Zero-Crossing Detection

The zero-crossing detection feature allows for building models that have discontinuous dynamics. An example for a dynamical system where this feature is used would be a ball bouncing up and down on a hard surface.

Theory

The goal of a zero-crossing detector is to solve the following equation

\[0 = z(x(t), p, t)\]

for $x(t)$ and $t$.

In simulation, this is done by checking for a change of sign in $z$.

If

\[z(x(t), p, t) < 0\\ z(x(t + T), p, t + T) > 0,\]

then a zero-crossing has happened. In that case, a bisection algorithm is applied to find $T^*$, such that

\[\lvert z(x(t + T^*), p, t + T^*)\rvert < \varepsilon\]

where $\varepsilon$ is a small tolerance.

Implementation

SimpleSim.jl supports the optional fields zc and zc_exec for continuous-time models that are used to implement the zero-crossing detection feature.

• zc tells us when the zero-crossing is happening
• zc_exec tells us what to do, once a zero-crossing is found

zc function

The function passed to zc takes the current state, the model parameters and the current time as an input and returns a Number. This number represents the quantity that is critical for a zero-crossing.

function zc_my_model(x, p, t)
    my_zc_indicator = # ...
    return my_zc_indicator
end

If the number returned by zc changes its sign, the simulation engine will try to figure out when exactly the zero-crossing has happened. In order to do so, a bisection algorithm is applied to find the right simulation step size, so that the next simulation step occurs right at the time of the zero-crossing.

zc_exec function

This function tells the simulation engine what to do, once it has figured out when a zero-crossing is happening (as defined by zc).

The zc_exec works similar to a discrete-time state update. When a zero-crossing is happening, it is called exactly once and should return a new, post-zero-crossing state of the system.

function zc_exec_my_model(x, u, p, t)
    my_new_state = # ...
    return my_new_state
end

The new state should be constructed in a way that resolves the zero-crossing. Otherwise, the simulation can get stuck in a loop.

In summary, a model with zero-crossing detection could look something like this.

my_ct_model = (
    p = nothing,
    fc = fc_my_model,
    gc = gc_my_model,
    zc = zc_my_model,
    zc_exec = zc_exec_my_model,
    xc0 = my_initial_state,
    uc0 = my_initial_input,
)

Since varying the simulation step size is not allowed for discrete-time systems, zero-crossing detection is only supported for continuous-time systems.