From 4549c5283e1f50c12f3a9db9bda75d8b1fad25a1 Mon Sep 17 00:00:00 2001 From: "Documenter.jl" Date: Fri, 12 Jan 2024 21:26:44 +0000 Subject: [PATCH] build based on f9b0046 --- dev/.documenter-siteinfo.json | 2 +- dev/about/index.html | 2 +- dev/generated_examples/Brusselator/index.html | 2 +- dev/generated_examples/Building/index.html | 2 +- .../DuffingOscillator/index.html | 2 +- dev/generated_examples/EMBrake/index.html | 2 +- dev/generated_examples/ISS/index.html | 2 +- dev/generated_examples/LaubLoomis/index.html | 2 +- dev/generated_examples/Lorenz/index.html | 2 +- .../LotkaVolterra/index.html | 2 +- dev/generated_examples/OpAmp.ipynb | 8 - dev/generated_examples/OpAmp/index.html | 2 +- dev/generated_examples/Platoon/index.html | 2 +- .../ProductionDestruction/index.html | 2 +- dev/generated_examples/Quadrotor/index.html | 2 +- dev/generated_examples/SEIR/index.html | 2 +- dev/generated_examples/Spacecraft/index.html | 2 +- .../SquareWaveOscillator/index.html | 2 +- dev/generated_examples/TransmissionLine.ipynb | 460 ++++++++-------- .../{7712e0e3.svg => b8ea2e8e.svg} | 230 ++++---- .../TransmissionLine/index.html | 4 +- dev/generated_examples/VanDerPol/index.html | 2 +- dev/index.html | 2 +- dev/lib/algorithms/A20/index.html | 2 +- dev/lib/algorithms/ASB07/index.html | 2 +- dev/lib/algorithms/BFFPSV18/index.html | 2 +- dev/lib/algorithms/BOX/index.html | 2 +- dev/lib/algorithms/CARLIN/index.html | 6 +- dev/lib/algorithms/GLGM06/index.html | 2 +- dev/lib/algorithms/INT/index.html | 2 +- dev/lib/algorithms/LGG09/index.html | 4 +- dev/lib/algorithms/ORBIT/index.html | 2 +- dev/lib/algorithms/QINT/index.html | 2 +- dev/lib/algorithms/TMJets/index.html | 2 +- dev/lib/algorithms/VREP/index.html | 2 +- dev/lib/clustering/index.html | 2 +- dev/lib/discretize/index.html | 8 +- dev/lib/distributed/index.html | 4 +- dev/lib/flowpipes/index.html | 2 +- dev/lib/internals/index.html | 2 +- dev/lib/operations/index.html | 2 +- dev/lib/projections/index.html | 2 +- dev/lib/reachsets/index.html | 4 +- dev/lib/solutions/index.html | 2 +- dev/lib/systems/index.html | 8 +- dev/man/algorithms/A20/index.html | 2 +- dev/man/algorithms/ASB07/index.html | 2 +- dev/man/algorithms/BFFPSV18/index.html | 2 +- dev/man/algorithms/BOX/index.html | 2 +- dev/man/algorithms/GLGM06/index.html | 2 +- dev/man/algorithms/INT/index.html | 2 +- dev/man/algorithms/LGG09/index.html | 2 +- dev/man/algorithms/ORBIT/index.html | 2 +- dev/man/algorithms/VREP/index.html | 2 +- dev/man/benchmarks/benchmarks/index.html | 2 +- .../benchmarks/filtered_oscillator/index.html | 2 +- dev/man/benchmarks/model_library/index.html | 2 +- dev/man/benchmarks/repeatability/index.html | 2 +- dev/man/examples_overview/index.html | 2 +- dev/man/faq/807473fd.png | Bin 269965 -> 0 bytes dev/man/faq/ca6f9925.png | Bin 0 -> 277763 bytes dev/man/faq/index.html | 4 +- dev/man/systems/index.html | 2 +- dev/references/index.html | 2 +- .../dense_time/{483b091b.svg => ecc66c19.svg} | 492 +++++++++--------- .../linear_methods/dense_time/index.html | 4 +- .../{4434c712.svg => 5d2983ee.svg} | 202 +++---- .../linear_methods/discrete_time/index.html | 4 +- .../{1483cb33.svg => 1261b965.svg} | 458 ++++++++-------- .../{50fd9efb.svg => 474e8f67.svg} | 462 ++++++++-------- .../{9f6fe39b.svg => 72af323b.svg} | 484 ++++++++--------- .../{990d3251.svg => 7be5f6af.svg} | 476 ++++++++--------- .../linear_methods/introduction/index.html | 8 +- .../set_representations/distances/index.html | 2 +- .../polyhedral_computations/index.html | 2 +- .../zonotopes/{54b72cc4.svg => 612d57dc.svg} | 90 ++-- .../set_representations/zonotopes/index.html | 2 +- .../domain_splitting/index.html | 2 +- .../taylor_methods/gotchas/index.html | 2 +- .../taylor_methods/introduction/index.html | 2 +- .../taylor_model_reachsets/index.html | 2 +- 81 files changed, 1764 insertions(+), 1772 deletions(-) rename dev/generated_examples/TransmissionLine/{7712e0e3.svg => b8ea2e8e.svg} (77%) delete mode 100644 dev/man/faq/807473fd.png create mode 100644 dev/man/faq/ca6f9925.png rename dev/tutorials/linear_methods/dense_time/{483b091b.svg => ecc66c19.svg} (95%) rename dev/tutorials/linear_methods/discrete_time/{4434c712.svg => 5d2983ee.svg} (76%) rename dev/tutorials/linear_methods/introduction/{1483cb33.svg => 1261b965.svg} (73%) rename dev/tutorials/linear_methods/introduction/{50fd9efb.svg => 474e8f67.svg} (74%) rename dev/tutorials/linear_methods/introduction/{9f6fe39b.svg => 72af323b.svg} (75%) rename dev/tutorials/linear_methods/introduction/{990d3251.svg => 7be5f6af.svg} (75%) rename dev/tutorials/set_representations/zonotopes/{54b72cc4.svg => 612d57dc.svg} (79%) diff --git a/dev/.documenter-siteinfo.json b/dev/.documenter-siteinfo.json index f8d119811..a745594fb 100644 --- a/dev/.documenter-siteinfo.json +++ b/dev/.documenter-siteinfo.json @@ -1 +1 @@ -{"documenter":{"julia_version":"1.10.0","generation_timestamp":"2024-01-12T15:46:17","documenter_version":"1.2.1"}} \ No newline at end of file +{"documenter":{"julia_version":"1.10.0","generation_timestamp":"2024-01-12T21:26:36","documenter_version":"1.2.1"}} \ No newline at end of file diff --git a/dev/about/index.html b/dev/about/index.html index 5a7ca6615..14fc0f4ec 100644 --- a/dev/about/index.html +++ b/dev/about/index.html @@ -1,4 +1,4 @@ How to contribute · ReachabilityAnalysis.jl
GitHub

About

This page contains some general information about this project, and recommendations about contributing.

Contributing

If you like this package, consider contributing! We welcome bug reports, examples that can be added to the documentation, or new feature proposals.

Below we list some conventions that we follow when contributing to this package. For specific guidelines on documentation see the JuliaReach Developer's Documentation.

Branches

Each pull request (PR) should be pushed in a new branch with the name of the author followed by a descriptive name, e.g. mforets/my_feature. If the branch is associated to a previous discussion in an issue, we use the name of the issue for easier lookup, e.g. mforets/7.

Unit testing and continuous integration (CI)

This project is synchronized with GitHub Actions such that each PR gets tested before merging (and the build is automatically triggered after each new commit). For the maintainability of this project, it is important to make all unit tests pass.

To run the unit tests locally, you can do:

julia> using Pkg
 
-julia> Pkg.test("ReachabilityAnalysis")

We also advise adding new unit tests when adding new features to ensure long-term support of your contributions.

Contributing to the documentation

This documentation is written in Markdown, and it relies on Documenter.jl to produce the HTML layout. To build the documentation, run make.jl:

$ julia --color=yes docs/make.jl
+julia> Pkg.test("ReachabilityAnalysis")

We also advise adding new unit tests when adding new features to ensure long-term support of your contributions.

Contributing to the documentation

This documentation is written in Markdown, and it relies on Documenter.jl to produce the HTML layout. To build the documentation, run make.jl:

$ julia --color=yes docs/make.jl
diff --git a/dev/generated_examples/Brusselator/index.html b/dev/generated_examples/Brusselator/index.html index c904aa060..c68939c12 100644 --- a/dev/generated_examples/Brusselator/index.html +++ b/dev/generated_examples/Brusselator/index.html @@ -32,4 +32,4 @@ plot!(sol_1; vars=(1, 2), lw=0.2, color=:red, lab="r = 0.1", alpha=0.4) -plot!(sol_01; vars=(1, 2), lw=0.2, color=:blue, lab="r = 0.01")Example block output

The volume at time $T = 9.0$ can be obtained by evaluating the flowpipe and computing the volume of the hyperrectangular overapproximation:

vol_01 = volume(set(overapproximate(sol_01(9.0), Hyperrectangle)))
1.7054828078460202e-5
vol_1 = volume(set(overapproximate(sol_1(9.0), Hyperrectangle)))
0.0009159502603777963

References

+plot!(sol_01; vars=(1, 2), lw=0.2, color=:blue, lab="r = 0.01")Example block output

The volume at time $T = 9.0$ can be obtained by evaluating the flowpipe and computing the volume of the hyperrectangular overapproximation:

vol_01 = volume(set(overapproximate(sol_01(9.0), Hyperrectangle)))
1.7054828078460202e-5
vol_1 = volume(set(overapproximate(sol_1(9.0), Hyperrectangle)))
0.0009159502603777963

References

diff --git a/dev/generated_examples/Building/index.html b/dev/generated_examples/Building/index.html index 5383b6689..2768c9c66 100644 --- a/dev/generated_examples/Building/index.html +++ b/dev/generated_examples/Building/index.html @@ -55,4 +55,4 @@ alg=LGG09(; δ=0.01, vars=(25), n=49, approx_model=NoBloating())) fig = plot(sol_BLDC01_discrete; vars=(0, 25), linecolor=:blue, color=:blue, alpha=0.8, lw=1.0, - xlab="t", ylab="x25")Example block output

Safety properties

ρ(x25e, sol_BLDC01_discrete)
0.004412266117562392
ρ(x25e, sol_BLDC01_discrete) <= 5.1e-3 # BLDC01 - BDS01
true
ρ(x25e, sol_BLDC01_discrete)
0.004412266117562392
ρ(x25e, sol_BLDC01_discrete) <= 4e-3 # BLDC01 - BDU01
false
ρ(x25e, sol_BLDC01_discrete(20.0))
5.582597411711546e-7
ρ(x25e, sol_BLDC01_discrete(20.0)) <= -0.78e-3 # BLDC01 - BDU02
false

References

+ xlab="t", ylab="x25")Example block output

Safety properties

ρ(x25e, sol_BLDC01_discrete)
0.004412266117562392
ρ(x25e, sol_BLDC01_discrete) <= 5.1e-3 # BLDC01 - BDS01
true
ρ(x25e, sol_BLDC01_discrete)
0.004412266117562392
ρ(x25e, sol_BLDC01_discrete) <= 4e-3 # BLDC01 - BDU01
false
ρ(x25e, sol_BLDC01_discrete(20.0))
5.582597411711546e-7
ρ(x25e, sol_BLDC01_discrete(20.0)) <= -0.78e-3 # BLDC01 - BDU02
false

References

diff --git a/dev/generated_examples/DuffingOscillator/index.html b/dev/generated_examples/DuffingOscillator/index.html index 092cf8bdd..54dbde264 100644 --- a/dev/generated_examples/DuffingOscillator/index.html +++ b/dev/generated_examples/DuffingOscillator/index.html @@ -17,4 +17,4 @@ ω = 1.2 T = 2 * pi / ω X0 = Singleton([1.0, 0.0]) ⊕ BallInf(zeros(2), 0.1) -prob = @ivp(x' = duffing!(x), x(0) ∈ X0, dim = 2);

Results

sol = solve(prob; tspan=(0.0, 20 * T), alg=TMJets21a());
fig = plot(sol; vars=(0, 1), xlab="t", ylab="x", lw=0.2, color=:blue)
Example block output
fig = plot(sol; vars=(0, 2), xlab="t", ylab="v", lw=0.2, color=:blue)
Example block output
fig = plot(sol; vars=(1, 2), xlab="x", ylab="v", lw=0.5, color=:red)
Example block output +prob = @ivp(x' = duffing!(x), x(0) ∈ X0, dim = 2);

Results

sol = solve(prob; tspan=(0.0, 20 * T), alg=TMJets21a());
fig = plot(sol; vars=(0, 1), xlab="t", ylab="x", lw=0.2, color=:blue)
Example block output
fig = plot(sol; vars=(0, 2), xlab="t", ylab="v", lw=0.2, color=:blue)
Example block output
fig = plot(sol; vars=(1, 2), xlab="x", ylab="v", lw=0.5, color=:red)
Example block output diff --git a/dev/generated_examples/EMBrake/index.html b/dev/generated_examples/EMBrake/index.html index 526095070..8a5ea2e0b 100644 --- a/dev/generated_examples/EMBrake/index.html +++ b/dev/generated_examples/EMBrake/index.html @@ -110,4 +110,4 @@ # hybrid system with clocked linear dynamics ha = HACLD1(EMbrake, reset_map, Tsample, ζ) return IVP(ha, X₀) -end
embrake_pv_2 (generic function with 1 method)

References

[]:

+end
embrake_pv_2 (generic function with 1 method)

References

[]:

diff --git a/dev/generated_examples/ISS/index.html b/dev/generated_examples/ISS/index.html index c78e92455..3ffb2b15e 100644 --- a/dev/generated_examples/ISS/index.html +++ b/dev/generated_examples/ISS/index.html @@ -53,4 +53,4 @@ xtick=[0, 5, 10, 15, 20.0], ytick=[-0.0002, -0.0001, 0.0, 0.0001, 0.0002], xlims=(0.0, 20.0), ylims=(-0.0002, 0.0002), bottom_margin=6mm, left_margin=2mm, right_margin=4mm, top_margin=3mm) -figExample block output

References

+figExample block output

References

diff --git a/dev/generated_examples/LaubLoomis/index.html b/dev/generated_examples/LaubLoomis/index.html index 54aa1d1a4..3ded81459 100644 --- a/dev/generated_examples/LaubLoomis/index.html +++ b/dev/generated_examples/LaubLoomis/index.html @@ -56,4 +56,4 @@ size=(600, 600)) plot!(fig, x -> x, x -> 4.5, 0.0, 20.0; line=2, color="red", linestyle=:dash, legend=nothing) -plot!(fig, x -> x, x -> 5.0, 0.0, 20.0; line=2, color="red", linestyle=:dash, legend=nothing)Example block output

References

+plot!(fig, x -> x, x -> 5.0, 0.0, 20.0; line=2, color="red", linestyle=:dash, legend=nothing)Example block output

References

diff --git a/dev/generated_examples/Lorenz/index.html b/dev/generated_examples/Lorenz/index.html index e18a5cbe5..2b7559c2f 100644 --- a/dev/generated_examples/Lorenz/index.html +++ b/dev/generated_examples/Lorenz/index.html @@ -21,4 +21,4 @@ solz = overapproximate(sol, Zonotope);
fig = plot(solz; vars=(0, 1), xlab="t", ylab="x")
Example block output

It is apparent by inspection that variable $x(t)$ does not exceed 20.0 in the computed time span:

fig = plot(solz(0.0 .. 1.5); vars=(0, 1), xlab="t", ylab="x", lw=0.0)
 plot!(x -> 20.0; c=:red, xlims=(0.0, 1.5), lab="")
Example block output

We can prove that it is the case by computing the support function of the flowpipe along direction [1.0, 0.0, 0.0]:

ρ([1.0, 0.0, 0.0], solz(0.0 .. 1.5))
19.991791450503634

In a similar fashion, we can compute extremal values of variable $y(t)$:

fig = plot(solz; vars=(0, 2), xlab="t", ylab="y")
Example block output

Since we have computed overapproximations of the exact flowipe, the following quantities are a lower bound on the exact minimum (resp. an upper bound on the exact maximum):

-ρ([0.0, -1.0, 0.0], solz)
 
-ρ([0.0, 1.0, 0.0], solz)
28.05101124282887
fig = plot(solz; vars=(0, 3), xlab="t", ylab="z")
Example block output

Below we plot the flowpipe projected on the (1, 3) plane.

fig = plot(solz; vars=(1, 3))
Example block output +ρ([0.0, 1.0, 0.0], solz)
28.05101124282887
fig = plot(solz; vars=(0, 3), xlab="t", ylab="z")
Example block output

Below we plot the flowpipe projected on the (1, 3) plane.

fig = plot(solz; vars=(1, 3))
Example block output diff --git a/dev/generated_examples/LotkaVolterra/index.html b/dev/generated_examples/LotkaVolterra/index.html index 7c40e411a..6640706a1 100644 --- a/dev/generated_examples/LotkaVolterra/index.html +++ b/dev/generated_examples/LotkaVolterra/index.html @@ -48,4 +48,4 @@ solz = overapproximate(sol, Zonotope) plot!(solz; vars=(1, 2), color=:blue, lw=0.3, lab="eps = 0.01", title="Uncertain u0 and uncertain params", - xlab="u1", ylab="u2")Example block output + xlab="u1", ylab="u2")Example block output diff --git a/dev/generated_examples/OpAmp.ipynb b/dev/generated_examples/OpAmp.ipynb index 7707e318e..fb3d7a44d 100644 --- a/dev/generated_examples/OpAmp.ipynb +++ b/dev/generated_examples/OpAmp.ipynb @@ -304,14 +304,6 @@ }, { "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "WARNING: using Symbolics.infimum in module ReachabilityAnalysis conflicts with an existing identifier.\n", - "WARNING: using Symbolics.supremum in module ReachabilityAnalysis conflicts with an existing identifier.\n" - ] - }, { "output_type": "execute_result", "data": { diff --git a/dev/generated_examples/OpAmp/index.html b/dev/generated_examples/OpAmp/index.html index 3d5ab109f..86a9d46d2 100644 --- a/dev/generated_examples/OpAmp/index.html +++ b/dev/generated_examples/OpAmp/index.html @@ -73,4 +73,4 @@ sol_exp = solve(prob_exp; T=0.1, alg=BOX(; δ=1e-4));
fig = plot(sol_lin; vars=(0, 2), xlab=L"t", ylab=L"e_{in}(t)", title="Linear input w/saturation",
            lw=0.2)
Example block output
fig = plot(sol_exp; vars=(0, 2), xlab=L"t", ylab=L"e_{in}(t)", title="Exponential input", lw=0.2)
Example block output
fig = plot(sol_lin; vars=(0, 1), xlab=L"t", ylab=L"e_{out}(t)", title="Solution for linear input",
            lw=0.2)
Example block output
fig = plot(sol_exp; vars=(0, 1), xlab=L"t", ylab=L"e_{out}(t)",
-           title="Solution for exponential input", lw=0.2)
Example block output

References

+ title="Solution for exponential input", lw=0.2)Example block output

References

diff --git a/dev/generated_examples/Platoon/index.html b/dev/generated_examples/Platoon/index.html index 572f998e1..7987b0411 100644 --- a/dev/generated_examples/Platoon/index.html +++ b/dev/generated_examples/Platoon/index.html @@ -129,4 +129,4 @@ intersection_method=TemplateHullIntersection(octdirs), intersect_source_invariant=false, tspan=(0.0 .. 20.0));

Verifying that the specification holds:

dmin_specification(sol_PLAD01_BND30, 30)
true

Let's check in more detail how close is the flowpipe to the safety conditions:

Minimum of $x_1(t)$:

-ρ(sparsevec([1], [-1.0], 10), sol_PLAD01_BND30)
-29.862721575258067

Minimum of $x_4(t)$:

-ρ(sparsevec([4], [-1.0], 10), sol_PLAD01_BND30)
-26.167902001889445

Minimum of $x_7(t)$:

-ρ(sparsevec([7], [-1.0], 10), sol_PLAD01_BND30)
-12.696657569596857
fig = plot(sol_PLAD01_BND30; vars=(0, 1), xlab=L"t", ylab=L"x_1", title="PLAD01 - BND30", lw=0.1)
-plot!(x -> x, x -> -30.0, 0.0, 20.0; linewidth=2, color="red", linestyle=:dash, leg=nothing)
Example block output

References

[^]:

+plot!(x -> x, x -> -30.0, 0.0, 20.0; linewidth=2, color="red", linestyle=:dash, leg=nothing)Example block output

References

[^]:

diff --git a/dev/generated_examples/ProductionDestruction/index.html b/dev/generated_examples/ProductionDestruction/index.html index 0c577d00a..e0424005e 100644 --- a/dev/generated_examples/ProductionDestruction/index.html +++ b/dev/generated_examples/ProductionDestruction/index.html @@ -79,4 +79,4 @@ du[3] = aux2 du[4] = zero(x) return du -end

With respect to parameter tuning, we have observed that for all three cases a good parameter choice is $n_T = 7$, $n_Q = 1$, and an adaptive absolute tolerance with initial value $10^{-11}$ (resp $10^{-12}$) for $I$ and $I$ and $P$ (resp. $P$). This choice offers a good balance between runtime and precision.

References

+end

With respect to parameter tuning, we have observed that for all three cases a good parameter choice is $n_T = 7$, $n_Q = 1$, and an adaptive absolute tolerance with initial value $10^{-11}$ (resp $10^{-12}$) for $I$ and $I$ and $P$ (resp. $P$). This choice offers a good balance between runtime and precision.

References

diff --git a/dev/generated_examples/Quadrotor/index.html b/dev/generated_examples/Quadrotor/index.html index 2f2f5d033..5e945143a 100644 --- a/dev/generated_examples/Quadrotor/index.html +++ b/dev/generated_examples/Quadrotor/index.html @@ -201,4 +201,4 @@ plot!(solz1; vars=(0, 3), linecolor="yellow", color=:yellow, alpha=0.8, xlab="t", ylab="x3", xtick=[0.0, 1.0, 2.0, 3.0, 4.0, 5.0], ytick=[-1.0, -0.5, 0.0, 0.5, 1.0, 1.5], - xlims=(0.0, 5.0), ylims=(-1.0, 1.5))Example block output

References

+ xlims=(0.0, 5.0), ylims=(-1.0, 1.5))Example block output

References

diff --git a/dev/generated_examples/SEIR/index.html b/dev/generated_examples/SEIR/index.html index e5f0c989e..aef03bd0d 100644 --- a/dev/generated_examples/SEIR/index.html +++ b/dev/generated_examples/SEIR/index.html @@ -42,4 +42,4 @@ plot!(fig, solz; vars=(0, 2), color=:green, lw=0.0, lab="E") plot!(fig, solz; vars=(0, 3), color=:red, lw=0.0, lab="I") plot!(solz; vars=(0, 4), color=:grey, lw=0.0, lab="R") -figExample block output +figExample block output diff --git a/dev/generated_examples/Spacecraft/index.html b/dev/generated_examples/Spacecraft/index.html index 95b48c132..24c3de15c 100644 --- a/dev/generated_examples/Spacecraft/index.html +++ b/dev/generated_examples/Spacecraft/index.html @@ -269,4 +269,4 @@ plot!(fig, solz[idx_aborting[k]]; vars=(1, 2), lw=0.0, color=:cyan, alpha=1.0) end -figExample block output

References

[1] N. Chan and S. Mitra. Verifying safety of an autonomous spacecraft rendezvous mission. In ARCH17. 4th International Workshop on Applied Verification of Continuous and Hybrid Systems, collocated with Cyber-Physical Systems Week (CPSWeek) on April 17, 2017 in Pittsburgh, PA, USA, pages 20–32, 2017.

+figExample block output

References

[1] N. Chan and S. Mitra. Verifying safety of an autonomous spacecraft rendezvous mission. In ARCH17. 4th International Workshop on Applied Verification of Continuous and Hybrid Systems, collocated with Cyber-Physical Systems Week (CPSWeek) on April 17, 2017 in Pittsburgh, PA, USA, pages 20–32, 2017.

diff --git a/dev/generated_examples/SquareWaveOscillator/index.html b/dev/generated_examples/SquareWaveOscillator/index.html index 26472d7bf..13a0e71ec 100644 --- a/dev/generated_examples/SquareWaveOscillator/index.html +++ b/dev/generated_examples/SquareWaveOscillator/index.html @@ -83,4 +83,4 @@ ReachSet{Float64, Hyperrectangle{Float64, Vector{Float64}, Vector{Float64}}}(Hyperrectangle{Float64, Vector{Float64}, Vector{Float64}}([6.7258286982437765], [0.024171301756223507]), [0.000316999, 0.000320001])

Plotting Xc matches with the flowpipe after the jump:

fig = plot(sol[1][(end - 10):end]; vars=(0, 1))
 plot!(sol[2][1:10]; vars=(0, 1))
 plot!(x -> 6.75; xlims=(3.1e-4, 3.3e-4), lab="Guard", lw=2.0, color=:red)
-plot!(Xc[1]; vars=(0, 1), c=:grey)
Example block output

Finally, we note that the algorithm finds an invariant of the system after the first period. To activate such check pass the fixpoint_check=true flag to the hybrid solve API.

sol = solve(prob; T=100e-4, alg=INT(; δ=1.E-6), fixpoint_check=true);
fig = plot(sol; vars=(0, 1), xlab="t", ylab="v-")
Example block output

When the fixpoint check is activated, the computation terminates as soon as the last reach-set is contained in a previously explored initial state.

tspan(sol)
[0, 0.00145601]

References

+plot!(Xc[1]; vars=(0, 1), c=:grey)Example block output

Finally, we note that the algorithm finds an invariant of the system after the first period. To activate such check pass the fixpoint_check=true flag to the hybrid solve API.

sol = solve(prob; T=100e-4, alg=INT(; δ=1.E-6), fixpoint_check=true);
fig = plot(sol; vars=(0, 1), xlab="t", ylab="v-")
Example block output

When the fixpoint check is activated, the computation terminates as soon as the last reach-set is contained in a previously explored initial state.

tspan(sol)
[0, 0.00145601]

References

diff --git a/dev/generated_examples/TransmissionLine.ipynb b/dev/generated_examples/TransmissionLine.ipynb index 3383dc1f3..3f163a6be 100644 --- a/dev/generated_examples/TransmissionLine.ipynb +++ b/dev/generated_examples/TransmissionLine.ipynb @@ -208,264 +208,264 @@ "\n", "\n", "\n", - " \n", + " \n", " \n", " \n", "\n", - "\n", + "\n", "\n", - " \n", + " \n", " \n", " \n", "\n", - "\n", + "\n", "\n", - " \n", + " \n", " \n", " \n", "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", "\n" ], "image/svg+xml": [ "\n", "\n", "\n", - " \n", + " \n", " \n", " \n", "\n", - "\n", + "\n", "\n", - " \n", + " \n", " \n", " \n", "\n", - "\n", + "\n", "\n", - " \n", + " \n", " \n", " \n", "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", "\n" ] }, diff --git a/dev/generated_examples/TransmissionLine/7712e0e3.svg b/dev/generated_examples/TransmissionLine/b8ea2e8e.svg similarity index 77% rename from dev/generated_examples/TransmissionLine/7712e0e3.svg rename to dev/generated_examples/TransmissionLine/b8ea2e8e.svg index 4507cbda7..d8a0dc936 100644 --- a/dev/generated_examples/TransmissionLine/7712e0e3.svg +++ b/dev/generated_examples/TransmissionLine/b8ea2e8e.svg @@ -1,130 +1,130 @@ - + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/dev/generated_examples/TransmissionLine/index.html b/dev/generated_examples/TransmissionLine/index.html index ea6f853fa..6a814ec14 100644 --- a/dev/generated_examples/TransmissionLine/index.html +++ b/dev/generated_examples/TransmissionLine/index.html @@ -14,7 +14,7 @@ end
tline (generic function with 1 method)

We can visualize the structure of the coefficients matrix $A$ for the case $\eta = 20$ with spy plot:

using Plots
 
 A, _ = tline(; η=20)
-spy(A; legend=nothing, markersize=2.0, title="Sparsity pattern of A", xlab="columns", ylab="rows")
Example block output

Notice that the matrix coefficients are rather big, hence it is convenient to rescale the system for numerical stability. Let $α > 0$ be a scaling factor, and let $\tilde{x}(t) = x(\alpha t)$. By the chain rule,

\[\tilde{x}'(t) = \alpha x'(\alpha t) = \alpha A x(\alpha t) + \alpha B U_{in}(\alpha t) = \tilde{A} \tilde{x}(t) + \tilde{B} \tilde{U}_{in}(t),\]

where $\tilde{A} := \alpha A$ and $\tilde{B} := \alpha B$.

function scale!(s, α=1.0)
+spy(A; legend=nothing, markersize=2.0, title="Sparsity pattern of A", xlab="columns", ylab="rows")
Example block output

Notice that the matrix coefficients are rather big, hence it is convenient to rescale the system for numerical stability. Let $α > 0$ be a scaling factor, and let $\tilde{x}(t) = x(\alpha t)$. By the chain rule,

\[\tilde{x}'(t) = \alpha x'(\alpha t) = \alpha A x(\alpha t) + \alpha B U_{in}(\alpha t) = \tilde{A} \tilde{x}(t) + \tilde{B} \tilde{U}_{in}(t),\]

where $\tilde{A} := \alpha A$ and $\tilde{B} := \alpha B$.

function scale!(s, α=1.0)
     s.A .*= α
     s.B .*= α
     return s
@@ -32,4 +32,4 @@
 scale!(s, α);
 
 T = 0.7 * 1e-11 # time horizon
-P = InitialValueProblem(s, X0);

Results

We solve system using a step size of δ=1e-3 and the box algorithm.

sol = solve(P; T=0.7, alg=BOX(; δ=1e-3));

To get the variable $U_{out}$ we have to project onto the $η$-th coordinate and invert the sign of the flowpipe.

Uout_vs_t = @. (-1.0) * project(sol, η);
fig = plot(Uout_vs_t; vars=(0, η), color=:blue, xlab="t", ylab="Uout", alpha=0.5, lw=0.5)
Example block output
Technical note

Since we are only interested in the behavior of $U_{out}$, we could use a support function based algorithm (see e.g. [GLG08] which can be used as LGG09 in this library), or the decomposition-based algorithm BFFPSV18 with the options alg=BFFPSV18(δ=1e-3, dim=statedim(P), vars=[η])). Those options will use an interval (1D) decomposition of the state space and only compute the flowpipe associated with variable $\eta$.

