# initial

Initial condition response of state-space model

## Syntax

initial(sys,x0)initial(sys,x0,Tfinal)initial(sys,x0,t)initial(sys1,sys2,...,sysN,x0)initial(sys1,sys2,...,sysN,x0,Tfinal)initial(sys1,sys2,...,sysN,x0,t)[y,t,x] = initial(sys,x0)[y,t,x] = initial(sys,x0,Tfinal)[y,t,x] = initial(sys,x0,t)

## Description

initial(sys,x0) calculates the unforced response of a state-space (ss) model sys with an initial condition on the states specified by the vector x0:

$\begin{array}{cc}\stackrel{˙}{x}=Ax,& x\left(0\right)={x}_{0}\\ y=Cx& \end{array}$

This function is applicable to either continuous- or discrete-time models. When invoked without output arguments, initial plots the initial condition response on the screen.

initial(sys,x0,Tfinal) simulates the 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), initial interprets Tfinal as the number of sampling periods to simulate.

initial(sys,x0,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 0:Ts:Tf, where Ts is the sample time. For continuous-time models, t should be of the form 0:dt:Tf, where dt becomes the sample time of a discrete approximation to the continuous system (see impulse).

To plot the initial condition responses of several LTI models on a single figure, use

initial(sys1,sys2,...,sysN,x0)

initial(sys1,sys2,...,sysN,x0,Tfinal)

initial(sys1,sys2,...,sysN,x0,t)

(see impulse for details).

When invoked with output arguments,

[y,t,x] = initial(sys,x0)

[y,t,x] = initial(sys,x0,Tfinal)

[y,t,x] = initial(sys,x0,t)

return the output response y, the time vector t used for simulation, and the state trajectories x. No plot is drawn on the screen. The array y has as many rows as time samples (length of t) and as many columns as outputs. Similarly, x has length(t) rows and as many columns as states.

## Examples

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### Response of State-Space Model to Initial Condition

Plot the response of the following state-space model:

Take the following initial condition:

a = [-0.5572, -0.7814; 0.7814, 0]; c = [1.9691 6.4493]; x0 = [1 ; 0]; sys = ss(a,[],c,[]); initial(sys,x0) 

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### Tips

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.

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