Stability analysis of time-varying delay systems in quadratic separation framework
by
Frederic Gouaisbaut
University of Toulouse, LAAS-CNRS
Coauthors: Yassine Ariba, Dimitry Peaucelle

Time-delay systems and especially their stability have been intensively studied since several decades. The reasons are not only the challenging theoretical issues of this problem, but also because the delay phenomenon is an important applied problem [9], inherent to data transportation, propagation time as well as processing time [2]. Moreover, time-delays are commonly used to model neglected dynamics. In the case of constant delay and unperturbed linear systems, efficient criteria based on roots location [10] allow to find the exact region of stability with respect to the value of the delay. For the case of uncertain linear systems, i.e. for proving robust stability, the problem has been partially solved, either by using Lyapunov functionals [5, 12] or robustness tools (small gain theory [5], IQCs [8] or quadratic separation [4]). All resulting stability conditions are based on convex optimization (Linear Matrix Inequality framework [1]) and allow to conclude on stability region with respect to the delay and/or the uncertainty. These conditions are conservative (produce inner approximations of the stability regions). To reduce conservatism new techniques have been recently proposed [4]. Conservatism reduction is then obtained by introducing new decision variables in the optimization problems, thus at the expense of increasing the numerical burden.

But the upper cited results with reduced conservatism are up to now limited to constant delays. For time-varying delays, some results based on either Lyapunov-Krasovskii or IQC methodologies have been successfully exploited [3][8][6] but, however, reveal to be very conservative in practice. This papers’ objective is to assess delay dependent stability of time-varying delay systems based on the quadratic separation framework [7][11]. Criteria are derived and expressed in terms of Linear Matrix Inequalities (LMIs) which may be solved efficiently with Semi-Definite Programming (SDP) solvers. The derivation of results is based on redundant system modeling. Indeed, based on known interactions between delays, their variations and derivatives, redundant equations are introduced to construct a new modeling of the delay systems. To this end, an augmented state is considered which is composed of the original state vector and its derivatives. The properties of the the augmented, descriptor form, system is that when applying robustness techniques criteria with reduced conservatism are produced. Since robustness techniques work by embedding infinite dimensional or non-linear operations in L2-norm bounded domains, whatever their dynamic characteristics, the conservatism reduction is achieved by extracting the information on dynamics before doing the embedding. Produced results for time-varying delay systems are achieved by extracting information on relationships between the delay h, its derivative h and the derivatives of the states x, x. The paper concludes with illustrative academic examples. Conservatism reduction is discussed and numerical complexity commented.

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Date received: April 1, 2008