Closed Loop

Closed-loop (feedback) control measures the output and corrects the error. Learn the feedback equation, loop gain, sensitivity, disturbance rejection, steady-state error, and the stability trade-off feedback introduces.

A closed-loop system continuously compares the measured output with the desired value and acts on the difference — the error. A thermostat measures room temperature and switches the heater; cruise control measures speed and adjusts the throttle.

Feedback makes systems accurate despite imperfect models and disturbances, can make slow systems faster, and can even stabilise unstable plants. The price is that too much feedback gain can cause oscillation.

An analogy

Driving a car is closed-loop control: your eyes measure the lane position, your brain computes the error and your hands correct the steering — continuously, so wind and road camber barely matter.

The feedback loop

Reference → summing junction → controller → plant → output, with a sensor feeding the output back to the summing junction (usually subtracted — negative feedback). The controller acts on the error, not on the reference directly.

Why high loop gain helps

When GH ≫ 1, T ≈ 1/H: the output depends on the sensor, not the plant. Plant variations and disturbances are divided by (1 + GH). This is why op-amp circuits with huge open-loop gain behave exactly as their resistor ratios dictate.

Steady-state error and system type

With proportional control of a plant without integrators (type 0), a step leaves a residual error 1/(1 + K_p). Each integrator in the loop removes error for one more class of input: type 1 has zero step error, type 2 zero ramp error. This is the reason controllers add integral action.

The cost: stability and noise

Feedback adds phase lag around the loop. When the loop gain is ≥ 1 at a frequency where the phase lag reaches 180°, the feedback becomes positive and the system oscillates. High gain also feeds sensor noise into the actuator. Control design balances performance, robustness and noise — the 'waterbed effect' says you can't reduce sensitivity at all frequencies at once.

Formulas

• Closed-loop transfer function: T = Y/R = G / (1 + G·H)
• Sensitivity: S = 1 / (1 + G·H)
• Disturbance at output: Y_d = D / (1 + G·H)
• Step error, type-0 loop: e_ss = 1 / (1 + K_p), K_p = lim_{s→0} G(s)H(s)

Worked example: Desensitising gain changes

Given: Forward gain A = 100; Feedback β = 0.1; A drops to 50 (−50 %)

Find: Closed-loop gain before and after.

• T = 100/(1 + 100 × 0.1) = 9.09
• T' = 50/(1 + 50 × 0.1) = 8.33
• Change = −8.3 % (vs −50 % open loop)

Answer: 9.09 → 8.33: feedback cut the effect six-fold

Worked example: Steady-state error

Given: G(s) = 10/(s + 1); Unity feedback, unit step

Find: Steady-state error.

• K_p = G(0) = 10
• e_ss = 1/(1 + 10) = 0.091

Answer: 9.1 % error — add integral action to remove it

Common mistakes

• Wrong: Assuming more gain is always better. — Right: Higher gain reduces error but eventually causes overshoot and instability.
• Wrong: Using positive feedback in a regulator. — Right: Regulation needs negative feedback; positive feedback is used deliberately only for oscillators and latches.

Applications

• Thermostats & HVAC: Temperature sensors close the loop around heaters and chillers.
• Voltage regulators: Every power supply compares its output with a reference and corrects the duty cycle.
• Grid frequency: Governors and AGC are huge feedback loops balancing generation with load.

Key takeaways

• Closed loop measures the output and acts on the error.
• T = G/(1 + GH); high loop gain makes T ≈ 1/H.
• Disturbances and plant variations are divided by (1 + GH).
• Integrators remove steady-state error; excessive gain threatens stability.

Going further

• Bode's sensitivity integral: For stable open-loop systems, ∫₀^∞ ln|S(jω)| dω = 0: pushing sensitivity down at low frequencies forces it up elsewhere. This fundamental limit shapes every loop design.
• Two-degree-of-freedom control: A reference pre-filter shapes the response to setpoint changes separately from the feedback controller, which is tuned for disturbance rejection and robustness.
• Cascade control: An inner fast loop (current, flow) inside an outer slow loop (speed, level) rejects inner disturbances quickly and linearises the plant for the outer loop — standard in motor drives.