AMAZON multi-meters discounts AMAZON oscilloscope discounts
<< cont. from part 1
5. Integral control
Integral control is the control mode where the controller output is proportional to the integral of the error with respect to time, i.e.:
controller output ∝ integral of error with time
and so we can write:
I controller output = Kl x integral of error with time
where Kl is the constant of proportionality and, when the controller output is expressed as a percentage and the error as a percentage, has units of s^-1.
To illustrate what is meant by the integral of the error with respect to time, consider a situation where the error varies with time in the way shown in FIG. 16. The value of the integral at some time t is the area under the graph between t = 0 and t. Thus we have:
controller output x area under the error graph between t = 0 and I
Thus as t increases, the area increases and so the controller output increases. Since, in this example, the area is proportional to t then the controller output is proportional to t and so increases at a constant rate. Note that this gives an alternative way of describing integral control as:
rate of change of controller output ∝ error
A constant error gives a constant rate of change of controller output.
FIG. 16 Integral control
FIG. 17 Example
FIG. 18 PI control
Example:
An integral controller has a value of K1 of 0.10 s^-1. What will be the output after times of (a) 1 s, (b) 2 s, if there is a sudden change to a constant error of 20%, as illustrated in FIG. 17? We can use the equation:
controller output = Ki x integral of error with time
(a) The area under the graph between a time of 0 and 1 s is 20%s.
Thus the controller output is 0.10 x 20 = 2%.
(b) The area under the graph between a time of 0 and 2 s is 40%s.
Thus the controller output is 0.10 x 40 = 4%.
5.1 PI control
The integral mode I of control is not usually used alone but generally in conjunction with the proportional mode P. When integral action is added to a proportional control system the controller output is given by:
PI controller output = Kp error + K i integral of error with time
where Kp is the proportional control constant and Ki the integral control constant.
FIG. 18 shows how a system with PI control reacts when there is an abrupt change to a constant error. The error gives rise to a proportional controller output which remains constant since the error does not change. There is then superimposed on this a steadily increasing controller output due to the integral action.
The combination of integral mode with proportional mode has one great advantage over the proportional mode alone: the steady state error can be eliminated. This is because the integral part of the control can provide a controller output even when the error is zero. The controller output is the sum of the area all the way back to time / = 0 and thus even when the error has become zero, the controller will give an output due to previous errors and can be used to maintain that condition. FIG. 19 illustrates this.
The above equation for PI controller output is often written as:
PI controller output = Kp [error + -1/ T integral of error ]
Kp/Ki is called the integral action time T1 and so:
Because of the lack of a steady state error, a PI controller can be used where there are large changes in the process variable.
However, because the integration part of the control takes time, the changes must be relatively slow to prevent oscillations.
FIG. 19 Controller output when error becomes zero
6. PID control
Combining all three modes of control (proportional, integral and derivative) enables a controller to be produced which has no steady state error and reduces the tendency for oscillations. Such a controller is known as a three-mode controller or PID controller. The equation describing its action is:
controller output = Kp x error + Ki x integral of error + KD x rate of change of error
where Kp is the proportionality constant, Ki the integral constant and KD the derivative constant. The above equation can be written as:
PID controller output =
Kp [error + -1/T integral of error + TD rate of change of error]
A PID controller can be considered to be a proportional controller which has integral control to eliminate the offset error and derivative control to reduce time lags.
