experiment 81 - design of a feedback control system 81 - design of a feedback control system...

Experiment 81 - Design of a Feedback Control System

201139030 (Group 44)ELEC273

May 9, 2016

Abstract

This report discussed the establishment of open-loop system using FOPDT medel whichis usually used to approximate high-order system, closed-loop system with different types ofcontrollers, and systems under disturbance signal. The plant transfer function was generatedusing 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 madeby different values of their gain values (proportional, integral, and derivative gain) wereinvestigated and discussed.

Declaration

I confirm that I have read and understood the Universitys definitions of plagiarism and collusion fromthe Code of Practice on Assessment. I confirm that I have neither committed plagiarism in the com-pletion of this work nor have I colluded with any other party in the preparation and production of thiswork. The work presented here is my own and in my own words except where I have clearly indicatedand acknowledged that I have quoted or used figures from published or unpublished sources (includingthe web). I understand the consequences of engaging in plagiarism and collusion as described in theCode of Practice on Assessment (Appendix L).

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Contents

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3.1 Parameters describing the system performance . . . . . . . . . . . . . . . 21.3.2 Characteristics of P, I and D controller . . . . . . . . . . . . . . . . . . . . 3

2 Part One 62.1 Open-loop response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.1 Method and procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.2 Result and Comment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 FOPDT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.1 Method and procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.2 Result and Comment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Part II 133.1 Method and procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Result and Comment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 Part III 164.1 Method and procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.2 Result and Comment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5 Part IV 215.1 Method and procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.2 Result and Comment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

6 Bonus 266.1 Method and procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.2 Result and Comment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

7 Discussion and Conclusion 297.1 Error Analysis and Suggestions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

References 30

Appendices 31

A Figures 31

1 Introduction

This lab intends to enable engineering students to practice the knowledge of control system bydesigning and simulating different systems with different models or different types of controllersapplied using MATLAB simulink (ver. 2015a).

1.1 Background

Systems are everywhere in industry, and it is control that makes the systems generate expectedoutcomes, in other words, control makes machines do what they are supposed to do [1]. Gen-erally, there are open-loop systems and closed-loop systems. A simplified conceptual structureof 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 togetherwith the reference signal. Concretely, the reference signal is the expected output, and the errorsignal is the difference between the expected output and the actual output. This error signalis also the input of the controller so that the controller can generate commands based on theerror signal to adjust the action of the plant (the object being controlled by the system). Thisdynamic process of exerting changing commands to the plant enables the closed-loop system tocontinuously fix its error thus producing better output.

Typically, closed-loop systems are more widely used in industrial applications primarily be-cause of its ability of utilizing feedback. The controller in a closed-loop system is the componentwhich converts the error signal into a command that can be understood and processed by theplant 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 andclosed-loop system design.

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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

1.3.1 Parameters describing the system performance

Typically, systems are approximately modelled as first-order or second-order system so thatthey can be analysed analytically. For both first-order systems and second-order systems, thereare four crucial parameters that can help describe the performance of a system which are respec-tively the percentage overshoot %OS, rising time Tr, the settling time Ts, and the steady-stateerror. They can be obtained both by direct calculation based on the general form of expressionsof 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 responsegraph of a system are listed in Table 1 below [3].

Table 1: Design Specification

item OS% Tr Ts steady error

Definition the amount thatthe waveformovershoots thesteady-state

the time it takesa system to movebetween 10% and90% of its steady-state response.

the time it takesa system to re-main within 2%of its steady-stateresponse

the differencebetween thissteady-state re-sponse and theinput

Method (A − S) × 100%,where A is peakpoint value, S issteady state

T90% - T10%,where two Tare time when10% and 90% ofsteady-state re-sponse is reached

draw two linesparallel lines on98% and 102% ofsteady response,then find the timeafter which therest curve lieswithin this band

Sr − S, where Sris the value wherethe curve remainsflat, S is steadystate response

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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 beexpressed using equation below [4], where KP , KI , and KD represent the proportional gain,integral gain and the derivative gain respectively:

u(t) = KP × e(t) +KI ×∫ t

0e(τ)dτ +KD ×

d

dte(t) (1)

It can be seen that for the P controller, the error is multiplied by a constant KP ; for the Icontroller, the error is multiplied by the product of a constant KI and the integration of theerror; and for the D controller, the error is multiplied by the product of a constant KD andthe derivative of the error.These three gain values are crucial parameters which apparently effect the systemperformance with under different values since the input command received by the plantchanges if this expression changes. therefore, these K can be adjusted to enable the plant tosatisfy certain requirement [2].

