ModelingToolkit.jl
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Published
20 hours ago •
SciML
SciML
Linearization of improper (differentiating / inverse) models
Describe the bug 🐞
Consider the well-known, academic "Four Tank" model. Simulation of the input-output (valve-signal to output levels, case = :io in MRE) works. Likewise, linearization (symbolic and numeric) of the input-output model works (case = :lin_io in MRE). Also simulation of the zero dynamics works (specifying the outputs; case = :zero in MRE).
HOWEVER, both symbolic and numeric linearization fails for the zero dynamics case (case = :lin_zero in MRE).
Expected behavior
Both symbolic and numeric linearization of the zero dynamics case (case = :lin_zero) should work. The eigenvalues of the zero dynamics system matrix should equal the transmission zeros of case = :lin_io.
Minimal Reproducible Example 👇
Here is my case, set up with both symbolic and numeric linearization of the zero dynamics.
# Packages
using ModelingToolkit
using ModelingToolkit: t_nounits as t, D_nounits as Dt
using DifferentialEquations
#
# Input functions
# -- control inputs
@register_symbolic u_u1(t)
@register_symbolic u_u2(t)
#
u_v_1(t) = 3.0
u_v_2(t) = 3.0
#
# -- output references
@register_symbolic u_r1(t)
@register_symbolic u_r2(t)
#
u_r_1(t) = 12.4*0.5
u_r_2(t) = 12.7*0.5
#
# Model FourTank
@mtkmodel FourTank begin
# Ref: Johansson, K.H. (2000): "The Quadruple-Tank Process: A Multivariable
# Laboratory Process with an Adjustable Zero",
# IEEE Trans. Ctrl. Tech., Vol. 8, No. 3, May 2000
# Structure parameters
@structural_parameters begin
# :io = standard input-output, :zero = zero dynamics
# :lin_io = linearizing input-output, :lin_zero = linearizing zero dynamics
case = :io
end
#
# Model parameters
@parameters begin
# Constant
g = 981, [description = "Acceleration of gravity, cm/s^2"]
# Parameters
# -- tank parameters
A_1 = 28, [description = "Tank 1, cross-section area, cm^2"]
A_2 = 32, [description = "Tank 2, cross-section area, cm^2"]
A_3 = 28, [description = "Tank 3, cross-section area, cm^2"]
A_4 = 32, [description = "Tank 4, cross-section area, cm^2"]
a_1 = 0.071, [description = "Tank 1, cross-section area of outlet hole, cm^2"]
a_2 = 0.057, [description = "Tank 2, cross-section area of outlet hole, cm^2"]
a_3 = 0.071, [description = "Tank 3, cross-section area of outlet hole, cm^2"]
a_4 = 0.057, [description = "Tank 4, cross-section area of outlet hole, cm^2"]
# -- valve parameters, case of LHP zeros/stable zero dynamics
γ_1 = 0.70, [description = "Valve 1 split setting, -"]
γ_2 = 0.60, [description = "Valve 2 split setting, -"]
k_1 = 3.33, [description = "Valve 1 gain, cm3/V.s"]
k_2 = 3.35, [description = "Valve 2 gain, cm3/V.s"]
# -- sensor parameter
k_c = 0.5, [description = "Sensor gain, V/cm"]
end
#
# Model variables, with initial values needed
@variables begin
# Variables
# -- states
h_1(t)=12.4, [description = "Tank 1 level, cm"]
h_2(t)=12.7, [description = "Tank 2 level, cm"]
h_3(t)=1.8, [description = "Tank 3 level, cm"]
h_4(t)=1.4, [description = "Tank 4 level, cm"]
# -- sensor signals
y_1(t), [description = "Output 1: level in Tank 1, cm"]
y_2(t), [description = "Output 2: level in Tank 2, cm"]
# -- actuator signals
v_1(t), [description = "Valve 1 signal, V"]
v_2(t), [description = "Valve 2 signal, V"]
end
#
# Model equations
@equations begin
# ODEs
Dt(h_1) ~ -a_1/A_1*sqrt(2g*h_1) + a_3/A_1*sqrt(2g*h_3) + γ_1*k_1/A_1*v_1
Dt(h_2) ~ -a_2/A_2*sqrt(2g*h_2) + a_4/A_2*sqrt(2g*h_4) + γ_2*k_2/A_2*v_2
