# What is proportional, integral and derivative control?

## Introduction

The aim of this unit is to describe the use of proportional, integral and derivative control. The course also introduces the newer methods of control; cascade, ratio, feedforward, adaptive and multi-variable.

## Proportional control action

The basic continuous control mode is “proportional control”. With proportional control the controller output is algebraically proportional to the error input signal to the controller. The simple block diagram model of the controller shows this.

In this case the controller output is the gain of the controller (K) times the error signal (E), or

O/P = KE

This equation is called the control algorithm. The value of K can be set manually on older pneumatic equipment. On modern DCS systems it is set using a computer programme.

The mechanism which adjusts the gain on many industrial controllers is expressed in terms of proportional band (PB). Proportional band is defined as the span of values of the input which corresponds to a full or complete change in the output. This is usually expressed as a percentage and is related to proportional gain by:

PB = x 100%

In this case the controller output is the gain of the controller (K) times the error signal (E), or

O/P = KE

This equation is called the control algorithm. The value of K can be set manually on older pneumatic equipment. On modern DCS systems it is set using a computer programme.

The mechanism which adjusts the gain on many industrial controllers is expressed in terms of proportional band (PB). Proportional band is defined as the span of values of the input which corresponds to a full or complete change in the output. This is usually expressed as a percentage and is related to proportional gain by:

PB = (1/Gain) x 100%

In practice, wide bands (high percentages of PB) have low gain and narrow bands have high gain. There are many ways to show the effects of varying proportional band. One example is shown in Figure

Proportional control is quite simple. It is the easiest of the continuous control modes to tune, as there is only one value to adjust. It is very stable and responds quickly to changes.

However, proportional control has one big disadvantage. At steady state, it shows “offset”. This means there is a difference at steady state between the set point (SP) and the actual value of the Measured Variable (MV).

## Integral (reset) control action

Reset (integral) action provides a signal which depends on the size of the error signal. It is different from proportional control because it will continue to cancel any error until the offset is zero.

Reset (integral) control action is combined with proportional control action. This combination is called proportional-reset or proportional-integral action (PI control). This combination provides a control action which is stable and responds quickly with no offset.

Figure above shows the action of P & I control. The rate at which integral action is applied depends on the reset time adjustment. This is measured in either repeats per minute or minutes per repeat, depending on the manufacturer. The simple diagram below is used to show what this means.

In the diagram above the reset action repeats the proportional action twice in one minute. The reset time is thus either2 repeats per minute or 0.5 minutes per repeat

## Derivative (rate) control action

Derivative (rate) control action produces an output signal which is proportional to how fast the error signal changes (its rate). This type of control is only used when the loop response is very slow. Using derivative control on a loop which responds to changes quickly is dangerous. The output moves too quickly to a maximum or a minimum and can produce shock waves in the process being controlled.

Derivative control action is only used with proportional and integral action. Together, the three control modes provide what is called a Proportional-Integral-Derivative control action, (PID control).

Figure shows the effect of PID control for a step change in the error signal.

The output signal is a combination of the three control actions. Note that the rate adjustment changes how long the derivative signal is applied. Some manufacturers call derivative action “pre-act” as it only produces a signal at the start to quicken the response time.

The older types of controller (e.g. Foxboro Pneumatic type 43AP or SPEC 200 analog electrical/electronic) combine the PID control into a single unit which operates on the error signal. Modern micro-processor controllers, however, use a lot of PID control but in a different way. The block diagrams below show the difference.

## The difference between old and modern controller

### 1.Old Type Controller

The block diagram shows a typical older type PID controller. If the set point is changed the derivative action can cause large and unstable changes in output. So, D is only used for very slow loops.

### 2.Modern DCS Controller

The block diagram shows a typical micro-processor based **DCS **controller. The derivative action is only applied to changes in the measured value. Changes in the set point are not affected by the derivative action. This method provides a better response to process changes and more accurate control. Note that the PID settings are changed using a computer programme. This programme must have an “algorithm” of the controller characteristics

## Main Features of control systems

__ON/OFF CONTROL__**.**

__ON/OFF CONTROL__

- Inexpensive
- Extremely simple
- Excellent for control of large capacity (volume) systems.
- Process variable cycles about set point.
**Control valves**are easily worn out.- Cannot be used for small capacity systems.

__PROPORTIONAL.__

__PROPORTIONAL.__

- Simple
- Good for small capacities.
- Stable when set up (tuned) correctly.
- Rapid response.
- Offset at steady state.
- Easy to tune.

__PROPORTIONAL plus RESET (P & ____I____)__**:**

__PROPORTIONAL plus RESET (P &__

__I__

__)__

- No offset
- Better response time than reset alone
- P & I can reduce the stability of the loop. The gain may need to be reduced when reset is added.

__PROPORTIONAL plus RESET plus RATE (PID).__

__PROPORTIONAL plus RESET plus RATE (PID).__

- Most complex
- Most expensive
- Rapid response
- No offset
- Difficult to tune
- Best control if properly tuned.