References

+P = InitialValueProblem(s, X0);

Results

We solve system using a step size of δ=1e-3 and the box algorithm.

sol = solve(P; T=0.7, alg=BOX(; δ=1e-3));

To get the variable $U_{out}$ we have to project onto the $η$-th coordinate and invert the sign of the flowpipe.

Uout_vs_t = @. (-1.0) * project(sol, η);
fig = plot(Uout_vs_t; vars=(0, η), color=:blue, xlab="t", ylab="Uout", alpha=0.5, lw=0.5)
Example block output
Technical note

Since we are only interested in the behavior of $U_{out}$, we could use a support function based algorithm (see e.g. [GLG08] which can be used as LGG09 in this library), or the decomposition-based algorithm BFFPSV18 with the options alg=BFFPSV18(δ=1e-3, dim=statedim(P), vars=[η])). Those options will use an interval (1D) decomposition of the state space and only compute the flowpipe associated with variable $\eta$.

References

diff --git a/dev/generated_examples/VanDerPol/index.html b/dev/generated_examples/VanDerPol/index.html index 5e6737b47..5c31b1067 100644 --- a/dev/generated_examples/VanDerPol/index.html +++ b/dev/generated_examples/VanDerPol/index.html @@ -33,4 +33,4 @@ return LineSegment(vl[1], vl[2]) end
cross_section (generic function with 1 method)

Then we can get the cross section of the first five sets and the last set, calling them i1 and i2 respectively.

ifirst = cross_section(line, solz, 1:13)
 ilast = cross_section(line, solz, [388])
LineSegment{Float64, Vector{Float64}}([1.3848606487034292, 2.1924303243517147], [1.4836620847695106, 2.2418310423847556])

We can also calculate the length of each cross section, remember that the system is 2D, so the cross section will be a line segment.

lfirst = norm(ifirst.q - ifirst.p)
0.28930757953811087
llast = norm(ilast.q - ilast.p);
fig = plot(ifirst; lw=3.0, alpha=1.0, label="First subsets", legend=:bottomright)
-plot!(ilast; lw=5.0, alpha=1.0, label="Last subset")
Example block output

The inclusion check succeeds:

ilast ⊆ ifirst
true

We can see, the cross section of the last subset is a subset of the first few sets, thus, the cycle will continue, presumably getting smaller each revolution.

References

+plot!(ilast; lw=5.0, alpha=1.0, label="Last subset")Example block output

The inclusion check succeeds:

ilast ⊆ ifirst
true

We can see, the cross section of the last subset is a subset of the first few sets, thus, the cycle will continue, presumably getting smaller each revolution.

References

diff --git a/dev/index.html b/dev/index.html index b65df2107..d326a0f01 100644 --- a/dev/index.html +++ b/dev/index.html @@ -1,2 +1,2 @@ -Overview · ReachabilityAnalysis.jl
GitHub

ReachabilityAnalysis.jl

What is reachability analysis?

Reachability analysis is a numerical method that aims to compute sets of states reachable by dynamical systems from all initial states and for all admissible inputs and parameters.

Where is reachability being applied?

Reachability analysis is of interest in diverse domains including automotive, aerospace, power systems, system biology, analog/mixed-signal circuits, robotics, and safe machine learning.

Below we briefly comment on a few remarkable problems where reachability is being applied. We refer to the technical literature for further applications; see Bibliography for a selection of papers to get you started! An up-to-date review with relevant references can be found in [AFG20].

Formal verification. Determining whether a system is safe, i.e. to verify if it does not enter into a region of unsafe sets. Typical applications are assessing the critical distance between autonomous vehicles or robots, or critical concentration of chemicals in a reactor.

Validation of control strategies. Checking if the system trajectories stay in a region around a reference trajectory, or reach a goal region around a setpoint, for any admissible value of the non-deterministic inputs, initial conditions or noise.

Controller synthesis. The controller synthesis problem consists in finding parameter sets of controllers that satisfy safety or performance constraints.

Deep neural network verification. Providing formal guarantees for the network behavior subject to perturbations in the inputs, e.g. detecting that small changes in an input image do not cause the network o misclassify it.

How to read this manual

Please note that the manual contains a Frequently Asked Questions (FAQ) section with a selection of different questions and answers, many of them with working code so that you can also experiment with the answers.

Note

If you have questions, want to chat about how to apply this package, or if you found a bug or have trouble running an example, feel free to open an issue or make a pull request on the project's GitHub page. You can also find us at the #reachability-analysis stream on JuliaLang's Zulip channel, and at the JuliaReach Gitter channel.

Features

This library implements reachability analysis methods for systems of ordinary differential equations (ODEs), for both continuous and hybrid dynamical systems.

The following types of systems are currently supported:

  • Continuous ODEs with linear dynamics.
  • Continuous ODEs with non-linear dynamics.
  • Hybrid systems with piecewise-affine dynamics.
  • Hybrid systems with non-linear dynamics.
  • Hybrid systems with space-triggered or time-triggered transitions.

The library can handle uncertainties in the sets of initial states, inputs, or parameter variation.

How to install the package?

You can easily install ReachabilityAnalysis.jl from your Julia console (press ] in the Julia REPL to enter the pkg> mode):

pkg> add ReachabilityAnalysis
+Overview · ReachabilityAnalysis.jl
GitHub

ReachabilityAnalysis.jl

What is reachability analysis?

Reachability analysis is a numerical method that aims to compute sets of states reachable by dynamical systems from all initial states and for all admissible inputs and parameters.

Where is reachability being applied?

Reachability analysis is of interest in diverse domains including automotive, aerospace, power systems, system biology, analog/mixed-signal circuits, robotics, and safe machine learning.

Below we briefly comment on a few remarkable problems where reachability is being applied. We refer to the technical literature for further applications; see Bibliography for a selection of papers to get you started! An up-to-date review with relevant references can be found in [AFG20].

Formal verification. Determining whether a system is safe, i.e. to verify if it does not enter into a region of unsafe sets. Typical applications are assessing the critical distance between autonomous vehicles or robots, or critical concentration of chemicals in a reactor.

Validation of control strategies. Checking if the system trajectories stay in a region around a reference trajectory, or reach a goal region around a setpoint, for any admissible value of the non-deterministic inputs, initial conditions or noise.

Controller synthesis. The controller synthesis problem consists in finding parameter sets of controllers that satisfy safety or performance constraints.

Deep neural network verification. Providing formal guarantees for the network behavior subject to perturbations in the inputs, e.g. detecting that small changes in an input image do not cause the network o misclassify it.

How to read this manual

Please note that the manual contains a Frequently Asked Questions (FAQ) section with a selection of different questions and answers, many of them with working code so that you can also experiment with the answers.

Note

If you have questions, want to chat about how to apply this package, or if you found a bug or have trouble running an example, feel free to open an issue or make a pull request on the project's GitHub page. You can also find us at the #reachability-analysis stream on JuliaLang's Zulip channel, and at the JuliaReach Gitter channel.

Features

This library implements reachability analysis methods for systems of ordinary differential equations (ODEs), for both continuous and hybrid dynamical systems.

The following types of systems are currently supported:

  • Continuous ODEs with linear dynamics.
  • Continuous ODEs with non-linear dynamics.
  • Hybrid systems with piecewise-affine dynamics.
  • Hybrid systems with non-linear dynamics.
  • Hybrid systems with space-triggered or time-triggered transitions.

The library can handle uncertainties in the sets of initial states, inputs, or parameter variation.

How to install the package?

You can easily install ReachabilityAnalysis.jl from your Julia console (press ] in the Julia REPL to enter the pkg> mode):

pkg> add ReachabilityAnalysis
diff --git a/dev/lib/algorithms/A20/index.html b/dev/lib/algorithms/A20/index.html index 3dd4f9d24..cf0f4ef88 100644 --- a/dev/lib/algorithms/A20/index.html +++ b/dev/lib/algorithms/A20/index.html @@ -1,2 +1,2 @@ -A20 · ReachabilityAnalysis.jl
GitHub
ReachabilityAnalysis.A20 — Type
A20{N} <: AbstractContinuousPost

Implementation of the reachability method for large linear systems with uncertain inputs in the Krylov subspace from [A20].

Fields

  • δ – step-size of the discretization
  • max_order – (optional, default: 5) maximum zonotope order

References

See [A20] and references therein.

source
+A20 · ReachabilityAnalysis.jl
GitHub
ReachabilityAnalysis.A20 — Type
A20{N} <: AbstractContinuousPost

Implementation of the reachability method for large linear systems with uncertain inputs in the Krylov subspace from [A20].

Fields

  • δ – step-size of the discretization
  • max_order – (optional, default: 5) maximum zonotope order

References

See [A20] and references therein.

source
diff --git a/dev/lib/algorithms/ASB07/index.html b/dev/lib/algorithms/ASB07/index.html index 95acca858..6a85d23d6 100644 --- a/dev/lib/algorithms/ASB07/index.html +++ b/dev/lib/algorithms/ASB07/index.html @@ -1,2 +1,2 @@ -ASB07 · ReachabilityAnalysis.jl
GitHub
ReachabilityAnalysis.ASB07 — Type
ASB07{N, AM, RM, S, R} <: AbstractContinuousPost

Implementation of Althoff - Stursberg - Buss algorithm for reachability of linear systems with uncertain parameters and inputs using zonotopes.

Fields

  • δ – step-size of the discretization
  • approx_model – (optional, default: CorrectionHull(; order=10, exp=:interval)) approximation model; see Notes below for possible options
  • max_order – (optional, default: 5) maximum zonotope order
  • reduction_method – (optional, default: GIR05()) zonotope order reduction method used
  • static – (optional, default: false) if true, convert the problem data to statically sized arrays
  • recursive – (optional default: true) if true, use the implementation that recursively computes each reach-set; otherwise, use the implementation that unwraps the sequence until the initial set

Notes

The type fields are:

  • N – number type of the step-size
  • AM – type of the approximation model
  • RM – type associated to the reduction method
  • S – value type associated to the static option
  • R – value type associated to the recursive option

The sole parameter which doesn't have a default value is the step-size, associated to the type parameter N.

The default approximation model is

approx_model=CorrectionHull(order=10, exp=:base)

Here, CorrectionHull refers to an implementation of the interval matrix approximation method described in [ASB07]. For technicalities on interval matrix operations, we refer to the package IntervalMatrices.jl.

References

The main ideas behind this algorithm can be found in [ASB07]. These methods are discussed at length in the dissertation [ALT10].

Regarding the zonotope order reduction methods, we refer to [COMB03], [GIR05] and the review article [YS18].

source
+ASB07 · ReachabilityAnalysis.jl
GitHub
ReachabilityAnalysis.ASB07 — Type
ASB07{N, AM, RM, S, R} <: AbstractContinuousPost

Implementation of Althoff - Stursberg - Buss algorithm for reachability of linear systems with uncertain parameters and inputs using zonotopes.

Fields

  • δ – step-size of the discretization
  • approx_model – (optional, default: CorrectionHull(; order=10, exp=:interval)) approximation model; see Notes below for possible options
  • max_order – (optional, default: 5) maximum zonotope order
  • reduction_method – (optional, default: GIR05()) zonotope order reduction method used
  • static – (optional, default: false) if true, convert the problem data to statically sized arrays
  • recursive – (optional default: true) if true, use the implementation that recursively computes each reach-set; otherwise, use the implementation that unwraps the sequence until the initial set

Notes

The type fields are:

  • N – number type of the step-size
  • AM – type of the approximation model
  • RM – type associated to the reduction method
  • S – value type associated to the static option
  • R – value type associated to the recursive option

The sole parameter which doesn't have a default value is the step-size, associated to the type parameter N.

The default approximation model is

approx_model=CorrectionHull(order=10, exp=:base)

Here, CorrectionHull refers to an implementation of the interval matrix approximation method described in [ASB07]. For technicalities on interval matrix operations, we refer to the package IntervalMatrices.jl.

References

The main ideas behind this algorithm can be found in [ASB07]. These methods are discussed at length in the dissertation [ALT10].

Regarding the zonotope order reduction methods, we refer to [COMB03], [GIR05] and the review article [YS18].

source
diff --git a/dev/lib/algorithms/BFFPSV18/index.html b/dev/lib/algorithms/BFFPSV18/index.html index bcec54b9d..0285554f9 100644 --- a/dev/lib/algorithms/BFFPSV18/index.html +++ b/dev/lib/algorithms/BFFPSV18/index.html @@ -1,2 +1,2 @@ -BFFPSV18 · ReachabilityAnalysis.jl
GitHub

BFFPSV18

ReachabilityAnalysis.BFFPSV18 — Type
BFFPSV18{N, ST, AM, IDX, BLK, RBLK, CBLK} <: AbstractContinuousPost

Implementation of the reachability method for linear systems using block decompositions.

Fields

  • δ – step-size of the discretization
  • approx_model – (optional, default: Forward) approximation model; see Notes below for possible options
  • vars – vector with the variables of interest
  • block_indices – vector of integers to index each block that contains a variable of interest
  • row_blocks – vector of integer vectors to index variables associated to blocks of interest
  • column_blocks – vector of integer vectors to index variables in the partition
  • lazy_initial_set – (optional, default: false) if true, use a lazy decomposition of the initial states after discretization
  • lazy_input – (optional, default: false) if true, use a lazy decomposition of the input set after discretization
  • sparse – (optional, default: false) if true, assume that the state transition matrix is sparse
  • view – (optional, default: false) if true, use implementation that uses arrays views

matrix is sparse

See the Examples section below for some concrete examples of these options.

Notes

This algorithm solves the set-based recurrence equation $X_{k+1} = ΦX_k ⊕ V_k$ by using block decompositions. The algorithm was introduced in [BFFPSV18].

Comments about some fields:

  • N – number type of the step-size, e.g. Float64
  • ST – set representation used; this is either a concrete LazySet subtype, eg. Interval{Float64}, or a tuple of concrete LazySet subtypes that is commensurate with the partition

The default approximation model is:

Forward(sih=:concrete, exp=:base, setops=:lazy)

TODO:

  • clarify assumption about contiguous blocks

Examples

References

This algorithm is essentially an extension of the method in [BFFPSV18]. Blocks can have different dimensions and the set representation can be different for each block.

For a general introduction we refer to the dissertation [SCHI18].

Regarding the approximation model, by default we use an adaptation of the method presented in [FRE11].

source
+BFFPSV18 · ReachabilityAnalysis.jl
GitHub

BFFPSV18

ReachabilityAnalysis.BFFPSV18 — Type
BFFPSV18{N, ST, AM, IDX, BLK, RBLK, CBLK} <: AbstractContinuousPost

Implementation of the reachability method for linear systems using block decompositions.

Fields

  • δ – step-size of the discretization
  • approx_model – (optional, default: Forward) approximation model; see Notes below for possible options
  • vars – vector with the variables of interest
  • block_indices – vector of integers to index each block that contains a variable of interest
  • row_blocks – vector of integer vectors to index variables associated to blocks of interest
  • column_blocks – vector of integer vectors to index variables in the partition
  • lazy_initial_set – (optional, default: false) if true, use a lazy decomposition of the initial states after discretization
  • lazy_input – (optional, default: false) if true, use a lazy decomposition of the input set after discretization
  • sparse – (optional, default: false) if true, assume that the state transition matrix is sparse
  • view – (optional, default: false) if true, use implementation that uses arrays views

matrix is sparse

See the Examples section below for some concrete examples of these options.

Notes

This algorithm solves the set-based recurrence equation $X_{k+1} = ΦX_k ⊕ V_k$ by using block decompositions. The algorithm was introduced in [BFFPSV18].

Comments about some fields:

  • N – number type of the step-size, e.g. Float64
  • ST – set representation used; this is either a concrete LazySet subtype, eg. Interval{Float64}, or a tuple of concrete LazySet subtypes that is commensurate with the partition

The default approximation model is:

Forward(sih=:concrete, exp=:base, setops=:lazy)

TODO:

  • clarify assumption about contiguous blocks

Examples

References

This algorithm is essentially an extension of the method in [BFFPSV18]. Blocks can have different dimensions and the set representation can be different for each block.

For a general introduction we refer to the dissertation [SCHI18].

Regarding the approximation model, by default we use an adaptation of the method presented in [FRE11].

source
diff --git a/dev/lib/algorithms/BOX/index.html b/dev/lib/algorithms/BOX/index.html index 97bee3a31..ee1f0024e 100644 --- a/dev/lib/algorithms/BOX/index.html +++ b/dev/lib/algorithms/BOX/index.html @@ -1,2 +1,2 @@ -BOX · ReachabilityAnalysis.jl
GitHub
ReachabilityAnalysis.BOX — Type
BOX{N, AM, S, D, R} <: AbstractContinuousPost

Implementation of a reachability method for linear systems using box approximations.

Fields

  • δ – step-size of the discretization
  • approx_model – (optional, default: Forward) approximation model; see Notes below for possible options
  • static
  • dim – (optional default: missing) ambient dimension
  • recursive – (optional default: false) if true, use the implementation that recursively computes each reach-set; otherwise, use the implementation that unwraps the sequence until the initial set

Notes

The type fields are:

  • N – number type of the step-size
  • AM – approximation model
  • S – value type for the static option
  • D – value type for the dimension
  • R – value type for the recursive option

The default approximation model is:

Forward(sih=:concrete, exp=:base, setops=:lazy)

This algorithm solves the set-based recurrence equation $X_{k+1} = ΦX_k ⊕ V_k$ by computing a tight hyperrectangular over-approximation of $X_{k+1}$ at each step $k ∈ \mathbb{N}$. The recursive implementation uses the previously computed set $X_k$ to compute $X_{k+1}$. However, it is known that this method incurs wrapping effects. The non-recursive implementation instead computes $X_{k+1}$ by unwrapping the discrete recurrence until $X_0 = Ω₀$, at the expense of computing powers of the matrix $Φ$. These ideas are discussed in [BFFPSV18].

References

This algorithm is essentially a non-decomposed version of the method in [BFFPSV18], using hyperrectangles as set representation. For a general introduction we refer to the dissertation [LG09].

Regarding the approximation model, by default we use an adaptation of the method presented in [FRE11].

source
+BOX · ReachabilityAnalysis.jl
GitHub
ReachabilityAnalysis.BOX — Type
BOX{N, AM, S, D, R} <: AbstractContinuousPost

Implementation of a reachability method for linear systems using box approximations.

Fields

  • δ – step-size of the discretization
  • approx_model – (optional, default: Forward) approximation model; see Notes below for possible options
  • static
  • dim – (optional default: missing) ambient dimension
  • recursive – (optional default: false) if true, use the implementation that recursively computes each reach-set; otherwise, use the implementation that unwraps the sequence until the initial set

Notes

The type fields are:

  • N – number type of the step-size
  • AM – approximation model
  • S – value type for the static option
  • D – value type for the dimension
  • R – value type for the recursive option

The default approximation model is:

Forward(sih=:concrete, exp=:base, setops=:lazy)

This algorithm solves the set-based recurrence equation $X_{k+1} = ΦX_k ⊕ V_k$ by computing a tight hyperrectangular over-approximation of $X_{k+1}$ at each step $k ∈ \mathbb{N}$. The recursive implementation uses the previously computed set $X_k$ to compute $X_{k+1}$. However, it is known that this method incurs wrapping effects. The non-recursive implementation instead computes $X_{k+1}$ by unwrapping the discrete recurrence until $X_0 = Ω₀$, at the expense of computing powers of the matrix $Φ$. These ideas are discussed in [BFFPSV18].

References

This algorithm is essentially a non-decomposed version of the method in [BFFPSV18], using hyperrectangles as set representation. For a general introduction we refer to the dissertation [LG09].

Regarding the approximation model, by default we use an adaptation of the method presented in [FRE11].

source
diff --git a/dev/lib/algorithms/CARLIN/index.html b/dev/lib/algorithms/CARLIN/index.html index 40d3e611e..148aedfde 100644 --- a/dev/lib/algorithms/CARLIN/index.html +++ b/dev/lib/algorithms/CARLIN/index.html @@ -1,9 +1,9 @@ -CARLIN · ReachabilityAnalysis.jl
GitHub

CARLIN

ReachabilityAnalysis.CARLIN — Type
CARLIN <: AbstractContinuousPost

Implementation of the reachability method using Carleman linearization from [1].

[1] Forets, Marcelo, and Christian Schilling. "Reachability of weakly nonlinear systems using Carleman linearization." International Conference on Reachability Problems. Springer, Cham, 2021.

source
ReachabilityAnalysis.kron_pow — Function
kron_pow(x::IA.Interval, pow::Int)

Given an interval x and an integer pow, compute x^pow.

Input

  • x – interval
  • pow – integer

Output

An interval enclosure of x^pow.

Examples

julia> [kron_pow(2 .. 3, i) for i in 1:3]
+CARLIN · ReachabilityAnalysis.jl
GitHub
CARLIN
ReachabilityAnalysis.CARLIN — Type
CARLIN <: AbstractContinuousPost
Implementation of the reachability method using Carleman linearization from [1].
[1] Forets, Marcelo, and Christian Schilling. "Reachability of weakly nonlinear systems using Carleman linearization."     International Conference on Reachability Problems. Springer, Cham, 2021.
source
ReachabilityAnalysis.kron_pow — Function
kron_pow(x::IA.Interval, pow::Int)
Given an interval x and an integer pow, compute x^pow.
Input
  • x   – interval
  • pow – integer
Output
An interval enclosure of x^pow.
Examples
julia> [kron_pow(2 .. 3, i) for i in 1:3]
 3-element Array{IntervalArithmetic.Interval{Float64},1}:
   [2, 3]
   [4, 9]
- [8, 27]
source
kron_pow(x::Interval, pow::Int)
Given an interval x and an integer pow, compute x^pow.
Input
  • x   – interval
  • pow – integer
Output
An interval enclosure of x^pow as a LazySets Interval.
source
kron_pow(H::AbstractHyperrectangle, pow::Int)
Given hyperrectangular set H and an integer pow, compute the Kronecker power H^{⊗ pow}.
Input
  • H   – hyperrectangular set
  • pow – integer power
Output
A hyperrectangle.
Algorithm
We compute H^{⊗ pow} where H is a hyperrectangular set by working with H as a product of intervals.
See also kron_pow which requires DynamicPolynomials.jl.
source
kron_pow(x::Vector{<:AbstractVariable}, pow::Int)
Compute the higher order concrete Kronecker power: x ⊗ x ⊗ ... ⊗ x, pow times for a vector of symbolic monomials.
Input
  • x   – polynomial variable
  • pow – integer
Output
Vector of multivariate monomial corresponding to x^{⊗ pow}.
Examples
julia> using DynamicPolynomials
+ [8, 27]
source
kron_pow(x::Interval, pow::Int)
Given an interval x and an integer pow, compute x^pow.
Input
  • x   – interval
  • pow – integer
Output
An interval enclosure of x^pow as a LazySets Interval.
source
kron_pow(H::AbstractHyperrectangle, pow::Int)
Given hyperrectangular set H and an integer pow, compute the Kronecker power H^{⊗ pow}.
Input
  • H   – hyperrectangular set
  • pow – integer power
Output
A hyperrectangle.
Algorithm
We compute H^{⊗ pow} where H is a hyperrectangular set by working with H as a product of intervals.
See also kron_pow which requires DynamicPolynomials.jl.
source
kron_pow(x::Vector{<:AbstractVariable}, pow::Int)
Compute the higher order concrete Kronecker power: x ⊗ x ⊗ ... ⊗ x, pow times for a vector of symbolic monomials.
Input
  • x   – polynomial variable
  • pow – integer
Output
Vector of multivariate monomial corresponding to x^{⊗ pow}.
Examples
julia> using DynamicPolynomials
 
 julia> @polyvar x[1:2]
 (PolyVar{true}[x₁, x₂],)
@@ -18,4 +18,4 @@
  x₁²
  x₁x₂
  x₁x₂
- x₂²
source
ReachabilityAnalysis.kron_pow_stack — Function
kron_pow_stack(x::IA.Interval, pow::Int)
Return a hyperrectangle with the interval powers [x, x^2, …, x^pow].
Input
  • x   – interval
  • pow – integer power
Output
A hyperrectangle such that the i-th dimension is the interval x^i.
source
kron_pow_stack(x::Interval, pow::Int)
Return a hyperrectangle with the interval powers [x, x^2, …, x^pow].
Input
  • x   – interval
  • pow – integer power
Output
A hyperrectangle such that the i-th dimension is the interval x^i.
source
kron_pow_stack(H::AbstractHyperrectangle, pow::Int)
Return the Cartesian product array $H × H^{⊗2} × ⋯ × H^{⊗pow}$ where $H$ is a hyperrectangular set and $H^{⊗ i}$ is the i-th Kronecker power of $H$.
Input
  • H   – hyperrectangular set
  • pow – integer power
Output
A Cartesian product array of hyperrectangles.
source

+ x₂²
source
ReachabilityAnalysis.kron_pow_stack — Function
kron_pow_stack(x::IA.Interval, pow::Int)

Return a hyperrectangle with the interval powers [x, x^2, …, x^pow].

Input

  • x – interval
  • pow – integer power

Output

A hyperrectangle such that the i-th dimension is the interval x^i.

source
kron_pow_stack(x::Interval, pow::Int)

Return a hyperrectangle with the interval powers [x, x^2, …, x^pow].

Input

  • x – interval
  • pow – integer power

Output

A hyperrectangle such that the i-th dimension is the interval x^i.

source
kron_pow_stack(H::AbstractHyperrectangle, pow::Int)

Return the Cartesian product array $H × H^{⊗2} × ⋯ × H^{⊗pow}$ where $H$ is a hyperrectangular set and $H^{⊗ i}$ is the i-th Kronecker power of $H$.

Input

  • H – hyperrectangular set
  • pow – integer power

Output

A Cartesian product array of hyperrectangles.

source
diff --git a/dev/lib/algorithms/GLGM06/index.html b/dev/lib/algorithms/GLGM06/index.html index cd47b725c..1e8ab5c69 100644 --- a/dev/lib/algorithms/GLGM06/index.html +++ b/dev/lib/algorithms/GLGM06/index.html @@ -1,2 +1,2 @@ -GLGM06 · ReachabilityAnalysis.jl
GitHub
ReachabilityAnalysis.GLGM06 — Type
GLGM06{N, AM, S, D, NG, P, RM} <: AbstractContinuousPost

Implementation of Girard - Le Guernic - Maler algorithm for reachability of linear systems using zonotopes.

Fields

  • δ – step-size of the discretization
  • approx_model – (optional, default: FirstOrderZonotope) approximation model; see Notes below for possible options
  • max_order – (optional, default: 5) maximum zonotope order
  • static – (optional, default: false) if true, convert the problem data to statically sized arrays
  • dim – (optional default: missing) ambient dimension
  • ngens – (optional, default: missing) number of generators
  • preallocate – (optional, default: true) if true, use the implementation which preallocates the zonotopes prior to applying the update rule
  • reduction_method – (optional, default: GIR05()) zonotope order reduction method used
  • disjointness_method – (optional, default: NoEnclosure()) method to check disjointness between the reach-set and the invariant

Notes

The type fields are:

  • N – number type of the step-size
  • AM – approximation model
  • S – value type associated to the static option
  • D – value type associated to the dimension of the system
  • NG – value type associated to the number of generators
  • P – value type associated to the preallocate option
  • RM – type associated to the reduction method

The sole parameter which doesn't have a default value is the step-size, associated to the type parameter N. Parameters D and NG are optionally specified (default to Missing). These parameters are needed for implementations that require the size of the zonotopes to be known (fixed) at compile time, namely the static=true version of this algorithm. Otherwise, the number of generators is not necessarily fixed.

The default approximation model is

approx_model=FirstOrderZonotope()

Here, FirstOrderZonotope refers to the forward-time adaptation of the approximation model from Lemma 3 in [FRE11]. Some of the options to compute this approximation can be specified, see the documentation of FirstOrderZonotope for details.

References

The main ideas behind this algorithm can be found in [GIR05] and [GLGM06]. These methods are discussed at length in the dissertation [LG09].

Regarding the zonotope order reduction methods, we refer to [COMB03], [GIR05] and the review article [YS18].

Regarding the approximation model, we use an adaptation of a result in [FRE11].

source
+GLGM06 · ReachabilityAnalysis.jl
GitHub
ReachabilityAnalysis.GLGM06 — Type
GLGM06{N, AM, S, D, NG, P, RM} <: AbstractContinuousPost

Implementation of Girard - Le Guernic - Maler algorithm for reachability of linear systems using zonotopes.

Fields

  • δ – step-size of the discretization
  • approx_model – (optional, default: FirstOrderZonotope) approximation model; see Notes below for possible options
  • max_order – (optional, default: 5) maximum zonotope order
  • static – (optional, default: false) if true, convert the problem data to statically sized arrays
  • dim – (optional default: missing) ambient dimension
  • ngens – (optional, default: missing) number of generators
  • preallocate – (optional, default: true) if true, use the implementation which preallocates the zonotopes prior to applying the update rule
  • reduction_method – (optional, default: GIR05()) zonotope order reduction method used
  • disjointness_method – (optional, default: NoEnclosure()) method to check disjointness between the reach-set and the invariant

Notes

The type fields are:

  • N – number type of the step-size
  • AM – approximation model
  • S – value type associated to the static option
  • D – value type associated to the dimension of the system
  • NG – value type associated to the number of generators
  • P – value type associated to the preallocate option
  • RM – type associated to the reduction method

The sole parameter which doesn't have a default value is the step-size, associated to the type parameter N. Parameters D and NG are optionally specified (default to Missing). These parameters are needed for implementations that require the size of the zonotopes to be known (fixed) at compile time, namely the static=true version of this algorithm. Otherwise, the number of generators is not necessarily fixed.

The default approximation model is

approx_model=FirstOrderZonotope()

Here, FirstOrderZonotope refers to the forward-time adaptation of the approximation model from Lemma 3 in [FRE11]. Some of the options to compute this approximation can be specified, see the documentation of FirstOrderZonotope for details.

References

The main ideas behind this algorithm can be found in [GIR05] and [GLGM06]. These methods are discussed at length in the dissertation [LG09].

Regarding the zonotope order reduction methods, we refer to [COMB03], [GIR05] and the review article [YS18].

