Experiment 81 - Design of a Feedback Control System

Transcription

1 Experiment 81 - Design of a Feedback Control System (Group 44) ELEC273 May 9, 2016 Abstract This report discussed the establishment of open-loop system using FOPDT medel which is usually used to approximate high-order system, closed-loop system with different types of controllers, and systems under disturbance signal. The plant transfer function was generated using a provided formula with birthday substituted. The proportional (P), proportional-integral (PI), and proportional-integral-derivative (PID) controllers were all used and analyzed. The effects on the system performance made by different values of their gain values (proportional, integral, and derivative gain) were investigated and discussed. Declaration I confirm that I have read and understood the Universitys definitions of plagiarism and collusion from the Code of Practice on Assessment. I confirm that I have neither committed plagiarism in the completion of this work nor have I colluded with any other party in the preparation and production of this work. The work presented here is my own and in my own words except where I have clearly indicated and acknowledged that I have quoted or used figures from published or unpublished sources (including the web). I understand the consequences of engaging in plagiarism and collusion as described in the Code of Practice on Assessment (Appendix L). 1

2 Contents 1 Introduction Background Objective Theory Parameters describing the system performance Characteristics of P, I and D controller Part One Open-loop response Method and procedure Result and Comment FOPDT Method and procedure Result and Comment Part II Method and procedure Result and Comment Part III Method and procedure Result and Comment Part IV Method and procedure Result and Comment Bonus Method and procedure Result and Comment Discussion and Conclusion Error Analysis and Suggestions Conclusions References 30 Appendices 31 A Figures 31

3 1 Introduction This lab intends to enable engineering students to practice the knowledge of control system by designing and simulating different systems with different models or different types of controllers applied using MATLAB simulink (ver. 2015a). 1.1 Background Systems are everywhere in industry, and it is control that makes the systems generate expected outcomes, in other words, control makes machines do what they are supposed to do [1]. Generally, there are open-loop systems and closed-loop systems. A simplified conceptual structure of a closed-loop system is presented in Figure 1. Figure 1: Closed-loop system [1] As can be seen in Figure 1, the actual output is sent back to produce the error signal together with the reference signal. Concretely, the reference signal is the expected output, and the error signal is the difference between the expected output and the actual output. This error signal is also the input of the controller so that the controller can generate commands based on the error signal to adjust the action of the plant (the object being controlled by the system). This dynamic process of exerting changing commands to the plant enables the closed-loop system to continuously fix its error thus producing better output. Typically, closed-loop systems are more widely used in industrial applications primarily because of its ability of utilizing feedback. The controller in a closed-loop system is the component which converts the error signal into a command that can be understood and processed by the plant so that expected output can be generated after the adjustment made correspondingly [2]. Therefore, it is essential for engineering students to have a good understand of controllers and closed-loop system design. 1

4 1.2 Objective Overall, systems using first-order flus time delay (FOPDT) model and control systems using P, PI, or PID controllers are required be designed, built and understood. 1. Approximate a high order open-loop system using FOPDT model 2. Design and build control system with different controllers (a) proportional (P) controller (b) proportional-integral (PI) controller (c) proportional-integral-derivative (PID) controller 3. Investigate and understand the simulation results 1.3 Theory Parameters describing the system performance Typically, systems are approximately modelled as first-order or second-order system so that they can be analysed analytically. For both first-order systems and second-order systems, there are four crucial parameters that can help describe the performance of a system which are respectively the percentage overshoot %OS, rising time T r, the settling time T s, and the steady-state error. They can be obtained both by direct calculation based on the general form of expressions of the transfer function or by observing the response graph of a system. The descriptions of these four parameters and methods of obtaining them from a response graph of a system are listed in Table 1 below [3]. Table 1: Design Specification item OS% T r T s steady error Definition the amount that the waveform overshoots the steady-state the time it takes a system to move between 10% and 90% of its steadystate response. the time it takes a system to remain within 2% of its steady-state response the difference between this steady-state response and the input Method (A S) 100%, where A is peak point value, S is steady state T 90% - T 10%, where two T are time when 10% and 90% of steady-state response is reached draw two lines parallel lines on 98% and 102% of steady response, then find the time after which the rest curve lies within this band S r S, where S r is the value where the curve remains flat, S is steady state response 2

