# Symbols and variables

**URL:** https://discourse.julialang.org/t/symbols-and-variables/105306
**Category:** Modelling & Simulations
**Tags:** question, package, modelingtoolkit
**Created:** [October 23, 2023, 10:04am UTC](https://discourse.julialang.org/t/symbols-and-variables/105306 "2023-10-23T10:04:51Z")
**Posts on this page:** 7
**Page:** 1

<div class="post-metadata">

### Author: ![ufechner7](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/ufechner7/32/51363_2.png) [@ufechner7](https://discourse.julialang.org/u/ufechner7)
#### Post date: [October 23, 2023, 10:04am UTC](https://discourse.julialang.org/t/symbols-and-variables/105306/1 "2023-10-23T10:04:51Z")

</div>

The following code fails:

```julia
using ModelingToolkit

@variables t
@syms ω(t) Pgc(t)
@variables R a b ρ U J A s

struct Equilibrium
    a_bar
    b_bar
    pg_bar # generator power
    ω_bar # rotor speed
end

mutable struct Settings
    ρ0::Float64
    U0::Float64
    A0::Float64
    R0::Float64
    J0::Float64
end

se::Settings = Settings(1.225, 7.0, 12469.0, 63.0, 4.047e7)
eq = Equilibrium(-0.012273163568697454, 0.5344744027433362, 1132254, 0.88612)
op_ = Dict([R => se.R0, ρ => se.ρ0, A => se.A0, J => se.J0, U => se.U0, 
            Pgc(t) => eq.pg_bar, ω(t) => eq.ω_bar, a => eq.a_bar, b => eq.b_bar])

function model_turbine_symbolic()
    @variables t ω(t) Pr(t) λ(t) Pgc(t)
    @parameters J R ρ A U a b
    D = Differential(t)
    eqs = [D(ω) ~ (Pr - Pgc) / (J * ω),
            0 ~ ω * R/U - λ,
            0 ~ 0.5 * ρ * A * U^3 * (a*λ + b) - Pr]
    @named sys = ODESystem(eqs, t)
    sys, Pgc, ω
end

function calc_lin_numeric()
    sys, Pgc, ω = model_turbine_symbolic()
    matrices, simplified_sys = ModelingToolkit.linearize(sys, [Pgc], [ω], op=op_)
    A_, B, C, D = matrices
    A_[2]
end

println("Numeric linearization: ", calc_lin_numeric())

