PID & Feedback

Posted on Edited on In
  • proportional term (P) depends on the present error
  • integral term (I) depends on past errors
  • derivative term (D) depends on anticipated future errors

PID controller makes use of linear extrapolation of the measured output

PI controller does not make use of any prediction of the future state of the system

The prediction by linear extrapolation (D) can generate large undesired control signals because measurement noise is amplified, that's why D is not used widely

The Problem of "Sinusoids Running Around Loops"

The representative of Fourier transform \(\frac{1}{j\omega+j\omega_0}\) back in the time domain \(e^{-j\omega_0 t}\) is infinite extent in time

Running around a loop, chasing one's tail — these are thought pictures that only work in a discretized, time-sequenced conceptual framework that has a beginning and an end

Fix in your mind that oscillations are a type of resonance

rearrangement

loop-refactor.drawio

The closed loop transfer function of \(Y/X\) and \(Y_1/X_1\) are almost same, except sign

\[\begin{align} \frac{Y}{X} &= +\frac{H_1(s)H_2(s)}{1+H_1(s)H_2(s)} \\ \frac{Y_1}{X_1} &= -\frac{H_1(s)H_2(s)}{1+H_1(s)H_2(s)} \end{align}\]

loop-refactor-partion.drawio

define \(-Y_1=Y_n\), then \[ \frac{Y_n}{X_1} = \frac{H_1(s)H_2(s)}{1+H_1(s)H_2(s)} \] loop-refactor-partion-general.drawio

image-20240805231921946

Saurabh Saxena, IIT Madras. CICC2022 Clocking for Serial Links - Frequency and Jitter Requirements, Phase-Locked Loops, Clock and Data Recovery

reference

Gene F. Franklin, J. David Powell, and Abbas Emami-Naeini. Feedback Control of Dynamic Systems, Global Edition (8th Edition). Pearson. [pdf]

Åström, K.J. & Murray, Richard. (2021). Feedback Systems: An Introduction for Scientists and Engineers Second Edition [pdf]

Dawson, Joel L. A Guide to Feedback Theory. Cambridge: Cambridge University Press, 2021.

Yan Lu, ISSCC2021 T10: Fundamentals of Fully Integrated Voltage Regulators [https://www.nishanchettri.com/isscc-slides/2021%20ISSCC/TUTORIALS/ISSCC2021-T10.pdf]

Jens Anders (University of Stuttgart, DE). ESSERC2025 Circuits Insights: The Magic of Feedback in Analog Circuit Design [https://youtu.be/NyXuA6WZ8Hg]