# How can I find the transfer function of the following block diagram?

I've the following image and I want to find the transfer function from input $$x(t)$$ to output $$y(t)$$. I know that I have to apply Laplace Transform, so the integrator becomes $$\dfrac{1}{s}$$, but I don't know what to do with the numbers $$a$$ and $$b$$. Should it still be the same?

If a and b still the same, then I found that $$H(s) = \dfrac{Y(s)}{X(s)} = s^2-as-b$$.

In such cases it can be useful to introduce auxiliary variables at the input of the integrators. For the given diagram you could use a signal $$u(t)$$ at the input of the second integrator. The equation for its Laplace transform $$U(s)$$ becomes
$$U(s)=\frac{1}{s}\big[X(s)-bY(s)\big]-aY(s)\tag{1}$$
You need another equation relating $$U(s)$$ to $$Y(s)$$, but that one is trivial. From those two equations you can express $$Y(s)$$ in terms of $$X(s)$$, and in this way you can obtain the transfer function $$H(s)=Y(s)/X(s)$$. The solution in your question is not correct. The correct solution must be a rational function, not a polynomial.
• I got it. So the correct solution should be the inverse of the one I gave changing some signals, right? $H(s) = \dfrac{1}{s^2+as+b}$. Another question regards the numbers $a$ and $b$, when using Laplace Transform the numbers still the same or I should use Laplace on them? – FY Gamer Jul 10 '20 at 12:40
• @FYGamer: Try to answer this yourself: what is the Laplace transform of $ax(t)$? – Matt L. Jul 10 '20 at 12:48