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A Guide To Proportional, Integral and Derivative (PID) Control

A Guide to Proportional, Integral and Derivative (PID) Control



A really short explanation

Proportional action (P)ArrestsIt arrest the change of the Measured Value but always with an Offset from the Measured Value
Integral action (I)Restores It removes the Offset ‚Äč
Derivative action (D)Accelerates It speeds up the removal of the Offset

A more detailed explanation

The explanation is based on a simple system similar to that used in a toilet cistern.
CONFIG


The system consists of a water tank which is filled with water controlled by a float operated Supply Valve (known as a ball cock in the UK).There is also a Load Valve which drains water from the tank.
The system has two controls:
A The Fulcrum can be adjusted horizontally by turning knob A
BThe Float can be adjusted vertically by turning knob B

The float is adjusted by knob B so that the Supply Valve is shut when the water level is at the required height.
Some definitions:
Desired Value (DV ) =The required level of water in the tank
Measured Value (MV)=The actual level of water in the tank.
Offset or Error (E)
also known as Droop
=The difference between the required and actual level (DV-MV)
Gain (K)=The ratio of float movement to valve movement

Proportional Action

A system under Proportional only control can be in a number of states:

State 1 - Steady State - No Load

CONFIG


The Load Valve is shut so there is no load on the system.
The tank has been filled to the desired water level which is equal to the measured value (DV = MV and Offset = 0).
The float is at its highest position and has shut the Supply Valve .
The water level is at the desired level and there is no flow into or out of the tank.

State 2 - Steady State - Under Load

CONFIG


The Load Valve has been partially opened so there is now a flow of water out of the tank i.e. there is Load on the system.
The tank level has dropped taking the Float with it.
The Supply Valve has been partially opened, by the action of the Float dropping, allowing water to flow into the tank.
The amount the Supply Valve opens is directly proportional to the level in the tank and therefore to flow of water out of the tank i.e. the load.
The water level and the Float continue to drop and the Supply Valve continues to open until the water flow into the tank equals the flow out of the tank at which point the water level stops falling.
The system has now reached steady state but the tank level is now lower than the required level (DV>MV) and there is an Offset.
In a Proportional only system under load there will always be an Offset and that offset will vary dependant on the size of the load.
The size of the Offset can be reduced (but not removed) by increasing the sensitivity of the system by:
  • Increasing the water supply pressure
  • Increasing the size of the inlet valve,
  • Increasing the gain of the system by moving the fulcrum towards the Float.
If the sensitivity or gain is increased too much the system will become unstable and the water level will hunt up and down.

State 3 - Unstable or Hunting

CONFIG


In this case knob A has been adjusted to move the Fulcrum closer to the Float so that a small change in the level of the Float results in a large change in the position of the input valve i.e. the gain has been increased and the system is more sensitive.
The Load Valve has been opened partially so there is a Load on the system.
The tank level has dropped taking the Float with it.
Due to the increased gain the Supply Valve is fully opened and the flow into the tank is much greater than the flow out.
The water level increase rapidly and overshoots the Desired Value.
The Float rises rapidly and shuts off the flow into the tank.
The water level drops, the Float sinks and the cycle could continue, ad infinitum, between level MV1 and MV2.
The system is unstable and is said to be hunting.
The gain should be reduced until the Offset is the smallest achievable with a stable water level.

State 4 - Saturation

CONFIG


The Load Valve has been opened such that the flow out is greater than the flow in through the wide open Supply Valve.
Further increases in load by adjusting the Load Valve until it is wide open will have no effect on the float and the system is said to be Saturated.
In this case the system has become saturated with the Inlet valve fully open however it can also become saturated with the valve fully shut.

Gain (K)

The sensitivity of a system under Proportional control is dependent on the gain of the system.
In this case the gain is set by adjusting the position of the Float relative to the fulcrum.
Low Gain - Insensitive system
CONFIG


In diagram above the fulcrum has been moved as far away from the float as possible and the sensitivity or gain of the system has been set to its lowest.
A large movement of the float will result in a small movement of the valve and the steady state will be achieved with a large Offset.
The lower the sensitivity of the system the longer it will take to achieve stability.
High Gain - Sensitive system
CONFIG


In diagram above the fulcrum has been moved as close to the float as possible and the sensitivity or gain of the system has been set to its highest.
A small movement of the float will result in a large movement of the valve and steady state can be achieved with a smaller Offset.
However it is likely that a small change in water level could result in the Input valve being fully opened or fully shut causing a rapid fall or rise in the water level.
The system would become unstable and result in a "hunting" water level.

Proportional Action Summary

Proportional control will always result in an Offset between Measured Value and Desired Value and for every load there will there will be a different steady state water level.
As the Gain increases the Offset decreases.
As the Gain increases the stability decreases until the system becomes unstable
With Proportional only control a compromise must be reached between size of Offset and stability by adjusting the Gain.
In some systems an Offset is acceptable, as in the water tank described above, and Proportional only control is acceptable..
However in other systems an offset of any size is unacceptable and some other form of control is required.

Integral Action

With the system we described above under load, assuming the Gain of has been adjusted to its optimum value, the water level will settle with an Offset from the Desired Value.
By adjusting knob B so that the float moves upwards, relative to the water level, the Supply Valve will open more, the flow in will increase and the Offset will reduce.
Eventually a new height of the Float will be found where the flow into the tank equals the flow out , the Measured value equals the Desired Value and Offset will be zero.
The speed at which the Float height is adjusted can be fast or slow.
If it is too fast the system can become unstable (hunting) and if it is too slow time will be wasted.
With Integral control the speed at which Offset is removed is made directly proportional to the size of the Offset.
In our water tank system we could achieve this by operating knob B with a variable speed servo motor.
The amount of integral action applied would be controlled by adjusting the ratio between Motor speed and size of Offset.

Integral Action Summary

Integral action restores the Measured Value to equal the Desired Value, i.e. the Offset is removed, under all loads.
The speed at which the Offset is removed is proportional to the size of the Offset.
The use of Integral action makes the system less stable and can produce serious hunting if care is not exercised.

Derivative Action

Not all systems can be controlled by Proportional and Integral control only.
In our water tank system an increase in load results in an immediate drop in water level and the Float.
The Supply Valve is immediately opened allowing water into the tank.
In some systems there is a delay or lag in response to a change in load.
For example a wind tunnel has a large heavy fan.
If more power is applied to increase the fan's speed there will be a significant delay before the new speed is achieved due to the time needed to overcome the inertia of the fan.
To overcome the inertia more power, than is required to maintain the desired speed (DV), is applied to accelerate the fans speed change.
The additional power is then reduced to the level required to maintain the required speed.
In our water tank system, under Proportional and Integral control, knob B is operated by a variable speed servo motor.
If there was inertia in the system, due to say friction in the linkage between the Float and the Supply Valve, Derivative action would temporarily apply a higher speed to the servo motor than was necessary to remove the Offset.

Derivative Action Summary

Derivative action speeds up the removal of the Offset.
It is required in systems which have large time delays due to Inertia or large capacities
It tends to make a system more stable as it is increased it can cause hunting and instability
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