Robust Semidefinite Programming and Its Application to Sampled-Data Control

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Robust Semidefinite Programming andIts
Application to Sampled-Data Control
Workshop on Uncertain Dynamical Systems

• Yasuaki Oishi (Nanzan University)
• Udine, Italy
• August 26, 2011

Joint work with Teodoro Alamo
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1. Introduction
Robust semidefinite programming problems

• Optimization problems constrained by uncertain
• linear matrix inequalities

• Many applications in robust control

Robust SDP problem
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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--Fia
cchini--Camacho 07
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2. Preparations
Problem

• Assumption

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DC-representation
convex
convex
Example
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Example
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3. Proposed approach

• Key step obtaining bounds

concave
convex
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Obtaining bounds
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• Approximate solution

• Number of LMIs

cf. NP-hardness
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Reduction of conservatism

• Adaptive division

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• Quality of the approximation
• depends on the choice

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• Measure of conservatism

Theorem
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Example
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Example
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4. Application to sampled-data control

• Analysis and design of such sampled-data systems

Fridman et al. 04Hetel et al. 06Mirkin
07Naghshtabrizi et al. 08 Suh 08Fujioka
09Skaf--Boyd 09O.--Fujioka 10Seuret 11...
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O.--Fujioka 10

• Formulation into a robust SDP
• Avoiding a numerical problem for a small sampling

interval
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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

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