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

LMIs and LMEs are matrix dependent expressions.

Optimisation tools use vectors of decision variables

E.g.

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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

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max

'''

IDC

DCBBBBAA

γYQ

YQYYQQQQ

γ

1YQK γ

9

Step 1 : Name the LM problem

The LM problem = a Matlab object

>> quiz = sdmpb(‘Hinfty state feedback’)LM problem: Hinfty state feedbackno matrix variableno equality constraintno inequality constraintno linear objectiveUnsolved>> whos quiz Name Size Bytes Class quiz 1x1 3358 sdmpb object

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Step 2 : declare matrix variables

Symmetric, diagonal, hermitian, structured…

>> [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 1 Y 2 gamma 3 Q=Q’ no equality constraintno inequality constraintno linear objectiveunsolved

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Step 3 : declare inequalities

Symmetric terms automatically addedPossible to define Kronecker products

>> [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], 0, 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);

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Step 4 : declare the objective

Possible to define :

>> quiz = sdmobj(quiz, gindex, 1, 1, ‘gamma’)LM problem: Hinfty state feedback matrix variables: index name 1 Y 2 gamma 3 Q=Q’ no equality constraintinequality constraints: index meig name 1 -Inf Q>0 2 -Inf Hinftymaximise objective: gammaunsolved

)(or XX Mtraceee rl

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Step 5 : solve the LM problem

>> quiz = sdmsol(quiz) ... computation ...LM problem: Hinfty state feedback matrix variables: index name 1 Y 2 gamma 3 Q=Q’ no equality constraintinequality constraints: index meig name 1 eps Q>0 2 eps Hinftymaximise objective: gamma = -27.3feasible

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Step 6 : analyse the result

inequality constraints: index meig name 1 eps Q>0 2 eps Hinftymaximise objective: gamma = -27.3feasible

Feasibility margins on each constraints:

Matrix formatted result:>> K = quiz(Yindex)/quiz(Qindex)K = 1.0e+03 * -2.5989 -8.5124 -0.1898 -4.0176

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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– …

H 2H

17

Conclusion

http://www.laas.fr/~peaucell/sedumiint

mailto:peaucelle@laas.fr