TAILIEUCHUNG - DELAY-DEPENDENT ROBUST STABILITY OF TIME DELAY SYSTEMS

Abstract: In this note, we provided an improved way of constructing a LyapunovKrasovskii functional for a linear time delay system. This technique is based on the reformulation of the original system and a discretization scheme of the delay. A hierarchy of Linear Matrix Inequality based results with increasing number of variables is given and is proved to have convergence properties in terms of conservatism reduction. | Author manuscript published in 5th IFAC Symposium on Robust Control Design Toulouse France 2006 DELAY-DEPENDENT ROBUST STABILITY OF TIME DELAY SYSTEMS Frederic Gouaisbaut Dimitri Peaucelle LAAS-CNRS 7 av. du colonel Roche 31077 Toulouse FRANCE Email gouaisbaut peaucelle @ hal-00401025 version 2 - 2 Jul 2009 Abstract In this note we provided an improved way of constructing a Lyapunov-Krasovskii functional for a linear time delay system. This technique is based on the reformulation of the original system and a discretization scheme of the delay. A hierarchy of Linear Matrix Inequality based results with increasing number of variables is given and is proved to have convergence properties in terms of conservatism reduction. Examples are provided which show the effectiveness of the proposed conditions. Keywords Linear time delay systems Stability Robustness 1. INTRODUCTION During the last decades stability of linear time delay systems have attracted a lot of attention see Moon et al. 2001 Park 1999 Xu and Lam 2005 Fridman and Shaked 2002a and references therein. The main approach relies on the use of a Lyapunov Krasovskii functional or a Lyapunov Razumikhin function. It leads to the so called delay dependent criteria which are expressed in terms of LMIs linear matrix inequalities and then easily solved using dedicated solvers. Generally all these approach have to tackle with two main difficulties. The first one is the choice of the model transformation which is closely related to a choice of Lyapunov Krasovskii functional see Kolmanovskii and Richard 1999 for a complete classification. The second problem lies on the bound of some cross terms which appears in the derivative of the Lyapunov functional see Park 1999 Moon et al. 2001 Gu et al. 2003 . The present paper brings a contribution to the first issue by appropriate redundant modeling it introduces new types of Lyapunov Krasovskii functionals. The methodology may be seen as similar to that in Peaucelle et al.

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