control systems lab report
Jess-93
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Experiment 1
1.1. Requirements of Experiment1
This Experiment aims to have a clear understanding of the System Response Characteristics in Time Domain.
Firstly, In the Experiment 1Athe students were required to plot and analyze a second order systems in MATLAB. Also they had to calculate the damping ratio, natural frequency, settling time, peak time and the overshoot percentagethen compare it with the actual graph readings from MATLAB. They also had to calculate the dominant Poles pair and compare it versus the actual MATLAB plot. Timetable was given to the students in the lab manual with specificParameters and Characteristics that should be filled with calculations Table 1 row(a) and (b)respectively. The systems that were used in the whole experiment are:
Secondly, the students also were required to plot the step response and extract the response characteristicssettling time, peak time, rise time and percentage of overshoot from the plotted curve and list them in Table 1 row (b).
Thirdly, In Experiment 1B the students were asked to usethe accuracy ELECTROMECHANICAL SERVOMECHANISM VIRTUAL LABORATORY (ESVL) – CLASSICAL DESIGN program to generate the systems T1(s), T2(s), T3(s)and find how the MATLAB and calculation accurate comparing with more accurate program.
1.2. Introduction
The output of a system is the sum of the forced and natural responses. A second order system illustrates a wide range of responses. Changing of a first order response will only change the speed of the response and never change the form of the response whereas changing of a second order response can definitely and dramatically change the form of the response.
Below figure 1, was given to the students in the lab manual and it shows the changing response curves of second order response systems.
( Figure 1 : T he varying response curves of second order response systems )
The transfer function of a second order system has two finite poles and no zero poles. They can be determined by using the function given in (equation 1.1). Whereas R(s) is the input signal, C(s) is the output signal and T(s) is the transfer function.
(Equation1.1)
(Formula1.2)
Moreover, Based on the location of the poles that can be found from the (formula 1.2), it can be figured out if the systems transient responsesareOverdamped, Underdamped, Undamped or Critically Damped. Figure 2 below shows examples of different transient responses.
‘
( Figure 2 : Different Transient Responses )
When a step input is introduced to the system then the output response and characteristics of the system can be easily measured. The most important parameters associated with a second order systems are damping ratio and the Natural Frequency n. Once they have been found, it is easy to calculate and find the other parameters related to the time response; percent overshoot, peak time, settling time, and rise time. Also, it can be determined by plotting the time responsesonce the Transfer function is found by using the steady state response and the natural response as inputs to the function. It is also important to define the other parameters:
· Peak time (Tp) which is the times required to reach the first, or maximum peak,
· Percent Overshoot (%OS) which is the amount the waveform overshoots the steady state.
· Settling time (Ts) which is the time required for the transient’s damped oscillations to reach and stay within +-2% of the steady state value.
· Rise time (Tr) which is the time required for the waveform to go from 0.1 of the final value to 0.9 of the final value.
Figure 3 explains all the above terms.
Figure 3: Determination of finding Peak, Rise, Settling Time and %OverShoot
These values can also be calculated using the formulas shown below.
1.3. Solution Description
To find the results of Tp, Ts, Tr and %OS from the transfer function plot, students can use MATLAB function called (tf)
Also, to use the virtual lab, students need to calculate Kp in order to generate the response of the transfer function. The transfer function formula can be used to calculate Kp for the given systems as shown below.
In order to run the virtual lab, firstly students need to set Kp values for each system. Secondly, set the input voltage at 1V as it is constant until the systems reach their steady state value. Thirdly, the voltage should be stepped up manually while the program running until the systems reach to the steady state value again. Finally, the transient response will be displayed on the inbuilt oscilloscope. After finishing all these steps, students can easily find the data Ts, Tp, Tr, and %OS by using the time and voltage cursors.
