Stability and Root Locus

Turn up the gain of a feedback controller and the closed-loop poles move. The root locus is the path they trace. It shows at a glance which gains give a fast, calm response and the gain at which the system tips into instability.

Controller gain K Plant pole a Plant pole b

The closed loop

With plant $G(s) = \dfrac{1}{s(s+a)(s+b)}$ under proportional gain $K$, the closed-loop poles are the roots of

$$s^3 + (a+b)\,s^2 + ab\,s + K = 0$$

By the Routh-Hurwitz test, the system stays stable while $K < (a+b)\,ab$.

What to notice

At low gain all three poles sit on the left, and the response is slow but calm. Raise the gain and two poles meet, break away from the real axis, and curve toward the right, so the response rings more. At the critical gain they cross the imaginary axis and the system oscillates without end, then grows. That is why simply turning up the gain eventually makes any real robot joint shake: every real system has extra lags like these.