Feedback Systems

A feedback system feeds a portion of the output back to the input, modifying what the system actually processes. Feedback is the mechanism behind proportional-integral-derivative controllers, phase-locked loops, and operational amplifier circuits, all of which achieve precision by continuously correcting the error between desired and actual output.

Core Concept

Open-loop systems process the input once through the forward path. Closed-loop systems route the output back to the input summing junction. In negative feedback, the fed-back signal is subtracted from the reference, creating an error signal that the forward-path system tries to minimize.

The closed-loop transfer function T(s) equals G(s) divided by [1 + G(s)H(s)]. The denominator [1 + G(s)H(s)] is called the characteristic equation. Setting it to zero gives the closed-loop poles. These poles determine stability, transient response speed, and damping. Negative feedback moves poles and can stabilize an otherwise unstable open-loop system.

Unity feedback is the special case where H(s) = 1. Then T(s) = G(s)/[1+G(s)]. This simplification appears in most introductory control examples. Positive feedback, where the fed-back signal is added rather than subtracted, tends to destabilize systems and appears in oscillators and latches.

Key Formula

Closed-loop transfer function with negative feedback: T(s) = G(s) / [1 + G(s)H(s)].

G(s): forward-path transfer function. H(s): feedback-path transfer function. G(s)*H(s): open-loop transfer function (loop gain). 1 + G(s)H(s) = 0: characteristic equation, roots are closed-loop poles.

Unity feedback steady-state error for step input: e_ss = 1 / [1 + Kp], where Kp = lim_{s->0} G(s) is the position constant.

Problem: Find closed-loop T(s) for G(s) = 10/(s+1), H(s) = 1 (unity feedback).

Step 1 — Apply formula:
  T(s) = G(s) / [1 + G(s)*H(s)]
       = [10/(s+1)] / [1 + 10/(s+1)]

Step 2 — Simplify denominator:
  1 + 10/(s+1) = (s+1+10)/(s+1) = (s+11)/(s+1)

Step 3 — Divide:
  T(s) = [10/(s+1)] * [(s+1)/(s+11)]
       = 10 / (s+11)

Step 4 — Closed-loop pole:
  Pole at s = -11 (shifted left from open-loop pole at s=-1)

Step 5 — DC gain:
  T(0) = 10/11 ≈ 0.909 (not exactly 1; there is steady-state error)

Final Answer: T(s) = 10/(s+11)
Feedback moved the pole from -1 to -11, speeding up the response tenfold.

Exam Tip: The closed-loop characteristic equation is 1 + G(s)H(s) = 0. For GATE stability questions using Routh-Hurwitz, always expand this polynomial completely before building the Routh array. High gain G can push poles into the right-half plane, causing instability. For Anna University problems, note that increasing forward gain always reduces steady-state error for type-1 systems but can cause oscillations if gain is too high.

Properties Summary

• Closed-loop T(s) = G(s) / [1 + G(s)H(s)] for negative feedback.
• Characteristic equation: 1 + G(s)H(s) = 0; roots determine closed-loop pole locations.
• Negative feedback reduces sensitivity: dT/T = (1/(1+GH)) * dG/G.
• Unity feedback DC gain: T(0) = G(0)/(1+G(0)); equals 1 only when G(0) -> infinity (integrator in forward path).
• Positive feedback: T(s) = G(s) / [1 - G(s)H(s)]; used in oscillator design.
• Bandwidth of closed-loop increases with feedback gain; transient speed improves.

Quick Revision

• Feedback subtracts (negative) or adds (positive) a scaled output back to the input.
• Closed-loop poles are roots of 1 + G(s)H(s) = 0.
• Negative feedback can stabilize open-loop unstable systems.
• High loop gain reduces steady-state error but risks instability.
• Unity feedback T(0) < 1 unless the forward path contains an integrator.
• Sensitivity to plant parameter changes drops by factor (1+GH) with feedback.
• Exam trap: confusing the open-loop transfer function G(s)H(s) with the closed-loop T(s); the characteristic equation uses G(s)H(s), not T(s).