Functions of P, I and D

Generally, after investigating the transfer function form of the expression of the controlleroutput, it would be found that P controller is to reach the expected steady output as fast aspossible, D controller is to restrain this approaching process to be too fast thus exceed theexpected output, and I controller is to remove steady state error [2].

The reason for these conclusions can be understood based on the mathematical relationshipexpressed in equation (1) above:

1. If we draw a curve expressing the output of P path, it would be the scaled error whichmeans that output is proportional to the error. Therefore, if the difference between theexpected output and actual output is big, the controller will increase the output toquickly pursue the expected level of output.

2. Similarly for I path, the output would be the area under the error curve, therefore nomatter how small a constant error is, after a period of time, its integration should be bigenough to be adjust the controller output, and this is the reason why it can removesteady-state error.

3. And for D path, it is the rate of the change of the error that contributes to the outputsignal, 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.

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Gain parameter effects

The effects on four controller parameters by increasing the value of the gains (KP , KI , andKD) are summarized in Table 2 below, where ↑ represents increase, ↓ represents decrease.

Table 2: Effects on controller parameters [5]

C-L response OS% Tr Ts steady error

KP ↓ ↑ small change ↓

KI ↓ ↑ ↑ eliminate

KD small change ↓ ↓ no change

From a macro perspective, these effects can then used to deduce the advantages anddisadvantages of these controllers. The effects on error type made by different controllers aresummarized 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 presentedin equation below, where G(s) represents plant transfer function, K equals the month and Tequals the day of birthday.

G(s) =K

(Ts+ 1)2(2)

The particular function used in for our group is equation below, which was generated based onmy birthday: K=12 and T=3

P (s) =12

9× s2 + 6× s+ 1(3)

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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].


Controller KP KI KD

P TL - -

PI 0.9× TL 0.27× T

L2 -

PID 1.2× TL 0.6× T

L2 0.6× T

The parameters T and L are the time constant and time delay respectively, which areobtained 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 maximumslope. 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 parametersdisplayed in this Figure: Tp = T , td = L, and Kp = K.

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2 Part One

2.1 Open-loop response

2.1.1 Method and procedure

Calculation

For the following analysis based on the time domain expression, we should first work out thetime domain format of the transfer function, which is:

Y (s) = U(s)× P (s) =K

T 2 × s3 + 2T × s2 + s(4)

We first apply the methods taught in lectures of module ELEC207 to factorize the function byintroducing three constants:

Y (s) =A

s+

Bs+ C

T 2s2 + 2Ts+ 1(5)

Rearranging this expression, we obtain:

Y (s) =(T 2A+B)s2 + (2AT )s+A

s(T 2s2 + 2Ts+ 1)(6)

Since T 2A+B)s2 + (2AT )s+A = K, we can obtain A = K, B = −T 2K, and C = −2TK.Substitute these three constants and rearrange it we obtain factorized form:

Y (s) =K

s+−T 2Ks− 2TK

(Ts+ 1)2=K

s−K × 1

s+ 1T

− K

T× 1

(s+ 1T )

2 (7)

Then using transform table provided in lecture notes, we obtain the time domain expression:

y(t) = k − ke−1Tt − K

Te−

1Ttt (8)

The used transform relation were:

L−1[1

s] = u(t) (9)

L−1[1

s+ a] = e−atu(t) (10)

L−1[ω

s2 + ω2] = sin(ωt)u(t) (11)

L−1[G(s+ a)] = e−atL−1[G(s)] (12)

Finally, we substitute K = 12, and T = 3, we obtain factorized expression:

y(t) = 12− 12e−13t − 4e−

13tt (13)

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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 bythe step block which was expected to give value 1 from t=0. However, the default set of thisblock was that the value 1 signal is given after 1 second, which means there is a time delay of1. This delay was cancelled by manually changing the “step time” to 0. The setup windowcan be seen in Figure 4.