Regarding the approximation model, we use an adaptation of a result in [FRE11].

source
diff --git a/dev/lib/algorithms/INT/index.html b/dev/lib/algorithms/INT/index.html index 0bc0fd75b..502c66242 100644 --- a/dev/lib/algorithms/INT/index.html +++ b/dev/lib/algorithms/INT/index.html @@ -1,2 +1,2 @@ -INT · ReachabilityAnalysis.jl
GitHub
ReachabilityAnalysis.INT — Type
INT{N, AM} <: AbstractContinuousPost

Implementation of a reachability method for linear one-dimensional systems interval arithmetic.

Fields

  • δ – step-size of the discretization
  • approx_model – (optional, default: Forward) approximation model; see Notes below for possible options

Notes

The type fields are:

  • N – number type of the step-size
  • AM – approximation model

The default approximation model used in this algorithm is:

Forward(sih=:concrete, exp=:base, setops=:Interval)

In particular, the setops=:Interval flag specifies that intermediate computations in the discretization are done using interval arithmetic. This allows for some optimizations.

References

This algorithm is essentially a non-decomposed version of the method in [BFFPSV18], using intervals as set representation. For a general introduction we refer to the dissertation [LG09].

Regarding the approximation model, by default we use an adaptation of the method presented in [FRE11].

Interval arithmetic operations are performed using the IntervalArithmetic.jl package. Hence, the results are guaranteed to comply to the IEE754 standard with respect to the floating-point operations using intervals.

source
+INT · ReachabilityAnalysis.jl
GitHub
ReachabilityAnalysis.INT — Type
INT{N, AM} <: AbstractContinuousPost

Implementation of a reachability method for linear one-dimensional systems interval arithmetic.

Fields

  • δ – step-size of the discretization
  • approx_model – (optional, default: Forward) approximation model; see Notes below for possible options

Notes

The type fields are:

  • N – number type of the step-size
  • AM – approximation model

The default approximation model used in this algorithm is:

Forward(sih=:concrete, exp=:base, setops=:Interval)

In particular, the setops=:Interval flag specifies that intermediate computations in the discretization are done using interval arithmetic. This allows for some optimizations.

References

This algorithm is essentially a non-decomposed version of the method in [BFFPSV18], using intervals as set representation. For a general introduction we refer to the dissertation [LG09].

Regarding the approximation model, by default we use an adaptation of the method presented in [FRE11].

Interval arithmetic operations are performed using the IntervalArithmetic.jl package. Hence, the results are guaranteed to comply to the IEE754 standard with respect to the floating-point operations using intervals.

source
diff --git a/dev/lib/algorithms/LGG09/index.html b/dev/lib/algorithms/LGG09/index.html index 47e6b993f..75fb031d0 100644 --- a/dev/lib/algorithms/LGG09/index.html +++ b/dev/lib/algorithms/LGG09/index.html @@ -1,5 +1,5 @@ -LGG09 · ReachabilityAnalysis.jl
GitHub
ReachabilityAnalysis.LGG09 — Type
LGG09{N, AM, TN<:AbstractDirections} <: AbstractContinuousPost

Implementation of Girard - Le Guernic algorithm for reachability analysis using support functions.

Fields

  • δ – step-size of the discretization
  • approx_model – (optional, default: Forward) approximation model; see Notes below for possible options
  • template – (alias: dirs) struct that holds the directions (either lazily or concretely) for each support function evaluation defining the template
  • vars – (optional, default: all variables are computed) an integer or a vector of integers specifying the variables of interest to automatically construct a template using canonical directions; requires that n (or dim) is specified as well
  • static – (optional, default: false) if true, use statically sized arrays
  • threaded – (optional, default: false) if true, use multi-threading to compute different template directions in parallel
  • sparse – (optional, default: false) if true, convert the matrix exponential obtained after discretization to a sparse matrix
  • cache – (optional, default: true) if true, use a cache for intermediate computations in the set recurrence loop
  • vars – (optional, default: missing) used to specify the variables instead of passing the template

Notes

The type fields are:

  • N – number type of the step-size
  • AM – type of the approximation model
  • TN – type of the abstract directions that define the template

The flag threaded=true specifies that the support functions along different directions should be computed in parallel. See the section on Multi-threading for details on how to setup the number of threads.

References

This is an implementation of the algorithm from [LGG09].

These methods are described at length in the dissertation [LG09].

source
ReachabilityAnalysis.reach_homog_krylov_LGG09! — Function
reach_homog_krylov_LGG09!(out, Ω₀::LazySet, Aᵀδ::AbstractMatrix,
+LGG09 · ReachabilityAnalysis.jl
GitHub
ReachabilityAnalysis.LGG09 — Type
LGG09{N, AM, TN<:AbstractDirections} <: AbstractContinuousPost
Implementation of Girard - Le Guernic algorithm for reachability analysis using support functions.
Fields
  • δ            – step-size of the discretization
  • approx_model – (optional, default: Forward) approximation model;                   see Notes below for possible options
  • template     – (alias: dirs) struct that holds the directions (either lazily or concretely)                   for each support function evaluation defining the template
  • vars         – (optional, default: all variables are computed) an integer or a vector of integers                   specifying the variables of interest to automatically construct a template                   using canonical directions; requires that n (or dim) is specified as well
  • static       – (optional, default: false) if true, use statically sized arrays
  • threaded     – (optional, default: false) if true, use multi-threading                   to compute different template directions in parallel
  • sparse       – (optional, default: false) if true, convert the matrix exponential                   obtained after discretization to a sparse matrix
  • cache        – (optional, default: true) if true, use a cache for intermediate                   computations in the set recurrence loop
  • vars         – (optional, default: missing) used to specify the variables instead                   of passing the template
Notes
The type fields are:
  • N        – number type of the step-size
  • AM       – type of the approximation model
  • TN       – type of the abstract directions that define the template
The flag threaded=true specifies that the support functions along different directions should be computed in parallel. See the section on Multi-threading for details on how to setup the number of threads.
References
This is an implementation of the algorithm from [LGG09].
These methods are described at length in the dissertation [LG09].
source
ReachabilityAnalysis.reach_homog_krylov_LGG09! — Function
reach_homog_krylov_LGG09!(out, Ω₀::LazySet, Aᵀδ::AbstractMatrix,
                           ℓ::AbstractVector, NSTEPS;
                           hermitian=false, m=min(30, size(Aᵀδ, 1)), tol=1e-7)
Algorithm
We compute the sequence:
\[    ρ(ℓ, Ω₀), ρ(ℓ, Φ Ω₀), ρ(ℓ, Φ^2 Ω₀), ρ(ℓ, Φ^3 Ω₀), ...\]
Using Krylov subspace approximations to compute the action of Φ := exp(Aδ) over the direction ℓ.
The method is (see [1]):
out[1] <- ρ(ℓ, Ω₀)
 
@@ -7,4 +7,4 @@
 
 out[3] <- ρ(ℓ, Φ^2 Ω₀) = ρ((Φᵀ)^2 ℓ, Ω₀)
 
-out[4] <- ρ(ℓ, Φ^3 Ω₀) = ρ((Φᵀ)^3 ℓ, Ω₀)
and so on.
References
[1] Reach Set Approximation through Decomposition with Low-dimensional Sets and     High-dimensional Matrices. Sergiy Bogomolov, Marcelo Forets, Goran Frehse,     Frédéric Viry, Andreas Podelski and Christian Schilling (2018) HSCC'18     Proceedings of the 21st International Conference on Hybrid Systems: Computation     and Control: 41–50.
source

+out[4] <- ρ(ℓ, Φ^3 Ω₀) = ρ((Φᵀ)^3 ℓ, Ω₀)

and so on.

References

[1] Reach Set Approximation through Decomposition with Low-dimensional Sets and High-dimensional Matrices. Sergiy Bogomolov, Marcelo Forets, Goran Frehse, Frédéric Viry, Andreas Podelski and Christian Schilling (2018) HSCC'18 Proceedings of the 21st International Conference on Hybrid Systems: Computation and Control: 41–50.

source
diff --git a/dev/lib/algorithms/ORBIT/index.html b/dev/lib/algorithms/ORBIT/index.html index 8b04cffe7..ca799de9f 100644 --- a/dev/lib/algorithms/ORBIT/index.html +++ b/dev/lib/algorithms/ORBIT/index.html @@ -1,2 +1,2 @@ -ORBIT · ReachabilityAnalysis.jl
GitHub
ReachabilityAnalysis.ORBIT — Type
ORBIT{N, VT, AM} <: AbstractContinuousPost

Implementation of discrete-time integration for deterministic linear ODEs for singleton initial conditions and proper input sets.

Fields

  • δ – step-size of the discretization
  • approx_model – (optional, default: NoBloating) approximation model

The type fields are:

  • N – number type of the step-size
  • AM – type of the approximation model

Notes

If the input set is a singleton, the result is the discrete-time sequence of states that correspond to the analytic solution. If the input set is a dense set, we sample a new input signal in every step.

Reachability solutions computed with the ORBIT algorithm have its own plot recipe, producing a scatter plot. Use the markershape option to change the shape of the markers used for the scatter plot. To "connect" the points, use the option seriestype=:path. For additional options see the Plots.jl documentation.

source
+ORBIT · ReachabilityAnalysis.jl
GitHub
ReachabilityAnalysis.ORBIT — Type
ORBIT{N, VT, AM} <: AbstractContinuousPost

Implementation of discrete-time integration for deterministic linear ODEs for singleton initial conditions and proper input sets.

Fields

  • δ – step-size of the discretization
  • approx_model – (optional, default: NoBloating) approximation model

The type fields are:

  • N – number type of the step-size
  • AM – type of the approximation model

Notes

If the input set is a singleton, the result is the discrete-time sequence of states that correspond to the analytic solution. If the input set is a dense set, we sample a new input signal in every step.

Reachability solutions computed with the ORBIT algorithm have its own plot recipe, producing a scatter plot. Use the markershape option to change the shape of the markers used for the scatter plot. To "connect" the points, use the option seriestype=:path. For additional options see the Plots.jl documentation.

source
diff --git a/dev/lib/algorithms/QINT/index.html b/dev/lib/algorithms/QINT/index.html index f76d30e31..e7ae1db61 100644 --- a/dev/lib/algorithms/QINT/index.html +++ b/dev/lib/algorithms/QINT/index.html @@ -1,2 +1,2 @@ -QINT · ReachabilityAnalysis.jl
GitHub
ReachabilityAnalysis.QINT — Type
QINT{N, AM} <: AbstractContinuousPost

Reachability method for one-dimensional quadratic ODEs with non-deterministic inputs.

Fields

The type fields are:

  • N – number type of the step-size
  • AM – type of the approximation model

Notes

source
+QINT · ReachabilityAnalysis.jl
GitHub
ReachabilityAnalysis.QINT — Type
QINT{N, AM} <: AbstractContinuousPost

Reachability method for one-dimensional quadratic ODEs with non-deterministic inputs.

Fields

The type fields are:

  • N – number type of the step-size
  • AM – type of the approximation model

Notes

source
diff --git a/dev/lib/algorithms/TMJets/index.html b/dev/lib/algorithms/TMJets/index.html index ab144d18c..9a21cec9e 100644 --- a/dev/lib/algorithms/TMJets/index.html +++ b/dev/lib/algorithms/TMJets/index.html @@ -1,2 +1,2 @@ -TMJets · ReachabilityAnalysis.jl
GitHub
ReachabilityAnalysis.TMJets20 — Type
TMJets20{N, DM<:AbstractDisjointnessMethod} <: AbstractContinuousPost

Validated integration using Taylor models.

Fields

  • orderQ – (optional, default: 2) order of the Taylor models for jet transport variables
  • orderT – (optional, default: 8) order of the Taylor model in time
  • abstol – (optional, default: 1e-10) absolute tolerance
  • maxsteps – (optional, default: 2000) maximum number of steps in the validated integration $x' = f(x)$
  • adaptive – (optional, default: true) if true, try decreasing the absolute tolerance each time step validation fails, until min_abs_tol is reached
  • minabstol – (optional, default: 1e-29) minimum absolute tolerance for the adaptive algorithm
  • disjointness – (optional, default: ZonotopeEnclosure()) defines the method to perform the disjointness check between the taylor model flowpipe and the invariant

Notes

The argument disjointness allows to control how are disjointness checks computed, in the case where the invariant is not universal. In particular, ZonotopeEnclosure() pre-processes the taylor model with a zonotopic overapproximation, then performs the disjointness check with that zonotope and the invariant. For other options, see the documentation of AbstractDisjointnessMethod.

This algorithm is an adaptation of the implementation in TaylorModels.jl (see copyright license in the file reach.jl of the current folder). The package TaylorIntegration.jl is used for jet-transport of ODEs using the Taylor method, and TaylorSeries.jl is used to work with truncated Taylor series.

source
ReachabilityAnalysis.TMJets21a — Type
TMJets21a{N, DM<:AbstractDisjointnessMethod} <: AbstractContinuousPost

Set propagation using Taylor models with the function validated_integ in TaylorModels.jl.

source
ReachabilityAnalysis.TMJets21b — Type
TMJets21b{N, DM<:AbstractDisjointnessMethod} <: AbstractContinuousPost

Set propagation using Taylor models with the function validated_integ in TaylorModels.jl.

source
ReachabilityAnalysis.iscontractive — Function
iscontractive(Δ, Δx)

Checks if Δ .⊂ Δx is satisfied. If `Δ ⊆ Δx is satisfied, it returns true if all cases where == holds corresponds to the zero IA.Interval.

source
ReachabilityAnalysis.absorb_remainder — Function
absorb_remainder(a::TaylorModelN)

Returns a TaylorModelN, equivalent to a, such that the remainder is mostly absorbed in the constant and linear coefficients. The linear shift assumes that a is normalized to the IntervalBox(-1..1, Val(N)).

Ref: Xin Chen, Erika Abraham, and Sriram Sankaranarayanan, "Taylor Model Flowpipe Construction for Non-linear Hybrid Systems", in Real Time Systems Symposium (RTSS), pp. 183-192 (2012), IEEE Press.

source
ReachabilityAnalysis.remainder_taylorstep! — Function
remainder_taylorstep!(f!, t, x, dx, xI, dxI, δI, δt, params, adaptive)

Returns a remainder for the integration step for the dependent variables (x) checking that the solution satisfies the criteria for existence and uniqueness.

source
+TMJets · ReachabilityAnalysis.jl
GitHub
ReachabilityAnalysis.TMJets20 — Type
TMJets20{N, DM<:AbstractDisjointnessMethod} <: AbstractContinuousPost

Validated integration using Taylor models.

Fields

  • orderQ – (optional, default: 2) order of the Taylor models for jet transport variables
  • orderT – (optional, default: 8) order of the Taylor model in time
  • abstol – (optional, default: 1e-10) absolute tolerance
  • maxsteps – (optional, default: 2000) maximum number of steps in the validated integration $x' = f(x)$
  • adaptive – (optional, default: true) if true, try decreasing the absolute tolerance each time step validation fails, until min_abs_tol is reached
  • minabstol – (optional, default: 1e-29) minimum absolute tolerance for the adaptive algorithm
  • disjointness – (optional, default: ZonotopeEnclosure()) defines the method to perform the disjointness check between the taylor model flowpipe and the invariant

Notes

The argument disjointness allows to control how are disjointness checks computed, in the case where the invariant is not universal. In particular, ZonotopeEnclosure() pre-processes the taylor model with a zonotopic overapproximation, then performs the disjointness check with that zonotope and the invariant. For other options, see the documentation of AbstractDisjointnessMethod.

This algorithm is an adaptation of the implementation in TaylorModels.jl (see copyright license in the file reach.jl of the current folder). The package TaylorIntegration.jl is used for jet-transport of ODEs using the Taylor method, and TaylorSeries.jl is used to work with truncated Taylor series.

source
ReachabilityAnalysis.TMJets21a — Type
TMJets21a{N, DM<:AbstractDisjointnessMethod} <: AbstractContinuousPost

Set propagation using Taylor models with the function validated_integ in TaylorModels.jl.

source
ReachabilityAnalysis.TMJets21b — Type
TMJets21b{N, DM<:AbstractDisjointnessMethod} <: AbstractContinuousPost

Set propagation using Taylor models with the function validated_integ in TaylorModels.jl.

source
ReachabilityAnalysis.iscontractive — Function
iscontractive(Δ, Δx)

Checks if Δ .⊂ Δx is satisfied. If `Δ ⊆ Δx is satisfied, it returns true if all cases where == holds corresponds to the zero IA.Interval.

source
ReachabilityAnalysis.absorb_remainder — Function
absorb_remainder(a::TaylorModelN)

Returns a TaylorModelN, equivalent to a, such that the remainder is mostly absorbed in the constant and linear coefficients. The linear shift assumes that a is normalized to the IntervalBox(-1..1, Val(N)).

Ref: Xin Chen, Erika Abraham, and Sriram Sankaranarayanan, "Taylor Model Flowpipe Construction for Non-linear Hybrid Systems", in Real Time Systems Symposium (RTSS), pp. 183-192 (2012), IEEE Press.

source
ReachabilityAnalysis.remainder_taylorstep! — Function
remainder_taylorstep!(f!, t, x, dx, xI, dxI, δI, δt, params, adaptive)

Returns a remainder for the integration step for the dependent variables (x) checking that the solution satisfies the criteria for existence and uniqueness.

source
diff --git a/dev/lib/algorithms/VREP/index.html b/dev/lib/algorithms/VREP/index.html index 949dfa6fd..a3bf5bc9d 100644 --- a/dev/lib/algorithms/VREP/index.html +++ b/dev/lib/algorithms/VREP/index.html @@ -1,2 +1,2 @@ -VREP · ReachabilityAnalysis.jl
GitHub
ReachabilityAnalysis.VREP — Type
VREP{N, AM, S, D} <: AbstractContinuousPost

Implementation of a linear reachability solver using the vertex representation.

source
+VREP · ReachabilityAnalysis.jl
GitHub
ReachabilityAnalysis.VREP — Type
VREP{N, AM, S, D} <: AbstractContinuousPost

Implementation of a linear reachability solver using the vertex representation.

source
diff --git a/dev/lib/clustering/index.html b/dev/lib/clustering/index.html index ff85ed794..24af54683 100644 --- a/dev/lib/clustering/index.html +++ b/dev/lib/clustering/index.html @@ -1,2 +1,2 @@ -Clustering · ReachabilityAnalysis.jl
GitHub

Clustering

Abstract interface

ReachabilityAnalysis.AbstractClusteringMethod — Type
AbstractClusteringMethod{P}

Abstract supertype for all clustering types, with partition of type P.

Notes

A clustering method defines a function which maps reach-sets to one or several reach-sets. The mapping can possibly be over-approximative, i.e. such that the set union of the input reach-sets is included in the set union of the output reach-sets.

By taking the convex hull of the input reach-sets one can reduce the number of outputs sets to a single one, overapproximately. This is the method that corresponds to the LazyClustering type. However, in some cases it is convenient to do other types of transformations, such as:

  • Return several reach-sets that are obtained by grouping the input in a way defined by a partition. For example, a vector of ten input sets can be clustered in two groups of five sets each, or five groups of two sets each, etc.
  • Use different set representations such as boxes or zonotopes.
  • Further post-process the output of the convexification by splitting into smaller sets, eg. box splitting.

Each concrete subtype of AbstractClusteringMethod has a parameter P that defines what type of clustering strategy is applied. The method should be accessed with the partition getter function.

The following strategies are implemented at the interface level:

  • If P is of type Missing: no partition is applied
  • If P is of type integer: the partition corresponds to grouping the into the given integer number of sets (or as close as possible)
  • If P is of type vector of vectors: the given partition is applied

Examples

LazyClustering([1:2, 3:10]) groups the first two reach-sets in one cluster and the third to tenth reach-sets in another cluster.

source

No clustering

The following methods are available.

Lazy convexification

Box clustering

ReachabilityAnalysis.BoxClustering — Type
BoxClustering{PI, PO} <: AbstractClusteringMethod{P}

Notes

This method first takes a lazy convex hull for the given partition, then computes a tight hyperrectangular approximation for each element in the partition.

source

Zonotope clustering

ReachabilityAnalysis.ZonotopeClustering — Type
ZonotopeClustering{P} <: AbstractClusteringMethod{P}

Notes

This method first takes a lazy convex hull for the given partition, then computes a zonotope overapproximation of the convex hull.

source

Lazy union

+Clustering · ReachabilityAnalysis.jl
GitHub

Clustering

Abstract interface

ReachabilityAnalysis.AbstractClusteringMethod — Type
AbstractClusteringMethod{P}

Abstract supertype for all clustering types, with partition of type P.

Notes

A clustering method defines a function which maps reach-sets to one or several reach-sets. The mapping can possibly be over-approximative, i.e. such that the set union of the input reach-sets is included in the set union of the output reach-sets.

By taking the convex hull of the input reach-sets one can reduce the number of outputs sets to a single one, overapproximately. This is the method that corresponds to the LazyClustering type. However, in some cases it is convenient to do other types of transformations, such as:

  • Return several reach-sets that are obtained by grouping the input in a way defined by a partition. For example, a vector of ten input sets can be clustered in two groups of five sets each, or five groups of two sets each, etc.
  • Use different set representations such as boxes or zonotopes.
  • Further post-process the output of the convexification by splitting into smaller sets, eg. box splitting.

Each concrete subtype of AbstractClusteringMethod has a parameter P that defines what type of clustering strategy is applied. The method should be accessed with the partition getter function.

The following strategies are implemented at the interface level:

  • If P is of type Missing: no partition is applied
  • If P is of type integer: the partition corresponds to grouping the into the given integer number of sets (or as close as possible)
  • If P is of type vector of vectors: the given partition is applied

Examples

LazyClustering([1:2, 3:10]) groups the first two reach-sets in one cluster and the third to tenth reach-sets in another cluster.

source

No clustering

The following methods are available.

Lazy convexification

Box clustering

ReachabilityAnalysis.BoxClustering — Type
BoxClustering{PI, PO} <: AbstractClusteringMethod{P}

Notes

This method first takes a lazy convex hull for the given partition, then computes a tight hyperrectangular approximation for each element in the partition.

source

Zonotope clustering

ReachabilityAnalysis.ZonotopeClustering — Type
ZonotopeClustering{P} <: AbstractClusteringMethod{P}

Notes

This method first takes a lazy convex hull for the given partition, then computes a zonotope overapproximation of the convex hull.

source

Lazy union

diff --git a/dev/lib/discretize/index.html b/dev/lib/discretize/index.html index 87eb2f5fc..9a08403a8 100644 --- a/dev/lib/discretize/index.html +++ b/dev/lib/discretize/index.html @@ -1,17 +1,17 @@ -Discretization · ReachabilityAnalysis.jl
GitHub

Discretization

Discretize API

LinearAlgebra.normalize — Function
normalize(system::AbstractSystem)

Transform a mathematical system to a normalized (or canonical) form.

Input

  • system – system; it can be discrete or continuous

Output

Either the same system if it already conforms to a canonical form, or a new system otherwise.

Notes

The normalization procedure consists of transforming a given system type into a "canonical" format that is used internally. More details are given below.

Algorithm

The implementation of normalize exploits MathematicalSystems's' types, which carry information about the problem as a type parameter.

Homogeneous ODEs of the form $x' = Ax, x ∈ \mathcal{X}$ are canonical if the associated problem is a ConstrainedLinearContinuousSystem and A is a matrix. This type does not handle non-deterministic inputs.

Note that a LinearContinuousSystem does not consider constraints on the state-space (such as an invariant); to specify state constraints, use a ConstrainedLinearContinuousSystem. If the passed system is a LinearContinuousSystem (i.e. no constraints) then the normalization fixes a universal set (Universe) as the constraint set.

The generalization to canonical systems with constraints and possibly time-varying non-deterministic inputs is considered next. These systems are of the form $x' = Ax + u, u ∈ \mathcal{U}, x ∈ \mathcal{X}$. The system type is ConstrainedLinearControlContinuousSystem, where A is a matrix, X is a set and U is an input, that is, any concrete subtype of AbstractInput.

If U is not given as an input, normalization accepts either a LazySet, or a vector of LazySets. In these cases, the sets are wrapped around an appropriate concrete input type.

If the system does not conform to a canonical form, the implementation tries to make the transformation; otherwise an error is thrown. In particular, ODEs of the form $x' = Ax + Bu$ are mapped into $x' = Ax + u, u ∈ B\mathcal{U}$, where now $u$ has the same dimensions as $x$.

The transformations described above are analogous in the discrete case, i.e. $x_{k+1} = A x_k$ and $x_{k+1} = Ax_{k} + u_k, u_k ∈ \mathcal{U}, x_k ∈ \mathcal{X}$ for the linear and affine cases respectively.

source
normalize(ivp::InitialValueProblem)

Transform an initial-value problem into a normalized (or canonical) form.

Input

  • ivp – initial-value problem

Output

Either the same initial-value problem if it already conforms to a canonical form, or a new one otherwise.

Notes

This function extends normalize for initial-value problems.

source
MathematicalSystems.discretize — Function
discretize(ivp::IVP, δ, alg::AbstractApproximationModel)

Set-based conservative discretization of a continuous-time initial value problem into a discrete-time problem.

Input

  • ivp – initial value problem for a linear ODE in canonical form (see Notes below)
  • δ – step size
  • alg – algorithm used to compute the approximation model

Output

The initial value problem of a discrete system.

Notes

Different approximation algorithms and their respective options are described in the docstring of each method, e.g. Forward. Here is a list of all the available approximation models:

