LTI Control with Unknown Coefficients
We jointly optimize a scalar input and three constant coefficients in $\dot{x} = A x + b u$, with $x(0)=0$ and $x(10)=[3,1,-2]^T$. The objective is minimum control energy, $\frac12\int_0^{10}u(t)^2\,dt$.
using Corleone, ModelingToolkit
using ModelingToolkit: t_nounits as t, D_nounits as D
using OrdinaryDiffEqTsit5
using ComponentArrays, ForwardDiff, Optimization, OptimizationMOI, Ipopt
using LuxCore, Random, LinearAlgebraModel
The state is $x\in\mathbb{R}^3$ and the input is scalar. The matrix and input vector are
\[A=\begin{bmatrix}0&1&0\\a_1&0&1\\0&0&a_2\end{bmatrix}, \qquad b=\begin{bmatrix}b_1\\0\\1\end{bmatrix}.\]
Bounds are $-0.5\le a_i\le-0.05$, $0.1\le b_1\le1$, and $|u|\le3$. Only A[2,1], A[3,3], and b[1] are unknown. All other entries are fixed. The physical initial state is fixed; interior shooting values remain tunable. The MTK array a contains the two unknown entries of A. Corleone scalarizes its components during setup and stores each in its own scalar vector buffer.
@variables begin
x₁(t) = 0.0, [tunable = false]
x₂(t) = 0.0, [tunable = false]
x₃(t) = 0.0, [tunable = false]
u(t) = 0.0, [input = true, bounds = (-3.0, 3.0)]
end
@parameters begin
a[1:2] = [-0.2, -0.1], [tunable = true, bounds = (-0.5, -0.05)]
b₁ = 0.5, [tunable = true, bounds = (0.1, 1.0)]
end
@constants begin
x_t[1:3] = [3.0, 1.0, -2.0]
end
A = [0.0 1.0 0.0; a[1] 0.0 1.0; 0.0 0.0 a[2]]
b = [b₁, 0.0, 1.0]
x = [x₁, x₂, x₃]
equations = D.(x) .~ A * x + b * u
t_inf = 10.0
terminal_constraint = EvalAt(t_inf).(x) .~ scalarize(x_t)
energy = Symbolics.Integral(t in (0.0, t_inf))(0.5 * u^2)
@named lti = System(equations, t; costs = [energy], constraints = terminal_constraint)Model lti:
Equations (3):
3 standard: see equations(lti)
Unknowns (4): see unknowns(lti)
x₁(t)
x₂(t)
x₃(t)
u(t)
Parameters (3): see parameters(lti)
b₁
a
x_tShooting formulation
Twenty piecewise-constant input intervals share one lookup. The three unknown coefficients have no switching grid, so they remain constant over the horizon and share another lookup. Each coefficient keeps its own scalar vector buffer. Every two time units we lift all states, including the energy quadrature. Corleone adds their continuity equalities to the three terminal equalities.
step = 0.5
starts = collect(0.0:step:(t_inf - step))
dynopt = CorleoneDynamicOptProblem(
scalarize(lti), [], u => starts;
algorithm = Tsit5(), tspan = (0.0, t_inf),
shooting = collect(2.0:2.0:8.0), rng = MersenneTwister(42),
problem_kwargs = (; abstol = 1.0e-10, reltol = 1.0e-10),
)CorleoneDynamicOptProblem{ShootingLayer{}, Corleone.ParsedTrajectoryExpression{RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(Symbol("##samples#1006"),), CorleoneRuntimeGeneratedFunctionsExtension.var"#_RGF_ModTag", CorleoneRuntimeGeneratedFunctionsExtension.var"#_RGF_ModTag", (0x76320c4e, 0x79a56b6b, 0xa58fa0ae, 0xa351a2a1, 0x13979cc0), Expr}, Vector{Tuple{Symbol, Int64, Union{Nothing, Real}}}, Nothing}, Tuple{Corleone.PreparedTrajectoryConstraint{Corleone.ParsedTrajectoryExpression{RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(Symbol("##samples#1007"),), CorleoneRuntimeGeneratedFunctionsExtension.var"#_RGF_ModTag", CorleoneRuntimeGeneratedFunctionsExtension.var"#_RGF_ModTag", (0xbe25d8bc, 0x7cb46c51, 0x8d09c9e4, 0xd51b29da, 0x32b4cf47), Expr}, Vector{Tuple{Symbol, Int64, Union{Nothing, Real}}}, Nothing}, false}, Corleone.PreparedTrajectoryConstraint{Corleone.ParsedTrajectoryExpression{RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(Symbol("##samples#1008"),), CorleoneRuntimeGeneratedFunctionsExtension.var"#_RGF_ModTag", CorleoneRuntimeGeneratedFunctionsExtension.var"#_RGF_ModTag", (0xdbd34a46, 0x979c808b, 0x738c081a, 0x9a277a4d, 0xfaa3329e), Expr}, Vector{Tuple{Symbol, Int64, Union{Nothing, Real}}}, Nothing}, false}, Corleone.PreparedTrajectoryConstraint{Corleone.ParsedTrajectoryExpression{RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(Symbol("##samples#1009"),), CorleoneRuntimeGeneratedFunctionsExtension.var"#_RGF_ModTag", CorleoneRuntimeGeneratedFunctionsExtension.var"#_RGF_ModTag", (0x108e0e10, 