LMIs in Control/pages/S Procedure

LMIs in Control/pages/S Procedure

S-Procedure

The Optimization Problem

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

$\begin{array}{c}{F}_0(x)+\sum _{i=1}^m{\tau }_i{F}_i(x)\le 0\end{array}$

Where $x\in {ℝ}^n$, and ${\tau }_i\in {ℝ}_{\ge 0}$

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

$\begin{array}{c}F-\tau G⪰0\end{array}$

Where the scalar $\tau \ge 0$.

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 $\tau \ge 0$ where:

$\begin{array}{c}F-\tau G⪰0\end{array}$

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

External Links

• 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.