1.4. Test Results
Table 1: Results
Figure 4:Step Response plot of T1(s)
( Figure 5 : Step Response plot of T2(s) )
Figure 5: Step Response plot of T2(s)
Figure 6: Step Response plot of T3(s)
Figure 7: Virtual Lab Response plot of T1(s)
Figure 8: Approx. Settling Time T1(s) = 0.861 seconds
Figure 9: Peak Time T1(s) = 0.806s
Figure 10: Virtual Lab Response plot of T2(s)
Figure11: Approx. Settling Time T2(s) = 0.944 seconds
Figure 12: Rise Time T2(s) = 0.389 seconds
Figure 13: Virtual Lab Response plot of T3(s)
Figure 14: T1(s) after setting the calculated natural frequency and damping ratio values into ESVT Program
Figure 15:T2(s) after setting the calculated natural frequency and damping ratio values into ESVT Program
Figure 16:T3(s) after increasing the natural frequency
Figure 17:T3(s) after increasing the damping ratio
Figure 8: T3(s) after increasing the damping ratio
1.5. Discussion& Conclusions
The formula proves that our figuration of what have been tested is proven to be accurate.
Also, table 1 shows that when the natural frequency was increasing the damping ratio decreases. Whereas the settling time was maintained at approximately 0.8 seconds. Because of the highest frequency, T3(s) has the quickest rise time and peak time, Tp. Also, because T3(s) has the lowest damping ratio as a result it will have the highest percentage of overshoot. T1(s) shows that its characteristics is closer to the critically damped system.
When the value of Kp increased the dominant pole pair changed significantly. The poles in T1(s) is close to the origin when it started off and the dominant poles of T3(s) were far away from the origin.
Also because of the increasing in the value of Kp the percentage of overshoot (%OS) was affected. According to the MATLAB calculations and as table 1 shows that the percentage of overshoot (%OS) was increased starting off T1(s) at 1.7% and ending to the maximum in T3(s) of 67%.
The recorded results were about the same values. The values calculated by MATLAB were the most accurate. However, the comparison between the MATLAB and the Virtual Lab responses was very closely. The plotted MATLAB graphs were smoother with respect of the transient response. Also the results was very similar to the calculated values, with allowances for some error. Theoretically, the virtual Lab responses were the exact same responses as the MATLAB plots. However, the accuracy of the results was affected by a large amount of noise in the signal
In the servo mechanism system, the variable Kp was changing in each of the three systems. It was effecting the transient response very fast. Increasing the value of Kp will increase the natural frequency, decrease the damping ratio and increase the pole location on the imaginary axis without any change in its location on the real axis.
However, increasing the natural frequency in a system will cause a decreasing the settling time, rise time and peak time. According to the formula below, the Damping ratio and the natural frequency have an inverse relationship, and only the value of D’ can make change on their relationship.
1.6. References
· Norman S.Nise – Control Systems Engineering (sixth edition)
1.7. Appendix:
MATLAB Code:
Experiment 2
2.1. Requirements of Experiment 2
The aim of this laboratory is to understand system response characteristics in the time domain.
Part 2A – Matlab
In this section, each of tested systems is investigated to determine whether or not it will be possible to approximate pole cancellation. In each system there were three things that needed to be accomplished. Firstly, it was required to find the response characteristics of all second order systems. Secondly, for every function in at least a second order system it was required to plot the unit step response. Finally, a comparison of all results needs to be made. Throughout the process it is necessary to explain and justify any differences between the actual systems and the calculated systems.
Part 2B – Electromechanical Servomechanism Virtual Laboratory (ESVL)
In this section, specific values (such as ) were observed in a different context to enhance the understanding of how they actually applied to the systems. The given system for this section is a third order system. By setting specific parameters such as the PID block, the block and the block to correct values, this third order system can actually be made equivalent to the systems in part 1. After these values are observed they will need to be compared with the values obtained for the second order systems.
2.2. Introduction
This report begins by defining a number of key terms. Firstly, it is important to know what a block diagram is. Block diagrams are a method of visualising a programming script by drawing each step of the process as a different block. Block diagrams should ideally show the input type that is being entered into the algorithm as well as the values that will be output by the algorithm. An example of this can be seen in Figure 2.1.
It is also important to know about the program ESVL as written above in part 2B. This program is used to emulate the servomechanism of a DC motor that would be used in a feedback control system. A screenshot of the physical servomechanism emulated by ESVL is shown in Figure 14. Additionally, a block diagram of the code relevant to the system is given in Figure 15.
