{ "cells": [ { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "# - Hybrid automaton -\n", "\n", "using Polyhedra, CDDLib\n", "using MathematicalSystems, HybridSystems\n", "\n", "rX = HalfSpace([-1, 0], 10) ∩ HalfSpace([1, 0], 1.85) ∩ HalfSpace([0, -1], 10) ∩ HalfSpace([0, 1], 2)\n", "pX = polyhedron(rX, CDDLib.Library())\n", "rU = HalfSpace([-1], 2) ∩ HalfSpace([1], 2)\n", "pU = polyhedron(rU, CDDLib.Library())\n", "\n", "rT = HalfSpace([-1, 0], 0.5) ∩ HalfSpace([1, 0], 0.5) ∩ HalfSpace([0, -1], 0.5) ∩ HalfSpace([0, 1], 0.5)\n", "pT = polyhedron(rT, CDDLib.Library())\n", "rO1 = HalfSpace([-1, 0], 10) ∩ HalfSpace([1, 0], -5) ∩ HalfSpace([0, -1], 10) ∩ HalfSpace([0, 1], -5)\n", "pO1 = polyhedron(rO1, CDDLib.Library())\n", "rO2 = HalfSpace([-1, 0], 5) ∩ HalfSpace([1, 0], 1.85) ∩ HalfSpace([0, -1], 4) ∩ HalfSpace([0, 1], -3)\n", "pO2 = polyhedron(rO2, CDDLib.Library());\n", "\n", "rA = HalfSpace([-1, 0], 6) ∩ HalfSpace([1, 0], -5) ∩ HalfSpace([0, -1], -1) ∩ HalfSpace([0, 1], 2)\n", "pA = polyhedron(rA, CDDLib.Library())\n", "rB = HalfSpace([-1, 0], 5) ∩ HalfSpace([1, 0], -4) ∩ HalfSpace([0, -1], 3) ∩ HalfSpace([0, 1], -2)\n", "pB = polyhedron(rB, CDDLib.Library());\n", "\n", "#Convex subsets:\n", "rC1 = HalfSpace([-1, 0], 6) ∩ HalfSpace([1, 0], -5) ∩ HalfSpace([0, -1], 5) ∩ HalfSpace([0, 1], 1)\n", "pC1 = polyhedron(rC1, CDDLib.Library())\n", "rC2 = HalfSpace([-1, 0], 10) ∩ HalfSpace([1, 0], -6) ∩ HalfSpace([0, -1], 5) ∩ HalfSpace([0, 1], 2)\n", "pC2 = polyhedron(rC2, CDDLib.Library())\n", "rC3 = HalfSpace([-1, 0], 5) ∩ HalfSpace([1, 0], 1.85) ∩ HalfSpace([0, -1], 10) ∩ HalfSpace([0, 1], -4)\n", "pC3 = polyhedron(rC3, CDDLib.Library())\n", "rC4 = HalfSpace([-1, 0], 4) ∩ HalfSpace([1, 0], -0.5) ∩ HalfSpace([0, -1], 3) ∩ HalfSpace([0, 1], 2)\n", "pC4 = polyhedron(rC4, CDDLib.Library())\n", "rC5 = HalfSpace([-1, 0], 5) ∩ HalfSpace([1, 0], -4) ∩ HalfSpace([0, -1], 2) ∩ HalfSpace([0, 1], 2)\n", "pC5 = polyhedron(rC5, CDDLib.Library())\n", "rC6 = HalfSpace([-1, 0], 0.5) ∩ HalfSpace([1, 0], 0.5) ∩ HalfSpace([0, -1], 3) ∩ HalfSpace([0, 1], -0.5)\n", "pC6 = polyhedron(rC6, CDDLib.Library())\n", "rC7 = HalfSpace([-1, 0], 0.5) ∩ HalfSpace([1, 0], 0.5) ∩ HalfSpace([0, -1], -0.5) ∩ HalfSpace([0, 1], 2)\n", "pC7 = polyhedron(rC7, CDDLib.Library())\n", "rC8 = HalfSpace([-1, 0], -0.5) ∩ HalfSpace([1, 0], 1.85) ∩ HalfSpace([0, -1], 3) ∩ HalfSpace([0, 1], 2)\n", "pC8 = polyhedron(rC8, CDDLib.Library())\n", "\n", "domains = [pC1 pC2 pC3 pC4 pC5 pC6 pC7 pC8 pT pA pB pC1 pC2 pC3 pC4 pC5 pC6 pC7 pC8 pT] #terminal first\n", "\n", "Modes = [ConstrainedContinuousIdentitySystem(2, i) for i in domains] #vector of modes\n", "\n", "#Automaton:\n", "Automaton = LightAutomaton(length(domains))\n", "\n", "#Resetmaps: (guards + reset)\n", "A = [1.0 1.0; 0.0 1.0]\n", "B = reshape([0.5, 1.0], 2, 1)\n", "\n", "ResetMaps = Vector{ConstrainedLinearControlDiscreteSystem}(undef, length(domains)^2)\n", "cU = polyhedron(HalfSpace([0, 0, -1], 2) ∩ HalfSpace([0, 0, 1], 2), CDDLib.Library())\n", "k = 1\n", "for i in 1:11\n", " for j in 1:11\n", " if j != 10 && j != 11\n", " guard = project(([A B]\\domains[i]) ∩ cU, 1:2) ∩ domains[j] #guard from j to i s.t. in i in 1 step (∩ cU)\n", " if isempty(guard) == false\n", " ResetMaps[k] = ConstrainedLinearControlDiscreteSystem(A, B, guard, pU) #vector of reset maps\n", " add_transition!(Automaton, j, i, k)\n", " k += 1\n", " end\n", " end\n", " end\n", "end\n", "for i in 12:length(domains)\n", " for j in 10:length(domains)\n", " if i != 10 && i != 11\n", " guard = project(([A B]\\domains[i]) ∩ cU, 1:2) ∩ domains[j] #guard from j to i s.t. in i in 1 step (∩ cU)\n", " if isempty(guard) == false\n", " ResetMaps[k] = ConstrainedLinearControlDiscreteSystem(A, B, guard, pU) #vector of reset maps\n", " add_transition!(Automaton, j, i, k)\n", " k += 1\n", " end\n", " end\n", " end\n", "end\n", " \n", "#Switchings: (controlled switchings)\n", "Switchings = Vector{ControlledSwitching}(undef, k) #vector of switchings\n", "\n", "#Hybrid system:\n", "HySys = HybridSystem(Automaton, vec(Modes), ResetMaps, Switchings);\n", "\n", "\n", "#Manually removing transitions that would violate obstacle region constraint:\n", "toRem = []\n", "for t in out_transitions(HySys, 3)\n", " if target(HySys, t) != 1 && target(HySys, t) != 2 && target(HySys, t) != 3\n", " push!(toRem, t)\n", " end\n", "end\n", "for t in in_transitions(HySys, 3)\n", " if source(HySys, t) != 1 && source(HySys, t) != 2 && source(HySys, t) != 3\n", " push!(toRem, t)\n", " end\n", "end\n", "for t in out_transitions(HySys, 14)\n", " if target(HySys, t) != 12 && target(HySys, t) != 13 && target(HySys, t) != 14\n", " push!(toRem, t)\n", " end\n", "end\n", "for t in in_transitions(HySys, 14)\n", " if source(HySys, t) != 12 && source(HySys, t) != 13 && source(HySys, t) != 14\n", " push!(toRem, t)\n", " end\n", "end\n", "for t in toRem\n", " rem_transition!(HySys.automaton, t)\n", "end" ] 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"Compose.Context(Measures.BoundingBox{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}},Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}((0.0w, 0.0h), (1.0w, 1.0h)), Compose.UnitBox{Float64,Float64,Float64,Float64}(-1.2, -1.2, 2.4, 2.4, 0.0mm, 0.0mm, 0.0mm, 0.0mm), nothing, nothing, nothing, List([Compose.Context(Measures.BoundingBox{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}},Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}((0.0w, 0.0h), (1.0w, 1.0h)), nothing, nothing, nothing, nothing, List([]), List([Compose.Form{Compose.LinePrimitive}(Compose.LinePrimitive[LinePrimitive{Tuple{Measure,Measure}}(Tuple{Measures.Measure,Measures.Measure}[(-0.408868cx, -0.303094cy), (-0.520671cx, -0.303094cy)]), LinePrimitive{Tuple{Measure,Measure}}(Tuple{Measures.Measure,Measures.Measure}[(-0.48532cx, -0.251106cy), (-0.482548cx, -0.258116cy)]), LinePrimitive{Tuple{Measure,Measure}}(Tuple{Measures.Measure,Measures.Measure}[(-0.427933cx, -0.345142cy), (-0.260301cx, -0.536496cy)]), LinePrimitive{Tuple{Measure,Measure}}(Tuple{Measures.Measure,Measures.Measure}[(-0.409328cx, -0.29593cy), (-0.183015cx, -0.266686cy)]), LinePrimitive{Tuple{Measure,Measure}}(Tuple{Measures.Measure,Measures.Measure}[(-0.42332cx, -0.265584cy), (-0.0814277cx, 