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1. Introduction
Process controllers are control system components which basically have an input of the error signal, i.e. the difference between the required value signal and the feedback signal, and an output of a signal to modify the system output. The ways in which such controllers react to error changes are termed the control laws, or more often, the control modes. The simplest form of controller is an on-off device which switches on some correcting device when there is an error and switches it off when the error ceases. However, such a method of control has limitations and often more sophisticated controllers are used. While there are many ways a controller could be designed to react to an error signal, a form of controller which can give satisfactory control in a wide number of situations is the three-term or PID controller. The three basic control modes are proportional (P), integral (I) and derivative (D); the three term controller is a combination of all three modes.
1.1 Direct and reverse action
In discussing the elements of a control system, the term direct action is used for an element that for an increase in its input gives an increase in its output, e.g. a domestic central heating furnace where an increase in the controlled input to the system results in an increase in temperature.
The term reverse action is used when an increase in input gives a decrease in output, e.g. an air conditioner where an increase in the energy input to it results in a decrease in temperature.
Application --- A source of dead time in a control system is the response time of the measurement sensor. Thus, for a system using a temperature sensor, a resistance temperature detector (RTD) has a slower response time than a thermocouple. A thermocouple has typically a response time of about 0.5 s while a RTD is a few seconds.
Application --- An example of a transfer delay is where a hopper is loading material onto a conveyor belt moving with a velocity v. The rate at which the material leaves the hopper is controlled by a valve with feedback from a weight sensor. If the weight of deposited material per unit length of belt is monitored a distance L from the hopper discharge point, then there v\nl l be a time delay of Uv in the control system.
1.2 Dead time
In any control system with feedback the system cannot respond instantly to any change and thus there are delays while the system takes time to accommodate the change. Such delays are referred to as dead time or lags. For example, in the control of the temperature in a room by means of a central heating system, if a window is suddenly opened and the temperature drops or the thermostat is suddenly set to a new value, a lag will occur before the control system responds, switches on the heater and gets the temperature back to its set value.
Transfer delays are a common event with control systems where flow is concerned, e.g. water flowing along a pipe from point A where the control valve is to point B where the rate of flow is required and monitored. Any change made at some point A will take some time before its affects are apparent at a point B, the time delay depending on the distance between A and B and the rate of flow between them. The term distance-velocity lag is sometimes used to describe such delays.
Dead time effectively hides a disturbance from the control system until its well into the system and needs to be made as small as possible.
Application --- Consider an electrical circuit with a capacitor C and resistor R in series.
When a voltage V is switched on, the voltage across the capacitor increases v\^h time until it eventually reaches a steady-state value. The initial rate of change of voltage across the capacitor is V/RC. Thus the bigger the capacitance the smaller the rate of change and so the longer it takes for the capacitor to become fully charged.
FIG. 1 On-off control
1.3 Capacitance
In the level control of the level of water in a tank, an important attribute of the system is its capacitance. If we have water leaving the tank and the control signal used to determine the rate of flow of water into the tank, then the greater the surface area of the water in the tank the longer it will take the controlled inflow of water to respond and restore a drop in level. We talk of the system having capacitance and the greater the capacitance the longer it takes to react to changes. If the capacitance were decreased then the system would react quicker to make the changes necessary to restore the required level.
Another example is a domestic heating system controlled by a thermostat. The larger the space being heated the longer it will take the controller to respond and restore a drop in temperature. Again we talk of the capacitance of the system.
Capacitance has the tendency to dampen out disturbances.
2. On-off control
With on-off control, the controller is essentially a switch which is activated by the error signal and supplies just an on-off correcting signal (FIG. 1). The controller output has just two possible values, equivalent to on and off. For this reason the controller is sometimes termed a two-step controller.
An example of such a controller is the bimetallic thermostat (FIG. 2) used with a simple temperature control system, ff the actual temperature is above the required temperature, the bimetallic strip is in an off position and the heater is switched off; if the actual temperature is below the required temperature, the bimetallic strip moves into the on position and the heater is switched on. The controller output is thus just on or off and so the correcting signal on or off.