Example:
Determine the controller output of a three-mode controller having Kp as 4, Ti as 0.2 s, To as 0.5 s at time (a) r = 0 and (b) r = 2 s when there is an error input which starts at 0 at time / = 0 and increases at 1%/ s (FIG. 20). (a) Using the equation: controller output = Arp(error + (\/T\) x integral of error
+ TD X rate of change of error)
we have for time / = 0 an error of 0, a rate of change of error with time of 1 s"^ and an area between this value of / and / = 0 of 0. Thus: controller output = 4(0 + 0 + 0.5 x 1) = 2.0% (b) When / = 2 s, the error has become 1%, the rate of change of the error with time is 1%/ s and the area under between r = 2 and r = 0 is l%s. Thus: controller output = 4(1 + (1/0.2) x 1 + 0.5 x 1) = 26%
FIG. 20 Example
6.1 PID process controller
FIG. 21 shows the basic elements that tend to figure on the front face of a typical three-term process controller. The controller can be operated in three modes by pressing the relevant key:
FIG. 21 Typical controller front panel
1. Manual mode
The operator directly controls the operation and can increase or decrease the controller output signal by holding down the M key and pressing the up or down keys. A LED above the key shows when this mode has been selected. The output is shown on the digital display and on the bar graph display.
2. Automatic mode
The controller operates as a 3-term controller with a set point specified by the operator. A LED above the key shows when this mode has been selected. The digital display shows the set point value when the SP key is depressed and the value changed by pressing the up or down keys. The digital display shows the set point value in units such as °C, the unit previously having been set up to give such values in the set up procedure. The set point is also displayed on the vertical bar graph as a percentage.
3. Remote automatic mode
The controller is operated in a similar manner to the automatic mode but with the set point established by an external signal. A LED above the key shows when this mode has been selected.
When no key is depressed, the process variable is shown on the digital display and on the vertical bar graph.
The procedure adopted when using the controller is to initially set the mode as manual. The set point is then set to the required value and the controller output manually adjusted until the deviation is zero and the plant thus operating at the required set point. FIG. 22 shows the block diagram of the control system when it is being operated in manual mode and the operator adjusting the controller output by adding in a signal. The controller can then be switched to automatic control. When this happens, the manual input signal is held constant at the value that was set in manual mode.
FIG. 22 Control system in manual mode
Switching back to manual mode from automatic mode to make some adjustment and then back to automatic mode gain can present a problem.
There can be a sudden change in controller output on the transition from manual to automatic modes, this being termed a *bump' in the plant operation. This arises because of the integral element in the controller which bases its error on the duration of the error signal input to the controller and does not take account of any manually introduced signals.
Thus, changing the manually introduced signal can lead to the output from the controller in the automatic mode not being the same as that in the manual mode. To avoid this 'bump' and give a bumpless transfer, modem controllers automatically adjust the contribution to the control law from the integral element.
Modem process controllers are likely to be microprocessor-based controllers, though operating as though they are conventional analogue controllers. They can be programmed by connecting a hand-held terminal to them so that the parameters of the PID controller can be set.
7. Tuning
The design of a controller for a particular situation involves selecting the control modes to be used and the control mode settings. This means determining whether proportional control, proportional plus derivative, proportional plus integral or proportional plus integral plus derivative is to be used and selecting the values of Kp, Kl and KD . These determine how the system reacts to a disturbance or a change in the set value, how fast it will respond to changes, how long it will take to settle down after a disturbance or change to the set value, and whether there will be a steady state error.
FIG. 23 illustrates the types of response that can occur with the different modes of control when subject to a step input, i.e. a sudden change to a different constant set value or perhaps a sudden constant disturbance. Proportional control gives a fast response with oscillations which die away to leave a steady state error. Proportional plus integral control has no steady state error but is likely to show more oscillations before settling down. Proportional but integral plus derivative control has also no steady state error, because of the integral element, and is likely to show less oscillations than the proportional plus integral control. The inclusion of derivative control reduces the oscillations.
The term tuning is used to describe methods used to select the best controller setting to obtain a particular form of performance, e.g. the component being where an error signal results in the controlled variable oscillating about the required value with an oscillation which decays quite rapidly so that each successive amplitude is a quarter of the preceding one. The following is a description of some of the methods used for tuning. Two methods that are widely used, are the process reaction method and the ultimate cycle method, both by Ziegler and Nichols.
FIG. 23 Responses to PI, (c) PID control (a)P,(b)
7.1 Process reaction tuning method
This method uses certain measurements made from testing the system with the control loop open so that no control action occurs. Generally the break is made between the controller and the correction unit (FIG. 24). A test input signal is then applied to the correction unit and the response of the controlled variable determined.
