---
date: '2024-12-18'
description: proportional-integral-derivative
id: PID controller
modified: 2026-06-05 15:08:42 GMT-04:00
tags:
- sfwr4aa4
title: PID controller
created: '2024-12-18'
published: '2024-12-18'
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slug: thoughts/university/twenty-four-twenty-five/sfwr-4aa4/PID-controller
permalink: https://aarnphm.xyz/thoughts/university/twenty-four-twenty-five/sfwr-4aa4/PID-controller.md
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---
## proportional control
> \[!definition\] Definition 1.
>
> $$
> K_p e(t) = K_p [u(t) - y(t)]
> $$
![[thoughts/university/twenty-four-twenty-five/sfwr-4aa4/prop-control.webp]]
Example: Given the closed-loop transfer function is $T(s) = \frac{G_p(s)}{1+G_p(s)} = \frac{1}{s+2}$
### adding proportional
closed-loop transfer function is:
$$
T(s) = \frac{K_p G_p}{1 + K_p G_p}
$$
## integral control
source code
Closed loop here yields
$$
T(s) = \frac{K_I}{s^2 + s + K_I}
$$
steady state error is 0, while steady-state output is 1
## PI control
_proportional-integral_
source code
Closed-loop transfer function
$$
T(s) = \frac{K_I + sK_p}{s^{2} + (1+K_p)s + K_I}
$$
- PC: impact on speed of response
- IC: force steady-state error to 0
## derivative control
source code
$$
T(s) = \frac{K_D s}{(1+K_D)s + 1}
$$
- introduces an open-loop zero
- $K_D$ increases system might not be stable
> \[!tip\] in second-order system
>
> $$
> T(s) = \frac{s K_D P \omega_n^2}{s^{2}+ 2(\zeta + \frac{K_D}{2} P\omega_n)\omega_n s + \omega_n^2}
> $$
>
> damping effect $\zeta^{'} = \zeta + \frac{K_D}{2}P\omega_n$
## PID control
$$
G_C(s) = K_p + \frac{K_I}{s} + K_D s
$$
in time domain:
$$
u(t) = K_P e(t) + K_I \int_{0}^{t} e(\eta) d\eta + K_D \frac{d(e(t))}{dt}
$$
| Component | Discrete-Time Equation |
| ------------ | ------------------------------------- |
| Proportional | $u(k) = K_P e(k)$ |
| Integral | $u(k) = K_I T \sum_{i=1}^{k} e(i)$ |
| Derivative | $u(k) = \frac{K_D}{T}[e(k) - e(k-1)]$ |
> approximate of PID controller: $u(k) = K_P e(k) + K_I T \sum_{i=1}^{n} e(i) + \frac{K_D}{T}[e(k) - e(k-1)]$