Objectives
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Describe the purpose of closed-loop negative feedback.
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Introduce proportional, proportional-integral, and proportional-integral-derivative control systems.
Chapter 12 Control Systems and Feedback
When controlling the output of a device (for example, the brightness of an LED or the speed of a motor), it is important to use some form of feedback to monitor the current status and see if changes need to be made. (If the LED is too bright, it may need to be dimmed. If the motor is spinning too slowly, it may need to be sped up.)
Example 12.0.1. Open-loop negative feedback.
Consider that you are indoors on a cold winter day in a room with heat provided by a radiator. There is no thermostat or other control available to change the amount of heat output by the radiator. The only way to control the temperature is to open or close a window. The window can only be opened and closed completely (there’s no way to get it to stay put if it’s halfway open). If it gets too warm in the room, you open the window. When it gets too cold, you close the window. This leads to a temperature response as shown in Figure 12.0.2. The dashed sinusoid depicts the actual temperature of the room. Each time the temperature gets too much larger than the desired temperature (represented by the solid line), you open the window. This causes the temperature to decrease. Once the temperature gets too much lower than desired, you close the window. This causes the temperature to increase. (The representation of the room temperature as a sinusoid is arbitrary, this may not be the outcome if you were to conduct this experiment in real life.)
Example 12.0.1 represents a negative feedback system. The term negative feedback comes from the subtraction operation denoting the difference between the actual and desired conditions. Example 12.0.1 represents open-loop feedback, where there is no direct control over the output based on the difference between actual and desired conditions. Open-loop feedback can lead to a constant fluctuation in output values without necessarily every reaching the desired value. Open-loop feedback also requires the user to continue monitoring and making changes, rather than having this process automated.
Aside
In a closed-loop negative feedback system, the desired value of some parameter (temperature of a room, brightness of an LED, speed of a motor, etc.) is compared in real-time to the actual value of that same parameter. The difference between the two values is used to adjust the signal sent to the device controlling that parameter.
The desired value of the parameter is known as the setpoint. The current value of the parameter is known as the process variable. The difference between these two quantities is known as the error and is defined by (12.0.1) where \(e(t)\) is the error, \(r(t)\) is the setpoint, and \(y(t)\) is the process variable.
\begin{equation}
e(t) = r(t) - y(t)\tag{12.0.1}
\end{equation}
A closed-loop negative feedback system will increase the value sent to the device (in an attempt to increase the value of the process variable \(y(t)\)) if the error becomes large and positive (in other words: the setpoint is greater than the actual value). The feedback system will decrease the value sent to the device (in an attempt to decrease \(y(t)\)) if the error becomes large and negative (in other words: the setpoint is less than the actual value.)
Example 12.0.3. Adaptive cruise control.
Consider an adaptive cruise control system in a car. The setpoint (desired speed) is programmed by the driver at 60 mph. The speed of the car at any moment in time is equal to the process variable \(y(t)\text{.}\) If the speed of the car becomes less than the setpoint, the error becomes greater than zero and the adaptive cruise control will increase the power sent to the car’s engine to speed the car up to the setpoint.
If the speed of the car becomes greater than the setpoint, the error becomes less than zero and the adaptive cruise control will decrease the power sent to the car’s engine or brake the car to slow the car down to the setpoint.
If the speed of the car is exactly equal to the setpoint, the error is zero and no change to the car’s engine power will be made.
