---
date: '2024-02-28'
description: root locus control for improving transient response and steady-state error using pid controllers, ideal integral compensation, and lag compensators.
id: Root locus control
modified: 2026-06-05 15:08:43 GMT-04:00
tags:
- sfwr3dx4
title: Root locus control
created: '2024-02-28'
published: '2024-02-28'
pageLayout: default
slug: thoughts/university/twenty-three-twenty-four/sfwr-3dx4/Root-locus-control
permalink: https://aarnphm.xyz/thoughts/university/twenty-three-twenty-four/sfwr-3dx4/Root-locus-control.md
generator:
quartz: v4.6.0
hostedProvider: Cloudflare
baseUrl: aarnphm.xyz
full: https://aarnphm.xyz/llms-full.txt
---
See also [[thoughts/university/twenty-three-twenty-four/sfwr-3dx4/root_locus_control.pdf|slides]] and [[thoughts/Root locus]]
closed-loop properties of the function of $K_1 G(s)$
## improving transient response
![[thoughts/university/twenty-three-twenty-four/sfwr-3dx4/images/transient-response-root-locus.webp]]
> \[!question\] Question
>
> How to calculate K?
- Product of distances from open-loop pole to point in question
> Second-order poles for the second-order system.
## improving steady state error (SSE)
adding PID (compensator) with an integrator ($\frac{1}{s}$) in feed forward path.
### ideal integral compensation
_proportional-plus-integral (PI) controller_ ⇒ causing error to go to zero.
![[thoughts/university/twenty-three-twenty-four/sfwr-3dx4/ideal-integral-compensator.webp]]
> \[!tip\] Important
>
> Add zero! on the pole near the origin at $s=-a$
![[thoughts/university/twenty-three-twenty-four/sfwr-3dx4/images/zero-add-compensator.webp]]
$$
\frac{K}{s}(s+a) = K_p + \frac{K_i}{s}
$$
where $K_p$ is the proportional gain, and $K_i$ is the integral gain.
> \[!tip\] Implementation `G_c(s)`
>
> $$
> G_c(s) = K_p + \frac{K_i}{s} = \frac{K_p(s+\frac{K_i}{K_p})}{s}
> $$
![[thoughts/university/twenty-three-twenty-four/sfwr-3dx4/images/idea-integral-compensator-impl.webp]]
### lag compensation