Copyright 0 IFAC Linear Time Delay Systems, Grenoble, France, 1998 QUASI-FINITE LINEAR DELAY SYSTEMS: THEORY AND APPLICATIONS M. Fliess· H. Mounier·· • Laboratoire des Signaux et Systemes, C.N.R.S.-Supelec, Plateau de Moulon, 91192 Gif-sur- Yvette, France, E-mail: fliessOlss.supelec.fr •• Laboratoire d'Automatique-Productique, Ecole des Mines de Nantes, 4, rue Alfred Kastler, La Chantrerie, 44307 Nantes, France, E-mail: mounierOlss.supelec.fr Resume: Dne classe particulierement simple de systemes lineaires a retards, qui se rattachent aux predicteurs de Smith, est definie, Oll la commandabilite et la stabilisation sont aisees. Un exemple elementaire pour les reseaux haut debits est examine. Abstract: A particularily simple class of linear delay systems, which are related to Smith predictors, is defined, on which controllability and stabilization are easy. An example for high speed networks is provided. Copyright ©J998 IFAC Keywords: Delay systems, modules, Smith predictors, high speed networks. INTRODUCTION Transfer matrices of the form x(t) = Ax(t) + Bu(t) y(t) = Gx(t - h) (2) x = Ax + Bu y = ex Then. P(s)e- h $ may be given the following three state-variable representations P(s)e- h $, h > 0 where P(s) is a rational transfer matrix, often occur in practice. The simplest and most common example certainly is the transfer function b:=:', which yields the classic Smith predictor (Smith, 1957) and also corresponds to the Broida-Stretch model (see, e.g., (Richalet, 1993)). Assume that P(s) is strictly proper and write its minimal realization (3) x(t) = Ax(t) + Bu(t - h) y(t) = Gx(t) (1) 169 and x(t) = Ax(t) + Bu(t - hd y(t) = Gx(t - h 2 ) where hi, h 2 > 0, hI + h 2 = h. Those three systems are quite different from an engineering or physical standpoint: the delays concern ei- ther the input, the output, or both. The con- trol, i.e., the stabilization of (1) requires new tools when compared to the well known theory of finite-dimensional systems, whereas constructing an asymptotic observer is easy. The situation is reversed with (2); (3) combines all the difficulties. We see that the transfer description of delay sys- tems is inherently ambiguous, although the same situation occurs with finite-dimensional systems only in more involved block diagrams (see, e.g., (Fliess and Bourles, 1996)). See also (Mounier and