0xc7bef6c8, 0x3a26b154, 0x50f151f0, 0xefd45409), Expr}, Vector{Tuple{Symbol, Int64, Union{Nothing, Real}}}, Nothing}, false}}, @NamedTuple{initialcondition::@NamedTuple{initialconditions::@NamedTuple{var"x₃(t)"::@NamedTuple{lookup::@NamedTuple{}, parameter::@NamedTuple{var"x₃(t)"::@NamedTuple{ps::Vector{Float64}}, var"x₂(t)"::@NamedTuple{ps::Vector{Float64}}, var"x₁(t)"::@NamedTuple{ps::Vector{Float64}}, var"var\"##quadrature#911\"(t)"::@NamedTuple{ps::Vector{Float64}}}}}}, parameter::@NamedTuple{parameters::@NamedTuple{b₁::@NamedTuple{lookup::@NamedTuple{}, parameter::@NamedTuple{b₁::@NamedTuple{ps::Vector{Float64}}, var"a[1]"::@NamedTuple{ps::Vector{Float64}}, var"a[2]"::@NamedTuple{ps::Vector{Float64}}, var"x_t[1]"::@NamedTuple{}, var"x_t[2]"::@NamedTuple{}, var"x_t[3]"::@NamedTuple{}}}, var"u(t)"::@NamedTuple{lookup::@NamedTuple{}, parameter::@NamedTuple{ps::Vector{Float64}}}}}}, @NamedTuple{initialcondition::@NamedTuple{initialconditions::@NamedTuple{var"x₃(t)"::@NamedTuple{lookup::@NamedTuple{collection::Dict{Float64, Int64}, inserted::Vector{Int64}, cache::Nothing}, parameter::@NamedTuple{var"x₃(t)"::@NamedTuple{}, var"x₂(t)"::@NamedTuple{}, var"x₁(t)"::@NamedTuple{}, var"var\"##quadrature#911\"(t)"::@NamedTuple{}}, active::Bool}}, active::Vector{Bool}, permutation::Vector{Int64}}, parameter::@NamedTuple{parameters::@NamedTuple{b₁::@NamedTuple{lookup::@NamedTuple{collection::Nothing, inserted::Vector{Int64}, cache::Nothing}, parameter::@NamedTuple{b₁::@NamedTuple{}, var"a[1]"::@NamedTuple{}, var"a[2]"::@NamedTuple{}, var"x_t[1]"::@NamedTuple{states::@NamedTuple{}, parameters::@NamedTuple{ps::Vector{Float64}}}, var"x_t[2]"::@NamedTuple{states::@NamedTuple{}, parameters::@NamedTuple{ps::Vector{Float64}}}, var"x_t[3]"::@NamedTuple{states::@NamedTuple{}, parameters::@NamedTuple{ps::Vector{Float64}}}}}, var"u(t)"::@NamedTuple{lookup::@NamedTuple{collection::Vector{Float64}, inserted::Vector{Int64}, cache::Nothing}, parameter::@NamedTuple{}}}, permutation::Vector{Int64}}, timestops::Vector{Float64}}, Dict{Symbol, Vector{Real}}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}}(ShootingLayer{}(nothing, SciMLBase.ODEProblem{Vector{Float64}, Tuple{Float64, Float64}, true, Vector{Float64}, SciMLBase.ODEFunction{true, SciMLBase.AutoDespecialize, ModelingToolkitBase.GeneratedFunctionWrapper{Tuple{2, 3, false}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:___mtkunknowns___, :__mtk_arg_2, :__argₛᵧₘ7628082562247908723), ModelingToolkitBase.var"#_RGF_ModTag", ModelingToolkitBase.var"#_RGF_ModTag", (0xa11de0fd, 0xd0d22996, 0xef560269, 0x8df2b13e, 0x5d1a9367), Nothing}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:__argₛᵧₘ1187374493630987813, :___mtkunknowns___, :__mtk_arg_2, :__argₛᵧₘ7628082562247908723), ModelingToolkitBase.var"#_RGF_ModTag", ModelingToolkitBase.var"#_RGF_ModTag", (0x85edc87a, 0x591dee3b, 0x76116080, 0xf678d492, 0x70f314aa), Nothing}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, ModelingToolkitBase.ObservedFunctionCache{ModelingToolkitBase.System, Nothing}, Nothing, ModelingToolkitBase.System, Union{Nothing, SciMLBase.OverrideInitData}, Union{Nothing, SciMLBase.ODENLStepData}}, Base.Pairs{Symbol, Any, Nothing, @NamedTuple{reltol::Float64, sensealg::SciMLBase.NoAD, abstol::Float64}}, SciMLBase.StandardODEProblem}(SciMLBase.ODEFunction{true, SciMLBase.AutoDespecialize, ModelingToolkitBase.GeneratedFunctionWrapper{Tuple{2, 3, false}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:___mtkunknowns___, :__mtk_arg_2, :__argₛᵧₘ7628082562247908723), ModelingToolkitBase.var"#_RGF_ModTag", ModelingToolkitBase.var"#_RGF_ModTag", (0xa11de0fd, 0xd0d22996, 0xef560269, 0x8df2b13e, 0x5d1a9367), Nothing}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:__argₛᵧₘ1187374493630987813, :___mtkunknowns___, :__mtk_arg_2, :__argₛᵧₘ7628082562247908723), ModelingToolkitBase.var"#_RGF_ModTag", ModelingToolkitBase.var"#_RGF_ModTag", (0x85edc87a, 