Scope of application and limitations of commonly used system modeling methods

Scope of application and limitations of commonly used system modeling methods? System Modeling Methods

2.1 System Abstraction and Mathematical Description

2.1.1 Abstraction of the Actual System

In essence, a mathematical model of a system is an abstract "image" of a small portion or a few aspects of the real world from the concept of a system.

To do this, the creation of a mathematical model of a system requires the creation of the following abstractions: inputs, outputs, state variables and their functional relationships. This abstraction process is called model construction. In the abstraction, it is necessary to relate the real system to the modeling objectives, where the descriptive variables play a very important role, which can be observed, or unobservable.

Observable variables that affect or disturb the system from outside are called input variables. The result of the system's response to the input variable is called the output variable.

The set of pairs of input and output variables characterizes the "input-output" nature (relationship) of a real system.

In summary, the real system can be regarded as a source of information that produces certain trait data, while the model is a collection of rules and instructions that produces the same trait data as the real system, in which abstraction plays a mediating role. Mathematical modeling of the system is to abstract the real system into a corresponding mathematical expression (a collection of rules and instructions).

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(Observable)

Input variables (observable) Output variables

ωt) Black box

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Gray box

White box ω(t), ρ(t) - the process of abstraction of the input and output variables to the

real system modeling

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2.1.2 General description of system model and description level (level)

2.1.2.1 General description of system model:

A mathematical model of a system can be described by the following set of seven tuples:

S?T,X,? ,Q,Y,? ,?

Where:

T:Time base, describes the time coordinates of the system changes, T is called discrete time system if it is an integer, and continuous time system if it is a real number;

X:Input set, represents the action of the external environment on the system.

:The set of input segments, which describes the pattern of inputs over some time interval, is a subset of ?X,T?.

Q:Internal state set, which describes the internal state quantities of the system and is central to modeling the internal structure of the system. ? :state transfer function, defines how the internal state of the system changes and is a mapping. Y:Output set, through which the system acts on the environment.

:The output function, a mapping, gives a set of output segments.

2.1.2.2 System Model Description Levels:

According to system theory, real systems can be decomposed at some level (level), and therefore a mathematical model of a system can have different description levels (levels):

Trait Description Level

The trait description level, or what is called the behavioral description level (behavioral level). Describing the system at this level involves taking the

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System is comparable to a black box and applying an input signal while measuring the output response, which results in an input-output pair:(ω, ρ) and its relationship Rs={(ω, ρ):Ω, ω, ρ}. - 3 -

The trait-level description of the system thus gives only input-output observations. It is modeled as a five-tuple ensemble structure:

S=(T, X, Ω, Y, R)

When ω, ρ satisfy the ρ = f(ω) functional relationship, the ensemble structure becomes: S=(T, X, Ω, Y, F)

Black box

Trait-level descriptions

At the trait-level (state-structured level) the model of the system should be able to reflect not only the input -output relationships, but also should be able to reflect the internal state of the system, and the relationship between the state and the inputs and outputs. That is, not only the inputs and outputs of the system are defined, but also the set of states and the state transfer function within the system

The mathematical model of the system can be described by a set of seven tuples for the dynamic structure:

S=(T, X, Ω, Q, Y, δ, λ)

For the static structure there are:

S=(X, Q, Y, λ)

White box

Composite structure level

Systems generally consist of a number of subsystems, each of which is given a behavioral level description and is considered a "component" of the system. These components have their own input and output variables, as well as inter-component connections and interfaces. Thus, a mathematical model of the system at the composite structure level (decomposition structure

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level) can be developed.

This composite structure-level description is the basis for modeling complex and large systems.

It should be emphasized that:

the decomposition of systems into composite structures is endless, i.e., each subsystem will also have its own composite structure;

a meaningful composite structure description can give only a unique description of the state structure, - 4 -

and a meaningful state structure description itself has only a unique description of the traits (behaviors);

the above notion of a system must allow decomposition to stop and further decomposition to be allowed, and contain both recursive decomposability.

Gray box

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2.2 Introduction to Similarity Concepts

2.2.1 Similarity Concepts and Meanings

Theoretical basis of simulation:similarity theory.

The concept of "similarity" exists widely in nature, the most common are:

Geometric similarity: the simplest and most intuitive, such as multiple deformation, triangle similarity;

Phenomenal similarity: the expansion of geometric similarity, such as the proportionality of the relationship between physical quantities. The use of similarity technology to establish a similar model of the actual system, which is the fundamental embodiment of the role of similarity theory in the foundation of the system simulation.

2.2.2 Classification of similarity

Absolute similarity: the two systems (such as the system prototype and the model) all the geometric dimensions and other corresponding parameters in the spatial and temporal domain of all the changes produced (or all the process) are similar;

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Complete similarity: the two systems are similar in a corresponding aspect of the process, such as the generator of the current voltage problem, it is sufficient that the model and the prototype are exactly similar in terms of electromagnetic phenomena, without having to consider the thermal and mechanical similarities;

Incomplete similarity (partial similarity):the similarity of the system is guaranteed only for the part under study, whereas the processes in the non-studied and undemanding part may be distorted, as permitted for the purpose of the study;

Approximate similarity:the similarity of the phenomena under certain simplifying assumptions, for which the mathematical modeling To ensure validity.

Similarities in different fields have their own characteristics and different levels of understanding of the field: environmental similarities (geometrical similarities, similarity of parametric proportions, etc.):models obtained by scaling down the dimensions of a structure - scaled-down models, such as those used in wind and water tunnel experiments.

Discrete similarity:Differential method, discrete similarity method to discretize a continuous time system into an equivalent discrete time system.

Performance similarity (equivalence, dynamics similarity, control response similarity, etc.):The principle of similarity in which the mathematical description is the same or the frequency characteristics are the same, and which is used to construct various types of simulations.

Sensory similarity (motion sensation, visual, acoustic sensation, etc.): ear, eye, nose, tongue, - 6 -

Body and other senses and experience, MIL simulation of the sensory similarity into the sensory information source similarity, the training simulator, VR are utilized to this similarity principle.

Thinking similarity:logical thinking similarity and image thinking similarity (comparison, synthesis, induction, etc.), expert systems, artificial neuron networks.

Systems have internal structure and external behavior, so there are two basic levels of system similarity:structural level and behavioral level.

Isomorphism must have the property of behavioral equivalence, but two systems with behavioral equivalence do not necessarily have isomorphism.

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