1. SeDuMi Interface A 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
LMIs and LMEs are matrix dependent expressions.
Optimisation tools use vectors of decision variables
E.g.

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
>> quiz = sdmpb(‘Hinfty state feedback’)
LM problem: Hinfty state feedback
no matrix variable
no equality constraint
no inequality constraint
no linear objective
Unsolved
>> whos quiz
Name Size Bytes Class
quiz 1x sdmpb object

10. Step 2 : declare matrix variables
>> [quiz, Yindex] = sdmvar(quiz, m, n, 'Y');
>> [quiz, gindex] = sdmvar(quiz, 1, 1, 'gamma');
>> [quiz, Qindex] = sdmvar(quiz, n, 's', 'Q=Q’’')
LM problem: Hinfty state feedback
matrix variables: index name Y
gamma
Q=Q’
no equality constraint
no inequality constraint
no linear objective
unsolved
Symmetric, diagonal, hermitian, structured…

11. Step 3 : declare inequalities
>> [quiz, lmi1] = sdmlmi(quiz, n, 'Q>0');
>> quiz = sdmineq(quiz, -lmi1, Qindex);
>> [quiz, lmi2] = sdmlmi(quiz, [n p], 'Hinfty');
>> quiz = sdmineq(quiz, [lmi2 1 1], Qindex, A, 1);
>> quiz = sdmineq(quiz, [lmi2 1 1], Yindex, B, 1);
>> quiz = sdmineq(quiz, [lmi2 1 1], , Bi);
>> quiz = sdmineq(quiz, [lmi2 2 1], Qindex, Ci, 1);
>> quiz = sdmineq(quiz, [lmi2 2 1], Yindex, Di, 1);
>> quiz = sdmineq(quiz, [lmi2 2 2], gindex);
Symmetric terms automatically added
Possible to define Kronecker products

12. Step 4 : declare the objective
>> quiz = sdmobj(quiz, gindex, 1, 1, ‘gamma’)
LM problem: Hinfty state feedback
matrix variables: index name Y
gamma
Q=Q’
no equality constraint
inequality constraints: index meig name
Inf Q>0
Inf Hinfty
maximise objective: gamma
unsolved
Possible to define :

13. Step 5 : solve the LM problem
>> quiz = sdmsol(quiz)
... computation ...
LM problem: Hinfty state feedback
matrix variables: index name Y
gamma
Q=Q’
no equality constraint
inequality constraints: index meig name
eps Q>0
eps Hinfty
maximise objective: gamma = -27.3
feasible

14. Step 6 : analyse the result
Feasibility margins on each constraints:
inequality constraints: index meig name
eps Q>0
eps Hinfty
maximise objective: gamma = -27.3
feasible
Matrix formatted result:
>> K = quiz(Yindex)/quiz(Qindex)
K = 1.0e+03 *

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