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The experiment card 'Controlled system simulations'

SO4201-5U

General information

The UNI-TRAIN-I experiment card

The UNI-TRAIN-I experiment card Controlled System Simulations Controlled System Simulations contains a series ofcontains a series of typical transfer elements, which are used to assemble practical controlled systems. The typical transfer elements, which are used to assemble practical controlled systems. The transfer elements are designed as electronic analog circuits (wired operational

transfer elements are designed as electronic analog circuits (wired operational amplifiers) or digital algorithms (with A/D and D/A converters connected in series amplifiers) or digital algorithms (with A/D and D/A converters connected in series upstream and downstream). The transfer elements can practically be combined at will upstream and downstream). The transfer elements can practically be combined at will for the assembly of sophisticated controlled systems.

for the assembly of sophisticated controlled systems.

Move the cursor over the graphic of the experiment Move the cursor over the graphic of the experiment card to find out more details.

card to find out more details.

Below you find a detailed list of the transfer elements at your disposal: Below you find a detailed list of the transfer elements at your disposal:

Proportional-action element (P element) with adjustable proportional-action Proportional-action element (P element) with adjustable proportional-action coefficient.

coefficient.

Integral-action element (I element) with adjustable integral time constant T

Integral-action element (I element) with adjustable integral time constant TII = 1/K= 1/KII (K(KII::

integral-action coefficient). In conjunction with the PID controller this time constant is integral-action coefficient). In conjunction with the PID controller this time constant is referred to also as integral-action time T

referred to also as integral-action time TNN..

Two time-delay elements of the 1st order (P-T

Two time-delay elements of the 1st order (P-T11elements) with varying timeelements) with varying time

constants. constants.

Non-linear characteristic f(x). Non-linear characteristic f(x).

Programmable digital algorithm (e.g. for simulating lag). Programmable digital algorithm (e.g. for simulating lag).

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Summation point (e.g. to feed disturbance signals forward). A fixed signal level of Summation point (e.g. to feed disturbance signals forward). A fixed signal level of 2.5V is applied to the upper Z socket. This signal can be fed forward to the

2.5V is applied to the upper Z socket. This signal can be fed forward to the

summation point either via a connecting jumper, or it can be connected automatically summation point either via a connecting jumper, or it can be connected automatically using the "Reference variable/disturbance variable" function in L@Bsoft. The

using the "Reference variable/disturbance variable" function in L@Bsoft. The disturbance variable relevant for this experiment card is the disturbance variable 1 disturbance variable relevant for this experiment card is the disturbance variable 1 (see also L@Bsoft-Help for the subject "Reference variable/disturbance variable"). (see also L@Bsoft-Help for the subject "Reference variable/disturbance variable"). The experiment card can be combined with both the PID controller card SO4201-5R as The experiment card can be combined with both the PID controller card SO4201-5R as well as with the two-position/three-position card SO4201-5S, to assemble closed control well as with the two-position/three-position card SO4201-5S, to assemble closed control loops. Within the framework of this course you will be dealing with both controller types loops. Within the framework of this course you will be dealing with both controller types in the appropriate chapters. For the simulation lag you can use the block labelled

in the appropriate chapters. For the simulation lag you can use the block labelled Algorithm

Algorithm which can be configured for lag using the virtual instrumentwhich can be configured for lag using the virtual instrument Lag element Lag element (see(see following screenshot). The desired lag can then be adjusted with a resolution given by following screenshot). The desired lag can then be adjusted with a resolution given by the time currently set.

the time currently set.