julia> subtypes(ReachabilityAnalysis.AbstractApproximationModel)
+Discretization · ReachabilityAnalysis.jl
GitHub
Discretization
Discretize API
LinearAlgebra.normalize — Function
normalize(system::AbstractSystem)
Transform a mathematical system to a normalized (or canonical) form.
Input
  • system – system; it can be discrete or continuous
Output
Either the same system if it already conforms to a canonical form, or a new system otherwise.
Notes
The normalization procedure consists of transforming a given system type into a "canonical" format that is used internally. More details are given below.
Algorithm
The implementation of normalize exploits MathematicalSystems's' types, which carry information about the problem as a type parameter.
Homogeneous ODEs of the form $x' = Ax, x ∈ \mathcal{X}$ are canonical if the associated problem is a ConstrainedLinearContinuousSystem and A is a matrix. This type does not handle non-deterministic inputs.
Note that a LinearContinuousSystem does not consider constraints on the state-space (such as an invariant); to specify state constraints, use a ConstrainedLinearContinuousSystem. If the passed system is a LinearContinuousSystem (i.e. no constraints) then the normalization fixes a universal set (Universe) as the constraint set.
The generalization to canonical systems with constraints and possibly time-varying non-deterministic inputs is considered next. These systems are of the form $x' = Ax + u, u ∈ \mathcal{U}, x ∈ \mathcal{X}$. The system type is ConstrainedLinearControlContinuousSystem, where A is a matrix, X is a set and U is an input, that is, any concrete subtype of AbstractInput.
If U is not given as an input, normalization accepts either a LazySet, or a vector of LazySets. In these cases, the sets are wrapped around an appropriate concrete input type.
If the system does not conform to a canonical form, the implementation tries to make the transformation; otherwise an error is thrown. In particular, ODEs of the form $x' = Ax + Bu$ are mapped into $x' = Ax + u, u ∈ B\mathcal{U}$, where now $u$ has the same dimensions as $x$.
The transformations described above are analogous in the discrete case, i.e. $x_{k+1} = A x_k$ and $x_{k+1} = Ax_{k} + u_k, u_k ∈ \mathcal{U}, x_k ∈ \mathcal{X}$ for the linear and affine cases respectively.
source
normalize(ivp::InitialValueProblem)
Transform an initial-value problem into a normalized (or canonical) form.
Input
  • ivp – initial-value problem
Output
Either the same initial-value problem if it already conforms to a canonical form, or a new one otherwise.
Notes
This function extends normalize for initial-value problems.
source
MathematicalSystems.discretize — Function
discretize(ivp::IVP, δ, alg::AbstractApproximationModel)
Set-based conservative discretization of a continuous-time initial value problem into a discrete-time problem.
Input
  • ivp   – initial value problem for a linear ODE in canonical form (see Notes below)
  • δ     – step size
  • alg   – algorithm used to compute the approximation model
Output
The initial value problem of a discrete system.
Notes
Different approximation algorithms and their respective options are described in the docstring of each method, e.g. Forward. Here is a list of all the available approximation models:
julia> subtypes(ReachabilityAnalysis.AbstractApproximationModel)
 5-element Vector{Any}:
  Backward
  CorrectionHull
  Forward
  NoBloating
- StepIntersect
Initial-value problems considered in this function are of the form
\[x' = Ax(t) + u(t),\qquad x(0) ∈ \mathcal{X}_0,\qquad (1)\]
and where $u(t) ∈ U(k)$ add where $\{U(k)\}_k$ is a sequence of sets of non-deterministic inputs and $\mathcal{X}_0$ is the set of initial states. Other problems, e.g. $x' = Ax(t) + Bu(t)$ can be brought to the canonical form with the function normalize.
For references to the original papers introducing each algorithm, see the docstrings, e.g. ?Forward.
source
Approximation models
ReachabilityAnalysis.NoBloating — Type
NoBloating{EM, SO, IT} <: AbstractApproximationModel
No bloating, or discrete-time, approximation model.
Fields
  • exp     – exponentiation method
  • setops  – set operations method
  • inv     – (optional, default: false) if true, assume that the state matrix              is invertible and use its inverse in the Φ functions
Algorithm
The transformations are:
  • $Φ ← \exp(Aδ)$
  • $Ω_0 ← \mathcal{X}_0$
  • $V(k) ← Φ₁(A, δ)U(k)$, $k ≥ 0$.
The function $Φ₁(A, δ)$ is defined in Φ₁. We allow $U$ to be a sequence of time varying non-deterministic input sets.
See also Eqs.(14) in [BFFPSV18].
source
ReachabilityAnalysis.Forward — Type
Forward{EM, SO, SI, IT, BT} <: AbstractApproximationModel
Forward approximation model.
Fields
  • exp     – exponentiation method
  • setops  – set operations method
  • sih     – symmetric interval hull
  • inv     – (optional, default: false) if true, assume that the state matrix              is invertible and use its inverse in the Φ functions
  • backend – (optional, default: nothing) used if the algorithm needs to apply              concrete polyhedral computations
Algorithm
The transformations are:
  • $Φ ← \exp(Aδ)$,
  • $Ω_0 ← CH(\mathcal{X}_0, Φ\mathcal{X}_0 ⊕ δU(0) ⊕ E_ψ(U(0), δ) ⊕ E^+(\mathcal{X}_0, δ))$,
  • $V(k) ← δU(k) ⊕ E_ψ(U(k), δ)$.
Here we allow $U$ to be a sequence of time varying non-deterministic input sets.
For the definition of the sets $E_ψ$ and $E^+$ see [FRE11]. The Backward method uses $E^-$.
source
ReachabilityAnalysis.Backward — Type
Backward{EM, SO, SI, IT, BT} <: AbstractApproximationModel
Backward approximation model.
Fields
  • exp     – exponentiation method
  • setops  – set operations method
  • sih     – symmetric interval hull
  • inv     – (optional, default: false) if true, assume that the state matrix              is invertible and use its inverse in the Φ functions
  • backend – (optional, default: nothing) used if the algorithm needs to apply              concrete polyhedral computations
Algorithm
The transformations are:
  • $Φ ← \exp(Aδ)$,
  • $Ω_0 ← CH(\mathcal{X}_0, Φ\mathcal{X}_0 ⊕ δU(0) ⊕ E_ψ(U(0), δ) ⊕ E^-(\mathcal{X}_0, δ))$,
  • $V(k) ← δU(k) ⊕ E_ψ(U(k), δ)$.
Here we allow $U$ to be a sequence of time varying non-deterministic input sets.
For the definition of the sets $E_ψ$ and $E^-$ see [FRE11]. The Forward method uses $E^+$.
source
ReachabilityAnalysis.FirstOrder — Type
FirstOrder{EM} <: AbstractApproximationModel
First-order approximation model.
Fields
  • exp – (optional, default: BaseExp) exponentiation method
Algorithm
The transformations are:
  • $Φ ← \exp(Aδ)$,
  • $Ω_0 ← CH(\mathcal{X}_0, Φ\mathcal{X}_0 ⊕ δU ⊕ B_ε)$
where $B_ε$ is the input ball of radius $ε$ centered in the origin.
Reference
C. Le Guernic and A. Girard: Reachability analysis of linear systems using support functions. NAHS 2010.
source
ReachabilityAnalysis.FirstOrderZonotope — Type
FirstOrderZonotope{EM} <: AbstractApproximationModel
First order approximation model that works with zonotopes.
Fields
  • exp – (optional, default: BaseExp) exponentiation method
Algorithm
The transformations are:
  • $Φ ← \exp(Aδ)$,
  • $Ω_0 ← bloat(zono(CH(\mathcal{X}_0, Φ\mathcal{X}_0)), α + β)$
where $bloat(\mathcal{X}, ε)$ bloats the set $\mathcal{X}$ with the value $ε$, $zono(·)$ overapproximates its argument with a zonotope, and $α$ and $β$ are factors computed for the homogeneous system and the inputs, respectively.
Reference
A. Girard: Reachability of uncertain linear systems using zonotopes. HSCC 2005.
source
ReachabilityAnalysis.SecondOrderddt — Type
SecondOrderddt{EM, SO, SI, IT, BT} <: AbstractApproximationModel
Second-order approximation model used in the tool d/dt. It can be used for overapproximation and underapproximation.
Fields
  • oa      – (optional, default: true) flag to choose between              overapproximation and underapproximation
  • exp     – (optional, default: BaseExp) exponentiation method
  • setops  – (optional, default: :lazy) set operations method
  • sih     – (optional, default: :concrete) way to compute the symmetric              interval hull
  • inv     – (optional, default: false) if true, assume that the state              matrix is invertible and use its inverse in the Φ functions
  • backend – (optional, default: nothing) used if the algorithm needs to              apply concrete polyhedral computations
Algorithm
The transformations are:
  • $Φ ← \exp(Aδ)$,
  • $Ω_0 ← bloat(CH(\mathcal{X}_0, Φ\mathcal{X}_0), ε)$
where $bloat(\mathcal{X}, ε)$ bloats the set $\mathcal{X}$ with the value $ε$. If oa == false, the bloating acts in an inverted way and shrinks the set.
Reference
E. Asarin, T. Dang, O. Maler, O. Bournez: Approximate reachability analysis of piecewise-linear dynamical systems. HSCC 2000.
source
ReachabilityAnalysis.CorrectionHull — Type
CorrectionHull{EM} <: AbstractApproximationModel
Discretization using the correction hull of the matrix exponential.
Fields
  • exp   – exponentiation method
  • order – order of the Taylor series expansion of the matrix exponential
Algorithm
For the homogeneous case, this method implements the transformation:
\[Ω_0 = CH(X_0, e^{Aδ}  X_0) ⊕ FX_0\]
where $F$ is the correction (interval) matrix.
For the inhomogeneous case, $x' = Ax + u$,  $x ∈ X, u ∈ U$, implements $Ω_0 = CH(X_0, exp(Aδ)  X0) ⊕ FX0$ where $F$ is the correction (interval) matrix.
In both cases, if $A$ is an interval matrix, the exponential is overapproximated using methods from IntervalMatrices.jl.
source
ReachabilityAnalysis.StepIntersect — Type
StepIntersect{DM<:AbstractApproximationModel} <: AbstractApproximationModel
Approximation model that composes (intersecting) one step Forward of a given model with one step backward of the same model.
Fields
  • model  – approximation model
  • setops – set operations method
Notes
Let $x' = Ax$ with $x(0) ∈ X₀$. This methods consists of:
  • Compute the discretized system with step-size $δ$ obtaining $Ω0$ and the given approximation model method.
  • Compute the (lazy) linear map $ΦX₀$. This set consists of the (exact) reachable states at the time point $δ$.
  • Apply the approximation model method with initial condition $ΦX₀$ one step backward in time, with the state transition matrix $-A$, call this reach-set $Ω₀₊$
  • Intersect $Ω₀₋$ and $Ω₀₊$ and return such set. The intersection is done either lazily or concretely depending on the specified setops field.
source
Exponentiation
The state transition matrix of the linear ODE $x'(t) = Ax(t) + u(t)$ at time $\delta > 0$ is $\Phi = e^{A\delta}$, hence the algorithms usually require to compute exponential matrices. There are distinct ways to compute the matrix exponential $e^{A\delta}$ depending on the type of $A$ (see e.g. [HIH08]). The available methods can be used through the (unexported) function _exp.
For high dimensional systems (typically n > 2000), computing the matrix exponential is expensive hence it is preferable to compute the action of the matrix exponential over vectors when needed, that is, $e^{δA} v$ for each $v$. This method is particularly well-suited if A is vert sparse. Use the option exp=:krylov (or exp=:lazy) for this purpose.
ReachabilityAnalysis.Exponentiation — Module
Interface to matrix exponential backends of different kinds.
Includes common integral computations arising in the discretization of linear differential equations using matrix methods. For applications see e.g. [1] and references therein.
[1] Conservative Time Discretization: A Comparative Study.     Marcelo Forets and Christian Schilling (2022).     Proceedings of the 17th International Conference on integrated Formal Methods (iFM),     LNCS, vol. 13274, pp. 149-167. doi: 10.1007/978-3-031-07727-2_9, arXiv: 2111.01454.
source
ReachabilityAnalysis.Exponentiation._exp — Function
_exp(A::AbstractMatrix, δ, [alg]::AbstractExpAlg=BaseExp)
Compute the matrix exponential $e^{Aδ}$.
Input
  • A    – matrix
  • δ    – step size
  • alg  – (optional, default: BaseExp) the algorithm used to take the matrix           exponential of Aδ, possible options are BaseExp, LazyExp, PadeExp and IntervalExp           see details in the Algorithm section below
Output
A matrix or a lazy wrapper of the matrix exponential, depending on alg.
Algorithm
  • BaseExp – (alias: :base) use Higham's scaling and squaring method implemented              in Julia standard library; see ?exp for details; if A is a static array,              uses the implementation in StaticArrays.jl
  • LazyExp – (alias: :lazy) return a lazy wrapper type around the matrix exponential              using the implementation LazySets.SparseMatrixExp
  • PadeExp – (alias: pade) apply Pade approximant method to compute the matrix              exponential of a sparse matrix (requires Expokit.jl)
  • IntervalExp – (alias: interval, taylor) apply the Taylor series expansion of the matrix                   exponential with an interval remainder; works if A is an interval matrix
Notes
If the algorithm LazyExp is used, actions of the matrix exponential are evaluated with an external library such as ExponentialUtilities.jl or Expokit.jl.
source
ReachabilityAnalysis.Exponentiation.Φ₁ — Function
Φ₁(A::AbstractMatrix, δ, [alg]::AbstractExpAlg=BaseExp, [isinv]::Bool=false, [Φ]=nothing)
Evaluate the series
\[Φ₁(A, δ) = ∑_{i=0}^∞ \dfrac{δ^{i+1}}{(i+1)!}A^i,\]
where $A$ is a square matrix of order $n$ and $δ > 0$.
Input
  • A      – coefficients matrix
  • δ      – step size
  • alg    – (optional, default: BaseExp) the method used to take the matrix              exponential of the coefficient matrix; see the documentation of              _exp for available options
  • isinv  – (optional, default: false) if true, assume that the coefficients             matrix is invertible and compute $A^{-1}$
  • Φ      – (optional, default: nothing) optionally pass the matrix exponential $e^{Aδ}$
Output
A matrix.
Algorithm
If $A$ is invertible, $Φ₁$ can be computed as
\[Φ₁(A, δ) = A^{-1}(e^{Aδ} - I_n),\]
where $I_n$ is the identity matrix of order $n$.
In the general case, implemented in this function, it can be computed as submatrices of the block matrix
\[P_{2n} = \exp \begin{pmatrix}
+ StepIntersect

Initial-value problems considered in this function are of the form

\[x' = Ax(t) + u(t),\qquad x(0) ∈ \mathcal{X}_0,\qquad (1)\]

and where $u(t) ∈ U(k)$ add where $\{U(k)\}_k$ is a sequence of sets of non-deterministic inputs and $\mathcal{X}_0$ is the set of initial states. Other problems, e.g. $x' = Ax(t) + Bu(t)$ can be brought to the canonical form with the function normalize.

For references to the original papers introducing each algorithm, see the docstrings, e.g. ?Forward.

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Approximation models

ReachabilityAnalysis.NoBloating — Type
NoBloating{EM, SO, IT} <: AbstractApproximationModel

No bloating, or discrete-time, approximation model.

Fields

  • exp – exponentiation method
  • setops – set operations method
  • inv – (optional, default: false) if true, assume that the state matrix is invertible and use its inverse in the Φ functions

Algorithm

The transformations are:

  • $Φ ← \exp(Aδ)$
  • $Ω_0 ← \mathcal{X}_0$
  • $V(k) ← Φ₁(A, δ)U(k)$, $k ≥ 0$.

The function $Φ₁(A, δ)$ is defined in Φ₁. We allow $U$ to be a sequence of time varying non-deterministic input sets.

See also Eqs.(14) in [BFFPSV18].

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ReachabilityAnalysis.Forward — Type
Forward{EM, SO, SI, IT, BT} <: AbstractApproximationModel

Forward approximation model.

Fields

  • exp – exponentiation method
  • setops – set operations method
  • sih – symmetric interval hull
  • inv – (optional, default: false) if true, assume that the state matrix is invertible and use its inverse in the Φ functions
  • backend – (optional, default: nothing) used if the algorithm needs to apply concrete polyhedral computations

Algorithm

The transformations are:

  • $Φ ← \exp(Aδ)$,
  • $Ω_0 ← CH(\mathcal{X}_0, Φ\mathcal{X}_0 ⊕ δU(0) ⊕ E_ψ(U(0), δ) ⊕ E^+(\mathcal{X}_0, δ))$,
  • $V(k) ← δU(k) ⊕ E_ψ(U(k), δ)$.

Here we allow $U$ to be a sequence of time varying non-deterministic input sets.

For the definition of the sets $E_ψ$ and $E^+$ see [FRE11]. The Backward method uses $E^-$.

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ReachabilityAnalysis.Backward — Type
Backward{EM, SO, SI, IT, BT} <: AbstractApproximationModel

Backward approximation model.

Fields

  • exp – exponentiation method
  • setops – set operations method
  • sih – symmetric interval hull
  • inv – (optional, default: false) if true, assume that the state matrix is invertible and use its inverse in the Φ functions
  • backend – (optional, default: nothing) used if the algorithm needs to apply concrete polyhedral computations

Algorithm

The transformations are:

  • $Φ ← \exp(Aδ)$,
  • $Ω_0 ← CH(\mathcal{X}_0, Φ\mathcal{X}_0 ⊕ δU(0) ⊕ E_ψ(U(0), δ) ⊕ E^-(\mathcal{X}_0, δ))$,
  • $V(k) ← δU(k) ⊕ E_ψ(U(k), δ)$.

Here we allow $U$ to be a sequence of time varying non-deterministic input sets.

For the definition of the sets $E_ψ$ and $E^-$ see [FRE11]. The Forward method uses $E^+$.

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ReachabilityAnalysis.FirstOrder — Type
FirstOrder{EM} <: AbstractApproximationModel

First-order approximation model.

Fields

  • exp – (optional, default: BaseExp) exponentiation method

Algorithm

The transformations are:

  • $Φ ← \exp(Aδ)$,
  • $Ω_0 ← CH(\mathcal{X}_0, Φ\mathcal{X}_0 ⊕ δU ⊕ B_ε)$

where $B_ε$ is the input ball of radius $ε$ centered in the origin.

Reference

C. Le Guernic and A. Girard: Reachability analysis of linear systems using support functions. NAHS 2010.

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ReachabilityAnalysis.FirstOrderZonotope — Type
FirstOrderZonotope{EM} <: AbstractApproximationModel

First order approximation model that works with zonotopes.

Fields

  • exp – (optional, default: BaseExp) exponentiation method

Algorithm

The transformations are:

  • $Φ ← \exp(Aδ)$,
  • $Ω_0 ← bloat(zono(CH(\mathcal{X}_0, Φ\mathcal{X}_0)), α + β)$

where $bloat(\mathcal{X}, ε)$ bloats the set $\mathcal{X}$ with the value $ε$, $zono(·)$ overapproximates its argument with a zonotope, and $α$ and $β$ are factors computed for the homogeneous system and the inputs, respectively.

Reference

A. Girard: Reachability of uncertain linear systems using zonotopes. HSCC 2005.

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ReachabilityAnalysis.SecondOrderddt — Type
SecondOrderddt{EM, SO, SI, IT, BT} <: AbstractApproximationModel

Second-order approximation model used in the tool d/dt. It can be used for overapproximation and underapproximation.

Fields

  • oa – (optional, default: true) flag to choose between overapproximation and underapproximation
  • exp – (optional, default: BaseExp) exponentiation method
  • setops – (optional, default: :lazy) set operations method
  • sih – (optional, default: :concrete) way to compute the symmetric interval hull
  • inv – (optional, default: false) if true, assume that the state matrix is invertible and use its inverse in the Φ functions
  • backend – (optional, default: nothing) used if the algorithm needs to apply concrete polyhedral computations

Algorithm

The transformations are:

  • $Φ ← \exp(Aδ)$,
  • $Ω_0 ← bloat(CH(\mathcal{X}_0, Φ\mathcal{X}_0), ε)$

where $bloat(\mathcal{X}, ε)$ bloats the set $\mathcal{X}$ with the value $ε$. If oa == false, the bloating acts in an inverted way and shrinks the set.

Reference

E. Asarin, T. Dang, O. Maler, O. Bournez: Approximate reachability analysis of piecewise-linear dynamical systems. HSCC 2000.

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ReachabilityAnalysis.CorrectionHull — Type
CorrectionHull{EM} <: AbstractApproximationModel

Discretization using the correction hull of the matrix exponential.

Fields

  • exp – exponentiation method
  • order – order of the Taylor series expansion of the matrix exponential

Algorithm

For the homogeneous case, this method implements the transformation:

\[Ω_0 = CH(X_0, e^{Aδ} X_0) ⊕ FX_0\]

where $F$ is the correction (interval) matrix.

For the inhomogeneous case, $x' = Ax + u$, $x ∈ X, u ∈ U$, implements $Ω_0 = CH(X_0, exp(Aδ) X0) ⊕ FX0$ where $F$ is the correction (interval) matrix.

In both cases, if $A$ is an interval matrix, the exponential is overapproximated using methods from IntervalMatrices.jl.

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ReachabilityAnalysis.StepIntersect — Type
StepIntersect{DM<:AbstractApproximationModel} <: AbstractApproximationModel

Approximation model that composes (intersecting) one step Forward of a given model with one step backward of the same model.

Fields

  • model – approximation model
  • setops – set operations method

Notes

Let $x' = Ax$ with $x(0) ∈ X₀$. This methods consists of:

  • Compute the discretized system with step-size $δ$ obtaining $Ω0$ and the given approximation model method.

  • Compute the (lazy) linear map $ΦX₀$. This set consists of the (exact) reachable states at the time point $δ$.

  • Apply the approximation model method with initial condition $ΦX₀$ one step backward in time, with the state transition matrix $-A$, call this reach-set $Ω₀₊$

  • Intersect $Ω₀₋$ and $Ω₀₊$ and return such set. The intersection is done either lazily or concretely depending on the specified setops field.

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Exponentiation

The state transition matrix of the linear ODE $x'(t) = Ax(t) + u(t)$ at time $\delta > 0$ is $\Phi = e^{A\delta}$, hence the algorithms usually require to compute exponential matrices. There are distinct ways to compute the matrix exponential $e^{A\delta}$ depending on the type of $A$ (see e.g. [HIH08]). The available methods can be used through the (unexported) function _exp.

For high dimensional systems (typically n > 2000), computing the matrix exponential is expensive hence it is preferable to compute the action of the matrix exponential over vectors when needed, that is, $e^{δA} v$ for each $v$. This method is particularly well-suited if A is vert sparse. Use the option exp=:krylov (or exp=:lazy) for this purpose.

ReachabilityAnalysis.Exponentiation — Module

Interface to matrix exponential backends of different kinds.

Includes common integral computations arising in the discretization of linear differential equations using matrix methods. For applications see e.g. [1] and references therein.

[1] Conservative Time Discretization: A Comparative Study. Marcelo Forets and Christian Schilling (2022). Proceedings of the 17th International Conference on integrated Formal Methods (iFM), LNCS, vol. 13274, pp. 149-167. doi: 10.1007/978-3-031-07727-2_9, arXiv: 2111.01454.

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ReachabilityAnalysis.Exponentiation._exp — Function
_exp(A::AbstractMatrix, δ, [alg]::AbstractExpAlg=BaseExp)

Compute the matrix exponential $e^{Aδ}$.

Input

  • A – matrix
  • δ – step size
  • alg – (optional, default: BaseExp) the algorithm used to take the matrix exponential of Aδ, possible options are BaseExp, LazyExp, PadeExp and IntervalExp see details in the Algorithm section below

Output

A matrix or a lazy wrapper of the matrix exponential, depending on alg.

Algorithm

  • BaseExp – (alias: :base) use Higham's scaling and squaring method implemented in Julia standard library; see ?exp for details; if A is a static array, uses the implementation in StaticArrays.jl

  • LazyExp – (alias: :lazy) return a lazy wrapper type around the matrix exponential using the implementation LazySets.SparseMatrixExp

  • PadeExp – (alias: pade) apply Pade approximant method to compute the matrix exponential of a sparse matrix (requires Expokit.jl)

  • IntervalExp – (alias: interval, taylor) apply the Taylor series expansion of the matrix exponential with an interval remainder; works if A is an interval matrix

Notes

If the algorithm LazyExp is used, actions of the matrix exponential are evaluated with an external library such as ExponentialUtilities.jl or Expokit.jl.

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ReachabilityAnalysis.Exponentiation.Φ₁ — Function
Φ₁(A::AbstractMatrix, δ, [alg]::AbstractExpAlg=BaseExp, [isinv]::Bool=false, [Φ]=nothing)

Evaluate the series

\[Φ₁(A, δ) = ∑_{i=0}^∞ \dfrac{δ^{i+1}}{(i+1)!}A^i,\]

where $A$ is a square matrix of order $n$ and $δ > 0$.

Input

  • A – coefficients matrix
  • δ – step size
  • alg – (optional, default: BaseExp) the method used to take the matrix exponential of the coefficient matrix; see the documentation of _exp for available options
  • isinv – (optional, default: false) if true, assume that the coefficients matrix is invertible and compute $A^{-1}$
  • Φ – (optional, default: nothing) optionally pass the matrix exponential $e^{Aδ}$

Output

A matrix.

Algorithm

If $A$ is invertible, $Φ₁$ can be computed as

\[Φ₁(A, δ) = A^{-1}(e^{Aδ} - I_n),\]

where $I_n$ is the identity matrix of order $n$.

In the general case, implemented in this function, it can be computed as submatrices of the block matrix

\[P_{2n} = \exp \begin{pmatrix} Aδ && δI_n \\ 0 && 0 \end{pmatrix}.\]

It can be shown that

\[\exp(P_{2n}) = \begin{pmatrix} Φ(A, δ) && Φ₁(A, δ) \\ 0 && δI_n -\end{pmatrix}.\]

where $Φ(A, δ) = e^{Aδ}$. In particular, Φ₁(A, δ) = P[1:n, (n+1):2*n]. This method can be found in [FRE11].

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ReachabilityAnalysis.Exponentiation.Φ₂ — Function
Φ₂(A::AbstractMatrix, δ, [alg]::AbstractExpAlg=BaseExp, [isinv]::Bool=false, [Φ]=nothing)

Evaluate the series

\[Φ₂(A, δ) = ∑_{i=0}^∞ \dfrac{δ^{i+2}}{(i+2)!}A^i,\]

where $A$ is a square matrix of order $n$ and $δ > 0$.

Input

  • A – coefficients matrix
  • δ – step size
  • alg – (optional, default: BaseExp) the method used to take the matrix exponential of the coefficient matrix; see the documentation of _exp for available options
  • isinv – (optional, default: false) if true, assume that the coefficients matrix is invertible and compute $A^{-1}$
  • Φ – (optional, default: nothing) optionally pass the matrix exponential $e^{Aδ}$

Output

A matrix.

Algorithm

If $A$ is invertible, $Φ₂$ can be computed as

\[Φ₂(A, δ) = A^{-2}(e^{δA} - I_n - δA).\]

In the general case, implemented in this function, it can be computed as submatrices of the block matrix

\[P_{3n} = \exp \begin{pmatrix} +\end{pmatrix}.\]

where $Φ(A, δ) = e^{Aδ}$. In particular, Φ₁(A, δ) = P[1:n, (n+1):2*n]. This method can be found in [FRE11].

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ReachabilityAnalysis.Exponentiation.Φ₂ — Function
Φ₂(A::AbstractMatrix, δ, [alg]::AbstractExpAlg=BaseExp, [isinv]::Bool=false, [Φ]=nothing)

Evaluate the series

\[Φ₂(A, δ) = ∑_{i=0}^∞ \dfrac{δ^{i+2}}{(i+2)!}A^i,\]

where $A$ is a square matrix of order $n$ and $δ > 0$.

Input

  • A – coefficients matrix
  • δ – step size
  • alg – (optional, default: BaseExp) the method used to take the matrix exponential of the coefficient matrix; see the documentation of _exp for available options
  • isinv – (optional, default: false) if true, assume that the coefficients matrix is invertible and compute $A^{-1}$
  • Φ – (optional, default: nothing) optionally pass the matrix exponential $e^{Aδ}$

Output

A matrix.

Algorithm

If $A$ is invertible, $Φ₂$ can be computed as

\[Φ₂(A, δ) = A^{-2}(e^{δA} - I_n - δA).\]

In the general case, implemented in this function, it can be computed as submatrices of the block matrix

\[P_{3n} = \exp \begin{pmatrix} Aδ && δI_n && 0 \\ 0 && 0 && δI_n \\ 0 && 0 && 0 @@ -19,4 +19,4 @@ Φ(A, δ) && Φ₁(A, δ) && Φ₂(A, δ) \\ 0 && I_n && δI_n \\ 0 && 0 && I_n -\end{pmatrix}.\]

where $Φ(A, δ) = e^{Aδ}$. In particular, Φ₂ = P_{3n}[1:n, (2*n+1):3*n]. This method can be found in [FRE11].

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ReachabilityAnalysis.Exponentiation.BaseExpAlg — Type
BaseExpAlg <: AbstractExpAlg

Matrix exponential using the scaling and squaring algorithm implemented in Julia Base.

Notes

If the array is static, the method implemented in StaticArrays.jl is applied. This algorithm admits the alias :base.

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ReachabilityAnalysis.Exponentiation.LazyExpAlg — Type
LazyExpAlg <: AbstractExpAlg

Lazy wrapper for the matrix exponential, operations defined in LazySets.jl.

Fields

  • m – (optional, default: 30) size of the Krylov subspace
  • tol – (optional, default: 1e-10) tolerance

Notes

This algorithm admits the alias :lazy and also :krylov(using default options).

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ReachabilityAnalysis.Exponentiation.IntervalExpAlg — Type
IntervalExpAlg <: AbstractExpAlg

Matrix exponential using an interval enclosure of the Taylor series remainder.

Fields

  • order – order of the Taylor series expansion of the matrix exponential

Notes

This method allows to overapproximate $exp(Aδ)$ with an interval matrix. It also accepts an interval matrix $A$ given as input. This method admits the alias :interval and :taylor.

source

References

  • HIH08Higham, Nicholas J. Functions of matrices: theory and computation. Society for Industrial and Applied Mathematics, 2008.
+\end{pmatrix}.\]

where $Φ(A, δ) = e^{Aδ}$. In particular, Φ₂ = P_{3n}[1:n, (2*n+1):3*n]. This method can be found in [FRE11].

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ReachabilityAnalysis.Exponentiation.AbstractExpAlg — Type
AbstractExpAlg

Abstract supertype for all exponentiation algorithms.

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ReachabilityAnalysis.Exponentiation.BaseExpAlg — Type
BaseExpAlg <: AbstractExpAlg

Matrix exponential using the scaling and squaring algorithm implemented in Julia Base.

Notes

If the array is static, the method implemented in StaticArrays.jl is applied. This algorithm admits the alias :base.

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ReachabilityAnalysis.Exponentiation.LazyExpAlg — Type
LazyExpAlg <: AbstractExpAlg

Lazy wrapper for the matrix exponential, operations defined in LazySets.jl.

Fields

  • m – (optional, default: 30) size of the Krylov subspace
  • tol – (optional, default: 1e-10) tolerance

Notes

This algorithm admits the alias :lazy and also :krylov(using default options).

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ReachabilityAnalysis.Exponentiation.IntervalExpAlg — Type
IntervalExpAlg <: AbstractExpAlg

Matrix exponential using an interval enclosure of the Taylor series remainder.

Fields

  • order – order of the Taylor series expansion of the matrix exponential

Notes

This method allows to overapproximate $exp(Aδ)$ with an interval matrix. It also accepts an interval matrix $A$ given as input. This method admits the alias :interval and :taylor.

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ReachabilityAnalysis.Exponentiation.PadeExpAlg — Type
PadeExpAlg <: AbstractExpAlg

Matrix exponential for sparse matrices using Pade approximants.

Notes

Requires Expokit.jl. This algorithm admits the alias :pade.

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References

diff --git a/dev/lib/distributed/index.html b/dev/lib/distributed/index.html index a93da4a44..4a5ffe843 100644 --- a/dev/lib/distributed/index.html +++ b/dev/lib/distributed/index.html @@ -3,7 +3,7 @@ intersection_method::AbstractIntersectionMethod=nothing, check_invariant=false, intersect_invariant=false, - ) where {HS<:HybridSystem, ST<:AdmissibleSet}

Distribute the set of initial states to each mode of a hybrid system.

Input

(and possibly overapproximate to keep WaitingList concretely typed)

Output

A new initial value problem with the same hybrid system but where the set of initial states is the list of tuples (state, X0), for each state in the hybrid system.

source

Julia's documentation on multi-threading describes how to check the number of threads available in the current session, Threads.nthreads(), and how to control the number of threads e.g. by starting Julia with $ julia --threads 4 to use four threads. Please note that to make the number of threads persistent across different Julia sessions you should export an environment variable, export JULIA_NUM_THREADS=4 to be added in your .bashrc or .bash_profile files, or setting ENV["JULIA_NUM_THREADS"]=4 in your .julia/config/startup.jl file.

GPGPU computing

The recommended entry point for using general-purpose graphical processing units (GPGPU) in Julia is the library CUDA.jl. The package exports an array type CuArray used as an abstraction to perform array operations on the GPU device; common linear algebra operations are readily available through dispatch on CuArray. Apart from convenient high-level syntax to do basic linear algebra on GPUs; more advanced kernel functions can be implemented as well as it is explained in the CUDA.jl documentation.

Parallel solve

Reachability computations can be computed in parallel using Julia's built-in multithreaded parallelism. This feature is available formulating and solving an initial-value problem with an array of initial conditions.

About BLAS threads

To control the number of threads used by your BLAS library, use the function BLAS.set_num_threads(n), where n is an integer. Furthermore, the function BLAS.get_num_threads() defined below will return the current value.

Note. If you are using Julia v"0.7-" (run the command VERSION to find this), instead of Base.LinAlg below use LinearAlgebra, and this module should have been loaded in the current scope with using LinearAlgebra.

#
+            ) where {HS<:HybridSystem, ST<:AdmissibleSet}

Distribute the set of initial states to each mode of a hybrid system.

Input

(and possibly overapproximate to keep WaitingList concretely typed)

Output

A new initial value problem with the same hybrid system but where the set of initial states is the list of tuples (state, X0), for each state in the hybrid system.

source

Julia's documentation on multi-threading describes how to check the number of threads available in the current session, Threads.nthreads(), and how to control the number of threads e.g. by starting Julia with $ julia --threads 4 to use four threads. Please note that to make the number of threads persistent across different Julia sessions you should export an environment variable, export JULIA_NUM_THREADS=4 to be added in your .bashrc or .bash_profile files, or setting ENV["JULIA_NUM_THREADS"]=4 in your .julia/config/startup.jl file.

GPGPU computing

The recommended entry point for using general-purpose graphical processing units (GPGPU) in Julia is the library CUDA.jl. The package exports an array type CuArray used as an abstraction to perform array operations on the GPU device; common linear algebra operations are readily available through dispatch on CuArray. Apart from convenient high-level syntax to do basic linear algebra on GPUs; more advanced kernel functions can be implemented as well as it is explained in the CUDA.jl documentation.

Parallel solve

Reachability computations can be computed in parallel using Julia's built-in multithreaded parallelism. This feature is available formulating and solving an initial-value problem with an array of initial conditions.

About BLAS threads

To control the number of threads used by your BLAS library, use the function BLAS.set_num_threads(n), where n is an integer. Furthermore, the function BLAS.get_num_threads() defined below will return the current value.