FIG. 24 Test arrangement
The test signal is a step signal with a step size expressed as the percentage change P in the correction unit (FIG. 25). The output response of the controlled variable, as a percentage of the full-scale range, to such an input is monitored and a graph (FIG. 26) of the variable plotted against time. This graph is called the process reaction curve. A tangent is drawn to give the maximum gradient of the graph.
The time between the start of the test signal and the point at which this tangent intersects the graph time axis is termed the lag L. If the value of the maximum gradient is M, expressed as the percentage change of the set value of the variable per minute, Table 1 shows the criteria given by Ziegler and Nichols to determine the controller settings. The basis behind these criteria is to give a closed-loop response for the system which exhibits a quarter amplitude decay (FIG. 27), i.e. the amplitude of the response of the system shows oscillations which decay with time so that the amplitude decreases by a quarter on each oscillation.
FIG. 25 Test signal
FIG. 26 The process reaction graph
FIG. 27 Quarter amplitude decay
Table 1 Settings from the process reaction curve method
7.2 Ultimate cycle tuning method
With this method, the integral and derivative actions are first reduced to their least effective values. The proportional constant Kp is then set low and gradually increased until oscillations in the controlled variable start to occur. The critical value of the proportional constant Kp at which this occurs is noted and the periodic time of the oscillations Tc measured. The procedure is thus:
1. Set the controller to manual operation and the plant near to its normal operating conditions.
2. Turn off all control modes but proportional.
3. Set AT p to a low value, i.e. the proportional band to a wide value.
4. Switch the controller to automatic mode and then introduce a small set-point change, e.g. 5 to 10%.
5. Observe the response.
6. Set Kp to a slightly higher value, i.e. make the proportional band narrower.
7. Introduce a small set-point change, e.g. 5 to 10%.
8. Observe the response.
9. Keep on repeating 6, 7 and 8, until the response shows sustained oscillations which neither grow nor decay. Note the value of Kp giving this condition (Kpu ) and the period (Tu) of the oscillation.
10. The Ziegler and Nichols recommended criteria for controller settings for a system to have quarter amplitude decay are given by Table 2. For a PID system with some overshoot or with no overshoot, the criteria are given by Table 3.
Table 2 Settings for the ultimate cycle method for quarter amplitude
decay
Table 3 Settings for the ultimate cycle method for PI control
Example:
When tuning a three-mode control system by the ultimate cycle method it was found that, with derivative and integral control switched off, oscillations begin when the proportional gain is increased to 3.3. The oscillations have a periodic time of 500 s.
What are the suitable values of Kp, Ki and KD for quarter amplitude decay?
Using the equations given above:
KP = 0.6^Pc = 0.6x3.3 =1.98
Ti = 500/2 = 250 s and so A: , = Kp/Ti = 1.98/200 = 0.0099 s^-1
TD = 500/8 = 62.5 s and so Ko = KPTD = 1.98 x 62.5 = 123.75 s
7.3 Quarter amplitude decay
A variation on the ultimate cycle method involves no adjusting the control system for sustained oscillations but for oscillations which have a quarter amplitude decay. The controller is set to proportional only. Then, with a step input to the control system, the output is monitored and the amplitude decay determined. If the amplitude decay is less than a quarter the proportional gain is increased, if less than a quarter it is decreased.
The step input is then repeated and the amplitude decay again determined. By a method of trial and error, the test input is repeated until a quarter wave amplitude decay is obtained. We then have the value for the proportional gain constant. The integral time constant is then set to be T/1.5 and the derivative time constant to T/6.
