2. basic control theory

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Basic Control Theory

Use the quick links below to take you to the main sections of this tutorial:

This tutorial looks at on/off and continuous control modes. It introduces proportional, integral andderivitive control actions and explains some of the terminology.

Modes of control

On/off control

Continuous control

Summary of modes of control

Further terminology

Modes of control

An automatic temperature control might consist of a valve, actuator, controller and sensor detecting the spacetemperature in a room. The control system is said to be 'in balance' when the space temperature sensor doesnot register more or less temperature than that required by the control system. What happens to the controlvalve when the space sensor registers a change in temperature (a temperature deviation) depends on the typeof control system used. The relationship between the movement of the valve and the change of temperature inthe controlled medium is known as the mode of control or control action.

There are two basic modes of control:

On/Off - The valve is either fully open or fully closed, with no intermediate state.

Continuous - The valve can move between fully open or fully closed, or be held at any intermediateposition.

Variations of both these modes exist, which will now be examined in greater detail.

On/off controlOccasionally known as two-step or two-position control, this is the most basic control mode. Considering thetank of water shown in Figure 5.2.1, the objective is to heat the water in the tank using the energy given off asimple steam coil. In the flow pipe to the coil, a two port valve and actuator is fitted, complete with a thermostat,placed in the water in the tank.

Fig. 5.2.1 On/off temperature control of water in a tank

The thermostat is set to 60C, which is the required temperature of the water in the tank. Logic dictates that ifthe switching point were actually at 60C the system would never operate properly, because the valve wouldnot know whether to be open or closed at 60C. From then on it could open and shut rapidly, causing wear.

For this reason, the thermostat would have an upper and lower switching point. This is essential to preventover-rapid cycling. In this case the upper switching point might be 61C (the point at which the thermostat tellsthe valve to shut) and the lower switching point might be 59C (the point when the valve is told to open). Thusthere is an in-built switching difference in the thermostat of 1C about the 60C set point.

This 2C (1C) is known as the switching differential. (This will vary between thermostats). A diagram of theswitching action of the thermostat would look like the graph shown in Figure 5.2.2. The temperature of the tankcontents will fall to 59C before the valve is asked to open and will rise to 61C before the valve is instructed toclose.

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continuous control.

Continuous controlContinuous control is often called modulating control. It means that the valve is capable of moving continuallyto change the degree of valve opening or closing. It does not just move to either fully open or fully closed, aswith on-off control.

There are three basic control actions that are often applied to continuous control:

Proportional (P)

Integral (I)

Derivative (D)It is also necessary to consider these in combination such as P + I, P + D, P + I + D. Although it is possible tocombine the different actions, and all help to produce the required response, it is important to remember thatboth the integral and derivative actions are usually corrective functions of a basic proportional control action.

The three control actions are considered below.

Proportional controlThis is the most basic of the continuous control modes and is usually referred to by use of the letter 'P'. Theprinciple aim of proportional control is to control the process as the conditions change.

This section shows that:

The larger the proportional band, the more stable the control, but the greater the offset.

The narrower the proportional band, the less stable the process, but the smaller the offset.The aim, therefore, should be to introduce the smallest acceptable proportional band that will always keep theprocess stable with the minimum offset.

In explaining proportional control, several new terms must be introduced.

To define these, a simple analogy can be considered - a cold water tank is supplied with water via a floatoperated control valve and with a globe valve on the outlet pipe valve 'V', as shown in Figure 5.2.4. Both valvesare the same size and have the same flow capacity and flow characteristic. The desired water level in the tankis at point B (equivalent to the set point of a level controller).

It can be assumed that, with valve 'V' half open, (50% load) there is just the right flowrate of water entering viathe float operated valve to provide the desired flow out through the discharge pipe, and to maintain the waterlevel in the tank at point at B.

Fig. 5.2.4 Valve 50% open

The system can be said to be in balance (the flowrate of water entering and leaving the tank is the same);under control, in a stable condition (the level is not varying) and at precisely the desired water level ( B ); giving

the required outflow.

With the valve 'V' closed, the level of water in the tank rises to point A and the float operated valve cuts off thewater supply (see Figure 5.2.5 below).

The system is still under control and stable but control is above level B. The difference between level B and theactual controlled level, A, is related to the proportional band of the control system.

Once again, if valve 'V' is half opened to give 50% load, the water level in the tank will return to the desiredlevel, point B.




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Fig. 5.2.5 Valve closed

In Figure 5.2.6 below, the valve 'V' is fully opened (100% load). The float operated valve will need to drop toopen the inlet valve wide and admit a higher flowrate of water to meet the increased demand from thedischarge pipe. When it reaches level C, enough water will be entering to meet the discharge needs and thewater level will be maintained at point C .

