# LMIs in Control/pages/S Procedure

LMIs in Control/pages/S Procedure

**S-Procedure**

**The Optimization Problem**Edit

In general procedures, considering following quadratic function , where . The inequality is satisfied when all .

Where , and

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

Where the scalar .

**The Data**Edit

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-ProcedureEdit

There exists a scalar where:

**Conclusion:**Edit

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**Edit

## External LinksEdit

- 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.