FIG. 2 Bimetallic thermostat
Because the control action is discontinuous and there are time lags in the system, oscillations, i.e. cycling, of the controlled variable occur about the required condition. Thus, with temperature control using the bimetallic thermostat, when the room temperature drops below the required level there is a significant time before the heater begins to have an effect on the room temperature and, in the meantime, the temperature has fallen even more. When the temperature rises to the required temperature, since time elapses before the control system reacts and switches the heater off and it cools, the room temperature goes beyond the required value. The result is that the room temperature oscillates above and below the required temperature (FIG. 3). There is also a problem with the simple on-off system in that when the room temperature is hovering about the set value the thermostat might be reacting to very slight changes in temperature and almost continually switching on or off . Thus, when it is at its set value a slight draught might cause it to operate. This problem can be reduced if the heater is switched on at a lower temperature than the one at which it is switched of f (FIG. 4). The term dead band or neutral zone is used for the values between the on and of f values. For example, if the set value on a thermostat is 20°C, then a dead band might mean it switches on when the temperature falls to 19.5° and off when it is 20.5°. The temperature has thus to change by one degree for the controller to switch the heater on or of f and thus smaller changes do not cause the thermostat to switch.
A large dead band results in large fluctuations of the temperature about the set temperature; a small dead band will result in an increased frequency of switching. The bimetallic thermostat shown in FIG. 2 has a permanent magnet on one switch contact and a small piece of soft iron on the other; this has the effect of producing a small dead band in that, when the switch is closed, a significant rise in temperature is needed for the bimetallic element to produce sufficient force to separate the contacts.
On-off control is not too bad at maintaining a constant value of the variable when the capacitance of the system is very large, e.g. a central heating system heating a large air volume, and so the effect of changes in, say, a heater output results in slow changes in the variable. It also involves simple devices and so is fairly cheap. On-off control can be implemented by mechanical switches such as bimetallic strips or relays, with more rapid switching being achieved with electronic circuits, e.g. thyristors or transistors used to control the speed of a motor.
On-off control is simple and inexpensive and is often used where cycling can be reduced to an acceptable level.
2.1 Relays
A widely used form of on-off controller is a relay. FIG. 5 shows the basic form of an electromagnetic relay. A small current at a low voltage applied to the solenoid produces a magnetic field and so an electromagnet. When the current is high enough, the electromagnet attracts the armature towards the pole piece and in doing so operates the relay contacts. A much larger current can then be switched on. When the current through the solenoid drops below the critical level, the springy nature of the strip on which the contacts are mounted pushes the armature back to the of f position. Thus if the error signal is applied to the relay, it trips on when the error reaches a certain size and can then be used to switch on a much larger current in a correction element such as a heater or a motor.
FIG. 3 Fluctuation of temperature about set value
FIG. 4 On-off controller with a dead band
Application --- The normal domestic central heating system has an on-off controller, though a modem one is more likely to be an electronic on-off sensor rather than a bimetallic strip.
Another example of on-off control is the control of a car radiator cooling fan using a temperature-sensitive switch.
FIG. 5 Relay
3. Proportional control
With the on-off method of control, the controller output is either an on or an off signal and so the output is not related to the size of the error.
With proportional control the size of the controller output is proportional to the size of the error (FIG. 6), i.e. the controller input.
Thus we have: controller output o c controller input. We can write this as:
controller output = KpX controller input
where Kp is a constant called the gain. This means the correction element of the control system will have an input of a signal which is proportional to the size of the correction required.
The float method of controlling the level of water in a cistern (FIG. 7) is an example of the use of a proportional controller. The control mode is determined by the lever.
FIG. 7 The float-lever proportional controller
Application---An example of a control system using proportional control Is that of control of engine idling speed in a car. The idle air control valve is closed by an amount which depends on the extent the idling speed is above the required value or opened by an amount which depends on how far the speed is below the required value.
The output is proportional to the error, the gain being x/y. The error signal is the input to the ball end of the lever, the output is the movement of the other end of the lever. Thus, we have output movement = (x/y) x the error.
Another example of a proportional mode controller is an amplifier which gives an output which is proportional to the size of the input.
FIG. 8 illustrates, for the control of temperature of the outflow of liquid from a tank, the use of a differential amplifier as a comparison element and another amplifier as supplying the proportional control mode.
