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:

$$\begin{array}{cc}\dot{x}=Ax,& x(0)=x_0\\ y=Cx& \end{array}$$

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

Response of State-Space Model to Initial Condition

Plot the response of the following state-space model:

$$\begin{array}{rcl}\left[\begin{array}{l}{\underset{}{\overset{\dot{}}{x}}}_1\\ {\underset{}{\overset{\dot{}}{x}}}_2\end{array}\right]& =& \left[\begin{array}{cc}-0.5572& -0.7814\\ 0.7814& 0\end{array}\right]\left[\begin{array}{l}{x}_1\\ x_2\end{array}\right]\\ y& =& \left[\begin{array}{cc}1.9691& 6.4493\end{array}\right]\left[\begin{array}{l}{x}_1\\ x_2\end{array}\right].\end{array}$$

Take the following initial condition:

$$x(0)=\left[\begin{array}{l}1\\ 0\end{array}\right].$$

a = [-0.5572, -0.7814; 0.7814, 0];
c = [1.9691  6.4493];
x0 = [1 ; 0];

sys = ss(a,[],c,[]);
initial(sys,x0)

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.