Classifying control loop elements

The idea of a

"transfer element"

All of the components of a control loop can be seen as

All of the components of a control loop can be seen as transfer elements transfer elements , which take, which take the predetermined signal characteristic(s) of their input variable(s) and generate

the predetermined signal characteristic(s) of their input variable(s) and generate characteristic(s) of output variable(s) in line with specific

characteristic(s) of output variable(s) in line with specific physical relationships** physical relationships** . Of. Of particular importance to automatic control engineers are the linear transfer elements, particular importance to automatic control engineers are the linear transfer elements, which excel at applying the

which excel at applying the principle of superpositioning principle of superpositioning . The complexity of a linear. The complexity of a linear transfer system is defined by its order (i.e. the number of energy storage elements transfer system is defined by its order (i.e. the number of energy storage elements included in the system). The RC element depicted in the subsequent figure has one included in the system). The RC element depicted in the subsequent figure has one energy storage element (namely the capacitor) and thus constitutes a system of the 1st energy storage element (namely the capacitor) and thus constitutes a system of the 1st order. If a constant input voltage

order. If a constant input voltage u u inin is applied to the network at timeis applied to the network at time t t = 0, the result for= 0, the result for

the output voltage

the output voltage u u outoutis the charging curve well known for a capacitor and takes theis the charging curve well known for a capacitor and takes the

form of an exponential function. form of an exponential function.

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The following graphic shows an electrical series resonant circuit consisting of a resistor The following graphic shows an electrical series resonant circuit consisting of a resistor R, inductor L and capacitor C.

R, inductor L and capacitor C.

What is the order of this electrical network? Enter your answer with your reasons in What is the order of this electrical network? Enter your answer with your reasons in the answer box below.

the answer box below.

In automatic control technology transfer elements are normally represented as a block In automatic control technology transfer elements are normally represented as a block structure regardless of their actual physical structure (electrical, mechanical...). Refer to structure regardless of their actual physical structure (electrical, mechanical...). Refer to the following graphic. Such a system tends to have one or more input variables (below the following graphic. Such a system tends to have one or more input variables (below y

y ) and one or more output variables (below) and one or more output variables (below x x ). By combining individual system blocks). By combining individual system blocks (series or parallel connections) any number of complex system structures can be (series or parallel connections) any number of complex system structures can be depicted in a clear and straightforward fashion.

depicted in a clear and straightforward fashion.

Types of linear transfer elements

All linear transfer elements can be made up of basic elements of a lower order (i.e All linear transfer elements can be made up of basic elements of a lower order (i.e zeroth, first and second orders). Here a distinction must be drawn between elements zeroth, first and second orders). Here a distinction must be drawn between elements with

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a step-shaped input variables, and elements without compensation, whose output variables increase at a constant rate. One example of the latter is a water tank whose water level continues to rise at a constant rate when being filled at a constant flow rate per unit of time until the tank finally overflows.

Right:

Typical step response of a transfer element with compensation (left) and without compensation (right).

Give at least one additional example of a transfer element without compensation. Enter your answer in the answer box below.

An additional distinguishing feature for linear transfer elements is the time delay effect of the element. Here a distinction is drawn between transfer elements with and without time delay. The following graphic provides an overview of one possible breakdown of linear transfer elements.

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Is the RC element depicted above a system with or without compensation? Enter your answer including your reasons in the answer box below.

Static and dynamic response of control loop elements

Static system response

When analyzing control loop elements a distinction is drawn between dynamic response (response over time) and static response (steady-state response) of the element. If you consider, for example, the system's response to a step change of the input variable, commonly referred to as the step response (see the following graphic), a static response is characterised by steady end state x0 of the output variable, i.e. the

respective value assumed by the system assumes after the transient response has faded.

The relationship of the output amplitude to the input amplitude is designated the proportional coefficient KP of the controlled system (also frequently called "system

gain"). The following expression holds true

If you determine the ratio x0 /y0 for various operating points (i.e. step amplitudes) y0 and

enter the results on a graph, the result you obtain is the so-called static characteristic of the system. In a linear system the proportional coefficient is independent of the

operating point; here this results in a linear characteristic whose slope corresponds to the proportional coefficient of the system.

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Right hand figure:

Static characteristic of a linear system.

You should now record the static characteristic of the left-hand PT 1element. To do this

apply a series of DC voltages from 0 to 10 V (in 1 V increments) to the input y = y3of the PT1element

and determine and use a voltmeter to determine the corresponding steady-state output voltage x = x3.

Enter the values obtained in the table below and determine the static characteristic.

