LMIs in Control/LMI for Polytopic Uncertainity/Polytopic Quadratic Stability

LMIs in Control/LMI for Polytopic Uncertainity/Polytopic Quadratic Stability

The System: Consider the system with Affine Time-Varying

uncertainty (No input)

$\begin{array}{cc}\dot{x}(t)& =(A_0+\mathrm{\Delta }A(t))x(t)\\ \end{array}$

where

$\begin{array}{c}\mathrm{\Delta }A(t)=A_1{\delta }_1(t)+....+A_k{\delta }_k(t)\end{array}$

where ${\delta }_i(t)$ lies in either the intervals

$\begin{array}{c}{\delta }_i\in [{\delta }_i^{-},{\delta }_i^{+}]\end{array}$

or the simplex

$\begin{array}{c}\delta (t)\in \delta :\mathrm{\Sigma }{\alpha }_i=1,\alpha \ge 0\end{array}$

The LMI: Polytopic Quadratic Stability

The system is Quadratically Stable over $\mathrm{\Delta }$ if there exists a P > 0

$(A+\mathrm{\Delta }A{)}^T+P(A+\mathrm{\Delta }A)<0$ for all $\mathrm{\Delta }A\in \mathrm{\Delta }$

Conclusion: 

Interpretation of the results of the LMI

Implementation A link to CodeOcean or other online implementation of the LMI

Related LMIs

Links to other closely-related LMIs

External Links

A list of references documenting and validating the LMI.

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

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