# LMIs in Control/pages/S Procedure

LMIs in Control/pages/S Procedure

S-Procedure

## The Optimization Problem

In general procedures, considering following quadratic function ${\displaystyle F_{0}(x):\mathbb {R} ^{n}\to \mathbb {R} }$ , ${\displaystyle F_{i}(x):\mathbb {R} ^{n}\to \mathbb {R} }$  where ${\displaystyle i=1,...,m}$ . The inequality ${\displaystyle F_{0}(x)\leq 0}$  is satisfied when all ${\displaystyle F_{i}(x)\geq 0}$ .

{\displaystyle {\begin{aligned}F_{0}(x)+\sum _{i=1}^{m}\tau _{i}F_{i}(x)\leq 0\end{aligned}}}

Where ${\displaystyle x\in \mathbb {R} ^{n}}$ , and ${\displaystyle \tau _{i}\in \mathbb {R} _{\geq 0}}$

This type of procedure is used to help solve problems that were originally NP-hard problems. An example of this is the following inequality: ${\displaystyle x^{T}Fx\geq 0:\forall x\geq 0}$ . By using the defined problem above, an LMI can be constructed using the S-Procedure:

{\displaystyle {\begin{aligned}F-\tau G\succeq 0\end{aligned}}}

Where the scalar ${\displaystyle \tau \geq 0}$ .

## The Data

The data is dependent on the type of problem being solved, and is used more as a tool to solve complex problems that were difficult to solve before.

## The LMI: S-Procedure

There exists a scalar ${\displaystyle \tau \geq 0}$  where:

{\displaystyle {\begin{aligned}F-\tau G\succeq 0\end{aligned}}}

## Conclusion:

The results from this LMI will help construct quadratic stability as quadratic stability requires matrix positivity on a subset. Examples of this implementation include creating a controller based on parametric, norm-bounded uncertainties for robust problems.