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Advances in Robust Semidefinite Programming for Uncertain Dynamical Systems

This workshop presentation by Yasuaki Oishi from Nanzan University focuses on robust semidefinite programming (SDP) applied to uncertain dynamical systems. The talk introduces robust SDP problems constrained by uncertain linear matrix inequalities, emphasizing the significance of affine and nonlinear parameter dependencies. Key techniques discussed include DC-representations and methods for reducing conservatism in conditions. Applications to sampled-data control systems are also explored, illustrating how these mathematical frameworks enhance system stability and performance in realistic scenarios.

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Advances in Robust Semidefinite Programming for Uncertain Dynamical Systems

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  1. 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

  2. 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.

  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]

  4. 2. Preparations Problem nonlinear fn. • Assumption

  5. DC-representation convex convex Example

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

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

  8. Obtaining bounds :concave :convex

  9. concave convex

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

  11. Reduction of conservatism • Adaptive division

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

  13. Measure of conservatism Theorem

  14. Example

  15. Example

  16. 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]...

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

  18. 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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