Print the value of t whenever x[t] crosses 1/2:

Print the value of t whenever x[t] becomes greater than 1/2:

Print the value of t whenever x[t] becomes less than 1/2:

Sample the solution at regular time intervals :

Specify several events in a list to trigger an event when any of them occur:

Use an event with the condition to detect crossings of the positive axis:

In general, the event can be any expression that can be tested for becoming True:

Trigger an event to stop integration when a button is clicked:

If the Stop button is clicked soon enough, the integration will be terminated before :

Specify several actions in a list to be executed in consecutive order:

Use "StopIntegration" to end the solution at an event:

Use "RemoveEvent" to remove an event after the first occurrence:

Remove an event after the third occurrence:

Use "RestartIntegration" to force restarting integration after an event:

A better way to handle this change is to use a discrete state variable a[t]:

Rules can be used to modify DependentVariables or DiscreteVariables:

Use "CrossDiscontinuity" to manage a discontinuity crossing:

Use "CrossSlidingDiscontinuity" to indicate that Filippov continuation can be used:

Use DiscreteVariables and "DiscontinuitySignature" to efficiently cross a discontinuity:

In the Filippov sliding mode case, the "DiscontinuitySignature" will be 0:

WhenEvent can be used in NDSolve:

DSolve:

ParametricNDSolve:

NDSolveValue:

ParametricNDSolveValue:

DSolveValue:

The default event detection method is "Sign", which checks for sign changes between steps:

The above is equivalent to:

"Sign" can miss events if there is an even number of crossings between steps:

"DerivativeSign" halves the step if the sign is the same, but the derivative sign changes:

For rapidly varying event functions, use the "Interpolation" method to locate the events:

The default location method "Brent" accurately locates an event:

"LinearInterpolation" method will locate the event reasonably well:

If high accuracy of the event location is not essential, use "StepBegin" or "StepEnd":

When an event function is varying more rapidly than the solution, crossings may be missed:

Plot the event function between and , with the time steps and crossings found:

With "IntegrateEvent"->True, the steps taken are much smaller:

Define a function for comparing event-detection performance:

With "DetectionMethod"->"Interpolation", the event is integrated by default for robustness:

Compare performance:

Set priority for events when they happen at the same time, in this case for every integer :

Automatic priority will occur after any finite priorities, but before priority Infinity:

Priorities are ordered in the canonical order used by Sort:

Model a ball bouncing down steps:

Plot the ball's kinetic, potential, and total energy:

Model 20 bouncing balls dropped simultaneously from different heights:

In a square box, model a ball that changes direction upon impact with the side walls:

With an irrational initial velocity vector, there are no longer periodic solutions:

Solving a differential equation with event detection can be used to find all local minimums and maximums for a function in a limited range. Consider the function :

Set the range to search:

Solve the differential equation where the derivate of the function is 0 and store the points:

Plot the function and the minimum and maximum values:

Mark the points of arc length 1 for a parametric curve:

Mark the points where a parametric solution enters or leaves an annular region:

Mark only the points where a parametric solution leaves a region:

Mark only the points where a parametric solution enters a region:

Mark the points where the solution enters or leaves a region:

Each time a linear oscillator solution crosses the negative axis, reflect it across the axis:

The solution of this reset oscillator exhibits chaotic behavior:

Plot the solution on the negative axis with a histogram of the reflection points:

Model a one-degree-of-freedom impact oscillator with sinusoidal forcing:

Model a damped oscillator that gets a kick at regular time intervals:

The trajectory eventually settles into a consistent orbit:

Use the same oscillator, but give a random kick at regular time intervals:

Model a block on a moving conveyor belt anchored to a wall by a spring using different models for the friction force , including viscous, Coulomb, Stribeck, and static. Compare positions and velocities for the different models:

Newton's equation for the block:

Viscous friction is proportional to the relative velocity :

The block stabilizes just above the spring's natural length of 1:

Coulomb friction is proportional to the sign of relative velocity :

The block moves with the belt until the spring force is strong enough:

Stribeck friction is a refined Coulomb friction :

The variation at low velocities is slightly reduced:

Static friction holds the block in place until the spring force exceeds some value μ depending on roughness of surfaces. Use the discrete variable stuck set to 1 when the block is stuck and 0 otherwise:

Check whether the spring force is smaller than μ, and if the block is not moving relative to the belt:

The block repeatedly sticks to the belt, then slips away due to the spring force:

Comparing the different models:

