Delayed feedback and bounded control

Prerequisite & prediction

Does clipping force to 5 N guarantee no overshoot?

Complete implementation & test runner

Download all four files into one folder. Run node check.mjs. The checks exercise all three projects; no installation or API key is required.

// Explicit Euler point-mass simulator. Units: kg, m, s, N. No hardware control.
export function simulate({delay=0,mass=1,kp=10,kd=3,limit=5,dt=.01,steps=400}={}) {
  if (!Number.isInteger(delay)||delay<0||delay>100||!Number.isInteger(steps)||steps<1||steps>10000||![mass,kp,kd,limit,dt].every(Number.isFinite)||mass<=0||dt<=0||limit<0) throw Error('Invalid configuration');
  let x=0,v=0,peak=0,clamps=0;const history=[{x,v}],trace=[];
  for(let t=0;t<steps;t++){
    const sensed=history[Math.max(0,history.length-1-delay)];
    const proposed=kp*(1-sensed.x)-kd*sensed.v;
    const force=Math.max(-limit,Math.min(limit,proposed));
    if(force!==proposed)clamps++;
    const nextX=x+dt*v;v+=dt*force/mass;x=nextX;peak=Math.max(peak,x);
    history.push({x,v});trace.push({time:(t+1)*dt,x,v,force});
  }
  return {finalError:Math.abs(1-x),overshoot:Math.max(0,peak-1),saturationSteps:clamps,
    maxForce:Math.max(...trace.map(r=>Math.abs(r.force))),trace};
}
export function run(){const summarize=config=>{const {trace,...metrics}=simulate(config);return {config,...metrics,first:trace[0],last:trace.at(-1)}};return {baseline:summarize({delay:0}),delayed:summarize({delay:10}),massShift:summarize({delay:0,mass:2}),
  limitation:'Teaching simulator; no collision geometry, contacts, sensor noise or hardware safety validation.'}}

import assert from 'node:assert/strict';
import {run as attention,softmax,attend,full,cached} from './attention.mjs';
import {run as retrieval,retrieve} from './retrieval.mjs';
import {run as control,simulate} from './control.mjs';
assert.ok(attention().passes);assert.throws(()=>softmax([-Infinity,-Infinity]));assert.throws(()=>attend([1,2],[[1]],[[1]]));
assert.equal(retrieval().baseline.accuracy,.4);assert.equal(retrieval().treatment.accuracy,1);assert.equal(retrieve('unseen vocabulary'),null);
const result=control();assert.ok(result.baseline.maxForce<=5);assert.ok(result.delayed.maxForce<=5);assert.equal(simulate({steps:1}).trace[0].x,0);assert.equal(simulate({steps:1}).trace[0].v,.05);assert.throws(()=>simulate({mass:0}));

// Full and cached paths must reject inconsistent shapes, not just the single-query helper.
assert.throws(()=>full([[1,2]],[[1]],[[1]]),/Shape mismatch/);
assert.throws(()=>cached([[1,2]],[],[]),/sequence shape/);

console.log('PASS: 12 capstone checks. Toy arithmetic and invariants only; no GPU or hardware benchmark.');

Transfer task & ablation

Sweep delay 0/5/10/20 while fixing mass, gains and integration step. Then repeat with half the step to detect numerical artifacts.

Rubric

Declare units and Euler convention; enforce actuator bounds; isolate delay from mass; report overshoot, error and model limitations.

Score each of the four criteria 0 (missing), 1 (with help), 2 (independent evidence). Passing requires 2 on every criterion. A self-check is not a certification.