0.0438116cy)]), LinePrimitive{Tuple{Measure,Measure}}(Tuple{Measures.Measure,Measures.Measure}[(-0.482548cx, -0.258116cy), (-0.48532cx, -0.251106cy)]), LinePrimitive{Tuple{Measure,Measure}}(Tuple{Measures.Measure,Measures.Measure}[(-0.447197cx, -0.206129cy), (-0.559cx, -0.206129cy)]), LinePrimitive{Tuple{Measure,Measure}}(Tuple{Measures.Measure,Measures.Measure}[(-0.469533cx, -0.250832cy), (-0.257031cx, -0.533842cy)]), LinePrimitive{Tuple{Measure,Measure}}(Tuple{Measures.Measure,Measures.Measure}[(-0.447754cx, -0.213998cy), (-0.18292cx, -0.251653cy)]), LinePrimitive{Tuple{Measure,Measure}}(Tuple{Measures.Measure,Measures.Measure}[(-0.455602cx, -0.176649cy), (-0.087475cx, 0.0518409cy)]) … LinePrimitive{Tuple{Measure,Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.146338cx, 0.877168cy), (0.268114cx, 0.613803cy)]), LinePrimitive{Tuple{Measure,Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.176232cx, 0.944589cy), (0.300123cx, 0.98332cy)]), LinePrimitive{Tuple{Measure,Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.178752cx, 0.92619cy), (0.510396cx, 0.915986cy)]), LinePrimitive{Tuple{Measure,Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.178778cx, 0.927909cy), (0.066975cx, 0.927909cy)]), LinePrimitive{Tuple{Measure,Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.175729cx, 0.909697cy), (0.291364cx, 0.869853cy)]), LinePrimitive{Tuple{Measure,Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.334184cx, 0.796648cy), (0.301607cx, 0.618057cy)]), LinePrimitive{Tuple{Measure,Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.347699cx, 0.907435cy), (0.349995cx, 0.944207cy)]), LinePrimitive{Tuple{Measure,Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.398019cx, 0.866816cy), (0.512469cx, 0.899094cy)]), LinePrimitive{Tuple{Measure,Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.291364cx, 0.869853cy), (0.175729cx, 0.909697cy)]), LinePrimitive{Tuple{Measure,Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.400118cx, 0.851642cy), (0.288314cx, 0.851642cy)])], Symbol(\"\"))]), List([Compose.Property{Compose.LineWidthPrimitive}(Compose.LineWidthPrimitive[LineWidthPrimitive(0.67082mm)]), Compose.Property{Compose.FillPrimitive}(Compose.FillPrimitive[FillPrimitive(RGBA{Float64}(0.0,0.0,0.0,0.0))]), Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[StrokePrimitive(RGBA{Float64}(0.827451,0.827451,0.827451,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\")), Compose.Context(Measures.BoundingBox{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}},Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}((0.0w, 0.0h), (1.0w, 1.0h)), nothing, nothing, nothing, nothing, List([]), List([Compose.Form{Compose.LinePrimitive}(Compose.LinePrimitive[LinePrimitive{Tuple{Measure,Measure}}(Tuple{Measures.Measure,Measures.Measure}[(-0.61464cx, -0.337296cy), (-0.520671cx, -0.303094cy), (-0.61464cx, -0.268892cy)]), LinePrimitive{Tuple{Measure,Measure}}(Tuple{Measures.Measure,Measures.Measure}[(-0.416197cx, -0.332933cy), (-0.482548cx, -0.258116cy), (-0.479811cx, -0.358079cy)]), LinePrimitive{Tuple{Measure,Measure}}(Tuple{Measures.Measure,Measures.Measure}[(-0.347948cx, -0.48835cy), (-0.260301cx, -0.536496cy), (-0.296495cx, -0.443276cy)]), LinePrimitive{Tuple{Measure,Measure}}(Tuple{Measures.Measure,Measures.Measure}[(-0.271827cx, -0.312649cy), (-0.183015cx, -0.266686cy), (-0.280593cx, -0.244809cy)]), LinePrimitive{Tuple{Measure,Measure}}(Tuple{Measures.Measure,Measures.Measure}[(-0.128154cx, -0.0446005cy), (-0.0814277cx, 0.0438116cy), (-0.174052cx, 0.00611879cy)]), LinePrimitive{Tuple{Measure,Measure}}(Tuple{Measures.Measure,Measures.Measure}[(-0.551671cx, -0.17629cy), (-0.48532cx, -0.251106cy), (-0.488057cx, -0.151144cy)]), LinePrimitive{Tuple{Measure,Measure}}(Tuple{Measures.Measure,Measures.Measure}[(-0.65297cx, -0.240331cy), (-0.559cx, -0.206129cy), (-0.65297cx, -0.171927cy)]), LinePrimitive{Tuple{Measure,Measure}}(Tuple{Measures.Measure,Measures.Measure}[(-0.340804cx, -0.479234cy), (-0.257031cx, -0.533842cy), (-0.286104cx, -0.438161cy)]), LinePrimitive{Tuple{Measure,Measure}}(Tuple{Measures.Measure,Measures.Measure}[(-0.280768cx, -0.272287cy), (-0.18292cx, -0.251653cy), (-0.271139cx, -0.204564cy)]), LinePrimitive{Tuple{Measure,Measure}}(Tuple{Measures.Measure,Measures.Measure}[(-0.149279cx, -0.026774cy), (-0.087475cx, 0.0518409cy), (-0.185352cx, 0.031345cy)]) … LinePrimitive{Tuple{Measure,Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.197632cx, 0.684741cy), (0.268114cx, 0.613803cy), (0.25972cx, 0.71345cy)]), LinePrimitive{Tuple{Measure,Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.220639cx, 0.922637cy), (0.300123cx, 0.98332cy), (0.200229cx, 0.987925cy)]), LinePrimitive{Tuple{Measure,Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.41542cx, 0.88469cy), (0.510396cx, 0.915986cy), (0.417523cx, 0.953062cy)]), LinePrimitive{Tuple{Measure,Measure}}(Tuple{Measures.Measure,Measures.Measure}[(-0.0269942cx, 0.893707cy), (0.066975cx, 0.927909cy), (-0.0269942cx, 0.962111cy)]), LinePrimitive{Tuple{Measure,Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.191379cx, 0.868129cy), (0.291364cx, 0.869853cy), (0.213663cx, 0.932802cy)]), LinePrimitive{Tuple{Measure,Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.284823cx, 0.716638cy), (0.301607cx, 0.618057cy), (0.352117cx, 0.704363cy)]), LinePrimitive{Tuple{Measure,Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.378275cx, 0.848289cy), (0.349995cx, 0.944207cy), (0.310004cx, 0.852551cy)]), LinePrimitive{Tuple{Measure,Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.431311cx, 0.840669cy), (0.512469cx, 0.899094cy), (0.412744cx, 0.906505cy)]), LinePrimitive{Tuple{Measure,Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.275714cx, 0.911421cy), (0.175729cx, 0.909697cy), (0.25343cx, 0.846749cy)]), LinePrimitive{Tuple{Measure,Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.194345cx, 0.81744cy), (0.288314cx, 0.851642cy), (0.194345cx, 0.885844cy)])], Symbol(\"\"))]), List([Compose.Property{Compose.LineWidthPrimitive}(Compose.LineWidthPrimitive[LineWidthPrimitive(0.67082mm)]), Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[StrokePrimitive(RGBA{Float64}(0.827451,0.827451,0.827451,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\")), Compose.Context(Measures.BoundingBox{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}},Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}((0.0w, 0.0h), (1.0w, 1.0h)), nothing, nothing, nothing, nothing, List([]), List([]), List([Compose.Property{Compose.FontSizePrimitive}(Compose.FontSizePrimitive[FontSizePrimitive(4.0mm)]), Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[StrokePrimitive(RGBA{Float64}(0.0,0.0,0.0,0.0))]), Compose.Property{Compose.FillPrimitive}(Compose.FillPrimitive[FillPrimitive(RGBA{Float64}(0.0,0.0,0.0,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\")), Compose.Context(Measures.BoundingBox{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}},Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}((0.0w, 