How to derive the state-space model in which the second derivative of the output of the system is needed?

Dear all:

My transfer function is:

Y/U = (a1s+a0)/(b4s^4+b3s^3+b2s^2+b1s); s is the laplace symbol, U is the single input, and Y is the single output.

Now I want a state-space model where I can have the second derivative of Y, which is dotdot(y) (dotdot is second derivative with respect to time).

Any help is appreciated!

Answer 1

[Sebastian Castro]

If the above is the TF for Y, then the second derivative of Y is just Ys^2, so the TF would be:

Ys^2/U = (a1s^3+a0s^2)/(b4s^4+b3s^3+b2s^2+b1s);

In MATLAB terms, you could either mathematically rework these into a state-space, or if you're lazy like me, make both the transfer functions and then convert to State-Space. NOTE: This requires Control System Toolbox.

Y = tf([a1 a0],[b4 b3 b2 b1 0]);
Ydd = Y*tf('s')^2;

G = [Y;Ydd];

Gss = ss(G);

... and there you have it, a state-space with 2 outputs: The first being Y, the second being Y doubel dot. Hope this helped.

- Sebastian

[Tao]

Thanks! I will first have a try and let you know.

[Tao]

You wrote Ydd = Y*tf('s')^2, is this right? Why is it the square of the transfer function? Thank you.

[Sebastian Castro]

Oh no, I just multiplied Y by s^2. To define "s", I made it into its own transfer function tf('s').

[Tao]

Hi, thanks! It works! But do you know anyway that only need one state-space model that can still derive the dotdoty? Thank you!

[Sebastian Castro]

Oh yes, I thought you wanted both Y and Ydotdot.

In that case, just do G = ss(Ydd);

[Tao]

Thank you a lot.