0x591dee3b, 0x76116080, 0xf678d492, 0x70f314aa), Nothing}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, ModelingToolkitBase.ObservedFunctionCache{ModelingToolkitBase.System, Nothing}, Nothing, ModelingToolkitBase.System, Union{Nothing, SciMLBase.OverrideInitData}, Union{Nothing, SciMLBase.ODENLStepData}}(ModelingToolkitBase.GeneratedFunctionWrapper{Tuple{2, 3, false}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:___mtkunknowns___, :__mtk_arg_2, :__argₛᵧₘ7628082562247908723), ModelingToolkitBase.var"#_RGF_ModTag", ModelingToolkitBase.var"#_RGF_ModTag", (0xa11de0fd, 0xd0d22996, 0xef560269, 0x8df2b13e, 0x5d1a9367), Nothing}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:__argₛᵧₘ1187374493630987813, :___mtkunknowns___, :__mtk_arg_2, :__argₛᵧₘ7628082562247908723), ModelingToolkitBase.var"#_RGF_ModTag", ModelingToolkitBase.var"#_RGF_ModTag", (0x85edc87a, 0x591dee3b, 0x76116080, 0xf678d492, 0x70f314aa), Nothing}}(RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:___mtkunknowns___, :__mtk_arg_2, :__argₛᵧₘ7628082562247908723), ModelingToolkitBase.var"#_RGF_ModTag", ModelingToolkitBase.var"#_RGF_ModTag", (0xa11de0fd, 0xd0d22996, 0xef560269, 0x8df2b13e, 0x5d1a9367), Nothing}(nothing), RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:__argₛᵧₘ1187374493630987813, :___mtkunknowns___, :__mtk_arg_2, :__argₛᵧₘ7628082562247908723), ModelingToolkitBase.var"#_RGF_ModTag", ModelingToolkitBase.var"#_RGF_ModTag", (0x85edc87a, 0x591dee3b, 0x76116080, 0xf678d492, 0x70f314aa), Nothing}(nothing)), LinearAlgebra.UniformScaling{Bool}(true), nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, ModelingToolkitBase.ObservedFunctionCache{ModelingToolkitBase.System, Nothing}(Model lti:
Equations (4):
4 standard: see equations(lti)
Unknowns (4): see unknowns(lti)
x₃(t)
x₂(t)
x₁(t)
var"##quadrature#911"(t)
Parameters (7): see parameters(lti)
b₁
a[1]
a[2]
x_t[1]
x_t[2]
x_t[3]
u(t) [defaults to missing], Dict{Any, Any}(), false, false, ModelingToolkitBase, false, nothing), nothing, Model lti:
Equations (4):
4 standard: see equations(lti)
Unknowns (4): see unknowns(lti)
x₃(t)
x₂(t)
x₁(t)
var"##quadrature#911"(t)
Parameters (7): see parameters(lti)
b₁
a[1]
a[2]
x_t[1]
x_t[2]
x_t[3]
u(t) [defaults to missing], nothing, nothing), [0.0, 0.0, 0.0, 0.0], (0.0, 10.0), [0.5, -0.2, -0.1, 3.0, 1.0, -2.0, 0.0], Base.Pairs{Symbol, Any, Nothing, @NamedTuple{reltol::Float64, sensealg::SciMLBase.NoAD, abstol::Float64}}(:reltol => 1.0e-10, :sensealg => SciMLBase.NoAD(), :abstol => 1.0e-10), SciMLBase.StandardODEProblem()), OrdinaryDiffEqTsit5.Tsit5{typeof(OrdinaryDiffEqCore.trivial_limiter!), typeof(OrdinaryDiffEqCore.trivial_limiter!), FastBroadcast.Serial}(OrdinaryDiffEqCore.trivial_limiter!, OrdinaryDiffEqCore.trivial_limiter!, FastBroadcast.Serial()), SciMLBase.EnsembleSerial(), InitialConditions{}(nothing, SciMLStructures.ArrayRepack{Vector{Float64}}([0.0, 0.0, 0.0, 0.0]), (var"x₃(t)" = InitialCondition{}(Symbol("x₃(t)"), LookUp{}(Dict(2.0 => 1, 6.0 => 3, 4.0 => 2, 8.0 => 4), nothing), (var"x₃(t)" = ParameterCollection{}(Corleone.var"#118#119"{Tuple{Float64, Float64}, Float64}((-Inf, Inf), 0.0, Core.Box([2.0, 4.0, 6.0, 8.0])), 4), var"x₂(t)" = ParameterCollection{}(Corleone.var"#118#119"{Tuple{Float64, Float64}, Float64}((-Inf, Inf), 0.0, Core.Box([2.0, 4.0, 6.0, 8.0])), 4), var"x₁(t)" = ParameterCollection{}(Corleone.var"#118#119"{Tuple{Float64, Float64}, Float64}((-Inf, Inf), 0.0, Core.Box([2.0, 4.0, 6.0, 8.0])), 4), var"var\"##quadrature#911\"(t)" = ParameterCollection{}(Corleone.var"#118#119"{Tuple{Float64, Float64}, Float64}((-Inf, Inf), 0.0, Core.Box([2.0, 4.0, 6.0, 8.0])), 4))),), ([1, 2, 3, 4],)), Parameter{}(nothing, SciMLStructures.ArrayRepack{Vector{Float64}}([0.5, -0.2, -0.1, 3.0, 1.0, -2.0, 0.0]), (b₁ = PiecewiseConstant{}(:b₁, LookUp{}(nothing, nothing), (b₁ = ParameterCollection{}(Corleone.var"#173#174"{Vector{Float64}}([0.0], Core.Box((0.1, 1.0)), Core.Box(0.5)), 