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Step response plot of dynamic system; step response data

`step(sys) `

step(sys,Tfinal)

step(sys,t)

step(sys1,sys2,...,sysN)

step(sys1,sys2,...,sysN,Tfinal)

step(sys1,sys2,...,sysN,t)

y = step(sys,t)

[y,t] = step(sys)

[y,t] = step(sys,Tfinal)

[y,t,x] = step(sys)

[y,t,x,ysd]
= step(sys)

[y,...] = step(sys,...,options)

`step`

calculates the step
response of a dynamic system. For the state-space case, zero initial
state is assumed. When it is invoked with no output arguments, this
function plots the step response on the screen.

`step(sys) `

plots the step
response of an arbitrary dynamic
system model, `sys`

. This model
can be continuous- or discrete-time, and SISO or MIMO. The step response
of multi-input systems is the collection of step responses for each
input channel. The duration of simulation is determined automatically,
based on the system poles and zeros.

`step(sys,Tfinal)`

simulates the step response
from `t = 0`

to the final time `t = Tfinal`

.
Express `Tfinal`

in the system time units, specified
in the `TimeUnit`

property of `sys`

.
For discrete-time systems with unspecified sample time (```
Ts
= -1
```

), `step`

interprets `Tfinal`

as
the number of sampling periods to simulate.

`step(sys,t)`

uses the user-supplied
time vector `t`

for simulation. Express `t`

in
the system time units, specified in the `TimeUnit`

property
of `sys`

. For discrete-time models, `t`

should
be of the form `Ti:Ts:Tf`

, where `Ts`

is
the sample time. For continuous-time models, `t`

should
be of the form `Ti:dt:Tf`

, where `dt`

becomes
the sample time of a discrete approximation to the continuous system
(see Algorithms). The `step`

command
always applies the step input at `t=0`

, regardless
of `Ti`

.

To plot the step response of several models `sys1`

,..., `sysN`

on
a single figure, use

`step(sys1,sys2,...,sysN)`

`step(sys1,sys2,...,sysN,Tfinal)`

`step(sys1,sys2,...,sysN,t)`

All of the systems plotted on a single plot must have the same number of inputs and outputs. You can, however, plot a mix of continuous- and discrete-time systems on a single plot. This syntax is useful to compare the step responses of multiple systems.

You can also specify a distinctive color, linestyle, marker, or all three for each system. For example,

step(sys1,'y:',sys2,'g--')

plots the step response of `sys1`

with a dotted
yellow line and the step response of `sys2`

with
a green dashed line.

When invoked with output arguments:

`y = step(sys,t)`

`[y,t] = step(sys)`

`[y,t] = step(sys,Tfinal)`

`[y,t,x] = step(sys) `

`step`

returns the output response `y`

,
the time vector `t`

used for simulation (if not supplied
as an input argument), and the state trajectories `x`

(for
state-space models only). No plot generates on the screen. For single-input
systems, `y`

has as many rows as time samples (length
of `t`

), and as many columns as outputs. In the multi-input
case, the step responses of each input channel are stacked up along
the third dimension of `y`

. The dimensions of `y`

are
then

$$(length\text{\hspace{0.17em}}of\text{\hspace{0.17em}}t)\times (number\text{\hspace{0.17em}}of\text{\hspace{0.17em}}outputs)\times (number\text{\hspace{0.17em}}of\text{\hspace{0.17em}}inputs)$$

and `y(:,:,j)`

gives the response to a unit
step command injected in the `j`

th input channel.
Similarly, the dimensions of `x`

are

$$(length\text{\hspace{0.17em}}of\text{\hspace{0.17em}}t)\times (number\text{\hspace{0.17em}}of\text{\hspace{0.17em}}states)\times (number\text{\hspace{0.17em}}of\text{\hspace{0.17em}}inputs)$$

For identified models (see `idlti`

and `idnlmodlel`

) ```
[y,t,x,ysd]
= step(sys)
```

also computes the standard deviation `ysd`

of
the response y (`ysd`

is empty if `sys`

does
not contain parameter covariance information).

`[y,...] = step(sys,...,options)`

specifies
additional options for computing the step response, such as the step
amplitude or input offset. Use `stepDataOptions`

to
create the option set `options`

.

You can change the properties of your plot, for example the units. For information on the ways to change properties of your plots, see Ways to Customize Plots.

Continuous-time models without internal delays are converted
to state space and discretized using zero-order hold on the inputs.
The sample time is chosen automatically based on the system dynamics,
except when a time vector `t = 0:dt:Tf`

is supplied
(`dt`

is then used as sampling period). The resulting
simulation time steps `t`

are equisampled with spacing `dt`

.

For systems with internal delays, Control System Toolbox™ software
uses variable step solvers. As a result, the time steps `t`

are
not equisampled.

[1] L.F. Shampine and P. Gahinet, "Delay-differential-algebraic
equations in control theory," *Applied Numerical Mathematics*,
Vol. 56, Issues 3–4, pp. 574–588.

`impulse`

| `initial`

| Linear System Analyzer | `lsim`

| `stepDataOptions`

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