Event segment - Alchetron, The Free Social Encyclopedia

A segment of a system variable shows a homogenous status of system dynamics over a time period. Here, a homogenous status of a variable is a state which can be described by a set of coefficients of a formula. For example, of homogenous statuses, we can bring status of constant ('ON' of a switch) and linear (60 miles or 96 km per hour for speed). Mathematically, a segment is a function mapping from a set of times which can be defined by an real interval, to the set Z [Zeigler76],[ZPK00], [Hwang13]. A trajectory of a system variable is a sequence of segments concatenated. We call a trajectory constant (respectively linear) if its concatenating segments are constant (respectively linear).

Time base

The time base of the concerning systems is denoted by T , and defined

T = [ 0 , ∞ )

as the set of non-negative real numbers.

Event and null event

An event is a label that abstracts a change. Given an event set Z , the null event denoted by ϵ ∉ Z stands for nothing change.

Timed event

A timed event a pair ( t , z ) where t ∈ T and z ∈ Z denotes that an event z ∈ Z occurs at time t ∈ T .

Null segment

The null segment over time interval [ t l , t u ] ⊂ T is denoted by ϵ [ t l , t u ] which means nothing in Z occurs over [ t l , t u ] .

Unit event segment

A unit event segment is either a null event segment or a timed event.

Concatenation

Given an event set Z , concatenation of two unit event segments ω over [ t 1 , t 2 ] and ω ′ over [ t 3 , t 4 ] is denoted by ω ω ′ whose time interval is [ t 1 , t 4 ] , and implies t 2 = t 3 .

Event trajectory

An event trajectory ( t 1 , z 1 ) ( t 2 , z 2 ) ⋯ ( t n , z n ) over an event set Z and a time interval [ t l , t u ] ⊂ T is concatenation of unit event segments ϵ [ t l , t 1 ] , ( t 1 , z 1 ) , ϵ [ t 1 , t 2 ] , ( t 2 , z 2 ) , … , ( t n , z n ) , and ϵ [ t n , t u ] where t l ≤ t 1 ≤ t 2 ≤ ⋯ ≤ t n − 1 ≤ t n ≤ t u .

Mathematically, an event trajectory is a mapping ω a time period [ t l , t u ] ⊆ T to an event set Z . So we can write it in a function form :

ω : [ t l , t u ] → Z ∗ .

Timed language

The universal timed language Ω Z , [ t l , t u ] over an event set Z and a time interval [ t l , t u ] ⊂ T , is the set of all event trajectories over Z and [ t l , t u ] .

A timed language L over an event set Z and a timed interval [ t l , t u ] is a set of event trajectories over Z and [ t l , t u ] if L ⊆ Ω Z , [ t l , t u ] .

References

Event segment Wikipedia

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