Note. If you are using Julia v"0.7-" (run the command VERSION to find this), instead of Base.LinAlg below use LinearAlgebra, and this module should have been loaded in the current scope with using LinearAlgebra.

#
 # This function is a part of Julia. License is MIT: https://julialang.org/license
 #
 function get_num_threads() # anonymous so it will be serialized when called
@@ -25,4 +25,4 @@
     end
 
     return nothing
-end
+end diff --git a/dev/lib/flowpipes/index.html b/dev/lib/flowpipes/index.html index 22950395c..c203aa767 100644 --- a/dev/lib/flowpipes/index.html +++ b/dev/lib/flowpipes/index.html @@ -1,2 +1,2 @@ -Flowpipes · ReachabilityAnalysis.jl
GitHub

Flowpipes

Abstract interface

The following functions are available at the interface level.

ReachabilityAnalysis.basetype — Method
basetype(T::Type{<:AbstractFlowpipe})

Return the base type of the given flowpipe type (i.e., without type parameters).

Input

  • T – flowpipe type, used for dispatch

Output

The base type of T.

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LazySets.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.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.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.

source

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.

Notes

See also Shift for the lazy counterpart.

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Hybrid flowpipe

ReachabilityAnalysis.HybridFlowpipe — Type
HybridFlowpipe{N, D, FT<:AbstractFlowpipe, VOA<:VectorOfArray{N, D, Vector{FT}}} <: 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, D, FT<:AbstractFlowpipe, VOA<:VectorOfArray{N, D, Vector{FT}}} <: 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.

source

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).

A convenience constructor alias Shift is given.

source

Partitioned flowpipe

ReachabilityAnalysis.PartitionedFlowpipe — Type
PartitionedFlowpipe{N, D, FT<:AbstractFlowpipe, VOA<:VectorOfArray{N, D, Vector{FT}}} <: AbstractFlowpipe

Type that wraps a vector of flowpipes of possibly different types.

Fields

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

Notes

source

Mapped flowpipe

+Flowpipes · ReachabilityAnalysis.jl
GitHub

Flowpipes

Abstract interface

The following functions are available at the interface level.

ReachabilityAnalysis.basetype — Method
basetype(T::Type{<:AbstractFlowpipe})

Return the base type of the given flowpipe type (i.e., without type parameters).

Input

  • T – flowpipe type, used for dispatch

Output

The base type of T.

source
LazySets.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.

source
LazySets.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.

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

Input

  • fp – flowpipe

Output

An integer representing the ambient dimension of the flowpipe.

source
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.

source
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.

source
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).

source
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.

source
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.

source
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.

source
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.

source

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.

source
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.

source

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.

source
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.

Notes

See also Shift for the lazy counterpart.

source

Hybrid flowpipe

ReachabilityAnalysis.HybridFlowpipe — Type
HybridFlowpipe{N, D, FT<:AbstractFlowpipe, VOA<:VectorOfArray{N, D, Vector{FT}}} <: 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.

source

Mixed flowpipe

ReachabilityAnalysis.MixedFlowpipe — Type
MixedFlowpipe{N, D, FT<:AbstractFlowpipe, VOA<:VectorOfArray{N, D, Vector{FT}}} <: 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.

source

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).

A convenience constructor alias Shift is given.

source

Partitioned flowpipe

ReachabilityAnalysis.PartitionedFlowpipe — Type
PartitionedFlowpipe{N, D, FT<:AbstractFlowpipe, VOA<:VectorOfArray{N, D, Vector{FT}}} <: AbstractFlowpipe

Type that wraps a vector of flowpipes of possibly different types.

Fields

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

Notes

source

Mapped flowpipe

diff --git a/dev/lib/internals/index.html b/dev/lib/internals/index.html index 4a829efe5..b7807088c 100644 --- a/dev/lib/internals/index.html +++ b/dev/lib/internals/index.html @@ -2,4 +2,4 @@ Internal functions and macros · ReachabilityAnalysis.jl
GitHub

Internal functions and macros

This section of the manual includes some internal (i.e. unexported) functions and macros used within the library.

Function annotations

ReachabilityAnalysis.@requires — Macro
@requires(module_name)

Convenience macro to annotate that a package is required to use a certain function.

Input

  • module_name – name of the required package

Output

The macro expands to an assertion that checks whether the module module_name is known in the calling scope.

Notes

Usage:

function foo(...)
     @require MyPackage
     ... # functionality that requires MyPackage to be loaded
-end
source

Algorithm defaults

ReachabilityAnalysis._default_cpost — Function
_default_cpost(ivp::IVP{<:AbstractContinuousSystem}, tspan; kwargs...)

Input

  • ivp – initial-value problem
  • tspan – time-span

Output

A continuous post-operator that can handle the given initial-value problem.

Notes

If the system is affine, then:

  • If it is one-dimensional, algorithm INT is used, otherwise,
  • Algorithm GLGM06 is used.

If the system is not affine, then the algorithm TMJets is used.

source

Documentation fixes

The following duplicated documentation is necessary due to a problem in the Documenter package.

ReachabilityAnalysis.basetype — Function
basetype(T::Type{<:AbstractReachSet})

Return the base type of the given reach-set type (i.e., without type parameters).

Input

  • T – reach-set type, used for dispatch

Output

The base type of T.

source
basetype(T::Type{<:AbstractFlowpipe})

Return the base type of the given flowpipe type (i.e., without type parameters).

Input

  • T – flowpipe type, used for dispatch

Output

The base type of T.

source
+endsource

Algorithm defaults

ReachabilityAnalysis._default_cpost — Function
_default_cpost(ivp::IVP{<:AbstractContinuousSystem}, tspan; kwargs...)

Input

  • ivp – initial-value problem
  • tspan – time-span

Output

A continuous post-operator that can handle the given initial-value problem.

Notes

If the system is affine, then:

  • If it is one-dimensional, algorithm INT is used, otherwise,
  • Algorithm GLGM06 is used.

If the system is not affine, then the algorithm TMJets is used.

source

Documentation fixes

The following duplicated documentation is necessary due to a problem in the Documenter package.

ReachabilityAnalysis.basetype — Function
basetype(T::Type{<:AbstractReachSet})

Return the base type of the given reach-set type (i.e., without type parameters).

Input

  • T – reach-set type, used for dispatch

Output

The base type of T.

source
basetype(T::Type{<:AbstractFlowpipe})

Return the base type of the given flowpipe type (i.e., without type parameters).

Input

  • T – flowpipe type, used for dispatch

Output

The base type of T.

source
diff --git a/dev/lib/operations/index.html b/dev/lib/operations/index.html index fc0b08017..b9fb82b26 100644 --- a/dev/lib/operations/index.html +++ b/dev/lib/operations/index.html @@ -1,2 +1,2 @@ -Further set operations · ReachabilityAnalysis.jl
GitHub

Further set operations

Convexification

ReachabilityAnalysis.convexify — Function
convexify(fp::Flowpipe{N, <:AbstractLazyReachSet}) where {N}

Return a reach-set representing the convex hull array of the flowpipe.

Input

  • fp – flowpipe

Output

A reach-set that contains the convex hull array, ConvexHullArray, of the given flowpipe.

Notes

The time span of this reach-set is the same as the time-span of the flowpipe.

This function allocates an array to store the sets of the flowpipe.

source
convexify(fp::AbstractVector{<:AbstractLazyReachSet{N}}) where {N}

Return a reach-set representing the convex hull array of the array of the array of reach-sets.

Input

  • fp – array of reach-sets

Output

A reach-set that contains the convex hull array, ConvexHullArray, of the given flowpipe.

Notes

The time span of this reach-set corresponds to the minimum (resp. maximum) of the time span of each reach-set in fp.

This function allocates an array to store the sets of the flowpipe.

The function doesn't assume that the reach-sets are time ordered.

source

Quality measures

ReachabilityAnalysis.relative_error — Function
relative_error(x, x_ref)

Compute the relative error between interval x and a reference interval xref.

Input

  • x – interval
  • xref – reference interval

Output

An interval representing the relative error (in percentage) of x with respect to the reference interval xref.

Algorithm

If $x = [x_L, x_H]$andxref = [xref_L, xref_H], the output is the intervaly = 100 * [y_L, y_H]computed asy_L = -(x_L - xref_L) / denandy_H = (x_H - xref_H) / den, whereden = xref_H - xref_L`.

This function measures the relative error between an interval x and a reference interval x_ref accounting for it the lower and the upper range bounds separately (see Eq. (20) in [1]).

References

  • [1] Althoff, Matthias, Dmitry Grebenyuk, and Niklas Kochdumper. "Implementation of Taylor models in CORA 2018." Proc. of the 5th International Workshop on Applied Verification for Continuous and Hybrid Systems. 2018. pdf
source
+Further set operations · ReachabilityAnalysis.jl
GitHub

Further set operations

Convexification

ReachabilityAnalysis.convexify — Function
convexify(fp::Flowpipe{N, <:AbstractLazyReachSet}) where {N}

Return a reach-set representing the convex hull array of the flowpipe.

Input

  • fp – flowpipe

Output

A reach-set that contains the convex hull array, ConvexHullArray, of the given flowpipe.

Notes

The time span of this reach-set is the same as the time-span of the flowpipe.

This function allocates an array to store the sets of the flowpipe.

source
convexify(fp::AbstractVector{<:AbstractLazyReachSet{N}}) where {N}

Return a reach-set representing the convex hull array of the array of the array of reach-sets.

Input

  • fp – array of reach-sets

Output

A reach-set that contains the convex hull array, ConvexHullArray, of the given flowpipe.

Notes

The time span of this reach-set corresponds to the minimum (resp. maximum) of the time span of each reach-set in fp.

This function allocates an array to store the sets of the flowpipe.

The function doesn't assume that the reach-sets are time ordered.

source

Quality measures

ReachabilityAnalysis.relative_error — Function
relative_error(x, x_ref)

Compute the relative error between interval x and a reference interval xref.

Input

  • x – interval
  • xref – reference interval

Output

An interval representing the relative error (in percentage) of x with respect to the reference interval xref.

Algorithm

If $x = [x_L, x_H]$andxref = [xref_L, xref_H], the output is the intervaly = 100 * [y_L, y_H]computed asy_L = -(x_L - xref_L) / denandy_H = (x_H - xref_H) / den, whereden = xref_H - xref_L`.

This function measures the relative error between an interval x and a reference interval x_ref accounting for it the lower and the upper range bounds separately (see Eq. (20) in [1]).

References

  • [1] Althoff, Matthias, Dmitry Grebenyuk, and Niklas Kochdumper. "Implementation of Taylor models in CORA 2018." Proc. of the 5th International Workshop on Applied Verification for Continuous and Hybrid Systems. 2018. pdf
source
diff --git a/dev/lib/projections/index.html b/dev/lib/projections/index.html index 167c2dc37..8abbcb53c 100644 --- a/dev/lib/projections/index.html +++ b/dev/lib/projections/index.html @@ -1,2 +1,2 @@ -Projections · ReachabilityAnalysis.jl
GitHub

Projections

Concrete projection

TODO: document project

Lazy projection

TODO: document Projection

LazySets.Projection — Function
Projection(fp::AbstractFlowpipe, vars::NTuple{D, T}) where {D, T<:Integer}

Return the lazy projection of a flowpipe.

Input

Output

Notes

The projection is lazy, and consists of mapping each set X in the flowpipe to MX, where M is the projection matrix associated with the given variables vars.

source
+Projections · ReachabilityAnalysis.jl
GitHub

Projections

Concrete projection

TODO: document project

Lazy projection

TODO: document Projection

LazySets.Projection — Function
Projection(fp::AbstractFlowpipe, vars::NTuple{D, T}) where {D, T<:Integer}

Return the lazy projection of a flowpipe.

Input

Output

Notes

The projection is lazy, and consists of mapping each set X in the flowpipe to MX, where M is the projection matrix associated with the given variables vars.

source
diff --git a/dev/lib/reachsets/index.html b/dev/lib/reachsets/index.html index b5377038a..e21eb3e9b 100644 --- a/dev/lib/reachsets/index.html +++ b/dev/lib/reachsets/index.html @@ -1,3 +1,3 @@ -Reach-sets · ReachabilityAnalysis.jl
GitHub

Reach-sets

Abstract interface

ReachabilityAnalysis.AbstractReachSet — Type
AbstractReachSet{N}

Abstract type for all reach-sets types.

Notes

A reach-set is a set representation X associated to a given time span Δt.

In its simplest form, we represent reach-sets with a struct that wraps the tuple (X, Δt), where X corresponds to a geometric set, eg. a polytope, and Δt is the interval with the time span associated to this reach-set.

This type is parametric in N. The parameter N represents for the type of numerical coefficient used in the representation (typically, N = Float64).

Although concrete subtypes of AbstractReachSet may represent the set X in different ways, or carry additional information as an extra type field, they should all implement the AbstractReachSet interface to enable shared functionality for reach-set types. In particular, each concrete subtype should implement the following methods:

  • set – return the geometric set
  • setrep – return the type of the set representation
  • tspan – return the time interval span
  • tstart – return the initial time
  • tend – return the final time
  • dim – return ambient dimension of the reach-set
source

The functions are available at the interface level.

ReachabilityAnalysis.basetype — Method
basetype(T::Type{<:AbstractReachSet})

Return the base type of the given reach-set type (i.e., without type parameters).

Input

  • T – reach-set type, used for dispatch

Output

The base type of T.

source
LazySets.set — Method
set(R::AbstractReachSet)

Return the geometric set represented by this reach-set.

Input

  • R – reach-set

Output

The set wrapped by the given reach-set.

source
ReachabilityAnalysis.setrep — Method
setrep(R::AbstractReachSet)

Return the type of the set representation of this reach-set.

Input

  • R – reach-set

Output

Type of the set representation of the given reach-set.

source
ReachabilityAnalysis.tspan — Method
tspan(R::AbstractReachSet)

Return time span of this reach-set.

Input

  • R – reach-set

Output

The interval representing the time span of the given reach-set.

source
ReachabilityAnalysis.tstart — Method
tstart(R::AbstractReachSet)

Return the initial time of this reach-set.

Input

  • R – reach-set

Output

A float representing the initial time of the given reach-set.

source
ReachabilityAnalysis.tend — Method
tend(R::AbstractReachSet)

Return the final time of this reach-set.

Input

  • R – reach-set

Output

A float representing the final time of the given reach-set.

source
LazySets.dim — Method
dim(R::AbstractReachSet)

Return the ambient dimension of the reach-set.

Input

  • R – reach-set

Output

An integer corresponding to the ambient dimension of the given reach-set.

source
Base.copy — Method
copy(R::AbstractReachSet)

Return a copy of the given reach-set.

Input

  • R – reach-set

Output

A new reach-set of the sam type and the same field values as R.

source
ReachabilityAnalysis.shift — Method
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.

Notes

See also Shift for the lazy counterpart.

source
ReachabilityAnalysis.AbstractLazyReachSet — Type
AbstractLazyReachSet{N} <: AbstractReachSet{N}

Abstract type for all reach-set types that use a LazySet for the underlying set representation.

Notes

An AbstractLazyReachSet is the interface for reach-sets such that the geometric set is represented by any subtype of LazySet.

This types implements the LazySets interface, namely support function (ρ), support vector (σ) and ambient dimension (dim) functions. Hence, these functions directly apply to concrete subtypes of an AbstractLazyReachSet. The set wrapped by this type is obtained through set(R).

The following functions should be implemented by any concrete subtype:

  • reconstruct – create a new instance of the given reach-set with a different set representation but sharing the other fields, i.e. the same time span (and the same for other fields, if applicable)

In addition to the functions inherited from AbstractReachSet, the following are available:

  • linear_map – concrete linear map of a reach-set
  • project – projection of a reach-set
  • shift – time-shift of a reach-set
  • vars – tuple of integers associated to the variables of the given reach-set
source
LazySets.project — Method
project(R::AbstractLazyReachSet, variables::NTuple{D, M};
-        check_vars::Bool=true) where {D, M<:Integer}

Projects a reach-set onto the subspace spanned by the given variables.

Input

  • R – reach-set
  • vars – tuple of variables for the projection
  • check_vars – (optional, default: true) if true, check that the given variable indices vars are a subset of the variables of R

Output

A SparseReachSet whose variable indices are given by vars.

The type of the new reach-set depends on the type of the reach-set R:

  • If R contains a hyperrectangular set, the output is a hyperrectangle.
  • If R contains a zonotopic set, the output is a zonotope.
  • Otherwise, the return type is a polytope either in constraint representation or in vertex representation, depending on the dimension and the properties of M. For details, see LazySets.project.

Notes

This function can be used to project a reach-set onto a lower-dimensional sub-space. The projection is concrete, and it consists of mapping the reach-set X = set(R) to a new reach-set through to MX, where M is the projection matrix associated with the given variables vars.

To project onto the time variable, use the index 0. For instance, (0, 1) projects onto the time variable and the first variable in R.

source

Reachable set

ReachabilityAnalysis.ReachSet — Type
ReachSet{N, ST<:LazySet{N}} <: AbstractLazyReachSet{N}

Type that wraps a reach-set using a LazySet as underlying representation.

Fields

  • X – set
  • Δt – time interval

Notes

A ReachSet is a struct representing (an approximation of) the reachable states for a given time interval. The type of the representation is ST, which may be any subtype LazySet. For efficiency reasons, ST should be concretely typed.

By assumption the coordinates in this reach-set are associated to the integers 1, …, n. The function vars returns such tuple.

source

Sparse reachable set

ReachabilityAnalysis.SparseReachSet — Type
SparseReachSet{N, ST<:LazySet{N}, D} <: AbstractReachSet{N}

Type that wraps a reach-set using a LazySet as underlying representation, together with a tuple of variables associated to this reach-set.

Fields

  • X – set
  • Δt – time interval
  • vars – tuple of variable indices represented by the set X

Notes

A SparseReachSet is a struct representing (an approximation of) the reachable states for a given time interval. The type of the representation is ST, which may be any subtype of LazySet (ideally, concrete). Moreover, this type also stores information about the variables (also named coordinates, or by abuse of notation, dimensions) corresponding to the set X.

For instance in the ambient space n=5, one may have a SparseReachSet whose variables tuple is vars = (4, 5, 6), i.e. representing a three-dimensional projection of the full-dimensional reach-set. In consequence, the dimension of X doesn't match the length of vars, in general

In this type, the parameter N represents the numerical type of the LazySet (typically, Float64), the type ST represents the set representation used, and D denotes the dimension of this sparse reach set. Note that, in contrast to ReachSet, for SparseReachSet the number of dimensions is part of the type information.

source

Taylor model reach-sets

ReachabilityAnalysis.TaylorModelReachSet — Type
TaylorModelReachSet{N, S} <: AbstractTaylorModelReachSet{N}

Reach-set representation consisting of a vector of taylor models in one variable (the "time" variable) whose coefficients are multivariate polynomials (the "space" variables).

Notes

The parameters N and S refer to the numerical type of the representation (used for the Taylor model in time and Taylor series polynomials respectively). The space variables are assumed to be normalized to the interval [-1, 1]. It is assumed that the time domain is the same for all components.

source

Template reach-sets

ReachabilityAnalysis.TemplateReachSet — Type
TemplateReachSet{N, VN, TN<:AbstractDirections{N, VN}, SN<:AbstractVector{N}} <: AbstractLazyReachSet{N}

Reach set that stores the support function of a set at a give set of directions.

Notes

The struct has the following parameters:

  • N – refers to the numerical type of the representation.
  • VN – refers to the vector type of the template
  • TN – refers to the template type
  • SN – vector type that holds the support function evaluations

Concrete subtypes of AbstractDirections are defined in the LazySets library.

This reach-set implicitly represents a set by a set of directions and support functions. set(R::TemplateReachSet) returns a polyhedron in half-space representation.

Apart from the getter functions inherited from the AbstractReachSet interface, the following methods are available:

  • directions(R) – return the set of directions normal to the faces of this reach-set
  • support_functions(R) – return the vector of support function evaluations
  • support_functions(R, i) – return the i-th coordinate of the vector of support function evaluatons
source
+Reach-sets · ReachabilityAnalysis.jl
GitHub

Reach-sets

Abstract interface

ReachabilityAnalysis.AbstractReachSet — Type
AbstractReachSet{N}

Abstract type for all reach-sets types.

Notes

A reach-set is a set representation X associated to a given time span Δt.

In its simplest form, we represent reach-sets with a struct that wraps the tuple (X, Δt), where X corresponds to a geometric set, eg. a polytope, and Δt is the interval with the time span associated to this reach-set.

This type is parametric in N. The parameter N represents for the type of numerical coefficient used in the representation (typically, N = Float64).

Although concrete subtypes of AbstractReachSet may represent the set X in different ways, or carry additional information as an extra type field, they should all implement the AbstractReachSet interface to enable shared functionality for reach-set types. In particular, each concrete subtype should implement the following methods:

  • set – return the geometric set
  • setrep – return the type of the set representation
  • tspan – return the time interval span
  • tstart – return the initial time
  • tend – return the final time
  • dim – return ambient dimension of the reach-set
source

The functions are available at the interface level.

ReachabilityAnalysis.basetype — Method
basetype(T::Type{<:AbstractReachSet})

Return the base type of the given reach-set type (i.e., without type parameters).

Input

  • T – reach-set type, used for dispatch

Output

The base type of T.

source
LazySets.set — Method
set(R::AbstractReachSet)

Return the geometric set represented by this reach-set.

Input

  • R – reach-set

Output

The set wrapped by the given reach-set.

source
ReachabilityAnalysis.setrep — Method
setrep(R::AbstractReachSet)

Return the type of the set representation of this reach-set.

Input

  • R – reach-set

Output

Type of the set representation of the given reach-set.

source
ReachabilityAnalysis.tspan — Method
tspan(R::AbstractReachSet)

Return time span of this reach-set.

Input

  • R – reach-set

Output

The interval representing the time span of the given reach-set.

source
ReachabilityAnalysis.tstart — Method
tstart(R::AbstractReachSet)

Return the initial time of this reach-set.

Input

  • R – reach-set

Output

A float representing the initial time of the given reach-set.

source
ReachabilityAnalysis.tend — Method
tend(R::AbstractReachSet)

Return the final time of this reach-set.

Input

  • R – reach-set

Output

A float representing the final time of the given reach-set.

source
LazySets.dim — Method
dim(R::AbstractReachSet)

Return the ambient dimension of the reach-set.

Input

  • R – reach-set

Output

An integer corresponding to the ambient dimension of the given reach-set.

source
Base.copy — Method
copy(R::AbstractReachSet)

Return a copy of the given reach-set.

Input

  • R – reach-set

Output

A new reach-set of the sam type and the same field values as R.

source
ReachabilityAnalysis.shift — Method
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.

Notes

See also Shift for the lazy counterpart.

source
ReachabilityAnalysis.AbstractLazyReachSet — Type
AbstractLazyReachSet{N} <: AbstractReachSet{N}

Abstract type for all reach-set types that use a LazySet for the underlying set representation.

Notes

An AbstractLazyReachSet is the interface for reach-sets such that the geometric set is represented by any subtype of LazySet.

This types implements the LazySets interface, namely support function (ρ), support vector (σ) and ambient dimension (dim) functions. Hence, these functions directly apply to concrete subtypes of an AbstractLazyReachSet. The set wrapped by this type is obtained through set(R).

The following functions should be implemented by any concrete subtype:

  • reconstruct – create a new instance of the given reach-set with a different set representation but sharing the other fields, i.e. the same time span (and the same for other fields, if applicable)

In addition to the functions inherited from AbstractReachSet, the following are available:

  • linear_map – concrete linear map of a reach-set
  • project – projection of a reach-set
  • shift – time-shift of a reach-set
  • vars – tuple of integers associated to the variables of the given reach-set
source
LazySets.project — Method
project(R::AbstractLazyReachSet, variables::NTuple{D, M};
+        check_vars::Bool=true) where {D, M<:Integer}

Projects a reach-set onto the subspace spanned by the given variables.

Input

  • R – reach-set
  • vars – tuple of variables for the projection
  • check_vars – (optional, default: true) if true, check that the given variable indices vars are a subset of the variables of R

Output

A SparseReachSet whose variable indices are given by vars.

The type of the new reach-set depends on the type of the reach-set R:

  • If R contains a hyperrectangular set, the output is a hyperrectangle.
  • If R contains a zonotopic set, the output is a zonotope.
  • Otherwise, the return type is a polytope either in constraint representation or in vertex representation, depending on the dimension and the properties of M. For details, see LazySets.project.

Notes

This function can be used to project a reach-set onto a lower-dimensional sub-space. The projection is concrete, and it consists of mapping the reach-set X = set(R) to a new reach-set through to MX, where M is the projection matrix associated with the given variables vars.

To project onto the time variable, use the index 0. For instance, (0, 1) projects onto the time variable and the first variable in R.

source

Reachable set

ReachabilityAnalysis.ReachSet — Type
ReachSet{N, ST<:LazySet{N}} <: AbstractLazyReachSet{N}

Type that wraps a reach-set using a LazySet as underlying representation.

Fields

  • X – set
  • Δt – time interval

Notes

A ReachSet is a struct representing (an approximation of) the reachable states for a given time interval. The type of the representation is ST, which may be any subtype LazySet. For efficiency reasons, ST should be concretely typed.

By assumption the coordinates in this reach-set are associated to the integers 1, …, n. The function vars returns such tuple.

source

Sparse reachable set

ReachabilityAnalysis.SparseReachSet — Type
SparseReachSet{N, ST<:LazySet{N}, D} <: AbstractReachSet{N}

Type that wraps a reach-set using a LazySet as underlying representation, together with a tuple of variables associated to this reach-set.

Fields

  • X – set
  • Δt – time interval
  • vars – tuple of variable indices represented by the set X

Notes

A SparseReachSet is a struct representing (an approximation of) the reachable states for a given time interval. The type of the representation is ST, which may be any subtype of LazySet (ideally, concrete). Moreover, this type also stores information about the variables (also named coordinates, or by abuse of notation, dimensions) corresponding to the set X.

For instance in the ambient space n=5, one may have a SparseReachSet whose variables tuple is vars = (4, 5, 6), i.e. representing a three-dimensional projection of the full-dimensional reach-set. In consequence, the dimension of X doesn't match the length of vars, in general

In this type, the parameter N represents the numerical type of the LazySet (typically, Float64), the type ST represents the set representation used, and D denotes the dimension of this sparse reach set. Note that, in contrast to ReachSet, for SparseReachSet the number of dimensions is part of the type information.

source

Taylor model reach-sets

ReachabilityAnalysis.TaylorModelReachSet — Type
TaylorModelReachSet{N, S} <: AbstractTaylorModelReachSet{N}

Reach-set representation consisting of a vector of taylor models in one variable (the "time" variable) whose coefficients are multivariate polynomials (the "space" variables).

Notes

The parameters N and S refer to the numerical type of the representation (used for the Taylor model in time and Taylor series polynomials respectively). The space variables are assumed to be normalized to the interval [-1, 1]. It is assumed that the time domain is the same for all components.

source

Template reach-sets

ReachabilityAnalysis.TemplateReachSet — Type
TemplateReachSet{N, VN, TN<:AbstractDirections{N, VN}, SN<:AbstractVector{N}} <: AbstractLazyReachSet{N}

Reach set that stores the support function of a set at a give set of directions.

Notes

The struct has the following parameters:

  • N – refers to the numerical type of the representation.
  • VN – refers to the vector type of the template
  • TN – refers to the template type
  • SN – vector type that holds the support function evaluations

Concrete subtypes of AbstractDirections are defined in the LazySets library.

This reach-set implicitly represents a set by a set of directions and support functions. set(R::TemplateReachSet) returns a polyhedron in half-space representation.

Apart from the getter functions inherited from the AbstractReachSet interface, the following methods are available:

  • directions(R) – return the set of directions normal to the faces of this reach-set
  • support_functions(R) – return the vector of support function evaluations
  • support_functions(R, i) – return the i-th coordinate of the vector of support function evaluatons
source
diff --git a/dev/lib/solutions/index.html b/dev/lib/solutions/index.html index e64514ae0..612d46524 100644 --- a/dev/lib/solutions/index.html +++ b/dev/lib/solutions/index.html @@ -1,2 +1,2 @@ -Solutions · ReachabilityAnalysis.jl
GitHub

Solutions

Abstract interface

Solution of a reachability problem

ReachabilityAnalysis.ReachSolution — Type
ReachSolution{FT<:AbstractFlowpipe, ST<:AbstractPost} <: AbstractSolution

Type that wraps the solution of a reachability problem as a sequence of lazy sets, and a dictionary of options.

Fields

  • Xk – the list of AbstractReachSets
  • alg – algorithm used
  • options – the dictionary of options
source

Solution of a verification problem

ReachabilityAnalysis.CheckSolution — Type
CheckSolution{T} <: AbstractSolution

Type that wraps the solution of a verification problem.

Fields

TODO

Notes

This type contains the answer if the property is satisfied, and if not, it contains the index at which the property might be violated for the first time.

source

Solving a reachability problem

CommonSolve.solve — Function
solve(ivp::IVP{<:AbstractContinuousSystem}, tspan, alg; kwargs...)

Solves the initial-value problem defined by ivp over the time span tspan, using the algorithm alg. If no algorithm is given, a default algorithm is chosen.

Input

  • ivp – initial-value problem
  • tspan – time span for this initial-value problem
  • alg – reachability algorithm

Additional options are passed as arguments or keyword arguments; see the notes below for details. See the online documentation for examples.

Output

The solution of a reachability problem, as an instance of a ReachSolution.

Notes

  • Use the alg, algorithm or opC keyword arguments to specify the algorithm to solve the initial-value problem. Algorithm-specific options should be passed to the algorithm constructor as well.

  • Use the tspan keyword argument to specify the time span; it can be:

    • a tuple,
    • an interval, or
    • a vector with two components.

  • Use the T keyword argument to specify the time horizon; the initial time is then assumed to be zero.

  • Use the static keyword argument to force conversion to static arrays in the algorithm (should be supported by the algorithm).

  • Use the NSTEPS keyword argument to specify the number of discrete steps solved in the set-based recurrence.

  • Use the threading option to use multi-threading parallelism. This option applies for initial-value problems whose initial condition is a vector of sets.

source

Solving a hybrid reachability problem

ReachabilityAnalysis.WaitingList — Type
WaitingList{TN, ST, M, QT<:StateInLocation{ST, M}} <: AbstractWaitingList

Iterable container representing a list of pairs (set, mode) of a hybrid automaton.

