**Reference:**

B. De Schutter,
*Max-Algebraic System Theory for Discrete Event Systems*. PhD
thesis, Faculty of Applied Sciences, K.U.Leuven, Leuven, Belgium, ISBN
90-5682-016-8, 331 pp., Feb. 1996.

**Abstract:**

Discrete event systems (DESs) are systems in which the state changes
only at discrete points in time in response to the occurrence of
particular events. Typical examples of DESs are: flexible
manufacturing systems, telecommunication networks, parallel processing
systems and traffic control systems. One of the frameworks that can be
used to model and to analyze certain types of DESs is the max-plus
algebra, which has maximization and addition as basic operations. In
this thesis we develop tools for solving some fundamental problems in
the max-algebraic system theory for DESs.

First we introduce a mathematical programming problem: the Extended
Linear Complementarity Problem (ELCP). We develop an algorithm to find
all the solutions of an ELCP.

We show that the problem of solving a system of multivariate
max-algebraic polynomial equalities and inequalities is equivalent to
an ELCP. This enables us to solve many other max-algebraic problems
such as computing max-algebraic matrix factorizations, performing
max-algebraic state space transformations, computing state space
realizations of the impulse response of a max-linear time-invariant
DES, computing max-algebraic singular value decompositions and QR
decompositions, and so on.

We also study the max-algebraic characteristic polynomial and state
space transformations for max-linear time-invariant DESs. Next we
develop a method to solve the minimal state space realization problem
for max-linear time-invariant DESs. First we use our results on the
max-algebraic characteristic polynomial to develop a procedure to
determine a lower bound for the minimal system order of a max-linear
time-invariant DES. Then we show that the ELCP can be used to compute
all fixed order partial state space realizations and all minimal state
space realizations of the impulse response of a max-linear
time-invariant DES.

Finally we prove the existence of max-algebraic analogues of two basic
matrix decompositions from linear algebra: the singular value
decomposition and the QR decomposition.

PhD thesis: pdf file (1.72 MB)

List of errata (as of November 30, 2008): pdf file (146 KB)

@phdthesis{DeS:96-02,

author={B. {De Schutter}},

title={Max-Algebraic System Theory for Discrete Event Systems},

school={Faculty of Applied Sciences, K.U.Leuven},

address={Leuven, Belgium},

month=feb,

year={1996}

}

@misc{DeS:96-02-errata,

author={B. De Schutter},

title={Errata for the PhD thesis of Bart De Schutter ``Max-Algebraic System Theory for Discrete Event Systems''},

note={Last update: November 30, 2008.}

}

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