Q2Control System

Consider a system with forward-path gain G(s) and feedback gain H(s), where the feedback signal is added to (rather than subtracted from) the reference. The error signal is E = R + H·C (instead of R - H·C for negative feedback), and C = G·E. Hence:

Comparing with the negative-feedback case, C/R = G/(1+GH), the sign of the GH term is reversed. This has a critical effect on stability: as GH(s) approaches +1 at some frequency, the denominator 1 - GH(s) approaches zero, causing the closed-loop gain to become extremely large (theoretically infinite), which corresponds to the system becoming marginally stable or unstable at that operating condition. Physically this means positive feedback regenerates (reinforces) the error signal rather than cancelling it — a small increase in output further increases the input to the forward path, creating a self-reinforcing (regenerative) loop. If GH ≥ 1, the system output grows without bound (instability), unlike negative feedback where increasing loop gain generally improves accuracy and bandwidth while retaining stability (up to the point of excessive phase lag). For this reason, positive feedback is deliberately used only in specialized applications such as oscillators (where sustained, undamped oscillation is desired, i.e. GH = 1 exactly satisfies the Barkhausen criterion) and Schmitt trigger comparators, while conventional control systems almost universally employ negative feedback to guarantee stability, disturbance rejection and reduced sensitivity to parameter variations.

Sensitivity comparison. The sensitivity of the closed-loop transfer function T = C/R to variations in the forward gain G, defined as S = (∂T/T)/(∂G/G), is S = 1/(1-GH) for positive feedback versus S = 1/(1+GH) for negative feedback. For large loop gain (|GH| ≫ 1), negative feedback drives sensitivity toward zero (S → 1/GH, very small), making the closed-loop response almost independent of drift or uncertainty in G — this is precisely why negative feedback is prized in amplifier and control-system design. Positive feedback does the opposite: as GH approaches 1, sensitivity S = 1/(1-GH) grows without bound, meaning the closed-loop response becomes extremely sensitive to even tiny changes in G or H, compounding the stability risk already noted above.