CPU automated control systems and industrial safety. Discrete and continuous models Classification of models and simulations

model material discontinuous discrete

We will assume that it is possible, at least in principle, to establish and characterize in some description language (for example, by means of mathematics) the dependence of each of the output variables on the input ones. The relationship between the input and output variables of the modeled object can, in principle, be characterized graphically, analytically, i.e. through some general formula, or algorithmically. Regardless of the form of representation of the construct that describes this relationship, we will call it an input-output operator and denote it by B.

Let M=M(X,Y,Z), where X is the set of inputs, Y is the set of outputs, Z is the system states. This can be represented schematically: X Z Y.

Let us now consider the most significant internal properties of objects of different classes from a modeling point of view. In this case, you will have to use the concept of structure and parameters of the modeled object. Structure is understood as a set of components and connections taken into account in the model, contained within an object, and after formalizing the description of the object, a type of mathematical expression that connects its input and output variables (for example: y=au+bv). Parameters are quantitative characteristics of the internal properties of an object, which are reflected by the adopted structure, and in a formalized mathematical model they are coefficients (constant variables) included in the expressions that describe the structure (a and b).

Continuity and discreteness.

All those objects whose variables (including, if necessary, time) can take on an uncountable set of values ​​that are arbitrarily close to each other are called continuous or continual. The vast majority of real physical and theoretical objects, the state of which is characterized only by macroscopic physical quantities (temperature, pressure, speed, acceleration, current strength, electric or magnetic field strength, etc.) have the property of continuity. Mathematical structures that adequately describe such objects must also be continuous. Therefore, in the model description of such objects, the apparatus of differential and integro-differential equations is used mainly. Objects whose variables can take on a certain, almost always finite number of previously known values ​​are called discrete. Examples: relay contact switching circuits, PBX switching systems. The basis for a formalized description of discrete objects is the apparatus of mathematical logic (logical functions, Boolean algebra apparatus, algorithmic languages). In connection with the development of computers, discrete methods of analysis have also become widespread for the description and study of continuous objects.

The property of continuity and discreteness is expressed in the structure of sets (collections) to which state parameters, process parameters and inputs and outputs of the system belong. Thus, the discreteness of the sets Z, T, X, Y leads to a model called discrete, and their continuity leads to a model with continuous properties. Discreteness of inputs (impulses of external forces, stepwise influences, etc.) in the general case does not lead to discreteness of the model as a whole. An important characteristic of a discrete model is the finiteness or infinity of the number of system states and the number of values ​​of output characteristics. In the first case, the model is called discrete finite. The discreteness of the model can also be either a natural condition (the system abruptly changes its state and output properties) or an artificially introduced feature. A typical example of the latter is replacing a continuous mathematical function with a set of its values ​​at fixed points.

Continuous mathematical models

To implement MMs represented by PDEs or ODE systems, numerical methods of continuous mathematics are used, therefore the considered MMs are called continuous.

In Fig. Figure 1 shows the transformations of continuous MMs in the process of transition from the original formulations of problems to working programs, which are sequences of elementary arithmetic and logical operations. Arrows 1, 2 and 3 show transitions from a description of the structure of objects at the corresponding hierarchical level to the mathematical formulation of the problem. Discretization (4) and algebraization (5) of PDEs in spatial variables are carried out using finite difference methods (FDM) or finite element methods (FEM). The application of MKR or FEM to stationary PDEs leads to a system of algebraic equations (AE), and to non-stationary PDEs to a system of ODEs. Algebraization and discretization of the ODE system with respect to the variable t are carried out by methods of numerical integration. For nonlinear ODEs (6), this transformation leads to a system of nonlinear AEs; for linear ODEs (7), to a system of linear algebraic equations (LAEs). Nonlinear EAs are solved by iterative methods. Arrow 8 corresponds to the solution by the Newton method, based on the linearization of equations, arrow 9 - by the Seidel, Jacobi, simple iteration methods, etc. The solution of the LAU system is reduced to a sequence of elementary operations (10) using the Gaussian methods or LU decomposition.

