16
1 Modeling Systems
given by the system of equations
dx
dt
= A · x + b
where A is some non-degenerate matrix and b is the vector of given functions of
time.
Linear models can be classified according to different types of behavior of dynamical systems, using information either only on the coefficients of the matrix A or on
its eigenvalues, that is, on the roots of the characteristic equation
det(A − λ · E) = 0,
where E is the dimension unit matrix n × n.
Singular and fixed points can be found in a known manner [1]. Moreover, each
type of a particular point can be associated with a certain phase portrait.
To classify the singular points, one can also use the trace of the matrix Sp(A) and
the determinant det(A).
The solution of linear ODE systems is not difficult for most modern simulation
environments. To solve nonlinear systems in modeling packages, as a rule, the possibility of both automatic and manual selections of numerical methods is provided.
At the same time, it is recommended that the user himself checks the existence and
uniqueness of the solution.
1.10 Component-Oriented Approach in Modeling
As mentioned earlier, component, or multi-component, models, unlike elementary
ones, have an internal structure. This means, along with state variables, parameters,
constants, and equations linking them, they include local components, which in turn
can be both elementary and again component models. Components interact with
each other through connections. Modern computer simulation environments provide
a graphical interface for component modeling. The components of the model are
schematic blocks in a graphic diagram, and the connections between them are defined
by lines connecting the corresponding external variables.
The component-oriented approach to modeling is primarily convenient for the
user to perceive the model. It also allows you to avoid many errors at the stage of
building a model. Naturally, any model can be simply described by a system of
equations. However, if the model is complex, such a system may consist of tens
and hundreds of equations, and there is always the risk of making a mistake if you
compose the system manually. For example, when modeling a complex electrical
circuit, it will be necessary, according to Kirchhoff’s laws, to describe all the cycles
of this circuit, and there may be a lot of them. Component modeling usually involves
the creation of libraries of some typical components for a given subject area (current
1 Modeling Systems
given by the system of equations
dx
dt
= A · x + b
where A is some non-degenerate matrix and b is the vector of given functions of
time.
Linear models can be classified according to different types of behavior of dynamical systems, using information either only on the coefficients of the matrix A or on
its eigenvalues, that is, on the roots of the characteristic equation
det(A − λ · E) = 0,
where E is the dimension unit matrix n × n.
Singular and fixed points can be found in a known manner [1]. Moreover, each
type of a particular point can be associated with a certain phase portrait.
To classify the singular points, one can also use the trace of the matrix Sp(A) and
the determinant det(A).
The solution of linear ODE systems is not difficult for most modern simulation
environments. To solve nonlinear systems in modeling packages, as a rule, the possibility of both automatic and manual selections of numerical methods is provided.
At the same time, it is recommended that the user himself checks the existence and
uniqueness of the solution.
1.10 Component-Oriented Approach in Modeling
As mentioned earlier, component, or multi-component, models, unlike elementary
ones, have an internal structure. This means, along with state variables, parameters,
constants, and equations linking them, they include local components, which in turn
can be both elementary and again component models. Components interact with
each other through connections. Modern computer simulation environments provide
a graphical interface for component modeling. The components of the model are
schematic blocks in a graphic diagram, and the connections between them are defined
by lines connecting the corresponding external variables.
The component-oriented approach to modeling is primarily convenient for the
user to perceive the model. It also allows you to avoid many errors at the stage of
building a model. Naturally, any model can be simply described by a system of
equations. However, if the model is complex, such a system may consist of tens
and hundreds of equations, and there is always the risk of making a mistake if you
compose the system manually. For example, when modeling a complex electrical
circuit, it will be necessary, according to Kirchhoff’s laws, to describe all the cycles
of this circuit, and there may be a lot of them. Component modeling usually involves
the creation of libraries of some typical components for a given subject area (current