0.0h), (1.0w, 1.0h)), nothing, nothing, nothing, nothing, List([]), List([Compose.Form{Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}}(Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}[CirclePrimitive{Tuple{Measure,Measure},Measure}((-0.464769cx, -0.303094cy), 0.0223607w), CirclePrimitive{Tuple{Measure,Measure},Measure}((-0.503099cx, -0.206129cy), 0.0223607w), CirclePrimitive{Tuple{Measure,Measure},Measure}((-1.0cx, -0.274716cy), 0.0223607w), CirclePrimitive{Tuple{Measure,Measure},Measure}((-0.223465cx, -0.578545cy), 0.0223607w), CirclePrimitive{Tuple{Measure,Measure},Measure}((-0.127574cx, -0.259522cy), 0.0223607w), CirclePrimitive{Tuple{Measure,Measure},Measure}((-0.44169cx, -0.866763cy), 0.0223607w), CirclePrimitive{Tuple{Measure,Measure},Measure}((-0.0849974cx, -0.888755cy), 0.0223607w), CirclePrimitive{Tuple{Measure,Measure},Measure}((-0.174614cx, -1.0cy), 0.0223607w), CirclePrimitive{Tuple{Measure,Measure},Measure}((-0.419298cx, -0.98729cy), 0.0223607w), CirclePrimitive{Tuple{Measure,Measure},Measure}((-0.0399785cx, 0.0813211cy), 0.0223607w), CirclePrimitive{Tuple{Measure,Measure},Measure}((0.210298cx, -0.102797cy), 0.0223607w), CirclePrimitive{Tuple{Measure,Measure},Measure}((0.485808cx, 0.175273cy), 0.0223607w), CirclePrimitive{Tuple{Measure,Measure},Measure}((0.437547cx, 0.26105cy), 0.0223607w), CirclePrimitive{Tuple{Measure,Measure},Measure}((1.0cx, 0.243566cy), 0.0223607w), CirclePrimitive{Tuple{Measure,Measure},Measure}((0.291575cx, 0.563063cy), 0.0223607w), CirclePrimitive{Tuple{Measure,Measure},Measure}((0.174981cx, 0.248876cy), 0.0223607w), CirclePrimitive{Tuple{Measure,Measure},Measure}((0.353478cx, 1.0cy), 0.0223607w), CirclePrimitive{Tuple{Measure,Measure},Measure}((0.566272cx, 0.914267cy), 0.0223607w), CirclePrimitive{Tuple{Measure,Measure},Measure}((0.122877cx, 0.927909cy), 0.0223607w), CirclePrimitive{Tuple{Measure,Measure},Measure}((0.344216cx, 0.851642cy), 0.0223607w)], Symbol(\"\"))]), List([Compose.Property{Compose.LineWidthPrimitive}(Compose.LineWidthPrimitive[LineWidthPrimitive(0.0mm)]), Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[StrokePrimitive(RGBA{Float64}(0.0,0.0,0.0,0.0))]), Compose.Property{Compose.FillPrimitive}(Compose.FillPrimitive[FillPrimitive(RGBA{Float64}(0.25098,0.878431,0.815686,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\")), Compose.Context(Measures.BoundingBox{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}},Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}((0.0w, 0.0h), (1.0w, 1.0h)), nothing, nothing, nothing, nothing, List([]), List([Compose.Form{Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}}(Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}[TextPrimitive{Tuple{Length{:cx,Float64},Length{:cy,Float64}},Rotation{Tuple{Length{:w,Float64},Length{:h,Float64}}},Tuple{Length{:mm,Float64},Length{:mm,Float64}}}((-0.464769cx, -0.303094cy), \"1\", HCenter(), VCenter(), Rotation{Tuple{Length{:w,Float64},Length{:h,Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), TextPrimitive{Tuple{Length{:cx,Float64},Length{:cy,Float64}},Rotation{Tuple{Length{:w,Float64},Length{:h,Float64}}},Tuple{Length{:mm,Float64},Length{:mm,Float64}}}((-0.503099cx, -0.206129cy), \"2\", 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"\n", "Algorithm general parameters: 15 iterations\n", "Forward pass parameters: desired max 20 steps, min horizon = 3\n", "Initial state: (q_0,x_0) = (8,[1.25, -2.5])\n", "Terminal state: q_T = 20\n", "Cost function: 1 ⋅ ∑ u⋅u + 1\n", "\n", "============== Forward pass #1: Model Predictive Control ============== \n", "(q_1,x_1,u_,1)=(4,[-1.25, -2.5], -0.0)\n", "(q_2,x_2,u_,2)=(4,[-3.75, -2.5], -0.0)\n", "(q_3,x_3,u_,3)=(2,[-6.07143, -2.14286], 0.3571)\n", "(q_4,x_4,u_,4)=(2,[-7.28571, -0.285714], 1.8571)\n", "(q_5,x_5,u_,5)=(2,[-7.30357, 0.25], 0.5357)\n", "(q_6,x_6,u_,6)=(2,[-6.92857, 0.5], 0.25)\n", "(q_7,x_7,u_,7)=(2,[-6.30357, 0.75], 0.25)\n", "(q_8,x_8,u_,8)=(10,[-5.42857, 1.0], 0.25)\n", "(q_9,x_9,u_,9)=(16,[-4.42857, 1.0], 0.0)\n", "(q_10,x_10,u_,10)=(15,[-3.19286, 1.47143], 0.4714)\n", "(q_11,x_11,u_,11)=(15,[-1.60357, 1.70714], 0.2357)\n", "(q_12,x_12,u_,12)=(20,[-0.5, 0.5], -1.2071)\n", "\n", "Cost continuous control = 5.786020434390103\n", "Cost discrete control = 12.0\n", "Total cost = 17.786020434390103\n", "============== Backward pass #1: Dual Dynamic Programming ==============\n", "(q_12,x_12)=(15,[-1.60357, 1.70714]) - New cut for q=15: [0.0, 0.0]⋅x + 0.0\n", "(q_11,x_11)=(15,[-3.19286, 1.47143]) - New cut for q=15: [0.0, 0.0]⋅x + 0.0\n", "(q_10,x_10)=(16,[-4.42857, 1.0]) - New cut for q=16: [0.285714, 0.285714]⋅x + 0.285714286\n", "(q_9,x_9)=(10,[-5.42857, 1.0]) - New cut for q=10: [0.222222, 0.222222]⋅x + 0.222222224\n", "(q_8,x_8)=(2,[-6.30357, 0.75]) - New cut for q=2: [-0.0, -0.0]⋅x + -0.0\n", "(q_7,x_7)=(2,[-6.92857, 0.5]) - New cut for q=2: [-0.25, -0.25]⋅x + -0.249999997\n", "(q_6,x_6)=(2,[-7.30357, 0.25]) - New cut for q=2: [-0.374023, -0.374023]⋅x + -0.374023444\n", "(q_5,x_5)=(2,[-7.28571, -0.285714]) - New cut for q=2: [-0.434826, -0.434826]⋅x + -0.434825884\n", "(q_4,x_4)=(2,[-6.07143, -2.14286]) - New cut for q=2: [-1.2254, -1.66023]⋅x + -1.660229826\n", "(q_3,x_3)=(4,[-3.75, -2.5]) - New cut for q=4: [-0.464459, -0.464459]⋅x + -0.464458667\n", "(q_2,x_2)=(4,[-1.25, -2.5]) - New cut for q=4: [-0.0, -0.0]⋅x + -0.0\n", "(q_1,x_1)=(8,[1.25, -2.5]) - New cut for q=8: [-0.285714, -0.285714]⋅x + -0.285714286\n", "\n", "============== Forward pass #2: Model Predictive Control ============== \n", "(q_1,x_1,u_,1)=(4,[-1.21429, -2.42857], 0.0714)\n", "(q_2,x_2,u_,2)=(4,[-3.28571, -1.71429], 0.7143)\n", "(q_3,x_3,u_,3)=(5,[-4.65306, -1.02041], 0.6939)\n", "(q_4,x_4,u_,4)=(1,[-5.41582, -0.505102], 0.5153)\n", "(q_5,x_5,u_,5)=(1,[-5.7517, -0.166667], 0.3384)\n", "(q_6,x_6,u_,6)=(1,[-5.66752, 0.335034], 0.5017)\n", "(q_7,x_7,u_,7)=(10,[-5.0, 1.0], 0.665)\n", "(q_8,x_8,u_,8)=(15,[-3.625, 1.75], 0.75)\n", "(q_9,x_9,u_,9)=(15,[-1.75, 2.0], 0.25)\n", "(q_10,x_10,u_,10)=(20,[-0.5, 0.5], -1.5)\n", "\n", "Cost continuous control = 4.945734412341814\n", "Cost discrete control = 10.0\n", "Total cost = 14.945734412341814\n", "============== Backward pass #2: Dual Dynamic Programming ==============\n", "(q_10,x_10)=(15,[-1.75, 2.0]) - New cut for q=15: [0.0, 9.7357e-5]⋅x + 9.7357e-5\n", "(q_9,x_9)=(15,[-3.625, 1.75]) - New cut for q=15: [0.0, 1.0e-9]⋅x + 1.0e-9\n", "(q_8,x_8)=(10,[-5.0, 1.0]) - New