Figure 4: Set step

After connection and step setup adjustment, the “run” button was clicked and then the scopeblock was clicked and the simulation graph was obtained.

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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 curverepresents the change of the derivative of the blue curve, and the yellow curve represents thestep input, which remains 1 for t > 0.The derivative curve (red curve) first increases to a maximum value at t = 3.031, and thedecreases 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). Andthe blue curve fits this described change process.

Experimental value of steady-state responseAlso, when time increases to a certain extend, the blue curve remains flat which gives us thesteady-state response 12 as can be seen from the graph.

Theoretical value of steady-state responseFor the time domain theoretical expression of the output, when t approaches infinity, yapproaches steady-state response. The theoretical value of the steady-state response can beworked out:

y(t) = 12− 12e−13t − 4e−

13tt ≈ 12 (14)

Therefore, the theoretical value of the steady-state response is the same as the experimentalvalue obtained from the simulation.

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2.2 FOPDT

2.2.1 Method and procedure

To apply the FOPDT model, we had to first work out the values of those parameters. Themethod 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 theconnection 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)

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There are two points marked on the tangent curve. The lower one is the intersection of thetangent line and the line y = 0 and the upper one is the intersection of the tangent line andthe line y = 12.

Experimental values of FOPDT model parametersThe horizontal component of the intersections marked in Figure ?? are t = 0.8448 andt = 8.954 respectively.Therefore, we obtained that: process delay td = 0.8448, process timeconstant Tp = 8.954− 0.8448 = 8.1092, and the process gain Kp = 12.

Theoretical values of FOPDT model parametersWe first worked out the expressions for the first and second order derivative.

dy =K

T 2e−

1Ttt (15)

ddy = −KT 3e−

1Ttt+

K

T 2e−

1Tt (16)

Then, we worked out the solution of ddy = 0, which is a = T , and then we substituted this ainto equation (15) to find maximum slope kmax = K

T 2 e−1.

After that, we substituted a into equation (13) to find b = K − 2Ke−1. Then, we used formulabelow to obtained tangent line.

kmax(t− a) = y − b (17)

The tangent line expression obtained is:

y =K

Te−1t+K − 3Ke−1 = 4e−1 + 12− 36e−1 (18)

Then, we worked out the solutions to 4e−1 + 12− 36e−1 = 0 and 4e−1 + 12− 36e−1 = 12,which are t = 0.845 and t = 9 respectively.

Finally, we obtain: process delay td = 0.845, process time constant Tp = 9− 0.845 = 8.155,and the process gain Kp = 12.

Since 0.8448 ≈ 0.845, and 8.954 ≈ 9, we can say that te theoretical results and the experimentresults are virtually the same which verifies the result reliance. The values we used in the latersections are the theoretical values. The values of model parameters are listed:

Tp td Kp

8.155 0.845 12

After obtaining these parameters, the FOPDT model was built.

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The block connection picture for obtaining the response using FOPDT model can be seen inFigure 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 delayvalue can be seen in Figure 9.

Figure 9: Set delay

Finally, “run” button was clicked and then the scope block was clicked to obtain thesimulation result.

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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 Tr, settling time Ts, overshoot percentage(OS%), and the steady error of bothtwo model responses are listed in Table 5 below.


Model OS% Tr Ts steady error

Open-loop 0 10.1 18.5 0FOPDT 0 17.9 33.7 0

This curve of the open-loop response and that of the FOPDT model response are quitesimilar. Therefore, the FOPDT model is generally suitable be used to approximate the highorder system [7].

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3 Part II

3.1 Method and procedure

The block connection picture for obtaining the responses of P controllers with different gainvalues 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 simulationresult window to display three curves with different proportional gain KP ; All the three pathshave feedback which makes this system a closed-loop system.

The proper gain value was worked out using methods introduced in Theory sectionKP = T

L = 9.65. The three gain values for the three controllers were set respectively with adifference of 2 successively as listed below (the setup window for this step can be seen inFigure 33 in Appendix A).

PID controller PID controller1 PID controller2

9.56 7.56 11.56

After building the blocks, before triggering the simulation, the maximum step size was set to0.01 (setup window can be seen in Figure 32 in Appendix A) so that more concentrated valuescan be obtained in the result curve. The reason for this was to ensure that the points whichcan display controller parameters such as settling time can be found successfully.