Fields

  • times – vector with a time interval associated to each state
  • array – vector of StateInLocation

Notes

A WaitingList is a list of pairs `(set_i, loc_i) for i in 1..k where times is a vector with a time interval associated to each state.

This waiting list allows for a unique set representation (ST) for all elements of the list.

source

TODO: document other methods in solutions.jl.

+Solutions · ReachabilityAnalysis.jl
GitHub

Solutions

Abstract interface

Solution of a reachability problem

ReachabilityAnalysis.ReachSolution — Type
ReachSolution{FT<:AbstractFlowpipe, ST<:AbstractPost} <: AbstractSolution

Type that wraps the solution of a reachability problem as a sequence of lazy sets, and a dictionary of options.

Fields

  • Xk – the list of AbstractReachSets
  • alg – algorithm used
  • options – the dictionary of options
source

Solution of a verification problem

ReachabilityAnalysis.CheckSolution — Type
CheckSolution{T} <: AbstractSolution

Type that wraps the solution of a verification problem.

Fields

TODO

Notes

This type contains the answer if the property is satisfied, and if not, it contains the index at which the property might be violated for the first time.

source

Solving a reachability problem

CommonSolve.solve — Function
solve(ivp::IVP{<:AbstractContinuousSystem}, tspan, alg; kwargs...)

Solves the initial-value problem defined by ivp over the time span tspan, using the algorithm alg. If no algorithm is given, a default algorithm is chosen.

Input

  • ivp – initial-value problem
  • tspan – time span for this initial-value problem
  • alg – reachability algorithm

Additional options are passed as arguments or keyword arguments; see the notes below for details. See the online documentation for examples.

Output

The solution of a reachability problem, as an instance of a ReachSolution.

Notes

  • Use the alg, algorithm or opC keyword arguments to specify the algorithm to solve the initial-value problem. Algorithm-specific options should be passed to the algorithm constructor as well.

  • Use the tspan keyword argument to specify the time span; it can be:

    • a tuple,
    • an interval, or
    • a vector with two components.

  • Use the T keyword argument to specify the time horizon; the initial time is then assumed to be zero.

  • Use the static keyword argument to force conversion to static arrays in the algorithm (should be supported by the algorithm).

  • Use the NSTEPS keyword argument to specify the number of discrete steps solved in the set-based recurrence.

  • Use the threading option to use multi-threading parallelism. This option applies for initial-value problems whose initial condition is a vector of sets.

source

Solving a hybrid reachability problem

ReachabilityAnalysis.WaitingList — Type
WaitingList{TN, ST, M, QT<:StateInLocation{ST, M}} <: AbstractWaitingList

Iterable container representing a list of pairs (set, mode) of a hybrid automaton.

Fields

  • times – vector with a time interval associated to each state
  • array – vector of StateInLocation

Notes

A WaitingList is a list of pairs `(set_i, loc_i) for i in 1..k where times is a vector with a time interval associated to each state.

This waiting list allows for a unique set representation (ST) for all elements of the list.

source

TODO: document other methods in solutions.jl.

diff --git a/dev/lib/systems/index.html b/dev/lib/systems/index.html index c834a9c12..482653263 100644 --- a/dev/lib/systems/index.html +++ b/dev/lib/systems/index.html @@ -1,5 +1,5 @@ -Systems · ReachabilityAnalysis.jl
GitHub

Systems

Systems types are defined in the library MathematicalSystems.jl. Apart from purely discrete or continuous, hybrid automata (i.e. those mixing discrete-continuous dynamics) are defined in HybridSystems.jl.

Types and macros

The API reference for systems types and macros can be found in the MathematicalSystems.jl documentation. Two commonly used macros are @system and @ivp, used to define a system and an initial-value problem respectively.

Normalization

LinearAlgebra.normalize — Method
normalize(system::AbstractSystem)

Transform a mathematical system to a normalized (or canonical) form.

Input

  • system – system; it can be discrete or continuous

Output

Either the same system if it already conforms to a canonical form, or a new system otherwise.

Notes

The normalization procedure consists of transforming a given system type into a "canonical" format that is used internally. More details are given below.

Algorithm

The implementation of normalize exploits MathematicalSystems's' types, which carry information about the problem as a type parameter.

Homogeneous ODEs of the form $x' = Ax, x ∈ \mathcal{X}$ are canonical if the associated problem is a ConstrainedLinearContinuousSystem and A is a matrix. This type does not handle non-deterministic inputs.

Note that a LinearContinuousSystem does not consider constraints on the state-space (such as an invariant); to specify state constraints, use a ConstrainedLinearContinuousSystem. If the passed system is a LinearContinuousSystem (i.e. no constraints) then the normalization fixes a universal set (Universe) as the constraint set.

The generalization to canonical systems with constraints and possibly time-varying non-deterministic inputs is considered next. These systems are of the form $x' = Ax + u, u ∈ \mathcal{U}, x ∈ \mathcal{X}$. The system type is ConstrainedLinearControlContinuousSystem, where A is a matrix, X is a set and U is an input, that is, any concrete subtype of AbstractInput.

If U is not given as an input, normalization accepts either a LazySet, or a vector of LazySets. In these cases, the sets are wrapped around an appropriate concrete input type.

If the system does not conform to a canonical form, the implementation tries to make the transformation; otherwise an error is thrown. In particular, ODEs of the form $x' = Ax + Bu$ are mapped into $x' = Ax + u, u ∈ B\mathcal{U}$, where now $u$ has the same dimensions as $x$.

The transformations described above are analogous in the discrete case, i.e. $x_{k+1} = A x_k$ and $x_{k+1} = Ax_{k} + u_k, u_k ∈ \mathcal{U}, x_k ∈ \mathcal{X}$ for the linear and affine cases respectively.

source
LinearAlgebra.normalize — Method
normalize(ivp::InitialValueProblem)

Transform an initial-value problem into a normalized (or canonical) form.

Input

  • ivp – initial-value problem

Output

Either the same initial-value problem if it already conforms to a canonical form, or a new one otherwise.

Notes

This function extends normalize for initial-value problems.

source
ReachabilityAnalysis.add_dimension — Function
add_dimension(A::AbstractMatrix, m=1)

Append one or more zero rows and columns to a matrix.

Input

  • A – matrix
  • m – (optional, default: 1) the number of extra dimensions

Examples

julia> A = [0.4 0.25; 0.46 -0.67]
+Systems · ReachabilityAnalysis.jl
GitHub
Systems
Systems types are defined in the library MathematicalSystems.jl. Apart from purely discrete or continuous, hybrid automata (i.e. those mixing discrete-continuous dynamics) are defined in HybridSystems.jl.
Types and macros
The API reference for systems types and macros can be found in the MathematicalSystems.jl documentation. Two commonly used macros are @system and @ivp, used to define a system and an initial-value problem respectively.
Normalization
LinearAlgebra.normalize — Method
normalize(system::AbstractSystem)
Transform a mathematical system to a normalized (or canonical) form.
Input
  • system – system; it can be discrete or continuous
Output
Either the same system if it already conforms to a canonical form, or a new system otherwise.
Notes
The normalization procedure consists of transforming a given system type into a "canonical" format that is used internally. More details are given below.
Algorithm
The implementation of normalize exploits MathematicalSystems's' types, which carry information about the problem as a type parameter.
Homogeneous ODEs of the form $x' = Ax, x ∈ \mathcal{X}$ are canonical if the associated problem is a ConstrainedLinearContinuousSystem and A is a matrix. This type does not handle non-deterministic inputs.
Note that a LinearContinuousSystem does not consider constraints on the state-space (such as an invariant); to specify state constraints, use a ConstrainedLinearContinuousSystem. If the passed system is a LinearContinuousSystem (i.e. no constraints) then the normalization fixes a universal set (Universe) as the constraint set.
The generalization to canonical systems with constraints and possibly time-varying non-deterministic inputs is considered next. These systems are of the form $x' = Ax + u, u ∈ \mathcal{U}, x ∈ \mathcal{X}$. The system type is ConstrainedLinearControlContinuousSystem, where A is a matrix, X is a set and U is an input, that is, any concrete subtype of AbstractInput.
If U is not given as an input, normalization accepts either a LazySet, or a vector of LazySets. In these cases, the sets are wrapped around an appropriate concrete input type.
If the system does not conform to a canonical form, the implementation tries to make the transformation; otherwise an error is thrown. In particular, ODEs of the form $x' = Ax + Bu$ are mapped into $x' = Ax + u, u ∈ B\mathcal{U}$, where now $u$ has the same dimensions as $x$.
The transformations described above are analogous in the discrete case, i.e. $x_{k+1} = A x_k$ and $x_{k+1} = Ax_{k} + u_k, u_k ∈ \mathcal{U}, x_k ∈ \mathcal{X}$ for the linear and affine cases respectively.
source
LinearAlgebra.normalize — Method
normalize(ivp::InitialValueProblem)
Transform an initial-value problem into a normalized (or canonical) form.
Input
  • ivp – initial-value problem
Output
Either the same initial-value problem if it already conforms to a canonical form, or a new one otherwise.
Notes
This function extends normalize for initial-value problems.
source
ReachabilityAnalysis.add_dimension — Function
add_dimension(A::AbstractMatrix, m=1)
Append one or more zero rows and columns to a matrix.
Input
  • A – matrix
  • m – (optional, default: 1) the number of extra dimensions
Examples
julia> A = [0.4 0.25; 0.46 -0.67]
 2×2 Array{Float64,2}:
  0.4    0.25
  0.46  -0.67
@@ -13,7 +13,7 @@
  0.4    0.25  0.0  0.0
  0.46  -0.67  0.0  0.0
  0.0    0.0   0.0  0.0
- 0.0    0.0   0.0  0.0
source
add_dimension(X::LazySet, m=1)
Adds an extra dimension to a LazySet through a Cartesian product.
Input
  • X – a lazy set
  • m – (optional, default: 1) the number of extra dimensions
Examples
julia> X = BallInf(ones(9), 0.5);
+ 0.0    0.0   0.0  0.0
source
add_dimension(X::LazySet, m=1)
Adds an extra dimension to a LazySet through a Cartesian product.
Input
  • X – a lazy set
  • m – (optional, default: 1) the number of extra dimensions
Examples
julia> X = BallInf(ones(9), 0.5);
 
 julia> dim(X)
 9
@@ -32,7 +32,7 @@
 ZeroSet{Float64}
More than one dimension can be added passing the second argument:
julia> Xext = add_dimension(BallInf(zeros(10), 0.1), 4);
 
 julia> dim(Xext)
-14
Notes
In the special case that the given set is a zero set, instead of cartesian product a new zero set with extended dimensions is returned.
source
add_dimension(cs, m=1)
Adds an extra dimension to a continuous system.
Input
  • cs – continuous system
  • m – (optional, default: 1) the number of extra dimensions
Examples
julia> using MathematicalSystems, SparseArrays
+14
Notes
In the special case that the given set is a zero set, instead of cartesian product a new zero set with extended dimensions is returned.
source
add_dimension(cs, m=1)
Adds an extra dimension to a continuous system.
Input
  • cs – continuous system
  • m – (optional, default: 1) the number of extra dimensions
Examples
julia> using MathematicalSystems, SparseArrays
 
 julia> A = sprandn(3, 3, 0.5);
 
@@ -69,4 +69,4 @@
 10
 
 julia> dim(next_set(inputset(sext), 1))
-10
source
Homogeneization
ReachabilityAnalysis.homogenize — Function
homogenize(ivp::IVP{CLCCS{N,MT,IdentityMultiple{N},XT,ConstantInput{SI}},ST}) where {N, MT<:AbstractMatrix{N}, XT<:LazySet{N}, SI<:Singleton{N}, ST<:LazySet{N}}
Transform an inhomogeneous linear initial-value problem into an homogeneous one by introducing auxiliary state variables.
Input
  • ivp – initial-value problem
Output
Homogeneous initial-value problem.
Notes
This function transforms the canonical initial-value problem $x' = Ax + u$, $x ∈ X$ with $u(0) ∈ U = {u}$ (a singleton) into an homogeneous problem without inputs $y' = Â * y$, $y ∈ Y$.
source
homogenize(sys::SOACS)
Transform an inhomogeneous second order system into an homogeneous one by introducing auxiliary state variables.
Input
  • sys – second order system
Output
First-order homogeneous system.
Notes
This function transforms the second-order system $Mx'' + Cx' + Kx = b$ into a first-order, homogeneous one, $y' = Â * y$. It is assumed that the matrix $M$ is invertible.
source
Hybrid systems
ReachabilityAnalysis.HACLD1 — Type
HACLD1{T<:AbstractSystem, MT, N, J} <: AHACLD
Single-mode hybrid automaton with clocked linear dynamics.
Fields
  • sys       – system
  • rmap      – reset map
  • Tsample   – sampling time
  • ζ         – jitter
  • switching – (optional) value that
Notes
This type is parametric in:
  • T  – system type
  • MT – type of the reset map
  • N  – numeric type, applies to the sampling time and jitter
  • J  – type associated to the jitter
The type associated to the jitter, J, can be one of the following:
  • Missing     – no jitter, i.e. switchings are deterministic
  • Number      – symmetric jitter, i.e. non-deterministic switchings occur in the                  intervals [Tsample - ζ, Tsample + ζ]
  • IA.Interval – non-symmetric jitter, i.e. non-deterministic switchings occur in the                  intervals [Tsample + inf(ζ), Tsample + sup(ζ)]; note that                  the infimum is expected to be negative for most use cases, i.e.                  when the jitter interval is centered at zero
The following getter functions are available:
  • initial_state  – initial state of the continuous mode
  • jitter         – return the jitter
  • reset_map      – return the reset map
  • sampling_time  – return the sampling time
  • statedim       – dimension of the state-space
  • system         – return the continuous mode
  • switching      – return the type of switching
Non-deterministic switching:
tstart       Ts-ζ⁻         tend  [––––––-|––––––-]
In the following, suppose that the continuous post-operator has fixed step-size δ > 0. If F denotes the flowpipe, then
F[1]    F[2]    F[3]    F[4]    F[5]        F[k] [–––][–––][–––][–––][–––]  ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅  [–––]
R = array(F) denote the array of reach-sets time-span for reach reach-set is of the form:
Similarly we compute tstart and tend for the supremum part Ts-ζ⁺
source
ReachabilityAnalysis.DiscreteTransition — Type
DiscreteTransition{RT, GT, IT⁻, IT⁺, WT} <: AbstractTransition
Type that encodes a discrete transition with an affine assignment of the form:
\[    post_d(X) = (R(X ∩ G ∩ I⁻) ⊕ W) ∩ I⁺\]
where $I⁻$ and $I⁺$  are invariants at the source and the target locations respectively, $G ⊆ \mathbb{R}^n$ is the guard set of the transition, the assignment is of the form $x^+ := Rx + w$, $w ∈ W$, $x^+ ∈ \mathbb{R}^m$ are the values after the transition, $R ∈ \mathbb{R}^{m\times n}$ is the assignment map (or reset map) and $W ⊆ \mathbb{R}^m$ is a closed and bounded convex set of non-deterministic inputs.
Fields
  • R  – assignment map
  • W  – non-deterministic inputs
  • G  – guard set of the transition from the source location to the target location
  • I⁻ – invariant at the source location
  • I⁺ – invariant at the target location
source
LazySets.constrained_dimensions — Method
constrained_dimensions(HS::HybridSystem)::Dict{Int,Vector{Int}}
For each location, compute all dimensions that are constrained in the invariant or the guard of any outgoing transition.
Input
  • HS  – hybrid system
Output
A dictionary mapping (::Dict{Int,Vector{Int}}) the index of each location $ℓ$ to the dimension indices that are constrained in $ℓ$.
source

+10
source

Homogeneization

ReachabilityAnalysis.homogenize — Function
homogenize(ivp::IVP{CLCCS{N,MT,IdentityMultiple{N},XT,ConstantInput{SI}},ST}) where {N, MT<:AbstractMatrix{N}, XT<:LazySet{N}, SI<:Singleton{N}, ST<:LazySet{N}}

Transform an inhomogeneous linear initial-value problem into an homogeneous one by introducing auxiliary state variables.

Input

  • ivp – initial-value problem

Output

Homogeneous initial-value problem.

Notes

This function transforms the canonical initial-value problem $x' = Ax + u$, $x ∈ X$ with $u(0) ∈ U = {u}$ (a singleton) into an homogeneous problem without inputs $y' = Â * y$, $y ∈ Y$.

source
homogenize(sys::SOACS)

Transform an inhomogeneous second order system into an homogeneous one by introducing auxiliary state variables.

Input

  • sys – second order system

Output

First-order homogeneous system.

Notes

This function transforms the second-order system $Mx'' + Cx' + Kx = b$ into a first-order, homogeneous one, $y' = Â * y$. It is assumed that the matrix $M$ is invertible.

source

Hybrid systems

ReachabilityAnalysis.HACLD1 — Type
HACLD1{T<:AbstractSystem, MT, N, J} <: AHACLD

Single-mode hybrid automaton with clocked linear dynamics.

Fields

  • sys – system
  • rmap – reset map
  • Tsample – sampling time
  • ζ – jitter
  • switching – (optional) value that

Notes

This type is parametric in:

  • T – system type
  • MT – type of the reset map
  • N – numeric type, applies to the sampling time and jitter
  • J – type associated to the jitter

The type associated to the jitter, J, can be one of the following:

  • Missing – no jitter, i.e. switchings are deterministic
  • Number – symmetric jitter, i.e. non-deterministic switchings occur in the intervals [Tsample - ζ, Tsample + ζ]
  • IA.Interval – non-symmetric jitter, i.e. non-deterministic switchings occur in the intervals [Tsample + inf(ζ), Tsample + sup(ζ)]; note that the infimum is expected to be negative for most use cases, i.e. when the jitter interval is centered at zero

The following getter functions are available:

  • initial_state – initial state of the continuous mode
  • jitter – return the jitter
  • reset_map – return the reset map
  • sampling_time – return the sampling time
  • statedim – dimension of the state-space
  • system – return the continuous mode
  • switching – return the type of switching

Non-deterministic switching:

tstart Ts-ζ⁻ tend [––––––-|––––––-]

In the following, suppose that the continuous post-operator has fixed step-size δ > 0. If F denotes the flowpipe, then

F[1] F[2] F[3] F[4] F[5] F[k] [–––][–––][–––][–––][–––] ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ [–––]

R = array(F) denote the array of reach-sets time-span for reach reach-set is of the form:

Similarly we compute tstart and tend for the supremum part Ts-ζ⁺

source
ReachabilityAnalysis.DiscreteTransition — Type
DiscreteTransition{RT, GT, IT⁻, IT⁺, WT} <: AbstractTransition

Type that encodes a discrete transition with an affine assignment of the form:

\[ post_d(X) = (R(X ∩ G ∩ I⁻) ⊕ W) ∩ I⁺\]

where $I⁻$ and $I⁺$ are invariants at the source and the target locations respectively, $G ⊆ \mathbb{R}^n$ is the guard set of the transition, the assignment is of the form $x^+ := Rx + w$, $w ∈ W$, $x^+ ∈ \mathbb{R}^m$ are the values after the transition, $R ∈ \mathbb{R}^{m\times n}$ is the assignment map (or reset map) and $W ⊆ \mathbb{R}^m$ is a closed and bounded convex set of non-deterministic inputs.

Fields

  • R – assignment map
  • W – non-deterministic inputs
  • G – guard set of the transition from the source location to the target location
  • I⁻ – invariant at the source location
  • I⁺ – invariant at the target location
source
LazySets.constrained_dimensions — Method
constrained_dimensions(HS::HybridSystem)::Dict{Int,Vector{Int}}

For each location, compute all dimensions that are constrained in the invariant or the guard of any outgoing transition.

Input

  • HS – hybrid system

Output

A dictionary mapping (::Dict{Int,Vector{Int}}) the index of each location $ℓ$ to the dimension indices that are constrained in $ℓ$.

source
diff --git a/dev/man/algorithms/A20/index.html b/dev/man/algorithms/A20/index.html index e4488bc08..b232b49b3 100644 --- a/dev/man/algorithms/A20/index.html +++ b/dev/man/algorithms/A20/index.html @@ -1,2 +1,2 @@ -A20 · ReachabilityAnalysis.jl
GitHub
+A20 · ReachabilityAnalysis.jl
GitHub
diff --git a/dev/man/algorithms/ASB07/index.html b/dev/man/algorithms/ASB07/index.html index 717e3b2ca..2957a29c2 100644 --- a/dev/man/algorithms/ASB07/index.html +++ b/dev/man/algorithms/ASB07/index.html @@ -1,2 +1,2 @@ -ASB07 · ReachabilityAnalysis.jl
GitHub
+ASB07 · ReachabilityAnalysis.jl
GitHub
diff --git a/dev/man/algorithms/BFFPSV18/index.html b/dev/man/algorithms/BFFPSV18/index.html index b22ba4b0a..112722159 100644 --- a/dev/man/algorithms/BFFPSV18/index.html +++ b/dev/man/algorithms/BFFPSV18/index.html @@ -1,2 +1,2 @@ -BFFPSV18 · ReachabilityAnalysis.jl
GitHub

Decomposition-based method (BFFPSV18)

Method

We overapproximate the reachable states of an affine system by solving a set-based recurrence. The key idea is that we first decompose the system into (low-dimensional) subsystems and later compose the results as a Cartesian product. Thus we have to solve many cheap problems instead of one hard problem. Since solving the recurrence scales superlinearly with the dimension, this approach is very scalable. To represent sets of states, we use the LazySets package which provides exact but lazy (i.e. symbolic) representations of common sets.

Decomposition error

Decomposition typically involves a loss in precision, and so does this approach. The good thing is that we can decompose the recurrence as well, which allows us to analyze each of the subsystems independently by only referring to the initial states of the other subsystems. Consequently, there are two main sources for precision loss:

  1. Decomposition of the initial states: If two subsystems are interdependent initially.
  2. Representation of the reachable states as a Cartesian product: If two subsystems are interdependent in the dynamics.
  3. Representation of the reachable states in general: The reachable states of affine systems cannot be represented precisely in all cases. This is a problem that all approaches suffer from. We overapproximate the reachable states by (unions of) convex polytopes.

Safety verification

The problem of checking a safety property can be reduced to a reachability problem. We provide special support for this reduction by inlining the property check into the reachable states computation. This has two benefits:

  1. We fail fast when the property is violated in our abstraction.
  2. The check is usually cheaper than computing the full reachable states. This is because we are often only interested in an upper or lower bound of a variable.
+BFFPSV18 · ReachabilityAnalysis.jl
GitHub

Decomposition-based method (BFFPSV18)

Method

We overapproximate the reachable states of an affine system by solving a set-based recurrence. The key idea is that we first decompose the system into (low-dimensional) subsystems and later compose the results as a Cartesian product. Thus we have to solve many cheap problems instead of one hard problem. Since solving the recurrence scales superlinearly with the dimension, this approach is very scalable. To represent sets of states, we use the LazySets package which provides exact but lazy (i.e. symbolic) representations of common sets.

Decomposition error

Decomposition typically involves a loss in precision, and so does this approach. The good thing is that we can decompose the recurrence as well, which allows us to analyze each of the subsystems independently by only referring to the initial states of the other subsystems. Consequently, there are two main sources for precision loss:

  1. Decomposition of the initial states: If two subsystems are interdependent initially.
  2. Representation of the reachable states as a Cartesian product: If two subsystems are interdependent in the dynamics.
  3. Representation of the reachable states in general: The reachable states of affine systems cannot be represented precisely in all cases. This is a problem that all approaches suffer from. We overapproximate the reachable states by (unions of) convex polytopes.

Safety verification

The problem of checking a safety property can be reduced to a reachability problem. We provide special support for this reduction by inlining the property check into the reachable states computation. This has two benefits:

  1. We fail fast when the property is violated in our abstraction.
  2. The check is usually cheaper than computing the full reachable states. This is because we are often only interested in an upper or lower bound of a variable.
diff --git a/dev/man/algorithms/BOX/index.html b/dev/man/algorithms/BOX/index.html index cfdd10156..87a97a83f 100644 --- a/dev/man/algorithms/BOX/index.html +++ b/dev/man/algorithms/BOX/index.html @@ -1,2 +1,2 @@ -BOX · ReachabilityAnalysis.jl
GitHub
+BOX · ReachabilityAnalysis.jl
GitHub
diff --git a/dev/man/algorithms/GLGM06/index.html b/dev/man/algorithms/GLGM06/index.html index 33eed4f32..19cf413a9 100644 --- a/dev/man/algorithms/GLGM06/index.html +++ b/dev/man/algorithms/GLGM06/index.html @@ -1,2 +1,2 @@ -GLGM06 · ReachabilityAnalysis.jl
GitHub
+GLGM06 · ReachabilityAnalysis.jl
GitHub
diff --git a/dev/man/algorithms/INT/index.html b/dev/man/algorithms/INT/index.html index 13869584c..7cd329188 100644 --- a/dev/man/algorithms/INT/index.html +++ b/dev/man/algorithms/INT/index.html @@ -1,2 +1,2 @@ -INT · ReachabilityAnalysis.jl
GitHub
+INT · ReachabilityAnalysis.jl
GitHub
diff --git a/dev/man/algorithms/LGG09/index.html b/dev/man/algorithms/LGG09/index.html index 35fc4c07e..6b8348204 100644 --- a/dev/man/algorithms/LGG09/index.html +++ b/dev/man/algorithms/LGG09/index.html @@ -11,4 +11,4 @@ \quad \rho(d, X_3) &= \rho((\Phi^T)^3 d, X_0) + \rho((\Phi^T)^2 d, V_0) + \rho(\Phi^T d, V_1) + \rho(d, V_2) \\[1mm] \quad &\vdots \\[1mm] \quad \rho(d, X_k) &= \rho((\Phi^T)^k d, X_0) + \sum_{i=0}^{k-1} \rho( (\Phi^T)^{k-i-1} d, V_i). -\end{aligned}\]

In a similar fashion to the homogeneous case, the method allows to efficiently reason about the the sequence $\{X_0, X_1, X_2, \ldots, X_N\}$ by evaluating the support function of the initial set $X_0$ and the input sets $\{V_k\}_k$ along the directions $\{d, \Phi^T d, (\Phi^T)^2 d, \ldots, (\Phi^T)^N d\}$. Implementation-wise, we update two sequences, one that accounts for the homogeneous term, and another sequence that accounts for the effect of the accumulated inputs.

Implementation details

The reach-set representation used is a TemplateReachSet, which stores the directions used (vector of vectors) and the support function evaluated at each direction (matrix, see below). The set representation, set(R::TemplateReachSet), is either a polyhedron in constraint form (HPolyhedron), or a polytope (HPolytope) if the directions are bounding, i.e. the template directions define a bounded set.

The computed support function values can accessed directly through the field sf::SN of each template reach-set. Here sf is an array view of type ::Matrix{N}(undef, length(dirs), NSTEPS): each row corresponds to one of the template directions and each column corresponds to a fixed iteration index $k \geq 0$.

If you use directions from the canonical basis of $\mathbb{R}^n$, it is recommended to define LazySets.Arrays.SingleEntryVector or "one-hot" arrays as they are commonly called, because there are methods that dispatch on such type of arrays efficiently.

Parallel formulation

The support functions of the sequence $\{X_k\}_k$ along different directions can be computed in parallel. Indeed, if $d_1$ and $d_2$ are two given template directions, two different processes can independently compute $\rho(d_1, X_k)$ and $\rho(d_2, X_k)$ for all $k = 0, 1, \ldots, N$ using the methods described above. Once both computations have finished, we can store the resulting support functions in the same array. Use the flag threaded=true to use this method.

Implementation-wise the function _reach_homog_dir_LGG09! spawns different threads which populate the matrix ρℓ::Matrix{N}(undef, length(dirs), NSTEPS) with the computed values. Hence each thread computes a subset of distinct rows of ρℓ.

Real eigenvalues

If the spectrum of the state matrix only has real eigenvalues, the sequence of support functions can be computed efficiently if we work with a template consisting of eigenvectors of $\Phi^T$. This idea is described in [LGG09] and we recall it here for convenience.

The method stems from the fact that if $(\lambda, d)$ is an eigenvalue-eigenvector pair of the matrix $\Phi^T$, with $\lambda \in \mathbb{R}$, then $\Phi^T d = \lambda d$, and if we apply $\Phi^T$ on both sides of this identity, we get $(\Phi^T)^2 d = \Phi^T (\Phi^T d) = \Phi^T(\lambda d) = \lambda^2 d$. In more generality, it holds that $(\Phi^T)^k d = \lambda^k d$ for all $k \ge 1$. Applying this relation to the support function recurrence described above, we get for the general inhomogeneous and possibly time-varying inputs case:

\[\rho(d, X_k) = \rho(\lambda^k d, X_0) + \sum_{i=0}^{k-1} \rho(\lambda^{k-i-1} d, V_i).\]

To further simplify this formula, we analyze different cases of $\lambda$. If $\lambda = 0$, then $\rho(d, X_k) = \rho(d, V_k)$ for all $k \geq 1$, so we focus on either $\lambda$ being positive or negative. To further simplify the computation of $\rho(d, X_k)$, we can use the property $\rho(\lambda d, X) = \lambda \rho(d, X)$ if $\lambda \geq 0$. We now consider the cases $\lambda > 0$ and $\lambda < 0$.

Case $\lambda > 0$. Then $\lambda^k > 0$ for all $k \geq 1$, and

\[\rho(d, X_k) = \lambda^k \rho(d, X_0) + \sum_{i=0}^{k-1} \lambda^{k-i-1} \rho(d, V_i).\]

We are left with evaluating the support function only at $\rho(d, X_0)$ and $\rho(d, V_i)$ to construct the full sequence $\{\rho(d, X_k)\}_{k}$. Moreover, if the $V_i$'s are constant we can extract them from the right-hand side sum and use that

\[\sum_{i=0}^{k-1} \lambda^{k-i-1} = 1 + \lambda + \ldots + \lambda^{k-1} = \dfrac{1 - \lambda^k}{1 - \lambda}.\]

Case $\lambda < 0$. Since $\lambda^k = (-1)^k (-\lambda)^k$ and $\lambda < 0$, then $\lambda^k$ is positive if $k$ is even, otherwise it is negative. So we can write:

\[\rho(d, X_k) = (-\lambda)^k \rho((-1)^k d, X_0) + \sum_{i=0}^{k-1} (-\lambda)^{k-i-1} \rho((-1)^{k-i-1} d, V_i).\]

The main difference between this case and the previous one is that now we have to evaluate support functions $\rho(\pm d, X_0)$ and $\rho(\pm d, V_i)$. Again, simplification takes place if the $V_i$'s are constant and such special case is considered in the implementation.