Rice. 1

Continuous MMs and the methods of computational mathematics used for their analysis have become widespread in CAD systems of various industries.

The creation of a technique for automatically generating mathematical models of systems made it possible to automate the procedures for analysis and verification of a wide class of technical objects. The invariant nature of this technique led to the development on its basis of methods and algorithms implemented in many PMCs for the design of electronic, mechanical, hydraulic, thermal power devices and systems. Such methods of MM formation are known as the nodal method, the contour method, and the method of state variables.

Discrete mathematical models

A discrete mathematical model is a model in which certain variables are discretized. Let us consider MMs in which the dependent variables characterizing the state of the modeled object are discrete.

System design at the functional-logical and system levels is based on the use of discrete MMs. When modeling in functional-logical design subsystems, the same assumptions are made as when modeling analog systems at the upper levels. In addition, the modeled object is represented by a set of interconnected logical elements, the states of which are characterized by variables that take values ​​in a finite set. In the simplest case, this is the set (0, 1). Continuous time t is replaced by a discrete sequence of moments in time tк, with the duration of the tact. Consequently, the mathematical model of the object is a finite automaton (FA). The functioning of the spacecraft is described by the system of logical equations of the spacecraft

At the system level of system design, queuing system (QS) models are predominantly common. Such models are characterized by the fact that they display objects of two types - service requests and service devices (OA). When designing an aircraft, the applications are the tasks to be solved, and the servicing devices are the aircraft equipment. The request may be in the “maintenance” or “waiting” state, and the servicing device may be in the “free” or “busy” state. The state of the QS is characterized by the states of its OA and requests. A change of state is called an event. QS models are used to study the processes occurring in this system when flows of applications are submitted to the inputs. These processes are represented as sequences of events. Based on the results of the study, the most important output parameters of the system are determined: productivity, throughput, probability and average time for solving problems, equipment load factors.

The emergence of parallel and pipeline systems, the need to simulate the functioning processes of not only hardware, but also software, led to the emergence of a class of discrete MMs called Petri nets. Petri nets can be used to model the design of a wide range of systems and networks at the functional, logical and system levels.

Petri nets and QS systems are widely used to describe the functioning of production sites, lines and workshops aimed at multi-product production. Petri nets are an effective tool for developing CAD systems themselves. These networks can serve as models of algorithms for the functioning of various discrete automation devices.

Discrete and continuous models.

Structural and functional models.

If models of the first type reflect the structure (structure) of the system being studied, which is a set of interconnected elements of the system, then in functional models attention is paid not to describing the structure of the system, but to a quantitative description of how this system reacts to external influences. In this case, the resulting model is called a “black box”. Structural models are usually built for well-structured systems. Functional models are built mainly for well-structured processes. It is also possible to combine these two types of models, resulting in a hybrid model that allows describing weakly structured systems and processes. An example of such models are system-dynamic models designed to describe environmental and economic processes. Structural models are used, for example, in the theory of the firm to study monopoly or consumer choice. An example of the application of functional models is the theory of production functions.

This division of models comes from dividing all quantities into discrete, taking values ​​at a finite number of points of the selected interval, and continuous, taking values ​​over the entire interval. Of course, an intermediate case is also possible. As a rule, most mathematical models allow both discrete and continuous interpretation. If in the discrete case the models are described in the language of sums and finite differences, then in continuous models - in the language of integrals and infinitesimal increments. As an example of discrete economic and mathematical models, we can cite widely used models associated with integer programming, mathematical game theory, and network planning. Continuous models include various models of mathematical economics, including market equilibrium, and many optimization models.

Linear and nonlinear models. This division of models comes from the nature of the relationships between the elements of the system. If in linear models a linear relationship is assumed between the variables that describe the model, then in nonlinear models there are connections between elements specified by nonlinear functions. An example of the use of linear and nonlinear models in economics is the solution of linear and, accordingly, nonlinear programming problems. If linear models, as a rule, describe simple systems, then nonlinear models, which include the majority of system dynamic models, describe complex systems. It is also possible to identify mixed models, an example of which are weakly nonlinear models.

