1.8 Continuous, Discrete, and Hybrid Models
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orders of magnitude shorter than the time of operation of the system under study, as
well as the idealization of parametric dependencies. An example of the idealization
of the first type is the neglect of the time of an absolutely elastic ball rebound from
an absolutely solid plane, and an example of the idealization of the second type is the
idealization of a real current–voltage characteristic for an ideal diode. In addition,
the researcher may simply abstract away from a detailed description of the dynamics
of a nonlinear transient process (or a detailed description may simply be unknown),
replacing it with some integral dependencies. It should be noted that, due to its
artificial nature, hybrid models of this type may exhibit paradoxical behaviors that
are not characteristic of the original continuous objects, and the researcher must be
extremely careful in formalizing the hybrid model.
Hybrid behavior due to system composition changes.
If continuous objects during operation can appear within the boundaries of the
system under study and leave it, then the composition of the total state vector of the
entire system x and its dimension will change. Examples of such systems are: an
airport (airplanes appear within the airport zone from the outside, land at the airport,
and take off from the runway), an air defense complex (targets appear in the detection
zone, exit it, and are destroyed by missiles), a system of emerging and disappearing
charged particles, etc.
1.9 Linear and Nonlinear Systems
Linear models of dynamic systems are based on systems of linear differential equations. These models are important from the viewpoint of modeling due to the fact
that:
• In many cases, linear models are sufficient to reflect the most important properties
of the object being modeled, and linear systems are well studied and amenable to
qualitative and quantitative analysis.
• In a sufficiently small neighborhood of any point of the solution of a nonlinear
equation, one can construct an approximate linear model, the analysis of which
makes it possible to judge the local properties of the nonlinear model.
Consider a system of ordinary differential equations:
dx
dt
= F(x, μ, t)
where t is the time, x = (x 1 , x 2 , . . . x n ) are the time-dependent state variables, μ is
the time-independent vector, and F = (F, F 2 , . . . F n ) are some given functions.
In the general case, the functions F, F 2 , . . . F n are nonlinear functions of the
state variables x 1 , x 2 , . . . x n . Such a model is called nonlinear. In the case when all
functions F, F 2 , . . . F n are linear with state variables, then the model is linear and is
15
orders of magnitude shorter than the time of operation of the system under study, as
well as the idealization of parametric dependencies. An example of the idealization
of the first type is the neglect of the time of an absolutely elastic ball rebound from
an absolutely solid plane, and an example of the idealization of the second type is the
idealization of a real current–voltage characteristic for an ideal diode. In addition,
the researcher may simply abstract away from a detailed description of the dynamics
of a nonlinear transient process (or a detailed description may simply be unknown),
replacing it with some integral dependencies. It should be noted that, due to its
artificial nature, hybrid models of this type may exhibit paradoxical behaviors that
are not characteristic of the original continuous objects, and the researcher must be
extremely careful in formalizing the hybrid model.
Hybrid behavior due to system composition changes.
If continuous objects during operation can appear within the boundaries of the
system under study and leave it, then the composition of the total state vector of the
entire system x and its dimension will change. Examples of such systems are: an
airport (airplanes appear within the airport zone from the outside, land at the airport,
and take off from the runway), an air defense complex (targets appear in the detection
zone, exit it, and are destroyed by missiles), a system of emerging and disappearing
charged particles, etc.
1.9 Linear and Nonlinear Systems
Linear models of dynamic systems are based on systems of linear differential equations. These models are important from the viewpoint of modeling due to the fact
that:
• In many cases, linear models are sufficient to reflect the most important properties
of the object being modeled, and linear systems are well studied and amenable to
qualitative and quantitative analysis.
• In a sufficiently small neighborhood of any point of the solution of a nonlinear
equation, one can construct an approximate linear model, the analysis of which
makes it possible to judge the local properties of the nonlinear model.
Consider a system of ordinary differential equations:
dx
dt
= F(x, μ, t)
where t is the time, x = (x 1 , x 2 , . . . x n ) are the time-dependent state variables, μ is
the time-independent vector, and F = (F, F 2 , . . . F n ) are some given functions.
In the general case, the functions F, F 2 , . . . F n are nonlinear functions of the
state variables x 1 , x 2 , . . . x n . Such a model is called nonlinear. In the case when all
functions F, F 2 , . . . F n are linear with state variables, then the model is linear and is