5 1.3.2 Characteristics of P, I and D controller Controller output For a closed-loop system with a PID controller, the controller output of the system can be expressed using equation below [4], where K P, K I, and K D represent the proportional gain, integral gain and the derivative gain respectively: u(t) = K P e(t) + K I t 0 e(τ)dτ + K D d e(t) (1) dt It can be seen that for the P controller, the error is multiplied by a constant K P ; for the I controller, the error is multiplied by the product of a constant K I and the integration of the error; and for the D controller, the error is multiplied by the product of a constant K D and the derivative of the error. These three gain values are crucial parameters which apparently effect the system performance with under different values since the input command received by the plant changes if this expression changes. therefore, these K can be adjusted to enable the plant to satisfy certain requirement [2]. Functions of P, I and D Generally, after investigating the transfer function form of the expression of the controller output, it would be found that P controller is to reach the expected steady output as fast as possible, D controller is to restrain this approaching process to be too fast thus exceed the expected output, and I controller is to remove steady state error [2]. The reason for these conclusions can be understood based on the mathematical relationship expressed in equation (1) above: 1. If we draw a curve expressing the output of P path, it would be the scaled error which means that output is proportional to the error. Therefore, if the difference between the expected output and actual output is big, the controller will increase the output to quickly pursue the expected level of output. 2. Similarly for I path, the output would be the area under the error curve, therefore no matter how small a constant error is, after a period of time, its integration should be big enough to be adjust the controller output, and this is the reason why it can remove steady-state error. 3. And for D path, it is the rate of the change of the error that contributes to the output signal, which enables the controller to change output more stable than simply using a P, and this is the reason why it can help avoid the P controller from functioning excessively. 3

6 Gain parameter effects The effects on four controller parameters by increasing the value of the gains (K P, K I, and K D ) are summarized in Table 2 below, where represents increase, represents decrease. Table 2: Effects on controller parameters [5] C-L response OS% T r T s steady error K P small change K I eliminate K D small change no change From a macro perspective, these effects can then used to deduce the advantages and disadvantages of these controllers. The effects on error type made by different controllers are summarized in Table 3 below, where represents decrease, represents heavily decrease, + represents increase, and ++ represents heavily increase. Table 3: Effects on controller parameters [4] Component stability fast transient response Zero steady-state error Small overshoot Proportional + Integral + + Derivative + As required by the lab script, the transfer function used in this lab for our group is presented in equation below, where G(s) represents plant transfer function, K equals the month and T equals the day of birthday. K G(s) = (T s + 1) 2 (2) The particular function used in for our group is equation below, which was generated based on my birthday: K=12 and T=3 12 P (s) = 9 s 2 (3) + 6 s + 1 4

7 Method to work out parameters For part II, III, and Bonus section, there were several gain parameter should be worked out. The method of obtaining proper gain parameters for P, PI, and PID are listed in Table 4 [6]. Table 4: Design Specification Controller K P K I K D P T L - - PI 0.9 T L 0.27 T L 2 - PID 1.2 T L 0.6 T L T The parameters T and L are the time constant and time delay respectively, which are obtained using graphical method as presented in Figure 2. Figure 2: To obtain T and L In the Figure above, the line plotted is the tangent line of output curve with the maximum slope. The T and L are obtained using two intersection points as demonstrated in Figure 2. Also, for FOPDT model, its feature parameters have direct relationships with parameters displayed in this Figure: T p = T, t d = L, and K p = K. 5

