How can I set the initial value for the output, "y[1]", when using the LSIM function in Control Systems Toolbox 8.1 (R2008a)?

I am using the function LSIM with a transfer function model. The following code produces an output that initializes at zero even when the x0 argument is set to three:

t = linspace(0,100*pi,1024);

u = 3 + sin(t);

H = tf([1 1],[1 3 3 2]);

lsim(H,u,t,3,'foh')

Answer 1

[MathWorks Support Team]

When using a transfer function system model, the ability to set an initial value for the output is not available in Control System Toolbox 8.1 (R2008a).

To work around this issue, use a state-space system model. Set the initial state "x0" so that your initial output takes the desired value based on the following state-space equation:

y[0] = C*x[0] + D*u[0]

Consider the following example:

t = linspace(0,100*pi,1024);

u = 3 + sin(t);

H = tf([1 1],[1 3 3 2]);

sys = ss(H)

The model "sys" has vales [0 0.5 0.5] for "C" and zero for "D". To achieve an initial output of "3", you can use an initial state of [0;3;3].

lsim(H,u,t,[0;3;3],'foh')

[Alberto Mora]

How do you define the initial condition of the states?

According to your example, the initial value state could be also [0; 12; -6].

I think that there are infinity combination of the initial value of the states that gives y(0)=3.