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

The System

Consider the set of matrices
$𝐀=\{{𝐀}_d(\alpha )\in {ℝ}^{n\times n}|{𝐀}_d(\alpha )={\mathrm{\Sigma }}_{i=1}^n\alpha {𝐀}_{d,i},{𝐀}_{d,i}\in {ℝ}^{n\times n},{\alpha }_i\in {ℝ}_{\ge 0}\},{\mathrm{\Sigma }}_{i=1}^n{\alpha }_i=1\}$,

The Data

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

$\left[\begin{array}{cc}{𝐏}_i& 𝐏A_{d,i}^T{𝐆}^T\\ *& 𝐆+{𝐆}^T-{𝐏}_i\end{array}\right]<0.$
, $i=1,...,n$

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