LMIs in Control/pages/S Procedure

S-Procedure

The Optimization Problem
In general procedures, considering following quadratic function $$ F_0(x) : \mathbb{R}^{n} \to \mathbb{R} $$, $$ F_i(x) : \mathbb{R}^{n} \to \mathbb{R} $$ where $$ i = 1,...,m $$. The inequality $$ F_0(x) \leq 0 $$ is satisfied when all $$ F_i(x) \geq 0 $$.



\begin{align} F_0(x)+\sum^{m}_{i=1}\tau_i F_i(x) \leq 0 \end{align}$$

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

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



\begin{align} F - \tau G \succeq 0 \end{align}$$

Where the scalar $$ \tau \geq 0 $$.

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 $$ \tau \geq 0 $$ where:



\begin{align} F - \tau G \succeq 0 \end{align}$$

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.

Implementation

 * S-Procedure Example