8. Digital systems
The term direct digital control is used to describe the use of digital computers in the control system to calculate the control signal that is applied to the actuators to control the plant. Such a system is of the form shown in FIG. 30. At each sample instant the computer samples, via the analogue-to-digital converter (ADC), the plant output to produce the sampled output value. This, together with the discrete input value is then processed by the computer according to the required control law to give the required correction signal which is then sent via the digital-to analogue converter (DAC) to provide the correcting action to the plant to give the required control. Direct digital control laws are computer programs that take the set value and feedback signals and operate on them to give the output signal to the actuator. The program might thus be designed to implement PID control.
FIG. 29 Quarter amplitude decay
FIG. 30 Direct digital control
The program involves the computer carrying out operations on the fed back measurement value occurring at the instant it is sampled and also using the values previously obtained. The program for proportional control thus takes the form of setting initial values to be used in the program and then a sequence of program instructions which are repeated every sampling period:
Initialize
Set the initial value of the error (this will be zero if the program is to start at the measurement value then occurring)
Set the value of the proportional gain
Loop
Input the error at the instant concerned Calculate the output by multiplying the error by the set value of the proportional gain Output the value of the calculated output Wait for the end of the sampling period Go back to Loop and repeat the program For PD control the program is:
Initialize
Set the initial value of the error (this will be zero if the program is to start at the measurement value then occurring)
Set the initial value of the error that is assumed to have occurred in the previous sampling period Set the value of the proportional gain
Set the value of the derivative gain
Loop
Input the error at the instant concerned Calculate the proportional part of the output by multiplying the error by the set value of the proportional gain Calculate the derivative part of the output by subtracting the value of the error at the previous sampling instant from the value at the current sampling instant (the difference is a measure of the rate of change of the error since the signals are sampled at regular intervals of time) and multiply it by the set value of the derivative gain.
Calculate the output by adding the proportional and derivative output elements
Output the value of the calculated output Wait for the end of the sampling period Go back to Loop and repeat the program For PI control the program is:
Initialize
Set the initial value of the error (this will be zero if the program is to start at the measurement value then occurring)
Set the value of the output that is assumed to have occurred in the previous sampling period Set the value of the proportional gain Set the value of the integral gain
Loop
Input the error at the instant concerned Calculate the proportional part of the output by multiplying the error by the set value of the proportional gain Calculate the integral part of the output by multiplying the value of the error at the current sampling instant by the sampling period and the set value of the integral gain (this assumes that the output has remained constant over the previous sampling period and so multiplying its value by the sampling period gives the area under the output-time graph) and add to it the previous value of the output.
Calculate the output by adding the proportional and integral output elements
Output the value of the calculated output
Wait for the end of the sampling period
Go back to Loop and repeat the program
8.1 Embedded systems
The term embedded system is used for control systems involving a microprocessor being used as the controller and located as an integral element, i.e. embedded, in the system. Such a system is used with engine management control systems in modem cars, exposure and focus control in modem cameras, the controlling of the operation of modem washing machines and indeed is very widely used in modem consumer goods.
Problems
Questions 1 to 20 have four answer options: A, B, C and D. Choose the correct answer from the answer options.
1. Decide whether each of these statements is True (T) or False (F). An on-off temperature controller must have: (i) An error signal input which switches the controller on or off . (ii) An output signal to switch on or of f the correction element.
A (i)T (ii)T
B (i)T (ii)F
C (i)F (ii)T
D (i)F (ii)F
2. Decide whether each of these statements is True (T) or False (F). Oscillations of the variable being controlled occur with on-off temperature controller because:
(i) There is a time delay in switching off the correction element when the variable reaches the set value.
(ii) There is a time delay in switching on the correction element when the variable falls below the set value.
A (i)T (ii)T
B (i)T (ii)F
C (i)F (ii)T
D (i)F (ii)F
3. Decide whether each of these statements is True (T) or False (F). With a proportional controller: (i) The controller output is proportional to the error.
(ii) The controller gain is proportional to the error.
A (i)T (ii)T
B (i)T (ii)F
C (i)F (ii)T
D (i)F (ii)F
4. A steady state error will not occur when there is a change to the set value with a control system operating in the mode: A Proportional B Proportional plus derivative C Derivative D Proportional plus integral Questions 5 to 8 concern the error input to a controller shown in FIG. 31(a) and the possible controller outputs shown in FIG. 31(b).
