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: Report on The Theory Behind the Different Controller Approaches

Category: Business Paper Type: Powerpoint Presentation (PPT) Reference: APA Words: 7907

Game controller - WikipediaGame controller - WikipediaControl Engineering Final Lab-Report

Candidate  Number: 181524

 

 


 

CONTENTS

I        Introduction The Theory Behind The Different ControllerApproaches        2

I-A1

Proportional . . . . . . . .

.3

I-A2

Integral . . . . . . . . . . .

.3

I-A3

Derivative . . . . . . . . .

.3

I-A4

The PID Controller . . . .

.3

I-A5

Ziegler Nichols tuning . .

.3

I-B

State Space theory . . . . . . . . . . .

.4

 

I-B1            Definition of SystemState

.4

 

I-B2            State Equation BasedMod-

eling procedure . . . . . .

 

.4

II         Continuous PID ControllerDesign

II-A         Controller Modeling and Simulation Results . . . . . . . . . . . . . . . ..

 

 

.6

 

5

II-B         Discussion . . . . . . . . . . . . . ..

.6

 

II-C         Tuning . . . . . . . . . . . . . . . . .

.6

 

II-D         Disturbance . . . . . . . . . . . . ..

.6

 

III       Digital PID ControllerDesign

III-A       Controller Modeling and Simulation Results . . . . . . . . . . . . . . . ..

 

 

.7

6

III-B       Discussion . . . . . . . . . . . . . ..

.7

 

 

I-A       PID Controller Characteristics Theory.3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

IV           Lead Compensators For Our CaseStudyDesign            7

IV-A                                  Backgroundinformation                7

IV-A1    BodePlots                       8

IV-A2    Steadystateerror              8

IV-B                                  Procedure                                       8

IV-C                                  Modeling andSimulationResults   8

IV-D                                  Discussion                                      9

IV-E                                   Summaryoftheleadcompensatordeign9

V              LagCompensatorsForOurCaseStudyDesign      9

V-A                                    Background information of the lead compensator            9

V-B                                    Procedure                                       9

V-C                                    Modeling andSimulationResults   10

V-D                                    Discussion                                      10

V-E                                     Summery of the lagcompensator   11

VI           State  SpacecontrollerDesign                       11

VI-A               simulation  modelandresults          11

VI-A1... How to find the value of y- out-ss          11

VI-B               What is state observer and why do  we needit      12


VII         FurtherInvestigation                                   13

VII-A              Noise                                             13

VII-B              Disturbance                                   13

VIII       Conclusion                                                    13

References                                                             13

IX            Appendix                                                      14

I.    INTRODUCTION of The Theory Behind the Different ControllerApproaches

In the previous decade, the control signal regulation was performed physically, but the efficiency as well as effectiveness of such kind of systems were low at the time of compared to automated controller designs like state space controllers, digital, PID as well as the lead-lag controllers. Such kind of latest controllers deliver automated system which is commonly used in the latest day control engineering. A benefit of such controllers that it can provide over older, manual systems contain the enhanced robustness as well as accuracy like system gives. Furthermore, the manual systems try along with some kind of errors into feedback of systems which robotically control the systems may manage with.Although, errors related insignificant feedback generate a problematic task for physical stabilization for achieving. In the latest control systems, the major thing of control system engineers include across controlling as well as understanding of environment to give products with the high performance for the society, which generally known as systems. Furthermore, it also needs that the system needs to model and understand for achieving the effective system controller. The control system is basically multiple components developing a system which will allow us for the achievement of wanted response of the system. On the other hand, a liner system theory demonizes the process to represent through a block which has relationship of input and output the as well as shows the effect and cause relationship of the major process. There are two different techniques that may be used in system controlling such as open loop and closed loop. They are same but main difference is feedback. The open loop control system uses an triggering component for the management of process directly without feedback. While the closed loop system concerns with feedback and uses it to estimate feedback signal for regulation of output to withstand recommended relation of system variables with each other, which can be performed by analysis of working of variable as well as using difference as a means of control.To control whether closed or open loop system, it can be called which use different kind of controllers, and might be useful to give immediate response and better refined performance. The report concerns with four kind of variables which are: state space controller, PID, lead lag compensator and digital controller. The workability of such application illustrate that which controllerwill be suitable for using to increase the performance of the system as well as wanted response. The goals of such experiment are to plan, design, simulate as well as examine the distinguished designs of controller
for the plant which is show below.