1), var"a[1]" = ParameterCollection{}(Corleone.var"#173#174"{Vector{Float64}}([0.0], Core.Box((-0.5, -0.05)), Core.Box(-0.2)), 1), var"a[2]" = ParameterCollection{}(Corleone.var"#173#174"{Vector{Float64}}([0.0], Core.Box((-0.5, -0.05)), Core.Box(-0.1)), 1), var"x_t[1]" = FixedLayer{}(ParameterCollection{}(Corleone.var"#173#174"{Vector{Float64}}([0.0], Core.Box((-Inf, Inf)), Core.Box(3.0)), 1)), var"x_t[2]" = FixedLayer{}(ParameterCollection{}(Corleone.var"#173#174"{Vector{Float64}}([0.0], Core.Box((-Inf, Inf)), Core.Box(1.0)), 1)), var"x_t[3]" = FixedLayer{}(ParameterCollection{}(Corleone.var"#173#174"{Vector{Float64}}([0.0], Core.Box((-Inf, Inf)), Core.Box(-2.0)), 1)))), var"u(t)" = PiecewiseConstant{}(Symbol("u(t)"), LookUp{}([0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0, 5.5, 6.0, 6.5, 7.0, 7.5, 8.0, 8.5, 9.0, 9.5], nothing), ParameterCollection{}(Corleone.var"#173#174"{Vector{Float64}}([0.0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0, 5.5, 6.0, 6.5, 7.0, 7.5, 8.0, 8.5, 9.0, 9.5], Core.Box((-3.0, 3.0)), Core.Box(0.0)), 20))), ([1, 2, 3, 4, 5, 6], [7])), ControlSymbolCache{}(Model lti:
Equations (4):
4 standard: see equations(lti)
Unknowns (4): see unknowns(lti)
x₃(t)
x₂(t)
x₁(t)
var"##quadrature#911"(t)
Parameters (7): see parameters(lti)
b₁
a[1]
a[2]
x_t[1]
x_t[2]
x_t[3]
u(t) [defaults to missing], SymbolicUtils.BasicSymbolicImpl.var"typeof(BasicSymbolicImpl)"{SymbolicUtils.SymReal}[u(t)], SymbolicUtils.BasicSymbolicImpl.var"typeof(BasicSymbolicImpl)"{SymbolicUtils.SymReal}[var"##quadrature#911"(t)], Dict{Any, Any}(var"##quadrature#911"(t) => (-Inf, Inf), x_t[2] => (-Inf, Inf), x₁(t) => (-Inf, Inf), x_t[1] => (-Inf, Inf), u(t) => (-3.0, 3.0), a[2] => (-0.5, -0.05), x₂(t) => (-Inf, Inf), x_t[3] => (-Inf, Inf), a[1] => (-0.5, -0.05), b₁ => (0.1, 1.0), x₃(t) => (-Inf, Inf)))), Corleone.ParsedTrajectoryExpression{RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(Symbol("##samples#1006"),), CorleoneRuntimeGeneratedFunctionsExtension.var"#_RGF_ModTag", CorleoneRuntimeGeneratedFunctionsExtension.var"#_RGF_ModTag", (0x76320c4e, 0x79a56b6b, 0xa58fa0ae, 0xa351a2a1, 0x13979cc0), Expr}, Vector{Tuple{Symbol, Int64, Union{Nothing, Real}}}, Nothing}(RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(Symbol("##samples#1006"),), CorleoneRuntimeGeneratedFunctionsExtension.var"#_RGF_ModTag", CorleoneRuntimeGeneratedFunctionsExtension.var"#_RGF_ModTag", (0x76320c4e, 0x79a56b6b, 0xa58fa0ae, 0xa351a2a1, 0x13979cc0), Expr}(:((+)(Base.getindex((var"##samples#1006").states, 4, 25), (*)(-1, Base.getindex((var"##samples#1006").states, 4, 1))))), Tuple{Symbol, Int64, Union{Nothing, Real}}[(:states, 4, 10.0), (:states, 4, 0.0)], nothing), (Corleone.PreparedTrajectoryConstraint{Corleone.ParsedTrajectoryExpression{RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(Symbol("##samples#1007"),), CorleoneRuntimeGeneratedFunctionsExtension.var"#_RGF_ModTag", CorleoneRuntimeGeneratedFunctionsExtension.var"#_RGF_ModTag", (0xbe25d8bc, 0x7cb46c51, 0x8d09c9e4, 0xd51b29da, 0x32b4cf47), Expr}, Vector{Tuple{Symbol, Int64, Union{Nothing, Real}}}, Nothing}, false}(Corleone.ParsedTrajectoryExpression{RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(Symbol("##samples#1007"),), CorleoneRuntimeGeneratedFunctionsExtension.var"#_RGF_ModTag", CorleoneRuntimeGeneratedFunctionsExtension.var"#_RGF_ModTag", (0xbe25d8bc, 0x7cb46c51, 0x8d09c9e4, 0xd51b29da, 0x32b4cf47), Expr}, Vector{Tuple{Symbol, Int64, Union{Nothing, Real}}}, Nothing}(RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(Symbol("##samples#1007"),), CorleoneRuntimeGeneratedFunctionsExtension.var"#_RGF_ModTag", CorleoneRuntimeGeneratedFunctionsExtension.var"#_RGF_ModTag", (0xbe25d8bc, 0x7cb46c51, 0x8d09c9e4, 0xd51b29da, 0x32b4cf47), Expr}(:((+)(Base.getindex((var"##samples#1007").states, 3, 25), (*)(-1, Base.getindex((var"##samples#1007").parameters, 4, 25))))), Tuple{Symbol, Int64, Union{Nothing, Real}}[(:states, 3, 10.0), (:parameters, 4, 10.0)], nothing), 