FIG. 8 Proportional controller for the control of temperature
FIG. 9 Water level control system
FIG. 10 Percentages
3.1 Proportional band
Note that it is customary to express the output of a controller as a percentage of the full range of output that it is capable of passing on to the correction element. Thus, with a valve as a correction element, as in the float operated control of level in FIG. 9, we might require it to be completely closed when the output from the controller is 0% and fully open when it is 100% (FIG. 10). Because the controller output is proportional to the error, these percentages correspond to a zero value for the error and the maximum possible error value. When the error is 50% of its maximum value then the controller output will be 50% of its full range.
Some terminology that is used in describing controllers:
1. Range
The range is the two extreme values between which the system operates. A common controller output range is 4 to 20 mA.
2. Span
The span is the difference between the two extreme values within which the system operates, e.g. a temperature control system might operate between 0°C and 30°C and so have a span of 30°C.
3. Absolute deviation
The set-point is compared to the measured value to give the error signal, this being generally termed the deviation. The term absolute deviation is used when the deviation is just quoted as the difference between the measured value and the set value, e.g. a temperature control system might operate between 0 C and 30°C and have an absolute deviation of 3 C.
4. Fractional deviation
The deviation is often quoted as a fractional or percentage deviation, this being the absolute deviation as a fraction or percentage of the span. Thus, a temperature control system operating between 0°C and 30°C with an error of 3°C has a percentage deviation of (3/30) x 100 = 10%. When there is no deviation then the percentage deviation is 0% and when the deviation is the maximum permitted by the span it is 100%.
FIG. 11 Proportional band
Generally with process controllers, the proportional gain is described in terms of its proportional band (PB). The proportional band is the fractional or percentage deviation that will produce a 100% change in controller output (FIG. 11):
%PB = [% deviation / % change in controller output]x 100
The 100% controller output might be a signal that fully opens a valve, the 0% being when it fully closes it. A 50% proportional band means that a 50% error will produce a 100% change in controller output; 100% proportional band means that a 100% error will produce a 100% change in controller output.
Since the percentage deviation is the error e as a percentage of the span and the percentage change in the controller output is the controller output yc as a percentage of the output span of the controller:
%PB = [e / measurement span] x [controller output span/yc] x 100
Since the controller gain Kp is yc/e:
%PB = 1/Kp [controller output span / measurement span] x 100
Example:
What is the controller gain of a temperature controller with a 60% PB if its input range is 0C to 50 C and its output is 4 mA to 20 mA?
%PB = 1/ Kp controller output span/ measurement span
and so:
=0.53 mA/degrees C
3.2 Limitations of proportional control
Proportional controllers have limitations. Consider the above example in FIG. 8 of the amplifier as the proportional controller. Initially, take the temperature of the liquid in the bath to be at the set value. There is then no error signal and consequently no current to the heating element.
Now suppose the temperature of the inflowing liquid changes to a constant lower value (FIG. 12). The temperature sensor will, after a time lag, indicate a temperature value which differs from the set value.
The greater the mass of the liquid in the tank, i.e. the capacitance, the longer will be the time taken for the sensor to react to the change. This is because it will take longer for the colder liquid to have mixed with the liquid in the tank and reached the sensor. The differential amplifier will then give an error signal and the power amplifier a signal to the heater which is proportional to the error. The current to the heater will be proportional to the error, the constant of proportionality being the gain of the amplifier. The higher the gain the larger will be the current to the heater for a particular error and thus the faster the system will respond to the temperature change. As indicated in FIG. 12, the inflow is constantly at this lower temperature. Thus, when steady state conditions prevail, we always need current passing through the heater. Thus there must be a continuing error signal and so the temperature can never quite be the set value. This error signal which persists under steady state conditions is termed the steady state error or the proportional offset.
The higher the gain of the amplifier the lower will be the steady state error because the system reacts more quickly.
In the above example, we could have obtained the same type of response if , instead of changing the temperature of the input liquid, we had made a sudden change of the set value to a new constant value.
There would need to be a steady state error or proportional offset from the original value. We can also obtain steady state errors in the case of a control system which has to, say, give an output of an output shaft rotating at a constant rate, the error results in a velocity-lag.
FIG. 12 Inflow change
All proportional control systems have a steady state error. The proportional mode of control tends to be used in processes where the gain Kp can be made large enough to reduce the steady state error to an acceptable level. However, the larger the gain the greater the chance of the system oscillating. The oscillations occur because of time lags in the system, the higher the gain the bigger will be the controlling action for a particular error and so the greater the chance that the system will overshoot the set value and oscillations occur.