Fig. 1. Graphical representation of the system to be controlled

 

The objectives of such experiment are to design, form as well as analyze distinguished controller designs for the plant which is given. The plant is formed as angle control pitch for a windmill rotor, as well as it can be defined as the function of transfer that shows who the plat is controlled in the mentioned figure. As continuous PID controller, the designs can be emulated and the discrete digital controller with the systems tuned for bringing the improvements accordingly, the lead lag compensator and

1.      Proportional: the proportional [2] of the function error is accepted in the sense where error is increased by the constant of proportional in equation 3. It shows the correction maintained by size of error at the given point in t.       

 

Pout = Kpe(t)                           (3)

e(t) is the instantaneous error at the given t as well as Pout is the output of the controller where Kp is the gain variable of proportional. A system that may be unstable but the gain is too low, is created by large value of the proportional gain, will not respond clearly for error came in the system.

 

1.      Integral: the integral [2] of the function error is also received in the account where the error up until a certain point at the time of integration that will mean that the errors up till the point is added. A factor is created by this which based on the total errors have connected in the equation 4.

 

t

 

Iout = Ki

e(t)dt                              (4)

Finally state space controllers. It is possible for the effectiveness of every type of the four types controllers the which is mentioned in the particular criteria of the design where the practical issues of real world must be collected into the account. The function of transfer show the plant to control, is measured by replacing particular parameter of the plant which are mentioned here parameters   into(Eq.1)

 

 

ng2                                (1)

(τs + 1)(s2 + 2ζωngs + ωng2)

 

It gives the function of transfer which show the rotor system of windmill to control into (Eq. 2)

 

  (2)

A.     PID Controller CharacteristicsTheory

The PID controllers maintain the stabilization of the system through conducting analysis of present the function of error into the system [2]. The function error (e(t)), is stated as the difference among the wanted actual value measured as well as the point of set that contain three most important factors of control concerning with the stabilization of error. Such kind of factors consist of the differential of error, integral of error as well as the proportional of the error. With the particular constant variable, such kind of factors are multiplied as well as finally the summation of all of them simultaneously which is shown in figure 2. Furthermore, it give the adjustment of error to make within a system.

 

Fig. 2. PID Controller design

e(t) is the instantaneous error at given t as well as the controller out is showing Pout where the value of Ki is the integral gain variable. Furthermore, to eliminate the error of steady state, the integra term is used to eliminate such as the difference among result and input of the system. Moreover, the factors which has an impact on the level of steady state error contain the system order in the addition for the type of input which is showing path control system and represented with P. Along with the term of proportional, an integral term is used in the sense of enhancing the speed at the output the comes to the wanted output. However, the function of integral control presents the overshoot in to the system. Therefore, the system of PID can have the effect in the negative sense on the stability of system as well as the response system.

1)      Derivative: the last factor is the derivation term [2]. On the certain amount of time, it is the gradient the of the error in the given equation.

d

Dout = Kd dte(t)                        (5)

 

Where Kd is the spinoff gain variable, e(t) is the in- stantaneous error at given t and Dout is the controller output. This term is introduced in addition to the PI controller in order to limit the dimension of overshoot. This issue is however extremely touchy to gadget trade such as noise as a result very small factors of Kd are often used.

1) The PID Controller: The PID controller is the sum of all of these elements (Eq. 6). It can be possible to create a easy two P controller by using ignoring the fundamental and spinoff factors. two Furthermore it is also possible to create a PI controller via ignoring the final, by-product factor. A PI controller can also be greater effective inside positive systems.

2) Ziegler Nichols tuning:

This technique was once established in order to approximationthrough managed error as well as trial, and the variable for tuning for a PID controller [3]. It is performed with the aid of calculating a value of critical gain Kcr (the obtain at which the output of a machine is stable), at which this fee can be used to calculate further parameters. A simple attain is added to the transfer characteristic (plant) of the gadget that is to be error corrected with the aid of PID. two two The reap is constantly multiplied from zero until the necessary

 

value is found. The ti.me period Pcr of accordingly output waveform two is also utilized.

 


The table which is given below for the estimation of tuning values PID controller by using the Pcr and Kcr.