Fig. 5.2.6 Valve open

The system is under control and stable, but there is an offset; the deviation in level between points B and C.Figure 5.2.7 combines the three conditions used in this example.

The difference in levels between points A and C is known as the Proportional Band or P-band, since this is thechange in level (or temperature in the case of a temperature control) for the control valve to move from fullyopen to fully closed.

One recognised symbol for Proportional Band is Xp.

The analogy illustrates several basic and important points relating to proportional control:

The control valve is moved in proportion to the error in the water level (or the temperature deviation, inthe case of a temperature control) from the set point.

The set point can only be maintained for one specific load condition.

Whilst stable control will be achieved between points A and C, any load causing a difference in level tothat of B will always provide an offset.

Fig. 5.2.7 Proportional band

Note: By altering the fulcrum position, the system Proportional Band changes. Nearer the float gives anarrower P-band, whilst nearer the valve gives a wider P-band. Figure 5.2.8 illustrates why this is so. Differentfulcrum positions require different changes in water level to move the valve from fully open to fully closed. In

both cases, It can be seen that level B represents the 50% load level, A represents the 0% load level, and Crepresents the 100% load level. It can also be seen how the offset is greater at any same load with the widerproportional band.




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The examples depicted in Figures 5.2.4 through to 5.2.8 describe proportional band as the level (or perhapstemperature or pressure etc.) change required to move the valve from fully open to fully closed. This isconvenient for mechanical systems, but a more general (and more correct) definition of proportional band is thepercentage change in measured value required to give a 100% change in output. It is therefore usuallyexpressed in percentage terms rather than in engineering units such as degrees centigrade.

For electrical and pneumatic controllers, the set value is at the middle of the proportional band. The effect ofchanging the P-band for an electrical or pneumatic system can be described with a slightly different example,by using a temperature control.

The space temperature of a building is controlled by a water (radiator type) heating system using a proportionalaction control by a valve driven with an electrical actuator, and an electronic controller and room temperaturesensor. The control selected has a proportional band (P-band or Xp) of 6% of the controller input span of 0-100C, and the desired internal space temperature is 18C. Under certain load conditions, the valve is 50%open and the required internal temperature is correct at 18C.

A fall in outside temperature occurs, resulting in an increase in the rate of heat loss from the building.Consequently, the internal temperature will decrease. This will be detected by the room temperature sensor,which will signal the valve to move to a more open position allowing hotter water to pass through the roomradiators.

The valve is instructed to open by an amount proportional to the drop in room temperature. In simplistic terms,if the room temperature falls by 1C, the valve may open by 10%; if the room temperature falls by 2C, thevalve will open by 20%.

In due course, the outside temperature stabilises and the inside temperature stops falling. In order to providethe additional heat required for the lower outside temperature, the valve will stabilise in a more open position;

but the actual inside temperature will be slightly lower than 18C.

Example 5.2.1 and Figure 5.2.9 explain this further, using a P-band of 6C.

Example 5.2.1 Consider a space heating application with the following characteristics:

1. The required temperature in the building is 18C.2. The room temperature is currently 18C, and the valve is 50% open.3. The proportional band is set at 6% of 100C = 6C, which gives 3C either side of the 18C set point.Figure 5.2.9 shows the room temperature and valve relationship:

Fig. 5.2.9 Room temperature and valve relationship - 6C proportional band

As an example, consider the room temperature falling to 16C. From the chart it can be seen that the newvalve opening will be approximately 83%.

With proportional control, if the load changes, so too will the offset:

A load of less than 50% will cause the room temperature to be above the set value.

A load of more than 50% will cause the room temperature to be below the set value.The deviation between the set temperature on the controller (the set point) and the actual room temperature iscalled the 'proportional offset'.

In Example 5.2.1, as long as the load conditions remain the same, the control will remain steady at a valveopening of 83.3%; this is called 'sustained offset'.

The effect of adjusting the P-band

In electronic and pneumatic controllers, the P-band is adjustable. This enables the user to find a settingsuitable for the individual application.

Increasing the P-band - For example, if the previous application had been programmed with a 12%proportional band equivalent to 12C, the results can be seen in Figure 5.2.10. Note that the wider P-bandresults in a less steep 'gain' line. For the same change in room temperature the valve movement will be




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smaller. The term 'gain' is discussed in a following section.

In this instance, the 2C fall in room temperature would give a valve opening of about 68% from the chart inFigure 5.2.10.