What is the proportional coefficient of the PT1element? Enter your answer into the answer

box below.

Dynamic system response

From the static characteristic you can derive which final value the output variable of a system will reach in response to a certain input variable. But this characteristic does not permit any conclusions as to the how (i.e. the how fast) this final value is reached.

Generally speaking technical systems can only follow changes of the input variable after a time lag; e.g. due to its mass inertia the speed of a motor only slowly increases after an abrupt jump in motor voltage. You notice inertia, for example, when driving and you try to accelerate to a certain speed: the vehicle can only follow an abrupt flooring of the gaspedal with a delayed response.

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It is the dynamic response of a system that describes the time characterisitc of the output variable of the system in response to a change of the input variable (transient process). This can be characterised by the system's step response already referred to above. Qualitative features used to assess the system's dynamism include especially the rate at which the final steady-state value is approximated, and the system's

oscillatory characteristics (asymptotic or oscillatory approach to the final value).

The step response of the left-hand PT1-element is to be recorded. To do this first assemble the

experiment circuit shown below.

Activate the step-response plotter and configure it as shown in the following Table. Settings Input

Channel A Meas. range: 10 V Coupling: DC Channel B Meas. range: 10 V Coupling: DC Other Range: 100 Offset: 0

Settings Output Step change from

... to ... 0 50% Wait time/ms 0 Measurements 300 Settings Diagram Display Channel A x-axis from ... to ... 0 0.1 s y-axis from ... to ... 0 100

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.

Step response of the PT1-element

Describe the characteristic of the step response in qualitative terms. To do this enter your explanation in the following answer box!

Parameters of the P element

Classification of the P element

The proportional-action element (P element) constitutes the simplest of all linear transfer elements. Output variable x and input variable y are combined using the mathematical expression

In the case of a P element the input variable y(t) has an immediate impact on the output variable - here we are dealing with a transfer element without delay . The parameter KP

is called the proportional coefficient .

The following graphic shows the step response and block symbol of the P element. The latter contains the step response inside it to ensure rapid identification of the control element within the control loop structure.

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At the system output you again obtain the input signal but amplified or attenuated by the factor KP. The P element is thus a transfer element with compensation (see the

following graph).

Example for a P element

The electrical network below constitutes an example P element in the technical sense.

Fig. right: If you select the current i as the input variable and the voltage u as the output variable, the network shown constitutes a P element

behaving in accordance with Ohm's law u = R x i .

What is the proportional coefficient KP of the network? Enter your answer in the

answer box below.

Experiment

In the following experiment you should determine the step response of the P element on the experiment card "controlled system simulation" (SO4201-5U). The proportional

coefficient KPis now determined from the step response with the potentiometer set to its

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First set up the experiment circuit below. Then adjust the control setting (potentiometer) for the P element to a medium setting.

Activate the step-response plotter and configure it in accordance with the settings in the Table below.

Settings Input

Channel A Meas. range: 10 V Coupling: DC Kanal B Meas. range: 10 V Coupling: DC Other Range: 100 Offset: 0

Settings Output Step change from ... to

... 0 50% Delay time/ms 0 Measurements 300 Settings Diagram Display Channel A x-axis from ... to ... 0 1 s y-axis from ... to ... 0 100

Now determine the step response and copy the plot into the space reserved for it below.

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Step response of the P element

What is the proportional coefficient of the P element at the selected setting? Enter your answer into the answer box below.

Parameters of the PT

1

element

Classification of the PT

1

element

A time delay element of the 1st order is called a PT 1-element. In this context the

relationship between the input variable y(t) and the output variable x(t) can be expressed by the differential equation

The parameter KP is referred to as the proportional coefficient , the parameter T is called

the time constant of the PT1element.

The following Figure shows the step response and the block symbol of the PT 1element.

Here the final steady-state value of the output variable is assumed to be only

asymptotic, i.e. time delayed. The time constant T specifies how fast the output variable tends towards the final value. In mathematical terms the following equation expresses the characteristic of the output variable for t > 0

A PT1element is thus a system with compensation and time delay (see the following

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Determining the time constants on the basis of the step response

Whereas the proportional coefficient KP of the PT1element for an input variable step

change of the height 1 can be read directly off the step response (as it corresponds to the final steady-state value of the output variable), finding out the time constant T is somewhat more complicated. It can be achieved in two different ways.