Example:
A proportional controller has a gain of 4. What will be the percentage steady state error signal required to maintain an output from the controller of 20% when the normal set value is 0%?
With a proportional controller we have:
% controller output = gain x % error
20 = 4 X % error
Hence the percentage error is 5%.
Example:
For the water level control system described in FIG. 9, the water level is at the required height when the linear control valve has a flow rate of 5 m^3/h and the outflow is 5 m^3/h. The controller output is then 50% and operates as a proportional controller with a gain of 10. What will be the controller output and the offset when the outflow changes to 6 m^3/h?
Since a controller output of 50% corresponds to 5 m^3/h from the linear control valve, then 6 m^3/h means that the controller output will need to be 60%. To give a change in output of 60 - 50 = 10% with a controller having a gain of 10 means that the error signal into the controller must be 1%. There is thus an offset of 1%.
4. Derivative control
With derivative control the change in controller output from the set point value is proportional to the rate of change with time of the error signal, i.e. controller output o c rate of change of error. Thus we can write:
D controller output = KD x rate of change of error
It is usual to express these controller outputs as a percentage of the full range of output and the error as a percentage of full range. KD is the constant of proportionality and is commonly referred to as the derivative time since it has units of time.
FIG. 13 Derivative control
FIG. 13 illustrates the type of response that occurs when there is a steadily increasing error signal. Because the rate of change of the error with time is constant, the derivative controller gives a constant controller output signal to the correction element. With derivative control, as soon as the error signal begins to change there can be quite a large controller output since it is proportional to the rate of change of the error signal and not its value. Thus with this form of control there can be rapid corrective responses to error signals that occur.
4.1 PD control
Derivative controllers give responses to changing error signals but do not, however, respond to constant error signals, since with a constant error the rate of change of error with time is zero. Because of this, derivative control D is combined with proportional control P. Then:
PD controller output = Kp x error + A'D x rate of change of error with time
FIG. 14 shows how, with proportional plus derivative control, the controller output can vary when there is a constantly changing error.
There is an initial quick change in controller output because of the derivative action followed by the gradual change due to proportional action. This form of control can thus deal with fast process changes better than just proportional control alone. It still, like proportional control alone, needs a steady state error in order to cope with a constant change in input conditions or a change in the set value.
The above equation for PD control is sometimes written as:
PD controller output = Kp [error+ KD/KP rate of change of error ]
KD/KP is called the derivative action time TD and so:
PD controller output = Kp(error + To x rate of change of error)
PD control can deal with fast process changes better than just proportional control alone. It still needs a steady state error in order to cope with a constant change in input conditions or a change in the set value.
FIG. 14 PD control
Example:
A derivative controller has a derivative constant Ko of 0.4 s. What will be the controller output when the error (a) changes at 2%/s, (b) is constant at 4%?
(a) Using the equation given above, i.e. controller output = AT D x rate of change of error, then we have: controller output = 0.4 x 2 = 0.8%
This is a constant output.
(b) With a constant error there is no change of error with time and thus the controller output is zero.
Example:
What will the controller output be for a proportional plus derivative controller (a) initially and (b) 2 s after the error begins to change from the zero error at the rate of 2%/s (FIG. 15). The controller has K p = 4 and TD = 0.4 s.
FIG. 15 Example
(a) Initially the error is zero and so there is no controller output due to proportional action. There will, however, be an output due to derivative action since the error is changing at 2%/s. Since the output of the controller, even when giving a response due to derivative action alone, is multiplied by the proportional gain, we have:
controller output = TD x rate of change of error
= 4 x 0.4x 2%= 3.2%
(b) Because the rate of change is constant, after 2 s the error will have become 4%. Hence, then the controller output due to the proportional mode will be given by:
controller output = Kp x error
and so that part of the output is:
controller output = 4 x 4% = 16%
The error is still changing and so there will still be an output due to the derivative mode. This will be given by:
controller output = KP TD x rate of change of error and so:
controller output = 4 x 0.4 x 2% = 3.2%
Hence the total controller output due to both modes is the sum of these two outputs and 16% + 3.2% = 19.2%.
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