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The experiment card 'Controlled system simulations' SO4201-5U General information The UNI-TRAIN-I experiment card Controlled System Simulations contains a series of typical transfer elements, which are used to assemble practical controlled systems. The transfer elements are designed as electronic analog circuits (wired operational 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 for the assembly of sophisticated controlled systems. Move the cursor over the graphic of the experiment card to find out more details.

Below you find a detailed list of the transfer elements at your disposal: Proportional-action element (P element) with adjustable proportional-action coefficient. Integral-action element (I element) with adjustable integral time constant T I = 1/KI (KI: integral-action coefficient). In conjunction with the PID controller this time constant is referred to also as integral-action time T N. Two time-delay elements of the 1st order (P-T 1 elements) with varying time constants. Non-linear characteristic f(x). Programmable digital algorithm (e.g. for simulating lag).

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 summation point either via a connecting jumper, or it can be connected automatically using the "Reference variable/disturbance variable" function in [email protected] The disturbance variable relevant for this experiment card is the disturbance variable 1 (see also [email protected] for the subject "Reference variable/disturbance variable"). 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 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 Algorithm which Algorithm which can be configured for lag using the virtual instrument Lag element (see element (see following screenshot). The desired lag can then be adjusted with a resolution given by 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 transfer elements , which take the predetermined signal characteristic(s) of their input variable(s) and generate characteristic(s) of output variable(s) in line with specific physical relationships** . relationships** . Of particular importance to automatic control engineers are the linear transfer elements, which excel at applying the principle of superpositioning . superpositioning . The complexity of a linear 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 energy storage element (namely the capacitor) and thus constitutes a system of the 1st order. If a constant input voltage u in t = 0, the result for in is applied to the network at time t = out is the charging curve well known for a capacitor and takes the the output voltage u out form of an exponential function.

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

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

In automatic control technology transfer elements are normally represented as a block 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 y ) and one or more output variables (below x ). ). By combining individual system blocks (series or parallel connections) any number of complex system structures can be 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 zeroth, first and second orders). Here a distinction must be drawn between elements with compensation, with compensation, whose output variables strive for finite output values in response to

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.

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 x 0 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 x 0 /y0 for various operating points (i.e. step amplitudes) y 0 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.

Right hand figure: Static characteristic of a linear system.

You should now record the static characteristic of the left-hand PT 1 element. To do this apply a series of DC voltages from 0 to 10 V (in 1 V increments) to the input y = y 3 of the PT 1 element and determine and use a voltmeter to determine the corresponding steady-state output voltage x = x 3. Enter the values obtained in the table below and determine the static characteristic.

What is the proportional coefficient of the PT 1 element? 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.

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 PT 1-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

Range: 100

Offset: 0

Other

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

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

. Step response of the PT 1-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 K P 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.

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 K P 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 KP is now determined from the step response with the potentiometer set to its medium setting.

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

Range: 100

Offset: 0

Other

Settings Output Step change from ... to ...

0

50%

Delay time/ms

0

Measurements

300 Settings Diagram

Display

Channel A

x-axis from ... to ...

0

1s

y-axis from ... to ...

0

100

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

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 PT1 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 K P is referred to as the proportional coefficient , the parameter T is called the time constant of the PT 1 element. The following Figure shows the step response and the block symbol of the PT 1 element.

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 PT1 element is thus a system with compensation and time delay (see the following Figure).

Determining the time constants on the basis of the step response Whereas the proportional coefficient K P of the PT 1 element 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.

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 PT1 behavior 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 element shown above, this mechanical system demonstrates PT 1 characteristics.

Fig. right: Shock absorber system for a mass as an example of a mechanical PT1 element.

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 K P 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 Channel A

Meas. range: 10 V

Coupling: DC

Channel B

Meas. range: 10 V

Coupling: DC

Range: 100

Offset: 0

Other

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 PT 1-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 PT 1 element, 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 PT 1 element

Step response of the right-hand PT 1 element

Proportional coefficients and time constants determined:

Now repeat the experiment using the right-hand PT 1 element, 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 PT 1 element for a change in the height of the input variable step

Answer:

Parameters of the PT2 element Classification of the PT 2 element The PT2 element (time delay element of the 2nd order) is expressed by the differential equation

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 PT2 element 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 PT 2 element 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 PT 2 element for the two cases.

