# Transfer function of a System Diagram

I would like to have some help on this system:

I have to consider that all initial conditions at t = 0 are null. I've tried some things and I have some answers but I don't know if they are true. Could someone shed some light on this?

Here is what I've tried:

Is this right? or wrong?

Also, is this system asymptotically stable if and only if a <0 ? Or does B effect on that?

• I can't see anything particularly wrong with it. Did you expect something else or more? Do you perhaps want to add another effort with a different answer and see where it went wrong? – Sven B Jun 17 '18 at 21:21
• I just want to know if this equation is the right way of representing this diagram, because I wasn't sure if it was, all I want is the H(s) of the diagram, if it is wrong, another option would be good :D – Matheus Ribeiro Jun 17 '18 at 21:25
• I think you're good. The input of the integrator is $bx(t) - ay(t)$ so the output is $y(t) = \int_0^t (bx(\tau) - ay(\tau)) d\tau$ which is a differential equation you then solve. The only comments I'd have are only minor. – Sven B Jun 17 '18 at 21:35
• The block diagram represents the classical active first-order lowpass. – LvW Jun 18 '18 at 7:32
• added a question to the main question – Matheus Ribeiro Jun 18 '18 at 16:09

• @Tri "sampling frequency of the control system" This is incorrect, the system is of continuous time. There is no sampling involved here, and the system is stable for any non-infinite $b$. You could argue that a discrete version of the system could be unstable, but then a discretization method (zoh, euler, tustin, ...) should be mentioned explicitly, as some will guarantee stability and some won't. – Vicente Cunha Jun 19 '18 at 12:11