Consider a hypothetical system with open-loop transfer function $G(s)$.
Place it in positive feedback with unit gain. (That is, take its output and directly add it to its input.)
The closed-loop system now satisfies $Y(s) = G(s) (X(s) + Y(s))$, i.e.:
$$\frac{Y(s)}{X(s)} = \frac{G(s)}{1 – G(s)}$$
This is, of course, simply Black's formula.
My questions:
If we have $G(s) = 2$, then this system stable? Why?
Intuitively, this system is (obviously!) unstable, because it is in positive unit feedback with gain > 1.
Yet according to Black's formula, it is actually a stable system with gain $2 / (1-2) = -2$!
(Notice that it has no poles in the right half-plane — in fact, it has no poles or zeros at all!)
What is there a discrepancy between the math and my intuition? Which one is correct and why?
What assumptions am I breaking exactly, if any?
Best Answer
This an an algebraic loop, not a dynamic one. The question of stability doesn't really mean much.