TABLE I

ZIEGLER NICHOLS TUNING

 

Control Type:

Kp

Ti

Td

P

0.5Kkr

infinity

0

PI

0.45kcr

1/12*pcr

0

PID

0.6kcr

0.5pcr

0.125pcr

 

The method of Ziegler Nichols gives the parameters which give good disturbance of PID system rejection. Therefore, It also can be seen that such kind of systems provide the overshoot as well as aggressive gain that the may affect the methodology for becoming the unsuitable for the particular situation.

 

Moreover, some other kind of tuning PID methods may be used for this purpose. Such kind of methods consist of some error as well as the manual trial in the addition to the particular softwaremethods tuning. They however share the difficulties like the manual tuning for this can take more time because of the error as well the nature of the trial along with the random variables. While training is required by the software tuning for being used. Zeigler Nichols method of tuning gives a fine compromise as it gives an estimated error as well as the trial.

 


 

 

 

 

 

 

 

 

 

 

 

Fig. 3. System inputs and outputsof The Theory Behind the Different Controller Approaches

 

The system which is shown in the figure 8 has two inputs with the names of u1(t) as well as u2(t), as well as four variables are shown in output such as y1(t), y2(t), y3(t) as well as y4(t).

At some initial time T0, the input u1(t) and u2(t) for ¿=t0, it is possible to recognize all of system behavior for the future, if the system state is recognized the as well as the information of the variables of the states recognized.

 

The internal state of variables can describe the system that characterize the state of the system completely at the given possible time, as well as from the output variables yi(t) may also be calculated. The states of the variables can represent the different kind of system such as the economic systems, big classes of engineering, social as well as biological systems.

System models constructed with the linear one plot elements are state-determined systems models (such as mass, spring and damper components). In particular systems,the


 

PIDoutput = Kp(e(t) +1/Ti

 

 

A.     State Spacetheory


t

e(t)dt + Td

0


de(t)

) (6)

dt


number of state variables,n, is equal to the number of the independent energy storage elements in the  system  itself.  The value of  the state  variables  at any  time t  can  be used  to determine the energy of the total system, and the time derivativesofthestatevariablesindicatetherateofchangeof


1)     The authentic manipulate concept consisting of root locus that has been in use is based on a easy input-output description of the plant [7], usually referred to as the switch function. one of the problems of this approach is that restrict us to signal input sign output systems that allow only limited manipulate of the closed loop behaviour when comments manage is in use. The current control idea can clear up among the barriers via using a far richer description of the plant dynamics. one of the methods of the superior manipulate idea is the kingdom space description. kingdom space offers the dynamics as a hard and fast of coupled first-order differential equations in a set of inner variables, known as nation variables, brought to a set of algebraic equations that integrate the kingdom variables into bodily output variables.

2)     Definition of System State:

The concept of the state of a dynamic machine refers to minimum set of variables, these set of variables completely describe the gadget and its reaction to any set of inputs. particularly, a country determined gadget models have the feature that: A mathematical description can illustrate the system in time period of a minimal set of variables Xi(t),i=1,...,n, mixed with expertise of these variables at an initial time t0 and the device inputs for t¿=t0, is enough to make prediction of the future system state and it's far output all the time t¿t0.

The definition means that the state determined system’s dynamic behavior is completely characterized by the set of n variables response xi(t), where n points out the order of the

 

system. In the figurer 8, it will show the theory of input system and output system. The framework vitality. Too, the esteem of the framework state factors at any time t gives adequate information to decide the values in the event that all other factors within the framework at that time.

 

There's no particular set of state factors that depict any given framework; a numerous distinctive of sets of factors can be chosen to total framework portrayal, but for a given framework the arrange n is special, and it depends on the specific set of state factors chosen. State variable depictions of a framework can be portrayed concerning physical and measurable factors, or in term of factors that are not straightforwardly quantifiable. It is conceivable to Convert a set of factors to another scientifically. The critical portion is that any set of state factors ought to give a total depiction of the framework. Based on this we concentrate on a specific set of state factors that are based on vitality capacity factors within the physical framework.

