Feedback Control Modes: Proportional, Integral, Derivative ~ Learning Instrumentation And Control Engineering Learning Instrumentation And Control Engineering

### Feedback Control Modes: Proportional, Integral, Derivative

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Feedback control is the most popular control system in use today in industrial control systems. The basic function of feedback control is to compute the error as the difference between the controlled variable and the set point; calculate and then output the signal to a control valve actuator based on the error.

To perform this function, feedback control can be operated in three modes:
(a) Proportional or Gain Mode
(b) The Integral or Reset Mode
(c) The Derivative or Rate Mode

Here we shall discuss the basic function and aspects of the Proportional, Integral and the Derivative modes of feedback control systems.

The Proportional Mode:
The purpose of the proportional mode is to cause an instantaneous response in the
controller output to changes in the error. The formula for the proportional mode is given by:

KC*ERR

Where KC is the controller gain and ERR is the error. The significance of the controller gain is that as it increases so does the change in the controller output. The controller output also increases as the error increases.
The proportional mode is used to set the basic gain value of the controller. The setting for the proportional mode may be expressed as either:
(1) Proportional Gain
(2) Proportional Band

Proportional Gain
In electronic controllers, proportional action is typically expressed as proportional gain. Proportional gain is expressed as:
Gain, (KC)       =  percentage change in controller output divided by the percentage change  in the controller         Input         = ΔOutput%/ΔInput%

Proportional Band
Proportional Band (PB) is another way of representing the proportional gain. PB is the percentage of change of the controller input span that will cause a 100% change in controller output:
PB = ΔInput (% Span) For 100% ΔOutput

The relationship between controller gain and its proportional band is then given by

PB = 100/KC

Where:
PB = Proportional Band
KC = Controller gain

Limitations of Proportional Action:
(1) Proportional action responds only to a change in error
(2) Proportional action will not return the PV to set point. It will, however, return the process variable (PV) to a value that is within a defined span (PB) around the PV.

Integral Mode
The purpose of integral action is to return the PV to SP. This is accomplished by repeating the action of the proportional mode as long as an error exists. The controller output from the integral or reset mode is a function of the duration of the error. With proportional control only, there will always be a error between the process variable (PV) and the set point (SP). This error is called an offset. The elimination of this offset or steady state error requires the integral mode. The integral mode does this by integrating the error over a period of time interval. It is defined by the formula:

KC/TI ∫edt

Where:
KC = controller gain
TI = Integral constant or Reset time
t  = time

Since integral control integrates the error over time, the control action grows larger the longer the error persists. This integration of the error takes place until no error exists. Every integral action has a phase lag of 90 degree and this phase shift has a destabilizing effect. For this reason, we rarely use I-control without P-control.

Integral, or reset action, may be expressed in terms of:
Repeats Per Minute - How many times the proportional action is repeated each minute.
Minutes Per Repeat - How many minutes are required for 1 repeat to occur

Derivative or Rate Mode
Though integral mode is effective in eliminating the offset, it is slower than the proportional mode in that it must act over a period of time. A faster than proportional mode is derivative or rate mode. This speeds up the controller action, compensating for some of the delays in the feedback loop. It can be expressed by the formula:

KC*TD* dERR/dt

Here TD is the derivative or rate time. The derivative time is the time it takes the proportional mode to match the instantaneous action of the derivative mode on an error that changes linearly with time (a ramp). It should be noted that the derivative mode acts only when there is a change in error with time.

One key advantage of the derivative mode is that it reduces the time required to return PV to SP in slow processes. However, it can dramatically amplify noisy signals in the process. One other disadvantage is that  it can lead to cycling in very fast processes.