LMIs in Control/pages/S Procedure

LMIs in Control/pages/S Procedure

S-Procedure

The Optimization Problem

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

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

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: $x^{T}Fx\geq 0:\forall x\geq 0$. By using the defined problem above, an LMI can be constructed using the S-Procedure:

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

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:

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

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.

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.