Discrete-Time Models

Discrete-time models are defined by a "step" function, that returns the next state, based on the current state of the system and a measurement function.

\[x_{k+1} = f(x_k, u_k, p, t)\\ y_k = g(x_k, u_k, p, t)\]

Both $f$ and $g$ are assumed to be explicit, algebraic functions. Implicit definitions of $x_{k+1}$ and $y_k$ are not supported.

Dynamics Model

The discrete-time dynamics

\[x_{k+1} = f(x_k, u_k, p, t)\]

are modeled in SimpleSim.jl using a simple Julia function that returns $x_{k+1}$ as an AbstractVector.

function fd_my_model(x, u, p, t)
    my_next_x = # ...
    return my_next_x
end

Note, that while working with integer time-steps $k$, the functions $f$ and $g$ still accept the actual time $t$ as an input. This is more practical for most applications. If you want to work with integer time steps $k$ instead, make sure to use a sampling time of 1 // 1 and convert the time t to an integer inside your functions $f$ and $g$.

Measurement Model

The measurement model

\[y_k = g(x_k, u_k, p, t)\]

is also given by a simple Julia function

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

(Pretty straighforward, right?)

Model Creation

Similar to continuous-time models, SimpleSim.jl supports all types of models that have named fields. So you can either define your own struct for each type of model, or simply use a NamedTuple.

my_dt_model = (
    p = nothing,
    fd = fd_my_model,
    gd = gd_my_model,
    Δt = Δt_my_model,
    xd0 = my_initial_state,
    ud0 = my_initial_input,
)

Mandatory fields for discrete-time models:

• p, set this to nothing if no parameters are needed
• fd, pass your dynamics function returning the next state
• gd, pass your measurement function
• Δt, the desired sampling time of the discrete-time model, use 1 // 1 for integer time steps $k$.

Optional fields for discrete-time models:

• xd0, the initial state of the system, nothing by default. Can be overriden by initial state directly passed to the simulate function.
• ud0, the initial input of the system, nothing by default.
• wd, function (x, u, p, t, rng) -> ... that returns a random draw. The random draw will be passed as a keyword argument w to fd and gd.
• wd_seed, the seed used for this model's random number generator

Hybrid Models

Models can have both continuous-time and discrete-time dynamics. These kind of models are considered HybridModels by SimpleSim.jl and they work very similar to continuous-time or discrete-time models.

Note, that for hybrid models, the mandatory fields for both continuous-time and discrete-time models must be given.

See @call_ct!/@call_dt!, @out_ct/@out_dt, and @state_ct/@state_dt for some comments about how do avoid ambiguity when working with hybrid models.