FIG. 31 Problems 5 to 8
5. Which one of the outputs could be given by a proportional controller?
6. Which one of the outputs could be given by a derivative controller?
7. Which one of the outputs could be given by an integral controller?
8. Which one of the outputs could be given by a proportional plus integral controller?
9. Decide whether each of these statements is True (T) or False (F). With a PID process control system tested at start-up using the ultimate cycle method with the derivative mode turned off and the integral mode set to its lowest setting, the period of oscillation was found to be 10 minutes with a proportional gain setting of 2. The optimum settings, using the criteria of Ziegler and Nichols, will be: (i) An integral constant of 0.2 min"'.
(ii) A proportional gain setting of 1.2.
A (i)T (ii)T
B (i)T (ii)F
C (i)F (ii)T
D (i)F (ii)F
10. Decide whether each of these statements is True (T) or False (F). With a PI process control system tested by the process reaction method and the controller output changed by 10%, the response graph obtained was as shown in FIG. 32. The optimum settings, using the criteria of Ziegler and Nichols, will be:
FIG. 32 Problem 10
(i) Proportional gain constant 7.2.
(ii) Integral constant 0.15 min"*.
A (i)T (ii)T
B (i)T (ii)F
C (i)F (ii)T
D (i)F (ii)F
11. Decide whether each of these statements is True (T) or False (F). With a PI process control system tested at start-up using the ultimate cycle method with the derivative mode turned off and the integral mode set to its lowest setting, the period of oscillation was found to be 20 minutes with a proportional gain setting of 1.2. The optimum settings, using the criteria of Ziegler and Nichols, will be: (i) An integral constant of 0.06 min''. (ii) A proportional gain setting of 1.2.
A (i)T (ii)T
B (i)T (ii)F
C (i)F (ii)T
D (i)F (ii)F
12. A control system is designed to control temperatures between -10° and +30 C. What is (a) the range, (b) the span?
13. A temperature control system has a set point of 20°C and the measured value is 18°C. What is (a) the absolute deviation, (b) the percentage deviation?
14. What is the controller gain of a temperature controller with a 80% PB if its input range is 40°C to 90° and its output is 4 mA to 20 mA?
15. A controller gives an output in the range 4 to 20 mA to control the speed of a motor in the range 140 to 600 rev/min. If the motor speed is proportional to the controller output, what will be the motor speed when the controller output is (a) 8 mA, (b) 40%?
16. FIG. 33 shows a control system designed to control the level of water in the container to a constant level. It uses a proportional controller with Kp equal to 10. The valve gives a flow rate of 10 m^3/h per percent of controller output, its flow rate being proportional to the controller input. If the controller output is initially set to 50% what will be the outflow from the container? If the outflow increases to 600 m^3/h, what will be the new controller output to maintain the water level constant?
17. Sketch graphs showing how the controller output will vary with time for the error signal shown in FIG. 34 when the controller is set initially at 50% and operates as (a) just proportional with AT p = 5, (b) proportional plus derivative with Kp = 5 and Kd = 1.0 s, (c) proportional plus integral with Kp = 5 and Ki = 0.5 s^-1 .
18. Using the Ziegler-Nichols ultimate cycle method for the determination of the optimum settings of a PID controller, oscillations began with a 30% proportional band and they had a period of 11 min. What would be the optimum settings for the PID controller?
19. Using the Ziegler-Nichols ultimate cycle method for the determination of the optimum settings of a PID controller, oscillations began with a gain of 2.2 with a period of 12 min. What would be the optimum settings for the PID controller?
20. FIG. 35 shows the open-loop response of a system to a unit step in controller output. Using the Ziegler-Nichols data, determine the optimum settings of the PID controller.
FIG. 33 Problem 16
FIG. 34 Problem 17
FIG. 35 Problem 20
Related Articles -- Top of Page -- Home