Closed-loop controller

The PID algorithm in the controller restores the actual speed to the desired speed in an optimum way, with minimal delay or overshoot, by controlling the power output of the vehicle's engine.

Control systems that include some sensing of the results they are trying to achieve are making use of feedback and can adapt to varying circumstances to some extent.

Open-loop control systems do not make use of feedback, and run only in pre-arranged ways.

In such systems, the open-loop control is termed feedforward and serves to further improve reference tracking performance.

In the case of the boiler analogy this would include a thermostat to monitor the building temperature, and thereby feed back a signal to ensure the controller maintains the building at the temperature set on the thermostat.

"[4] The output of the system y(t) is fed back through a sensor measurement F to a comparison with the reference value r(t).

In such cases variables are represented through vectors instead of simple scalar values.

For some distributed parameter systems the vectors may be infinite-dimensional (typically functions).

If we assume the controller C, the plant P, and the sensor F are linear and time-invariant (i.e., elements of their transfer function C(s), P(s), and F(s) do not depend on time), the systems above can be analysed using the Laplace transform on the variables.

, then Y(s) is approximately equal to R(s) and the output closely tracks the reference input.

A PID controller continuously calculates an error value e(t) as the difference between a desired setpoint and a measured process variable and applies a correction based on proportional, integral, and derivative terms.

If u(t) is the control signal sent to the system, y(t) is the measured output and r(t) is the desired output, and e(t) = r(t) − y(t) is the tracking error, a PID controller has the general form The desired closed loop dynamics is obtained by adjusting the three parameters KP, KI and KD, often iteratively by "tuning" and without specific knowledge of a plant model.

The integral term permits the rejection of a step disturbance (often a striking specification in process control).

However, in practice, a pure differentiator is neither physically realizable nor desirable[6] due to amplification of noise and resonant modes in the system.

Therefore, a phase-lead compensator type approach or a differentiator with low-pass roll-off are used instead.

Example of a single industrial control loop; showing continuously modulated control of process flow.
A basic feedback loop
An electromechanical timer, normally used for open-loop control based purely on a timing sequence, with no feedback from the process
A simple feedback control loop
A simple feedback control loop
A block diagram of a PID controller in a feedback loop, r ( t ) is the desired process value or "set point", and y ( t ) is the measured process value.