The discreteness of the spacecraft model in space is an advantage from the point of view of mathematics and computational procedures. But from the point of view of practical applications this is a disadvantage. Sometimes the focus of the study is changes in the width of the opening or corridor within 5-15 cm at the site. Due to the larger cell size, CA models are insensitive to such changes in the linear dimensions of the object. Problems arise with the “arrangement” of furniture in such a discrete space (for example, this is relevant for a kindergarten, where the dimensions of the furniture in most cases are not a multiple of the size of the cell, and the area of ​​the premises is very limited). Also in CA models it is difficult to assign different sizes and shapes to particles.

In addition, in the discrete model, the particle can move only in one of four directions, since the field is divided into cells.

The disadvantage of the continuous approach is that it is based on the fact that the movement of people is described using differential equations. Determining the right-hand sides of these equations is quite difficult.

In addition, there are positive aspects of these models. The discrete model makes it possible to reproduce various phenomena of the physical aspect of people's movement: merging, reformation (spreading, compaction), non-simultaneous merging of flows, formation and resorption of clusters, flow around turns, movement in rooms with a developed internal layout, countercurrents and intersecting flows. It is possible to take into account changes in visibility, people's awareness of the building's layout, avoiding obstacles in advance, and using various movement strategies (shortest path and shortest time). And continuous models allow you to take into account the mass and speed of an individual person (that is, his physical parameters). And in this model there are no restrictions on the direction and length of the step.

The content of the problems associated with the calculation of evacuation imposes certain requirements on the mathematical apparatus that should be used to model the evacuation process. Recently, design cases involving premises with developed internal infrastructure (lecture and auditoriums, classrooms, trading floors, etc.) have become a frequent occurrence; it is important to take into account unique physical parameters (including age).

Combining the advantages of both models made it possible to move to a new stage in the study of human flow. The new model that has emerged is called the field discrete-continuous evacuation model “SigMA.DC” (Stochastic field Movement of Artificially People Intelligent discrete-continuous model - stochastic field continuous-discrete model of people’s movement with elements of artificial intelligence).

This model takes into account the dependence of a person’s speed on density, age, emotional state, and mobility group. It is continuous in space in the chosen direction, but only a finite number of directions are assumed where a person can move from the current position.

Table 1 summarizes the most significant, in the opinion of many researchers, criteria for choosing a mathematical model, as well as a comparative analysis of three models from the Fire Risk Calculation Methodology (Appendix to Order of the Ministry of Emergency Situations of Russia N382 dated June 30, 2009) and the field evacuation model SigMA.DC. The above list arose based on the need to reproduce evacuation scenarios from scientific and educational institutions as closely as possible with their inherent specifics: the movement of people in premises with developed infrastructure, various roles (sequence of prescribed actions) of individual evacuees, unique physical parameters (including age), various level of awareness about fire safety rules and building layout, changing level of visibility. I was also interested in the issue of extensibility of the model for integration with models of the development of fire hazards.

Table 1 - Comparative analysis of simplified analytical, individual-flow, simulation-stochastic and field - SigMA.DC models of evacuation.

Analysis of the data from the table shows that the field model SigMA.DC has an overwhelming advantage.

It is this model that is the object of study in this work.

Lecture 1

The objects of study of this course are the processes and apparatus of chemical technology.

Chemical technology processes are physical and chemical systems that are characterized by complex interactions of phases and components. During the course of technological processes, a transfer of momentum, energy or mass occurs at each point of the phases and at their interface. Chemical technology processes take place in devices that have specific geometric characteristics, which in turn have a significant impact on the course of the process.

To study various physical and chemical processes, test scientific hypotheses and obtain experimental material, modeling of real objects has long been used.

Modeling called the study of an object by creating and studying its model.

Modeling is a method of studying objects, in which, instead of the original object, the research is carried out on a model, and the research results are extended to the original object.

There are two main types of models - physical models and mathematical models. Accordingly, there are two modeling methods: physical and mathematical.

