Continuous-Time Models

We assume every continuous-time model to consist of a dynamics model $f$ and a measurement model $g$.

\[\dot{x}(t) = f(x(t), u(t), p, t)\\ y(t) = g(x(t), u(t), p, t)\]

The dynamics model is given by a first-order ordinary differential equation. The measurement model is given by a simple algebraic function.

Dynamics Model

The SimpleSim.jl equivalent of the first-order ordinary differential equation

\[\dot{x}(t) = f(x(t), u(t), p, t)\]

is to define a function that returns the current derivative of $x$ as an AbstractVector.

function fc_my_model(x, u, p, t)
    my_x_derivative = # ...
    return my_x_derivative
end

Implicit differential equations $0 = F(\dot{x}, x, u, p, t)$ are not supported (yet...).

Measurement Model

The measurement model

\[y(t) = g(x(t), u(t), p, t)\]

is also implemented as a simple Julia function that returns the current output $y(t)$ given the current state $x(t)$, input $u(t)$ and time $t$, as well as the parameters $p$. The output $y$ should also be returned as an AbstractVector.

function gc_my_model(x, u, p, t)
    my_output = # ...
    return my_output
end

Model Creation

Every SimpleSim.jl model has to be a data structure with named fields. So, you can use a custom struct to define your models or simply use a NamedTuple. Structs may have some advantages when it comes to debugging your code, however, for simple examples, NamedTuples are more than sufficient.

my_ct_model = (
    p = nothing,
    fc = fc_my_model,
    gc = gc_my_model,
    xc0 = my_initial_state,
    uc0 = my_initial_input,
)

Mandatory fields for continuous-time models:

• p, set this to nothing if no parameters are needed
• fc, pass your dynamics function, returning the right-hand side of the ODE
• gc, pass your measurement function

Optional fields for continuous-time models:

• xc0, the initial state of the system, nothing by default. Can be overriden by an initial state directly passed to the simulate function.
• uc0, the initial input of the system, nothing by default.
• zc, function that returns a scalar and will be watched by SimpleSim.jl to detect zero-crossings.
• zc_exec, function that returns a post-zero-crossing state