+\end{aligned}\]

In a similar fashion to the homogeneous case, the method allows to efficiently reason about the the sequence $\{X_0, X_1, X_2, \ldots, X_N\}$ by evaluating the support function of the initial set $X_0$ and the input sets $\{V_k\}_k$ along the directions $\{d, \Phi^T d, (\Phi^T)^2 d, \ldots, (\Phi^T)^N d\}$. Implementation-wise, we update two sequences, one that accounts for the homogeneous term, and another sequence that accounts for the effect of the accumulated inputs.

Implementation details

The reach-set representation used is a TemplateReachSet, which stores the directions used (vector of vectors) and the support function evaluated at each direction (matrix, see below). The set representation, set(R::TemplateReachSet), is either a polyhedron in constraint form (HPolyhedron), or a polytope (HPolytope) if the directions are bounding, i.e. the template directions define a bounded set.

The computed support function values can accessed directly through the field sf::SN of each template reach-set. Here sf is an array view of type ::Matrix{N}(undef, length(dirs), NSTEPS): each row corresponds to one of the template directions and each column corresponds to a fixed iteration index $k \geq 0$.

If you use directions from the canonical basis of $\mathbb{R}^n$, it is recommended to define LazySets.Arrays.SingleEntryVector or "one-hot" arrays as they are commonly called, because there are methods that dispatch on such type of arrays efficiently.

Parallel formulation

The support functions of the sequence $\{X_k\}_k$ along different directions can be computed in parallel. Indeed, if $d_1$ and $d_2$ are two given template directions, two different processes can independently compute $\rho(d_1, X_k)$ and $\rho(d_2, X_k)$ for all $k = 0, 1, \ldots, N$ using the methods described above. Once both computations have finished, we can store the resulting support functions in the same array. Use the flag threaded=true to use this method.

Implementation-wise the function _reach_homog_dir_LGG09! spawns different threads which populate the matrix ρℓ::Matrix{N}(undef, length(dirs), NSTEPS) with the computed values. Hence each thread computes a subset of distinct rows of ρℓ.

Real eigenvalues

If the spectrum of the state matrix only has real eigenvalues, the sequence of support functions can be computed efficiently if we work with a template consisting of eigenvectors of $\Phi^T$. This idea is described in [LGG09] and we recall it here for convenience.

The method stems from the fact that if $(\lambda, d)$ is an eigenvalue-eigenvector pair of the matrix $\Phi^T$, with $\lambda \in \mathbb{R}$, then $\Phi^T d = \lambda d$, and if we apply $\Phi^T$ on both sides of this identity, we get $(\Phi^T)^2 d = \Phi^T (\Phi^T d) = \Phi^T(\lambda d) = \lambda^2 d$. In more generality, it holds that $(\Phi^T)^k d = \lambda^k d$ for all $k \ge 1$. Applying this relation to the support function recurrence described above, we get for the general inhomogeneous and possibly time-varying inputs case:

\[\rho(d, X_k) = \rho(\lambda^k d, X_0) + \sum_{i=0}^{k-1} \rho(\lambda^{k-i-1} d, V_i).\]

To further simplify this formula, we analyze different cases of $\lambda$. If $\lambda = 0$, then $\rho(d, X_k) = \rho(d, V_k)$ for all $k \geq 1$, so we focus on either $\lambda$ being positive or negative. To further simplify the computation of $\rho(d, X_k)$, we can use the property $\rho(\lambda d, X) = \lambda \rho(d, X)$ if $\lambda \geq 0$. We now consider the cases $\lambda > 0$ and $\lambda < 0$.

Case $\lambda > 0$. Then $\lambda^k > 0$ for all $k \geq 1$, and

\[\rho(d, X_k) = \lambda^k \rho(d, X_0) + \sum_{i=0}^{k-1} \lambda^{k-i-1} \rho(d, V_i).\]

We are left with evaluating the support function only at $\rho(d, X_0)$ and $\rho(d, V_i)$ to construct the full sequence $\{\rho(d, X_k)\}_{k}$. Moreover, if the $V_i$'s are constant we can extract them from the right-hand side sum and use that

\[\sum_{i=0}^{k-1} \lambda^{k-i-1} = 1 + \lambda + \ldots + \lambda^{k-1} = \dfrac{1 - \lambda^k}{1 - \lambda}.\]

Case $\lambda < 0$. Since $\lambda^k = (-1)^k (-\lambda)^k$ and $\lambda < 0$, then $\lambda^k$ is positive if $k$ is even, otherwise it is negative. So we can write:

\[\rho(d, X_k) = (-\lambda)^k \rho((-1)^k d, X_0) + \sum_{i=0}^{k-1} (-\lambda)^{k-i-1} \rho((-1)^{k-i-1} d, V_i).\]

The main difference between this case and the previous one is that now we have to evaluate support functions $\rho(\pm d, X_0)$ and $\rho(\pm d, V_i)$. Again, simplification takes place if the $V_i$'s are constant and such special case is considered in the implementation.

diff --git a/dev/man/algorithms/ORBIT/index.html b/dev/man/algorithms/ORBIT/index.html index 8638ae69d..ef69d8c45 100644 --- a/dev/man/algorithms/ORBIT/index.html +++ b/dev/man/algorithms/ORBIT/index.html @@ -1,2 +1,2 @@ -ORBIT · ReachabilityAnalysis.jl
GitHub

Singleton propagation (ORBIT)

Method

Consider the ODE

\[x'(t) = Ax(t) + u, \qquad x_0 \in \mathbb{R}^n,\qquad (1)\]

with $A \in \mathbb{R}^{n\times n}$ and $u \in \mathbb{R}^n$. The analytic solution at $\delta > 0$ is known to be

\[x(\delta) = e^{A\delta}x_0 + \int_0^\delta e^{A(\delta - s)} uds.\]

Let us introduce the matrices $\Phi(A, \delta) := e^{A\delta}$ and $\Phi_1(A, \delta) := \int_0^{\delta} e^{A(\delta - s)} ds$. The computation of these matrices will be discussed later.

Since $u$ is assumed constant,

\[x(\delta) = \Phi x_0 + v,\]

where we have defined $v := \Phi_1(A, \delta) u$. To get the exact solution at $t = 2\delta$, note that by time invariance of Eq. (1),

\[x(2\delta) = e^{A\delta} x(\delta) + \int_0^\delta e^{A(\delta - s)}u ds,\]

hence

\[x(2\delta) = \Phi (\Phi x_0 + v) + v = \Phi^2 x_0 + \Phi v + v.\]

The solution at subsequent multiples of the step-size $\delta$ is achieved by applying the preceding rule iteratlvely. For any $k \geq 1$, we obtain

\[x(k\delta) = \Phi^k x_0 + \sum_{i=0}^{k-1} \Phi^i v,\qquad k \geq 1.\]

Before considering the computation of $\Phi$ and $\Phi_1$, note that the method admits a straightforward generalization for non-constant inputs. Indeed, if $u : [0, T]\to \mathbb{R}^n$ is a piecewise-constant function, there always exists a sufficiently small step-size $\delta$ such that the following premise holds. Assume that $\{u_1, u_2, \ldots, u_N\} \subseteq \mathbb{R}^n$ is the range of values of the input, where $u(t) = u_k$ for $t \in [(k-1)\delta, k\delta)$, $k = 1, \ldots, N$, and let $v_k := \Phi_1(u_k, \delta)$ for all $k = 1,\ldots, N$. Then it holds that $x(\delta) = \Phi x_0 + v_{1}$, $x(2\delta) = \Phi^2 x_0 + \Phi v_1+ v_{2}$, and

\[x(k\delta) = \Phi^k x_0 + \sum_{i=0}^{k-1} \Phi^i v_{k-i},\qquad k = 1,\ldots, N\]

The matrix $\Phi = e^{A\delta}$ can be evaluated in different ways, using the function _exp:

(1) method=:base uses Julia's built-in implementation (if method = :base),

(2) method = :lazy uses a lazy wrapper of the matrix exponential which is then evaluated using Krylov subspace methods.

Method (1) is the default method. Method (2) is particularly useful to work with very large and sparse matrices (e.g. typically of order n > 2000). Evaluation of $\Phi_1(u, \delta)$ is available through the function Φ₁. Two implementations are available:

(1) If the coefficients matrix $A$ is invertible, then the integral is equivalent to computing $A^{-1}(e^{A\delta} - I)$.

(2) In general, $\Phi_1(u, \delta)$ can be obtained as a sub-block of a larger matrix. See [FRE11] for details.

Method (2) is the default method, although there are cases in which method (1) is more convenient. For example, if we are only interested in singleton inputs and $A$ is invertible, it is possible to compute $\Phi_1(A, \delta) u $ efficiently without actually inverting the matrix $A$ in full.

+ORBIT · ReachabilityAnalysis.jl
GitHub

Singleton propagation (ORBIT)

Method

Consider the ODE

\[x'(t) = Ax(t) + u, \qquad x_0 \in \mathbb{R}^n,\qquad (1)\]

with $A \in \mathbb{R}^{n\times n}$ and $u \in \mathbb{R}^n$. The analytic solution at $\delta > 0$ is known to be

\[x(\delta) = e^{A\delta}x_0 + \int_0^\delta e^{A(\delta - s)} uds.\]

Let us introduce the matrices $\Phi(A, \delta) := e^{A\delta}$ and $\Phi_1(A, \delta) := \int_0^{\delta} e^{A(\delta - s)} ds$. The computation of these matrices will be discussed later.

Since $u$ is assumed constant,

\[x(\delta) = \Phi x_0 + v,\]

where we have defined $v := \Phi_1(A, \delta) u$. To get the exact solution at $t = 2\delta$, note that by time invariance of Eq. (1),

\[x(2\delta) = e^{A\delta} x(\delta) + \int_0^\delta e^{A(\delta - s)}u ds,\]

hence

\[x(2\delta) = \Phi (\Phi x_0 + v) + v = \Phi^2 x_0 + \Phi v + v.\]

The solution at subsequent multiples of the step-size $\delta$ is achieved by applying the preceding rule iteratlvely. For any $k \geq 1$, we obtain

\[x(k\delta) = \Phi^k x_0 + \sum_{i=0}^{k-1} \Phi^i v,\qquad k \geq 1.\]

Before considering the computation of $\Phi$ and $\Phi_1$, note that the method admits a straightforward generalization for non-constant inputs. Indeed, if $u : [0, T]\to \mathbb{R}^n$ is a piecewise-constant function, there always exists a sufficiently small step-size $\delta$ such that the following premise holds. Assume that $\{u_1, u_2, \ldots, u_N\} \subseteq \mathbb{R}^n$ is the range of values of the input, where $u(t) = u_k$ for $t \in [(k-1)\delta, k\delta)$, $k = 1, \ldots, N$, and let $v_k := \Phi_1(u_k, \delta)$ for all $k = 1,\ldots, N$. Then it holds that $x(\delta) = \Phi x_0 + v_{1}$, $x(2\delta) = \Phi^2 x_0 + \Phi v_1+ v_{2}$, and

\[x(k\delta) = \Phi^k x_0 + \sum_{i=0}^{k-1} \Phi^i v_{k-i},\qquad k = 1,\ldots, N\]

The matrix $\Phi = e^{A\delta}$ can be evaluated in different ways, using the function _exp:

(1) method=:base uses Julia's built-in implementation (if method = :base),

(2) method = :lazy uses a lazy wrapper of the matrix exponential which is then evaluated using Krylov subspace methods.

Method (1) is the default method. Method (2) is particularly useful to work with very large and sparse matrices (e.g. typically of order n > 2000). Evaluation of $\Phi_1(u, \delta)$ is available through the function Φ₁. Two implementations are available:

(1) If the coefficients matrix $A$ is invertible, then the integral is equivalent to computing $A^{-1}(e^{A\delta} - I)$.

(2) In general, $\Phi_1(u, \delta)$ can be obtained as a sub-block of a larger matrix. See [FRE11] for details.

Method (2) is the default method, although there are cases in which method (1) is more convenient. For example, if we are only interested in singleton inputs and $A$ is invertible, it is possible to compute $\Phi_1(A, \delta) u $ efficiently without actually inverting the matrix $A$ in full.

diff --git a/dev/man/algorithms/VREP/index.html b/dev/man/algorithms/VREP/index.html index 730e424da..e1a1f7cb8 100644 --- a/dev/man/algorithms/VREP/index.html +++ b/dev/man/algorithms/VREP/index.html @@ -4,4 +4,4 @@ # ... define model ... # prob = @ivp(...) alg = VREP(δ=1e-3, static=true, dim=4, backend=CDDLib.Library()) -solve(prob, tspan=(0.0, 1.0), alg=alg)

Other backends are available e.g. QHull.jl.

+solve(prob, tspan=(0.0, 1.0), alg=alg)

Other backends are available e.g. QHull.jl.

diff --git a/dev/man/benchmarks/benchmarks/index.html b/dev/man/benchmarks/benchmarks/index.html index 78592c905..6734ecd86 100644 --- a/dev/man/benchmarks/benchmarks/index.html +++ b/dev/man/benchmarks/benchmarks/index.html @@ -1,2 +1,2 @@ -Benchmark repository · ReachabilityAnalysis.jl
GitHub

Benchmark repository

The benchmark suite of JuliaReach is available at the ReachabilityBenchmarks repository.

The repository includes:

  • Benchmarks for all models in the ReachabilityAnalysis.jl test suite.
  • Benchmarks for models which are not part of the ReachabilityAnalysis.jl test suite, and are used to detect regressions (i.e. examples that accidentally run slower, due to changes in core downstream libraries such as LazySets.jl).
  • The SLICOT models, from the SLICOT Model and Controller Reduction Toolbox, which reflect real world applications with dimensions ranging from several dozens to over 10.000.
+Benchmark repository · ReachabilityAnalysis.jl
GitHub

Benchmark repository

The benchmark suite of JuliaReach is available at the ReachabilityBenchmarks repository.

The repository includes:

  • Benchmarks for all models in the ReachabilityAnalysis.jl test suite.
  • Benchmarks for models which are not part of the ReachabilityAnalysis.jl test suite, and are used to detect regressions (i.e. examples that accidentally run slower, due to changes in core downstream libraries such as LazySets.jl).
  • The SLICOT models, from the SLICOT Model and Controller Reduction Toolbox, which reflect real world applications with dimensions ranging from several dozens to over 10.000.
diff --git a/dev/man/benchmarks/filtered_oscillator/index.html b/dev/man/benchmarks/filtered_oscillator/index.html index b05d14d76..de4d810ff 100644 --- a/dev/man/benchmarks/filtered_oscillator/index.html +++ b/dev/man/benchmarks/filtered_oscillator/index.html @@ -111,4 +111,4 @@ function fosc(X0; n0::Int=4, one_loop_iteration::Bool=false) H = filtered_oscillator_hybrid(n0, one_loop_iteration) return IVP(H, [(1, X0)]) -end
fosc (generic function with 1 method)

Results

+end
fosc (generic function with 1 method)

Results

diff --git a/dev/man/benchmarks/model_library/index.html b/dev/man/benchmarks/model_library/index.html index 274736994..aa4ed25c1 100644 --- a/dev/man/benchmarks/model_library/index.html +++ b/dev/man/benchmarks/model_library/index.html @@ -5,4 +5,4 @@ sol = solve(prob, T=5.0); # solve it using default options -plot(sol, vars=(0, 25)) # plot the solution +plot(sol, vars=(0, 25)) # plot the solution diff --git a/dev/man/benchmarks/repeatability/index.html b/dev/man/benchmarks/repeatability/index.html index f8ada4d85..0d2af5b09 100644 --- a/dev/man/benchmarks/repeatability/index.html +++ b/dev/man/benchmarks/repeatability/index.html @@ -1,2 +1,2 @@ -Repeatability evaluations · ReachabilityAnalysis.jl
GitHub

Repeatability evaluations

Traditionally, re-creation of computational results in research work is a challenging task because details of the implementation are unavoidably absent in the paper. Some authors post their code and data to their websites, but there is little formal incentive to do so and no easy way to determine whether others can actually use the result. As a consequence, computational results often become non reproducible – even by the research group which originally produced them – after just a few years.

More recently, scientific conferences encourage and sometimes require that authors improve the reproducibility of their computational results by providing suitable reproducibility evaluation (RE) packages. Such RE are self-contained codes that can be used to run and reproduce the results from the paper in a host machine. Fortunately, the tools available for the Julia language are very convenenient to prepare and distribute RE packages. The following RE packages are available:

  • ARCH2020 NLN RE – Repeatability Evaluation package for the ARCH2020 Nonlinear Continuous and Hybrid Systems Competition.

  • ARCH2020 AFF RE – Repeatability Evaluation package for the ARCH2020 Linear and Hybrid Systems Competition.

  • HSCC2019 RE – Repeatability Evaluation (RE) package for the paper JuliaReach: a Toolbox for Set-Based Reachability published at the HSCC'2019 conference.

  • ARCH2019 RE – Repeatability Evaluation package for the ARCH2019 Competition.

  • ARCH2018 RE – Repeatability Evaluation package for the ARCH2018 Competition.

For installation instructions, see the README file in each package. We have examples using virtual machines, and also using Docker containers for RE packages. The advantage of using a Docker container is that downloading the necessary requirements (including Julia itself) is an automated process.

+Repeatability evaluations · ReachabilityAnalysis.jl
GitHub

Repeatability evaluations

Traditionally, re-creation of computational results in research work is a challenging task because details of the implementation are unavoidably absent in the paper. Some authors post their code and data to their websites, but there is little formal incentive to do so and no easy way to determine whether others can actually use the result. As a consequence, computational results often become non reproducible – even by the research group which originally produced them – after just a few years.

More recently, scientific conferences encourage and sometimes require that authors improve the reproducibility of their computational results by providing suitable reproducibility evaluation (RE) packages. Such RE are self-contained codes that can be used to run and reproduce the results from the paper in a host machine. Fortunately, the tools available for the Julia language are very convenenient to prepare and distribute RE packages. The following RE packages are available:

  • ARCH2020 NLN RE – Repeatability Evaluation package for the ARCH2020 Nonlinear Continuous and Hybrid Systems Competition.

  • ARCH2020 AFF RE – Repeatability Evaluation package for the ARCH2020 Linear and Hybrid Systems Competition.

  • HSCC2019 RE – Repeatability Evaluation (RE) package for the paper JuliaReach: a Toolbox for Set-Based Reachability published at the HSCC'2019 conference.

  • ARCH2019 RE – Repeatability Evaluation package for the ARCH2019 Competition.

  • ARCH2018 RE – Repeatability Evaluation package for the ARCH2018 Competition.

For installation instructions, see the README file in each package. We have examples using virtual machines, and also using Docker containers for RE packages. The advantage of using a Docker container is that downloading the necessary requirements (including Julia itself) is an automated process.

diff --git a/dev/man/examples_overview/index.html b/dev/man/examples_overview/index.html index 79842e4cc..0e5cc3f31 100644 --- a/dev/man/examples_overview/index.html +++ b/dev/man/examples_overview/index.html @@ -1,2 +1,2 @@ -Overview · ReachabilityAnalysis.jl
GitHub

Overview

We organize the models by the type of nonlinearities (if there are some), and whether they are purely continuous or present discrete transitions, i.e. hybrid systems. We have added a column with the associated scientific domain, and another column with the number of state variables. Roughly speaking, a higher number of state variables usually corresponds to problems which are harder to solve, although strictly speaking, this usually depends on the property to be verified.

Column P.V. refers to the cases where the example is presented with at lest one instance with parameter variation.

Further examples

In addition to those present in this manual, a larger collection of examples can be found in the models library ReachabilityModels.jl. For further instructions see the section Model library.

Linear continuous

NameAreaState dim.
Damped oscillatorPhysics2
BuildingMechanical Engineering48
Transmission line circuitPower Systems Stability4 to 40
International Space StationAerospace Engineering270
Modified Nodal Analysis 1Electronics1002
Modified Nodal Analysis 2Electronics10913
Heat PDEPhysics125 to 125000

Linear hybrid

NameAreaState dim.
Amplifier circuitElectronic Engineering2
Electromechanic breakElectronic Engineering
GearboxMechanical Engineering
PlatoonAutonomous Driving
PowertrainMechanical Engineering

Nonlinear continuous

NameAreaState dim.
Brusselator
Laub-LoomisMolecular Biology7
Lorenz system
Lotka-Volterra
Production-DestructionElectrical engineering
Quadrotor
SEIR Model
Van der Pol

Nonlinear hybrid

NameAreaState dim.
Spacecraft
Lotka-Volterra w/crossingBiology, Nonlinear physics
+Overview · ReachabilityAnalysis.jl
GitHub

Overview

We organize the models by the type of nonlinearities (if there are some), and whether they are purely continuous or present discrete transitions, i.e. hybrid systems. We have added a column with the associated scientific domain, and another column with the number of state variables. Roughly speaking, a higher number of state variables usually corresponds to problems which are harder to solve, although strictly speaking, this usually depends on the property to be verified.

Column P.V. refers to the cases where the example is presented with at lest one instance with parameter variation.

Further examples

In addition to those present in this manual, a larger collection of examples can be found in the models library ReachabilityModels.jl. For further instructions see the section Model library.

Linear continuous

NameAreaState dim.
Damped oscillatorPhysics2
BuildingMechanical Engineering48
Transmission line circuitPower Systems Stability4 to 40
International Space StationAerospace Engineering270
Modified Nodal Analysis 1Electronics1002
Modified Nodal Analysis 2Electronics10913
Heat PDEPhysics125 to 125000

Linear hybrid

NameAreaState dim.
Amplifier circuitElectronic Engineering2
Electromechanic breakElectronic Engineering
GearboxMechanical Engineering
PlatoonAutonomous Driving
PowertrainMechanical Engineering

Nonlinear continuous

NameAreaState dim.
Brusselator
Laub-LoomisMolecular Biology7
Lorenz system
Lotka-Volterra
Production-DestructionElectrical engineering
Quadrotor
SEIR Model
Van der Pol

Nonlinear hybrid

NameAreaState dim.
Spacecraft
Lotka-Volterra w/crossingBiology, Nonlinear physics
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z0RWyb7##qhQN-Z8|8p2_Zf?LJHo$}Ib^@2_^6~QedvPHwBy`Y*4g~z)dc2NzcY{}4 ziW?h~3+SQ??74}Sq#*ubE)v3QQ-iiZO-&24v+TUQ!}iRD>g9EE4 z$w^6c(YtMQz_;+r7`!19&kfBU!j|LHdX1c(2p bEwL9QV$E6UJ4EG50Q{1ZQkJX{Hwpe9#Sk)t literal 0 HcmV?d00001 diff --git a/dev/man/faq/index.html b/dev/man/faq/index.html index fcdc47650..ef66dadc6 100644 --- a/dev/man/faq/index.html +++ b/dev/man/faq/index.html @@ -29,7 +29,7 @@ using Plots fig = plot(sol, vars=(0, 1), linewidth=0.2, xlab="t", ylab="x(t)") -plot!(ensemble(sol), vars=(0, 1), linealpha=1.0)Example block output

Please note that the latency (compilation time) of the first using line is long, typically one minute with Julia v1.5.3.

Can I solve a for a single initial condition?

To solve for a single initial condition, i.e. a "point", use Singleton as the initial set (singleton means a set with one element). For example, here we plot the free vibration solution of a standard single degree of freedom system without physical damping,

\[ x''(t) + 4x(t) = 0, \qquad x(0) = 1,\qquad x'(0) = 0.\]

In this initial-value problem, the initial condition is given as a point that we can model as X0 = Singleton([1.0, 0.0]), where we associate the first coordinate to position, x(t), and the second coordinate to velocity, x'(t).

Note

Usual Julia vectors such as X0 = [1.0, 0.0] are also valid input, and are treated as a singleton. It is also valid to use tuples in second order systems, e.g. prob = @ivp(X' = AX, X(0) ∈ ([1.0], [0.0])).

Below we plot the flowpipe for the same initial condition and different step sizes.

using ReachabilityAnalysis, Plots
+plot!(ensemble(sol), vars=(0, 1), linealpha=1.0)
Example block output

Please note that the latency (compilation time) of the first using line is long, typically one minute with Julia v1.5.3.

Can I solve a for a single initial condition?

To solve for a single initial condition, i.e. a "point", use Singleton as the initial set (singleton means a set with one element). For example, here we plot the free vibration solution of a standard single degree of freedom system without physical damping,

\[ x''(t) + 4x(t) = 0, \qquad x(0) = 1,\qquad x'(0) = 0.\]

In this initial-value problem, the initial condition is given as a point that we can model as X0 = Singleton([1.0, 0.0]), where we associate the first coordinate to position, x(t), and the second coordinate to velocity, x'(t).

Note

Usual Julia vectors such as X0 = [1.0, 0.0] are also valid input, and are treated as a singleton. It is also valid to use tuples in second order systems, e.g. prob = @ivp(X' = AX, X(0) ∈ ([1.0], [0.0])).

Below we plot the flowpipe for the same initial condition and different step sizes.

using ReachabilityAnalysis, Plots
 
 # x' = v
 # v' = -4x
@@ -69,4 +69,4 @@
 sol = solve(prob, T=12.0, alg=GLGM06(δ=0.02));
 fig = plot(sol, vars=(0, 2), c=:red, alpha=.5, lw=0.2, xlab="t", ylab="y")
Example block output

On the other hand, please note that in the example of above, you can compute with a single integration the flowpipe corresponding to the convex hull of the elements in the array B.

prob = @ivp(x' = Ax, x(0) ∈ ConvexHullArray(B))
 sol = solve(prob, T=12.0, alg=GLGM06(δ=0.02, approx_model=Forward()));
-plot!(sol, vars=(0, 2), c=:lightgreen, alpha=.5, lw=0.2, xlab="t", ylab="y")
Example block output

Modeling questions

Can I use ODE solvers with interval initial conditions?

Although it is in principle possible to ODE solvers for

How do I use the @taylorize macro?

The section Some common gotchas of the user manual details do's and dont's for the @taylorize macro to speedup reachability computations using Taylor models.

A note on interval types

When using intervals as set representation, ReachabilityAnalysis.jl relies on rigorous floating-point arithmetic implemented in pure Julia in the library IntervalArithmetic.jl (we often use const IA = IntervalArithmetic as an abbreviation). The main struct defined in the library is IA.Interval (and the corresponding multi-dimensional interval is IA.IntervalBox). Internally, the set LazySets.Interval is wrapper-type of IA.Interval and these two types should not be confused, although our user APIs extensively use duck typing, in the sense that x(0) ∈ 0 .. 1 (IA.Interval type) and x(0) ∈ Interval(0, 1) are valid.

On a technical level, the reason to have LazySets.Interval as a wrapper type of IA.Interval is that Julia doesn't allow multiple inheritance, but it was a design choice that intervals should belong to the LazySets type hierarchy.

References

  • v1Version 1.0 of the language was released in August 2018; see this blog post.
  • BEKS17Bezanson, J., Edelman, A., Karpinski, S., & Shah, V. B. (2017). Julia: A fresh approach to numerical computing. SIAM review, 59(1), 65-98.
+plot!(sol, vars=(0, 2), c=:lightgreen, alpha=.5, lw=0.2, xlab="t", ylab="y")Example block output

Modeling questions

Can I use ODE solvers with interval initial conditions?

Although it is in principle possible to ODE solvers for

How do I use the @taylorize macro?

The section Some common gotchas of the user manual details do's and dont's for the @taylorize macro to speedup reachability computations using Taylor models.

A note on interval types

When using intervals as set representation, ReachabilityAnalysis.jl relies on rigorous floating-point arithmetic implemented in pure Julia in the library IntervalArithmetic.jl (we often use const IA = IntervalArithmetic as an abbreviation). The main struct defined in the library is IA.Interval (and the corresponding multi-dimensional interval is IA.IntervalBox). Internally, the set LazySets.Interval is wrapper-type of IA.Interval and these two types should not be confused, although our user APIs extensively use duck typing, in the sense that x(0) ∈ 0 .. 1 (IA.Interval type) and x(0) ∈ Interval(0, 1) are valid.

On a technical level, the reason to have LazySets.Interval as a wrapper type of IA.Interval is that Julia doesn't allow multiple inheritance, but it was a design choice that intervals should belong to the LazySets type hierarchy.

References

  • v1Version 1.0 of the language was released in August 2018; see this blog post.
  • BEKS17Bezanson, J., Edelman, A., Karpinski, S., & Shah, V. B. (2017). Julia: A fresh approach to numerical computing. SIAM review, 59(1), 65-98.
diff --git a/dev/man/systems/index.html b/dev/man/systems/index.html index 31738def3..31105223a 100644 --- a/dev/man/systems/index.html +++ b/dev/man/systems/index.html @@ -3,4 +3,4 @@ 0 && I \\ -M^{-1}K && -M^{-1}C \end{pmatrix},\qquad B = \begin{pmatrix} 0 \\ M^{-1} -\end{pmatrix}\]

See the SecondOrder documentation in MathematicalSystems.jl for additional details in second order ODEs types.

Note

A similar relation can be obtained using the alternative convention $[v(t),~x(t)]^T$. Use derivatives_first=true in the normalize function to swap between these conventions (it is set to false by default).

Normalization

Homogeneization

Nonlinear systems

Parametric systems

+\end{pmatrix}\]

See the SecondOrder documentation in MathematicalSystems.jl for additional details in second order ODEs types.

Note

A similar relation can be obtained using the alternative convention $[v(t),~x(t)]^T$. Use derivatives_first=true in the normalize function to swap between these conventions (it is set to false by default).