cut for q=10: [0.222222, 0.222222]⋅x + 0.222222224\n", "(q_7,x_7)=(1,[-5.66752, 0.335034]) - New cut for q=1: [0.0, 0.0]⋅x + 0.0\n", "(q_6,x_6)=(1,[-5.7517, -0.166667]) - New cut for q=1: [-0.330978, -0.330978]⋅x + -0.33097849\n", "(q_5,x_5)=(1,[-5.41582, -0.505102]) - New cut for q=1: [-0.330978, -0.330978]⋅x + -0.33097849\n", "(q_4,x_4)=(5,[-4.65306, -1.02041]) - New cut for q=5: [-0.330978, -0.330978]⋅x + -0.33097849\n", "(q_3,x_3)=(4,[-3.28571, -1.71429]) - New cut for q=4: [-0.248973, -0.186675]⋅x + -0.186675276\n", "(q_2,x_2)=(4,[-1.21429, -2.42857]) - New cut for q=4: [-0.0, -0.0]⋅x + -0.0\n", "(q_1,x_1)=(8,[1.25, -2.5]) - New cut for q=8: [-0.285714, -0.285714]⋅x + -0.285714286\n", "\n", "============== Forward pass #3: Model Predictive Control ============== \n", "(q_1,x_1,u_,1)=(4,[-1.325, -2.65], -0.15)\n", "(q_2,x_2,u_,2)=(11,[-4.0, -2.7], -0.05)\n", "(q_3,x_3,u_,3)=(13,[-6.4, -2.1], 0.6)\n", "None of the QPs is optimal so momentarily extending the horizon\n", "(q_4,x_4,u_,4)=(13,[-7.67917, -0.458333], 1.6417)\n", "(q_5,x_5,u_,5)=(13,[-7.45417, 0.908333], 1.3667)\n", "(q_6,x_6,u_,6)=(13,[-6.35458, 1.29083], 0.3825)\n", "(q_7,x_7,u_,7)=(16,[-5.0, 1.41833], 0.1275)\n", "(q_8,x_8,u_,8)=(15,[-3.44083, 1.7], 0.2817)\n", "(q_9,x_9,u_,9)=(15,[-1.67042, 1.84083], 0.1408)\n", "(q_10,x_10,u_,10)=(20,[-0.5, 0.5], -1.3408)\n", "\n", "Cost continuous control = 7.0074138813476585\n", "Cost discrete control = 10.0\n", "Total cost = 17.00741388134766\n", "============== Backward pass #3: Dual Dynamic Programming ==============\n", "(q_10,x_10)=(15,[-1.67042, 1.84083]) - New cut for q=15: [0.0, 3.0e-9]⋅x + 3.0e-9\n", "(q_9,x_9)=(15,[-3.44083, 1.7]) - New cut for q=15: [0.0, 1.0e-9]⋅x + 1.0e-9\n", "(q_8,x_8)=(16,[-5.0, 1.41833]) - New cut for q=16: [0.191979, 0.328073]⋅x + 0.32807292\n", "(q_7,x_7)=(13,[-6.35458, 1.29083]) - New cut for q=13: [0.181818, 0.363636]⋅x + 0.363636354\n", "(q_6,x_6)=(13,[-7.45417, 0.908333]) - New cut for q=13: [-0.00155233, 0.00733254]⋅x + 0.007332537\n", "(q_5,x_5)=(13,[-7.67917, -0.458333]) - New cut for q=13: [-0.25116, -0.243828]⋅x + -0.243827612\n", "(q_4,x_4)=(13,[-6.4, -2.1]) - New cut for q=13: [-0.803238, -1.04707]⋅x + -1.047065236\n", "(q_3,x_3)=(11,[-4.0, -2.7]) - New cut for q=11: [-0.376539, -0.376539]⋅x + -0.376539309\n", "(q_2,x_2)=(4,[-1.325, -2.65]) - New cut for q=4: [-0.387604, -0.770896]⋅x + -0.770895566\n", "(q_1,x_1)=(8,[1.25, -2.5]) - New cut for q=8: [-0.348721, -0.418503]⋅x + -0.418502637\n", "\n", "============== Forward pass #4: Model Predictive Control ============== \n", "(q_1,x_1,u_,1)=(4,[-1.43269, -2.86538], -0.3654)\n", "(q_2,x_2,u_,2)=(11,[-4.0, -2.26923], 0.5962)\n", "(q_3,x_3,u_,3)=(12,[-5.58242, -0.895604], 1.3736)\n", "(q_4,x_4,u_,4)=(12,[-5.84478, 0.370879], 1.2665)\n", "(q_5,x_5,u_,5)=(12,[-5.36931, 0.580063], 0.2092)\n", "(q_6,x_6,u_,6)=(16,[-4.78925, 0.580063], -0.0)\n", "(q_7,x_7,u_,7)=(15,[-3.62461, 1.74922], 1.1692)\n", "(q_8,x_8,u_,8)=(15,[-1.75, 2.0], 0.2508)\n", "(q_9,x_9,u_,9)=(20,[-0.5, 0.5], -1.5)\n", "\n", "Cost continuous control = 7.7033031640349146\n", "Cost discrete control = 9.0\n", "Total cost = 16.703303164034914\n", "============== Backward pass #4: Dual Dynamic Programming ==============\n", "(q_9,x_9)=(15,[-1.75, 2.0]) - New cut for q=15: [-0.0, 0.000131549]⋅x + 0.000131549\n", "(q_8,x_8)=(15,[-3.62461, 1.74922]) - New cut for q=15: [-2.3e-6, 1.0349e-5]⋅x + 1.0349e-5\n", "(q_7,x_7)=(16,[-4.78925, 0.580063]) - New cut for q=16: [0.222222, 0.222222]⋅x + 0.222222222\n", "(q_6,x_6)=(12,[-5.36931, 0.580063]) - New cut for q=12: [0.222222, 0.222222]⋅x + 0.222222222\n", "(q_5,x_5)=(12,[-5.84478, 0.370879]) - New cut for q=12: [-0.00221292, 0.180873]⋅x + 0.180872923\n", "(q_4,x_4)=(12,[-5.58242, -0.895604]) - New cut for q=12: [-0.157871, -0.162962]⋅x + -0.162962277\n", "(q_3,x_3)=(11,[-4.0, -2.26923]) - New cut for q=11: [-0.530849, -1.00748]⋅x + -1.007477437\n", "(q_2,x_2)=(4,[-1.43269, -2.86538]) - New cut for q=4: [-0.396988, -0.794782]⋅x + -0.794782325\n", "(q_1,x_1)=(8,[1.25, -2.5]) - New cut for q=8: [-0.348721, -0.418503]⋅x + -0.418502664\n", "\n", "============== Forward pass #5: Model Predictive Control ============== \n", "(q_1,x_1,u_,1)=(4,[-0.960394, -1.92079], 0.5792)\n", "(q_2,x_2,u_,2)=(4,[-2.59158, -1.34158], 0.5792)\n", "(q_3,x_3,u_,3)=(4,[-3.64354, -0.762363], 0.5792)\n", "(q_4,x_4,u_,4)=(5,[-4.15374, -0.258036], 0.5043)\n", "(q_5,x_5,u_,5)=(5,[-4.69191, -0.818292], -0.5603)\n", "(q_6,x_6,u_,6)=(1,[-5.35231, -0.502522], 0.3158)\n", "(q_7,x_7,u_,7)=(1,[-5.70289, -0.198633], 0.3039)\n", "(q_8,x_8,u_,8)=(1,[-5.6511, 0.302208], 0.5008)\n", "(q_9,x_9,u_,9)=(10,[-5.0, 1.0], 0.6978)\n", "(q_10,x_10,u_,10)=(15,[-3.625, 1.75], 0.75)\n", "(q_11,x_11,u_,11)=(15,[-1.75, 2.0], 0.25)\n", "(q_12,x_12,u_,12)=(20,[-0.5, 0.5], -1.5)\n", "\n", "Cost continuous control = 5.379507035444277\n", "Cost discrete control = 12.0\n", "Total cost = 17.379507035444277\n", "============== Backward pass #5: Dual Dynamic Programming ==============\n", "(q_12,x_12)=(15,[-1.75, 2.0]) - New cut for q=15: [-0.0, 0.000121519]⋅x + 0.000121519\n", "(q_11,x_11)=(15,[-3.625, 1.75]) - New cut for q=15: [-0.134159, -0.121347]⋅x + -0.121347363\n", "(q_10,x_10)=(10,[-5.0, 1.0]) - New cut for q=10: [-0.00161383, 0.111275]⋅x + 0.111275312\n", "(q_9,x_9)=(1,[-5.6511, 0.302208]) - New cut for q=1: [-0.16538, -0.496359]⋅x + -0.496358801\n", "(q_8,x_8)=(1,[-5.70289, -0.198633]) - New cut for q=1: [-0.16538, -0.496359]⋅x + -0.496358801\n", "(q_7,x_7)=(1,[-5.35231, -0.502522]) - New cut for q=1: [-0.176818, -0.519461]⋅x + -0.519461004\n", "(q_6,x_6)=(5,[-4.69191, -0.818292]) - New cut for q=5: [-0.16538, -0.496359]⋅x + -0.496358804\n", "(q_5,x_5)=(5,[-4.15374, -0.258036]) - New cut for q=5: [-0.0414861, -0.185536]⋅x + -0.185536224\n", "(q_4,x_4)=(4,[-3.64354, -0.762363]) - New cut for q=4: [-0.364129, -0.711995]⋅x + -0.711995217\n", "(q_3,x_3)=(4,[-2.59158, -1.34158]) - New cut for q=4: [-0.396057, -0.792114]⋅x + -0.792114193\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "(q_2,x_2)=(4,[-0.960394, -1.92079]) - New cut for q=4: [-0.396057, -0.792114]⋅x + -0.792114193\n", "(q_1,x_1)=(8,[1.25, -2.5]) - New cut for q=8: [-0.360744, -0.442155]⋅x + -0.44215502\n", "\n", "============== Forward pass #6: Model Predictive Control ============== \n", "(q_1,x_1,u_,1)=(4,[-1.29408, -2.58815], -0.0882)\n", "(q_2,x_2,u_,2)=(4,[-3.50175, -1.82721], 0.7609)\n", "(q_3,x_3,u_,3)=(5,[-4.94849, -1.06626], 0.7609)\n", "(q_4,x_4,u_,4)=(1,[-5.66997, -0.3767], 0.6896)\n", "(q_5,x_5,u_,5)=(1,[-5.68572, 