Determining the value of T using the tangent method

The so-called tangent method uses the application of tangents on the step response to determine the time constant T. The point where the tangent intersects with the final steady-state value of the output variable and then drop a perpendicular line down to the time axis. The resulting segment of the time axis corresponds to the time constant.

Fig. on the left: Determining the time constant T according to the tangent method. The tangent is drawn as a red line to the step response from the time point t = 0.

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Determining the value of T according to the 63% method

The so-called 63% method is based on the fact that the time corresponding to the time constant T has elapsed when 63% of the final value has been reached. This can be derived directly from the equation given above by inserting the value T for the time variable t. We thus obtain the following for the output variable

The following graph illustrates how the time constant can be derived directly from the step response by this method.

Fig. left: Determining the time constant T according to the 63% method. This method gives relatively good results even when the signals are distorted.

Example for a PT

1

element

PT1behavior is evident wherever there is a system with precisely one energy storage

element. The Figure below shows a mechanical system comprising a mass m (energy storage element) and a shock absorber r, whose frictional force is assumed to be proportional to the velocity. Furthermore an external force F acts on the mass. If you take the sum of the forces, you arrive at the following expression for the motion

As can be seen from a comparison to the differential equation of the general PT 1

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Fig. right: Shock absorber system for a mass as an example of a mechanical PT1element.

What are the system's proportional coefficient and time constant? Enter your answer into the following answer box.

Experiment

In the following experiment you shall determine the step response of the two PT 1

elements of the P element of the experiment card "controlled system simulation" (SO4201-5U). Use the step response to determine the respective proportional coefficient KP and the time constant T.

First set up the following experiment circuit.

Activate the step response plotter and configure it as shown in the following Table.

Settings Input

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Settings Output Step response from ... to

... 0 50% Delay time/ms 0 Measurements 300 Settings Diagram Display Channel A x-axis from ... to ... 0 0.2 s y-axis from ... to ... 0 100

Now determine the step response of the left-hand PT1-element and copy the diagram into

the upper space reserved for the graph. Determine the proportional coefficient and time constant in accordance with both the tangent and the 63% method. Then repeat the experiment with the right-hand PT1element, copy the step response into the lower space

reserved for graphs and determine from this the proportional coefficient and time constants. Enter the numerical values obtained for the parameters in the answer box below.

Step response of the left-hand PT1element

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Proportional coefficients and time constants determined:

Now repeat the experiment using the right-hand PT1element, but for a different amplitude of

the input variable step change (alter the step change from 0 to 25%). Drag and drop the step response into the space reserved for the graphic below and use this to also determine

proportional coefficient and time constant. Do the parameters change because the height of the step response changes? Enter the your answer with your reasons into the answer box below!

Step response of the right-hand PT1element for a change in the height of the input variable step

Answer:

Parameters of the PT

2

element

Classification of the PT

2

element

The PT2element (time delay element of the 2nd order) is expressed by the differential

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The parameter T stands for the time constant of the element, D is its damping and again KP is the proportional coefficient. The system demonstrates various kinds of behavior

depending on the magnitude of its damping D:

D > 1: The PT2element can be understood in this case as a series connection of two

PT1-elements. The system's step response is aperiodic.

D < 1: In this case the PT2element is capable of oscillation. The step response of the

system is thus oscillatory. At the extreme D = 0 the oscillation is undamped. The following Figure shows the step response and the block symbol of the PT2element

for the two cases.

The PT2element like the PT1element is a transfer element with compensation and time

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Example for a PT

2

element

PT2elements contain exactly two energy storage elements. The Figure below illustrates

a mechanical PT2element consisting of a spring, a mass and shock-absorber.

Fig. right: Spring-mass-shock absorber system as an example of a PT2element.

Which two components constitute the energy storage elements? Is the system able to oscillate? How is the damping determined? Enter your answer into the following answer box.