```

Error message:

```julia
julia> include("mwes\\mwe16.jl")
ERROR: LoadError: ArgumentError: SymbolicUtils.BasicSymbolic[ω(t)] are missing from the variable map.
Stacktrace:
  [1] throw_missingvars(vars::Vector{SymbolicUtils.BasicSymbolic})
    @ ModelingToolkit C:\Users\uwefechner\.julia\packages\ModelingToolkit\rrbUl\src\variables.jl:121
  [2] _varmap_to_vars(varmap::Dict{Any, Any}, varlist::Vector{SymbolicUtils.BasicSymbolic{Real}}; defaults::Dict{Any, Any}, check::Bool, toterm::typeof(ModelingToolkit.default_toterm))
    @ ModelingToolkit C:\Users\uwefechner\.julia\packages\ModelingToolkit\rrbUl\src\variables.jl:115
  [3] varmap_to_vars(varmap::Dict{Any, Any}, varlist::Vector{Any}; defaults::Dict{Any, Any}, check::Bool, toterm::Function, promotetoconcrete::Nothing, tofloat::Bool, use_union::Bool)
    @ ModelingToolkit C:\Users\uwefechner\.julia\packages\ModelingToolkit\rrbUl\src\variables.jl:84
  [4] get_u0_p(sys::ODESystem, u0map::Dict{Any, Any}, parammap::SciMLBase.NullParameters; use_union::Bool, tofloat::Bool)
    @ ModelingToolkit C:\Users\uwefechner\.julia\packages\ModelingToolkit\rrbUl\src\systems\diffeqs\abstractodesystem.jl:591
  [5] linearize(sys::ODESystem, lin_fun::ModelingToolkit.var"#132#135"{Vector{Num}, UnitRange{Int64}, UnitRange{Int64}, UnitRange{Int64}, Vector{Any}, ODEFunction{true, SciMLBase.FullSpecialize, ModelingToolkit.var"#f#533"{RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋arg1, :ˍ₋arg2, :t), ModelingToolkit.var"#_RGF_ModTag", ModelingToolkit.var"#_RGF_ModTag", (0xd4d85126, 0x18aadede, 0x17a09b22, 0x02577da8, 0x2c5f5169), Nothing}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :ˍ₋arg1, :ˍ₋arg2, :t), ModelingToolkit.var"#_RGF_ModTag", ModelingToolkit.var"#_RGF_ModTag", (0x30918c58, 0xe8de3e77, 0xb7bd31f3, 0x4caf8f90, 0x996d4b0b), Nothing}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Vector{Symbol}, Symbol, Vector{Symbol}, ModelingToolkit.var"#583#generated_observed#541"{Bool, ODESystem, Dict{Any, Any}}, Nothing, ODESystem}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(Symbol("##arg#5403431765531858355"), Symbol("##arg#3669817888844577317"), :t), ModelingToolkit.var"#_RGF_ModTag", ModelingToolkit.var"#_RGF_ModTag", (0xc4eb96ba, 0x09fdb654, 0x41bf9ba7, 0xc49409ef, 0x508b1e9f), Nothing}, ForwardDiff.Chunk{1}}; t::Float64, op::Dict{Num, Float64}, allow_input_derivatives::Bool, p::SciMLBase.NullParameters)
    @ ModelingToolkit C:\Users\uwefechner\.julia\packages\ModelingToolkit\rrbUl\src\systems\abstractsystem.jl:1518
  [6] linearize(sys::ODESystem, inputs::Vector{Num}, outputs::Vector{Num}; op::Dict{Num, Float64}, t::Float64, allow_input_derivatives::Bool, zero_dummy_der::Bool, kwargs::Base.Pairs{Symbol, Union{}, Tuple{}, NamedTuple{(), Tuple{}}})
    @ ModelingToolkit C:\Users\uwefechner\.julia\packages\ModelingToolkit\rrbUl\src\systems\abstractsystem.jl:1570
  [7] calc_lin_numeric()
    @ Main C:\Users\uwefechner\repos\WindTurbines\mwes\mwe16.jl:40
  [8] top-level scope
    @ C:\Users\uwefechner\repos\WindTurbines\mwes\mwe16.jl:45
  [9] include(fname::String)
    @ Base.MainInclude .\client.jl:478
 [10] top-level scope
    @ REPL[1]:1
in expression starting at C:\Users\uwefechner\repos\WindTurbines\mwes\mwe16.jl:45

```

Any idea why?

I assume that I do not make the correct use of variables and symbols, but how can I use them correctly?

Is there a difference between the symbol ` ω(t)` and the variable `ω(t)`, and if yes what is the difference?

---

<div class="post-metadata">

### Author: ![ufechner7](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/ufechner7/32/51363_2.png) [@ufechner7](https://discourse.julialang.org/u/ufechner7)
#### Post date: [October 23, 2023, 11:29am UTC](https://discourse.julialang.org/t/symbols-and-variables/105306/2 "2023-10-23T11:29:10Z")

</div>

I was able to fix the example by not using @syms to define `ω(t)`. I used it in the first place because an error message suggested it when I used ` ω(t) => eq.ω_bar` in the dictionary. I have to use just ` ω` here. Still not fully clear to me when I have to use ` ω(t)` and when `ω` …

```julia
using ModelingToolkit, ControlSystemsBase

@variables t ω(t) Pgc(t)
@variables R a b ρ U J A s

struct Equilibrium
    a_bar
    b_bar
    pg_bar # generator power
    ω_bar # rotor speed
end

mutable struct Settings
    ρ0::Float64
    U0::Float64
    A0::Float64
    R0::Float64
    J0::Float64
end

se::Settings = Settings(1.225, 7.0, 12469.0, 63.0, 4.047e7)
eq = Equilibrium(-0.012273163568697454, 0.5344744027433362, 1132254, 0.88612)
op_ = Dict([R => se.R0, ρ => se.ρ0, A => se.A0, J => se.J0, U => se.U0, 
            Pgc => eq.pg_bar, ω => eq.ω_bar, a => eq.a_bar, b => eq.b_bar])

function model_turbine_symbolic()
    @variables t ω(t) Pr(t) λ(t) Pgc(t)
    @parameters J R ρ A U a b
    D = Differential(t)
    eqs = [D(ω) ~ (Pr - Pgc) / (J * ω),
            0 ~ ω * R/U - λ,
            0 ~ 0.5 * ρ * A * U^3 * (a*λ + b) - Pr]
    @named sys = ODESystem(eqs, t)
    sys, Pgc, ω
end

function calc_lin_numeric()
    sys, Pgc, ω = model_turbine_symbolic()
    matrices, simplified_sys = ModelingToolkit.linearize(sys, [Pgc], [ω], op=op_)
    A_, B, C, D = matrices
    A_[1]
end

function calc_lin_symbolic(num=true)
    sys, Pgc, ω = model_turbine_symbolic()
    matrices, simplified_sys = ModelingToolkit.linearize_symbolic(sys, [Pgc], [ω])
    A_, B, C, D = matrices
    if num; return substitute(A_[1], op_); else return A_[1] end
end

println("Symbolic linearization: ", calc_lin_symbolic(false))
println("Symbolic linearization, numeric result: ", calc_lin_symbolic())
println("Numeric linearization: ", calc_lin_numeric())

