Part A – Course 1 Pressure control loop
5. Limit of stability
The limit of stability KPcrit is determined by means of increasing the pro- portional coefficient KP and is reached when continuous oscillations oc- cur.
To demonstrate the dependence of the limit of stability on the reference variable, a small step of the reference variable is set. By offsetting the mean value, the entire range of potential reference variables is exam-
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Exercise 6
WORKSHEET
Pressure control loop 1. Pressure control loop
Construct the pressure control loop. Use the hydraulic and electrical circuit diagrams.
Set the controller card in the initial position: - Limiter to ± 10V,
- Offset to 0V,
- Proportional coefficient KP = 1, - Other controller coefficients = 0.
2. Control direction
Interrupt the closed control loop by not connecting the pressure sen- sor to the controller card.
Check the control direction:
Does the controlled variable x increase with rising reference variable w?
If “Yes”, then the control direction is correct: + w equals + x.
Nevertheless, carry out a check of the interfaces. Make sure that the following conditions are met:
+ w equals + y + y equals + x + w equals + x
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Exercise 6
3. Closed control loop
Close the control loop by connecting the pressure sensor to the con- troller card.
Check whether the system deviation e becomes smaller.
• If “Yes”, then the connection of the pressure sensor is also in or- der.
• If “No”, reverse the signal connections of the pressure sensor. Check the effects of the following polarity reversals:
Reverse polarity Change in controlled variable x with increasing reference variable w Reference variable w Correcting variable y Feedback r Value table
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Exercise 6
WORKSHEET
4. Control quality
Set a step-change reference variable:
w = 3V ± 2V f = 5Hz in square wave form
Select the following scales on the oscilloscope: Time t: 0,02 s/Div. Reference variable w: 1 V/Div. Controlled variable x: 1 V/Div.
Determine the characteristics of the control quality in relation to dif- ferent proportional coefficients KP:
- Overshoot amplitude xm,
- Steady-state system deviation estat, - Settling time Ta.
KP xm (V) estat (V) Ta (s) Oscillations Evaluation
1 3 5 8 10 12
Which controller setting do you consider to be an optimum setting?
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Exercise 6
What then are the characteristics of the controller quality:
Overshoot amplitude xm,opt = Steady-state system deviation estat,opt = Settling time Ta,opt = Stability:
5. Limit of stability
Determine the limit of stability by increasing KP until continuous os- cillations occur.
KPcrit = (with w = 3V ± 2V, 5 Hz)
Set a step of ± 0.5 V as reference variable and determine the limit of stability for different reference variables.
Reference variable w Limit of stability KPcrit Evaluation
1V ± 0.5V 2V ± 0.5V 3V ± 0.5V 4V ± 0.5V 5V ± 0.5V Value table
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Exercise 7
Closed-loop hydraulics
Injection moulding machine
To learn about the function of I and PI controllers
To be able to determine the characteristics of I and PI controllers
To be able to describe the purpose of using I controllers
Integral controller (I controller)
The behaviour of the I controller is determined by the integral element.
The I element adds the input signal e via the time t and
amplifies it by the factor KI to the output signal yI.
With a constant input signal this results in the following equation: yI = KI⋅ e ⋅ t
A complete I controller consists of
the comparator to form the system deviation e as input signal of the I element,
the I element and
the limiter to form a suitable correcting variable y.
Subject Title
Training aim
Technical knowledge
Fig. A7.1:
Block diagram and symbol of integral controller
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Exercise 7
Characteristics of the I element
The transition function of an I element displays a ramp-shaped pattern since the I element carries out a continuous summation (= integration) of the input signal. The ramp gradient is determined by the integral- action coefficient KI. The integration time TI elapses until the output sig- nal y has reached the same value as that of the input signal w, whereby the following applies:
TI = 1 KI
Use of an I controller:
The I controller reacts only slowly to changes in the reference vari- able (in comparison with the P controller) and are therefore rarely used alone.
Fig. A7.2: Transition function and block diagram of I element
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Exercise 7
The proportional integral controller (PI controller)
A parallel circuit consisting of a proportional and integral controller forms a PI controller. It combines the advantages of both types of controller, giving a controller which is able both to react quickly and to eliminate system deviations.