 

3)     State Equation Based Modeling procedure: For a linear time-invariant system, the complete system model contains two essential factors. Firstly, a number of set (n) of state equations, expressed in term of matrices A and B. Secondly,    a set of output equations that relate the output variables of interest to the state variables and the inputs, which are usually expressed as C and D matrices. The idea of the modelling the system is to derive the elements of the matrices and represent thesysteminthisformwhichisinequations(34-35).

 

X = Ax+Bu                              (7)


 

= Cx+Du                             (8)

The matrices A and B are determined by the system structure and elements. On the other hand, The output equation C and  D matrices are determined by the particular choice of output variables. The overall modelling procedure is based on the followingstepstoensurethecheapestpossiblecost.

  From the linear graph system representation, the system orderncanbedeterminedandthesetofthestate


parameters - table 2.


 

 

 

TABLE II

Control Type:

Kp

Ti

Td

P

0.98

infinity

0

PI

0.882

0.882

0

PID

1.176

1085

0.27125

 

FORM 1CALCULATIONS


variables.

  From the linear graph system description, the state equa- tion of ths system and the system A and B matrices can be generated using a well-definedmethodology.

  DeterminationofasuitablesetofoutputsequationsofC


III: Find parameters in PID form 2.of The Theory Behind the Different Controller Approaches

 

When such kind of value will be found, more values may also be measured by using the equation 4 for generating the table 3. The block controller of PID is also used along with the PID form the two different values for the creation of PID, P as well as PI Controller.


and D matrices and their derivation.

 

II.      CONTINUOUS PID CONTROLLERDESIGN


1

P + I

s


N

+ D

(1 +N/s)


 

(9)


This is an analog design that is able to use PID function     to correct errors. Three steps are required to design PID controller:

I: Finding the critical gain and period:

 

This is observed by connecting the plant switch characteristic directly into a attain block [3]. Trial and error where the reap of the machine is multiplied until the graphical wave form of the machine output is a perfect sine wave. When the attain is too excessive or low the wave is unstable. The vital achieve kcr can be located where furthermore the imperative time length pcr can also be measured. It can be seen in the discern (fig 3), when the price of the gain is 1.96 the amplitude is steady over time. On the different hand, when the obtain cost was once changed to a larger and smaller price than 1.96, the amplitude is now not constant. Kcr is measured as 1.96 and Pcr as 2.170 seconds.


Fig.4.   Outputgraphofdifferentgainvalues

 

II:FindparametersinPIDform1

With the values of Kcr and Pcr is is possible to calculate  usingtheZieglerNicholsmethod,estimationsforP,IandD


 

TABLE III

Control Type:

P

I

D

P

0.98

0

0

PI

0.882

0.4877509263

0

PID

1.176

1.083870968

0.31899

 

FORM 2 CALCULATIONS


A.     Controller Modeling and SimulationResults

A continuous PID controller is simulated in Simulink com- puter software (fig 4). as it can be seen in the figure, an step input is injected to the system, followed by the summer toAdd or subtract inputs, one from the set input and the other  one from the feedback. The summer determine the value ofthe signal supplied to the PID controller, then the scoop will beusedtoplottheoutputgraph.


 

 

Fig. 5. PID Controller design block diagram

 

Using PID variables obtained in table 3, the waveform of    P,PIandPIDcontrollerisfound(fig5).


 

Fig. 6. P, PI and PID waveform for continuous PID controller

 

 

B.     Discussionof The Theory Behind the Different Controller Approaches

In comparison with the PID controller, P has a steady state error from the above graph. The outcome steadies around 0.7s rather than the wanted reference that makes it equal to 1. It the big problem but it can be resolved through the use of bigger gain for the improvements in the state error. On the other hand, the PI controller reduces the steady state error as well as the forced oscillations.

C.     Tuningof The Theory Behind the Different Controller Approaches

For this purpose, the tune PID variables can possibly be used to provide the better outcome. The factors like overshoot in the controller of PID may be limited or excluded entirely that gives the better efficiency of the system. To changes in the error, the response time may also be reduced the system mean. The parameters of PID are also altered, so the output of the controller are also improved which can be seen in figure 6.


 

 

Fig. 7. Tuned PID graph waveformof The Theory Behind the Different Controller Approaches

 

 

D.     Disturbance

An input for addition function is the disturbance in the system which is inserted in the plant. To reject such kind of disturbance through minimizing the output of the system back to steady state when the disturbance is added, the PID controller is able. The real PID controller continuously is able to give the rejection of disturbance which can be shown in the figure 7.


