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1 Modeling Systems
the structural mathematical model. If the mathematical model reflects only how the
system works, given its internal structure, then it is called functional. There may also
be combined mathematical models that describe both the functional and structural
properties of the system.
Functional models can be classified by different properties. The following types
of models can be distinguished.
By the set of parameter values, the models are divided into continuous and discrete.
Each system parameter can be of two types—continuously changing in a certain
interval of its values or accepting only some discrete values. An intermediate situation
is also possible, when in one region the parameter takes all possible values and in the
other only discrete ones. The same definition can be applied to the description of the
behavior of models. If all parameters of the model are continuous, then the model is
called continuous; if the parameters are discrete, then the model is discrete. If some
parameters can change both continuously and discretely, then this model is hybrid.
By the type of dependencies between the parameters, the models are divided into
linear and nonlinear. If all output parameters of the model are linearly dependent on
the input (superposition principle), then the model is called linear. In the opposite
case, it is referred as a nonlinear model. Linear models are simpler and allow you
to get an analytical solution, but they do not always allow satisfactory description of
the object under study.
By the relation to external factors, models can be divided into open and closed
(isolated). A closed model is a model that functions without communication with
external variables. In a closed model, changes in model parameter values in time are
determined only by the internal interaction of the parameters themselves. An open
model associated with external variables.
By the presence or absence of random parameters in the system, the models are
divided into deterministic and stochastic. The deterministic model does not take into
account the influence of random factors; therefore, its behavior is predetermined by
a system of equations. With repeated experiments with the same initial values, the
result will be the same. There are random parameters in the stochastic model, so its
behavior cannot be predicted.
The essential feature of the classification of mathematical models is their ability
to describe the change in system parameters over time. If the state of the model does
not depend on time, then the model is called static. In contrast, a model in which
parameters change over time is called dynamic. Separately, it is worth highlighting
the class of stationary models. Stationary models describe systems in which socalled steady-state processes occur, i.e., processes in which the output parameters
are constant in time. Periodic processes are also established, in which some output
parameters remain unchanged and the rest oscillate.
By the degree of detail of the description of the processes occurring in the system, functional mathematical models can be divided into hierarchical levels: micro-,
macro-, and meta-level. Mathematical models of the micro-level describe processes
in systems with distributed parameters and mathematical models of the macro-level
in systems with concentrated parameters. In micro-level models, phase variables
can depend on both time and spatial coordinates, whereas in macro-level models,
1 Modeling Systems
the structural mathematical model. If the mathematical model reflects only how the
system works, given its internal structure, then it is called functional. There may also
be combined mathematical models that describe both the functional and structural
properties of the system.
Functional models can be classified by different properties. The following types
of models can be distinguished.
By the set of parameter values, the models are divided into continuous and discrete.
Each system parameter can be of two types—continuously changing in a certain
interval of its values or accepting only some discrete values. An intermediate situation
is also possible, when in one region the parameter takes all possible values and in the
other only discrete ones. The same definition can be applied to the description of the
behavior of models. If all parameters of the model are continuous, then the model is
called continuous; if the parameters are discrete, then the model is discrete. If some
parameters can change both continuously and discretely, then this model is hybrid.
By the type of dependencies between the parameters, the models are divided into
linear and nonlinear. If all output parameters of the model are linearly dependent on
the input (superposition principle), then the model is called linear. In the opposite
case, it is referred as a nonlinear model. Linear models are simpler and allow you
to get an analytical solution, but they do not always allow satisfactory description of
the object under study.
By the relation to external factors, models can be divided into open and closed
(isolated). A closed model is a model that functions without communication with
external variables. In a closed model, changes in model parameter values in time are
determined only by the internal interaction of the parameters themselves. An open
model associated with external variables.
By the presence or absence of random parameters in the system, the models are
divided into deterministic and stochastic. The deterministic model does not take into
account the influence of random factors; therefore, its behavior is predetermined by
a system of equations. With repeated experiments with the same initial values, the
result will be the same. There are random parameters in the stochastic model, so its
behavior cannot be predicted.
The essential feature of the classification of mathematical models is their ability
to describe the change in system parameters over time. If the state of the model does
not depend on time, then the model is called static. In contrast, a model in which
parameters change over time is called dynamic. Separately, it is worth highlighting
the class of stationary models. Stationary models describe systems in which socalled steady-state processes occur, i.e., processes in which the output parameters
are constant in time. Periodic processes are also established, in which some output
parameters remain unchanged and the rest oscillate.
By the degree of detail of the description of the processes occurring in the system, functional mathematical models can be divided into hierarchical levels: micro-,
macro-, and meta-level. Mathematical models of the micro-level describe processes
in systems with distributed parameters and mathematical models of the macro-level
in systems with concentrated parameters. In micro-level models, phase variables
can depend on both time and spatial coordinates, whereas in macro-level models,