Fig. 5.2.10 Room temperature and valve relationship - 12C Proportional band

Reducing the P-band - Conversely, if the P-band is reduced, the valve movement per temperature incrementis increased. However, reducing the P-band to zero gives an on/off control. The ideal P-band is as narrow aspossible without producing a noticeable oscillation in the actual room temperature.

GainThe term 'gain' is often used with controllers and is simply the reciprocal of proportional band.

The larger the controller gain, the more the controller output will change for a given error. For instance for again of 1, an error of 10% of scale will change the controller output by 10% of scale, for a gain of 5, an error of10% will change the controller output by 50% of scale, whilst for a gain of 10, an error of 10% will change the

output by 100% of scale.

The proportional band in 'degree terms' will depend on the controller input scale. For instance, for a controllerwith a 200C input scale:An Xp of 20% = 20% of 200C = 40CAn Xp of 10% = 10% of 200C = 20C

Example 5.2.2Let the input span of a controller be 100C.

If the controller is set so that full change in output occurs over a proportional band of 20% the controller gain is:

Equally it could be said that the proportional band is 20% of 100C = 20C and the gain is:

The controller in Example 5.2.1 had a gain of:

Therefore the relationship between P-band and Gain is:




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As a reminder:

A wide proportional band (small gain) will provide a less sensitive response, but a greater stability.

A narrow proportional band (large gain) will provide a more sensitive response, but there is a practicallimit to how narrow the Xp can be set.

Too narrow a proportional band (too much gain) will result in oscillation and unstable control.For any controller for various P-bands, gain lines can be determined as shown in Figure 5.2.11, where thecontrol ler input span is 100C.

Fig. 5.2.11 Proportional band and gain

Reverse or direct acting control signalA closer look at the figures used so far to describe the effect of proportional control shows that the output isassumed to be reverse acting. In other words, a rise in process temperature causes the control signal to falland the valve to close. This is usually the situation on heating controls. This configuration would not work on acooling control; here the valve must open with a rise in temperature. This is termed a direct acting controlsignal. Figures 5.2.12 and 5.2.13 depict the difference between reverse and direct acting control signals for thesame valve action.

Fig. 5.2.12 Reverse acting signal




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Fig. 5.2.13 Direct acting signal

On mechanical controllers (such as a pneumatic controller) it is usual to be able to invert the output signal ofthe controller by rotating the proportional control dial. Thus, the magnitude of the proportional band and thedirection of the control action can be determined from the same dial.

On electronic controllers, reverse acting (RA) or direct acting (DA) is selected through the keypad.

Gain line offset or proportional effectFrom the explanation of proportional control, it should be clear that there is a control offset or a deviation of theactual value from the set value whenever the load varies from 50%.

To further illustrate this, consider Example 5.2.1 with a 12C P-band, where an offset of 2C was expected. Ifthe offset cannot be tolerated by the application, then it must be eliminated.

This could be achieved by relocating (or resetting) the set point to a higher value. This provides the same valveopening after manual reset but at a room temperature of 18C not 16C.

Fig. 5.2.14 Gain line offset

Manual resetThe offset can be removed either manually or automatically. The effect of manual reset can be seen in Figure5.2.14, and the value is adjusted manually by applying an offset to the set point of 2C.

It should be clear from Figure 5.2.14 and the above text that the effect is the same as increasing the set valueby 2C. The same valve opening of 66.7% now coincides with the room temperature at 18C.

The effects of manual reset are demonstrated in Figure 5.2.15.

Fig. 5.2.15 Effect of manual reset

Integral control - automatic reset action'Manual reset' is usually unsatisfactory in process plant where each load change will require a reset action. It is




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Set value - What is on the dial.

Actual value - What the process value is.

Required value - The perfect process condition.Such problems are overcome by the reset action being contained within the mechanism of an automaticcontroller.

Such a controller is primarily a proportional controller. It then has a reset function added, which is called'integral action'. Automatic reset uses an electronic or pneumatic integration routine to perform the resetfunction. The most commonly used term for automatic reset is integral action, which is given the letter I.

The function of integral action is to eliminate offset by continuously and automatically modifying the controlleroutput in accordance with the control deviation integrated over time. The Integral Action Time ( IAT) is definedas the time taken for the controller output to change due to the integral action to equal the output change dueto the proportional action. Integral action gives a steadily increasing corrective action as long as an errorcontinues to exist. Such corrective action will increase with time and must therefore, at some time, be sufficientto eliminate the steady state error altogether, providing sufficient time elapses before another change occurs.The controller allows the integral time to be adjusted to suit the plant dynamic behaviour.

Proportional plus integral (P + I) becomes the terminology for a controller incorporating these features.