```

which has the output:

```julia
julia> include("mwes\\mwe16.jl")
Symbolic linearization: (0.5A*R*(U^2)*a*ρ) / (J*ω(t)) - J*((-Pgc(t) + 0.5A*(U^3)*(b + (R*a*ω(t)) / U)*ρ) / ((J^2)*(ω(t)^2)))
Symbolic linearization, numeric result: -0.008428790869465188
Numeric linearization: -0.008428790869465188

```

---

<div class="post-metadata">

### Author: ![isaacsas](https://avatars.discourse-cdn.com/v4/letter/i/f6c823/32.png) [@isaacsas](https://discourse.julialang.org/u/isaacsas)
#### Post date: [October 23, 2023, 2:31pm UTC](https://discourse.julialang.org/t/symbols-and-variables/105306/3 "2023-10-23T14:31:32Z")

</div>

If you declare a variable like

```julia
@variables t ω(t)

```

then you should generally use it in equations and mappings as just `ω`. It will implicitly know it is `ω(t)`.

The one time you might need to specify the arguments is if you declared it with unbounded arguments like

```julia
@variables ω(...)

```

then you will always need to say `ω(t)` when using it to make clear the arguments. This representation is only used by the PDE packages like MethodOfLines as far as I am aware (and is an unfortunate inconsistency).

---

<div class="post-metadata">

### Author: ![ufechner7](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/ufechner7/32/51363_2.png) [@ufechner7](https://discourse.julialang.org/u/ufechner7)
#### Post date: [October 23, 2023, 4:06pm UTC](https://discourse.julialang.org/t/symbols-and-variables/105306/4 "2023-10-23T16:06:23Z")

</div>

And when should I use `@syms` instead of `@variables` ?

---

<div class="post-metadata">

### Author: ![isaacsas](https://avatars.discourse-cdn.com/v4/letter/i/f6c823/32.png) [@isaacsas](https://discourse.julialang.org/u/isaacsas)
#### Post date: [October 23, 2023, 4:19pm UTC](https://discourse.julialang.org/t/symbols-and-variables/105306/5 "2023-10-23T16:19:04Z")

</div>

Never as far as I am aware. That is a much lower-level macro from SymbolicUtils that ModelingToolkit users should not ever need to use. Just use `@variables` and `@parameters`.

---

<div class="post-metadata">

### Author: ![ufechner7](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/ufechner7/32/51363_2.png) [@ufechner7](https://discourse.julialang.org/u/ufechner7)
#### Post date: [October 23, 2023, 4:28pm UTC](https://discourse.julialang.org/t/symbols-and-variables/105306/6 "2023-10-23T16:28:23Z")

</div>

But then the error message that suggests to use `@syms` if you accidentally write `ω(t)` instead of `ω` is misleading…

---

<div class="post-metadata">

### Author: ![isaacsas](https://avatars.discourse-cdn.com/v4/letter/i/f6c823/32.png) [@isaacsas](https://discourse.julialang.org/u/isaacsas)
#### Post date: [October 23, 2023, 6:03pm UTC](https://discourse.julialang.org/t/symbols-and-variables/105306/7 "2023-10-23T18:03:35Z")

</div>

Sounds like that is worth opening an issue on ModelingToolkit about then. Seems like a lower-level error from SymbolicUtils is being propagated up in this case instead of a more helpful message being generated in ModelingToolkit.