The first edition of this book was published in 1992 by Wiley (ISBN 0 471 93609 X). Since this book is now out of print, and to answer the request of several colleagues, the authors have decided to make it available freely on the Web, while retaining the copyright, for the benefit of the scientific community. Copyright Statement This electronic document is in PDF format. One needs Acrobat Reader (available freely for most platforms from the Adobe web site) to benefit from the full interactive machinery: using the package hyperref by Sebastian Rahtz, the table of contents and all LATEX cross-references are automatically converted into clickable hyperlinks, bookmarks are generated automatically, etc.. So, do not hesitate to click on references to equation or section numbers, on items of thetableofcontents and of the index, etc.. One may freely use and print this document for one’s own purpose or even distribute it freely, but not commercially, provided it is distributed in its entirety and without modifications, including this preface and copyright statement. Any use of thecontents should be acknowledged according to the standard scientific practice. The

Abstract. We show that the minimal dimension of a linear realization over the (max,+) semiring of a convex sequence is equal to the minimal size of a decomposition of the sequence as a supremum of discrete affine maps. The minimal-dimensional realization of any convex realizable sequence can thus be found in linear time. The result is based on a bound in terms of minors of the Hankel matrix.

One of the open problems in the max-plus-algebraic system theory for discrete event systems is the minimal realization problem. In this paper we present some results in connection with the minimal realization problem in the max-plus algebra. First we characterize the minimal system order of a max-linear discrete event system. We also introduce a canonical representation of the impulse response of a max-linear discrete event system. Next we consider a simpli#ed version of the general minimal realization problem: the boolean minimal realization problem, i.e., we consider models in which the entries of the system matrices are either equal to the max-plus-algebraic zero element or to the max-plus-algebraic identity element. We give a lower bound for the minimal system order of a max-plus-algebraic boolean discrete event system. We show that the decision problem that corresponds to the boolean realization problem (i.e., deciding whether or not a boolean realization of a given order exists) ...

We consider a system of uniform recurrence equations (URE) of dimension one. We show how its computation can be carried out using minimal memory size with several synchronous processors. This result is then applied to register minimization for digital circuits and parallel computation of task graphs.

We show that the set of realizations of dimension n of a max-plus linear sequence is a nite union of polyhedral sets, which can be computed from any realization of the sequence. This yields an (expensive) algorithm to solve the max-plus minimal realization problem. These results are derived from general facts on rational expressions over idempotent commutative semirings: we show more generally that the set of values of the coecients of a commutative rational expression in one letter that yield a given max-plus rational series, is a nite union of polyhedral sets.

The purpose of this paper is to examine the eigen-properties for a particular complex field mapping proposed in Bundell [2] to represent the 2-dimensional Time-Event State-space proposed by Cohen et al [5]. This work extends the definition of the characteristic equation developed by Olsder & Roos [7] for the Max algebra to the Information (Time-Event) State-space and examines an approach to its solution. An examples is given to illustrate the approach. 1. INTRODUCTION The specification and modelling of timing properties for real-time systems has become a major area of topical research. This heightened level of activity should benefit from the convergence of work by the real-time computation community and, more recently, the control engineering community. It is interesting to note the growing interest in this multi-disciplinary area of endeavour [1] and the importance that may researchers attach to the cross-fertilisation of ideas from one domain to another. This paper very much follow...

We consider a system of uniform recurrence equations (URE) of dimension one. We show how its computation can be carried out using minimal memory size with several synchronous processors. This result is then applied to register minimization for digital circuits and parallel computation of task graphs.