Robust Semidefinite Programming and Its Application to Sampled-Data Control Yasuaki Oishi (Nanzan University) Udine, Italy August 26, 2011 Workshop on Uncertain Dynamical Systems * Joint work with Teodoro Alamo

1. Introduction 2 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.

3 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 4 nonlinear fn. Problem Assumption

DC-representation 5 convex Example

6 cf. [Adjiman--Floudas 96] Mild enough to assume

3. Proposed approach 7 Assumption: DC-representation is available convex Key step: obtaining bounds concaveconvex

Obtaining bounds 8 :concave :convex

9 concaveconvex

10 Approximate problem Number of LMIs Approximate solution cf. NP-hardness Conservative

Reduction of conservatism 11 Adaptive division

12 Quality of the approximation depends on the choice Measure of conservatism

13 Measure of conservatism Theorem

Example 14

Example 15

4. Application to sampled-data control 16 Analysis and design of such sampled-data systems hold sampler discrete [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]...

17 [O.--Fujioka 10] hold sampler discrete  Formulation into a robust SDP  Avoiding a numerical problem for a small sampling [O.--Fujioka 10] interval

6. Summary 18 Robust SDP problems with nonlinear param. dep. Conservative approach using DC-representations  Concave and convex bounds  Approximate problem  Reduction of conservatism Combination with the polynomial-based methods [Chesi--Hung 08][Peaucelle--Sato 09][O. 09] Optimization of the bounds w.r.t. some measure Application to sampled-data control