Dt(h_3) ~ -a_3/A_3*sqrt(2g*h_3) + (1-γ_2)*k_2/A_3*v_2
Dt(h_4) ~ -a_4/A_4*sqrt(2g*h_4) + (1-γ_1)*k_1/A_4*v_1
# Outputs
y_1 ~ k_c*h_1
y_2 ~ k_c*h_2
# Inputs
if case == :io
v_1 ~ u_v_1(t)
v_2 ~ u_v_2(t)
elseif case == :lin_io
#v_1 ~ u_v_1(t)
#v_2 ~ u_v_2(t)
elseif case == :zero
y_1 ~ u_r_1(t)
y_2 ~ u_r_2(t)
elseif case == :lin_zero
#y_1 ~ u_r_1(t)
#y_2 ~ u_r_2(t)
end
end
end
#
# Instantiating case for linearizing zero dynamics
@named t4_LZ= FourTank(; case=:lin_zero)
t4_LZ= complete(t4_LZ)
#
# Symbolic linearization -- gives error
mats,t4z_ = ModelingToolkit.linearize_symbolic(t4_LZ, [t4_LZ.y_1, t4_LZ.y_2], [t4_LZ.h_3, t4_LZ.h_4])
#
# Numeric linearization -- gives error
mats,t4z_ = ModelingToolkit.linearize(t4_LZ, [t4_LZ.y_1, t4_LZ.y_2], [t4_LZ.h_3, t4_LZ.h_4];
op=Dict(t4_LZ.y_1 => 6.2, t4_LZ.y_2 => 6.35))
Error & Stacktrace ⚠️ Case of symbolic linearization
BoundsError: attempt to access 1-element Vector{Any} at index [2]
Stacktrace:
[1] getindex(A::Vector{Any}, i1::Int64)
@ Base .\essentials.jl:13
[2] expand_derivatives(O::SymbolicUtils.BasicSymbolic{Real}, simplify::Bool; occurrences::SymbolicUtils.BasicSymbolic{Real}) (repeats 5 times)
@ Symbolics C:\Users\Bernt_Lie\.julia\packages\Symbolics\6WqId\src\diff.jl:257
[3] expand_derivatives(O::SymbolicUtils.BasicSymbolic{Real}, simplify::Bool; occurrences::Nothing)
@ Symbolics C:\Users\Bernt_Lie\.julia\packages\Symbolics\6WqId\src\diff.jl:257
[4] expand_derivatives(O::SymbolicUtils.BasicSymbolic{Real}, simplify::Bool)
@ Symbolics C:\Users\Bernt_Lie\.julia\packages\Symbolics\6WqId\src\diff.jl:183
[5] jacobian(ops::Vector{SymbolicUtils.BasicSymbolic{Real}}, vars::Vector{Num}; simplify::Bool, scalarize::Bool)
@ Symbolics C:\Users\Bernt_Lie\.julia\packages\Symbolics\6WqId\src\diff.jl:470
[6] jacobian(ops::Vector{SymbolicUtils.BasicSymbolic{Real}}, vars::Vector{Num})
@ Symbolics C:\Users\Bernt_Lie\.julia\packages\Symbolics\6WqId\src\diff.jl:465
[7] linearize_symbolic(sys::ODESystem, inputs::Vector{Num}, outputs::Vector{Num}; simplify::Bool, allow_input_derivatives::Bool, eval_expression::Bool, eval_module::Module, kwargs::@Kwargs{})
@ ModelingToolkit C:\Users\Bernt_Lie\.julia\packages\ModelingToolkit\UXr3S\src\systems\abstractsystem.jl:2454
[8] linearize_symbolic(sys::ODESystem, inputs::Vector{Num}, outputs::Vector{Num})
@ ModelingToolkit C:\Users\Bernt_Lie\.julia\packages\ModelingToolkit\UXr3S\src\systems\abstractsystem.jl:2434
[9] top-level scope
@ c:\Users\Bernt_Lie\OneDrive\Documents\researchUSN\Supervisor\PhD\Kushila_Jayamanne\Notebooks\jl_notebook_cell_df34fa98e69747e1a8f8a730347b8e2f_X36sZmlsZQ==.jl:1
Case of (numeric) linearization:
{
"name": "ArgumentError",
"message": "ArgumentError: Cannot convert Sym to Float64 since Sym is symbolic and Float64 is concrete. Use `substitute` to replace the symbolic unwraps.",
"stack": "ArgumentError: Cannot convert Sym to Float64 since Sym is symbolic and Float64 is concrete. Use `substitute` to replace the symbolic unwraps.