Model an AC-to-DC full wave rectifier with four diodes and a capacitor for smoothing the output:

The input voltage vi and rectified voltage vr:

When vr increases, vo[t]=vr[t]; it charges the capacitor and supplies current to the load. When vr starts to decrease (vr'[t]<0), the capacitor discharges through the load and the output voltage follows vo'[t]=vo[t]/c r :

Simulate the system:

Plot the smoothened capacitor output voltage:

Model a DC-to-DC buck converter from input voltage level vi to desired output voltage level vd using a pulse-width modulated feedback control q[t]:

Use Kirchhoff's laws to get a model for the circuit above:

The control signal q[t] will switch the transistor on for a fraction vd/vi of each period τ:

Buck from a higher voltage vi=24 to a lower voltage vd=16:

Model a DC-to-DC boost converter from input voltage level vi to desired output voltage level vd using a pulse-width modulated feedback control q[t]:

Use Kirchhoff's laws to get a model for the circuit above:

The control signal q[t] will switch the transistor on for a fraction vi/vd of each period τ:

Boost from a lower voltage vi=24 to a higher voltage vd=36:

Model a DC-to-DC buck-boost converter from input voltage level vi to desired output voltage level vo using a pulse-width modulated feedback control q[t]:

Use Kirchhoff's laws to get a model for the circuit above:

The control signal q[t] will switch the transistor on for a fraction vd/(vi+vd) of each period:

Boost from a lower voltage vi=24 to a higher voltage vd=36:

Buck from a higher voltage vi=24 to a lower voltage vd=16:

Simulate the system stabilized with a discrete-time controller :

Simulate and visualize:

Control a double integrator using a dead-beat discrete-time controller:

Use a dead-beat digital feedback controller :

Simulate and visualize:

Model the position of a moving body with 1 kg mass:

Use a sampled proportional-derivative (PD) controller to keep the position constant:

Simulate and visualize:

Design a discrete-time controller for the plant with state equations , and output and simulate the closed-loop system:

Design a continuous-time controller by using a state estimator and state feedback:

Get a discrete-time controller by sampling with sample time :

Use the discrete-time controller:

Simulate the closed-loop system with the controller and plant together:

Plot the result:

Model a pendulum at angle θ anchored to a moving cart at position x undergoing a horizontal force f:

Design a controller to stabilize the pendulum about the unstable vertical fixed point:

Use WhenEvent to apply the feedback at regular sample times:

Simulate the system:

The pendulum's angle is quickly stabilized in the inverted position ():

Model thermostat-controlled heat generation in a room with three insulated walls and a glass front subject to the outside temperature:

A heater load is ramped up or down at an event:

The PDE models heat diffusion through air while generating heat inside a circle and losing heat through the glass window:

If the thermostat at position measures a temperature below/above a trigger, and if the discrete variable changed, the heater is switched on/off:

Monitor the time integration of the PDE with initial condition equal to the outside temperature:

Visualize the temperature measured at the thermostat, the outside temperature, and the triggers for the heater. A blue background is shown where the heater is on:

Inspect the time steps taken during the integration and note how NDSolveValue adjusted the time step size as needed during events:

Events given explicitly with conditions have a slightly different interpretation than a single predicate:

WhenEvent cannot find an explicit root function, so the event occurs when pred becomes True:

When an explicit function () is given, the event is considered as the root but is conditional on the predicate () that is False at , so it is never triggered:

Show the solution trajectory and the event location with the background colored green where the condition is True and red where it is False:

When an explicit function () is given, the event is considered as the root :

Show the solution trajectory and the event location with the background colored green where the condition is True and red where it is False:

With the default "DetectionMethod", some events may be missed:

Try "DerivativeSign" or "Interpolation" for better event detection:

Events corresponding to even-order roots may be missed:

Instead, rewrite with odd-order roots:

Some systems admit an infinite number of events in a finite span of time:

Eventually, an event is not detected:

Tightening the tolerances for locating events improves detection:

Events will not be detected if they are closer than this:

With detection by interpolation, event detection can be further improved:

The gap between the last two events is very close to this limit:

Stop the integration automatically before events are missed:

Events are detected up to the point at which integration is stopped:

Increasing the working precision allows detection of events on a finer time scale:

The highest derivative cannot be reset in an ODE:

When solving as a DAE, the highest derivative can be reset:

The corresponding value of is found by reinitializing so that the residual is 0:

With sequential event actions, the variables are modified in turn:

To swap the variable values, use simultaneous events: