---
readstatus: false
dateread:
title: "Regions of robust relative stability for PI controllers and LTI plants with unstructured multiplicative uncertainty: A second-order-based example"
year: 2023
authors:

  
  - "Matušů, Radek"
  
  - "Senol, Bilal"
  
  - "Pekař, Libor"
  

citekey: "matusuRegionsRobustRelative2023"

journal: "Heliyon"


volume: 9


issue: 8





---
# Indexing Information
## DOI
[10.1016/j.heliyon.2023.e18924](https://doi.org/10.1016/j.heliyon.2023.e18924)
## ISBN
[](https://www.isbnsearch.org/isbn/)
## Tags:
#Robust-control, #H-Infinity-norm, #PI-controllers, #Robust-performance, #Robust-relative-stability, #Unstructured-multiplicative-uncertainty

>[!Abstract]
>

>[!note] Markdown Notes
>This might be an example of sampling unstructured uncertainties

>[!seealso] Related Papers
>

# Annotations

>[!fail] Possibly Incorrect
> *With respect to the classical robust control literature, the robust stability condition under consideration of multiplicative uncertainty can be formulated as follows: Under the assumption of a nominally stable feedback control system (that is, for G0(s)), the related perturbed feedback control system (containing the plant affected by multiplicative uncertainty) is robustly stable if and only if [7,13, 14]:  ‖WM(s)T0(s)‖∞ < 1 (3)  Recently, the paper [7] has presented the relativized version of this condition as follows: Under the assumption of a nominally stable feedback control system, the related uncertain feedback control loop is robustly relatively stable, having a margin factor of α if and only if [7]:  ‖WM(s)T0(s)‖∞ < 1  α (4)  Typically, the margin factor α is assumed to be positive. For the special case of α = 1, both conditions (3) and (4) become identical. More information on the robust stability, robust performance, and robust relative stability conditions, including their graphical interpretations and their versions for the other sorts of models under unstructured uncertainty, can be found in Ref. [7].*
> 
> >[!note] Note
> >This is braindead. They invented a factor of safety... that was a whole publication?

>[!attention] Highlight
> *Then, the weight function WM(s), which must fulfill (2), was determined in Ref. [16] as the worst-case uncertainty member (with K = 2.2, T1 = 9, T2 = 0.9) of the plant family (5). This worst-case combination of parameters tallies with the uppermost magnitude characteristics of the normalized perturbations from Figs. 2 and 3 [16], and so the weight function was chosen exactly accordingly:*
> 

>[!quote] Other Highlight
> *The main purpose of ΔM(s) is (except for acting as a scaling factor on the perturbation magnitude) to account for phase uncertainty [13].*
> 

### Imported: 2024-10-22 9:44 am