The PT2 element like the PT 1 element is a transfer element with compensation and time delay (see the following graphic).

Example for a PT 2 element PT2 elements contain exactly two energy storage elements. The Figure below illustrates a mechanical PT 2 element consisting of a spring, a mass and shock-absorber.

Fig. right : Spring-mass-shock absorber system as an example of a PT2 element.

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. First set up the following experiment circuit.

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

Meas. range: 10 V

Coupling: DC

Channel B

Meas. range: 10 V

Coupling: DC

Range: 100

Offset: 0

Other

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

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

Step response of the PT 2 element

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 K P of the PT 2 element? 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 T I = 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.

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

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 Channel A

Meas. range: 10 V

Coupling: DC

Channel B

Meas. range: 10 V

Coupling: DC

Range: 100

Offset: 0

Other

Settings Output

Step change from ... to ...

0

100%

Delay time/ms

0

Measurements

300 Settings Diagram

Display

Channel A

x-axis from ... to ...

0

1s

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 K I 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 your answer into the answer box below.

Parameters of the lag element Classifying the lag element The lag element (T t element) like both the PT 1 and PT2 elements 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 T t 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 element (see the following flow chart).

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 x IN and the output variable x OUT. 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.

First assemble 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

Channel B

Meas. range: 10 V

Coupling: DC

Range: 100

Offset: 0

Other

Settings Output Step change from ... to ...

0

50%

Delay time/ms

0

Measurements

300 Settings Diagram

Display

Channel A

x-axis from ... to ...

0

1s

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.

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. PT 3 for 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.

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 K P ). 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, towards which the step response tends for prolonged time periods with the time axis.

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 PT 1 elements on the experiment card "controlled system simulation" (SO4201-5U). Based on the step response resolve the proportional coefficient K S, the delay time T u and the compensation time T g. 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

Range: 100

Offset: 0

Other

Settings Output Step change from ... to ...

0

50%

Delay time/ms

0

Measurements

300 Settings Diagram

Display x-axis from ... to ...

Channel A 0

0.3 s

y-axis from ... to ...

0

100

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

Step response of the series connection for the two PT 1 elements

Now reverse the sequence of the two PT 1 elements 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 PT 1 element (time constant T 1) 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.

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

View more... Below you find a detailed list of the transfer elements at your disposal: Proportional-action element (P element) with adjustable proportional-action coefficient. Integral-action element (I element) with adjustable integral time constant T I = 1/KI (KI: integral-action coefficient). In conjunction with the PID controller this time constant is referred to also as integral-action time T N. Two time-delay elements of the 1st order (P-T 1 elements) with varying time constants. Non-linear characteristic f(x). Programmable digital algorithm (e.g. for simulating lag).

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 summation point either via a connecting jumper, or it can be connected automatically using the "Reference variable/disturbance variable" function in [email protected] The disturbance variable relevant for this experiment card is the disturbance variable 1 (see also [email protected] for the subject "Reference variable/disturbance variable"). 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 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 Algorithm which Algorithm which can be configured for lag using the virtual instrument Lag element (see element (see following screenshot). The desired lag can then be adjusted with a resolution given by 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 transfer elements , which take the predetermined signal characteristic(s) of their input variable(s) and generate characteristic(s) of output variable(s) in line with specific physical relationships** . relationships** . Of particular importance to automatic control engineers are the linear transfer elements, which excel at applying the principle of superpositioning . superpositioning . The complexity of a linear 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 energy storage element (namely the capacitor) and thus constitutes a system of the 1st order. If a constant input voltage u in t = 0, the result for in is applied to the network at time t = out is the charging curve well known for a capacitor and takes the the output voltage u out form of an exponential function.

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

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

In automatic control technology transfer elements are normally represented as a block 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 y ) and one or more output variables (below x ). ). By combining individual system blocks (series or parallel connections) any number of complex system structures can be 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 zeroth, first and second orders). Here a distinction must be drawn between elements with compensation, with compensation, whose output variables strive for finite output values in response to

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.

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 x 0 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 x 0 /y0 for various operating points (i.e. step amplitudes) y 0 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.

Right hand figure: Static characteristic of a linear system.