Fig. 8. Waveform of PID controller with disturbance

 

 

 

III.       DIGITAL PID CONTROLLERDESIGNof The Theory Behind the Different Controller Approaches

The transfer function has converted in the digital controller design from the consecutive domain of time S to domain Z [4] so it can be represented into the digital design. Furthermore, the variables of the PID controller form the continuous controller are also utilized for making another function of transfer in the figure 8 which cause the control of PID. To use the MATLAB function c2d discretely, it is converted. It must select the specified time value as well as it will change the equation in the domain.

 

                                   (10)



To transfer the equation form time to domain Z, the MATLAB script code1 is formed as well as to insert the value of T.

 

A.     Controller Modeling and SimulationResults

Form MATLAB is inserted within the system earlier the path, a model of SIMULINK is developed through the use of the domain Z equation.


 

Fig. 9. Block diagram of simulated digital controller

 


 

Fig. 10. Digital PI controller output graphof The Theory Behind the Different Controller Approaches

 

 

 

B.     Discussion

As it has been viewed in the article that PID digital Controller can be assumed as reduction in the value of T and enhancement in the over shoot system despite from this the greater value of T has been utilized because its work faster. Furthermore, decrease in wave can be noticed as the decreased value of T. Moreover, in PI digital the T denotes as smaller and the results did not changed essentially. This will prove that digital PID were not be responsive to a little change for the value of T as well as in the combination with digital PID controller. IT had also be seen that enhancement in the value of T has been taken long replication this can be happen because the smaller value of T and this will happen because the high load in the control processing Unit(CPU). It is clearly said that smaller value of T will lead the system toward an improvement in the outcomes of PID through system and this will also increase the power of processing which has been necessary for the real world. The significant factor such as specification of Budget have to be assist in order to justify the most appropriate system.

 

 

 

 

 

 

 

 

 

 

 

 

 


Fig. 11. Digital PID controller output graph

 

 

IV.      LEAD COMPENSATORS FOR OUR CASE STUDY DESIGN

A.     Backgroundinformation

The leading compensator has been referred as another kind of system controller implications that can be able to produce the smooth activity of the waveform in the plant and provides the step input. The same function of transference can be used before when the compensator pays the primary role to increased the sustainability of a system and mainly summarized and improved the response of transient. This would be attained as in the from of factors in setting time as well as shorter large, short overshoot and quick transient response. The phase lead compensator can be able to perform the task and as well as changed the original function which has been modify the margin phase holding the representation of domain frequency of the system. The magnitude of the system will remain constant at all the values of frequency as well as they will also shift through gain crossover fe.

Lead compensator has been designed for the usage of method of roots locus or this can also be used through the frequency of response plots. The bases of system control process while using the methods of using this system with the modifier of system bode phase margins which was mention in the existence gain-crossover frequency. This procedure can be done without the troubling the magnitude curve plot of which was providing the same incidence which will also alter the value in to zero magnitude. The magnitude value would also sanction the system a change in the response of transits’ without altering the response of momentary value of factors including system. Without doing any amendment in the gain crossover would be noticeable in the point of resultants.

The function and amount of shift has been transferred in the system to meet the requirements of margins in the system.

·         The magnitude curve of bode plot frequency that can be passed from 0 dB with in the accepted range of frequency up to 2. The uncompromised phase has been transferred to the frequency of gain crossover that has been the most negative aspect then from all those values that has been required to meet the requirements of Margin phase.

In this investigation, he bode plot frequency curve would be utilized to display the outcomes of the responses of the system, and these examinations has been compared with the outcomes of gain dB as well as in the phase margins. Although the details of about bode plot and what were the instruments that would be useful to test and investigate about the performance.

1.      Bode Plots: In the trial, we were interested to find out the whole spectrum of the frequency, it is the essential first to know about the gain, there phase and their shifts in the complete spectrum, and implementation of these frequencies can be done through bode plot. The bode plot defines the response of the frequency which includes the axis of horizontal logarithm. That’s why the bode plot has been considered perfect for the investigation to understands the behavior of the system.