The integral action on a controller is often restricted to within the proportional band. A typical P + I response isshown in Figure 5.2.16, for a step change in load.

Fig. 5.2.16 P+I Function after a step change in load

The IAT is adjustable within the controller:

If it is too short, over-reaction and instability will result.

If it is too long, reset action will be very slow to take effect.IAT is represented in time units. On some controllers the adjustable parameter for the integral action is termed'repeats per minute', which is the number of times per minute that the integral action output changes by theproportional output change.

Repeats per minute = 1/(IAT in minutes)

IAT = Infinity - Means no integral action

IAT = 0 - Means infinite integral actionIt is important to check the controller manual to see how integral action is designated.

Overshoot and 'wind up'With P+ I controllers (and with P controllers), overshoot is likely to occur when there are time lags on thesystem.

A typical example of this is after a sudden change in load. Consider a process application where a processheat exchanger is designed to maintain water at a fixed temperature.

The set point is 80C, the P-band is set at 5C (2.5C), and the load suddenly changes such that the returningwater temperature falls almost instantaneously to 60C.

Figure 5.2.16 shows the effect of this sudden (step change) in load on the actual water temperature. Themeasured value changes almost instantaneously from a steady 80C to a value of 60C.

By the nature of the integration process, the generation of integral control action must lag behind theproportional control action, introducing a delay and more dead time to the response. This could have seriousconsequences in practice, because it means that the initial control response, which in a proportional systemwould be instantaneous and fast acting, is now subjected to a delay and responds slowly. This may cause theactual value to run out of control and the system to oscillate. These oscillations may increase or decrease

depending on the relative values of the controller gain and the integral action. If applying integral action it isimportant to make sure, that it is necessary and if so, that the correct amount of integral action is applied.

Integral control can also aggravate other situations. If the error is large for a long period, for example after alarge step change or the system being shut down, the value of the integral can become excessively large andcause overshoot or undershoot that takes a long time to recover. To avoid this problem, which is often called' '




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'integral wind-up', sophisticated controllers will inhibit integral action until the system gets fairly close toequilibrium.

To remedy these situations it is useful to measure the rate at which the actual temperature is changing; in otherwords, to measure the rate of change of the signal. Another type of control mode is used to measure how fastthe measured value changes, and this is termed Rate Action or Derivative Action.

Derivative control - rate actionA Derivative action (referred to by the letter D) measures and responds to the rate of change of process signal,and adjusts the output of the controller to minimise overshoot.

If applied properly on systems with time lags, derivative action will minimise the deviation from the set pointwhen there is a change in the process condition. It is interesting to note that derivative action will only applyitself when there is a change in process signal. If the value is steady, whatever the of fset, then derivative actiondoes not occur.

One useful function of the derivative function is that overshoot can be minimised especially on fast changes inload. However, derivative action is not easy to apply properly; if not enough is used, little benefit is achieved,and applying too much can cause more problems than it solves.

D action is again adjustable within the controller, and referred to as TD in time units:

T D = 0 - Means no D action.

T D = Infinity - Means infinite D action.

P + D controllers can be obtained, but proportional offset will probably be experienced. It is worth rememberingthat the main disadvantage with a P control is the presence of offset. To overcome and remove offset, 'I' actionis introduced. The frequent existence of time lags in the control loop explains the need for the third action D.The result is a P + I + D controller which, if properly tuned, can in most processes give a rapid and stableresponse, with no offset and without overshoot.

PID controllersP and I and D are referred to as 'terms' and thus a P + I + D controller is often referred to as a three termcontroller.

Summary of modes of controlA three-term controller contains three modes of control:

Proportional (P) action with adjustable gain to obtain stability.

Reset (Integral) (I) action to compensate for offset due to load changes.

Rate (Derivative) (D) action to speed up valve movement when rapid load changes take place.The various characteristics can be summarised, as shown in Figure 5.2.17.




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63.2% of 50C = 31.6C

The initial datum temperature was 25C, consequently the time constant for this simple example is the timerequired for the sensor to reach 56.6C, as shown below:

25C + 31.6C = 56.6C

HuntingOften referred to as instability, cycling or oscillation. Hunting produces a continuously changing deviation from

the normal operating point. This can be caused by:

Fig. 5.2.17 Summary of control modes and responses

Finally, the controls engineer must try to avoid the danger of using unnecessarily complicated controls for aspecific application. The least complicated control action, which will provide the degree of control required,should always be selected.

Further terminology

Time constantThis is defined as: 'The time taken for a controller output to change by 63.2% of its total due to a step (orsudden) change in process load'.

