Catálogo de publicaciones - libros

People observe various phenomena of nature and endeavor to comprehend them. The first step in that is a reflection of the phenomena by imagination and description. The reflexive process is a process of abstraction. In this process, the notion of “system” is of basic importance.

A system is defined to be a group of objects separated from the universe and having mutual relations.

Different physical entities can constitute system objects. If time is included among the system objects, their temporal properties or the system dynamics can be considered. The system dynamics is given by the time behavior of the system objects. The behavior is called the process.

A real physical system is represented by an ideal system created by human thinking and understanding. Mathematical representations of real systems are the most abstract and precise descriptions. Since the very beginning of its existence, mankind strives not only to know and to describe natural systems but also to govern and control them.

Control of a system is based on knowledge about the particular system. This knowledge is developed abstraction based on observation of the system. The observation is realized by measurements and if possible, by experimentation with the system.

The basic transition model and its derivatives can be very easily and transparently represented graphically. A platform for this representation is a mathematical graph.

We will follow the way of reasoning developed in the preceding chapters. Consider an event set Σ ={, ,...,} given for a DEDS. Further consider it to be in an initial state . The system behavior can be defined by all possible sequences (strings or words) of events that can occur in it starting from . It is assumed that an event occurs in a discrete point of time. Further it is assumed that just one event occurs in one discrete time point. The set of all finite and infinite sequences, which can be created from the elements of Σ including the empty sequence , is denoted as Σ*. The set that does not include is denoted as Σ , , Σ=Σ*\ where symbol “\” stands for the set subtraction.

Usually only a part of all possible sequences Σ* can occur in a given DEDS. Such a particular behavior of the DEDS is due to a subset of sequences from Σ* , , ⊆ Σ*. is supposed always to include the empty sequence (string, word) and is called a formal language. The formal language defines the behavior of a DEDS. Our attention is aimed at formal languages with respect to the above-introduced interpretation related to DEDS. A formal definition is useful in order to exactly communicate the idea of a formal language.

Consider the basic transition system Equation (1.19) by Manna and Pnueli described in Section 1.5 as a general model of a DEDS. As pointed out earlier, time is not explicitly considered in the model. The DEDS dynamics depends on events that appear in discrete time points. The events provoke changes of the system states. The relations and mutual influence of events and states with respect to control are studied in this chapter.

Finite automata are a classical tool used for many years for DEDS modeling. A finite automaton incorporates both principal system features - system states and system transitions in an abstract form. The basic definition of a finite automaton is given in the sequel.

Flow diagrams, sometimes called flow charts, are popular in programming. They are graphical tools for drawn-graphical visualization of algorithms to be programmed and executed by computers. Flow diagrams make final programming easier and help one minimize programming errors.

Flow diagrams have been developed and used for decades for transformation programs dealt with in Section 4.2. A flow diagram prescribes a sequence of computer operations forced by computer instructions. There are four basic elementary building blocks used in flow diagrams: operational block (Fig. 6.1a), decision block (Fig. 6.1b), start and end block (Fig. 6.1c), and subprogram block (6.1d). The blocks are connected with arrows determining the next operation block. The decision block is equipped with one or more conditions. A continuation of a program depends on the conditions. A cyclic repetition of a same group of operations can be specified by means of a decision block, too.

As described in Chapter 5, a finite automaton specifies a system by means of a set of states and a transition function. The arguments of the transition function are the state and event. We can speak about an actual state. The transition function assigns a state to an actual state. The assigned state is a next state while the actual state can be called the active present state. By repeating the assignments, a sequence of actual states is obtained. In the finite automaton there is always only one state active.

A system can often be broken down into subsystems. If it is required to describe activities of subsystems and their mutual relations, a finite automaton model can be cumbrous because each combination of subsystem states needs a separate state of the finite automaton. Another model known as a Petri net removes that inadequacy. Petri nets are named after a German mathematician C. A. Petri who first proposed a model of that kind (C. A. Petri, 1962). With Petri nets the main idea is to represent states of subsystems separately. Then, the distributed activities of a system can be represented very effectively. Many properties of the DEDS, , synchronization, concurrency, and choices can be well presented and analyzed using Petri nets. They can be used not only for the specification of the DEDS behavior but also the control design. However, Petri nets have various other uses.

Many important properties of Petri nets can be analyzed by means of Petri net reachability and coverability graphs. The reachability graph relates to the Petri net marking reachability. A frequently asked question is whether a given marking is reachable by a transition firing sequence. First let us define the transition firing sequence. The starting marking is important in that question.

Grafcet is designed as a specification tool for logic control to be implemented preferably on programmable logic controllers (PLC). It is a tool related closely to the binary safe Petri nets interpreted for control (Section 7.5).

Petri nets in the standard form as considered until now are an effective tool for DEDS modeling and control design. They enable one to specify powerfully the system function. Analysis methods are used for testing Petri net model properties and hence to check the correct system function (Desel 2000). Very often quantitative properties of the system behavior are another point of interest. In other words, a kind of system function performance or system efficiency is dealt with. In order to make the performance analysis feasible, additional values, parameters, and variables are used within the Petri nets (Čapkovič 1993, 1994, 1998). Another reason for additional values to be built in the Petri nets is to make the Petri net models more transparent and understandable even for large and complex DEDS. Such extensions are often denoted as high level Petri nets (Struhar 2000) or generalized Petri nets (Juhás 2000).

Standard Petri nets are not suitable for performance analysis. Undoubtedly, for performance analysis, an important system variable is time. Time enriches information by telling in what time or time interval a particular event occurs or should occur (Čapek and Hanzálek 2000). There are three ways to embed time into Petri nets. The first is to map the Petri net places into time intervals given as real or integer numbers; the second is to map them analogously into the Petri net transitions; and the last is to map into the arcs (Zhou and Venkatesh 1998). The options can be used separately or together.