Effect of feedback on stability

Feedback changes the location of the closed-loop poles (roots of $1 + G(s)H(s) = 0$), so it can make a system more stable or less stable.

• Stabilizing effect: an unstable plant $G(s) = \dfrac{1}{s-1}$ (pole at $+1$) with proportional feedback gain $K$ gives $\dfrac{K}{s-1+K}$; the pole moves to $s = 1-K$, which is stable for $K > 1$.
• Destabilizing effect: take $G(s) = \dfrac{K}{s(s+1)(s+2)}$ with unity feedback. The open-loop poles $0, -1, -2$ do not move with $K$, but the closed-loop characteristic equation is

$$s^3 + 3s^2 + 2s + K = 0$$

Routh array: row $s^1$ term $= \dfrac{3\times2 - K}{3}$. The system is stable only for $0 < K < 6$; at $K = 6$ it oscillates at $\omega = \sqrt{2}$ rad/s and for $K > 6$ it is unstable. So too much feedback gain can destabilize a system.