1), Corleone.PreparedTrajectoryConstraint{Corleone.ParsedTrajectoryExpression{RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(Symbol("##samples#1008"),), CorleoneRuntimeGeneratedFunctionsExtension.var"#_RGF_ModTag", CorleoneRuntimeGeneratedFunctionsExtension.var"#_RGF_ModTag", (0xdbd34a46, 0x979c808b, 0x738c081a, 0x9a277a4d, 0xfaa3329e), Expr}, Vector{Tuple{Symbol, Int64, Union{Nothing, Real}}}, Nothing}, false}(Corleone.ParsedTrajectoryExpression{RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(Symbol("##samples#1008"),), CorleoneRuntimeGeneratedFunctionsExtension.var"#_RGF_ModTag", CorleoneRuntimeGeneratedFunctionsExtension.var"#_RGF_ModTag", (0xdbd34a46, 0x979c808b, 0x738c081a, 0x9a277a4d, 0xfaa3329e), Expr}, Vector{Tuple{Symbol, Int64, Union{Nothing, Real}}}, Nothing}(RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(Symbol("##samples#1008"),), CorleoneRuntimeGeneratedFunctionsExtension.var"#_RGF_ModTag", CorleoneRuntimeGeneratedFunctionsExtension.var"#_RGF_ModTag", (0xdbd34a46, 0x979c808b, 0x738c081a, 0x9a277a4d, 0xfaa3329e), Expr}(:((+)(Base.getindex((var"##samples#1008").states, 2, 25), (*)(-1, Base.getindex((var"##samples#1008").parameters, 5, 25))))), Tuple{Symbol, Int64, Union{Nothing, Real}}[(:states, 2, 10.0), (:parameters, 5, 10.0)], nothing), 1), Corleone.PreparedTrajectoryConstraint{Corleone.ParsedTrajectoryExpression{RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(Symbol("##samples#1009"),), CorleoneRuntimeGeneratedFunctionsExtension.var"#_RGF_ModTag", CorleoneRuntimeGeneratedFunctionsExtension.var"#_RGF_ModTag", (0x108e0e10, 0xc7bef6c8, 0x3a26b154, 0x50f151f0, 0xefd45409), Expr}, Vector{Tuple{Symbol, Int64, Union{Nothing, Real}}}, Nothing}, false}(Corleone.ParsedTrajectoryExpression{RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(Symbol("##samples#1009"),), CorleoneRuntimeGeneratedFunctionsExtension.var"#_RGF_ModTag", CorleoneRuntimeGeneratedFunctionsExtension.var"#_RGF_ModTag", (0x108e0e10, 0xc7bef6c8, 0x3a26b154, 0x50f151f0, 0xefd45409), Expr}, Vector{Tuple{Symbol, Int64, Union{Nothing, Real}}}, Nothing}(RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(Symbol("##samples#1009"),), CorleoneRuntimeGeneratedFunctionsExtension.var"#_RGF_ModTag", CorleoneRuntimeGeneratedFunctionsExtension.var"#_RGF_ModTag", (0x108e0e10, 0xc7bef6c8, 0x3a26b154, 0x50f151f0, 0xefd45409), Expr}(:((+)(Base.getindex((var"##samples#1009").states, 1, 25), (*)(-1, Base.getindex((var"##samples#1009").parameters, 6, 25))))), Tuple{Symbol, Int64, Union{Nothing, Real}}[(:states, 1, 10.0), (:parameters, 6, 10.0)], nothing), 1)), (initialcondition = (initialconditions = (var"x₃(t)" = (lookup = NamedTuple(), parameter = (var"x₃(t)" = (ps = [0.0, 0.0, 0.0, 0.0],), var"x₂(t)" = (ps = [0.0, 0.0, 0.0, 0.0],), var"x₁(t)" = (ps = [0.0, 0.0, 0.0, 0.0],), var"var\"##quadrature#911\"(t)" = (ps = [0.0, 0.0, 0.0, 0.0],))),),), parameter = (parameters = (b₁ = (lookup = NamedTuple(), parameter = (b₁ = (ps = [0.5],), var"a[1]" = (ps = [-0.2],), var"a[2]" = (ps = [-0.1],), var"x_t[1]" = NamedTuple(), var"x_t[2]" = NamedTuple(), var"x_t[3]" = NamedTuple())), var"u(t)" = (lookup = NamedTuple(), parameter = (ps = [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0],))),)), (initialcondition = (initialconditions = (var"x₃(t)" = (lookup = (collection = Dict(2.0 => 1, 6.0 => 3, 4.0 => 2, 8.0 => 4), inserted = Int64[], cache = nothing), parameter = (var"x₃(t)" = NamedTuple(), var"x₂(t)" = NamedTuple(), var"x₁(t)" = NamedTuple(), var"var\"##quadrature#911\"(t)" = NamedTuple()), active = false),), active = Bool[0, 0, 0, 0], permutation = [1, 2, 3, 4]), parameter = (parameters = (b₁ = (lookup = (collection = nothing, inserted = Int64[], cache = nothing), parameter = (b₁ = NamedTuple(), var"a[1]" = NamedTuple(), var"a[2]" = NamedTuple(), var"x_t[1]" = (states = NamedTuple(), parameters = (ps = [3.0],)), var"x_t[2]" = (states = NamedTuple(), parameters = (ps = [1.0],)), var"x_t[3]" = (states = NamedTuple(), parameters = (ps = [-2.0],)))), var"u(t)" = (lookup = (collection = [0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0, 