Closed Loop Systems

A closed-loop control system is one in which the output is continuously measured, compared to the desired reference value, and the resulting difference (the error signal) is used to drive the controller and correct the output. This fundamental principle of feedback transforms a simple open-loop system into a self-correcting mechanism capable of handling disturbances, parameter variations, and modeling inaccuracies.

Closed-loop systems form the backbone of modern control engineering, from PID controllers in industrial plants to autopilots in aircraft. For GATE aspirants, the closed-loop transfer function derivation, the role of the error signal, and the trade-off between stability and performance are among the most frequently tested concepts.

Core Concept Explanation

The defining element of a closed-loop system is the feedback path. The actual output is measured by a sensor and returned to the input side where it is compared to the reference input at a summing junction (also called a comparator). The difference between the reference and the fed-back signal is the error signal e(t) = r(t) - b(t), where b(t) is the feedback signal. This error signal is what the controller acts upon to drive the plant.

The self-correcting behavior arises naturally from this structure. If a disturbance causes the output to rise above the desired value, the feedback signal b(t) increases, the error e(t) decreases, the controller reduces its output, and the plant output is brought back towards the reference. Conversely, if the output drops, the error increases, the controller commands more effort, and the output is restored. This automatic compensation is the fundamental advantage of feedback over open-loop control.

The feedback can be negative feedback (subtracted from reference, as in most control systems) or positive feedback (added to reference). Negative feedback provides stability and error reduction. Positive feedback amplifies deviations and leads to instability or is used deliberately in oscillator circuits.

Mathematical Expression

For a standard unity feedback system where H(s) = 1, the closed-loop transfer function (CLTF) is derived from the block diagram algebra:

E(s) = R(s) - C(s), and C(s) = Gc(s) * Gp(s) * E(s). Substituting:

C(s) = Gc(s)*Gp(s) * [R(s) - C(s)] which gives the closed-loop transfer function:

C(s)/R(s) = G(s) / [1 + G(s)] where G(s) = Gc(s)*Gp(s) is the open-loop transfer function. For a non-unity feedback system with sensor H(s):

C(s)/R(s) = G(s) / [1 + G(s)*H(s)]. The denominator 1 + G(s)*H(s) = 0 is the characteristic equation, and its roots (the closed-loop poles) determine system stability and transient response.

Practical Understanding

Negative feedback in closed-loop systems provides several key benefits: reduced steady-state error, improved disturbance rejection, reduced sensitivity to plant parameter variations, and the ability to control unstable plants (under appropriate conditions). The steady-state error for a type-1 system with a step input can be reduced to zero using integral control within the feedback loop.

However, feedback also introduces the risk of instability. Increasing the controller gain G improves speed of response and reduces error but can cause the closed-loop poles to move into the right half of the s-plane, making the system unstable. This trade-off between performance and stability is the central design challenge of classical control theory, analyzed using Bode plots, Nyquist criterion, and root locus techniques.

Given:
Open-loop transfer function G(s) = 10 / (s+2)
Unity feedback H(s) = 1
Input: unit step R(s) = 1/s

Why this formula applies:
CLTF = G(s) / [1 + G(s)*H(s)]
Steady-state output by final value theorem: lim s->0 of s*C(s)

Formula:
C(s)/R(s) = G(s) / [1 + G(s)]
Steady-state value = lim(s->0) [s * CLTF * R(s)]

Substitution:
CLTF = [10/(s+2)] / [1 + 10/(s+2)]
     = 10/(s+2) / [(s+2+10)/(s+2)]
     = 10 / (s + 12)

C(s) = [10/(s+12)] * (1/s)

Calculation:
Steady-state = lim(s->0) s * 10/[s(s+12)]
= 10/12 = 0.833

Final Answer:
Closed-loop pole is at s = -12 (stable, faster than open-loop pole at s = -2)
Steady-state output = 0.833 for unit step
Steady-state error = 1 - 0.833 = 0.167
Feedback increased speed (pole moved from -2 to -12) but error remains due to proportional control only.

Exam Tip: The characteristic equation for a unity feedback system is 1 + G(s) = 0, NOT G(s) = 0. GATE questions frequently test whether you correctly identify closed-loop poles versus open-loop poles. Also: increasing gain reduces steady-state error but can cause instability. The gain-stability trade-off is tested in root locus problems.

• The error signal e(t) = r(t) - b(t) is the key signal driving the controller. It represents the instantaneous deviation of the output from the desired reference.
• Closed-loop transfer function (unity feedback): C(s)/R(s) = G(s) / [1 + G(s)]. The characteristic equation is 1 + G(s) = 0.
• Negative feedback reduces steady-state error, improves disturbance rejection, and reduces sensitivity to plant variations.
• Increasing loop gain G improves speed and reduces error but moves closed-loop poles towards instability.
• The closed-loop pole locations determine transient response (damping, natural frequency) and stability.

Quick Revision

• Closed-loop: output is fed back and compared to reference. Error signal e(t) = r(t) - b(t) drives controller.
• CLTF (unity feedback): G(s)/[1+G(s)]. CLTF (non-unity): G(s)/[1+G(s)H(s)].
• Characteristic equation: 1 + G(s)H(s) = 0. Closed-loop poles from this equation determine stability.
• Advantages over open-loop: disturbance rejection, parameter insensitivity, reduced steady-state error.
• Disadvantage: risk of instability at high gain. Trade-off between performance (speed, accuracy) and stability.
• Negative feedback: stable, error-reducing. Positive feedback: used in oscillators, unstable for control.
• GATE trap: closed-loop pole is NOT the same as open-loop pole. Always form 1+G(s)=0 for closed-loop poles.