8 2 Part One 2.1 Open-loop response Method and procedure Calculation For the following analysis based on the time domain expression, we should first work out the time domain format of the transfer function, which is: Y (s) = U(s) P (s) = K T 2 s 3 + 2T s 2 + s We first apply the methods taught in lectures of module ELEC207 to factorize the function by introducing three constants: Y (s) = A s + Bs + C T 2 s 2 (5) + 2T s + 1 Rearranging this expression, we obtain: (4) Y (s) = (T 2 A + B)s 2 + (2AT )s + A s(t 2 s 2 + 2T s + 1) (6) Since T 2 A + B)s 2 + (2AT )s + A = K, we can obtain A = K, B = T 2 K, and C = 2T K. Substitute these three constants and rearrange it we obtain factorized form: Y (s) = K s + T 2 Ks 2T K (T s + 1) 2 = K s K 1 s + 1 T K T 1 (s + 1 T )2 (7) Then using transform table provided in lecture notes, we obtain the time domain expression: y(t) = k ke 1 T t K T e 1 T t t (8) The used transform relation were: L 1 [ 1 ] = u(t) (9) s L 1 1 [ s + a ] = e at u(t) (10) L 1 ω [ s 2 ] = sin(ωt)u(t) + ω2 (11) L 1 [G(s + a)] = e at L 1 [G(s)] (12) Finally, we substitute K = 12, and T = 3, we obtain factorized expression: y(t) = 12 12e 1 3 t 4e 1 3 t t (13) 6

9 Procedure The block connection pictures for obtaining the open-loop response can be seen in Figure 3, where an extra derivative block is added whose curve will be used in Part II. Figure 3: With derivative block As can be seen in Figure above, the input signal of this system is the a step signal provided by the step block which was expected to give value 1 from t=0. However, the default set of this block was that the value 1 signal is given after 1 second, which means there is a time delay of 1. This delay was cancelled by manually changing the step time to 0. The setup window can be seen in Figure 4. Figure 4: Set step After connection and step setup adjustment, the run button was clicked and then the scope block was clicked and the simulation graph was obtained. 7

10 2.1.2 Result and Comment The simulation graphs for the open-loop response in Figure 10. Figure 5: Open-loop response In the Figure above, the blue curve represents the change of the output y(t), the red curve represents the change of the derivative of the blue curve, and the yellow curve represents the step input, which remains 1 for t > 0. The derivative curve (red curve) first increases to a maximum value at t = 3.031, and the decreases to virtually zero but always remaining non-negative. This indicates that the output continuously increases with speed first increases then decreases, and finally remain unchanged when t is big enough (approximately t > 25 in this case). And the blue curve fits this described change process. Experimental value of steady-state response Also, when time increases to a certain extend, the blue curve remains flat which gives us the steady-state response 12 as can be seen from the graph. Theoretical value of steady-state response For the time domain theoretical expression of the output, when t approaches infinity, y approaches steady-state response. The theoretical value of the steady-state response can be worked out: y(t) = 12 12e 1 3 t 4e 1 3 t t 12 (14) Therefore, the theoretical value of the steady-state response is the same as the experimental value obtained from the simulation. 8

11 2.2 FOPDT Method and procedure To apply the FOPDT model, we had to first work out the values of those parameters. The method adopted and explained in this section was searched out from the Internet [7]. First, the tangent line with the maximum slope on the output curve was plotted using the connection as shown in Figure 6. Figure 6: With ramp block The simulation result with the tangent line (blue curve) displayed is presented in Figure 7. Figure 7: Open-loop response (tangent line) 9