\n\nStacktrace:\n [1] Float64(x::SymbolicUtils.BasicSymbolic{ForwardDiff.Dual{ForwardDiff.Tag{SciMLBase.UJacobianWrapper{true, ODEFunction{true, SciMLBase.FullSpecialize, ModelingToolkit.var\"#f#815\"{RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋arg1, :ˍ₋arg2, :t), ModelingToolkit.var\"#_RGF_ModTag\", ModelingToolkit.var\"#_RGF_ModTag\", (0x68d7fc22, 0x9cd0330d, 0x51936f95, 0xda8a0357, 0xa6ac43de), Nothing}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :ˍ₋arg1, :ˍ₋arg2, :t), ModelingToolkit.var\"#_RGF_ModTag\", ModelingToolkit.var\"#_RGF_ModTag\", (0x077ceea2, 0x254bf500, 0x74c8eda1, 0x54c835c5, 0x5d3dd8d7), Nothing}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, ModelingToolkit.ObservedFunctionCache{ODESystem}, Nothing, ODESystem, Nothing, Nothing}, Float64, MTKParameters{Vector{Float64}, Tuple{}, Tuple{}, Tuple{}}}, Float64}, Float64, 2}})\n @ Symbolics C:\\Users\\Bernt_Lie\\.julia\\packages\\Symbolics\\6WqId\\src\\Symbolics.jl:171\n [2] convert\n @ C:\\Users\\Bernt_Lie\\.julia\\packages\\ForwardDiff\\PcZ48\\src\\dual.jl:435 [inlined]\n [3] setindex!(A::Vector{ForwardDiff.Dual{ForwardDiff.Tag{SciMLBase.UJacobianWrapper{true, ODEFunction{true, SciMLBase.FullSpecialize, ModelingToolkit.var\"#f#815\"{RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋arg1, :ˍ₋arg2, :t), ModelingToolkit.var\"#_RGF_ModTag\", ModelingToolkit.var\"#_RGF_ModTag\", (0x68d7fc22, 0x9cd0330d, 0x51936f95, 0xda8a0357, 0xa6ac43de), Nothing}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :ˍ₋arg1, :ˍ₋arg2, :t), ModelingToolkit.var\"#_RGF_ModTag\", ModelingToolkit.var\"#_RGF_ModTag\", (0x077ceea2, 0x254bf500, 0x74c8eda1, 0x54c835c5, 0x5d3dd8d7), Nothing}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, ModelingToolkit.ObservedFunctionCache{ODESystem}, Nothing, ODESystem, Nothing, Nothing}, Float64, MTKParameters{Vector{Float64}, Tuple{}, Tuple{}, Tuple{}}}, Float64}, Float64, 2}}, x::SymbolicUtils.BasicSymbolic{ForwardDiff.Dual{ForwardDiff.Tag{SciMLBase.UJacobianWrapper{true, ODEFunction{true, SciMLBase.FullSpecialize, ModelingToolkit.var\"#f#815\"{RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋arg1, :ˍ₋arg2, :t), ModelingToolkit.var\"#_RGF_ModTag\", ModelingToolkit.var\"#_RGF_ModTag\", (0x68d7fc22, 0x9cd0330d, 0x51936f95, 0xda8a0357, 0xa6ac43de), Nothing}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :ˍ₋arg1, :ˍ₋arg2, :t), ModelingToolkit.var\"#_RGF_ModTag\", ModelingToolkit.var\"#_RGF_ModTag\", (0x077ceea2, 0x254bf500, 0x74c8eda1, 0x54c835c5, 0x5d3dd8d7), Nothing}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, ModelingToolkit.ObservedFunctionCache{ODESystem}, Nothing, ODESystem, Nothing, Nothing}, Float64, MTKParameters{Vector{Float64}, Tuple{}, Tuple{}, Tuple{}}}, Float64}, Float64, 2}}, i1::Int64)\n @ Base .\\array.jl:1021\n [4] macro expansion\n @ C:\\Users\\Bernt_Lie\\.julia\\packages\\SymbolicUtils\\5UmB7\\src\\code.jl:430 [inlined]\n [5] macro expansion\n @ C:\\Users\\Bernt_Lie\\.julia\\packages\\Symbolics\\6WqId\\src\\build_function.jl:546 [inlined]\n [6] macro expansion\n @ C:\\Users\\Bernt_Lie\\.julia\\packages\\SymbolicUtils\\5UmB7\\src\\code.jl:387 [inlined]\n [7] macro expansion\n @ C:\\Users\\Bernt_Lie\\.julia\\packages\\RuntimeGeneratedFunctions\\M9ZX8\\src\\RuntimeGeneratedFunctions.jl:163 [inlined]\n [8] macro expansion\n @ .\\none:0 [inlined]\n [9] generated_callfunc\n @ .\\none:0 [inlined]\n [10] (::RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :ˍ₋arg1, :ˍ₋arg2, :t), ModelingToolkit.var\"#_RGF_ModTag\", ModelingToolkit.var\"#_RGF_ModTag\", (0x077ceea2, 0x254bf500, 0x74c8eda1, 0x54c835c5, 0x5d3dd8d7), Nothing})(::Vector{ForwardDiff.Dual{ForwardDiff.Tag{SciMLBase.UJacobianWrapper{true, ODEFunction{true, SciMLBase.FullSpecialize, ModelingToolkit.var\"#f#815\"{RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋arg1, :ˍ₋arg2, :t), ModelingToolkit.var\"#_RGF_ModTag\", ModelingToolkit.var\"#_RGF_ModTag\", (0x68d7fc22, 0x9cd0330d, 0x51936f95, 0xda8a0357, 