0.34519], 0.7219)\n", "(q_6,x_6,u_,6)=(10,[-5.01313, 1.0], 0.6548)\n", "(q_7,x_7,u_,7)=(15,[-3.63156, 1.76313], 0.7631)\n", "(q_8,x_8,u_,8)=(15,[-1.75, 2.0], 0.2369)\n", "(q_9,x_9,u_,9)=(20,[-0.5, 0.5], -1.5)\n", "\n", "Cost continuous control = 5.4797105277549285\n", "Cost discrete control = 9.0\n", "Total cost = 14.479710527754929\n", "============== Backward pass #6: Dual Dynamic Programming ==============\n", "(q_9,x_9)=(15,[-1.75, 2.0]) - New cut for q=15: [-0.0, 0.000114458]⋅x + 0.000114458\n", "(q_8,x_8)=(15,[-3.63156, 1.76313]) - New cut for q=15: [-0.134159, -0.121347]⋅x + -0.121347363\n", "(q_7,x_7)=(10,[-5.01313, 1.0]) - New cut for q=10: [-0.00161383, 0.111275]⋅x + 0.111275313\n", "(q_6,x_6)=(1,[-5.68572, 0.34519]) - New cut for q=1: [-0.0, -0.0]⋅x + -0.0\n", "(q_5,x_5)=(1,[-5.66997, -0.3767]) - New cut for q=1: [-0.121527, -0.556353]⋅x + -0.556352996\n", "(q_4,x_4)=(5,[-4.94849, -1.06626]) - New cut for q=5: [-0.121527, -0.556353]⋅x + -0.556352997\n", "(q_3,x_3)=(4,[-3.50175, -1.82721]) - New cut for q=4: [-0.416847, -0.851673]⋅x + -0.851673254\n", "(q_2,x_2)=(4,[-1.29408, -2.58815]) - New cut for q=4: [-0.416847, -0.851673]⋅x + -0.851673254\n", "(q_1,x_1)=(8,[1.25, -2.5]) - New cut for q=8: [-0.360744, -0.442155]⋅x + -0.442155\n", "\n", "============== Forward pass #7: Model Predictive Control ============== \n", "(q_1,x_1,u_,1)=(4,[-1.29408, -2.58815], -0.0882)\n", "(q_2,x_2,u_,2)=(4,[-3.50175, -1.82721], 0.7609)\n", "(q_3,x_3,u_,3)=(5,[-4.94849, -1.06626], 0.7609)\n", "(q_4,x_4,u_,4)=(1,[-5.66997, -0.3767], 0.6896)\n", "(q_5,x_5,u_,5)=(1,[-5.68572, 0.34519], 0.7219)\n", "(q_6,x_6,u_,6)=(10,[-5.01313, 1.0], 0.6548)\n", "(q_7,x_7,u_,7)=(15,[-3.63156, 1.76313], 0.7631)\n", "(q_8,x_8,u_,8)=(15,[-1.75, 2.0], 0.2369)\n", "(q_9,x_9,u_,9)=(20,[-0.5, 0.5], -1.5)\n", "\n", "Cost continuous control = 5.479710352617289\n", "Cost discrete control = 9.0\n", "Total cost = 14.479710352617289\n", "============== Backward pass #7: Dual Dynamic Programming ==============\n", "(q_9,x_9)=(15,[-1.75, 2.0]) - New cut for q=15: [-0.0, 0.000114457]⋅x + 0.000114457\n", "(q_8,x_8)=(15,[-3.63156, 1.76313]) - New cut for q=15: [-0.134159, -0.121347]⋅x + -0.121347363\n", "(q_7,x_7)=(10,[-5.01313, 1.0]) - New cut for q=10: [-0.00161383, 0.111275]⋅x + 0.111275314\n", "(q_6,x_6)=(1,[-5.68572, 0.34519]) - New cut for q=1: [-0.0, -0.0]⋅x + -0.0\n", "(q_5,x_5)=(1,[-5.66997, -0.3767]) - New cut for q=1: [-0.121527, -0.556353]⋅x + -0.556352996\n", "(q_4,x_4)=(5,[-4.94849, -1.06626]) - New cut for q=5: [-0.121527, -0.556353]⋅x + -0.556353001\n", "(q_3,x_3)=(4,[-3.50175, -1.82721]) - New cut for q=4: [-0.416847, -0.851673]⋅x + -0.851673254\n", "(q_2,x_2)=(4,[-1.29408, -2.58815]) - New cut for q=4: [-0.416847, -0.851673]⋅x + -0.851673254\n", "(q_1,x_1)=(8,[1.25, -2.5]) - New cut for q=8: [-0.360744, -0.442155]⋅x + -0.442155\n", "\n", "============== Forward pass #8: Model Predictive Control ============== \n", "(q_1,x_1,u_,1)=(4,[-1.29408, -2.58815], -0.0882)\n", "(q_2,x_2,u_,2)=(4,[-3.50175, -1.82721], 0.7609)\n", "(q_3,x_3,u_,3)=(5,[-4.94849, -1.06626], 0.7609)\n", "(q_4,x_4,u_,4)=(1,[-5.66997, -0.3767], 0.6896)\n", "(q_5,x_5,u_,5)=(1,[-5.68572, 0.34519], 0.7219)\n", "(q_6,x_6,u_,6)=(10,[-5.01313, 1.0], 0.6548)\n", "(q_7,x_7,u_,7)=(15,[-3.63156, 1.76313], 0.7631)\n", "(q_8,x_8,u_,8)=(15,[-1.75, 2.0], 0.2369)\n", "(q_9,x_9,u_,9)=(20,[-0.5, 0.5], -1.5)\n", "\n", "Cost continuous control = 5.479710353185093\n", "Cost discrete control = 9.0\n", "Total cost = 14.479710353185093\n", "============== Backward pass #8: Dual Dynamic Programming ==============\n", "(q_9,x_9)=(15,[-1.75, 2.0]) - New cut for q=15: [-0.0, 0.000114457]⋅x + 0.000114457\n", "(q_8,x_8)=(15,[-3.63156, 1.76313]) - New cut for q=15: [-0.134159, -0.121347]⋅x + -0.121347363\n", "(q_7,x_7)=(10,[-5.01313, 1.0]) - New cut for q=10: [-0.00161383, 0.111275]⋅x + 0.111275312\n", "(q_6,x_6)=(1,[-5.68572, 0.34519]) - New cut for q=1: [-0.0, -0.0]⋅x + -0.0\n", "(q_5,x_5)=(1,[-5.66997, -0.3767]) - New cut for q=1: [-0.121527, -0.556353]⋅x + -0.556352996\n", "(q_4,x_4)=(5,[-4.94849, -1.06626]) - New cut for q=5: [-0.121527, -0.556353]⋅x + -0.556353001\n", "(q_3,x_3)=(4,[-3.50175, -1.82721]) - New cut for q=4: [-0.416847, -0.851673]⋅x + -0.851673254\n", "(q_2,x_2)=(4,[-1.29408, -2.58815]) - New cut for q=4: [-0.416847, -0.851673]⋅x + -0.851673254\n", "(q_1,x_1)=(8,[1.25, -2.5]) - New cut for q=8: [-0.360744, -0.442155]⋅x + -0.442155\n", "\n", "============== Forward pass #9: Model Predictive Control ============== \n", "(q_1,x_1,u_,1)=(4,[-1.29408, -2.58815], -0.0882)\n", "(q_2,x_2,u_,2)=(4,[-3.50175, -1.82721], 0.7609)\n", "(q_3,x_3,u_,3)=(5,[-4.94849, -1.06626], 0.7609)\n", "(q_4,x_4,u_,4)=(1,[-5.66997, -0.3767], 0.6896)\n", "(q_5,x_5,u_,5)=(1,[-5.68572, 0.34519], 0.7219)\n", "(q_6,x_6,u_,6)=(10,[-5.01313, 1.0], 0.6548)\n", "(q_7,x_7,u_,7)=(15,[-3.63156, 1.76313], 0.7631)\n", "(q_8,x_8,u_,8)=(15,[-1.75, 2.0], 0.2369)\n", "(q_9,x_9,u_,9)=(20,[-0.5, 0.5], -1.5)\n", "\n", "Cost continuous control = 5.479710353185093\n", "Cost discrete control = 9.0\n", "Total cost = 14.479710353185093\n", "============== Backward pass #9: Dual Dynamic Programming ==============\n", "(q_9,x_9)=(15,[-1.75, 2.0]) - New cut for q=15: [-0.0, 0.000114458]⋅x + 0.000114458\n", "(q_8,x_8)=(15,[-3.63156, 1.76313]) - New cut for q=15: [-0.134159, -0.121347]⋅x + -0.121347363\n", "(q_7,x_7)=(10,[-5.01313, 1.0]) - New cut for q=10: [-0.00161383, 0.111275]⋅x + 0.111275312\n", "(q_6,x_6)=(1,[-5.68572, 0.34519]) - New cut for q=1: [-0.0, -0.0]⋅x + -0.0\n", "(q_5,x_5)=(1,[-5.66997, -0.3767]) - New cut for q=1: [-0.121527, -0.556353]⋅x + -0.556352996\n", "(q_4,x_4)=(5,[-4.94849, -1.06626]) - New cut for q=5: [-0.121527, -0.556353]⋅x + -0.556353001\n", "(q_3,x_3)=(4,[-3.50175, -1.82721]) - New cut for q=4: [-0.416847, -0.851673]⋅x + -0.851673254\n", "(q_2,x_2)=(4,[-1.29408, -2.58815]) - New cut for q=4: [-0.416847, -0.851673]⋅x + -0.851673254\n", "(q_1,x_1)=(8,[1.25, -2.5]) - New cut for q=8: [-0.360744, -0.442155]⋅x + -0.442155\n", "\n", "============== Forward pass #10: Model Predictive Control ============== \n", "(q_1,x_1,u_,1)=(4,[-1.29408, -2.58815], -0.0882)\n", "(q_2,x_2,u_,2)=(4,[-3.50175, -1.82721], 0.7609)\n", "(q_3,x_3,u_,3)=(5,[-4.94849, -1.06626], 0.7609)\n", "(q_4,x_4,u_,4)=(1,[-5.66997, -0.3767], 0.6896)\n", "(q_5,x_5,u_,5)=(1,[-5.68572, 0.34519], 0.7219)\n", "(q_6,x_6,u_,6)=(10,[-5.01313, 1.0], 0.6548)\n", "(q_7,x_7,u_,7)=(15,[-3.63156, 1.76313], 0.7631)\n", "(q_8,x_8,u_,8)=(15,[-1.75, 2.0], 0.2369)\n", "(q_9,x_9,u_,9)=(20,[-0.5, 0.5], -1.5)\n", "\n", "Cost continuous control = 5.479710353185093\n", "Cost discrete control = 9.0\n", "Total cost = 14.479710353185093\n", "============== Backward pass #10: Dual Dynamic Programming ==============\n", "(q_9,x_9)=(15,[-1.75, 2.0]) - New cut for q=15: [-0.0, 0.000114458]⋅x + 0.000114458\n", "(q_8,x_8)=(15,[-3.63156, 1.76313]) - New cut for q=15: [-0.134159, -0.121347]⋅x + -0.121347363\n", "(q_7,x_7)=(10,[-5.01313, 1.0]) - New cut for q=10: [-0.00161383, 0.111275]⋅x + 0.111275313\n", "(q_6,x_6)=(1,[-5.68572, 0.34519]) - New cut for q=1: [-0.0, -0.0]⋅x + -0.0\n", "(q_5,x_5)=(1,[-5.66997, -0.3767]) - New cut for q=1: [-0.121527, -0.556353]⋅x + -0.556352996\n", "(q_4,x_4)=(5,[-4.94849, -1.06626]) - New cut for q=5: [-0.121527, -0.556353]⋅x + -0.556353001\n", "(q_3,x_3)=(4,[-3.50175, -1.82721]) - New cut for q=4: [-0.416847, -0.851673]⋅x + -0.851673254\n", "(q_2,x_2)=(4,[-1.29408, -2.58815]) - New cut for q=4: [-0.416847, -0.851673]⋅x + -0.851673254\n", "(q_1,x_1)=(8,[1.25, -2.5]) - New cut for q=8: [-0.360744, -0.442155]⋅x + -0.442155\n", "\n", "============== Forward pass #11: Model Predictive Control ============== \n", "(q_1,x_1,u_,1)=(4,[-1.29408, -2.58815], -0.0882)\n", "(q_2,x_2,u_,2)=(4,[-3.50175, -1.82721], 0.7609)\n", "(q_3,x_3,u_,3)=(5,[-4.94849, -1.06626], 0.7609)\n", "(q_4,x_4,u_,4)=(1,[-5.66997, -0.3767], 0.6896)\n", "(q_5,x_5,u_,5)=(1,[-5.68572, 0.34519], 0.7219)\n", "(q_6,x_6,u_,6)=(10,[-5.01313, 1.0], 0.6548)\n", "(q_7,x_7,u_,7)=(15,[-3.63156, 1.76313], 0.7631)\n", "(q_8,x_8,u_,8)=(15,[-1.75, 2.0], 0.2369)\n", "(q_9,x_9,u_,9)=(20,[-0.5, 0.5], -1.5)\n", "\n", "Cost continuous control = 5.479710353185093\n", "Cost discrete control = 9.0\n", "Total cost = 14.479710353185093\n", "============== Backward pass #11: Dual Dynamic Programming ==============\n", "(q_9,x_9)=(15,[-1.75, 2.0]) - New cut for q=15: [-0.0, 0.000114458]⋅x + 0.000114458\n", "(q_8,x_8)=(15,[-3.63156, 1.76313]) - New cut for q=15: [-0.134159, -0.121347]⋅x + -0.121347363\n", "(q_7,x_7)=(10,[-5.01313, 1.0]) - New cut for q=10: [-0.00161383, 0.111275]⋅x + 0.111275313\n", "(q_6,x_6)=(1,[-5.68572, 0.34519]) - New cut for q=1: [-0.0, -0.0]⋅x + -0.0\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "(q_5,x_5)=(1,[-5.66997, -0.3767]) - New cut for q=1: [-0.121527, -0.556353]⋅x + -0.556352996\n", "(q_4,x_4)=(5,[-4.94849, -1.06626]) - New cut for q=5: [-0.121527, -0.556353]⋅x + -0.556353001\n", "(q_3,x_3)=(4,[-3.50175, -1.82721]) - New cut for q=4: [-0.416847, -0.851673]⋅x + -0.851673254\n", "(q_2,x_2)=(4,[-1.29408, -2.58815]) - New cut for q=4: [-0.416847, -0.851673]⋅x + -0.851673254\n", "(q_1,x_1)=(8,[1.25, -2.5]) - New cut for q=8: [-0.360744, -0.442155]⋅x + -0.442155\n", "\n", "============== Forward pass #12: Model Predictive Control ============== \n", "(q_1,x_1,u_,1)=(4,[-1.29408, -2.58815], -0.0882)\n", "(q_2,x_2,u_,2)=(4,[-3.50175, -1.82721], 0.7609)\n", "(q_3,x_3,u_,3)=(5,[-4.94849, -1.06626], 0.7609)\n", "(q_4,x_4,u_,4)=(1,[-5.66997, -0.3767], 0.6896)\n", "(q_5,x_5,u_,5)=(1,[-5.68572, 0.34519], 0.7219)\n", "(q_6,x_6,u_,6)=(10,[-5.01313, 1.0], 0.6548)\n", "(q_7,x_7,u_,7)=(15,[-3.63156, 1.76313], 0.7631)\n", "(q_8,x_8,u_,8)=(15,[-1.75, 2.0], 0.2369)\n", "(q_9,x_9,u_,9)=(20,[-0.5, 0.5], -1.5)\n", "\n", "Cost continuous control = 5.479710353164572\n", "Cost discrete control = 9.0\n", "Total cost = 14.479710353164572\n", "============== Backward pass #12: Dual Dynamic Programming ==============\n", "(q_9,x_9)=(15,[-1.75, 2.0]) - New cut for q=15: [-0.0, 0.000114457]⋅x + 0.000114457\n", "(q_8,x_8)=(15,[-3.63156, 1.76313]) - New cut for q=15: [-0.134159, -0.121347]⋅x + -0.121347363\n", "(q_7,x_7)=(10,[-5.01313, 1.0]) - New cut for q=10: [-0.00161383, 0.111275]⋅x + 0.111275313\n", "(q_6,x_6)=(1,[-5.68572, 0.34519]) - New cut for q=1: [-0.0, -0.0]⋅x + -0.0\n", "(q_5,x_5)=(1,[-5.66997, -0.3767]) - New cut for q=1: [-0.121527, -0.556353]⋅x + -0.556352996\n", "(q_4,x_4)=(5,[-4.94849, -1.06626]) - New cut for q=5: [-0.121527, -0.556353]⋅x + -0.556353001\n", "(q_3,x_3)=(4,[-3.50175, -1.82721]) - New cut for q=4: [-0.416847, -0.851673]⋅x + -0.851673254\n", "(q_2,x_2)=(4,[-1.29408, -2.58815]) - New cut for q=4: [-0.416847, -0.851673]⋅x + -0.851673254\n", "(q_1,x_1)=(8,[1.25, -2.5]) - New cut for q=8: [-0.360744, -0.442155]⋅x + -0.442155\n", "\n", "============== Forward pass #13: Model Predictive Control ============== \n", "(q_1,x_1,u_,1)=(4,[-1.29408, -2.58815], -0.0882)\n", "(q_2,x_2,u_,2)=(4,[-3.50175, -1.82721], 0.7609)\n", "(q_3,x_3,u_,3)=(5,[-4.94849, -1.06626], 0.7609)\n", "(q_4,x_4,u_,4)=(1,[-5.66997, -0.3767], 0.6896)\n", "(q_5,x_5,u_,5)=(1,[-5.68572, 0.34519], 0.7219)\n", "(q_6,x_6,u_,6)=(10,[-5.01313, 1.0], 0.6548)\n", "(q_7,x_7,u_,7)=(15,[-3.63156, 1.76313], 0.7631)\n", "(q_8,x_8,u_,8)=(15,[-1.75, 2.0], 0.2369)\n", "(q_9,x_9,u_,9)=(20,[-0.5, 0.5], -1.5)\n", "\n", "Cost continuous control = 5.479710353164572\n", "Cost discrete control = 9.0\n", "Total cost = 14.479710353164572\n", "============== Backward pass #13: Dual Dynamic Programming ==============\n", "(q_9,x_9)=(15,[-1.75, 2.0]) - New cut for q=15: [-0.0, 0.000114457]⋅x + 0.000114457\n", "(q_8,x_8)=(15,[-3.63156, 1.76313]) - New cut for q=15: [-0.134159, -0.121347]⋅x + -0.121347363\n", "(q_7,x_7)=(10,[-5.01313, 1.0]) - New cut for q=10: [-0.00161383, 0.111275]⋅x + 0.111275313\n", "(q_6,x_6)=(1,[-5.68572, 0.34519]) - New cut for q=1: [-0.0, -0.0]⋅x + -0.0\n", "(q_5,x_5)=(1,[-5.66997, -0.3767]) - New cut for q=1: [-0.121527, -0.556353]⋅x + -0.556352996\n", "(q_4,x_4)=(5,[-4.94849, -1.06626]) - New cut for q=5: [-0.121527, -0.556353]⋅x + -0.556353001\n", "(q_3,x_3)=(4,[-3.50175, -1.82721]) - New cut for q=4: [-0.416847, -0.851673]⋅x + -0.851673254\n", "(q_2,x_2)=(4,[-1.29408, -2.58815]) - New cut for q=4: [-0.416847, -0.851673]⋅x + -0.851673254\n", "(q_1,x_1)=(8,[1.25, -2.5]) - New cut for q=8: [-0.360744, -0.442155]⋅x + -0.442155\n", "\n", "============== Forward pass #14: Model Predictive Control ============== \n", "(q_1,x_1,u_,1)=(4,[-1.29408, -2.58815], -0.0882)\n", "(q_2,x_2,u_,2)=(4,[-3.50175, -1.82721], 0.7609)\n", "(q_3,x_3,u_,3)=(5,[-4.94849, -1.06626], 0.7609)\n", "(q_4,x_4,u_,4)=(1,[-5.66997, -0.3767], 0.6896)\n", "(q_5,x_5,u_,5)=(1,[-5.68572, 0.34519], 0.7219)\n", "(q_6,x_6,u_,6)=(10,[-5.01313, 1.0], 0.6548)\n", "(q_7,x_7,u_,7)=(15,[-3.63156, 1.76313], 0.7631)\n", "(q_8,x_8,u_,8)=(15,[-1.75, 2.0], 0.2369)\n", "(q_9,x_9,u_,9)=(20,[-0.5, 0.5], -1.5)\n", "\n", "Cost continuous control = 5.479710353185093\n", "Cost discrete control = 9.0\n", "Total cost = 