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3.2 Result and Comment

The simulation graph for the responses with different gain values obtained can be seen inFigure 12.

Figure 12: P controller response

As can be seen in Figure above, the response is in under-damping condition, and the differentvalues 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 firstsub-figure is used to obtain %OS, the horizontal component of points in second and thirdsub-figure are used to obtain Tt, and the last sub-figure is used to obtain the steady-state error.

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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% to102% of the steady-state response. The right three points are times starting from when theremaining curves only lie within the band.

The rising time Tr, settling time Ts, overshoot percentage %OS, and the steady error ofresponse with P controller with different gain values are listed in Table 6 below.

Table 6: Parameters for different KP

Gain %OS Tr Ts steady error

7.65 70.2% 0.35 10.27 0.01089.65 73.2% 0.3 10.85 0.008511.65 75.4% 0.28 12.89 0.0071

It can be seen from the results that with the increase of the proportional gain value, thepercentage overshoot %OS increases, the rising time Tr decreases, the settling time Tsincreases and the steady error decreases.

The results agree with the theory as discussed in Theory section that P controller functions asreducing the time to reach the expected output, as a result of which, the rising time will bedecreased and the steady-error is decreased correspondingly. However, since there is no Icontroller working together with the P controller to avoid P from functioning excessively thusexceeding the expected output, the percentage overshoot rises a bit with the increase of the Pgain value. Also, for the increase of settling time, the reason can be that when the P gain istoo big, it tends to make controller output push the plant too much thus it exceeds the steadyresponse and waste some time to adjust the plant to drag back the output.

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4 Part III


The block connection picture for obtaining the responses with different gain values obtainedcan be seen in Figure 15.

Figure 15: PI controller

Here in this Figure, compared with the block connection for Part II, the PID controller blockswere replaced by PI controller blocks. Also, the simulation result window would display threecurves with different integral gain KI or different proportional gain KP .The proportional gain values were all set to be 8.686, which is obtained using method inTheory section KP = 0.9× T

L = 8.686. The proper integral gain is KI = 0.27× TL2 = 3.084.

The proportional gain was kept 8.686 while integral gain values were set differently as listedbelow (the setup window for this step can be seen in Figure 33 in Appendix A).


3.084 2.584 3.584

The integral gain was kept 3.084 while integral gain values were set differently as listed below.


8.686 7.686 9.686

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Change KI

The simulation graph for the responses with different integral gain values obtained can be seenin Figure 16.

Figure 16: PI controller response with different KI

As can be seen in Figure above, the response is in still under-damping condition, and thedifferent values of integral gain indeed affect the performance slightly.


Figure 17: PI controller parameters

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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 bandwhile the right three points are from which the settling time can be obtained.

The rising time Tr, settling time Ts, overshoot percentage(OS%), and the steady error ofresponse with P and I controller are listed in Table 7 below.

Table 7: Parameter for different KI

Gain OS% Tr Ts steady error

2.584 1.843 0.31 21.39 03.084 1.866 0.31 25.09 03.584 1.888 0.31 30.63 0

It can be seen from the results that with the increase of the integral gain value of PIcontroller, the percentage overshoot %OS increases, the rising time Tr remains the same, thesettling time Ts increases and the steady error decreases to zero.

The results agree with the theory as discussed in Theory section that I controller functions asfurther stabilize the performance of P by limiting the direct proportional controller outputthusthe 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 Pand I can make the system more stable since the steady error is zero after the addition of Icontroller.

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Change KP

The simulation graph for the responses with different proportional gain values obtained can beseen in Figure 19.

Figure 19: PI controller response for different KP

As can be seen in Figure above, the response is in still under-damping condition, and thedifferent values of integral gain indeed affect the performance slightly.


Figure 20: PI controller parameters

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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 bandwhile the right three points are from which the settling time can be obtained.

The rising time Tr, settling time Ts, overshoot percentage(OS%), and the steady error ofresponse with P and I controller are listed in Table 8 below.

