Linear control

Two classic mechanical examples are the toilet bowl float proportioning valve and the fly-ball governor.

Proportional control overcomes this by modulating the manipulated variable (MV), such as a control valve, at a gain level that avoids instability, but applies correction as fast as practicable by applying the optimum quantity of proportional correction.

The width of this range, in units of the error variable and therefore of the PV, is called the proportional band (PB).

Adjusting gain to achieve critically damped behavior is known as tuning the control system.

Once the setpoint is reached, stored heat within the heater sub-system and in the walls of the furnace will keep the measured temperature rising beyond what is required.

Any delay in reheating the heater sub-system allows the furnace temperature to fall further below the setpoint and the cycle repeats.

The PID controller addresses these final shortcomings by introducing a derivative (D) action to retain stability while responsiveness is improved.

A utility boiler "unit may be required to change load at a rate of as much as 5% per minute (IEA Coal Online - 2, 2007)".

By filtering out that frequency, stronger overall feedback can be applied before oscillation occurs, making the system more responsive without shaking itself apart.

Examples of such field applications include fly-by-wire aircraft control systems, chemical plants, and oil refineries.

Step responses for a second order system defined by the transfer function , where is the damping ratio and is the undamped natural frequency
A block diagram of a PID controller
Effects of varying PID parameters (K p ,K i ,K d ) on the step response of a system
Change of response of a second-order system to a step input for varying Ki values