5.5, 6.0, 6.5, 7.0, 7.5, 8.0, 8.5, 9.0, 9.5], inserted = Int64[], cache = nothing), parameter = NamedTuple())), permutation = [1, 2, 3, 4, 5, 6, 7]), timestops = [0.0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0, 5.5, 6.0, 6.5, 7.0, 7.5, 8.0, 8.5, 9.0, 9.5, 10.0]), Dict{Symbol, Vector{Real}}(Symbol("x₁(t)") => [10.0], Symbol("x₂(t)") => [10.0], Symbol("x_t[1]") => [10.0], Symbol("x_t[3]") => [10.0], Symbol("var\"##quadrature#911\"(t)") => [0.0, 10.0], Symbol("x_t[2]") => [10.0], Symbol("x₃(t)") => [10.0]), [0.0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0, 5.5, 6.0, 6.5, 7.0, 7.5, 8.0, 8.5, 9.0, 9.5, 10.0], [0.0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0, 5.5, 6.0, 6.5, 7.0, 7.5, 8.0, 8.5, 9.0, 9.5, 10.0], [0.0, 0.5, 1.0, 1.5, 2.0, 2.0, 2.5, 3.0, 3.5, 4.0, 4.0, 4.5, 5.0, 5.5, 6.0, 6.0, 6.5, 7.0, 7.5, 8.0, 8.0, 8.5, 9.0, 9.5, 10.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], 16)Feasible initialization
At the nominal coefficients, an exact zero-order-hold discretization gives $x_{k+1}=\Phi x_k+\Gamma u_k$. Its reachability matrix supplies a minimum-norm input that reaches the target. This initializes the control without changing the layer or its parameter layout. Forward initialization fills the lifted states and quadrature from that input.
A_nominal = [0.0 1.0 0.0; -0.2 0.0 1.0; 0.0 0.0 -0.1]
b_nominal = [0.5, 0.0, 1.0]
@assert rank(hcat(b_nominal, A_nominal * b_nominal, A_nominal^2 * b_nominal)) == 3
transition = exp([A_nominal b_nominal; zeros(1, 4)] * step)
Φ, Γ = transition[1:3, 1:3], transition[1:3, 4]
N = length(starts)
reachability = hcat([Φ^(N - k) * Γ for k in 1:N]...)
control_guess = reachability \ ModelingToolkit.getdefault(x_t)
@assert maximum(abs, control_guess) < 3.0
input_name = Symbol(u)
dynopt.ps.parameter.parameters[input_name].parameter.ps .= control_guess
forward_initialization(MersenneTwister(42), dynopt.layer, dynopt.ps, dynopt.st)
initial_energy = optimization_objective(dynopt)
@assert maximum(abs, optimization_constraints(dynopt)) < 1.0e-7Joint optimization
The target alone does not identify unique coefficients. We treat them as design variables: their bounds and the energy objective define the joint optimization. We keep them shared between shooting segments; no parameter splitting is used.
optprob = OptimizationProblem(dynopt, AutoForwardDiff())
result = solve(
optprob, Ipopt.Optimizer(); max_iter = 500, tol = 1.0e-7,
hessian_approximation = "limited-memory", bound_relax_factor = 0.0, print_level = 0,
)
@assert successful_retcode(result)
_, reconstruct = pack(default_vectorizer(), dynopt.ps)
optimized = reconstruct(result.u)
trajectory, _ = dynopt.layer(nothing, optimized, optprob.p)
terminal_state = [last(trajectory[state]) for state in x]
coefficients = (; A₂₁ = last(trajectory[a[1]]), A₃₃ = last(trajectory[a[2]]), b₁ = last(trajectory[b₁]))
@assert isapprox(terminal_state, ModelingToolkit.getdefault(x_t); atol = 1.0e-5, rtol = 0.0)
@assert maximum(abs, optimization_constraints(dynopt, optimized, optprob.p)) < 1.0e-5
@assert result.objective <= initial_energy + 1.0e-6Replaying the fitted constant system provides a check independent of lifted states.
A_fitted = [0.0 1.0 0.0; coefficients.A₂₁ 0.0 1.0; 0.0 0.0 coefficients.A₃₃]
b_fitted = [coefficients.b₁, 0.0, 1.0]
fitted_transition = exp([A_fitted b_fitted; zeros(1, 4)] * step)
Φ_fitted, Γ_fitted = fitted_transition[1:3, 1:3], fitted_transition[1:3, 4]
replayed_terminal = foldl(
(state, input) -> Φ_fitted * state + Γ_fitted * input,
trajectory[u]; init = zeros(3)
)
@assert isapprox(replayed_terminal, ModelingToolkit.getdefault(x_t); atol = 1.0e-5, rtol = 0.0)
(; coefficients, terminal_state, energy = result.objective)(coefficients = (A₂₁ = -0.4082192613181105, A₃₃ = -0.05000000000344472, b₁ = 0.10000000001017893), terminal_state = [3.0000000000001674, 1.000000000000076, -2.0000000000000013], energy = 1.4685725855497465)Plot
Plot the three physical states and the scalar input. The energy is reported above.