0xa6ac43de), Nothing}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :ˍ₋arg1, :ˍ₋arg2, :t), ModelingToolkit.var\"#_RGF_ModTag\", ModelingToolkit.var\"#_RGF_ModTag\", (0x077ceea2, 0x254bf500, 0x74c8eda1, 0x54c835c5, 0x5d3dd8d7), Nothing}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, ModelingToolkit.ObservedFunctionCache{ODESystem}, Nothing, ODESystem, Nothing, Nothing}, Float64, MTKParameters{Vector{Float64}, Tuple{}, Tuple{}, Tuple{}}}, Float64}, Float64, 2}}, ::Vector{ForwardDiff.Dual{ForwardDiff.Tag{SciMLBase.UJacobianWrapper{true, ODEFunction{true, SciMLBase.FullSpecialize, ModelingToolkit.var\"#f#815\"{RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋arg1, :ˍ₋arg2, :t), ModelingToolkit.var\"#_RGF_ModTag\", ModelingToolkit.var\"#_RGF_ModTag\", (0x68d7fc22, 0x9cd0330d, 0x51936f95, 0xda8a0357, 0xa6ac43de), Nothing}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :ˍ₋arg1, :ˍ₋arg2, :t), ModelingToolkit.var\"#_RGF_ModTag\", ModelingToolkit.var\"#_RGF_ModTag\", (0x077ceea2, 0x254bf500, 0x74c8eda1, 0x54c835c5, 0x5d3dd8d7), Nothing}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, ModelingToolkit.ObservedFunctionCache{ODESystem}, Nothing, ODESystem, Nothing, Nothing}, Float64, MTKParameters{Vector{Float64}, Tuple{}, Tuple{}, Tuple{}}}, Float64}, Float64, 2}}, ::Vector{Float64}, ::Float64)\n @ RuntimeGeneratedFunctions C:\\Users\\Bernt_Lie\\.julia\\packages\\RuntimeGeneratedFunctions\\M9ZX8\\src\\RuntimeGeneratedFunctions.jl:150\n [11] f\n @ C:\\Users\\Bernt_Lie\\.julia\\packages\\ModelingToolkit\\UXr3S\\src\\systems\\diffeqs\\abstractodesystem.jl:351 [inlined]\n [12] ODEFunction\n @ C:\\Users\\Bernt_Lie\\.julia\\packages\\SciMLBase\\j8jQg\\src\\scimlfunctions.jl:2335 [inlined]\n [13] UJacobianWrapper\n @ C:\\Users\\Bernt_Lie\\.julia\\packages\\SciMLBase\\j8jQg\\src\\function_wrappers.jl:34 [inlined]\n [14] vector_mode_dual_eval!\n @ C:\\Users\\Bernt_Lie\\.julia\\packages\\ForwardDiff\\PcZ48\\src\\apiutils.jl:24 [inlined]\n [15] vector_mode_jacobian(f::SciMLBase.UJacobianWrapper{true, ODEFunction{true, SciMLBase.FullSpecialize, ModelingToolkit.var\"#f#815\"{RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋arg1, :ˍ₋arg2, :t), ModelingToolkit.var\"#_RGF_ModTag\", ModelingToolkit.var\"#_RGF_ModTag\", (0x68d7fc22, 0x9cd0330d, 0x51936f95, 0xda8a0357, 0xa6ac43de), Nothing}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :ˍ₋arg1, :ˍ₋arg2, :t), ModelingToolkit.var\"#_RGF_ModTag\", ModelingToolkit.var\"#_RGF_ModTag\", (0x077ceea2, 0x254bf500, 0x74c8eda1, 0x54c835c5, 0x5d3dd8d7), Nothing}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, ModelingToolkit.ObservedFunctionCache{ODESystem}, Nothing, ODESystem, Nothing, Nothing}, Float64, MTKParameters{Vector{Float64}, Tuple{}, Tuple{}, Tuple{}}}, x::Vector{Float64}, cfg::ForwardDiff.JacobianConfig{ForwardDiff.Tag{SciMLBase.UJacobianWrapper{true, ODEFunction{true, SciMLBase.FullSpecialize, ModelingToolkit.var\"#f#815\"{RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋arg1, :ˍ₋arg2, :t), ModelingToolkit.var\"#_RGF_ModTag\", ModelingToolkit.var\"#_RGF_ModTag\", (0x68d7fc22, 0x9cd0330d, 0x51936f95, 0xda8a0357, 0xa6ac43de), Nothing}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :ˍ₋arg1, :ˍ₋arg2, :t), ModelingToolkit.var\"#_RGF_ModTag\", ModelingToolkit.var\"#_RGF_ModTag\", (0x077ceea2, 0x254bf500, 0x74c8eda1, 0x54c835c5, 0x5d3dd8d7), Nothing}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, ModelingToolkit.ObservedFunctionCache{ODESystem}, Nothing, ODESystem, Nothing, Nothing}, Float64, MTKParameters{Vector{Float64}, Tuple{}, Tuple{}, Tuple{}}}, Float64}, Float64, 