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

The System

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

The Data

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

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

Implementation

This is used to get open-loop stability.

Related Links

• LMIs in Control/Stability Analysis/Discrete Time/Transient Bound for Discrete-Time Non-Autonomous LTI Systems

References

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