12 There are two points marked on the tangent curve. The lower one is the intersection of the tangent line and the line y = 0 and the upper one is the intersection of the tangent line and the line y = 12. Experimental values of FOPDT model parameters The horizontal component of the intersections marked in Figure?? are t = and t = respectively.therefore, we obtained that: process delay t d = , process time constant T p = = , and the process gain K p = 12. Theoretical values of FOPDT model parameters We first worked out the expressions for the first and second order derivative. dy = K T 2 e 1 T t t (15) ddy = K T 3 e 1 T t t + K T 2 e 1 T t (16) Then, we worked out the solution of ddy = 0, which is a = T, and then we substituted this a into equation (15) to find maximum slope k max = K T 2 e 1. After that, we substituted a into equation (13) to find b = K 2Ke 1. Then, we used formula below to obtained tangent line. k max (t a) = y b (17) The tangent line expression obtained is: y = K T e 1 t + K 3Ke 1 = 4e e 1 (18) Then, we worked out the solutions to 4e e 1 = 0 and 4e e 1 = 12, which are t = and t = 9 respectively. Finally, we obtain: process delay t d = 0.845, process time constant T p = = 8.155, and the process gain K p = 12. Since , and , we can say that te theoretical results and the experiment results are virtually the same which verifies the result reliance. The values we used in the later sections are the theoretical values. The values of model parameters are listed: T p t d K p After obtaining these parameters, the FOPDT model was built. 10

13 The block connection picture for obtaining the response using FOPDT model can be seen in Figure 3, which was connected based on the lab script information. Figure 8: FOPDT model The delay block was clicked to set the value. The delay setup window for setting the delay value can be seen in Figure 9. Figure 9: Set delay Finally, run button was clicked and then the scope block was clicked to obtain the simulation result. 11

14 2.2.2 Result and Comment The simulation graph for the response obtained using FOPDT model can be seen in Figure 10. Figure 10: FOPDT model response The rising time T r, settling time T s, overshoot percentage(os%), and the steady error of both two model responses are listed in Table 5 below. Table 5: Design Specification Model OS% T r T s steady error Open-loop FOPDT This curve of the open-loop response and that of the FOPDT model response are quite similar. Therefore, the FOPDT model is generally suitable be used to approximate the high order system [7]. 12

15 3 Part II 3.1 Method and procedure The block connection picture for obtaining the responses of P controllers with different gain values can be seen in Figure 11. Figure 11: P controller Here in this Figure, the thick black bar on the right is a mux which enables the simulation result window to display three curves with different proportional gain K P ; All the three paths have feedback which makes this system a closed-loop system. The proper gain value was worked out using methods introduced in Theory section K P = T L = The three gain values for the three controllers were set respectively with a difference of 2 successively as listed below (the setup window for this step can be seen in Figure 33 in Appendix A). PID controller PID controller1 PID controller After building the blocks, before triggering the simulation, the maximum step size was set to 0.01 (setup window can be seen in Figure 32 in Appendix A) so that more concentrated values can be obtained in the result curve. The reason for this was to ensure that the points which can display controller parameters such as settling time can be found successfully. 13

16 3.2 Result and Comment The simulation graph for the responses with different gain values obtained can be seen in Figure 12. Figure 12: P controller response As can be seen in Figure above, the response is in under-damping condition, and the different values of gain indeed affect the performance slightly. Screen-shots of points for obtaining the controller parameters can be seen in Figure 13. Figure 13: P controller parameters Based on the method explained in Theory section: The vertical component of points in first sub-figure is used to obtain %OS, the horizontal component of points in second and third sub-figure are used to obtain T t, and the last sub-figure is used to obtain the steady-state error. 14

17 The method of obtaining the settling time can be seen in Figure 14. Figure 14: Find settling time for P controller The left four points were used to draw the two parallel line to define the range of from 98% to 102% of the steady-state response. The right three points are times starting from when the remaining curves only lie within the band. The rising time T r, settling time T s, overshoot percentage %OS, and the steady error of response with P controller with different gain values are listed in Table 6 below. Table 6: Parameters for different K P Gain %OS T r T s steady error % % % It can be seen from the results that with the increase of the proportional gain value, the percentage overshoot %OS increases, the rising time T r decreases, the settling time T s increases and the steady error decreases. The results agree with the theory as discussed in Theory section that P controller functions as reducing the time to reach the expected output, as a result of which, the rising time will be decreased and the steady-error is decreased correspondingly. However, since there is no I controller working together with the P controller to avoid P from functioning excessively thus exceeding the expected output, the percentage overshoot rises a bit with the increase of the P gain value. Also, for the increase of settling time, the reason can be that when the P gain is too big, it tends to make controller output push the plant too much thus it exceeds the steady response and waste some time to adjust the plant to drag back the output. 15

