# LMIs in Control/Matrix and LMI Properties and Tools/Quadratic Schur Stability

The following LMI can be used to discern whether or not a system of a particular form is Quadratically Schur Stable or not.

## The System

We consider the following system:

{\displaystyle {\begin{aligned}{\dot {x}}(t)&=A(\delta (t))x(t)\end{aligned}}}

This is the continuous time case. The system coefficient matrix takes the form:

{\displaystyle {\begin{aligned}A(\delta (t))=A_{0}+\Delta A(\delta (t))\end{aligned}}}

where ${\displaystyle A_{0}}$  is the known nominal system matrix while ${\displaystyle \Delta A(\delta (t))}$  is the system matrix perturbation.

## The Data

The data required is both the matrices ${\displaystyle A_{0}}$  and ${\displaystyle \Delta A(\delta (t))}$  as seen in the form above, or the combined ${\displaystyle A(\delta (t))}$ .

## The Optimization Problem

The goal of the optimization is to find the minimum P such that the following LMI is satisfied.

## The LMI: LMI for Quadratic Schur Stability

The LMI problem is to find a symmetric matrix P satisfying:

{\displaystyle {\begin{aligned}{\begin{bmatrix}-P&A(\delta (t))P\\PA^{T}(\delta (t))&-P\end{bmatrix}}<0\\\end{aligned}}} , ${\displaystyle \delta (t)\in \Delta }$

## Conclusion:

If the one of the above LMIs is found to be feasible, then the system is Quadratically Schur Stable, and has possibly already gone through the process of LMI usage for Quadratic Schur Stabilization.

## Implementation

A link to Matlab codes for this problem in the Github repository:

## Related LMIs

[1] - Schur Stabilization