To create S-functions, you need to understand how S-functions
work. Such knowledge requires an understanding of how the Simulink^{®} engine
simulates a model, including the mathematics of blocks. This section
begins by explaining the mathematical relationships between the inputs,
states, and outputs of a block.

A Simulink block consists of a set of inputs, a set of states, and a set of outputs, where the outputs are a function of the simulation time, the inputs, and the states.

The following equations express the mathematical relationships between the inputs, outputs, states, and simulation time

$$\begin{array}{l}y={f}_{0}(t,x,u)\text{(Outputs)}\\ \dot{x}={f}_{d}(t,x,u)\text{(Derivatives)}\\ {x}_{{d}_{k+1}}={f}_{u}(t,{x}_{c},{x}_{{d}_{k}},u),\text{(Update)}\end{array}$$

where $$x=[{x}_{c};{x}_{d}].$$

Execution
of a Simulink model proceeds in stages. First comes the initialization
phase. In this phase, the Simulink engine incorporates library
blocks into the model, propagates signal widths, data types, and sample
times, evaluates block parameters, determines block execution order,
and allocates memory. The engine then enters a *simulation loop*,
where each pass through the loop is referred to as a *simulation
step*. During each simulation step, the engine executes
each block in the model in the order determined during initialization.
For each block, the engine invokes functions that compute the block
states, derivatives, and outputs for the current sample time.

The following figure illustrates the stages of a simulation.
The inner integration loop takes place only if the model contains
continuous states. The engine executes this loop until the solver
reaches the desired accuracy for the state computations. The entire
simulation loop then continues until the simulation is complete. See Simulation Phases in Dynamic Systems in* Using Simulink* for
more detailed information on how the engine executes a model. See Simulink Engine Interaction with C S-Functions for
a description of how the engine calls the S-function API during initialization
and simulation.

**How the Simulink Engine Performs Simulation**

An S-function comprises a set of *S-function
callback methods *that perform tasks required at each simulation
stage. During simulation of a model, at each simulation stage, the Simulink engine
calls the appropriate methods for each S-Function block in the model.
Tasks performed by S-function
callback methods include:

Initialization — Prior to the first simulation loop, the engine initializes the S-function, including:

Initializing the

`SimStruct`

, a simulation structure that contains information about the S-functionSetting the number and dimensions of input and output ports

Setting the block sample times

Allocating storage areas

Calculation of next sample hit — If you created a variable sample time block, this stage calculates the time of the next sample hit; that is, it calculates the next step size.

Calculation of outputs in the major time step — After this call is complete, all the block output ports are valid for the current time step.

Update of discrete states in the major time step — In this call, the block performs once-per-time-step activities such as updating discrete states.

Integration — This applies to models with continuous states and/or nonsampled zero crossings. If your S-function has continuous states, the engine calls the output and derivative portions of your S-function at minor time steps. This is so the solvers can compute the states for your S-function. If your S-function has nonsampled zero crossings, the engine also calls the output and zero-crossings portions of your S-function at minor time steps so that it can locate the zero crossings.

**Note:**See Simulation Phases in Dynamic Systems for an explanation of major and minor time steps.

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