14.479710353185093\n", "============== Backward pass #14: Dual Dynamic Programming ==============\n", "(q_9,x_9)=(15,[-1.75, 2.0]) - New cut for q=15: [-0.0, 0.000114458]⋅x + 0.000114458\n", "(q_8,x_8)=(15,[-3.63156, 1.76313]) - New cut for q=15: [-0.134159, -0.121347]⋅x + -0.121347363\n", "(q_7,x_7)=(10,[-5.01313, 1.0]) - New cut for q=10: [-0.00161383, 0.111275]⋅x + 0.111275312\n", "(q_6,x_6)=(1,[-5.68572, 0.34519]) - New cut for q=1: [-0.0, -0.0]⋅x + -0.0\n", "(q_5,x_5)=(1,[-5.66997, -0.3767]) - New cut for q=1: [-0.121527, -0.556353]⋅x + -0.556352996\n", "(q_4,x_4)=(5,[-4.94849, -1.06626]) - New cut for q=5: [-0.121527, -0.556353]⋅x + -0.556353001\n", "(q_3,x_3)=(4,[-3.50175, -1.82721]) - New cut for q=4: [-0.416847, -0.851673]⋅x + -0.851673254\n", "(q_2,x_2)=(4,[-1.29408, -2.58815]) - New cut for q=4: [-0.416847, -0.851673]⋅x + -0.851673254\n", "(q_1,x_1)=(8,[1.25, -2.5]) - New cut for q=8: [-0.360744, -0.442155]⋅x + -0.442155\n", "\n", "============== Forward pass #15: Model Predictive Control ============== \n", "(q_1,x_1,u_,1)=(4,[-1.29408, -2.58815], -0.0882)\n", "(q_2,x_2,u_,2)=(4,[-3.50175, -1.82721], 0.7609)\n", "(q_3,x_3,u_,3)=(5,[-4.94849, -1.06626], 0.7609)\n", "(q_4,x_4,u_,4)=(1,[-5.66997, -0.3767], 0.6896)\n", "(q_5,x_5,u_,5)=(1,[-5.68572, 0.34519], 0.7219)\n", "(q_6,x_6,u_,6)=(10,[-5.01313, 1.0], 0.6548)\n", "(q_7,x_7,u_,7)=(15,[-3.63156, 1.76313], 0.7631)\n", "(q_8,x_8,u_,8)=(15,[-1.75, 2.0], 0.2369)\n", "(q_9,x_9,u_,9)=(20,[-0.5, 0.5], -1.5)\n", "\n", "Cost continuous control = 5.479710353185093\n", "Cost discrete control = 9.0\n", "Total cost = 14.479710353185093\n", "============== Backward pass #15: Dual Dynamic Programming ==============\n", "(q_9,x_9)=(15,[-1.75, 2.0]) - New cut for q=15: [-0.0, 0.000114458]⋅x + 0.000114458\n", "(q_8,x_8)=(15,[-3.63156, 1.76313]) - New cut for q=15: [-0.134159, -0.121347]⋅x + -0.121347363\n", "(q_7,x_7)=(10,[-5.01313, 1.0]) - New cut for q=10: [-0.00161383, 0.111275]⋅x + 0.111275314\n", "(q_6,x_6)=(1,[-5.68572, 0.34519]) - New cut for q=1: [-0.0, -0.0]⋅x + -0.0\n", "(q_5,x_5)=(1,[-5.66997, -0.3767]) - New cut for q=1: [-0.121527, -0.556353]⋅x + -0.556352996\n", "(q_4,x_4)=(5,[-4.94849, -1.06626]) - New cut for q=5: [-0.121527, -0.556353]⋅x + -0.556353001\n", "(q_3,x_3)=(4,[-3.50175, -1.82721]) - New cut for q=4: [-0.416847, -0.851673]⋅x + -0.851673254\n", "(q_2,x_2)=(4,[-1.29408, -2.58815]) - New cut for q=4: [-0.416847, -0.851673]⋅x + -0.851673254\n", "(q_1,x_1)=(8,[1.25, -2.5]) - New cut for q=8: [-0.360744, -0.442155]⋅x + -0.442155\n" ] } ], "source": [ "# - Hybrid dual dynamic programming algorithm -\n", "\n", "using LightGraphs\n", "using JuMP, ParameterJuMP\n", "using LinearAlgebra\n", "\n", "#If Gurobi is available:\n", "using Gurobi\n", "gurobi_env = Gurobi.Env()\n", "solver = with_optimizer(Gurobi.Optimizer, gurobi_env, OutputFlag=0)\n", "#else use another QP solver, like OSQP\n", "\n", "#Algorithm general parameters:\n", "max_n_iterations = 15 #Number of iterations of the algorithm\n", "println(\"\\nAlgorithm general parameters: \", max_n_iterations, \" iterations\")\n", "\n", "#Fwd pass parameters:\n", "min_horizon = 3 #Horizon of the MPC\n", "MAX_N_STEPS = 20 #Max number of steps for the forward pass to reach desired terminal state\n", "println(\"Forward pass parameters: desired max \", MAX_N_STEPS, \" steps, min horizon = \", min_horizon)\n", "\n", "#Initial hybrid state:\n", "#q0 = 3; x0 = [1.;-6] #Figures 4.6 and 4.7\n", "q0 = 8; x0 = [1.25; -2.5] #Figures 4.8 to 4.11\n", "println(\"Initial state: (q_0,x_0) = (\", q0, \",\", x0, \")\")\n", "\n", "#Terminal discrete state:\n", "qT = 20\n", "println(\"Terminal state: q_T = \", qT)\n", "\n", "#Cost function: Q ⋅ ∑ u^2 + C\n", "Q = 1\n", "C = 1\n", "println(\"Cost function: \", Q, \" ⋅ ∑ u⋅u + \", C)\n", "\n", "##########\n", "\n", "#First approximation of the value function:\n", "cuts = [[i;0;0;0.] for i in 1:nstates(HySys)]\n", "\n", "#Keeping track of the cost:\n", "cost_c = zeros(max_n_iterations)\n", "cost_d = zeros(max_n_iterations)\n", "\n", "#Forward iterates:\n", "FWD_Q = Vector{Vector{Int64}}(undef, max_n_iterations)\n", "FWD_X = Vector{Vector{Vector{Float64}}}(undef, max_n_iterations)\n", "FWD_U = Vector{Vector{Float64}}(undef, max_n_iterations)\n", "\n", "n_iterations = 0\n", "while n_iterations < max_n_iterations\n", " \n", " println(\"\\n============== Forward pass #\", n_iterations+1, \": Model Predictive Control ============== \")\n", " \n", " FWD_Q[n_iterations+1] = [q0]\n", " FWD_X[n_iterations+1] = [x0]\n", " FWD_U[n_iterations+1] = [0.]\n", " \n", " current_mode = q0\n", " terminal_mode = qT\n", " current_X = x0\n", " \n", " N = min_horizon #Effective MPC horizon, N ≥ min_horizon\n", " max_n_steps = MAX_N_STEPS\n", " \n", " counter = 0\n", " while (current_mode ≠ terminal_mode) && (counter < max_n_steps)\n", " T = max_n_steps-counter #Number of iterations left for the fwd pass to reach qT\n", " \n", " counter += 1 \n", " \n", " new_mode = current_mode\n", " new_X = current_X\n", " new_U = 0.\n", " obj_val = Inf\n", " \n", " perm = Iterators.product(ntuple(i -> 1:nstates(HySys), N)...)\n", " \n", " Θc = 0. #to compute the continuous cost\n", " Θd = 0. #to compute the discrete cost\n", " \n", " Flag_distance = false\n", " Flag_reach = false\n", " Flag_QP = false\n", " \n", " for modes in perm\n", " modes = collect(modes)\n", " \n", " #0. Path in the automaton should exist\n", " flag_path = has_transition(HySys, current_mode, modes[1]) #to avoid wasting time\n", " for i in 1:N-1\n", " if has_transition(HySys, modes[i], modes[i+1]) == false\n", " flag_path = false\n", " end\n", " end\n", " \n", " #1. (optional) Distance at the horizon should strictly decrease\n", " flag_distance = false\n", " if dijkstra_shortest_paths(HySys.automaton.G, modes[end]).dists[qT] < dijkstra_shortest_paths(HySys.automaton.G, current_mode).dists[qT]\n", " flag_distance = true\n", " Flag_distance = true #At least one candidate trajectory is OK\n", " end\n", " \n", " #2. Terminal state should be reached by the last iteration of the forward pass\n", " flag_reach = true\n", " if T-N ≤ dijkstra_shortest_paths(HySys.automaton.G, modes[end]).dists[qT]\n", " flag_reach = false\n", " else\n", " Flag_reach = true #At least one candidate trajectory is OK\n", " end\n", " \n", " if flag_path == true && flag_reach == true && flag_distance == true\n", " model = Model(solver)\n", " \n", " @variable(model, x[1:N+1,1:2])\n", " @variable(model, u[1:N,1:1])\n", " @variable(model, θ)\n", " \n", " @objective(model, Min, sum((Q ⋅ u[i,1]^2 + C)*(modes[i] != qT) for i in 1:N) + θ)\n", " \n", " @constraint(model, x[1,:] .