Feedback Control Basics
Open and closed loops, the block diagram, error, steady-state error and the shared oven example.
What a control system does
A control system makes a physical quantity (a temperature, a speed, a position, a pressure, a level) follow a target in spite of disturbances and uncertainty. A thermostat, a cruise control, a robot joint, a water-level float and a PLC loop are all the same idea: measure the output, compare it with the target, and act on the difference. This guide builds the subject from the first principles, using two examples throughout: an oven (a slow process with a dead time, the classic process-control loop) and a mass-spring-damper servo (a fast second-order mechanical system). Every response and number is computed from the equations.
The feedback loop
- Setpoint (reference) r: the target.
- Process variable (output) y: what is measured; here the oven temperature rise above ambient.
- Error e = r − y: the signal the controller works on.
- Controller: turns the error into a command u (on-off, PID, state feedback, model predictive control).
- Actuator and process: the heater and the oven, with gain K, time constant τ and a dead time θ.
- Disturbance d: anything else that moves the output: a door opened, a cold load, a supply-voltage change.
- Sensor: measures y, with its own lag, noise and error.
Open loop against closed loop
An open-loop controller sets the command from a model and never looks at the result (a toaster timer, a washing machine cycle, a stepper motor moving a known number of steps). It is simple and cannot be unstable, but it cannot correct for anything the model got wrong. A closed-loop (feedback) controller measures the output and corrects it, so it copes with disturbances and model errors, at the price of a sensor and of possible instability when the loop is too aggressive for the delays in it.
Worked example: the oven
Worked example: raise the oven by 20 °C
The oven is a first-order lag with gain K = 2 °C per percent of heater power, time constant τ = 120 s and a dead time θ = 15 s. The target is 20 °C above ambient.
- Open loop: the model says u = 20 ÷ 2 = 10 percent. But if the door seal ages and the real gain falls 20 percent to K = 1.6, the oven only reaches 1.6 × 10 = 16 °C: a permanent error of 4 °C that nobody notices.
- Proportional feedback u = Kp e with Kp = 2 %/°C: the output settles where the error just supplies the power, y = K Kp ÷ (1 + K Kp) × r = 0.80 × 20 = 16.0 °C, a steady error of r ÷ (1 + K Kp) = 4.0 °C. With the 20 percent lower gain the error only rises to 4.8 °C: feedback has made the result about five times less sensitive to the gain.
- A higher gain (Kp = 3.6) cuts the error to 2.4 °C but a high gain with a delay in the loop eventually makes the loop oscillate (see Stability: Routh, Root Locus and Margins).
- A disturbance of 5 percent of heater power (a cold load) moves the output by only K d ÷ (1 + K Kp) = 2.0 °C with the proportional controller, against 10 °C in open loop.
- Add integral action (a PI controller, PID Control and Tuning): the same disturbance causes a peak deviation of 2.1 °C and the controller removes it completely (final deviation 0.07 °C), with the setpoint step reaching 20 °C with an overshoot of 5 percent.
On-off control
The simplest feedback controller switches the heater fully on below the setpoint and off above it, with a small band (hysteresis) to stop it chattering. It is cheap and robust (a thermostat, a pressure switch) but the output always cycles. Dead time and lag make the cycle larger than the band:
Worked example: an on-off oven
- A 20 percent heater heats the oven at up to K u ÷ τ = 2 × 20 ÷ 120 = 0.33 °C/s.
- During the 15 s dead time the temperature keeps rising after the switch: 0.33 × 15 = 5.0 °C on each side.
- The temperature therefore swings about 2 + 2 × 5.0 = 12 °C peak to peak, though the hysteresis is only 2 °C. A PID loop of PID Control and Tuning holds the oven within a fraction of a degree.
Steady-state error and the integral
A proportional controller needs an error to produce its output, so it leaves an offset. The cure is integral action: the controller adds up the error over time, so it keeps increasing the output until the error is exactly zero. Any controller whose loop contains an integrator (type 1 or higher) has zero steady-state error to a step. The price is a phase lag that reduces stability, which is why the integral time is set long compared with the process dynamics.
Types of control and where they are used
| Method | Use | Notes |
|---|---|---|
| On-off | thermostats, pressure switches, low-cost devices | simple; the output cycles |
| PID | about 90 percent of industrial loops: temperature, flow, pressure, level, speed | simple, robust, well understood (PID Control and Tuning) |
| Feedforward and cascade | known disturbances, nested loops | added to PID (Practical Control: Windup, Feedforward and Cascade) |
| State feedback and observers | multivariable and fast systems: aircraft, robots, servo drives | needs a model; optimal (LQR) designs |
| Model predictive control (MPC) | refineries, chemical plants, batch processes with constraints | optimizes over a horizon; needs computing and a good model |
| Adaptive, fuzzy, learning controllers | changing or hard-to-model plants | more complex to validate and certify |
On the PLC side, loops are built from instructions such as PID blocks (see PLC and Logic Control) and the practical use of drives is in Motion Control and Actuators.
Related
- Next: Laplace Transforms and Transfer Functions.
- PID Tuner.
- Servo Motors and Tuning.
- Control Systems: the whole guide.
FAQ
Why not just use a high gain?
High gain reduces the error, but with delay or lag in the loop it makes the loop oscillate and eventually go unstable. The gain is limited by the phase margin.
When do I not need feedback?
When the process is accurate, repeatable and has no disturbances (a stepper moving a light load), or when feedback is too expensive. Even then a safety limit is a feedback.
What does steady-state error mean?
The remaining difference between the setpoint and the output after the transients die out. An integrator drives it to zero for a step setpoint.
What is the difference between control and automation?
Control keeps a quantity on target (continuous variables); automation sequences machines and events (discrete logic). Most real machines have both.
References
- Ogata, K., Modern Control Engineering, Pearson; Franklin, G. F., Powell, J. D., Emami-Naeini, A., Feedback Control of Dynamic Systems, Pearson.
- Åström, K. J., Murray, R. M., Feedback Systems: An Introduction for Scientists and Engineers, Princeton University Press (free online).
- Dorf, R. C., Bishop, R. H., Modern Control Systems; Seborg, D. E., et al., Process Dynamics and Control, Wiley.
Last reviewed: October 2026. Oven parameters are an illustrative model.