You should now record the static characteristic of the left-hand PT 1 element. To do this apply a series of DC voltages from 0 to 10 V (in 1 V increments) to the input y = y 3 of the PT 1 element and determine and use a voltmeter to determine the corresponding steady-state output voltage x = x 3. Enter the values obtained in the table below and determine the static characteristic.

What is the proportional coefficient of the PT 1 element? 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.

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 PT 1-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

Range: 100

Offset: 0

Other

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

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

. Step response of the PT 1-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 K P 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.

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 K P 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 KP is now determined from the step response with the potentiometer set to its medium setting.

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

Range: 100

Offset: 0

Other

Settings Output Step change from ... to ...

0

50%

Delay time/ms

0

Measurements

300 Settings Diagram

Display

Channel A

x-axis from ... to ...

0

1s

y-axis from ... to ...

0

100

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

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 PT1 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 K P is referred to as the proportional coefficient , the parameter T is called the time constant of the PT 1 element. The following Figure shows the step response and the block symbol of the PT 1 element.

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 PT1 element is thus a system with compensation and time delay (see the following Figure).

Determining the time constants on the basis of the step response Whereas the proportional coefficient K P of the PT 1 element 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.

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 PT1 behavior 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 element shown above, this mechanical system demonstrates PT 1 characteristics.

Fig. right: Shock absorber system for a mass as an example of a mechanical PT1 element.

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 K P 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 Channel A

Meas. range: 10 V

Coupling: DC

Channel B

Meas. range: 10 V

Coupling: DC

Range: 100

Offset: 0

Other

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 PT 1-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 PT 1 element, 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 PT 1 element

Step response of the right-hand PT 1 element

Proportional coefficients and time constants determined:

Now repeat the experiment using the right-hand PT 1 element, 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 PT 1 element for a change in the height of the input variable step

Answer:

Parameters of the PT2 element Classification of the PT 2 element The PT2 element (time delay element of the 2nd order) is expressed by the differential equation

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 PT2 element 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 PT 2 element 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 PT 2 element for the two cases.

The PT2 element like the PT 1 element is a transfer element with compensation and time delay (see the following graphic).

Example for a PT 2 element PT2 elements contain exactly two energy storage elements. The Figure below illustrates a mechanical PT 2 element consisting of a spring, a mass and shock-absorber.

Fig. right : Spring-mass-shock absorber system as an example of a PT2 element.

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. First set up the following experiment circuit.

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

Meas. range: 10 V

Coupling: DC

Channel B

Meas. range: 10 V

Coupling: DC

Range: 100

Offset: 0

Other

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

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

Step response of the PT 2 element

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 K P of the PT 2 element? 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 T I = 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.

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

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 Channel A

Meas. range: 10 V

Coupling: DC

Channel B

Meas. range: 10 V

Coupling: DC

Range: 100

Offset: 0

Other

Settings Output

Step change from ... to ...

0

100%

Delay time/ms

0

Measurements

300 Settings Diagram

Display

Channel A

x-axis from ... to ...

0

1s

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 K I 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 your answer into the answer box below.

Parameters of the lag element Classifying the lag element The lag element (T t element) like both the PT 1 and PT2 elements 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 T t 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 element (see the following flow chart).

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 x IN and the output variable x OUT. 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.

First assemble 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

Channel B

Meas. range: 10 V

Coupling: DC

Range: 100

Offset: 0

Other

Settings Output Step change from ... to ...

0

50%

Delay time/ms

0

Measurements

300 Settings Diagram

Display

Channel A

x-axis from ... to ...

0

1s

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.

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. PT 3 for 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.

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 K P ). 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, towards which the step response tends for prolonged time periods with the time axis.

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 PT 1 elements on the experiment card "controlled system simulation" (SO4201-5U). Based on the step response resolve the proportional coefficient K S, the delay time T u and the compensation time T g. 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

Range: 100

Offset: 0

Other

Settings Output Step change from ... to ...

0

50%

Delay time/ms

0

Measurements

300 Settings Diagram

Display x-axis from ... to ...

Channel A 0

0.3 s

y-axis from ... to ...

0

100

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

Step response of the series connection for the two PT 1 elements

Now reverse the sequence of the two PT 1 elements 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 PT 1 element (time constant T 1) 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.

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

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