Table 8: Parameters for diffrent KP

Gain OS% Tr Ts steady error

7.686 87.7% 0.33 27.72 08.686 86.6% 0.31 24.2 09.686 85.9% 0.3 21.2 0

It can be seen from the results that with the increase of the proportional gain value of the PIcontroller, the percentage overshoot %OS decreases, the rising time Tr decreases, the settlingtime Ts decreases and the steady error still remains zero.

The results agree with the theory as discussed in Theory section that proportional gaindecreases the rising time and settling time. Therefore, it can be deduced that when P and Icontroller are used together, this might achieve a combination of their advantages which areeliminating the steady error (I), decreases the time required to reach steady state.

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5 Part IV


Disturbance value is 1 and start time is 5

The block connection pictures for obtaining the responses under disturbance with P controllerand 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 windowcan be seen in Figure 35 in Appendix A) which is a time before reaching steady-state. Theexistence 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 (KP = 9.65 for Pcase, and KP = 8.686, KI = 3.084 for PI case), all the other steps were the same.

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Disturbance value is 5 and start time is 5

The block connection pictures for obtaining the responses under disturbance value of 5 with Pcontroller 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 thefirst case.

The disturbance was generated using step block in a value of 5 whose step time was set to 5which is a time before reaching steady-state. The block setup window can be seen in Figure 36in Appendix A.

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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 methodsintroduced in Theory section, and are listed below:

OS% Tr Ts steady error

73.2% 0.3 12.29 0.094

It can be seen clearly from the result, when there is only a P controller, under a disturbancegenerated by a step block whose step time is 5, the steady error exists. This means that thedisturbance 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 slightjump of the curve different from that without the disturbance. This means that for Pcontroller, the disturbance makes influence on the curve shape in a tiny range of time startingfrom 5 (the disturbance start time).

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Simulation graph for responses under disturbance with PI controller can be seen in Figure 27.

Figure 26: Response under disturbance with PI controller



86.6% 0.31 24.1 0

It can be seen clearly from the result, when there is only a PI controller, under a disturbancegenerated by a step block whose step time is 5, the steady error does not. This means that thedisturbance 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 aslight jump of the curve different from that without the disturbance. This means that for PIcontroller, the disturbance makes influence on the curve shape in a tiny range of time startingfrom 5 (the disturbance start time).

ComparisonComparing response with P controller and PI controller, it can be seen that PI controllerreaches stability later with a bigger settling time but it can achieve better steady responsewhich make no steady error. The maximum amplitude reaches of P controller is slightlysmaller than that of PI though.

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Disturbance value is 5

Simulation graph for responses under disturbance of value 5 with P and PI controllerrespectively can be seen in Figure 27.

Figure 27: Response under disturbance value of 5


Controller Maximum Ts steady error

P 1.977 13.19 0.506PI 1.877 26.54 0

It can be seen clearly from the result, for the P controller path, compared with that of the PIcontroller 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, sincethe disturbance value is big, the time when peak response is reached is not the first jump ofthe 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 greatlyincrease the percentage overshoot and make the peak response right at the time when it startsto give value bigger than zero.

Summary comparisonThe biggest difference between response of P and PI path in this case is basically the samethat that when disturbance value is 1. One more deduction can be found the the steady errorof 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 controllereliminated the constant error which agrees with the description of I controller in Theorysection.

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6 Bonus


The connection block picture for obtaining the response under disturbance with PIDcontroller 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 obtainedusing methods introduced in Theory section: KP = 1.2× T

L = 11.58, KI = 0.6× TL2 = 6.853

and KD = 0.6× T = 4.893.

The connection block picture for obtaining responses with P, PI, and PID respectively in asame scope can be seen in Figure 29.

Figure 29: Response under disturbance with different controllers

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The simulation graph for the response obtained with PID controller under disturbance can beseen in Figure 30.

Figure 30: Response under disturbance with PID controller



14.4% 0.2 7.79 0

It can be seen clearly from the result, when there is a PID controller, under a disturbancegenerated by a step block whose step time is 5, there is no steady error. This means that thedisturbance 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 aslight jump of the curve different from that without the disturbance. This means that for PIDcontroller, the disturbance makes influence on the curve shape in a tiny range of time startingfrom 5 (the disturbance start time).

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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, PIcontroller, and PID controller, it can be seen that the response with PID controller has theleast fluctuating curve and reaches stability without steady error the fastest.

