Lecture
The source information used in constructing mathematical models of the operating processes of systems consists of data on the purpose and operating conditions of the system S under study (being designed). This information determines the main goal of modelling the system S and makes it possible to formulate requirements for the mathematical model M being developed. Moreover, the level of abstraction depends on the range of questions to which the researcher of the system wants to obtain an answer with the help of the model, and to some extent determines the choice
of the mathematical scheme.
Introducing the concept of a «mathematical scheme» makes it possible to view mathematics not as a method of calculation but as a method of thinking, as a means of formulating concepts, which is most important when moving from a verbal
description of the system to a formal representation of the process of its operation in the form of some mathematical model (analytical or simulation). When using a mathematical scheme, the researcher of the system S is
primarily concerned with the question of the adequacy of the representation in
the form of specific schemes of the real processes in the system under study, rather than the possibility of obtaining an answer (the result of a solution) to a specific research question.
A mathematical scheme can be defined as a link in the transition from
a substantive to a formal description of the system's operating process, taking into account the influence of the external environment, i.e. there is a chain «descriptive model — mathematical scheme — mathematical [analytical
or (and) simulation] model».
Each specific system is characterised by a set of properties, understood as quantities reflecting the behaviour of the object being modelled
(the real system) and taking into account the conditions of its operation in inter-
action with the external environment (system) E. When constructing a mathematical model of a system, it is necessary to resolve the question of its completeness. The completeness of the model is governed mainly by the choice of the boundary «system S—environment E». It is also necessary
to solve the problem of simplifying the model, which helps identify the main
properties of the system, discarding the secondary ones. Moreover, whether properties of a system are classified as primary or secondary essentially depends on the goal of modelling the system (for example, analysis of the probabilistic-temporal characteristics of the system's operating process, synthesis of the system's structure, etc.).
The model of the object being modelled, i.e. of the system S, can be represented in
the form of a set of quantities describing the operating process of the real
system and forming, in the general case, the following subsets:
the set of input actions on the system
i nX
x ∈ X, i =1,
the set of actions of the external environment
i nV υ ∈V, i =1,
the set of internal (intrinsic) parameters of the system
k nH h ∈H, k =1,
the set of output characteristics of the system
j nY
y ∈Y, j =1,
In doing so, controllable and uncontrollable variables can be distinguished within the enumerated subsets. In the general case
i
x , υi
,
k h ,
j
y
are elements of non-intersecting subsets and contain both deterministic,
and stochastic components.
When modelling the system S, the input actions, the actions of the external environment E and the internal parameters of the system are independent (exogenous) variables, which in vector form have, respectively, the form
x(t) (x (t) x (t) x (t)) nX
, , ...,
= 1 2
, (t) ( (t) (t) (t)) nV
υ = υ1
,υ2
, ...,υ , h(t) (h (t) h (t) h (t)) nH
, , ...,
= 1 2
,
while the output characteristics of the system are dependent (endogenous) variables and in vector form have the form
y(t) (y (t) y (t) y (t)) nY
, , ...,
= 1 2
.
The operating process of the system S is described over time by the operator Fs
, which in the general case transforms exogenous variables into endogenous ones in accordance with relations of the form
y(t) F (x h t) S =
,υ, , . (1)
The set of dependences of the output characteristics of the system on time
y (t) j
for all kinds
nY
j =1,
is called the output trajectory. Relation (1) is called the law of functioning of the system S and is denoted by Fs
.
In the general case, the law of functioning of the system Fs can be given in the form
of a function, a functional, logical conditions, in algorithmic and tabular
forms, or in the form of a verbal correspondence rule.
Of great importance for describing and studying the system S is the concept of the functioning algorithm As
, by which is understood the method of obtaining the output characteristics taking into account the input actions
xt, the actions
of the external environment
υ(t)
and the intrinsic parameters of the system
ht. It is obvious that
one and the same law of functioning Fs of the system S can be implemented
in various ways, i.e. by means of a set of different algorithms
of functioning As
.
Relations (1) are a mathematical description of the behaviour of the object (system) of modelling over time t, i.e. they reflect its dynamic
properties. Therefore, mathematical models of this kind are conventionally called dynamic models (systems).
