LMIs in Control/pages/S Procedure
LMIs in Control/pages/S Procedure
S-Procedure
The Optimization Problem
[edit | edit source]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 | edit source]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
[edit | edit source]There exists a scalar where:
Conclusion:
[edit | edit source]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 | edit source]External Links
[edit | edit source]- 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.