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

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Tao
Tao 2015-8-11
评论： Tao 2015-8-11
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!

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Sebastian Castro
Sebastian Castro 2015-8-11
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
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