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1 Modeling Systems
T j = T j−1 · k j−1 ; j = 1, 2, 3, . . . ;
T 0 =
2 · v 0
g
; k 0 = 1.
Using such difference equations, it is possible to calculate only the moments of
time of rebounds, but it is impossible to obtain the law of change of state variables.
The third formulation of the problem: hybrid time.
The question arises: Is it possible to reproduce both long-term and instantaneous
processes within one model?
We use the concept of hybrid time. We divide the processes taking place into
long-term “flight” and instantaneous—“rebound,” and we will alternate long actions
with instantaneous ones. Namely, calculate the altitude change (continuous time),
suspend lengthy processes upon the occurrence of an event—rebound—perform the
specified sequence of instantaneous actions (discrete time), and then continue the
long actions.
The ability to alternate between purely continuous and discrete behaviors, clearly
separating them, gives undeniable advantages. This allows, within the framework of
one model, to combine the incompatible: flight, described by ordinary differential
equations, and body rebound from the surface is a task, the necessitating use of the
theory of elasticity.
There are three main factors contributing to the emergence of hybrid behavior.
Hybrid behavior due to the combined operation of continuous and discrete objects.
Such hybrid behavior is a characteristic of automatic control systems, in which
there are a continuous control object and a discrete control device (controller). The
simplest case is a conventional discrete controller, which with a certain tact forms
the control action.
For the upper levels of control in complex hierarchical control systems, typical
are processes of so-called logical control. In this case, the behavior of the control
device is set by an asynchronous process, in which the next discrete event depends in
general on the previous one, as well as on continuous variables of the control object.
For example, a logic control device for a rocket issues a command to cut off thrust
when a certain functional reaches a threshold value.
Hybrid behavior due to instantaneous qualitative changes in a continuous object.
Some systems that are continuous in nature may exhibit discrete behavioral traits
associated with the qualitative changes occurring in them. The qualitative changes
themselves are primarily due to the multi-mode operation of the system. It should be
noted that, unlike the first and third types of hybrid behavior, where hybrid behavior
is a natural initial property of the problem itself, the hybrid behavior of this type is
artificial to a certain extent and is associated exclusively with the convenient for the
researcher formalization of the phenomenon. In fact, discreteness appears here due
to the idealization of the initially continuous behavior of real physical systems. The
main idealization of this kind is the neglect of the time of transients, when this time is
1 Modeling Systems
T j = T j−1 · k j−1 ; j = 1, 2, 3, . . . ;
T 0 =
2 · v 0
g
; k 0 = 1.
Using such difference equations, it is possible to calculate only the moments of
time of rebounds, but it is impossible to obtain the law of change of state variables.
The third formulation of the problem: hybrid time.
The question arises: Is it possible to reproduce both long-term and instantaneous
processes within one model?
We use the concept of hybrid time. We divide the processes taking place into
long-term “flight” and instantaneous—“rebound,” and we will alternate long actions
with instantaneous ones. Namely, calculate the altitude change (continuous time),
suspend lengthy processes upon the occurrence of an event—rebound—perform the
specified sequence of instantaneous actions (discrete time), and then continue the
long actions.
The ability to alternate between purely continuous and discrete behaviors, clearly
separating them, gives undeniable advantages. This allows, within the framework of
one model, to combine the incompatible: flight, described by ordinary differential
equations, and body rebound from the surface is a task, the necessitating use of the
theory of elasticity.
There are three main factors contributing to the emergence of hybrid behavior.
Hybrid behavior due to the combined operation of continuous and discrete objects.
Such hybrid behavior is a characteristic of automatic control systems, in which
there are a continuous control object and a discrete control device (controller). The
simplest case is a conventional discrete controller, which with a certain tact forms
the control action.
For the upper levels of control in complex hierarchical control systems, typical
are processes of so-called logical control. In this case, the behavior of the control
device is set by an asynchronous process, in which the next discrete event depends in
general on the previous one, as well as on continuous variables of the control object.
For example, a logic control device for a rocket issues a command to cut off thrust
when a certain functional reaches a threshold value.
Hybrid behavior due to instantaneous qualitative changes in a continuous object.
Some systems that are continuous in nature may exhibit discrete behavioral traits
associated with the qualitative changes occurring in them. The qualitative changes
themselves are primarily due to the multi-mode operation of the system. It should be
noted that, unlike the first and third types of hybrid behavior, where hybrid behavior
is a natural initial property of the problem itself, the hybrid behavior of this type is
artificial to a certain extent and is associated exclusively with the convenient for the
researcher formalization of the phenomenon. In fact, discreteness appears here due
to the idealization of the initially continuous behavior of real physical systems. The
main idealization of this kind is the neglect of the time of transients, when this time is