2, Vector{ForwardDiff.Dual{ForwardDiff.Tag{SciMLBase.UJacobianWrapper{true, ODEFunction{true, SciMLBase.FullSpecialize, ModelingToolkit.var\"#f#815\"{RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋arg1, :ˍ₋arg2, :t), ModelingToolkit.var\"#_RGF_ModTag\", ModelingToolkit.var\"#_RGF_ModTag\", (0x68d7fc22, 0x9cd0330d, 0x51936f95, 0xda8a0357, 0xa6ac43de), Nothing}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :ˍ₋arg1, :ˍ₋arg2, :t), ModelingToolkit.var\"#_RGF_ModTag\", ModelingToolkit.var\"#_RGF_ModTag\", (0x077ceea2, 0x254bf500, 0x74c8eda1, 0x54c835c5, 0x5d3dd8d7), Nothing}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, ModelingToolkit.ObservedFunctionCache{ODESystem}, Nothing, ODESystem, Nothing, Nothing}, Float64, MTKParameters{Vector{Float64}, Tuple{}, Tuple{}, Tuple{}}}, Float64}, Float64, 2}}})\n @ ForwardDiff C:\\Users\\Bernt_Lie\\.julia\\packages\\ForwardDiff\\PcZ48\\src\\jacobian.jl:125\n [16] jacobian(f::Function, x::Vector{Float64}, cfg::ForwardDiff.JacobianConfig{ForwardDiff.Tag{SciMLBase.UJacobianWrapper{true, ODEFunction{true, SciMLBase.FullSpecialize, ModelingToolkit.var\"#f#815\"{RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋arg1, :ˍ₋arg2, :t), ModelingToolkit.var\"#_RGF_ModTag\", ModelingToolkit.var\"#_RGF_ModTag\", (0x68d7fc22, 0x9cd0330d, 0x51936f95, 0xda8a0357, 0xa6ac43de), Nothing}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :ˍ₋arg1, :ˍ₋arg2, :t), ModelingToolkit.var\"#_RGF_ModTag\", ModelingToolkit.var\"#_RGF_ModTag\", (0x077ceea2, 0x254bf500, 0x74c8eda1, 0x54c835c5, 0x5d3dd8d7), Nothing}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, ModelingToolkit.ObservedFunctionCache{ODESystem}, Nothing, ODESystem, Nothing, Nothing}, Float64, MTKParameters{Vector{Float64}, Tuple{}, Tuple{}, Tuple{}}}, Float64}, Float64, 2, Vector{ForwardDiff.Dual{ForwardDiff.Tag{SciMLBase.UJacobianWrapper{true, ODEFunction{true, SciMLBase.FullSpecialize, ModelingToolkit.var\"#f#815\"{RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋arg1, :ˍ₋arg2, :t), ModelingToolkit.var\"#_RGF_ModTag\", ModelingToolkit.var\"#_RGF_ModTag\", (0x68d7fc22, 0x9cd0330d, 0x51936f95, 0xda8a0357, 0xa6ac43de), Nothing}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :ˍ₋arg1, :ˍ₋arg2, :t), ModelingToolkit.var\"#_RGF_ModTag\", ModelingToolkit.var\"#_RGF_ModTag\", (0x077ceea2, 0x254bf500, 0x74c8eda1, 0x54c835c5, 0x5d3dd8d7), Nothing}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, ModelingToolkit.ObservedFunctionCache{ODESystem}, Nothing, ODESystem, Nothing, Nothing}, Float64, MTKParameters{Vector{Float64}, Tuple{}, Tuple{}, Tuple{}}}, Float64}, Float64, 2}}}, ::Val{true})\n @ ForwardDiff C:\\Users\\Bernt_Lie\\.julia\\packages\\ForwardDiff\\PcZ48\\src\\jacobian.jl:21\n [17] jacobian(f::Function, x::Vector{Float64}, cfg::ForwardDiff.JacobianConfig{ForwardDiff.Tag{SciMLBase.UJacobianWrapper{true, ODEFunction{true, SciMLBase.FullSpecialize, ModelingToolkit.var\"#f#815\"{RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋arg1, :ˍ₋arg2, :t), ModelingToolkit.var\"#_RGF_ModTag\", ModelingToolkit.var\"#_RGF_ModTag\", (0x68d7fc22, 0x9cd0330d, 0x51936f95, 0xda8a0357, 0xa6ac43de), Nothing}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :ˍ₋arg1, :ˍ₋arg2, :t), ModelingToolkit.var\"#_RGF_ModTag\", ModelingToolkit.var\"#_RGF_ModTag\", (0x077ceea2, 0x254bf500, 0x74c8eda1, 0x54c835c5, 