# initial

System response to initial states of state-space model

## Syntax

## Description

For state-space and sparse state-space models, `initial`

computes the unforced system response *y* to initial states
*x _{init}*.

Continuous time:

$$\begin{array}{cc}\dot{x}=d{x}_{0}+A(x-{x}_{0}),& x({t}_{0})={x}_{init}\\ y={y}_{0}+C(x-{x}_{0})& \end{array}$$

Discrete time:

$$\begin{array}{cc}x[k+1]=d{x}_{0}+A(x[k]-{x}_{0})& x[{k}_{0}]={x}_{init}\\ y={y}_{0}+C(x[k]-{x}_{0})& \end{array}$$

This is the system response when *u*(*t*) is maintained
at the offset value *u _{0}*.

For linear time-varying or linear parameter-varying state-space models,
`initial`

computes the response with initial state
*x _{init}*, initial parameters

*p*(LPV models), and input held to the offset value (

_{init}*u*(

*t*) =

*u*

_{0}(

*t*) or

*u*(

*t*) =

*u*

_{0}(

*t*,

*p*), which corresponds to the initial condition response of the local linear dynamics.

`initial(___)`

plots the initial condition response of
`sys`

with default plotting options for all of the previous input
argument combinations. For more plot customization options, use `initialplot`

.

To plot responses for multiple dynamic systems on the same plot, you can specify

`sys`

as a comma-separated list of models. For example,`initial(sys1,sys2,sys3)`

plots the responses for three models on the same plot.To specify a color, line style, and marker for each system in the plot, specify a

`LineSpec`

value for each system. For example,`initial(sys1,LineSpec1,sys2,LineSpec2)`

plots two models and specifies their plot style. For more information on specifying a`LineSpec`

value, see`initialplot`

.

## Examples

## Input Arguments

## Output Arguments

## Version History

**Introduced before R2006a**

## See Also

`initialplot`

| `impulse`

| `lsim`

| Linear System Analyzer | `step `