Name every signal
Define reference r, output y, error e = r − y and control effort u. A proportional controller uses u = Kp e. Units of gain matter: an error in kelvin and an output in watts require Kp in W/K.
Opening Lilerno
lilernoEngineeringlilernoEngineeringOpening Lilerno
Chapter 18 · Guided self-study
Intro + external course studyDescribe the trade-off between speed, overshoot, and stability.
Build the idea
An original introduction, worked example and practice, followed by a curated reading sequence. The external courses supply deeper teaching and problem sets.
This guide is an orientation, not a complete university module. Practical work needs suitable facilities, safety review and supervision.
Start with these prerequisites
Shared science foundations
Feedback compares a measured output with a target and changes the input. It can reduce error, but delay, noise and actuator limits can undermine a controller.
Module lens
A controller compares a reference with measured output and changes the input to the plant.
Define reference r, output y, error e = r − y and control effort u. A proportional controller uses u = Kp e. Units of gain matter: an error in kelvin and an output in watts require Kp in W/K.
Rise time, overshoot, stability, noise sensitivity and actuator effort can compete. Integral action can reduce steady error but can wind up during saturation. A larger gain is not automatically better.
See the relationship
Change one quantity, watch the graph respond, then explain the result in your own words.
Predict → change → explain
Does proportional gain remove all steady error in this model?
Axes: Time (s) → Normalized output. Bounds may rescale when inputs change.
Steady output = 0.6667; steady error = 0.3333; closed-loop time constant = 0.6667 s.
Dimensionless unity negative feedback: plant 1/(τs+1), proportional gain Kp, unit step, zero initial output. No delay, noise or saturation. Real systems may become unstable as gain increases.
Original Lilerno illustration. Inputs are illustrative; this is not experimental evidence or a design rating.
| Series | Time (s) | Normalized output |
|---|---|---|
| Reference | 0 | 1 |
| Reference | 0.125 | 1 |
| Reference | 0.25 | 1 |
| Reference | 0.375 | 1 |
| Reference | 0.5 | 1 |
| Reference | 0.625 | 1 |
| Reference | 0.75 | 1 |
| Reference | 0.875 | 1 |
| Reference | 1 | 1 |
| Reference | 1.125 | 1 |
| Reference | 1.25 | 1 |
| Reference | 1.375 | 1 |
| Reference | 1.5 | 1 |
| Reference | 1.625 | 1 |
| Reference | 1.75 | 1 |
| Reference | 1.875 | 1 |
| Reference | 2 | 1 |
| Reference | 2.125 | 1 |
| Reference | 2.25 | 1 |
| Reference | 2.375 | 1 |
| Reference | 2.5 | 1 |
| Reference | 2.625 | 1 |
| Reference | 2.75 | 1 |
| Reference | 2.875 | 1 |
| Reference | 3 | 1 |
| Reference | 3.125 | 1 |
| Reference | 3.25 | 1 |
| Reference | 3.375 | 1 |
| Reference | 3.5 | 1 |
| Reference | 3.625 | 1 |
| Reference | 3.75 | 1 |
| Reference | 3.875 | 1 |
| Reference | 4 | 1 |
| Reference | 4.125 | 1 |
| Reference | 4.25 | 1 |
| Reference | 4.375 | 1 |
| Reference | 4.5 | 1 |
| Reference | 4.625 | 1 |
| Reference | 4.75 | 1 |
| Reference | 4.875 | 1 |
| Reference | 5 | 1 |
| Reference | 5.125 | 1 |
| Reference | 5.25 | 1 |
| Reference | 5.375 | 1 |
| Reference | 5.5 | 1 |
| Reference | 5.625 | 1 |
| Reference | 5.75 | 1 |
| Reference | 5.875 | 1 |
| Reference | 6 | 1 |
| Reference | 6.125 | 1 |
| Reference | 6.25 | 1 |
| Reference | 6.375 | 1 |
| Reference | 6.5 | 1 |
| Reference | 6.625 | 1 |
| Reference | 6.75 | 1 |
| Reference | 6.875 | 1 |
| Reference | 7 | 1 |
| Reference | 7.125 | 1 |
| Reference | 7.25 | 1 |
| Reference | 7.375 | 1 |
| Reference | 7.5 | 1 |
| Reference | 7.625 | 1 |
| Reference | 7.75 | 1 |
| Reference | 7.875 | 1 |
| Reference | 8 | 1 |
| Reference | 8.125 | 1 |
| Reference | 8.25 | 1 |
| Reference | 8.375 | 1 |
| Reference | 8.5 | 1 |
| Reference | 8.625 | 1 |
| Reference | 8.75 | 1 |
| Reference | 8.875 | 1 |
| Reference | 9 | 1 |
| Reference | 9.125 | 1 |
| Reference | 9.25 | 1 |
| Reference | 9.375 | 1 |
| Reference | 9.5 | 1 |
| Reference | 9.625 | 1 |
| Reference | 9.75 | 1 |
| Reference | 9.875 | 1 |
| Reference | 10 | 1 |
| Closed loop | 0 | 0 |
| Closed loop | 0.125 | 0.114 |
| Closed loop | 0.25 | 0.2085 |
| Closed loop | 0.375 | 0.2868 |
| Closed loop | 0.5 | 0.3518 |
| Closed loop | 0.625 | 0.4056 |
| Closed loop | 0.75 | 0.4502 |
| Closed loop | 0.875 | 0.4872 |
