Introduction to control systems
A proportional control that is too gentle leaves the oven seven degrees below the setpoint. One that is too strong, with a twenty-second delay, makes it oscillate like a thermostat.
01Open loop and closed loop
A control system makes a physical quantity, the controlled variable, follow a desired value, the reference (or setpoint), by acting on a plant. In open loop, the controller applies an action calculated in advance, like a toaster that heats for a fixed time: it works if the model is exact and there are no disturbances. In closed loop, a sensor measures the controlled variable and the controller acts on the difference from the reference, the error. It is feedback applied to any plant: motors, ovens, tanks, aircraft.
| Element | Function | Example in an oven |
|---|---|---|
| Reference | The desired value | 100 °C |
| Controller | Computes the action from the error | A microcontroller or a temperature controller |
| Actuator | Applies the action to the plant | A solid-state relay and a heating element |
| Plant | The process being controlled | The oven chamber |
| Sensor | Measures the controlled variable | A thermocouple or a Pt100 |
| Disturbance | Whatever pushes the variable off unintentionally | Opening the door |
02On-off and proportional
The simplest controller is the on-off controller: it turns the heating on if the temperature is below the reference and off if it is above. So that the relay does not switch nonstop around the point, it has a band of hysteresis. The temperature always oscillates, and the more so the greater the delay between the action and the measurement.
The proportional controller grades the action according to the error: \( u = K_p\, e \). It does not oscillate if the gain is moderate, but it leaves a steady-state error: to maintain the required power there has to be an error. For a process with gain \( K \) and limited action:
An oven with a time constant of 300 s that at full power would heat 200 °C above the 20 °C ambient, with a reference of 100 °C. The sensor measures with a delay. The simulation lasts 30 minutes, with a step of 0.1 s.
03Industrial automation
In a plant, control loops are part of a pyramid. At the bottom are the sensors and actuators; above them, the programmable logic controllers (PLCs), rugged computers that read inputs, run a cyclic program and write outputs, programmed in the languages of the IEC 61131-3 standard, such as ladder diagram. Higher up are the SCADA systems, which supervise and log, and the production management systems. Loop controllers, PID, are blocks inside PLCs or dedicated devices. The mathematical models of the following units are what make it possible to calculate them instead of tuning them by trial and error.
04In the lab
Control the temperature of an aluminum block with a heating resistor, a sensor and a microcontroller, first on-off and then proportional. Record the curves and measure the oscillation and the error.
Move the sensor away from the resistor to increase the delay and observe how the oscillation grows.
Program in ladder diagram a motor start and stop with an interlock and a temperature alarm.
05Common mistakes
- Expecting the proportional controller to reach the reference. With a plant that needs a sustained action, it leaves an error.
- Narrowing the hysteresis to reduce the oscillation. If there is a delay, the oscillation is dominated by the delay, and the relay switches more.
- Raising the gain without measuring. With delay, the loop ends up oscillating.
- Confusing closed loop with automatic. An automatic system in open loop does not correct disturbances.
06Self-assessment
With K = 200 °C, Kp = 0.05 /°C, ambient 20 °C and reference 100 °C, what error remains?
\( 80/(1 + 10) \approx 7.3 \) °C.
What Kp would be needed to bring the error down to 1 °C, ignoring stability?
\( 1 + 200 K_p = 80 \): Kp ≈ 0.4 /°C.
Why does an on-off controller with delay oscillate more than its hysteresis band?
Because when the sensor detects the crossing, the plant has already kept heating or cooling during the delay.
What languages does the IEC 61131-3 standard define?
Ladder diagram, function block diagram, structured text, instruction list and sequential function chart.
07Further reading
- Katsuhiko Ogata. Modern Control Engineering. 5th ed., Prentice Hall, 2010 (Spanish edition: “Ingeniería de control moderna”, Pearson). The classic text of the subject, with models, analysis and design.
- Karl J. Åström and Richard M. Murray. Feedback Systems: An Introduction for Scientists and Engineers. 2nd ed., Princeton University Press, 2021. A modern introduction to feedback, freely available.
- Frank D. Petruzella. Programmable Logic Controllers. 5th ed., McGraw-Hill, 2016. PLCs and their programming in industrial automation.