Proceedings of the First Multidisciplinary International Symposium on Positive Systems: Theory and Applications (POSTA 2003), Rome, Italy, August 28–30, 2003
The theory of positive systems can be extended to random positive systems, along lines originally developed by Bellman and extended by Furstenberg and Kesten. This theory, in turn, can be extended to nonlinear random positive systems that are homogeneous of degree one. These results generalize the Frobenius–Perron theory which defines a maximal growth rate for linear positive systems.
The purpose of the lecture associated with this paper is to present problems, concepts, and theorems of control and system theory for a subclass of the rational positive systems of which examples have been published as models of biochemical cell reaction networks.
Network Calculus is a set of recent developments, which provide a deep insight into flow problems encountered in networking. It can be viewed as the system theory that applies to computer networks. Contrary to traditional system theory, it relies on max-plus and min-plus algebra. In this paper, we show how a simple but important fixed-point theorem (residuation theorem) in min-plus or max-plus algebra...
We present a synthesis of recent results concerning reachability and invariance problems for max-plus linear dynamical systems. Semigroup membership and orbit problems, reachable spaces, and A,B invariant spaces, are discussed.
We consider the modelling of urban bus networks in dioid algebras. In particular, we show that their dynamic behavior can be modeled by a Min-Max recursive equation.
This paper presents results concerning the relations between a propositional modal logic (NK-logic) and the algebra of dioids. The technique of analytic tableau, a well known proof technique in logic systems, is used in combination with the NK-logic to verify specifications written in dioid algebra. The concept of terminality, herein introduced, allows the establishment of important relations that...
Linear systems over naturally ordered dioids are other kinds of positive systems than the usual ones over semiring ($\mathbb{R}_+,+,\cdot$). In this short paper we study some monotonicity concepts of linear systems over dioids inspired by results on monotonicity of Markov chains which are also particular cases of positive systems. We derive a necessary and sufficient condition for monotonicity in...
This paper deals with control of (max,+)-linear systems when a disturbance acts on system state. In a first part we synthesize the greatest control which allows to match the disturbance action. Then, we look for an output feedback which makes the disturbance matching. Formally, this problem is very close to the disturbance decoupling problem for continuous linear systems.
Petri nets (PNs) are a well-known family of formalisms whose definition immediately sets them, in a broad sense, as positive systems. Although they are originally discrete event models, their relaxation through continuization transforms them in continuous models. In this paper one of the most relevant timing interpretations of continuous PNs, unforced infinite servers semantics continuous PNs, is...
An autonomous continuous Petri net is a model in which the time is not involved, the marking is a vector or non-negative real numbers, and a transition firing corresponds to some “quantity of firing” (positive number) compatible with the current marking. The paper presents the new concepts of OG-firing (standing for “at one go firing”) and macro-marking. From these concepts, a reachability...
In this paper the possibility of modeling positive systems by means of Hybrid Petri Nets (HPN) is discussed. Hybrid (or Fluid) Petri Nets are Petri net (PN) based model with two classes of places: discrete places that carry a natural number of distinct objects (tokens), and fluid places that hold a positive amount of fluid, represented by a real number. The HPN formalism we present in this work allows...
In this paper we show how First-Order Hybrid Petri nets, an hybrid positive model that combines fluid and discrete event dynamics, may be efficiently used to simulate the dynamic concurrent activities of manufacturing systems. In particular we deal with the performance analysis via simulation of a mineral water bottling plant according to the variations of the production controlling input parameters...
Parameters characterizing the internal behaviour of biological and physiological systems are usually not directly accessible to measurement. Their measurement is usually approached indirectly as a parameter estimation problem. A dynamic model describing the internal structure of the system is formulated and an input-output experiment is designed for model identification. Identifiability is a fundamental...
Based on recent developments for new measurement technologies that enable researches to get quantitative information on intracellular processes, the setup of very detailed models describing metabolism as well as regulatory networks becomes very popular. However, biochemical networks are rather complex including many feed-forward and feedback loops. In this contribution we propose an interdisciplinary...
The parameters of cooperative models are estimated in a boundederror context, i.e., all uncertain quantities are assumed to be bounded, with known bounds. Guaranteed estimation is then the characterization of the set of all parameter vectors that are consistent with the model and experimental data, given these bounds. Interval techniques provide an approximate but guaranteed enclosure of this set...
The analysis of genetic regulatory networks will much benefit from the recent upscaling to the genomic level of experimental methods in molecular biology. In addition to high-throughput experimental methods, mathematical and bioinformatics approaches are indispensable for the analysis of genetic regulatory networks. Given the size and complexity of most networks of biological interest, an intuitive...
Building upon the logical approach developed by the group of R. Thomas in Brussels, we are defining a rigorous mathematical framework to model genetic regulatory graphs. Referring to discrete mathematics and graph-theoretic notions, our formal approach supports the development of a software suite in Java, GIN-sim, which allows the qualitative simulation and the analysis of the dynamics of regulatory...
In this paper we propose a new gene network reconstruction (or identification) scheme which takes advantage of the sparseness of a gene network using a decomposition of the given linear dynamical system describing the network, into two positive linear systems. First, we will describe how gene networks can be modelled as linear systems and an “ideal” situation is considered in order to state an identification...
A directed graph with cities as vertices and arcs determined by outgoing (or return) travel represents the mobility component in a population of individuals who travel between n cities. A model with 4 epidemiological compartments in each city that describes the propagation of a disease in this population is formulated as a system of 4n 2 ordinary differential equations. Terms in the system...
This paper deals with the stability analysis of a simple metabolic system with feedback inhibition. The system is a sequence of monomolecular enzymatic reactions. The last metabolite acts as a feedback regulator for the first enzyme of the pathway. The enzymatic reactions of the pathway satisfy Michaelis-Menten kinetics. The inhibition is described by an hyperbolic model. Without inhibition, it is...