2.      Steady state Error: this error can be defined as the error in the value of signals when it comes consistent and goes infinite with the passage of time. This error has been used to identify the perfectness of the system as well as to find the reason behind the how the state error would be aroused. And the answer of this question has been defines as this will be occur because of the nature of input and the type of the system. In this type of error if the output would not b matched with the provided input the difference between them would be known as an error in the states.

Theory Based: The explanation can be done of the basis of theory, as in the code of MATLAB which has been created to analyze the parameters of lead compensatrs as well as to state the needs of bode plots. For example, actual plant as well as the plant which have gain-cross curve frequency.

B.     Procedure

The lead compensator has been accomplished through the making of MATLAB by including the steps which were given below:

1)  Find out the gain in the compensator kc with satisfying the error of constant.

2)  Attain the extra phase of lead through desired adjustment of grain.

3)  Obtained the equation from α


C.     Modeling and SimulationResults


 

Fig. 12. Block diagram of simulated Lead compensator controller


 

 

 


 

Fig. 13. Bode plot of the original system plant

 

 

 


 

Fig. 14. Bode plot of the original plant with the gain added to the system

 

sinαm = 1 −α

1+α


 

 

 

 

(11)


4)  DeterminethenewgaincrossoverfrequencyofGc(s).

1

T=αωc                           (12)


 

 

Fig. 15. Bode plot of the original plant of the system when the compensator  isadded


 

Fig. 16. Bode plot of the whole system

 


 

Fig. 17. step response output graph with the lead compensator

 

 

D.     Discussion

The gain has been found by doing trail and error and this can be done through increased and decreased in the values of gain cross curve frequency and this can be done continuously until the margin line has been reached upto the requirement of the design. The decrease and increased of values has been in the figure, when the frequency of gain has been added within the system the values of state error will remain constant as desired needs. That’s why the compensator has been added in this to handled the problem by including the negative phase of the margin lines as well as to decreased the value of other phases of margins.

The output of compensators reveals a quick response instead of uncompensated the output which take more time and high percentage of damping.

 

E.     Summaryoftheleadcompensatordeign

The controller of the system has been designed and placed into the software of MATLAB and this would also be viewed as the design of lead compensator which enhance the bode diagram and values were also be mentioned. The increased in the values of frequency can be changed in to the lower values of the frequency. Despite from this, the latest solution can be included in the final phase of the margin values up to the degree of 37 which includes the range of 40 degree and this will be range between the values of 0.05 and 1 and these ranges were also included as the requirements of this assignment. Moreover, this system has done struggle to maintain the basic needs as well as to maintain the range of the margins. The project also included some limitations where transient respond to the plant and increased the criteria of the project. The main response of the transient has been in largely improved with the response of transient as well as uncompensated responses to better the frequencies of settling time, levels of overshoot, and fluctuation which will lead toward the excellent quality of lead system. Furthermore, the state error in the system had negative influence on the responses of transient and this will give an effective outcome.

V.     LAGCOMPENSATORSFOROURCASESTUDYDESIGN

A.     Backgroundinformationoftheleadcompensator

The descriptive background of lag compensator has been related to the compensator of lead. The lead includes the improved consistency of systematic sacrificing which will produce the stable state error. The lag compensator provides the stable balance between the state error in the system. The lag compensator would also provide extra gain crossover frequencies and this will also generate low frequency among the regions. Despit from all these systems also gives the appropriate phase of frequencies. As well as the phase margins has the ability to increase the required levels of closed loop of bandwidth would be decreased.

 

B.     Procedureof The Theory Behind the Different Controller Approaches

1)      The compensator attains the kc β to achieve the needs of constant error.

2)      The limits of point of frequency has been changed as in the phase of open-loop system which is                             similar to the frequency of gain crossover.

3)      Select the zero compensator with ω= t and 1 can be denoted as low decade.

4)      These limitations has been further turned into the magnitude and this will provide the curve of 0dB to achieve the current gain crossover frequencies.

5)      This will further handover the functions of Gc (s).

The explanation has been based on the theory which describes the Code MATLAB as in the code of MATLAB which has been created to analyze the parameters of lead compensators as well as to state the needs of bode plots. For example, actual plant as well as the plant which have gain-cross curve frequency. A Matlab code was established to conduct the entire former steps (included in the Appendix). On the other hand, the value of  is identified by viewing the output graph, as well as detecting the constant state error.