Figure 9: Servomechanism
Figure 10: Block Diagram
This block diagram shows that a first order system can be used to represent the motor, power amplifier and gearbox. The value here represents the gain of the motor, which includes the gain from all components. As can be seen in Figure 2.3, , , and .
The servomechanism being emulated by ESVL is incorporated into a feedback control system for the shaft angle of the motor. Another screenshot is given in Figure 16 which shows this system, where the PID block is the controller being used.
Figure 11: ESVL Display
The meaning of the term feedback control system mentioned that the output value is controlled by feeding back the controlled value and by using it to alter or manipulate the input value. Also the output value will be the same or very similar to the desired output.
2.3. Solution Description
Given:
Kp = 0.5
KI = 1
GT = 0
Gp = 1
The given box diagram below will simplify the function Gc(s) and the values which was given.
The function is third order as seen above from the denominator. By using the given values this will let the value to be very alike to a second order function. The extra pole will be the only different.
The experiment 2A is required some explanation in order to place into MATLAB.
Compulsory activity:
By using the MATLAB it is very important to use the correct method of placing all the numbers into the formula for the correct function to output.
“T2=tf([77.66 155.33],[1 10 77.662 155.333]) “
As shown in the above line, the numerator of the function has to be correctly typed on the top line. Also the denominator has to be in a separate square bracket on the right of the first.
This method is similar to the incentive activity; just values will need to be swapped around to suit the function.
Incentive activity:
All functions are given by the formula:
2.4. Test Results
Table 2: Results of Experiment 2
Q2-A part (1)
Figure 12: Step Response for Q2-A part (i)
As shown above, there are only two graphs. The first one is the blue graph (top) which is the second order approximation. The second one is the green graph (bottom) which represents the higher order function.
Q2-A part(2)
Figure 13: Step Response for Q2-A part (2)
As shown above, there are only two graphs. The first one is the blue graph (top) which is the second order approximation. The second one is the green graph (bottom) which represents the higher order function.
2.5. Discussion& Conclusions
As seen in figure 5, pole cancellation does not always work well. The approximated step response is about 33% off therefore 33% error from the higher order system. This is because as seen in the T1(s) the brackets in the top (s+2) and (s+2.675) are different by about 33.75%. This is almost exactly the difference that was approximated by looking off the graph.
One conclusion that can be made here is that wherever the numerator of the brackets is very close to the denominator, the graph values will also appear very close. The accuracy of the graph could be improved even further by actually expanding the bracket, rather than using the pole cancellation method. To show this a simple example of expanding the brackets was accomplished and graphed. Observing this graph clearly shows that the two outputs are the same, even though the second order system can be seen to be more damped.
Figure 14: Step response for expanded system
It was stated earlier that a similarity in the numerator and denominator of the brackets would translate to a similarity in graph outputs. It can be observed here that the outputs for the second section were indeed very close. The two brackets in this case differ by less than 1%, showing the accuracy of the outputs. This provides an example of a method which is more accurate of an approximation than MATLAB’s pole cancellation method.
Comparing the results from MATLAB with the results from ESVL shows that MATLAB can be very useful for approximating and calculating the values for the system response. This can be shown not only in the compulsory task, but also in the incentive task, supporting the accuracy of this program. The differences between the results can be seen to be mostly a percentage overshoot. It can be postulated that this would be due to the restrictions imposed by needing to read values off of the graph when using MATLAB. Apart from the percentage overshoot, the accuracy of both the settling time and peak time is shown, as these lie within 4% of the actual values.
Altogether it can be seen that if the approximation method is used, then the consequence will be a slight error in the output values which will cause them to deviate from the desired output. It can be concluded that while MATLAB is not entirely accurate for this task, it does give a decent approximation for the expected output. The final values and graphs supported the expectation held before running the program, which is a strong result.
2.6. References
· Laboratory manual No.1 – System response in time domain Designed in line with textbook by N. Nise: Control systems Engineering, prescribed for the course
· Norman S.Nise – Control Systems Engineering (sixth edition)
2.7. Appendix
MATLAB Code
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