# Calculate transfer function of two parallel transfer functions in a feedback loop

I have rational transfer functions:

$$H_1(z) = \frac{ b_0 + b_1z^{-1} + b_2z^{-2} }{a_0 + a_1z^{-1} + a_2z^{-2}}$$

$$H_2(z) = \frac{ q_0 + q_1z^{-1} + q_2z^{-2} }{p_0 + p_1z^{-1} + p_2z^{-2}}$$

And they are being used in the following feedback loop:

Sometimes this system blows up, and other times it is stable depending $$H_1$$ and $$H_2$$. I'm guessing that when it blows up, the poles of the combined system are outside the unit circle.

How do I calculate the transfer function of the system so I can start re-arranging it?

Your two transfer functions are in parallel, i.e they simply add up. So your feedback transfer function is simply $$G(z) = H_1(z)+H_2(z)$$. You want to makes sure that the magnitude of $$G(z)$$ is smaller than one.
$$H(z) = \frac{Y(z)}{X(z)} = \frac{1}{1+H_1(z)+H_2(z)}$$
• Thanks, although I disagree on the validity of the condition $|G(z)| < 1$ for all $z$ on the unit circle. – onthefritz Sep 11 '19 at 20:59