In reality, the explanation is more involved because the time constant is really the time taken for a signal oroutput to achieve its final value from its initial value, had the original rate of increase been maintained. Thisconcept is depicted in Figure 5.12.18.

Fig. 5.2.18 Time constant

Example 5.2.2 A practical appreciation of the time constant

Consider two tanks of water, tank A at a temperature of 25C, and tank B at 75C. A sensor is placed in tank Aand allowed to reach equilibrium temperature. It is then quickly transferred to tank B. The temperaturedifference between the two tanks is 50C, and 63.2% of this temperature span can be calculated as shownbelow:

The proportional band being too narrow.

The integral time being too short.

The derivative time being too long.




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A combination of these.

Long time constants or dead times in the control system or the process itself.In Figure 5.2.19 the heat exchanger is oversized for the application. Accurate temperature control will bedifficult to achieve and may result in a large proportional band in an attempt to achieve stability.

If the system load suddenly increases, the two port valve will open wider, filling the heat exchanger with hightemperature steam. The heat transfer rate increases extremely quickly causing the water system temperatureto overshoot. The rapid increase in water temperature is picked up by the sensor and directs the two port valveto close quickly. This causes the water temperature to fall, and the two port valve to open again. This cycle isrepeated, the cycling only ceasing when the PID terms are adjusted. The following example (Example 5.2.3)

gives an idea of the effects of a hunting steam system.

Fig. 5.2.19 Hunting

Example 5.2.3 The effect of hunting on the system in Figure 5.2.19 Consider the steam to water heat exchanger system in Figure 5.2.19. Under minimum load conditions, the sizeof the heat exchanger is such that it heats the constant flowrate secondary water from 60C to 65C with asteam temperature of 70C. The controller has a set point of 65C and a P-band of 10C.

Consider a sudden increase in the secondary load, such that the returning water temperature almostimmediately drops by 40C. The temperature of the water flowing out of the heat exchanger will also drop by40C to 25C. The sensor detects this and, as this temperature is below the P-band, it directs the pneumaticallyactuated steam valve to open fully.

The steam temperature is observed to increase from 70C to 140C almost instantaneously. What is the effecton the secondary water temperature and the stability of the control system?

As demonstrated in Tutorial 13.2 (The heat load, heat exchanger and steam load relationship), the heatexchanger temperature design constant, TDC, can be calculated from the observed operating conditions andEquation 13.2.2:

Equation 13.2.2Where:

TDC = Temperature Design Constant

T s = Steam temperature

T 1 = Secondary fluid inlet temperature

T 2 = Secondary fluid outlet temperature

In this example, the observed conditions (at minimum load) are as follows:




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When the steam temperature rises to 140C, it is possible to predict the outlet temperature from Equation13.2.5:

Equation 13.2.5

Where:

Ts = 140C

T 1 = 60C - 40C = 20C temperature

TDC = 2

The heat exchanger outlet temperature is 80C, which is now above the P-band, and the sensor now signalsthe controller to shut down the steam valve.

The steam temperature falls rapidly, causing the outlet water temperature to fall; and the steam valve opensyet again. The system cycles around these temperatures until the control parameters are changed. Thesesymptoms are referred to as 'hunting'. The control valve and its controller are hunting to find a stable condition.In practice, other factors will add to the uncertainty of the situation, such as the system size and reaction to

temperature change and the position of the sensor.

Hunting of this type can cause premature wear of system components, in particular valves and actuators, andgives poor control.

Example 5.2.3 is not typical of a practical application. In reality, correct design and sizing of the control systemand steam heated heat exchanger would not be a problem.

LagLag is a delay in response and will exist in both the control system and in the process or system under control.

Consider a small room warmed by a heater, which is controlled by a room space thermostat. A large window isopened admitting large amounts of cold air. The room temperature will fall but there will be a delay while themass of the sensor cools down to the new temperature - this is known as control lag. The delay time is alsoreferred to as dead time.

Having then asked for more heat from the room heater, it will be some time before this takes effect and warmsup the room to the point where the thermostat is satisfied. This is known as system lag or thermal lag.

RangeabilityThis relates to the control valve and is the ratio between the maximum controllable flow and the minimumcontrollable flow, between which the characteristics of the valve (linear, equal percentage, quick opening) willbe maintained. With most control valves, at some point before the fully closed position is reached, there is nolonger a defined control over flow in accordance with the valve characteristics. Reputable manufacturers willprovide rangeability figures for their valves.

Turndown ratioTurndown ratio is the ratio between the maximum flow and the minimum controllable flow. It will besubstantially less than the valve's rangeability if the valve is oversized.

Although the definition relates only to the valve, it is a function of the complete control system.

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