In terms of achieving better steady response, the rank can be PID > PI > P . This is becausealthough both PID and PI controller achieve zero steady error, PID reaches steady reponsefaster, while P is the only one that still have steady error after reaching stability.

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7 Discussion and Conclusion

7.1 Error Analysis and Suggestions

The main limitations in this experiment is that when finding the points that help obtainingcontroller parameters such as settling time on the simulation graph using the cursor tool, itwas not always possible to find the exactly expected value. For example, we worked out thevalue (vertical component) of 10% of the response is 10%× 1 = 0.1, when we found the pointwith (t,0.1), we might only find a point quite close to the it but not exactly with the samevalue such as (t,0.102). This slight difference can cause a slight error in terms of the obtainedcorresponding time parameters.

To solve this problem, it is suggested that the maximum step size should be set smaller sothat the points display on the simulation result can be more concentrated and the possiblity offinding 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, andPID controllers were built and investigated comprehensively. Reasonable deduction based onthe obtained results were proposed which generally agreed with the related theoreticalknowledge provided in Theory sections.

The limitation that it was difficult to mark points with exact expected values using cursor wasassumed to be possible to be solved by decreasing the step size. For future improvement, itwas suggested that more types of controllers can be analyzed using similar approaches tobetter understand the controller design.

Overall, based on comprehensive analysis on the simulation results, it was found that theobtained simulated results using MATLAB simulink (ver.2015) generally agree with theknowledge of PID controller characteritics and close-loop system performance features learntfrom lectures of module ELEC207.

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References

[1] S. Maskell, “Lecture notes one,” https://vital.liv.ac.uk/webapps/blackboard/execute/content/file?cmd=view&content id= 959315 1&course id= 229521 1, University ofLiverpool, 2016.

[2] B. Douglas, “Pid control - a brief introduction,”https://www.youtube.com/watch?v=UR0hOmjaHp0, 2012.

[3] “lecture notes six,” https://vital.liv.ac.uk/bbcswebdav/pid-974699-dt-content-rid-6797002 1/courses/ELEC207-201516/6%20%28steady\discretionary-state%20response%20design%29%281%29.pdf, 2016.

[4] “lecture notes seven,”https://vital.liv.ac.uk/bbcswebdav/pid-980620-dt-content-rid-6799624 1/courses/ELEC273-201516/Experiment%2081%20Lab%20script.pdf, 2016.

[5] C. Tutorial, “Introduction: Pid controller design,” http://ctms.engin.umich.edu/CTMS/index.php?example=Introduction&section=ControlPID,2016.

[6] J. Zhong, “Pid controller tuning: A short tutorial,”http://saba.kntu.ac.ir/eecd/pcl/download/PIDtutorial.pdf.

[7] G. Reeves, “First-order plus deadtime (fopdt) model,”https://www.youtube.com/watch?v=4o4cqsu8JnE, 2013.

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https://vital.liv.ac.uk/webapps/blackboard/execute/content/file?cmd=view&content_id=_959315_1&course_id=_229521_1

https://vital.liv.ac.uk/webapps/blackboard/execute/content/file?cmd=view&content_id=_959315_1&course_id=_229521_1

https://www.youtube.com/watch?v=UR0hOmjaHp0

https://vital.liv.ac.uk/bbcswebdav/pid-974699-dt-content-rid-6797002_1/courses/ELEC207-201516/6%20%28steady\discretionary -state%20response%20design%29%281%29.pdf



https://vital.liv.ac.uk/bbcswebdav/pid-980620-dt-content-rid-6799624_1/courses/ELEC273-201516/Experiment%2081%20Lab%20script.pdf

https://vital.liv.ac.uk/bbcswebdav/pid-980620-dt-content-rid-6799624_1/courses/ELEC273-201516/Experiment%2081%20Lab%20script.pdf

http://ctms.engin.umich.edu/CTMS/index.php?example=Introduction&section=ControlPID

http://ctms.engin.umich.edu/CTMS/index.php?example=Introduction&section=ControlPID

http://saba.kntu.ac.ir/eecd/pcl/download/PIDtutorial.pdf

https://www.youtube.com/watch?v=4o4cqsu8JnE

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

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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

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experiment 81 - design of a feedback control system 81 - design of a feedback control system...

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