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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.
slide3

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

slide9

concave

convex

slide10

Approximate problem

  • Approximate solution
  • Number of LMIs

cf. NP-hardness

  • Conservative
reduction of conservatism
Reduction of conservatism
  • Adaptive division
slide12

Quality of the approximation

  • depends on the choice
  • 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]...

slide17

[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]