# LMIs in Control/Stability Analysis/Discrete Time/Polytopic Uncertainty/Open-Loop Robust Stability

## The System

Consider the set of matrices
${\displaystyle {\mathbf {A}}=\{{\mathbf {A}}_{d}(\alpha )\in \mathbb {R} ^{n\times n}|{\mathbf {A}}_{d}(\alpha )=\Sigma _{i=1}^{n}\alpha {\mathbf {A}}_{d,i},{\mathbf {A}}_{d,i}\in \mathbb {R} ^{n\times n},\alpha _{i}\in \mathbb {R} _{\geq 0}\},\Sigma _{i=1}^{n}\alpha _{i}=1\}}$,

## The Data

The discrete-time LTI system ${\displaystyle {\mathbf {x}}_{k+1}={\mathbf {A}}_{d}(\alpha ){\mathbf {x}}_{k+1}}$ is asymptotically stable for all ${\displaystyle {\mathbf {A}}_{d}(\alpha )\in \mathbb {\mathbf {A}} }$ if there exists ${\displaystyle {\mathbf {P}}\in \mathbb {S} ^{n}}$ ,${\displaystyle i=1,...,n}$ , and ${\displaystyle {\mathbf {G}}\in \mathbb {R} ^{n\times n}}$ , where ${\displaystyle {\mathbf {P}}_{i}>0,i=1,...,n}$ , such that

${\displaystyle {\begin{bmatrix}{\mathbf {P}}_{i}&{\mathbf {P}}A_{d,i}^{T}{\mathbf {G}}^{T}\\*&{\mathbf {G}}+{\mathbf {G}}^{T}-{\mathbf {P}}_{i}\end{bmatrix}}<0.}$
, ${\displaystyle i=1,...,n}$

## Implementation

This is used to get open-loop stability.

## References

Caverly, Ryan; Forbes, James (2021). LMI Properties and Applications in Systems, Stability, and Control Theory.