using CairoMakie
trajectoryplot(
trajectory; skips = quadrature_symbols(dynopt.layer.sys),
state_string = "x", control_string = "u", show_variable_residuals = false
)
Appendix
This example was built using these direct dependencies,
using Pkg
Pkg.status()Status `~/work/Corleone.jl/Corleone.jl/docs/Project.toml`
[13f3f980] CairoMakie v0.15.17
[b0b7db55] ComponentArrays v0.15.50
[2751e0db] Corleone v2.0.0 `~/work/Corleone.jl/Corleone.jl`
[6590a4b1] CorleoneOED v2.0.0 `../lib/CorleoneOED`
[e30172f5] Documenter v1.19.0
[daee34ce] DocumenterCitations v1.5.0
[f6369f11] ForwardDiff v1.4.6
[b6b21f68] Ipopt v1.16.0
[98b081ad] Literate v2.21.0
[bb33d45b] LuxCore v1.5.4
[961ee093] ModelingToolkit v11.46.1
[d147904a] OptimalControlBenchmarks v0.1.0 `../lib/OptimalControlBenchmarks`
[7f7a1694] Optimization v5.9.1
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[9a3f8284] Random v1.11.0
and using this machine and Julia version.
using InteractiveUtils
InteractiveUtils.versioninfo()Julia Version 1.13.3
Commit 67ac47b9cd8 (2026-10-25 15:28 UTC)
Build Info:
Official https://julialang.org release
Platform Info:
OS: Linux (x86_64-linux-gnu)
CPU: 4 × AMD EPYC 7763 64-Core Processor
WORD_SIZE: 64
LLVM: libLLVM-20.1.8 (ORCJIT, znver3)
GC: Built with stock GC
Threads: 1 default, 1 interactive, 1 GC (on 4 virtual cores)
A more complete overview of all dependencies and their versions is also provided.
using Pkg
Pkg.status(; mode = PKGMODE_MANIFEST)Status `~/work/Corleone.jl/Corleone.jl/docs/Manifest.toml`
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[2efcf032] SymbolicIndexingInterface v0.3.55
[19f23fe9] SymbolicLimits v1.2.2
[d1185830] SymbolicUtils v4.54.0
[0c5d862f] Symbolics v7.48.0
[3783bdb8] TableTraits v1.0.1
[bd369af6] Tables v1.14.0
[ed4db957] TaskLocalValues v0.1.3
[62fd8b95] TensorCore v0.1.1
[8ea1fca8] TermInterface v2.0.1
[5d786b92] TerminalLoggers v0.1.8
[1c621080] TestItems v1.1.0
[731e570b] TiffImages v0.11.9
[a759f4b9] TimerOutputs v1.2.2
[3bb67fe8] TranscodingStreams v0.11.3
[981d1d27] TriplotBase v0.1.0
[781d530d] TruncatedStacktraces v1.4.0
[5c2747f8] URIs v1.7.3
[3a884ed6] UnPack v1.0.2
[1cfade01] UnicodeFun v0.4.1
[1986cc42] Unitful v1.29.0
[d30d5f5c] WeakCacheSets v0.1.0
[e3aaa7dc] WebP v0.1.3
[efce3f68] WoodburyMatrices v1.1.0
[ddb6d928] YAML v0.4.17
[ae81ac8f] ASL_jll v0.1.5+0
[6e34b625] Bzip2_jll v1.0.9+0
[4e9b3aee] CRlibm_jll v1.0.1+0
[83423d85] Cairo_jll v1.18.8+1
[a38c48d9] CoreMath_jll v0.1.0+0
⌅ [5ae413db] EarCut_jll v2.2.4+0
[2e619515] Expat_jll v2.10.0+0
[b22a6f82] FFMPEG_jll v9.0.2+0
[a3f928ae] Fontconfig_jll v2.18.3+0
⌅ [d7e528f0] FreeType2_jll v2.14.3+1
[559328eb] FriBidi_jll v1.0.19+0
⌅ [b0724c58] GettextRuntime_jll v0.22.4+0
[59f7168a] Giflib_jll v6.1.3+0
[020c3dae] Git_LFS_jll v3.8.1+0
[f8c6e375] Git_jll v2.56.0+0
[7746bdde] Glib_jll v2.90.1+0
[3b182d85] Graphite2_jll v1.3.16+0
[2e76f6c2] HarfBuzz_jll v100.15000.0+0
[e33a78d0] Hwloc_jll v2.15.0+0
[905a6f67] Imath_jll v3.2.2+0
[1d5cc7b8] IntelOpenMP_jll v2025.2.0+0
⌅ [9cc047cb] Ipopt_jll v300.1400.1902+0
[aacddb02] JpegTurbo_jll v3.2.1+0
[c1c5ebd0] LAME_jll v3.100.3+0
[88015f11] LERC_jll v4.3.0+0
[1d63c593] LLVMOpenMP_jll v23.1.1+0
[81d17ec3] L_BFGS_B_jll v3.0.1+0
⌅ [e9f186c6] Libffi_jll v3.4.7+0
[7e76a0d4] Libglvnd_jll v1.7.1+1
