A purely discrete system is composed solely of discrete blocks
and can be modeled using either a fixed-step or a variable-step solver.
Simulating a discrete system requires that the simulator take a simulation
step at every sample time hit. For a *multirate discrete
system*—a system whose blocks Simulink^{®} samples
at different rates—the steps must occur at integer multiples
of each of the system sample times. Otherwise, the simulator might
miss key transitions in the states of the system. The step size that
the Simulink software chooses depends on the type of solver you
use to simulate the multirate system and on the fundamental sample
time.

The *fundamental sample time* of a multirate
discrete system is the largest double that is an integer divisor of
the actual sample times of the system. For example, suppose that a
system has sample times of 0.25 and 0.50 seconds. The fundamental
sample time in this case is 0.25 seconds. Suppose, instead, the sample
times are 0.50 and 0.75 seconds. The fundamental sample time is again
0.25 seconds.

The importance of the fundamental sample time directly relates to whether you direct the Simulink software to use a fixed-step or a variable-step discrete solver to solve your multirate discrete system. A fixed-step solver sets the simulation step size equal to the fundamental sample time of the discrete system. In contrast, a variable-step solver varies the step size to equal the distance between actual sample time hits.

The following diagram illustrates the difference between a fixed-step and a variable-step solver.

In the diagram, the arrows indicate simulation steps and circles represent sample time hits. As the diagram illustrates, a variable-step solver requires fewer simulation steps to simulate a system, if the fundamental sample time is less than any of the actual sample times of the system being simulated. On the other hand, a fixed-step solver requires less memory to implement and is faster if one of the system sample times is fundamental. This can be an advantage in applications that entail generating code from a Simulink model (using Simulink Coder™). In either case, the discrete solver provided by Simulink is optimized for discrete systems; however, you can simulate a purely discrete system with any one of the solvers and obtain equivalent results.

Consider the following example of a simple multirate system.
For this example, the DTF1 Discrete Transfer Fcn block's **Sample time** is set to [`1 0.1`

],
which gives it an offset of `0.1`

. The **Sample time** of the DTF2 Discrete Transfer Fcn
block is set to `0.7`

, with no offset. The solver
is set to a variable-step discrete solver.

Running the simulation and plotting the outputs using
the `stairs`

function

simOut = sim('ex_dtf','StopTime', '3'); t = simOut.find('tout') y = simOut.find('yout') stairs(t,y, '-*')

produces the following plot.

(For information on the `sim`

command. see Run Simulation Using the sim Command.
)

As the figure demonstrates, because the DTF1 block
has a `0.1`

offset, the DTF1 block
has no output until `t = 0.1`

. Similarly, the initial
conditions of the transfer functions are zero; therefore, the output
of DTF1, y(1), is zero before this time.

*Hybrid systems* contain both discrete and
continuous blocks and thus have both discrete and continuous states.
However, Simulink solvers treat any system that has both continuous
and discrete sample times as a hybrid system. For information on modeling
hybrid systems, see Modeling Hybrid Systems.

In block diagrams, the term hybrid applies to both hybrid systems
(mixed continuous-discrete systems) and systems with multiple sample
times (multirate systems). Such systems turn yellow in color when
you perform an **Update Diagram** with Sample
Time Display **Colors** turned '`on`

'.
As an example, consider the following model that contains an atomic
subsystem, "Discrete Cruise Controller", and a virtual
subsystem, "Car Dynamics". (See `ex_execution_order`

.)

**Car Model**

With the **Sample Time** option set to **All**,
an **Update Diagram** turns the virtual subsystem
yellow, indicating that it is a hybrid subsystem. In this case, the
subsystem is a true hybrid system since it has both continuous and
discrete sample times. As shown below, the discrete input signal,
D1, combines with the continuous velocity signal, v, to produce a
continuous input to the integrator.

**Car Model after an Update Diagram**

**Car Dynamics Subsystem after an Update Diagram**

Now consider a multirate subsystem that contains three Sine Wave source blocks, each of which has a unique sample time — 0.2, 0.3, and 0.4, respectively.

**Multirate Subsystem after an Update Diagram**

An **Update Diagram** turns the subsystem
yellow because the subsystem contains more than one sample time. As
shown in the block diagram, the Sine Wave blocks have
discrete sample times D1, D2, and D3 and the output signal is fixed
in minor step.

In assessing a system for multiple sample times, Simulink does not consider either constant [inf, 0] or asynchronous [–1, –n] sample times. Thus a subsystem consisting of one block that outputs constant value and one block with a discrete sample time will not be designated as hybrid.

The hybrid annotation and coloring are very useful for evaluating whether or not the subsystems in your model have inherited the correct or expected sample times.

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