Original teaching model
One discrete feedback response
Change the gain to see how a correction affects the next state. This deliberately simple model is not an equation from Maxwell or Ashby.
y[n+1] = y[n] + k(r − y[n])
e[n] = r − y[n] → e[n+1] = (1 − k)e[n]
Here the fixed target r is 0, so the initial state is y[0] = −e[0]. The correction k(r − y) has derivative −k with respect to y. That correction sign is distinct from 1 − k, the multiplier governing successive errors.
Change the conditions
Use arrow keys to adjust a focused slider. Home and End select its limits.
Error over time
Red points show error; the shaded band is the selected tolerance around zero. The vertical scale changes with the data.
Values at every observed step
The band test uses JavaScript floating-point values. Displayed values are rounded; y = −e because r = 0.
| Step n | State y[n] | Error e[n] | Within band? |
|---|
What the example assumes
Measurement is exact and immediate at each discrete step. The target and gain are fixed. The recurrence already contains an integrating state update. No additional plant or delay dynamics, noise, saturation, or disturbance are represented. The initial state counts in the band test. A finite observed run cannot by itself establish asymptotic stability, and asymptotic decay does not ensure every observed state lies in the band.