Analog-to-digital and digital-to-analog conversion techniques are found in process control applications. Consider the automatic pilot of an aircraft. At any instant the position (location and altitude) of an aircraft is measured, together with its performance (heading, speed, rate of climb, rate of turn and engine power). All these values are converted into digital form and fed into a digital computer that determines the best position for the controls (throttle, elevator, aileron, and rudder). The digital output from the computer is applied to digital-to-analog converters, whose analog outputs operate actuators that directly move the appropriate control surfaces.

The figure below describes a primitive control system. The input is an analog value that is digitized and processed by the computer. Real control systems are often much more sophisticated than this. Suppose you apply a new demand input to a system such as banking an aircraft’s wings. The aircraft rolls into the bank and attempts to attain the angle requested. However, the mechanical inertia of the aircraft might cause it to roll past the point it was aiming for (i.e., it overshoots). A practical control system should also be able to take account of rapidly changing conditions.

The crudest control mechanism is found in central heating systems where the desired temperature or setpoint is obtained from a control unit on the wall. The demand input is compared with the actual temperature measured by a sensor. If it is colder than the setpoint, the heater is turned on. Otherwise the heater is turned off.

The next figure demonstrates the operation of such a system. The temperature of the room rises and eventually the heater is turned off. Because of the heater's thermal inertia the room, the temperature will continue to rise after the current has been cut off. Eventually, the room begins to cool and the heater is turned on and the temperature starts rising again.

This type of on-off control system is also called a bang-bang control system to indicate its crude approach—bang the system goes on and bang it goes off. There is no intermediate point between on and off, and the room is never at the correct temperature because it's either slightly too hot or too cold.

A better method of controlling the temperature of a room is to measure the difference between the desired temperature and the actual temperature and use this value to determine how much power is to be fed to the heater. The colder the room, the more power sent to the heater. If the room is close to its desired temperature, less power if fed to the heater. This is an example of a proportional control system.  As the room temperature approaches its desired setpoint value, the power fed to the heater is progressively reduced; that is, the current supplied to the heater is K(t_setpoint - t_room).

The proportional control system can be improved further by taking into account changes in the variable you are trying to control. Suppose you're designing a camera with an automatic focusing mechanism for use at sporting events. The camera measures the distance of the subject from the camera uses the difference between the current point-of-focus and the desired point-of-focus to drive the motor that performs the focusing.  

Suppose the subject suddenly changes direction, speeds up, or slows down. A proportional control system can't deal with this situation well. 

If the subject is in focus and then begins accelerating away, a proportional control signal can't apply a correction until the target is actually out of focus. What we need is a control signal that doesn't depend on the magnitude of the error but on the rate at which the error is changing. 

A differential control system uses the rate of change of the error as a control signal; for example, a camera with auto-focusing can use any rapid change in the subject's position to control the focusing motor—even if the subject is approximately in focus and there's no "proportional error". A differential control system must also incorporate proportional control (if the subject were out of focus but not moving there would be no differential feedback signal). 

If we call the error between the setpoint in a control system and its output e, the control input in a proportional plus derivative (i.e., differential) control system is given by: 

y = K₁e + K₂de/dt, where K₁ and K₂ are the proportional and derivative control coefficients, respectively. 

Even this control algorithm isn't perfect. Suppose you design a radar-controlled docking system for spacecraft. One craft can track the other by using both proportional control and derivative control to minimize the difference between their trajectories. However, once their trajectories are closely (but not exactly) matched, there is neither a proportional error signal nor a derivative error signal to force exact tracking. What we need is a mechanism that takes account of a persistent small error. 

An integral control signal adds up the error signal over a period of time. Even the smallest error eventually generates a control signal to further reduce the error. Integral control ensures that any drift over time is corrected. 

A high-performance controller might combine proportional control, rate-of-change control, and integral control as the next diagram demonstrates. This system is called a PID (proportional, integral, and derivative) controller. The box marked differentiator calculates the rate-of-change of the system output being controlled. 

 The equation for a PID can be expressed in the form  y = K₁e + K₂de/dt + K₃òedt

 The control signal y now depends and the size of the error between the desired and actual outputs from the controller, the rate at which the error is changing, and the accumulated error over a period.  Designing a PID system is not easy. You have to choose the amounts of proportional, derivative, and integral feedback as well as the time constant of the integrator. If the system is not correctly designed it can become unstable and oscillate.