Robust semidefinite programming and its application to sampled data control
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Workshop on Uncertain Dynamical Systems. Robust Semidefinite Programming and Its Application to Sampled-Data Control. Yasuaki Oishi (Nanzan University) Udine, Italy August 26, 2011. * Joint work with Teodoro Alamo. 1. Introduction. Robust semidefinite programming problems.

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Robust semidefinite programming and its application to sampled data control

Workshop on Uncertain Dynamical Systems

Robust Semidefinite Programming andIts Application to Sampled-Data Control

Yasuaki Oishi (Nanzan University)

Udine, Italy

August 26, 2011

*Joint work with Teodoro Alamo


1 introduction
1. Introduction

Robust semidefinite programming problems

  • Optimization problems constrained by uncertain

  • linear matrix inequalities

  • Many applications in robust control

Robust SDP problem

  • Affine parameter dependence

  • Polynomial or rational par. dep.


This talk: general nonlinear parameter dependence

  • How to obtain the sufficient condition?

  • How to make the condition less conservative?

Key idea: DC-representations

“difference of two convex functions”

[Tuan--Apkarian--Hosoe--Tuy 00]

[Bravo--Alamo--Fiacchini--Camacho 07]


2 preparations
2. Preparations

Problem

nonlinear fn.

  • Assumption


Dc representation
DC-representation

convex

convex

Example


Example
Example

cf. [Adjiman--Floudas 96]

  • Mild enough to assume


3 proposed approach
3. Proposed approach

  • Assumption: DC-representation is available

convex

convex

  • Key step: obtaining bounds

concave

convex


Obtaining bounds
Obtaining bounds

:concave

:convex


concave

convex


Approximate problem

  • Approximate solution

  • Number of LMIs

cf. NP-hardness

  • Conservative


Reduction of conservatism
Reduction of conservatism

  • Adaptive division


  • Measure of conservatism





4 application to sampled data control
4. Application to sampled-data control

sampler

hold

discrete

discrete

  • Analysis and design of such sampled-data systems

[Fridman et al. 04][Hetel et al. 06][Mirkin 07][Naghshtabrizi et al. 08]

[Suh 08][Fujioka 09][Skaf--Boyd 09][O.--Fujioka 10][Seuret 11]...


[O.--Fujioka 10]

sampler

hold

discrete

discrete

  • Formulation into a robust SDP

  • Avoiding a numerical problem for a small sampling

interval

[O.--Fujioka 10]


6 summary
6. Summary

Robust SDP problems with nonlinear param. dep.

  • Conservative approach using DC-representations

    • Concave and convex bounds

    • Approximate problem

    • Reduction of conservatism

  • Optimization of the bounds w.r.t. some measure

  • Application to sampled-data control

  • Combination with the polynomial-based methods

[Chesi--Hung 08][Peaucelle--Sato 09][O. 09]


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