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Flowpipes

Abstract interface

The following functions are available at the interface level.

LazySets.API.ρ — Method
LazySets.ρ(d::AbstractVector, fp::AbstractFlowpipe)

Input

  • d – direction
  • fp – flowpipe

Output

The support function of the flowpipe along the given direction d.

Notes

In this fallback implementation, the flowpipe behaves like the union of the reach-sets, i.e. the implementation is analogue to that of a LazySet.UnionSetArray.

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LazySets.API.σ — Method
LazySets.σ(d::AbstractVector, fp::AbstractFlowpipe)

Input

  • d – direction
  • fp – flowpipe

Output

The support vector of the flowpipe along the given direction d.

Notes

In this fallback implementation, the flowpipe behaves like the union of the reach-sets, i.e. the implementation is analogue to that of a LazySet.UnionSetArray.

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LazySets.API.dim — Method
LazySets.dim(fp::AbstractFlowpipe)

Input

  • fp – flowpipe

Output

An integer representing the ambient dimension of the flowpipe.

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LazySets.set — Method
set(fp::AbstractFlowpipe, ind::Integer)

Return the geometric set represented by this flowpipe at the given index.

Input

  • fp – flowpipe
  • ind – index (from 1 to length(flowpipe))

Output

The set wrapped by the flowpipe at the given index.

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LazySets.set — Method
set(fp::AbstractFlowpipe, ind::AbstractVector)

Return the union of set represented by this flowpipe at the given indices.

Input

  • fp – flowpipe
  • ind – vector of indices

Output

The set union stored in the flowpipe at the given indices.

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LazySets.set — Method
set(fp::AbstractFlowpipe)

Return the geometric set represented by this flowpipe as the union of reach-sets.

Input

  • fp – flowpipe

Output

The set union of the array of reach-sets of the flowpipe.

Notes

To retrieve the array of sets stored in the flowpipe use array(fp). To get a set at a particular index, use set(F[ind]) or set(F, ind).

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ReachabilityAnalysis.tstart — Method
tstart(fp::AbstractFlowpipe)

Return the initial time of this flowpipe.

Input

  • fp – flowpipe

Output

A float representing the initial time of the given flowpipe. The fallback is computed by taking the initial time of the first reach-set.

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ReachabilityAnalysis.tend — Method
tend(fp::AbstractFlowpipe)

Return the final time of this flowpipe.

Input

  • fp – flowpipe

Output

A float representing the initial time of the given flowpipe. The fallback is computed by taking the final time of the last reach-set.

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ReachabilityAnalysis.tspan — Method
tspan(fp::AbstractFlowpipe)

Return time span of this flowpipe.

Input

  • fp – flowpipe

Output

The interval representing the time span of the given flowpipe. The fallback is computed as (tstart(fp), tend(fp)), see tstart(::AbstractFlowpipe) and tend(::AbstractFlowpipe) for details.

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ReachabilityAnalysis.vars — Method
vars(fp::AbstractFlowpipe)

Return the tuple of variable indices of the flowpipe.

Input

  • fp – flowpipe

Output

Tuple of integers with the variable indices of the flowpipe, typically $1, 2, …, n$ where $n$ is the dimension of the flowpipe.

Notes

The fallback implementation assumes first reach-set is representative.

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Flowpipe

ReachabilityAnalysis.Flowpipe — Type
Flowpipe{N, RT<:AbstractReachSet{N}, VRT<:AbstractVector{RT}} <: AbstractFlowpipe

Type that wraps a flowpipe, which is an iterable collection of reach-sets that behaves like their set union.

Fields

  • Xk – array of reach-sets
  • ext – extension dictionary; field used by extensions

Notes

The dimension of the flowpipe corresponds to the dimension of the underlying reach-sets; in this type, it is is assumed that the dimension is the same for the different reach-sets.

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ReachabilityAnalysis.location — Method
location(F::Flowpipe)

Return the location of a flowpipe within a hybrid system, or missing if it is not defined.

Input

  • F – flowpipe

Output

The :loc_id value of stored in the flowpipe's extension field.

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The following methods are available.

ReachabilityAnalysis.shift — Function
shift(R::AbstractReachSet, t0::Number)

Perform a time-shift of the given reach-set.

Input

  • R – reach-set
  • t0 – number that corresponds to the time-shift

Output

A new reach-set of the same type of R such that its time-span has been shifted by t0.

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shift(fp::Flowpipe{N, <:AbstractReachSet}, t0::Number) where {N}

Return the time-shifted flowpipe by the given number.

Input

  • fp – flowpipe
  • t0 – time shift

Output

A new flowpipe such that the time-span of each constituent reach-set has been shifted by t0.

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Hybrid flowpipe

ReachabilityAnalysis.HybridFlowpipe — Type
HybridFlowpipe{N,RT<:AbstractReachSet{N},FT<:AbstractFlowpipe} <: AbstractFlowpipe

Type that wraps a vector of flowpipes of the same type, such that they are contiguous in time.

Fields

  • Fk – vector of flowpipes
  • ext – (optional, default: empty) dictionary for extensions

Notes

The evaluation functions (in time) for this type do not assume that the flowpipes are contiguous in time. That is, the final time of the i-th flowpipe does not match the start time of the i+1-th flowpipe.

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Mixed flowpipe

ReachabilityAnalysis.MixedFlowpipe — Type
MixedFlowpipe{N,RT<:AbstractReachSet{N},FT<:AbstractFlowpipe} <: AbstractFlowpipe

Type that wraps a vector of flowpipes of the same time, such that they are not necessarily contiguous in time.

Fields

  • Fk – vector of flowpipes
  • ext – (optional, default: empty) dictionary for extensions

Notes

This type does not assume that the flowpipes are contiguous in time.

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Shifted flowpipe

ReachabilityAnalysis.ShiftedFlowpipe — Type
ShiftedFlowpipe{FT<:AbstractFlowpipe, NT<:Number} <: AbstractFlowpipe

Type that lazily represents a flowpipe that has been shifted in time.

Fields

  • F – original flowpipe
  • t0 – time shift

Notes

This type can wrap any concrete subtype of AbstractFlowpipe, and the extra field t0 is such that the time spans of each reach-set in F are shifted by the amount t0 (which should be a subtype of Number).

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Mapped flowpipe