# Calibration of an Unstable Circuit Model using Prediction Error Method

In this example, we present the creation of a custom component is demonstrated via the Chua's circuit. The circuit is a simple circuit that shows chaotic behaviour. Except for a non-linear resistor every other component already is part of ModelingToolkitStandardLibrary.Electrical. We can then seamlessly plug this model with JuliaSimModelOptimizer for calibration.

## Julia environment

For this example, we will need the following packages:

ModuleDescription
JuliaSimModelOptimizerThe high-level library used to formulate our problem and perform automated model discovery
ModelingToolkitThe symbolic modeling environment
ModelingToolkitStandardLibraryLibrary for using standard modeling components
OrdinaryDiffEqThe numerical differential equation solvers
DataFramesFor converting simulation into a Dataframe
PlotsThe plotting and visualization library
using JuliaSimModelOptimizer
using ModelingToolkit
import ModelingToolkit: D_nounits as D, t_nounits as t
using ModelingToolkitStandardLibrary.Electrical
using ModelingToolkitStandardLibrary.Electrical: OnePort
using OrdinaryDiffEq
using DataFrames
using Plots

## Model Setup

The first step is to use the pre defined components defined in the Electrical Toolkit in the ModelingToolkit library. We can also define custom components such as the "Non linear resistor" as defined below. One advantage of using ModelingToolkit is being directly able to use custom components out of the box. We can define a Resistor component, a Capacitor component, an Inductor component etc. ModelingToolkit defined models can be seamlessly integrated with the solvers from DifferentialEquations.jl.

function NonlinearResistor(; name, Ga, Gb, Ve)
@named oneport = OnePort()
@unpack v, i = oneport
pars = @parameters Ga=Ga Gb=Gb Ve=Ve
eqs = [
i ~ ifelse(v < -Ve,
Gb * (v + Ve) - Ga * Ve,
ifelse(v > Ve,
Gb * (v - Ve) + Ga * Ve,
Ga * v)),
]
extend(ODESystem(eqs, t, [], pars; name = name), oneport)
end

@named L = Inductor(L = 18)
@named Ro = Resistor(R = 12.5e-3)
@named G = Conductor(G = 0.565)
@named C1 = Capacitor(C = 10, v = 4)
@named C2 = Capacitor(C = 100)
@named Nr = NonlinearResistor(Ga = -0.757576, Gb = -0.409091, Ve = 1)
@named Gnd = Ground()

connections = [connect(L.p, G.p)
connect(G.n, Nr.p)
connect(Nr.n, Gnd.g)
connect(C1.p, G.n)
connect(L.n, Ro.p)
connect(G.p, C2.p)
connect(C1.n, Gnd.g)
connect(C2.n, Gnd.g)
connect(Ro.n, Gnd.g)]

@named model = ODESystem(connections, t, systems = [L, Ro, G, C1, C2, Nr, Gnd])
sys = structural_simplify(model)
\begin{align} \frac{\mathrm{d} L_{+}i\left( t \right)}{\mathrm{d}t} =& \frac{L_{+}v\left( t \right)}{L_{+}L} \\ \frac{\mathrm{d} C1_{+}v\left( t \right)}{\mathrm{d}t} =& \frac{C1_{+}i\left( t \right)}{C1_{+}C} \\ \frac{\mathrm{d} C2_{+}v\left( t \right)}{\mathrm{d}t} =& \frac{C2_{+}i\left( t \right)}{C2_{+}C} \end{align}

## Data Setup

Let us simulate using a stiff ODE solver Rodas4 and use this data for calibration.

@unpack L, C2 = model
prob = ODEProblem(sys, [L.i => 0.0, C2.v => 0.0], (0, 5e4), [Ro.R => 11e-3, C1.C => 9.3, C2.C => 102.5], saveat = 10)
sol = solve(prob, Rodas4())
data = DataFrame(sol)
first(data, 5)
5×4 DataFrame
RowtimestampL₊iC1₊vC2₊v
Float64Float64Float64Float64
10.00.04.00.0
210.00.05848023.788220.206174
320.00.2227473.71610.383145
430.00.4756593.745410.529685
540.00.7994313.843080.644382

## Defining Experiment and InverseProblem

This system is unstable and it can be difficult to simulate it for different sets of parameters. To mitigate this, we will use Prediction Error Method, where the simulation is guided by the data such that the trajectory won't diverge and this should help with the calibration process.

In order to create an Experiment, we will use the default initial values of the states and parameters of our model. These are our initial guesses which will be used to optimize the inverse problem in order to fit the given data. To use Prediction Error Method, we also need to pass it in the model_transformations keyword in the constructor.

experiment = Experiment(data, sys, initial_conditions = [L.i => 0.0, C2.v => 0.0], model_transformations = [DiscreteFixedGainPEM(0.2)], alg = Rodas4())
Experiment for model with no fixed parameters or initial conditions.
The simulation of this experiment is given by:
ODEProblem with uType Vector{Float64} and tType Float64. In-place: true
timespan: (0.0, 50000.0)

Argument passed to DiscreteFixedGainPEM is the amount of correction needed during simulation. 1.0 represents completely using the data and 0.0 represents completely ignoring the data. Typically, we should use this be about 0.2-0.3 to help guide the simulation.

The next step is to define an InverseProblem by specifying the parameters we want to optimize and the search space of those parameters.

prob = InverseProblem(experiment, [Ro.R => (9.5e-3, 13.5e-3), C1.C => (9, 11), C2.C => (95, 105)])
InverseProblem with one experiment with 3 elements in the search space.


## Calibration

We will use SingleShooting as our calibration algorithm. To calibrate, we simply call calibrate with our inverse problem and calibration algorithm.

alg = SingleShooting(maxiters = 10^3)
r = calibrate(prob, alg)
Calibration result computed in 1 minute and 26 seconds. Final objective value: 0.00675041.
Optimization ended with Success Return Code and returned.

┌──────────┬─────────┬─────────┐
│     Ro₊R │    C1₊C │    C2₊C │
├──────────┼─────────┼─────────┤
│ 0.010999 │ 9.29164 │ 102.568 │
└──────────┴─────────┴─────────┘


As we see above, the parameters recovered match the true parameters!

## Visualization

We can now plot the simulation using the calibrated parameters and compare it against the data.

plot(experiment, prob, r, show_data = true, legend = :best, ms = 0.2, layout = (3, 1), size = (1000, 900))