For static models, the mathematical model (1) represents
a mapping between two subsets of properties of the object being modelled Y
and {X, V, H}, which in vector form can be written as
y = f(x,υ, h). (2)
Relations (1) and (2) can be given in various ways: analytically (using formulas), graphically, in tabular form, etc. Such relations can in a number of cases be obtained via the properties of the system S at specif-
ic moments in time, called states. The state of the system S is characterised by the vectors
' ( '
,
'
, ...,
') 1 2 k
z = z z z , '' ( ''
,
''
, ...,
'') 1 2 k
z = z z z ,
where
' ( ') 1 1
z = z t , ' ( '),...,
' ( ') 2 2
z z t z z t = k = k
at the moment
t' (t ,T) ∈ 0
;
'' ( '') 1 1
z = z t , '' ( ''),...,
'' ( '') 2 2
z z t z z t = k = k
at the moment
t' ' (t ,T) ∈ 0
;
nZ
k =1, .
If the operating process of the system S is regarded as a sequential change of states
z (t) 1
, z (t) 2
, …, z (t) k
, then they can be interpreted as the coordinates of a point in
a k – dimensional phase space, wherein
each realisation of the process will correspond to a certain phase trajectory. The set of all possible values of the states is called the state space of the object being modelled Z, wherein
zk ∈ Z .
The states of the system S at the moment
t < t ≤T
*
0
are completely determined
by the initial conditions
( )
0 0
2
0
1
0
, , ..., k
z = z z z ,
where
( ) 1 0
0
1
z = z t , ( ) 2 0
0
2
z = z t , …, ( ) 0
0
z z t
k = k
,
by the input actions
x(t),
by the internal parameters
h(t)
and
by the actions of the external environment
υ(t),
which took place over the time interval
0
*
t t , by means of two vector
equations
z(t) (z , x, , h,t)
0
= Φ υ
; (3)
y(t) = F(z,t). (4)
The first equation, based on the initial state
0
z
and the exogenous variables
x,υ, h
determines the vector function
z(t)
, and the second, based on the obtained value
of the states,
zt — the endogenous variables at the output of the system
yt. Thus, the chain of object equations «input — states — output» makes it possible to determine the characteristics of the system
y(t) F[ (z , x, , h,t)]
0
= Φ υ . (5)
Thus, by the mathematical model of an object (a real system) is understood a finite subset of the variables
{x(t),υ(t), h(t)}
together with the mathematical relations between them and the characteristics
y(t).
If the mathematical description of the object being modelled does not contain
elements of randomness, or they are not taken into account, i.e. if it can be assumed that in
this case the stochastic actions of the external environment
υ(t)
and the stochastic
internal parameters
h(t)
are absent, then the model is called deterministic in the sense that the characteristics are uniquely determined by the deterministic input actions
y(t) = f(x,t)
(6)
It is obvious that a deterministic model is a special case
of a stochastic model.
The mathematical relations given above are mathematical schemes of general form and make it possible to describe a wide class of systems.
However, in the practice of modelling objects in the field of systems engineering and systems analysis, at the initial stages of studying a system it is more rational to use standard mathematical schemes: differential equations, finite and probabilistic automata, queueing systems,
Petri nets, etc.
Not possessing the same degree of generality as the models discussed above, standard mathematical schemes have the advantages of simplicity and clarity, but
with a significant narrowing of the possibilities of application. As deterministic models, when random factors are not taken into account in the study, for representing systems operating in continuous time, differential, integral, integro-differential and
other equations are used, while for representing systems operating in discrete time — finite automata and finite-difference schemes are used. As
stochastic models (taking random factors into account), for representing
systems with discrete time, probabilistic automata are used, while for
representing a system with continuous time — queueing systems, etc., are used.
The standard mathematical schemes listed above, naturally, cannot
claim the ability to describe, on their basis, all the processes occurring in large information-control systems. For such systems,
in a number of cases the use of aggregative models is more promising. Aggregative models (systems) make it possible to describe a wide range of objects of study while reflecting the systemic nature of these objects. It is precisely
in an aggregative description that a complex object (system) is divided into a finite number of parts (subsystems), while retaining the connections that ensure the interaction of the parts.
Thus, when constructing mathematical models of the processes
of the operation of systems, the following main approaches can be distinguished:
continuous-deterministic (for example, differential equations);
discrete-deterministic (finite automata);
discrete-stochastic (probabilistic automata);
continuous-stochastic (queueing systems);
generalised, or universal (aggregative systems).
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