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

The System: Consider the system with Affine Time-Varying

uncertainty (No input)

\begin{align}

\dot x(t)&=(A_0+\Delta A(t))x(t)\\ \end{align}$$ where



\begin{align} \Delta A(t) = A_1 \delta_1 (t)+....+A_k

\delta_k (t) \end{align}$$ where $$ \delta_i (t) $$ lies in either the intervals



\begin{align} \delta_i \in [\delta_i^-,\delta_i^+] \end{align} $$ or the simplex

\begin{align} \delta(t) \in {\delta : \Sigma \alpha_i = 1,

\alpha \geq 0} \end{align} $$

The LMI: Polytopic Quadratic Stability

The system is Quadratically Stable over $$

\Delta $$ if there exists a P > 0 $$ (A+\Delta A)^T + P(A+\Delta A) < 0

$$ for all $$ \Delta A \in \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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