A physical model in most cases is a scaled copy of a real object, which preserves the physical nature of the processes occurring in the object under study.

When using the physical modeling method, two basic requirements must be met:

1. An experiment conducted on a model should be simpler, more economical or safer than an experiment conducted on a real object.

2. The patterns connecting the model and the real object must be known.

For objects of chemical technology, such patterns are certain relationships called similarity criteria: Reynolds, Prandtl, Archimedes criteria, etc.

According to the theory of similarity, the necessary physical similarity of the model and the object is ensured when all similarity defining criteria I.

If the number of phenomena considered when studying an object is large, then the required number of defining similarity criteria increases accordingly. In this case, it can be practically impossible to ensure equality of values ​​of all defining criteria for the similarity of the model and the object.

It follows that the possibilities of physical modeling based on the theory of similarity are significantly limited by the complexity of the object being studied.

For objects in which physical modeling is limited by the difficulties of research, the danger of experiments, technical difficulties or the high cost of creating physical models, mathematical modeling is used.

Mathematical the model describes the processes occurring in a real object in symbolic form, i.e. in the form of mathematical expressions.

The study of an object by the method of mathematical modeling consists in solving a system of equations for the mathematical description of the object.

There are various types of mathematical models that can be roughly classified according to the following criteria:

1. By the nature of the temporary description:

continuous and discrete.

Continuous Models allow you to obtain the characteristics of an object at each current moment in time;

discrete models allow you to obtain the characteristics of an object in a fixed sequence of time intervals.

To describe the dynamics of the simulated processes in simulation, it is implemented mechanism for setting model time. These mechanisms are built into the control programs of any modeling system.

If the behavior of one component of the system were simulated on a computer, then the execution of actions in the simulation model could be carried out sequentially, by recalculating the time coordinate. To ensure the simulation of parallel events of a real system, some global variable is introduced (ensuring synchronization of all events in the system) t0, which is called model (or system) time.

There are two main ways to change t 0 :

• step-by-step(fixed intervals of model time changes are applied);
• no-event(variable intervals of change in model time are used, while the step size is measured by the interval until the next event).

In case step by step method time advances with the smallest possible constant step length (t principle). These algorithms are not very efficient in terms of using computer time for their implementation.

Event-based method (principle of “special states”). In it, time coordinates change only when the state of the system changes. In event-based methods, the length of the time shift step is the maximum possible. Model time changes from the current moment to the nearest moment of the next event. The use of the event-based method is preferable if the frequency of occurrence of events is low, then a large step length will speed up the progress of model time. In practice, the event-based method is most widespread.

The fixed step method is used:

if the law of change over time is described by integro-differential equations. A typical example: solving integro-differential equations using a numerical method. In such methods, the modeling step is equal to the integration step. When using them, the dynamics of the model is a discrete approximation of real continuous processes; when events are distributed evenly and the step of changing the time coordinate can be selected; when it is difficult to predict the occurrence of certain events; when there are a lot of events and they appear in groups.

In other cases, the event-based method is used. It is preferable when events are distributed unevenly on the time axis and appear at significant time intervals.

Thus, due to the sequential nature of information processing in a computer, parallel processes occurring in the model are transformed using the considered mechanism into sequential ones. This method of representation is called a quasi-parallel process.

The simplest classification into the main types of simulation models is associated with the use of these two methods of advancing model time. There are simulation models:

Continuous;

Discrete;

Continuous-discrete.

IN continuous simulation models variables change continuously, the state of the modeled system changes as a continuous function of time, and, as a rule, this change is described by systems of differential equations. Accordingly, the advancement of model time depends on numerical methods for solving differential equations.

IN discrete simulation models variables change discretely at certain moments of simulation time (the occurrence of events). The dynamics of discrete models is the process of transition from the moment of the onset of the next event to the moment of the onset of the next event.

Since in real systems continuous and discrete processes are often impossible to separate, continuous-discrete models, which combine the mechanisms of time progression characteristic of these two processes.