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SeDuMi Interface A tool for solving LMI problems with SeDuMi

SeDuMi Interface A tool for solving LMI problems with SeDuMi. Yann labit, Dimitri Peaucelle Didier Henrion LAAS-CNRS, Toulouse, France. Motivation. Importance of Linear matrix Inequalities (LMI) in Control theory and applications Limitations of existing tools: Problem size

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SeDuMi Interface A tool for solving LMI problems with SeDuMi

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  1. SeDuMi InterfaceA tool for solving LMI problems with SeDuMi Yann labit, Dimitri Peaucelle Didier Henrion LAAS-CNRS, Toulouse, France

  2. Motivation • Importance of Linear matrix Inequalities (LMI) in Control theory and applications • Limitations of existing tools: • Problem size • Computation time • Convivial, relevant display • Build together an adapted software

  3. Solver & Interface objectives • Selected solver : SeDuMi. • Necessity to write an Interface. • Not a GUI tool.

  4. SeDuMi : not ideal nor definitive but • Low computational complexity • Sparse data format • Works with Matlab • Free software • LMI and equality constraints • Complex valued constraints • Work in progress Jos F. Sturm

  5. Interface to build canonical expressions for optimisation tools

  6. Existing Interfaces/Parsers • LMIlab, LMITOOL, sdpsol, YALMIP • Some critiques • Lack of functionalities (LMIlab, sdpsol) • Slow conversion (LMITOOL,sdpsol) • Difficulties to create Matlab functions to generate LMI problems (LMITOOL, sdpsol) • Difficulties to analyse the obtained solution… not a GUI and no symbolic declarations

  7. Interface dedicated to LMIs • Simplified declaration • Structured variables : symmetric, diagonal, Hermitian… • Structured constraints : block decomposition, Kronecker… • Predefined objectives : trace, log(det)… • Analysis of the solution : margin on constraints, matrix format... • Software constraints • Speed : simple algebaric manipulations • Memory space : sparse format • Open to modifications : Matlab free source code

  8. Using SeDuMi Interface : An example - state feedback Optimal controller Optimal norm

  9. Step 1 : Name the LM problem • The LM problem = a Matlab object

  10. Step 2 : declare matrix variables Symmetric, diagonal, hermitian, structured…

  11. Step 3 : declare inequalities Symmetric terms automatically added Possible to define Kronecker products

  12. Step 4 : declare the objective Possible to define :

  13. Step 5 : solve the LM problem

  14. Step 6 : analyse the result Feasibility margins on each constraints: Matrix formatted result:

  15. Other features of the Interface • Matrix equalities • Complex valued constraints • Complex valued variables • Radius on the vector of decision variables • Adapted tuning of SeDuMi options Acknowledgements to K. Taitz

  16. Future evolutions • Warm-start with feasible solution • Pre-conditioning of the optimisation problem • Other predefined options (user’s feedback) • Platform incorporating other solvers • Predefined LM problems for Automatic control • Robust analysis • Performance (pole location, , …) • State and Output feedback • …

  17. Conclusion http://www.laas.fr/~peaucell/sedumiint mailto:peaucelle@laas.fr

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