[94ce4f54] Libiconv_jll v1.18.0+0
[4b2f31a3] Libmount_jll v2.42.4+0
[89763e89] Libtiff_jll v4.7.3+0
[38a345b3] Libuuid_jll v2.42.4+0
[d00139f3] METIS_jll v5.1.4+0
[856f044c] MKL_jll v2025.2.0+0
[d7ed1dd3] MUMPS_seq_jll v500.900.100+0
[e7412a2a] Ogg_jll v1.3.6+0
[656ef2d0] OpenBLAS32_jll v0.3.34+1
[6cdc7f73] OpenBLASConsistentFPCSR_jll v0.3.34+0
[18a262bb] OpenEXR_jll v3.5.3+0
⌅ [95196869] OpenJPH_jll v0.32.0+0
[9bd350c2] OpenSSH_jll v10.6.1+0
[efe28fd5] OpenSpecFun_jll v0.5.6+0
[91d4177d] Opus_jll v1.6.1+0
[36c8627f] Pango_jll v1.58.2+0
[30392449] Pixman_jll v0.46.4+0
[f50d1b31] Rmath_jll v0.5.2+0
[319450e9] SPRAL_jll v2025.9.18+1
[02c8fc9c] XML2_jll v2.15.4+0
[ffd25f8a] XZ_jll v5.8.4+0
[4f6342f7] Xorg_libX11_jll v1.8.14+0
[0c0b7dd1] Xorg_libXau_jll v1.0.13+0
[a3789734] Xorg_libXdmcp_jll v1.1.6+0
[1082639a] Xorg_libXext_jll v1.3.8+0
[d091e8ba] Xorg_libXfixes_jll v6.0.2+0
[ea2f1a96] Xorg_libXrender_jll v0.9.13+0
[a65dc6b1] Xorg_libpciaccess_jll v0.19.0+0
[c7cfdc94] Xorg_libxcb_jll v1.17.1+0
[c5fb5394] Xorg_xtrans_jll v1.6.0+0
[9a68df92] isoband_jll v0.2.3+0
[a4ae2306] libaom_jll v3.15.3+0
[0ac62f75] libass_jll v0.17.5+0
[46979653] libdeflate_jll v1.26.0+1
[8e53e030] libdrm_jll v2.4.134+0
⌅ [f638f0a6] libfdk_aac_jll v2.0.4+0
[b53b4c65] libpng_jll v1.6.59+1
[075b6546] libsixel_jll v1.80.702+0
[9a156e7d] libva_jll v2.24.1+0
[f27f6e37] libvorbis_jll v1.3.8+0
[c5f90fcd] libwebp_jll v1.6.1+0
[1317d2d5] oneTBB_jll v2023.1.0+0
⌅ [1270edf5] x264_jll v10164.0.1+0
⌅ [dfaa095f] x265_jll v4.1.0+0
[0dad84c5] ArgTools v1.1.2
[56f22d72] Artifacts v1.11.0
[2a0f44e3] Base64 v1.11.0
[8bf52ea8] CRC32c v1.11.0
[ade2ca70] Dates v1.11.0
[8ba89e20] Distributed v1.11.0
[f43a241f] Downloads v1.7.0
[7b1f6079] FileWatching v1.11.0
[9fa8497b] Future v1.11.0
[b77e0a4c] InteractiveUtils v1.11.0
[ac6e5ff7] JuliaSyntaxHighlighting v1.12.0
[4af54fe1] LazyArtifacts v1.11.0
[b27032c2] LibCURL v1.0.0
[76f85450] LibGit2 v1.11.0
[8f399da3] Libdl v1.11.0
[37e2e46d] LinearAlgebra v1.13.0
[56ddb016] Logging v1.11.0
[d6f4376e] Markdown v1.11.0
[a63ad114] Mmap v1.11.0
[ca575930] NetworkOptions v1.3.0
[44cfe95a] Pkg v1.13.0
[de0858da] Printf v1.11.0
[3fa0cd96] REPL v1.11.0
[9a3f8284] Random v1.11.0
[ea8e919c] SHA v1.0.0
[9e88b42a] Serialization v1.11.0
[1a1011a3] SharedArrays v1.11.0
[6462fe0b] Sockets v1.11.0
[2f01184e] SparseArrays v1.13.0
[f489334b] StyledStrings v1.11.0
[4607b0f0] SuiteSparse
[fa267f1f] TOML v1.0.3
[a4e569a6] Tar v1.10.0
[8dfed614] Test v1.11.0
[cf7118a7] UUIDs v1.11.0
[4ec0a83e] Unicode v1.11.0
[e66e0078] CompilerSupportLibraries_jll v1.5.5+2
[deac9b47] LibCURL_jll v8.22.0+0
[e37daf67] LibGit2_jll v1.9.7+0
[29816b5a] LibSSH2_jll v1.11.104+0
[14a3606d] MozillaCACerts_jll v2026.10.20
[4536629a] OpenBLAS_jll v0.3.30+0
[05823500] OpenLibm_jll v0.8.7+0
[458c3c95] OpenSSL_jll v3.5.8+0
[efcefdf7] PCRE2_jll v10.47.0+0
[bea87d4a] SuiteSparse_jll v7.10.1+0
[83775a58] Zlib_jll v1.3.2+0
[3161d3a3] Zstd_jll v1.5.7+1
[8e850b90] libblastrampoline_jll v5.15.0+0
[8e850ede] nghttp2_jll v1.70.0+0
[3f19e933] p7zip_jll v17.8.3+0
Info Packages marked with ⌅ have new versions available but compatibility constraints restrict them from upgrading. To see why use `status --outdated -m`
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