| Closed loop | 1 | 0.5179 |
| Closed loop | 1.125 | 0.5433 |
| Closed loop | 1.25 | 0.5644 |
| Closed loop | 1.375 | 0.5819 |
| Closed loop | 1.5 | 0.5964 |
| Closed loop | 1.625 | 0.6084 |
| Closed loop | 1.75 | 0.6184 |
| Closed loop | 1.875 | 0.6266 |
| Closed loop | 2 | 0.6335 |
| Closed loop | 2.125 | 0.6392 |
| Closed loop | 2.25 | 0.6439 |
| Closed loop | 2.375 | 0.6478 |
| Closed loop | 2.5 | 0.651 |
| Closed loop | 2.625 | 0.6537 |
| Closed loop | 2.75 | 0.6559 |
| Closed loop | 2.875 | 0.6577 |
| Closed loop | 3 | 0.6593 |
| Closed loop | 3.125 | 0.6605 |
| Closed loop | 3.25 | 0.6616 |
| Closed loop | 3.375 | 0.6624 |
| Closed loop | 3.5 | 0.6632 |
| Closed loop | 3.625 | 0.6638 |
| Closed loop | 3.75 | 0.6643 |
| Closed loop | 3.875 | 0.6647 |
| Closed loop | 4 | 0.665 |
| Closed loop | 4.125 | 0.6653 |
| Closed loop | 4.25 | 0.6655 |
| Closed loop | 4.375 | 0.6657 |
| Closed loop | 4.5 | 0.6659 |
| Closed loop | 4.625 | 0.666 |
| Closed loop | 4.75 | 0.6661 |
| Closed loop | 4.875 | 0.6662 |
| Closed loop | 5 | 0.6663 |
| Closed loop | 5.125 | 0.6664 |
| Closed loop | 5.25 | 0.6664 |
| Closed loop | 5.375 | 0.6665 |
| Closed loop | 5.5 | 0.6665 |
| Closed loop | 5.625 | 0.6665 |
| Closed loop | 5.75 | 0.6665 |
| Closed loop | 5.875 | 0.6666 |
| Closed loop | 6 | 0.6666 |
| Closed loop | 6.125 | 0.6666 |
| Closed loop | 6.25 | 0.6666 |
| Closed loop | 6.375 | 0.6666 |
| Closed loop | 6.5 | 0.6666 |
| Closed loop | 6.625 | 0.6666 |
| Closed loop | 6.75 | 0.6666 |
| Closed loop | 6.875 | 0.6666 |
| Closed loop | 7 | 0.6666 |
| Closed loop | 7.125 | 0.6667 |
| Closed loop | 7.25 | 0.6667 |
| Closed loop | 7.375 | 0.6667 |
| Closed loop | 7.5 | 0.6667 |
| Closed loop | 7.625 | 0.6667 |
| Closed loop | 7.75 | 0.6667 |
| Closed loop | 7.875 | 0.6667 |
| Closed loop | 8 | 0.6667 |
| Closed loop | 8.125 | 0.6667 |
| Closed loop | 8.25 | 0.6667 |
| Closed loop | 8.375 | 0.6667 |
| Closed loop | 8.5 | 0.6667 |
| Closed loop | 8.625 | 0.6667 |
| Closed loop | 8.75 | 0.6667 |
| Closed loop | 8.875 | 0.6667 |
| Closed loop | 9 | 0.6667 |
| Closed loop | 9.125 | 0.6667 |
| Closed loop | 9.25 | 0.6667 |
| Closed loop | 9.375 | 0.6667 |
| Closed loop | 9.5 | 0.6667 |
| Closed loop | 9.625 | 0.6667 |
| Closed loop | 9.75 | 0.6667 |
| Closed loop | 9.875 | 0.6667 |
| Closed loop | 10 | 0.6667 |
Save your place when you finish reading.
Original Lilerno example
An ideal proportional controller has Kp = 2 W/K and temperature error 3 K.
Use e = target − measured temperature.
Compute u = Kp e = 2 × 3 = 6 W.
This calculates effort only; it does not establish closed-loop stability.
Test the model
Open learning, traceable sources
Work in this order. These links open the publisher’s material; free access does not always permit republication.
MIT OpenCourseWare
Start with: System models, feedback and controller design
Work assignments with a clearly labelled block diagram; treat hardware labs as supervised work.
MIT OpenCourseWare
Start with: Linear system response
Relate model poles to time response.
Recall, then record
Close the explanation and answer these in your own words. Return tomorrow, then again later in the week.
What are the gain's units?
Can the actuator produce the requested effort?
Does smaller steady error imply stability?
Draw a complete feedback diagram including a sensor, noise, delay and actuator limits. Explain what the ideal model omits.
Self-reported tasks, not an assessment of mastery or university credit. Reading a page does not complete a chapter.
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Open learning, traceable sources
Original Lilerno lessons and diagrams, supported by these references. Free access does not always permit republication. Links open the publisher’s material.
First-order equations, second-order oscillators, Laplace transforms and systems. Start with the first-order unit.
CC BY-NC-SA 4.0 except separately credited material. Linked for external study, not reproduced. Some assigned textbooks/software require separate access.
Open source ↗Rights / publisher record ↗Feedback models, stability and controller design. Laboratory instructions are references, not unsupervised hardware tasks.
CC BY-NC-SA 4.0 except separately credited material. Linked for external study, not reproduced. Some assigned textbooks/software require separate access.
Open source ↗Rights / publisher record ↗