A PI controller operates according to the following equation:
t) T 1 + K ( e = t) K + (K e = y I P I P PI ⋅ ⋅ ⋅ ⋅ Characteristics of a PI element Fig. A7.3:
Block diagram and symbol of PI controller
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Exercise 7
The integral-action time can also be calculated from the coefficients of the PI element:
Output signal of the P element: yP = KP⋅ e Output signal of the I element: yI = KI⋅ e ⋅ t Integral-action time Tn: yP = yI
And thus: KP⋅ e = KI⋅ e ⋅ Tn
Hence the following applies for the integral-action time: T = K K n
P I The equation of the PI controller can thus be simplified to:
) T t + 1 ( K e = y n P PI ⋅ ⋅
The characteristics of the PI controller specified are:
either the controller coefficients KP and KI,
or the proportional coefficient KP and the integral-action time Tn. Fig. A7.4:
Transition function and block diagram of PI element
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Exercise 7
Controller type Advantage Disadvantage
P controller fast inaccurate
I controller accurate slow,
tendency towards oscillations PI controller fast and accurate tendency towards oscillations
Different pressures are to be set on an injection moulding machine: a low charging pressure to fill the mould, a slightly higher forming pres- sure to fill the entire cavity and a higher calibrating pressure for accurate hardening. A pressure control loop is to be constructed to be able to achieve the required pressure quickly and accurately and to maintain it for the required period of time. You are to examine whether a P control- ler is adequate or whether a PI controller would give certain advantages.
I controller Table A7.1: Advantages and disadvantages of P, I and PI controllers Problem description Positional sketch Exercise
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Exercise 7
1. Measuring circuit
The following are to be measured
the reference variable w as input signal of the controller and
the correcting variable y as output signal of the controller. The following devices are required for this:
the PID controller card with the I controller,
a generator for step-change test signals in a range of ± 10V,
an oscilloscope to record the time characteristic of the output vari- able,
a multimeter for commissioning,
a power supply unit for the voltage supply to the controller. The following settings are to be made prior to switching on:
Limiter to ± 10V,
Offset precisely to zero,
Integral-action coefficient KI = 1,
All other controller coefficients to zero.
The setting of the integral-action coefficient KI is the result of the value of the potentiometer and of the rotary switch.
Execution
Fig. A7.5: Setting of integral coefficient KI
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Exercise 7
2. I controller
The transition function of the I controller is as follows:
the step-change reference variable w
results in a ramp-shaped correcting variable y.
Various transition functions are illustrated on the worksheet. The follow- ing points must be observed in order to determine the characteristics of an I controller:
1. With a reference variable of w = 0V ± 10V, the magnitude of the step- change w equals = 10V (not 20V!).
2. The integration time TI is reached when zero of the correcting vari- able y has risen to the level of the step-change w.
3. The integration coefficient KI indicates the increase of the transition function. It is therefore a measure for the rate of change of the cor- recting variable y.
Only by following the above conditions, can the characteristics be cor- rectly established.
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Exercise 7
3. PI controller
The transition function of the PI controller differs from that of the I con- troller by displaying an initial step-change. After that, it results in the same ramp-shape as the I controller.
To determine the integral-action time Tn, the proportion of the P con- troller is calculated first:
yP = KP ⋅ w
The integral-action time Tn has been reached, when yI = yP.
When the signals are set, this does not result in the ideal transition function illustrated in the worksheet. This is due to different limitations of the P and I element and the downstream limitation of the correcting variable:
Limitation of P element yP = ± 10V, Limitation of I element yI = ± 14V, Limitation of correcting variable y = ± 10V.
If the output variable of the I element yI exceeds 10V, this does not be- come apparent in the correcting variable y. A subsequently resulting step-change of the P element yP is therefore incorrectly interpreted. In that case, the transition functions of the individual elements must be measured. The addition of the two signals produces the transition func- tion of the PI controller: y = yP + yI.
Fig. A7.6: Measuring of integral-action time Tn
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Exercise 7
Mathematically, the integral-action time is the quotient of the controller coefficient settings: T = K K n P I 4. P-, I and PI controller
Use the table to evaluate the different types of controller relative to the speed and adjustment characteristics of the system deviation.
Fig. A7.7: Calculating of integral-action time Tn
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Exercise 7
WORKSHEET
The I controller