0x5d3dd8d7), Nothing}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, ModelingToolkit.ObservedFunctionCache{ODESystem}, Nothing, ODESystem, Nothing, Nothing}, Float64, MTKParameters{Vector{Float64}, Tuple{}, Tuple{}, Tuple{}}}, Float64}, Float64, 2, Vector{ForwardDiff.Dual{ForwardDiff.Tag{SciMLBase.UJacobianWrapper{true, ODEFunction{true, SciMLBase.FullSpecialize, ModelingToolkit.var\"#f#815\"{RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋arg1, :ˍ₋arg2, :t), ModelingToolkit.var\"#_RGF_ModTag\", ModelingToolkit.var\"#_RGF_ModTag\", (0x68d7fc22, 0x9cd0330d, 0x51936f95, 0xda8a0357, 0xa6ac43de), Nothing}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :ˍ₋arg1, :ˍ₋arg2, :t), ModelingToolkit.var\"#_RGF_ModTag\", ModelingToolkit.var\"#_RGF_ModTag\", (0x077ceea2, 0x254bf500, 0x74c8eda1, 0x54c835c5, 0x5d3dd8d7), Nothing}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, ModelingToolkit.ObservedFunctionCache{ODESystem}, Nothing, ODESystem, Nothing, Nothing}, Float64, MTKParameters{Vector{Float64}, Tuple{}, Tuple{}, Tuple{}}}, Float64}, Float64, 2}}})\n @ ForwardDiff C:\\Users\\Bernt_Lie\\.julia\\packages\\ForwardDiff\\PcZ48\\src\\jacobian.jl:19\n [18] (::ModelingToolkit.var\"#358#367\"{UnitRange{Int64}, UnitRange{Int64}, Vector{ModelingToolkit.ParameterIndex{SciMLStructures.Tunable, Int64}}, Vector{SymbolicUtils.BasicSymbolic{Real}}, ModelingToolkit.var\"#356#364\"{MTKParameters{Vector{Float64}, Tuple{}, Tuple{}, Tuple{}}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(Symbol(\"##arg#10936873250485650685\"), Symbol(\"##arg#12853417078996959593\"), :t), ModelingToolkit.var\"#_RGF_ModTag\", ModelingToolkit.var\"#_RGF_ModTag\", (0xdb453d6f, 0x8ce078e3, 0xf0a30935, 0x139cbd07, 0xbba648b8), Nothing}, SymbolicIndexingInterface.ParameterHookWrapper{SymbolicIndexingInterface.MultipleSetters{Vector{SymbolicIndexingInterface.SetParameterIndex{ModelingToolkit.ParameterIndex{SciMLStructures.Tunable, Int64}}}}, Vector{SymbolicUtils.BasicSymbolic{Real}}}, ModelingToolkit.var\"#355#363\"}, ODEFunction{true, SciMLBase.FullSpecialize, ModelingToolkit.var\"#f#815\"{RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋arg1, :ˍ₋arg2, :t), ModelingToolkit.var\"#_RGF_ModTag\", ModelingToolkit.var\"#_RGF_ModTag\", (0x68d7fc22, 0x9cd0330d, 0x51936f95, 0xda8a0357, 0xa6ac43de), Nothing}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :ˍ₋arg1, :ˍ₋arg2, :t), ModelingToolkit.var\"#_RGF_ModTag\", ModelingToolkit.var\"#_RGF_ModTag\", (0x077ceea2, 0x254bf500, 0x74c8eda1, 0x54c835c5, 0x5d3dd8d7), Nothing}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, ModelingToolkit.ObservedFunctionCache{ODESystem}, Nothing, ODESystem, Nothing, Nothing}, NonlinearFunction{true, SciMLBase.FullSpecialize, ModelingToolkit.var\"#f#648\"{RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋arg1, :ˍ₋arg2), ModelingToolkit.var\"#_RGF_ModTag\", ModelingToolkit.var\"#_RGF_ModTag\", (0x11ce7c29, 0x6f65c7e8, 0x0fb6ac7c, 0x343656a1, 0xca307b6b), Nothing}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :ˍ₋arg1, :ˍ₋arg2), ModelingToolkit.var\"#_RGF_ModTag\", ModelingToolkit.var\"#_RGF_ModTag\", (0x58b56277, 0x73127dde, 0x68c0af47, 0x517d1f4b, 0x707f7aca), Nothing}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, ModelingToolkit.ObservedFunctionCache{NonlinearSystem}, Nothing, NonlinearSystem, Vector{Float64}}, ModelingToolkit.var\"#fn#366\"{RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(Symbol(\"##arg#5964805160111424296\"), Symbol(\"##arg#9353732160355970344\")), ModelingToolkit.var\"#_RGF_ModTag\", ModelingToolkit.var\"#_RGF_ModTag\", (0x6579d8b3, 