Experiment

In the following experiment the step responses of the two series connected PT 1

elements on the experiment card "controlled system simulation" (SO4201-5U) are determined.

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Activate the step response plotter and configure it as shown in the Table. Settings Input

Settings Output Step response from ... to

... 0 50% Delay time/ms 0 Measurements 300 Settings Diagram Display Channel A x-axis from ... to ... 0 0.3 s y-axis from ... to ... 0 100

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Step response of the PT2element

Do we obtain an aperiodic or oscillatory response? What is the slope of the step response at time t = 0? What is the proportional coefficient KP of the PT2element?

Enter your answer into the answer box below.

Parameters of the I-element

Classification of the I-element

In the case of the integral-action element (I element) the output variable x and the input variable y are combined using the expression

The output variable x(t) arises through the integration of the input variable y(t). The

parameter KI is referred to as the integral action coefficient , its inverse value TI = 1/KI is

called the time constant of the I element.

The following diagram shows the step response and the block symbol of the I element. From the constant input variable for t > 0 a linear characteristic is generated at the output of the I element, the slope of which is given by K I.

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Thus the step response does not tend towards a constant value. On the contrary the forward feed of a step change immediately produces an output variable which increases at a linear rate at the output (without additional delay). As a consequence the I element is a transfer element without compensation and without delay (see the following chart).

Examples for I elements

I elements are systems with time delay of the 1st order and thus contain only one

energy storage element. A typical example found in everday use is a tank or a bunker, whose input variable is the mass flow (mass per unit of time) and its output variable is the fill level. When there is a inlet flow which is constant in time, the fill level increases linearly until the maximum fill level is reached. An example for an electrical I element is a capacitor, in which the input variable selected is the charging current and the output variable is the capacitor's voltage: when the charge current is constant the capacitor voltage increases linearly (theoretically ad infinitum).

Even a DC motor used to position the slide carriage of a machine tool, constitutes an I element, if the armature voltage is selected as the system's input variable and the slide carriage's position as the output variable (see the following Figure). If the armature voltage is constant the motor operates at constant speed and resulting in the slide

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carriage moving forward at a constant speed. The slide carriage's position thus changes at a linear rate.

Fig. right: positioning drive as an example for an I element.

Experiment

In the following experiment you should determine the step response of the I element of the experiment card "controlled system simulation" (SO4201-5U). The step response should be used to determine the proportional coefficient K I with the potentiometer

adjusted to a medium setting.

First set up the following experiment circuit. Adjust the potentiometer for the I element to the medium setting.

Activate the step-response plotter and configure it as shown in the following Table. Settings Input

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Step change from ... to ... 0 100% Delay time/ms 0 Measurements 300 Settings Diagram Display Channel A x-axis from ... to ... 0 1 s y-axis from ... to ... 0 100

Reset the I element by pressing the reset button. Then determine the step response and copy the plot into the space reserved for it below.

Step response of the I element

What is the integral-action coefficient KI of the I element? Enter your response in the

answer box below.

Raise the integral-action coefficient by turning the knob to the left and repeat the experiment.

What do you notice in the step response which indicates the change in K I? Enter

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Parameters of the lag element

Classifying the lag element

The lag element (Ttelement) like both the PT1and PT2elements demonstrates time

delay, but of a totally different nature. The output variable x and input variable y are in fact identical in terms of their characteristic, but with this element there is a time shift in accordance with the relationship

The time shift Tt between the input variable and the output variable is termed lag or

dead time . According to this definition the proportional coefficient K P of the element is 1.

The following Figure shows the step response and the block symbol of the lag element.

In some cases the lag element can also have a proportional coefficient, which is not equal to 1. For that reason the lag element is also frequently referred to as a PT t

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Example - lag element

Lag elements can be found wherever running times exist (e.g. in the context of

conveying materials). The Figure below shows the example of a conveyor belt with the input variable xIN and the output variable xOUT. The belt is of length l and operates at

speed v .

Fig. right: conveyor belt example for a lag element.

