Equation-based modeling and simulations in Julia
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May 2012

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FunctionalModels.jl (formerly Sims.jl)

A Julia package for equation-based modeling and simulations. For more information, see the documentation:

NOTE: This is a work in progress to convert the package to use ModelingToolkit.

Some of the components and/or examples do not work, yet. This especially includes models requiring events and discrete systems.

FunctionalModels builds on top of ModelingToolkit. The following are exported:

  • t: independent variable
  • D and der: aliases for Differential(t)
  • system: flattens a set of hierarchical equations and returns a simplified ODESystem
  • Unknown: helper function to create variables
  • default_value: return the default (starting) value of a variable
  • compatible_values: return the base value from a variable to use when creating other variables
  • RefBranch and Branch: marks nodes and flow variables

Equations are standard ModelingToolkit equations. The main difference in FunctionalModels is that variables should be created with Unknown(val; name) or one of the helpers like Voltage(). Variables created this way include metadata to ensure that variable names don't clash. Multiple subcomponents can all have a v(t) variable for example. Once the model is flattened, the variable names will be normalized.

FunctionalModels uses a functional style as opposed to the more object-oriented approach of ModelingToolkit, Modia, and Modelica. Because system return an ODESystem, models can be built up of FunctionalModels components and standard ModelingToolkit components.


This package is for non-causal modeling in Julia. The idea behind non-causal modeling is that the user develops models based on components which are described by a set of equations. A tool can then transform the equations and solve the differential algebraic equations. Non-causal models tend to match their physical counterparts in terms of their specification and implementation.

Causal modeling is where all signals have an input and an output, and the flow of information is clear. Simulink is the highest-profile example. The problem with causal modeling is that it is difficult to build up models from components.

The highest profile noncausal modeling tools are in the Modelica family. The MathWorks company also has FunctionalModelscape that uses Matlab notation. Modelica is an object-oriented, open language with multiple implementations. It is a large, complex, powerful language with an extensive standard library of components.

This implementation follows the work of David Broman (thesis and code) and George Giorgidze (Hydra code and thesis) and Henrik Nilsson and their functional hybrid modeling. FunctionalModels is most similar to Modelyze by David Broman (report).


FunctionalModels is an installable package. To install FunctionalModels, use the following:


Model Libraries

FunctionalModels.jl has one main module named FunctionalModels and the following submodules:

  • FunctionalModels.Lib -- the standard library

  • FunctionalModels.Examples -- example models, including:

    • FunctionalModels.Examples.Basics
    • FunctionalModels.Examples.Lib
    • FunctionalModels.Examples.Neural

Basic example

FunctionalModels uses ModelingToolkit to build up models. All equations use the ModelingToolkit variables and syntax. In a simulation, the unknowns are to be solved based on a set of equations. Equations are built from device models.

A device model is a function that returns a vector of equations or other devices that also return lists of equations.

Electrical example

This example shows definitions of several electrical components. Each is again a function that returns a list of equations.

Arguments to each function are model parameters. These normally include nodes specifying connectivity followed by parameters specifying model characteristics.

Models can contain models or other functions that return equations. The function Branch is a special function that returns an equation specifying relationships between nodes and flows. It also acts as an indicator to mark nodes. In the flattening/elaboration process, equations are created to sum flows (in this case electrical currents) to zero at all nodes. RefBranch is another special function for marking nodes and flow variables.

Nodes passed as parameters are unknown variables. For these electrical examples, a node is simply an unknown voltage.

function Resistor(n1, n2; R::Real) 
    i = Current()
    v = Voltage()
        Branch(n1, n2, v, i)
        R * i ~ v

function Capacitor(n1, n2; C::Real) 
    i = Current()
    v = Voltage()
        Branch(n1, n2, v, i)
        D(v) ~ i / C

What follows is a top-level circuit definition. In this case, there are no input parameters. The ground reference "g" is assigned zero volts.

All of the equations returned in the list of equations are other models with various parameters.

In this example, the model components are named (:vs, :r1, ...). Unnamed components can also be used, but then variables used in components have anonymized naming (c1₊i(t) vs. var"##i#1057"(t)).

function Circuit()
    @named n1 = Voltage()
    @named n2 = Voltage()
    g = 0.0  # A ground has zero volts; it's not an unknown.
        :vs => SineVoltage(n1, g, V = 10.0, f = 60.0)
        :r1 => Resistor(n1, n2, R = 10.0)
        :r2 => Resistor(n2, g, R = 5.0)
        :c1 => Capacitor(n2, g, C = 5.0e-3)

ckt = Circuit()

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