== current_X)\n", " for i = 2:N+1\n", " @constraint(model, x[i,:] .== A*x[i-1,:]+B*u[i-1,:])\n", " for h in halfspaces(stateset(HySys, modes[i-1]))\n", " @constraint(model, x[i,:] ⋅ h.a ≤ h.β)\n", " end\n", " end\n", "\n", " for i in 1:N\n", " @constraint(model, -2 ≤ u[i,1] ≤ 2)\n", " end\n", " \n", " #Approximation of the value function at the horizon:\n", " for i in 1:size(cuts,1)\n", " if cuts[i][1] == modes[end]\n", " @constraint(model, x[end,:] ⋅ cuts[i][2:length(x0)+1] + cuts[i][end] ≤ θ) #a^T x + b <= θ, missing b (dual obj val?)\n", " end\n", " end\n", " \n", " optimize!(model)\n", "\n", " if (termination_status(model) == MOI.OPTIMAL)\n", " if (objective_value(model) < obj_val)\n", " Flag_QP = true #at least one QP is feasible\n", " \n", " obj_val = objective_value(model)\n", " \n", " Θc = Q*value(u[1,1])^2\n", " Θd = C\n", " \n", " new_mode = modes[1]\n", " new_X = value.(x[2,:])\n", " new_U = value(u[1,1])\n", " end\n", " end\n", " end\n", " end #end of trial of each candidate trajectory\n", " \n", " #Decision tree:\n", " if Flag_distance == false\n", " counter -= 1\n", " N += 1\n", " println(\"Impossible to lower the distance to T so momentarily extending the horizon\")\n", " else\n", " if Flag_reach == false\n", " counter -= 1\n", " max_n_steps += 1\n", " println(\"Impossible to finish on time so extending deadline\")\n", " else\n", " if Flag_QP == false\n", " if T-N < dijkstra_shortest_paths(HySys.automaton.G, current_mode).dists[qT]\n", " counter -= 1\n", " max_n_steps += 1\n", " println(\"None of the QPs is optimal so extending deadline\")\n", " else\n", " counter -= 1\n", " N += 1\n", " println(\"None of the QPs is optimal so momentarily extending the horizon\")\n", " end\n", " else \n", " N = min_horizon\n", " \n", " cost_c[n_iterations+1] += Θc\n", " cost_d[n_iterations+1] += Θd\n", " \n", " push!(FWD_Q[n_iterations+1], Int(new_mode))\n", " push!(FWD_X[n_iterations+1], new_X)\n", " push!(FWD_U[n_iterations+1], new_U)\n", " \n", " println(\"(q_\",counter,\",x_\",counter,\",u_,\",counter,\")=(\", Int(new_mode),\",\", new_X,\", \", round(new_U, digits=4),\")\")\n", " end\n", " end\n", " end\n", " \n", " current_X = new_X\n", " current_mode = new_mode\n", " end #end of forward pass\n", " \n", " print(\"\\nCost continuous control = \", cost_c[n_iterations+1])\n", " print(\"\\nCost discrete control = \", cost_d[n_iterations+1])\n", " print(\"\\nTotal cost = \", cost_c[n_iterations+1]+cost_d[n_iterations+1])\n", " \n", " println(\"\\n============== Backward pass #\", n_iterations+1, \": Dual Dynamic Programming ==============\")\n", "\n", " for bwd_step in length(FWD_Q[n_iterations+1])-1:-1:1 #for each forward iterate, solve the dual of the relaxed primal\n", " \n", " print(\"(q_\",bwd_step,\",x_\",bwd_step,\")=(\",FWD_Q[n_iterations+1][bwd_step],\",\",FWD_X[n_iterations+1][bwd_step],\")\")\n", "\n", " trans = out_transitions(HySys, FWD_Q[n_iterations+1][bwd_step])\n", " vertices = Dict()\n", " for t in trans\n", " model = Model()\n", " @variable(model, y[1:2,1:1])\n", " @variable(model, c ≥ 0)\n", " \n", " #Cuts:\n", " for k in 1:size(cuts, 1)\n", " if target(HySys, t) == Int(cuts[k][1])\n", " @constraint(model, c ≥ y ⋅ cuts[k][2:end-1] + cuts[k][end])\n", " end\n", " end\n", " \n", " #Domain:\n", " for k in halfspaces(stateset(HySys, target(HySys, t)))\n", " @constraint(model, y ⋅ k.a ≤ k.β)\n", " end\n", " \n", " pol = polyhedron(hrep(model), CDDLib.Library())\n", " removehredundancy!(pol)\n", " \n", " #Collecting the vertices of the polyhedron:\n", " vertices[t] = collect(points(pol))\n", " end \n", " trans = collect(trans)\n", " \n", " model = ModelWithParams(solver)\n", " \n", " params = Parameters(model, FWD_X[n_iterations+1][bwd_step])\n", " @variable(model, -2 ≤ u ≤ 2)\n", " @variable(model, θ ≥ 0)\n", " @variable(model, λ[t in trans, 1:length(vertices[t])] ≥ 0)\n", " \n", " @objective(model, Min, Q*u^2 + C + θ)\n", " \n", " @constraint(model, [params[1]+params[2]+0.5*u, params[2]+u] .== sum(λ[t,i].*vertices[t][i][1:2] for t in trans for i in 1:length(vertices[t])))\n", " @constraint(model, θ ≥ sum(λ[t,i]*vertices[t][i][3] for t in trans for i in 1:length(vertices[t])))\n", " \n", " @constraint(model, sum(λ[t,i] for t in trans for i in 1:length(vertices[t])) == 1)\n", " \n", " optimize!(model)\n", " \n", " if termination_status(model) == MOI.OPTIMAL\n", " #Adding the new cut:\n", " push!(cuts, [FWD_Q[n_iterations+1][bwd_step], round(dual(params[1]), digits=8), round(dual(params[2]), digits=9), round(objective_value(model)-dual(params[1])*value(params[1])-dual(params[2])*value(params[2]), digits=8)])\n", " #rounding to avoid numerical inconsistencies with CDD\n", " println(\" - New cut for q=\", Int(cuts[end][1]), \": \", cuts[end][2:end-1], \"⋅x + \", cuts[end][3])\n", " else\n", " println(\" - No new cut\")\n", " end \n", " end #end of backward pass\n", " n_iterations += 1\n", "end" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "p = Plot{Plots.GRBackend() n=182}\n" ] }, { "data": { "image/png": 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" }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# - Graphical representation -\n", "\n", "using Plots\n", "using Colors\n", "\n", "#Modes:\n", "p = plot(pX, fmt = :png, fillcolor = :white)\n", "for mode in states(HySys)\n", " plot!(p, stateset(HySys, mode), fillcolor = :white, linecolor = :black)\n", "end\n", "\n", "#Initial state:\n", "scatter!(p, [x0[1]],[x0[2]])\n", "annotate!(p, [(x0[1],x0[2]-0.5,text(\"x0\",10))])\n", "\n", "#Terminal state:\n", "annotate!(p, [(0,0,text(\"T\",12))])\n", "\n", "#Obstacles:\n", "plot!(p, pO1, fillcolor = :black, fillalpha = 0.1) \n", "plot!(p, pO2, fillcolor = :black, fillalpha = 0.1)\n", "annotate!(p, [(-7.5,-7.5,text(\"O1\",12))])\n", "annotate!(p, [(-1.5,-3.5,text(\"O2\",12))])\n", "\n", "#Regions to visit:\n", "annotate!(p, [(-5.5,1.5,text(\"A\",12))])\n", "annotate!(p, [(-4.5,-2.5,text(\"B\",12))])\n", "\n", "#Forward trajectories:\n", "colors = distinguishable_colors(max_n_iterations)\n", "for i in 1:max_n_iterations\n", " for j in 1:length(FWD_X[i])\n", " scatter!(p, [FWD_X[i][j][1]],[FWD_X[i][j][2]], markercolor=colors[i], markersize=3+2*(i==max_n_iterations))\n", " end\n", "end\n", "\n", "@show p" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Julia 1.1.0", "language": "julia", "name": "julia-1.1" }, "language_info": { "file_extension": ".jl", "mimetype": "application/julia", "name": "julia", "version": "1.1.0" } }, "nbformat": 4, "nbformat_minor": 2 }