Normalization

Homogeneization

Nonlinear systems

Parametric systems

diff --git a/dev/references/index.html b/dev/references/index.html index 79131e327..a0516c486 100644 --- a/dev/references/index.html +++ b/dev/references/index.html @@ -214,4 +214,4 @@ pages={378--384}, year={2018}, publisher={Elsevier} -}
+}
diff --git a/dev/tutorials/linear_methods/dense_time/483b091b.svg b/dev/tutorials/linear_methods/dense_time/ecc66c19.svg similarity index 95% rename from dev/tutorials/linear_methods/dense_time/483b091b.svg rename to dev/tutorials/linear_methods/dense_time/ecc66c19.svg index 0321555b5..d05961a04 100644 --- a/dev/tutorials/linear_methods/dense_time/483b091b.svg +++ b/dev/tutorials/linear_methods/dense_time/ecc66c19.svg @@ -1,260 +1,260 @@ - + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/dev/tutorials/linear_methods/dense_time/index.html b/dev/tutorials/linear_methods/dense_time/index.html index 2a625ddc5..60bd50cfe 100644 --- a/dev/tutorials/linear_methods/dense_time/index.html +++ b/dev/tutorials/linear_methods/dense_time/index.html @@ -34,7 +34,7 @@ [plot!(fig, xcoords, ycoords, seriestype=:path, lab="", c=:magenta, ratio=1.) for (xcoords, ycoords) in analytic_sol.(samples, Ref(tt))] figExample block output
xlims!(1.0, 1.5)
-ylims!(0.8, 1.2)
Example block output
discretize(ivp::IVP, δ, alg::AbstractApproximationModel) function. Set-based conservative discretization of a continuous-time initial value problem into a discrete-time problem. This function receives three inputs: the initial value problem (`ivp) for a linear ODE in canonical form, (e.g. the system returned by normalize); the step-size (δ), and the algorithm (alg) used to compute the approximation model. Do subtypes(ReachabilityAnalysis.AbstractApproximationModel) to see the available approximation models.

The output of a discretization is a new initial value problem of a discrete system. Different approximation algorithms and their respective options are described in the docstring of each method, e.g. Forward.

Initial-value problems considered in this function are of the form

\[x' = Ax(t) + u(t),\\qquad x(0) ∈ \\mathcal{X}_0,\\qquad (1)\]

and where $u(t) ∈ U(k)$ add where $\\{U(k)\\}_k$ is a sequence of sets of non-deterministic inputs and $\\mathcal{X}_0$ is the set of initial states. Recall that this initial-value problem is called homogeneous whenever U is the empty set. Other problems, e.g. $x' = Ax(t) + Bu(t)$ can be brought to the canonical form with the function normalize.

The initial value problem returned by this function consists of a set discretized (also called bloated) initial states $Ω₀$, together with the coefficient matrix $Φ = e^{Aδ}$ and a transformed sequence of inputs if $U$ is non-empty.

Two main variations of this algorithm are considered: dense time case and discrete time case.

  • In the dense time case, the transformation is such that the trajectories

of the given continuous system are included in the computed flowpipe of the discretized system. More precisely, given a step size $δ$ and the system (1) conservative set-based discretization function computes a set, $Ω₀$, that guarantees to contain all the trajectories of (1) starting at any $x(0) ∈ \\mathcal{X}_0$ and for any input function that satisfies $u(t) ∈ U(1)$, for any $t ∈ [0, δ]$. If $U$ is time-varying, this function also discretizes the inputs for $k ≥ 0$.

  • In the discrete time case, there is no bloating of the initial states and the

input is assumed to remain constant between sampled times. Use the algorithm NoBloating() for this setting. If $U$ is time-varying, this function also discretizes the inputs for $k ≥ 0$.

There are algorithms to obatin such transformations, called approximation models in the technical literature. For references to the original papers, see the docstring of each concrete subtype of AbstractApproximationModel.

+

Algorithms implementing conservative time discretization can be used from the discretize(ivp::IVP, δ, alg::AbstractApproximationModel) function. Set-based conservative discretization of a continuous-time initial value problem into a discrete-time problem. This function receives three inputs: the initial value problem (`ivp) for a linear ODE in canonical form, (e.g. the system returned by normalize); the step-size (δ), and the algorithm (alg) used to compute the approximation model. Do subtypes(ReachabilityAnalysis.AbstractApproximationModel) to see the available approximation models.

The output of a discretization is a new initial value problem of a discrete system. Different approximation algorithms and their respective options are described in the docstring of each method, e.g. Forward.

Initial-value problems considered in this function are of the form

\[x' = Ax(t) + u(t),\\qquad x(0) ∈ \\mathcal{X}_0,\\qquad (1)\]

and where $u(t) ∈ U(k)$ add where $\\{U(k)\\}_k$ is a sequence of sets of non-deterministic inputs and $\\mathcal{X}_0$ is the set of initial states. Recall that this initial-value problem is called homogeneous whenever U is the empty set. Other problems, e.g. $x' = Ax(t) + Bu(t)$ can be brought to the canonical form with the function normalize.

The initial value problem returned by this function consists of a set discretized (also called bloated) initial states $Ω₀$, together with the coefficient matrix $Φ = e^{Aδ}$ and a transformed sequence of inputs if $U$ is non-empty.

Two main variations of this algorithm are considered: dense time case and discrete time case.

  • In the dense time case, the transformation is such that the trajectories

of the given continuous system are included in the computed flowpipe of the discretized system. More precisely, given a step size $δ$ and the system (1) conservative set-based discretization function computes a set, $Ω₀$, that guarantees to contain all the trajectories of (1) starting at any $x(0) ∈ \\mathcal{X}_0$ and for any input function that satisfies $u(t) ∈ U(1)$, for any $t ∈ [0, δ]$. If $U$ is time-varying, this function also discretizes the inputs for $k ≥ 0$.

  • In the discrete time case, there is no bloating of the initial states and the

input is assumed to remain constant between sampled times. Use the algorithm NoBloating() for this setting. If $U$ is time-varying, this function also discretizes the inputs for $k ≥ 0$.

There are algorithms to obatin such transformations, called approximation models in the technical literature. For references to the original papers, see the docstring of each concrete subtype of AbstractApproximationModel.

diff --git a/dev/tutorials/linear_methods/discrete_time/4434c712.svg b/dev/tutorials/linear_methods/discrete_time/5d2983ee.svg similarity index 76% rename from dev/tutorials/linear_methods/discrete_time/4434c712.svg rename to dev/tutorials/linear_methods/discrete_time/5d2983ee.svg index 5e8577e35..de115d57d 100644 --- a/dev/tutorials/linear_methods/discrete_time/4434c712.svg +++ b/dev/tutorials/linear_methods/discrete_time/5d2983ee.svg @@ -1,116 +1,116 @@ - + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/dev/tutorials/linear_methods/discrete_time/index.html b/dev/tutorials/linear_methods/discrete_time/index.html index 833263bf2..6b7e98a8e 100644 --- a/dev/tutorials/linear_methods/discrete_time/index.html +++ b/dev/tutorials/linear_methods/discrete_time/index.html @@ -28,7 +28,7 @@ plot!(X0, c=:white) plot!(M(pi/4) * X0, c=:white) plot!(M(pi/2) * X0, c=:white) -plot!(M(4pi/3) * X0, c=:white)Example block output

Does propagating point clouds solve the problem? Practically speaking, while we can compute the successors of any $x_0 \in X_0$ we still lack a global description of the set according the the given discrete recurrence.. which brings us back to the original question: how to represent the solution of the recurrence for all points in simultaneous?

Propagating zonotopes

The set representation that is most effective to this problem are zonotopes since they are closed under linear maps. Moreover, since the initial states is a hyperrectangle (thus a zonotope), we can propagate the whole set exactly (and efficiently).

We now implement the solution by propagating zonotopes.

# map X0 according to the rotation matrix
+plot!(M(4pi/3) * X0, c=:white)
Example block output

Does propagating point clouds solve the problem? Practically speaking, while we can compute the successors of any $x_0 \in X_0$ we still lack a global description of the set according the the given discrete recurrence.. which brings us back to the original question: how to represent the solution of the recurrence for all points in simultaneous?

Propagating zonotopes

The set representation that is most effective to this problem are zonotopes since they are closed under linear maps. Moreover, since the initial states is a hyperrectangle (thus a zonotope), we can propagate the whole set exactly (and efficiently).

We now implement the solution by propagating zonotopes.

# map X0 according to the rotation matrix
 X0z = convert(Zonotope, X0)
 
 arr = [linear_map(M(θi), X0z) for θi in range(0, 2pi, length=50)]
@@ -79,4 +79,4 @@
 # plot the result
 fig = plot(H, alpha=.3, lab="H", c=:grey)
 plot!(F ∩ H, vars=(1, 2), ratio=1., lab="F ∩ H")
-xlims!(-2.0, 2.0); ylims!(-2, 2.)
Example block output

Using the solve interface for discrete problems

By default, the solve interface propagates sets in dense time (as explained in the next section). However, it is possible to propagate sets in discrete time by specifying that the approximation model should not add any bloating to the initial states, approx_model=NoBloating() to the solver used.

Exercise. Solve the linear set based recurrence in the current section using the solve interface. Hint: use the algorithm alg=GLGM06(δ=0.05, approx_model=NoBloating()) and the state matrix as specified in the following section.

+xlims!(-2.0, 2.0); ylims!(-2, 2.)Example block output

Using the solve interface for discrete problems

By default, the solve interface propagates sets in dense time (as explained in the next section). However, it is possible to propagate sets in discrete time by specifying that the approximation model should not add any bloating to the initial states, approx_model=NoBloating() to the solver used.

Exercise. Solve the linear set based recurrence in the current section using the solve interface. Hint: use the algorithm alg=GLGM06(δ=0.05, approx_model=NoBloating()) and the state matrix as specified in the following section.

diff --git a/dev/tutorials/linear_methods/introduction/1483cb33.svg b/dev/tutorials/linear_methods/introduction/1261b965.svg similarity index 73% rename from dev/tutorials/linear_methods/introduction/1483cb33.svg rename to dev/tutorials/linear_methods/introduction/1261b965.svg index 9c347aaa2..c193cbbae 100644 --- a/dev/tutorials/linear_methods/introduction/1483cb33.svg +++ b/dev/tutorials/linear_methods/introduction/1261b965.svg @@ -1,244 +1,244 @@ - + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/dev/tutorials/linear_methods/introduction/50fd9efb.svg b/dev/tutorials/linear_methods/introduction/474e8f67.svg similarity index 74% rename from dev/tutorials/linear_methods/introduction/50fd9efb.svg rename to dev/tutorials/linear_methods/introduction/474e8f67.svg index b1a3888b4..4c23cfb10 100644 --- a/dev/tutorials/linear_methods/introduction/50fd9efb.svg +++ b/dev/tutorials/linear_methods/introduction/474e8f67.svg @@ -1,246 +1,246 @@ - + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/dev/tutorials/linear_methods/introduction/9f6fe39b.svg b/dev/tutorials/linear_methods/introduction/72af323b.svg similarity index 75% rename from dev/tutorials/linear_methods/introduction/9f6fe39b.svg rename to dev/tutorials/linear_methods/introduction/72af323b.svg index fdaceaaad..563c9ca64 100644 --- a/dev/tutorials/linear_methods/introduction/9f6fe39b.svg +++ b/dev/tutorials/linear_methods/introduction/72af323b.svg @@ -1,256 +1,256 @@ - + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/dev/tutorials/linear_methods/introduction/990d3251.svg b/dev/tutorials/linear_methods/introduction/7be5f6af.svg similarity index 75% rename from dev/tutorials/linear_methods/introduction/990d3251.svg rename to dev/tutorials/linear_methods/introduction/7be5f6af.svg index 4b6b6d3d5..24610aeea 100644 --- a/dev/tutorials/linear_methods/introduction/990d3251.svg +++ b/dev/tutorials/linear_methods/introduction/7be5f6af.svg @@ -1,252 +1,252 @@ - + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/dev/tutorials/linear_methods/introduction/index.html b/dev/tutorials/linear_methods/introduction/index.html index 83ce64985..e8c53b3e9 100644 --- a/dev/tutorials/linear_methods/introduction/index.html +++ b/dev/tutorials/linear_methods/introduction/index.html @@ -8,19 +8,19 @@ sol = solve(p, T=5)

Here we have solved the differential equation

\[x'(t) = -x(t),\qquad x(0) \in X_0 = [1, 2] \subset \mathbb{R},\]

for $t \in [0, 5]$, whose solution is known to be the decaying exponential $x(t) = x_0e^{-t}$. Let's plot the result, which requires loading the Plots.jl package

variable index 0 denotes time).

using Plots
 
 # post-processing or plotting
-plot(sol, vars=(0, 1), xlab="t", ylab="x(t)")
Example block output

For comparison we can plot the trajectories at the endpoints of the initial interval. The example also shows that the black line segments can be hidden by setting the lw=0 argument (for linewidth).

trange = range(0, 5, length=100)
+plot(sol, vars=(0, 1), xlab="t", ylab="x(t)")
Example block output

For comparison we can plot the trajectories at the endpoints of the initial interval. The example also shows that the black line segments can be hidden by setting the lw=0 argument (for linewidth).

trange = range(0, 5, length=100)
 
 plot(sol, vars=(0, 1), xlab="t", ylab="x(t)", lw=0)
 plot!(trange, 1.0 * exp.(-trange), vars=(0, 1), c=:magenta, lab="")
-plot!(trange, 2.0 * exp.(-trange), vars=(0, 1), c=:magenta, lab="")
Example block output

The user-facing interface is designed to be intuitive and interactive, and it is inspired by of other Julia packages such as DifferentialEquations.jl. Moreover, the library internals are written in a modular and composable way, such that advanced users are able to modify and changed easily, or to compose with other algorithms, different steps of the solution process.

In the rest of this notebook we explore different problem specifications, different algorithm choices as well as some processing capabilities. As a preliminary, we introduce the concept of reach-set in the context of the toy model presented above.

Introducing reach-sets

Basically, a reach-set is a structure that holds a set and a time span associated to it. Here we pick the first reach-set and show that it is of an interval type.

# solutions implement the array interface
+plot!(trange, 2.0 * exp.(-trange), vars=(0, 1), c=:magenta, lab="")
Example block output

The user-facing interface is designed to be intuitive and interactive, and it is inspired by of other Julia packages such as DifferentialEquations.jl. Moreover, the library internals are written in a modular and composable way, such that advanced users are able to modify and changed easily, or to compose with other algorithms, different steps of the solution process.

In the rest of this notebook we explore different problem specifications, different algorithm choices as well as some processing capabilities. As a preliminary, we introduce the concept of reach-set in the context of the toy model presented above.

Introducing reach-sets

Basically, a reach-set is a structure that holds a set and a time span associated to it. Here we pick the first reach-set and show that it is of an interval type.

# solutions implement the array interface
 R = sol[1]
ReachSet{Float64, Interval{Float64}}(Interval{Float64}([0.94877, 2]), [0, 0.0500001])

Note that sol[1] works since solution structures implement Julia's array interface – meaning that slicing also works, and e.g. sol[end-3:end] returns the last three reach-sets computed.

The solution obtained by set propagation consists of a flowpipe, which is just an array of reach-sets, and behaves like their set union. Flowpipes are at the right level of abstraction concerning solutions obtained with set propagation methods.

We can plot the first reach-set as well.

trange = range(0, 5, length=100)
 
 plot(sol, vars=(0, 1), xlab="t", ylab="x(t)", lw=0)
 plot!(R, vars=(0, 1), xlab="t", ylab="x(t)", lw=0, alpha=1., c=:orange, lab="R = sol[1]")
 plot!(trange, 1.0 * exp.(-trange), vars=(0, 1), c=:magenta, lab="")
-plot!(trange, 2.0 * exp.(-trange), vars=(0, 1), c=:magenta, lab="")
Example block output

We can also plot the tenth reach-set in red:

plot!(sol[10], vars=(0, 1), xlab="t", ylab="x(t)", lw=0, alpha=1., c=:red, lab="sol[10]")
Example block output

The type of reach-sets specifies its numeric type as well as the set representation used; in this case, an interval.

typeof(R)
ReachSet{Float64, Interval{Float64}}

Observing the horizontal axis of the plots reveals that reach-sets have a non-zero along on the time axis. By construction, it is guaranteed that the flowpipe is an enclosure of the true solutions, i.e. no trajectory escapes each reach-set for a given time-span. The time span associated to a reach-set is obtained with the function tspan.

tspan(R)
[0, 0.0500001]

Actually, the algorithm that has been used to solve the problem has a fixed step size of $δ = 0.05$.

tspan(sol[end]) # time-span of the final reach-set
[4.94999, 5.00001]

We can check by extracting the algorithm information from the solution struct:

# methods' step-size
+plot!(trange, 2.0 * exp.(-trange), vars=(0, 1), c=:magenta, lab="")
Example block output

We can also plot the tenth reach-set in red:

plot!(sol[10], vars=(0, 1), xlab="t", ylab="x(t)", lw=0, alpha=1., c=:red, lab="sol[10]")
Example block output

The type of reach-sets specifies its numeric type as well as the set representation used; in this case, an interval.

typeof(R)
ReachSet{Float64, Interval{Float64}}

Observing the horizontal axis of the plots reveals that reach-sets have a non-zero along on the time axis. By construction, it is guaranteed that the flowpipe is an enclosure of the true solutions, i.e. no trajectory escapes each reach-set for a given time-span. The time span associated to a reach-set is obtained with the function tspan.

tspan(R)
[0, 0.0500001]

Actually, the algorithm that has been used to solve the problem has a fixed step size of $δ = 0.05$.

tspan(sol[end]) # time-span of the final reach-set
[4.94999, 5.00001]

We can check by extracting the algorithm information from the solution struct:

# methods' step-size
 sol.alg.δ
0.05

The functions tstart and tend return the starting and final time of the reach-set.

tstart(R)
-0.0
tend(R)
0.05

The set wrapped by $R$ is obtained with set:

set(R)
Interval{Float64}([0.94877, 2])

We see it is an interval (LazySets.Interval is just a thin wrapper around IntervalArithmetic.Interval; more on this in a note below).

It is interesting to observe that the infimum of sol[1] 0.9487, while we have specified that the initial interval is $X_0 = [1, 2]$. The explanation is that the computed reach-set contains the true solution for all intermediate times between 0 and 0.05 and for all initial states in $X_0$. Since the values of $x(t)$ decrease in the time interval $[0, 0.05]$ the width of sol[1] should be sufficiently big as to enclose those variations. We can make a quick check:

1.0 * exp(-0.05)
0.951229424500714

which shows that R indeed is a correct enclosure of the solution for all points in the initial set. It is easy to check that by decreasing the step-size, the enclosure of the solution at $[0, 0.05]$ converges to the true value (from outside), and similarly for other time intervals.

More about time intervals

The time span associated to a reach-sets is an interval satisfying the rules of interval arithmetic. We use IntervalArithmetic.jl to represent time intervals.

ReachabilityAnalysis.TimeInterval # alias
IntervalArithmetic.Interval{Float64}
# retrieve the time span associated to this reach-set
 tspan(R)
[0, 0.0500001]
# it is an interval (from IntervalArithmetic.jl)
 typeof(tspan(R))
IntervalArithmetic.Interval{Float64}

Time intervals are represented using intervals from IntervalArithmetic.jl. This choice guarantees that all calculations involving time are carried out using rigorous floating-point calculations with interval arithmetic: all quantities are treated as intervals, which are propagated throughout a calculation. The final result is an interval that is guaranteed to contain the correct result, starting from the given initial data.

If desired, it is also work with set using interval elements:

Bint = BallInf(interval.(ones(2)), interval(0.2))
BallInf{IntervalArithmetic.Interval{Float64}, Vector{IntervalArithmetic.Interval{Float64}}}(IntervalArithmetic.Interval{Float64}[[1, 1], [1, 1]], [0.2, 0.200001])

Finally, note that if you create a reach-set by passing a time point, it is automatically converted to an interval:

R = ReachSet(rand(BallInf, dim=5), 1.0)
 
-tspan(R)
[1, 1]
+tspan(R)
[1, 1]
diff --git a/dev/tutorials/set_representations/distances/index.html b/dev/tutorials/set_representations/distances/index.html index 4be9214fc..83cdb2e0d 100644 --- a/dev/tutorials/set_representations/distances/index.html +++ b/dev/tutorials/set_representations/distances/index.html @@ -1,2 +1,2 @@ -Metric notions · ReachabilityAnalysis.jl
GitHub

Hausdorff distance

The notion of Hausdorff distance can be used to measure the distance between sets. It constitutes a practical theoretical tool to quantify the quality of an approximation.

\[ d_H(\mathcal{X}, \mathcal{Y}) = \max \left( \sup_{x \in \mathcal{X}}\inf_{y \in \mathcal{Y}} \Vert x - y \Vert, \sup_{y \in \mathcal{Y}}\inf_{x \in \mathcal{X}} \Vert x - y \Vert \right)\]

+Metric notions · ReachabilityAnalysis.jl
GitHub

Hausdorff distance

The notion of Hausdorff distance can be used to measure the distance between sets. It constitutes a practical theoretical tool to quantify the quality of an approximation.

\[ d_H(\mathcal{X}, \mathcal{Y}) = \max \left( \sup_{x \in \mathcal{X}}\inf_{y \in \mathcal{Y}} \Vert x - y \Vert, \sup_{y \in \mathcal{Y}}\inf_{x \in \mathcal{X}} \Vert x - y \Vert \right)\]

diff --git a/dev/tutorials/set_representations/polyhedral_computations/index.html b/dev/tutorials/set_representations/polyhedral_computations/index.html index 204f4680e..2cec6dc7f 100644 --- a/dev/tutorials/set_representations/polyhedral_computations/index.html +++ b/dev/tutorials/set_representations/polyhedral_computations/index.html @@ -1,2 +1,2 @@ -Polyhedral computations · ReachabilityAnalysis.jl
GitHub

Concrete polyhedra

Polyhedra backends

Limitations

Computing with concrete polyhedra in high dimensions is generally expensive. In particular, converting between vertex and constraint representations (so-called dual representations) should be used only if it is strictly necessary. However, there are some operations that are cheap:

  • Intersecting two (or more) sets in constraint representation, or whose constraints_list can be computed efficiently. Such computation only requires concatenating the constraints and removing redundant inequalities (operation that requires the solution of linear programs).

  • Taking linear maps of sets in vertex representation, $MX$. This operation requires to map each vertex of $X$ under the transformation $M$. Linear transformations can also be done efficiently in constraint representation provided that the matrix $M$ is invertible. LazySets handles other cases ($M$ not invertible, and the sets either in constraint or in vertex representation), but they are generally expensive in high dimensions. However, using specific classes of sets (e.g. zonotopes).

+Polyhedral computations · ReachabilityAnalysis.jl
GitHub

Concrete polyhedra

Polyhedra backends

Limitations

Computing with concrete polyhedra in high dimensions is generally expensive. In particular, converting between vertex and constraint representations (so-called dual representations) should be used only if it is strictly necessary. However, there are some operations that are cheap:

  • Intersecting two (or more) sets in constraint representation, or whose constraints_list can be computed efficiently. Such computation only requires concatenating the constraints and removing redundant inequalities (operation that requires the solution of linear programs).

  • Taking linear maps of sets in vertex representation, $MX$. This operation requires to map each vertex of $X$ under the transformation $M$. Linear transformations can also be done efficiently in constraint representation provided that the matrix $M$ is invertible. LazySets handles other cases ($M$ not invertible, and the sets either in constraint or in vertex representation), but they are generally expensive in high dimensions. However, using specific classes of sets (e.g. zonotopes).

diff --git a/dev/tutorials/set_representations/zonotopes/54b72cc4.svg b/dev/tutorials/set_representations/zonotopes/612d57dc.svg similarity index 79% rename from dev/tutorials/set_representations/zonotopes/54b72cc4.svg rename to dev/tutorials/set_representations/zonotopes/612d57dc.svg index 53b287d14..361cc0501 100644 --- a/dev/tutorials/set_representations/zonotopes/54b72cc4.svg +++ b/dev/tutorials/set_representations/zonotopes/612d57dc.svg @@ -1,60 +1,60 @@ - + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/dev/tutorials/set_representations/zonotopes/index.html b/dev/tutorials/set_representations/zonotopes/index.html index cc68888e6..3c35015a9 100644 --- a/dev/tutorials/set_representations/zonotopes/index.html +++ b/dev/tutorials/set_representations/zonotopes/index.html @@ -8,4 +8,4 @@ 0.0 1.0 -1.0 0.0

The generators can be

Z3 = Zonotope([1.0, 0.0], [0.1 0.0; 0.0 0.1])
Zonotope{Float64, Vector{Float64}, Matrix{Float64}}([1.0, 0.0], [0.1 0.0; 0.0 0.1])

The order of a zonotope is the ratio between the number of dimensions and the number of generators; i.e. $o = \frac{k}{n}$. Use the function order(Z) to get the order of a given zonotope.

Z = Zonotope([1, 1.], [-1 0.3 1.5 0.3; 0 0.1 -0.3 0.3])
 plot(Z)
-quiver!(fill(1., 4), fill(1., 4), quiver=(genmat(Z)[1, :], genmat(Z)[2, :]), color=:black)
Example block output

Other characterizations

There are other useful characterization of zonotopes. A zonotope can be seen as the Minkowski addition of line segments resulting in centrally symmetric convex polytopes as shown in the following figure, which illustrates how each generator spans the zonotope.

Operations with zonotopes

The cost is measured in terms of the number of binary operations, $\mathrm{Op}(\cdot)$.

\[Z_1 = (c, \langle v_1, \dotsb, v_k \rangle), Z_2 = (d, \langle w_1, \dotsb, w_m \rangle) \subset \mathbb{R}^n, M \in \mathbb{R}^{m \times n}\]

\[Z_1 \oplus Z_2 = (c+d, \langle v_1, \dotsb, v_k, w_1, \dotsb, w_m \rangle)\]

\[MZ_1 = (Mc, \langle Mv_1, \dotsb, Mv_k \rangle)\]

\[CH(Z_1, e^{A\delta}Z_1) \subseteq \frac{1}{2}(c + e^{A\delta}c,\langle v_1 + e^{A\delta}v_1, \dotsb, v_k+e^{A\delta}v_k, v_1 - e^{A\delta}v_1, v_k - e^{A\delta}v_k, c - e^{A\delta}c \rangle )\]

OperationSimplification RuleCost
$Z_1 \oplus Z_2$$n$
$MZ_1$$2mn(k+1)$
$CH(Z_1, e^{A\delta}Z_1)$$2n^2(k+1)+2n(k+2)$
+quiver!(fill(1., 4), fill(1., 4), quiver=(genmat(Z)[1, :], genmat(Z)[2, :]), color=:black)Example block output

Other characterizations

There are other useful characterization of zonotopes. A zonotope can be seen as the Minkowski addition of line segments resulting in centrally symmetric convex polytopes as shown in the following figure, which illustrates how each generator spans the zonotope.

Operations with zonotopes

The cost is measured in terms of the number of binary operations, $\mathrm{Op}(\cdot)$.

\[Z_1 = (c, \langle v_1, \dotsb, v_k \rangle), Z_2 = (d, \langle w_1, \dotsb, w_m \rangle) \subset \mathbb{R}^n, M \in \mathbb{R}^{m \times n}\]

\[Z_1 \oplus Z_2 = (c+d, \langle v_1, \dotsb, v_k, w_1, \dotsb, w_m \rangle)\]

\[MZ_1 = (Mc, \langle Mv_1, \dotsb, Mv_k \rangle)\]

\[CH(Z_1, e^{A\delta}Z_1) \subseteq \frac{1}{2}(c + e^{A\delta}c,\langle v_1 + e^{A\delta}v_1, \dotsb, v_k+e^{A\delta}v_k, v_1 - e^{A\delta}v_1, v_k - e^{A\delta}v_k, c - e^{A\delta}c \rangle )\]

OperationSimplification RuleCost
$Z_1 \oplus Z_2$$n$
$MZ_1$$2mn(k+1)$
$CH(Z_1, e^{A\delta}Z_1)$$2n^2(k+1)+2n(k+2)$
diff --git a/dev/tutorials/taylor_methods/domain_splitting/index.html b/dev/tutorials/taylor_methods/domain_splitting/index.html index c8a3be9e2..c6acd6bed 100644 --- a/dev/tutorials/taylor_methods/domain_splitting/index.html +++ b/dev/tutorials/taylor_methods/domain_splitting/index.html @@ -1,2 +1,2 @@ -Domain splitting · ReachabilityAnalysis.jl
GitHub

Domain splitting

A common technique to reduce wrapping effects is to split the set of initial states. If an initial-value problem has been setup with an array of sets, then the flowpipe starting from each initial set scomputed in parallel, using Julia's built-in multithreaded support.

Note

To turn off multithreading, pass the multithreaded=false option flag to solve method. It is true by default.

Note

To change the number of threads being used, change the THREADS flag in . . .

+Domain splitting · ReachabilityAnalysis.jl
GitHub

Domain splitting

A common technique to reduce wrapping effects is to split the set of initial states. If an initial-value problem has been setup with an array of sets, then the flowpipe starting from each initial set scomputed in parallel, using Julia's built-in multithreaded support.

Note

To turn off multithreading, pass the multithreaded=false option flag to solve method. It is true by default.

Note

To change the number of threads being used, change the THREADS flag in . . .

diff --git a/dev/tutorials/taylor_methods/gotchas/index.html b/dev/tutorials/taylor_methods/gotchas/index.html index 7f52d4b26..ae2f0b97c 100644 --- a/dev/tutorials/taylor_methods/gotchas/index.html +++ b/dev/tutorials/taylor_methods/gotchas/index.html @@ -31,4 +31,4 @@ du[3] = two du[4] = u[4] return du -end +end diff --git a/dev/tutorials/taylor_methods/introduction/index.html b/dev/tutorials/taylor_methods/introduction/index.html index cb54a5ade..e28981e77 100644 --- a/dev/tutorials/taylor_methods/introduction/index.html +++ b/dev/tutorials/taylor_methods/introduction/index.html @@ -62,4 +62,4 @@ disjointness: ZonotopeEnclosure ZonotopeEnclosure()

The type of reach-set representation as well as the set representation can be obtained using rsetrep and setrep respectively.

# type of the reach-set representation
 rsetrep(sol)
TaylorModelReachSet{Float64}
# type of the set representation
-setrep(sol)
Vector{TaylorModel1{TaylorN{Float64}, Float64}} (alias for Array{TaylorModels.TaylorModel1{TaylorN{Float64}, Float64}, 1})

The following section discusses more details about those representations.

+setrep(sol)
Vector{TaylorModel1{TaylorN{Float64}, Float64}} (alias for Array{TaylorModels.TaylorModel1{TaylorN{Float64}, Float64}, 1})

The following section discusses more details about those representations.

diff --git a/dev/tutorials/taylor_methods/taylor_model_reachsets/index.html b/dev/tutorials/taylor_methods/taylor_model_reachsets/index.html index a748b38f2..362dd002d 100644 --- a/dev/tutorials/taylor_methods/taylor_model_reachsets/index.html +++ b/dev/tutorials/taylor_methods/taylor_model_reachsets/index.html @@ -1,2 +1,2 @@ -Taylor model reach-sets · ReachabilityAnalysis.jl
GitHub
+Taylor model reach-sets · ReachabilityAnalysis.jl
GitHub

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