What is the lag of the illustrated conveyor belt as a function of band length and speed? Enter your answer in the following answer box.

Experiment

The function block denoted Algorithm and located on the experiment card "controlled system simulation" (SO4201-5U) can be programmed as a lag element using the Lag Element virtual instrument. To do this the instrument only has to be activated, the

desired lag entered and applied to the Algorithm block.

Fig. right: Lag instrument. The set lag is

transferred to the Algorithm block using the Apply button.

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First assemble the following experiment circuit.

Activate the step response plotter and configure it as shown in the following Table. Settings Input

... 0 50% Delay time/ms 0 Measurements 300 Settings Diagram Display Channel A x-axis from ... to ... 0 1 s y-axis from ... to ... 0 100

Activate the lag element instrument and set a lag of 0.5 s. Then determine the step response and copy the plot into the space reserved for it below.

Step response of the lag element

What is the proportional coefficient of the lag element? Enter your answer into the following answer box.

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Combined controlled system elements

Parameters of elements with higher order time delay

System of more complexity (e.g. controlled systems) are normally elements of a higher order. However, these can frequently be formed by combining (series connection) of basic elements of a lower order. The systems cumulative order is results from the sum order of the subsystems. If one of the transfer elements contains an I-action component, the result is a system without compensation. The subsequent graph shows two

examples of combined elements.

Which of the combined elements is shown in the following Figure? Enter your answer in the following answer box.

Combined elements can be specified by their type (i.e. PT3for example) and the

parameters of their fundmental components. However, in reality this proves to be very difficult because it entails manipulating all of the internal variables of the system (i.e.

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intermediate variables). Therefore, in actual practice such systems are described particularly by their "basic", i.e. steady-state response (system with and without

compensation) and by so-called "surrogate" parameters, which can be derived directly from the system's step response. The following Figure illustrates the meaning of these parameters for systems with compensation (left) and without compensation (right).

Controlled systems with compensation are expressed by:

The proportional coefficient KS (also frequently termed KP ). This corresponds to the

final steady-state value of the step response for an input step change to a height of 1.

The delay time Tu. This corresponds to the intersecting point of the inflectional

tangent applied to the step response and dropped down to the time axis. The delay time is a measure for how long it takes for the output variable to respond noticeably to the input step change.

The compensation time Tg. To determine this you drop the intersecting point of the

inflectional tangent with the final steady-state value to the time axis and subtract from this the delay time previously obtained. The compensation time is a measure for how long it takes until the transient process has been completed.

Naturally controlled systems without compensation have no compensation time because a final steady-state is never reached. Thus two parameters suffice for their characterisation:

The integral-action coefficient KIS. It corresponds to the steady-state slope of the step

response.

The delay time Tu. It is found from the point of intersection of the straight lines,

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It remains to be said that a system capable of oscillating can not be described by these parameters! Equally impossible to describe in this way are controlled systems without compensation comprising more than one I element.

Experiments

In the first experiment the step response is to be determined from a system made up of the series connection of the two PT1elements on the experiment card "controlled

system simulation" (SO4201-5U). Based on the step response resolve the proportional coefficient KS, the delay time Tu and the compensation time Tg.

First set up the following experiment circuit.

Activate the step response plotter and configure it as shown in the following Table. Settings Input

Channel A Meas. range: 10 V Coupling: DC Kanal B Meas. range: 10 V Coupling: DC Other Range: 100 Offset: 0

... 0 50% Delay time/ms 0 Measurements 300 Settings Diagram Display Channel A x-axis from ... to ... 0 0.3 s

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y-axis from ... to ... 0 100

Step response of the series connection for the two PT1 elements

Now reverse the sequence of the two PT1elements and repeat the experiment!

Determine the parameters of the series connection for both cases. How do the results differ? How can this be explained? Enter your findings and answers into the answer box below.

In the second experiment a series connection comprising P element, Integral-action element and the left-hand PT1element (time constant T1) is to be investigated. Set

up the following experiment circuit. Adjust the potentiometer of the P element to the medium setting and the potentiometer of the I element to far left limit.