18 4 Part III 4.1 Method and procedure The block connection picture for obtaining the responses with different gain values obtained can be seen in Figure 15. Figure 15: PI controller Here in this Figure, compared with the block connection for Part II, the PID controller blocks were replaced by PI controller blocks. Also, the simulation result window would display three curves with different integral gain K I or different proportional gain K P. The proportional gain values were all set to be 8.686, which is obtained using method in Theory section K P = 0.9 T L = The proper integral gain is K I = 0.27 T L 2 = The proportional gain was kept while integral gain values were set differently as listed below (the setup window for this step can be seen in Figure 33 in Appendix A). PID controller PID controller1 PID controller The integral gain was kept while integral gain values were set differently as listed below. PID controller PID controller1 PID controller

19 4.2 Result and Comment Change K I The simulation graph for the responses with different integral gain values obtained can be seen in Figure 16. Figure 16: PI controller response with different K I As can be seen in Figure above, the response is in still under-damping condition, and the different values of integral gain indeed affect the performance slightly. Screen-shots of points for obtaining the controller parameters can be seen in Figure 17. Figure 17: PI controller parameters 17

20 The method of obtaining the settling time can be seen in Figure 18. Figure 18: Find settling time for PI controller Similarly as discussed in Part II, in the Figure, the left four points are for drawing the band while the right three points are from which the settling time can be obtained. The rising time T r, settling time T s, overshoot percentage(os%), and the steady error of response with P and I controller are listed in Table 7 below. Table 7: Parameter for different K I Gain OS% T r T s steady error It can be seen from the results that with the increase of the integral gain value of PI controller, the percentage overshoot %OS increases, the rising time T r remains the same, the settling time T s increases and the steady error decreases to zero. The results agree with the theory as discussed in Theory section that I controller functions as further stabilize the performance of P by limiting the direct proportional controller outputthus the settling time will be increased but the steady-error is eliminated. It can be deduced that though requiring more time to reach stable stage, the combination of P and I can make the system more stable since the steady error is zero after the addition of I controller. 18

21 Change K P The simulation graph for the responses with different proportional gain values obtained can be seen in Figure 19. Figure 19: PI controller response for different K P As can be seen in Figure above, the response is in still under-damping condition, and the different values of integral gain indeed affect the performance slightly. Screen-shots of points for obtaining the controller parameters can be seen in Figure 20. Figure 20: PI controller parameters 19

22 The method of obtaining the settling time can be seen in Figure 21. Figure 21: Find settling time for PI controller Similarly as discussed in Part II, in the Figure, the left four points are for drawing the band while the right three points are from which the settling time can be obtained. The rising time T r, settling time T s, overshoot percentage(os%), and the steady error of response with P and I controller are listed in Table 8 below. Table 8: Parameters for diffrent K P Gain OS% T r T s steady error % % % It can be seen from the results that with the increase of the proportional gain value of the PI controller, the percentage overshoot %OS decreases, the rising time T r decreases, the settling time T s decreases and the steady error still remains zero. The results agree with the theory as discussed in Theory section that proportional gain decreases the rising time and settling time. Therefore, it can be deduced that when P and I controller are used together, this might achieve a combination of their advantages which are eliminating the steady error (I), decreases the time required to reach steady state. 20

23 5 Part IV 5.1 Method and procedure Disturbance value is 1 and start time is 5 The block connection pictures for obtaining the responses under disturbance with P controller and that with PI controller can be seen in Figure 22 and Figure 23 respectively. Figure 22: P controller under disturbance Figure 23: PI controller under disturbance For both two block connections, compared to previous part, a disturbance block was added. The disturbance was generated using step block whose step time was set to 5 (setup window can be seen in Figure 35 in Appendix A) which is a time before reaching steady-state. The existence of this disturbance block means that after 5 seconds of normal closed-loop response, an extra step 1 would be added into the system. For scenarios of P and PI, except the PID block values were set differently (K P = 9.65 for P case, and K P = 8.686, K I = for PI case), all the other steps were the same. 21

