturally fills exactly to the target. With P-only, holding a drain requires error; I stores that missing demand and removes the offset.
Compute first: drain 0.18 plus load 0.12 equals 0.30. With Kp = 2.0, what steady error is needed to hold the valve open?
The P-only hold needs error = valve demand / Kp = 0.30 / 2.0 = 0.15, so a target of 0.70 parks near 0.55. PI does not make the valve stronger; it stores the persistent error until the output includes the standing valve command. The proof is steady error: after PI hold, the tank can stay near the target without preserving that 0.15 gap.
The servo is the cleanest ring story in this tool: P pulls hard toward the target, inertia carries the arm past it, and the loop rings. D adds damping so the same target move becomes crisp instead of twitchy.
Myth: if the servo overshoots, add more P. More P usually makes the ring faster and harder; D is the term that brakes fast motion.
Predict: if Snappy P overshoots, should the next move be more P, less target, or D?
The right next move is D. The servo is already getting enough proportional pull to move quickly; the failure signature is overshoot and ringing, not lack of authority. D reacts to angular speed and subtracts energy from the ring. The proof metric is overshoot: Smooth PD should cut the large Snappy P peak without needing a stronger first push.
The pendulum is different because upright is unstable. Small errors do not naturally fade; gravity pushes them larger. P must first beat the gravity threshold, and D must damp the recovery.
Myth: any feedback eventually stabilizes an unstable plant. Feedback below the threshold can still lose to the plant's own dynamics.
Predict: with Kp below the rough gravity threshold, will the pole slowly improve, wobble forever, or fall away?
Below the threshold, the corrective torque is too weak and the pole falls away instead of recovering. Once P is strong enough, D still matters because the pole arrives at upright with angular speed. Balanced PD supplies both authority and damping: enough push to fight gravity, and enough braking to avoid sailing through the balance point.
Explain: what changed when the controller had enough authority and damping?
Free play
Pick any non-hover plant, make one controller change that creates a clear failure, then copy the simulator URL. Name the failure signature before you fix it: steady error, overshoot, settling never appears, saturation, or falling away.
Keystone preview
The servo ring is the cleanest bridge from time-domain behavior to the next theory layer. Overshoot points at damping ratio and pole angle. Decay points at the negative real part. Oscillation points at the imaginary part. Instability means the pole is on the wrong side of the plane.
Scope: this page does not compute closed-loop poles from the live simulator gains. Treat the pole picture as a literal analytic second-order story for the servo-style response, and as qualitative language for the other plants until the tool grows pole computation.
If the servo response has a tall first peak and several crossings, imagine poles with low damping: a shallow angle into the left half-plane and a strong imaginary part. If the response decays quickly with little overshoot, the pole pair is more strongly damped: farther left, with less ringing. If a response grows instead of decays, the real part has gone right of zero.
Tuning recipe
Work these rungs in order. Each one removes a little support, so the last is yours alone.
Revisit
First, a spaced return. Re-open your D-control and I-control states and switch the profile to weak actuator or sensor lag. Before each run, answer from memory: what does this gain do, and what makes it fragile? Then run it and watch which metric degrades.
Now diagnose. Open each state below in mixed order, decide whether the limit is gain, actuator authority, or measurement quality, and name the one plot or metric that proves it.
Recall: raising Kp shrinks steady error. Name the two things it makes worse.
Diagnose the limit, then prove it with steady error and overshoot.
Raising Kp makes two things worse: it adds overshoot and oscillation (the loop rings before it settles), and it demands more output effort. The limit here is gain: a proportional-only loop can hold the load only by keeping a non-zero error, so a steady offset always remains. The proof is in the plot — the measurement settles short of the target and never reaches zero, and pushing Kp up to shrink that offset grows the overshoot. The honest fix is integral action, not infinite Kp.
Recall: when does D help, and what makes it fragile?
Diagnose the limit, then prove it with overshoot and settling time.
D adds damping that brakes the fast approach and calms overshoot. It turns fragile when the measurement is noisy or late — D amplifies noise into output jitter, and turns a delayed signal into mistimed, late damping. The limit here is gain: high Kp with no derivative (Kd) term drives a hard push that overshoots and rings on this lightly-damped loop. The proof is a peak above the target and a long, oscillating settling time. The fix is adding Kd to damp the ring — lowering Kp would calm it too, but only by trading the ring for a wider steady offset and a sluggish approach, which Kd avoids.
Recall: what is windup, and what does anti-windup change?
Diagnose the limit, then prove it with saturation time and recovery.
Windup is the integral term piling up error while the actuator is briefly pinned at its limit and cannot act on it; when the error finally reverses, that stored-up command unwinds into a large, late overshoot. Anti-windup stops the integrator from accumulating during saturation, so recovery is prompt instead of delayed. The limit here is not actuator authority — the actuator is strong enough to hold the target at steady state; it only saturates transiently during the climb. The real limit is the integrator winding up under that transient saturation. The proof: a burst of saturation during the rise, then a delayed overshoot and slow recovery — and turning anti-windup on shortens that recovery with no change in actuator strength, which proves authority was never the problem. The cure is anti-windup, and a smaller Ki if needed, not a stronger actuator.
Recall: why does widening the hysteresis band calm a chattering relay?
Diagnose the limit, then prove it with the relay's switching rate.
An on/off relay has no resting point inside the band: the output is either full-on or off, so the platform is always driven across the band and keeps switching. Widening the band lengthens the trip between the on and off thresholds, so it switches less often — that is why a wider band calms it. The tell is the switching rate at the relay output, not the effort: average effort (the duty cycle) barely moves, because it tracks the load and setpoint, not the band. Narrowing the band only makes the platform cross it faster — that faster switching is the chatter — so the band trades switching rate against how tightly the output tracks.
Recall: P-only does not fail the same way on every plant. Sort the proof metric before opening the links.
Diagnose each P-only failure signature: hover offset, water offset, beam hunting, servo ringing, and pendulum falling or wobbling.
Hover and water both expose steady error, but for different plant reasons: hover needs force against load; water needs valve opening against drain. Beam exposes settling failure because position error alone misses velocity. Servo exposes overshoot and ringing because inertia carries the arm past target. Pendulum exposes unstable recovery: below threshold it falls, and with P-only above threshold it wobbles without enough damping.
Make one stable tuning for a non-hover plant, copy the simulator link, and explain the tradeoff you accepted. Use the hover sine-load challenge only if you want a familiar warm-up.
Start challengeThis is a side branch off the core path. Rejoin it with electronics for sensors and actuators, phasors for signal intuition, and radar math for tracking loops once this feedback vocabulary is stable.
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