0x45aaa53d, 0xc45bd7e9, 0xae240d39, 0xf23e180a), Nothing}}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(Symbol(\"##arg#10936873250485650685\"), Symbol(\"##arg#12853417078996959593\"), :t), ModelingToolkit.var\"#_RGF_ModTag\", ModelingToolkit.var\"#_RGF_ModTag\", (0xbcec8039, 0xc13893b9, 0xe9fd9bf0, 0xd326ed21, 0x0dd80d24), Nothing}, ForwardDiff.Chunk{2}, MTKParameters{Vector{Float64}, Tuple{}, Tuple{}, Tuple{}}, Bool, GeneralizedFirstOrderAlgorithm{nothing, :TrustRegion, Missing, NonlinearSolve.GenericTrustRegionScheme{NonlinearSolve.RadiusUpdateSchemes.__Simple, Rational{Int64}, Rational{Int64}, Rational{Int64}, Rational{Int64}, Rational{Int64}, Nothing, Nothing, Nothing, Nothing}, Dogleg{NewtonDescent{Nothing, typeof(NonlinearSolve.DEFAULT_PRECS)}, SteepestDescent{Nothing, typeof(NonlinearSolve.DEFAULT_PRECS)}}, Nothing, Nothing, Nothing}, ODESystem})(u::Vector{Float64}, p::Dict{Num, Float64}, t::Float64)\n @ ModelingToolkit C:\\Users\\Bernt_Lie\\.julia\\packages\\ModelingToolkit\\UXr3S\\src\\systems\\abstractsystem.jl:2383\n [19] linearize(sys::ODESystem, lin_fun::ModelingToolkit.var\"#358#367\"{UnitRange{Int64}, UnitRange{Int64}, Vector{ModelingToolkit.ParameterIndex{SciMLStructures.Tunable, Int64}}, Vector{SymbolicUtils.BasicSymbolic{Real}}, ModelingToolkit.var\"#356#364\"{MTKParameters{Vector{Float64}, Tuple{}, Tuple{}, Tuple{}}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(Symbol(\"##arg#10936873250485650685\"), Symbol(\"##arg#12853417078996959593\"), :t), ModelingToolkit.var\"#_RGF_ModTag\", ModelingToolkit.var\"#_RGF_ModTag\", (0xdb453d6f, 0x8ce078e3, 0xf0a30935, 0x139cbd07, 0xbba648b8), Nothing}, SymbolicIndexingInterface.ParameterHookWrapper{SymbolicIndexingInterface.MultipleSetters{Vector{SymbolicIndexingInterface.SetParameterIndex{ModelingToolkit.ParameterIndex{SciMLStructures.Tunable, Int64}}}}, Vector{SymbolicUtils.BasicSymbolic{Real}}}, ModelingToolkit.var\"#355#363\"}, ODEFunction{true, SciMLBase.FullSpecialize, ModelingToolkit.var\"#f#815\"{RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋arg1, :ˍ₋arg2, :t), ModelingToolkit.var\"#_RGF_ModTag\", ModelingToolkit.var\"#_RGF_ModTag\", (0x68d7fc22, 0x9cd0330d, 0x51936f95, 0xda8a0357, 0xa6ac43de), Nothing}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :ˍ₋arg1, :ˍ₋arg2, :t), ModelingToolkit.var\"#_RGF_ModTag\", ModelingToolkit.var\"#_RGF_ModTag\", (0x077ceea2, 0x254bf500, 0x74c8eda1, 0x54c835c5, 0x5d3dd8d7), Nothing}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, ModelingToolkit.ObservedFunctionCache{ODESystem}, Nothing, ODESystem, Nothing, Nothing}, NonlinearFunction{true, SciMLBase.FullSpecialize, ModelingToolkit.var\"#f#648\"{RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋arg1, :ˍ₋arg2), ModelingToolkit.var\"#_RGF_ModTag\", ModelingToolkit.var\"#_RGF_ModTag\", (0x11ce7c29, 0x6f65c7e8, 0x0fb6ac7c, 0x343656a1, 0xca307b6b), Nothing}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :ˍ₋arg1, :ˍ₋arg2), ModelingToolkit.var\"#_RGF_ModTag\", ModelingToolkit.var\"#_RGF_ModTag\", (0x58b56277, 0x73127dde, 0x68c0af47, 0x517d1f4b, 0x707f7aca), Nothing}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, ModelingToolkit.ObservedFunctionCache{NonlinearSystem}, Nothing, NonlinearSystem, Vector{Float64}}, ModelingToolkit.var\"#fn#366\"{RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(Symbol(\"##arg#5964805160111424296\"), Symbol(\"##arg#9353732160355970344\")), ModelingToolkit.var\"#_RGF_ModTag\", ModelingToolkit.var\"#_RGF_ModTag\", (0x6579d8b3, 