24 Disturbance value is 5 and start time is 5 The block connection pictures for obtaining the responses under disturbance value of 5 with P controller and PI controller together in two paths can be seen in Figure 24. Figure 24: P controller under disturbance of 5 In the Figure above, the block Step1 and block Step3 serve as the disturbance blocks. However, the disturbance value here was set to 5 rather than the default value 1 used in the first case. The disturbance was generated using step block in a value of 5 whose step time was set to 5 which is a time before reaching steady-state. The block setup window can be seen in Figure 36 in Appendix A. 22

25 5.2 Result and Comment Disturbance value is 1 Simulation graph for responses under disturbance with P controller can be seen in Figure 25. Figure 25: Response under disturbance with P controller As can be seen in Figure above, the controller parameters can be obtained using methods introduced in Theory section, and are listed below: OS% T r T s steady error 73.2% It can be seen clearly from the result, when there is only a P controller, under a disturbance generated by a step block whose step time is 5, the steady error exists. This means that the disturbance makes influence on the steady response. Also, it can be seen that at t = 5 when the disturbance signal starts to be 1, there is a slight jump of the curve different from that without the disturbance. This means that for P controller, the disturbance makes influence on the curve shape in a tiny range of time starting from 5 (the disturbance start time). 23

26 Simulation graph for responses under disturbance with PI controller can be seen in Figure 27. Figure 26: Response under disturbance with PI controller As can be seen in Figure above, the controller parameters can be obtained using methods introduced in Theory section, and are listed below: OS% T r T s steady error 86.6% It can be seen clearly from the result, when there is only a PI controller, under a disturbance generated by a step block whose step time is 5, the steady error does not. This means that the disturbance makes no influence on the steady response. However, it can be seen that at t = 5 when the disturbance signal starts to be 1, there is a slight jump of the curve different from that without the disturbance. This means that for PI controller, the disturbance makes influence on the curve shape in a tiny range of time starting from 5 (the disturbance start time). Comparison Comparing response with P controller and PI controller, it can be seen that PI controller reaches stability later with a bigger settling time but it can achieve better steady response which make no steady error. The maximum amplitude reaches of P controller is slightly smaller than that of PI though. 24

27 Disturbance value is 5 Simulation graph for responses under disturbance of value 5 with P and PI controller respectively can be seen in Figure 27. Figure 27: Response under disturbance value of 5 As can be seen in Figure above, the controller parameters can be obtained using methods introduced in Theory section, and are listed below: Controller Maximum T s steady error P PI It can be seen clearly from the result, for the P controller path, compared with that of the PI controller path, it has bigger maximum response, smaller settling time and bigger steady error. Also, different from the scenario where the disturbance value was set to 1, in this case, since the disturbance value is big, the time when peak response is reached is not the first jump of the curve anymore but the time when the disturbance interferes in, which is t = 5. Therefore, it can be deduced that if the disturbance value is big enough, it can greatly increase the percentage overshoot and make the peak response right at the time when it starts to give value bigger than zero. Summary comparison The biggest difference between response of P and PI path in this case is basically the same that that when disturbance value is 1. One more deduction can be found the the steady error of P path tend to be 10% of the disturbance value. Also, apparently, for both two cases discussed, it can be seen that the addition of I controller eliminated the constant error which agrees with the description of I controller in Theory section. 25

28 6 Bonus 6.1 Method and procedure The connection block picture for obtaining the response under disturbance with PID controller can be seen in Figure 28. Figure 28: Response under disturbance with PID controller The PID block was set to be P, I, and D working together. The parameters were obtained using methods introduced in Theory section: K P = 1.2 T L = 11.58, K I = 0.6 T L 2 = and K D = 0.6 T = The connection block picture for obtaining responses with P, PI, and PID respectively in a same scope can be seen in Figure 29. Figure 29: Response under disturbance with different controllers 26

