# DifferentialEquations & inputs

**URL:** <https://discourse.julialang.org/t/differentialequations-inputs/25157>\
**Category:** General Usage\
**Created:** [June 11, 2019, 4:45pm UTC](https://discourse.julialang.org/t/differentialequations-inputs/25157 "2019-06-11T16:45:33Z")\
**Posts on this page:** 3\
**Page:** 1

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**Author:** ![BLI](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/bli/32/37206_2.png) [@BLI](https://discourse.julialang.org/u/BLI)\
**Post date:** [June 11, 2019, 4:45pm UTC](https://discourse.julialang.org/t/differentialequations-inputs/25157/1 "2019-06-11T16:45:33Z")

</div>

I’m trying to figure out the best way to pose a (control) system model of form:

\frac{dx}{dt} = f(x,u,t;\theta)

and use this model with `DifferentialEquations`. The “problem” is to merge two contradicting constraints:

- `DifferentialEquations` requires the model to be posed as \frac{dx}{dt} = f(x,t;\theta), and
- I would like to be able to pose the model in the form with input u, so that I can use Automatic Differentiation to linearize the model.

To test my idea, I used the “academic” model of level h [`h`] in a water tank, with input \dot{m}\_\mathrm{i} [`mdi`], and effluent \dot{m}\_\mathrm{e} driven by gravity, \dot{m}\_\mathrm{e} = K\sqrt{h/h^\varsigma} where K is some valve constant, and h^\varsigma [`hs`] is some scaling level to keep the square root argument dimensionless. With density \rho [`rho`] and cross sectional area A [`A`], the model for the system is:

\frac{dh}{dt} = \frac{\dot{m}\_\mathrm{i}-K\sqrt{h/h^\varsigma}}{\rho A}

I use the following parameters:

- K=5, h^\varsigma = 3, \rho = 1, A=5.

I use the following initial value:

- h(0) = 1.5.

At first, I want to set \dot{m}\_\mathrm{i} = 2. Later on, I want to change this input value, e.g., by setting \dot{m}\_\mathrm{i} = 2 + 0.1\sin(2\pi t/10), or by using a proportional controller \dot{m}\_\mathrm{i} = K\_\mathrm{p}(h\_\mathrm{ref} - h). The point is: I want to make it as flexible as possible, while also allowing for Automatic Differentiation.

So… first I created a parameter tuple:

```julia
p = (rho=1.,A=5.,K=5.,hs=3.)

```

Next, I created a model function with input:

```julia
function mod_i(h,mdi,p,t)
    dh = (mdi-p.K*sqrt(h/p.hs))/(p.rho*p.A)
end

```

With an operating point (h^\*=1.5,\dot{m}\_\mathrm{i}^\*=2), I can now linearize the model using `ForwardDiff`:

```julia
julia> ForwardDiff.derivative(h -> mod_i(h,2,p,0),1.5)
-0.2357022603955158
julia> ForwardDiff.derivative(mdi -> mod_i(1.5,mdi,p,0),2)
0.2

```

So far, so good. Next, I need to create a model function suitable for `DifferentialEquations`… and I attempt:

```julia
function mod(h,p,t)
    mdi = 2.
    return mod_i(h,mdi,p,t)
end

```

and set up the simulation case as:

```julia
h0 = 1.5
tspan = (0.,15.)
prob = ODEProblem(mod,h0,tspan)
sol = solve(prob)

```

Problem is: Julia crashes “big time”, with error message:

```julia
type Nothing has no field K
......

```

QUESTION:

- What have I done wrong? Why does Julia crash?

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<div class="post-metadata">

**Author:** ![BeastyBlacksmith](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/beastyblacksmith/32/4741_2.png) [@BeastyBlacksmith](https://discourse.julialang.org/u/BeastyBlacksmith)\
**Post date:** [June 11, 2019, 5:25pm UTC](https://discourse.julialang.org/t/differentialequations-inputs/25157/2 "2019-06-11T17:25:16Z")

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> [@BLI](#):
>
> prob = ODEProblem(mod,h0,tspan)

you just forgot to pass the parameters to the `ODEProblem`, like so

```julia
prob = ODEProblem(mod,h0,tspan,p)

```

---

<div class="post-metadata">

**Author:** ![BLI](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/bli/32/37206_2.png) [@BLI](https://discourse.julialang.org/u/BLI)\
**Post date:** [June 11, 2019, 6:18pm UTC](https://discourse.julialang.org/t/differentialequations-inputs/25157/3 "2019-06-11T18:18:01Z")

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Ooops… I did it again…