Events and Discrete Behavior

Physical models often mix continuous dynamics with discrete switching: impacts, controllers that sample, valves that open. Modelica expresses these with events, and the solver locates them precisely instead of stepping blindly past them.

When Equations and reinit

A when equation activates at the instant its condition becomes true. The classic example is the bouncing ball, which reverses its velocity at impact:

model Ball
  Real x(start = 10) "Height [m]";
  Real v(start = 1) "Velocity [m/s]";
  parameter Real g = 9.81;
  parameter Real e = 0.8 "Coefficient of restitution";
equation
  der(x) = v;
  der(v) = -g;
  when x < 0 then
    reinit(v, -e * pre(v));
  end when;
  annotation(experiment(StopTime = 10.0));
end Ball;

• when x < 0 then ... end when — fires once at the crossing instant, not continuously while the condition holds.
• pre(v) — the value of v immediately before the event.
• reinit(v, ...) — restarts the state v from a new value.

The solver detects the zero crossing of x, locates the impact time, applies the reinit, and continues. Run the example and zoom in on the velocity trace: each bounce is a clean jump, not a smoothed-over spike.

Sampled (Clocked-Style) Control

sample(start, period) fires periodically, which is the standard way to model a digital controller around a continuous plant:

model SampledControl "Continuous plant with a sampled P controller"
  parameter Real kp = 2.0 "Proportional gain";
  parameter Real Ts = 0.2 "Sample period [s]";
  parameter Real r = 1.0 "Reference";
  Real x(start = 0.0) "Plant state";
  discrete Real u(start = 0.0) "Held control signal";
equation
  der(x) = -x + u;
  when sample(0, Ts) then
    u = kp * (r - pre(x));
  end when;
  annotation(experiment(StopTime = 6.0));
end SampledControl;

u is discrete: it changes only when the when fires and is held constant (zero-order hold) in between. Try shortening Ts toward continuous control, or raising kp until the loop rings.

Hysteresis with elsewhen

elsewhen chains mutually exclusive switching conditions. A thermostat with a hysteresis band:

model Thermostat "Bang-bang temperature control with hysteresis"
  parameter Real T_set = 21.0 "Setpoint [degC]";
  parameter Real band = 1.0 "Hysteresis half-width [degC]";
  parameter Real T_amb = 5.0 "Outside temperature [degC]";
  parameter Real tau = 600.0 "Thermal time constant [s]";
  parameter Real heat = 0.02 "Heater authority [degC/s]";
  Real T(start = 15.0) "Room temperature [degC]";
  Boolean on(start = true) "Heater state";
equation
  der(T) = (T_amb - T) / tau + (if on then heat else 0.0);
  when T > T_set + band then
    on = false;
  elsewhen T < T_set - band then
    on = true;
  end when;
  annotation(experiment(StopTime = 7200.0, Solver = "rk-like"));
end Thermostat;

This model pins Solver = "rk-like" in its experiment annotation: the explicit solver handles its rapid relay switching robustly, while the default implicit solver can currently stall on it (see Solvers and Accuracy).

Conditional Expressions vs Events

An if-expression inside an equation (if on then heat else 0.0) also generates events at its switch points so the integrator never smears across a discontinuity. When a discontinuity is harmless and you want to suppress event handling, Modelica provides noEvent(...); smooth(...) asserts differentiability.

Things to Keep In Mind

• when bodies relate discrete values; use reinit to restart continuous states.
• pre(x) is only meaningful for discrete-valued variables or at event instants.
• Event-heavy models simulate best with explicit solvers today; pin Solver = "rk-like" in the experiment annotation or pass --solver rk-like.