29 6.2 Result and Comment The simulation graph for the response obtained with PID controller under disturbance can be seen in Figure 30. Figure 30: Response under disturbance with PID controller As can be seen in Figure above, the controller parameters can be obtained using methods introduced in Theory section, and are listed below: OS% T r T s steady error 14.4% It can be seen clearly from the result, when there is a PID controller, under a disturbance generated by a step block whose step time is 5, there is no steady error. This means that the disturbance makes no influence on the steady response. However, it can be seen that at t = 5 when the disturbance signal starts to be 1, there is a slight jump of the curve different from that without the disturbance. This means that for PID controller, the disturbance makes influence on the curve shape in a tiny range of time starting from 5 (the disturbance start time). 27

30 Simulation graph for response with different controllers respectively can be seen in Figure 31. Figure 31: Response under disturbance with different controllers As can be seen in the Figure above, comparing the the response with P controller, PI controller, and PID controller, it can be seen that the response with PID controller has the least fluctuating curve and reaches stability without steady error the fastest. In terms of achieving better steady response, the rank can be P ID > P I > P. This is because although both PID and PI controller achieve zero steady error, PID reaches steady reponse faster, while P is the only one that still have steady error after reaching stability. 28

31 7 Discussion and Conclusion 7.1 Error Analysis and Suggestions The main limitations in this experiment is that when finding the points that help obtaining controller parameters such as settling time on the simulation graph using the cursor tool, it was not always possible to find the exactly expected value. For example, we worked out the value (vertical component) of 10% of the response is 10% 1 = 0.1, when we found the point with (t,0.1), we might only find a point quite close to the it but not exactly with the same value such as (t,0.102). This slight difference can cause a slight error in terms of the obtained corresponding time parameters. To solve this problem, it is suggested that the maximum step size should be set smaller so that the points display on the simulation result can be more concentrated and the possiblity of finding the point with the exact value can be reached. 7.2 Conclusions The open-loop system using FOPDT model, and closed-loop control system with P, PI, and PID controllers were built and investigated comprehensively. Reasonable deduction based on the obtained results were proposed which generally agreed with the related theoretical knowledge provided in Theory sections. The limitation that it was difficult to mark points with exact expected values using cursor was assumed to be possible to be solved by decreasing the step size. For future improvement, it was suggested that more types of controllers can be analyzed using similar approaches to better understand the controller design. Overall, based on comprehensive analysis on the simulation results, it was found that the obtained simulated results using MATLAB simulink (ver.2015) generally agree with the knowledge of PID controller characteritics and close-loop system performance features learnt from lectures of module ELEC

32 References [1] S. Maskell, Lecture notes one, content/file?cmd=view&content id= &course id= , University of Liverpool, [2] B. Douglas, Pid control - a brief introduction, [3] lecture notes six, pid dt-content-rid /courses/ELEC /6%20%28steady\ discretionary{-}{}{}state%20response%20design%29%281%29.pdf, [4] lecture notes seven, 1/courses/ ELEC /Experiment%2081%20Lab%20script.pdf, [5] C. Tutorial, Introduction: Pid controller design, http: //ctms.engin.umich.edu/ctms/index.php?example=introduction&section=controlpid, [6] J. Zhong, Pid controller tuning: A short tutorial, [7] G. Reeves, First-order plus deadtime (fopdt) model,

33 Appendices A Figures Part II set window for setting the step size to 0.01 can be seen in Figure 32. Figure 32: Step size Part II set window for setting the gain values can be seen in Figure 33. Figure 33: P controller gain Part III set window for setting the gain values can be seen in Figure 34. Figure 34: PI controller gain 31

34 Part IV set window for setting disturbance of value 1 can be seen in Figure 32. Figure 35: Disturbance is 1 Part IV set window for setting disturbance of value 1 can be seen in Figure 32. Figure 36: Disturbance is 5 32