0x45aaa53d, 0xc45bd7e9, 0xae240d39, 0xf23e180a), Nothing}}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(Symbol(\"##arg#10936873250485650685\"), Symbol(\"##arg#12853417078996959593\"), :t), ModelingToolkit.var\"#_RGF_ModTag\", ModelingToolkit.var\"#_RGF_ModTag\", (0xbcec8039, 0xc13893b9, 0xe9fd9bf0, 0xd326ed21, 0x0dd80d24), Nothing}, ForwardDiff.Chunk{2}, MTKParameters{Vector{Float64}, Tuple{}, Tuple{}, Tuple{}}, Bool, GeneralizedFirstOrderAlgorithm{nothing, :TrustRegion, Missing, NonlinearSolve.GenericTrustRegionScheme{NonlinearSolve.RadiusUpdateSchemes.__Simple, Rational{Int64}, Rational{Int64}, Rational{Int64}, Rational{Int64}, Rational{Int64}, Nothing, Nothing, Nothing, Nothing}, Dogleg{NewtonDescent{Nothing, typeof(NonlinearSolve.DEFAULT_PRECS)}, SteepestDescent{Nothing, typeof(NonlinearSolve.DEFAULT_PRECS)}}, Nothing, Nothing, Nothing}, ODESystem}; t::Float64, op::Dict{Num, Float64}, allow_input_derivatives::Bool, p::SciMLBase.NullParameters)\n @ ModelingToolkit C:\\Users\\Bernt_Lie\\.julia\\packages\\ModelingToolkit\\UXr3S\\src\\systems\\abstractsystem.jl:2657\n [20] linearize(sys::ODESystem, inputs::Vector{Num}, outputs::Vector{Num}; op::Dict{Num, Float64}, t::Float64, allow_input_derivatives::Bool, zero_dummy_der::Bool, kwargs::@Kwargs{})\n @ ModelingToolkit C:\\Users\\Bernt_Lie\\.julia\\packages\\ModelingToolkit\\UXr3S\\src\\systems\\abstractsystem.jl:2708\n [21] top-level scope\n @ c:\\Users\\Bernt_Lie\\OneDrive\\Documents\\researchUSN\\Supervisor\\PhD\\Kushila_Jayamanne\\Notebooks\\jl_notebook_cell_df34fa98e69747e1a8f8a730347b8e2f_X24sZmlsZQ==.jl:1"
}
Environment (please complete the following information):
- Output of
using Pkg; Pkg.status()
Status `C:\Users\Bernt_Lie\.julia\environments\v1.10\Project.toml`
[6e4b80f9] BenchmarkTools v1.5.0
[336ed68f] CSV v0.10.14
[a6e380b2] ControlSystems v1.10.4
[687d7614] ControlSystemsMTK v2.2.1
[a93c6f00] DataFrames v1.6.1
[82cc6244] DataInterpolations v6.4.0
[0c46a032] DifferentialEquations v7.14.0
[b964fa9f] LaTeXStrings v1.3.1
[23fbe1c1] Latexify v0.16.5
[2fda8390] LsqFit v0.15.0
[add582a8] MLJ v0.20.7
[961ee093] ModelingToolkit v9.40.0
[16a59e39] ModelingToolkitStandardLibrary v2.12.0
[0987c9cc] MonteCarloMeasurements v1.2.1
[8913a72c] NonlinearSolve v3.14.0
[6fe1bfb0] OffsetArrays v1.14.1
[429524aa] Optim v1.9.4
[91a5bcdd] Plots v1.40.8
[98d1487c] PolyesterForwardDiff v0.1.2
[f27b6e38] Polynomials v4.0.11
[0c5d862f] Symbolics v6.11.0
[37e2e46d] LinearAlgebra
- Output of
using Pkg; Pkg.status(; mode = PKGMODE_MANIFEST)
[32m[1mStatus[22m[39m `C:\Users\Bernt_Lie\.julia\environments\v1.10\Manifest.toml`
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[90m[efcefdf7] [39mPCRE2_jll v10.42.0+1
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[36m[1mInfo[22m[39m Packages marked with [32m⌃[39m and [33m⌅[39m have new versions available. Those with [32m⌃[39m may be upgradable, but those with [33m⌅[39m are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`
- Output of
versioninfo()
Julia Version 1.10.5
Commit 6f3fdf7b36 (2024-08-27 14:19 UTC)
Build Info:
Official https://julialang.org/ release
Platform Info:
OS: Windows (x86_64-w64-mingw32)
CPU: 16 × Intel(R) Core(TM) i9-9900K CPU @ 3.60GHz
WORD_SIZE: 64
LIBM: libopenlibm
LLVM: libLLVM-15.0.7 (ORCJIT, skylake)
Threads: 1 default, 0 interactive, 1 GC (on 16 virtual cores)
Environment:
JULIA_NUM_THREADS =
Additional context
Add any other context about the problem here.
