2. 1 system abstraction and mathematical description
2. 1. 1 Summary of the actual system
In essence, the system mathematical model is an abstract "image" of a small part or several aspects of the real world from the system concept.
Therefore, the establishment of the mathematical model of the system needs to establish the following abstractions: input, output, state variables and their functional relationships. This abstract process is called model building. In the abstract, it is necessary to link the real system with the modeling goal, in which the description variables play a very important role, which can be observed or not.
Observable variables that affect or interfere with the system from the outside are called input variables. The response result of the system to the input variable is called the output variable.
The set of input and output variable pairs represents the "input-output" behavior (relationship) of a real system.
To sum up, the real system can be regarded as an information source that produces certain character data, while the model is a collection of rules and instructions that produce the same character data as the real system, in which abstraction plays a media role. System mathematical modeling is to abstract the real system into corresponding mathematical expressions (sets of rules and instructions).
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(Observable)
Input variables (observable) output variables
Ω t) black box
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Gray box
White box ω(t), ρ(t)- input and output variable pairs
Abstract process of real system modeling
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2. 1.2 General description and description level of system model (level)
2. 1.2. 1 system model overview:
The mathematical model of the system can be described by the following seven-tuple set:
ST，X，？ ，Q，Y，？ ,
These include:
T: Time base, which describes the time coordinates of system changes. If t is an integer, it is called a discrete-time system, and if it is a real number, it is called a continuous-time system.
X: input set, indicating the influence of external environment on the system.
: input segment set, which describes the input method within a certain time interval, right? x，T？ A subset of.
Q: The internal state set that describes the internal state quantity of the system is the core of the internal structure modeling of the system. ? : The state transition function that defines how the internal state of the system changes is a kind of mapping. Y: output set, through which the system acts on the environment.
: output function, which is a mapping that gives a set of output segments.
2. 1.2.2 system model description level:
According to the viewpoint of system theory, the actual system can be decomposed at a certain level, so the mathematical model of the system can have different description levels (levels):
Character description level
Character description level or behavior description level (behavior level). The system described at this level is integrated.
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The system can be called a black box, and the input signal is applied and the output response is measured. The result is an input-output pair: (ω, ρ) and its relationship Rs={(ω, ρ): ω, ω, ρ}. - 3 -
Therefore, the character-level description of the system only gives the input and output observation results. Its model is a five-tuple set structure:
S=(T，X，ω，Y，R)
When ω and ρ satisfy ρ= f(ω), their collective structure becomes: S=(T, x, ω, y, f).
black box
State description level
At the state structure level, the system model can not only reflect the input-output relationship, but also reflect the internal state of the system and the relationship between state and input-output. That is, not only the input and output of the system are defined, but also the state set and state transfer function in the system are defined.
The mathematical model of the system can be described as a dynamic structure by a group of seven tuples:
S=(T，X，ω，Q，Y，δ，λ)
For static structures, there are:
S=(X，Q，Y，λ)
SHIROBAKO
Composite structure level
A system is generally composed of several subsystems, each of which is given a behavior description and regarded as a "component" of the system. These components have their own input and output variables, as well as the connection relationships and interfaces between components. Therefore, it can be determined that the system is in a composite structure (decomposition structure
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Level) on the mathematical model.
This composite structure-level description is the basis of complex system and large-scale system modeling.
It should be emphasized that:
The decomposition of a system into a composite structure is endless, that is, each subsystem will have its own composite structure;
A meaningful composite structure description can only give a unique state structure description, -4-
However, the only meaningful description of the state structure itself is the unique description of behavior;
The above concepts of the system must allow decomposition to stop and further decomposition, including recursive decomposability.
Gray box
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2.2 Introduction of similar concepts
2.2. The concept and meaning of1similarity
The theoretical basis of simulation: similarity theory.
The concept of "similarity" exists widely in nature, and the most common ones are:
Geometric similarity: the simplest and most intuitive, such as multi-deformation, triangle similarity;
Phenomenal similarity: the expansion of geometric similarity, such as the proportional relationship between physical quantities. Using similarity technology to establish the similarity model of the actual system is the fundamental embodiment of the basic role of similarity theory in system simulation.
Similarity classification
Absolute similarity: all changes (or all processes) in time and space of all geometric dimensions and other corresponding parameters of two systems (such as system prototype and model) are similar;
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Completely similar: the two systems are similar in a certain aspect, such as the current and voltage of the generator, as long as the model and the prototype are completely similar on the electromagnetic phenomena, regardless of thermal and mechanical similarities;
Partial similarity: only the system of the research part is guaranteed to be similar, but the process of non-research and non-essential parts may be distorted and allowed to be used for research purposes;
Approximate similarity: some phenomena under simplified assumptions are similar, so the effectiveness of mathematical modeling should be ensured.
Similarity in different fields has its own characteristics, and the level of understanding of the field is also different: environmental similarity (geometric similarity, parameter proportion similarity, etc. ): scaled model obtained by scaling down the structural size, such as the model used in wind tunnel and water tunnel experiments.
Discrete similarity: the difference method and discrete similarity method discretize a continuous-time system into an equivalent discrete-time system.
Performance similarity (equivalence, dynamics similarity, control response similarity, etc. ): The same mathematical description or the same frequency characteristics are used to construct the similarity principle of various simulations.
Sense of similarity (sense of movement, vision, hearing, etc.). ): ears, eyes, nose, tongue, -6-
On the body level, sensory and experience, MIL simulation transforms sensory similarity into sensory information source similarity, and training simulators and VR all use this similarity principle.
Thinking similarity: logical thinking similarity and image thinking similarity (comparison, synthesis, induction, etc. ), expert system, artificial neural network.
The system has internal structure and external behavior, so the similarity of the system has two basic levels: structural level and behavioral level.
Isomorphism must have the characteristics of behavioral equivalence, but two systems with behavioral equivalence do not necessarily have isomorphic relationship.
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