C.     Modeling and SimulationResults


Fig. 18. Bode plot of the original system plant

 


 

Fig. 19. Bode plot of the original plant with the gain added to the the system

 

 

D.     Discussion


 

Fig. 20.  Bode plot of the original plant of the system with the lag compensator

 


 

Fig. 21. Bode plot of the whole system

 


 

Fig. 22. step response output graph with the lag compensator

 

 

 

The compensated output improves steady state error re- sponse sacrificing stability, the opposite of the uncompensated output which have less damping ratio and stabilize faster.


E.     Summery of the lagcompensatorof The Theory Behind the Different Controller Approaches

F.     The controller of Lag compensator has been applied successfully as well as this will be imitation in the software of Matlab, which will attain as in the form of wave which will be gained as in the phase of lag response controller. Despite from this the bode plot responses of the ag compensators would be characterized and represent as the reduced pass filters and the system will include the negative phase and will also provide the appropriate range of the frequency.

 

VI.       STATE SPACE CONTROLLERDESIGN

This model describes the simulated space with in the link. The perspectives of the design were as below.

1.      Identify the values of a2, b2, c2 and evaluate b2 from the number of plant.

2.      Identify the system poles of open-loop system that can be utilized by commanding the codes on Matlab.

3.      Utilized MATLAB and command them i=qr which is related to the system of closed-loop poles.

     Start the simulation to attain the state value of y-out, to do this process first we have to run the simulation       first to achieve the needs of ks, and the difference between the graphs of output as well as input to identify the     consistent state error.

4.      The current value of Kss can be defined as the 1/y-out that can be required to change the frequency again.

The code has been changed into the new values and then this will be represented on the graph after doing the value of gain crossover frequency to identify the change the frequency in the graphs.

 

A.     simulation model andresults

A Matlab code was established to conduct the entire former steps (included in the Appendix). On the other hand, the value of  is identified by viewing the output graph, as well as detecting the constant state error.

1)      How to identify the value of :

The first step is to set the gain value to 1, then plot the output graph, in order to identify the value of constant state error. In figure (27) we can see that the graph become stable at 0.4733, and therefore, the new value of  would be equivalent to

1

youtss=0.4722   (13)

 

 

 

Fig. 23. State space model


 

Fig. 24. State space model output graph

 

 

In order to add the new value of the , the code has been edited, then the output graph will be plotted right after the gain value has been modified.

 


 

Fig. 25. State space model output graph for two different value of gain

 

The application of a new gain value in  can be seen in the figure (28), in which the output graph was stabilized at one as preferred and there is a stable system in comparison at the time the gain was assigned to  1. Regardless of there is an improvement in the output graph stability, the time for response endure to be the same before and also after the gain value has been modified.


B.     Whatisstateobserverandwhydoweneedit


Observer: Defined as an input framework which has a criticism increase of . Expect the criticism controller is as of now has been planned and the following stage is to think about how to execute this controller, a  is required along with sensors to quantify the state factors, which are the contribution to the . On the other hand, in some genuine applications, states are hard to or quite costly to gauge. For this situation, we can utilize a software part which is known as the observer. The observer consists of two inputs, which are  and the . The observer output evaluates the state factors as opposed to utilizing expensive sensors, so for this situation, the observer can be measured as a sensor of software.

Fig. 26. State space model output graph for two different values of gain

 

 


 

 

Fig. 27. State space model simulated on Simulink

 

The figure (29) presents the comprehensive observer model, in which the A, B as well as C observer matrixes are indistinguishable from the A, B and C matrix of the initial plant. Likewise, there is the observer’s feedback addition which is reflected to as , and the output of observer is portrayed as . As found in the figure that there are two outputs which are provided to the whole, as the observer output as well as  as the output of the initial plant, the two values then will be contrasted along, in order to discover the error signal from the observer, that is pursuing the essential rule applied to the input control dependent on error. From that point forward, the  observer’s gain must be planned, as it referenced previously that the observer itself is an input framework it has poles which are equivalent to ,  require to follow evolving . For this reason, to pick the poles of the observer with the end goal that observer’s reaction will a lot quicker than the  framework ,i.e.,  are more left than c-p. Like state-space criticism task, the estimations of the spectator posts should be allocated utilizing the  on Matlab. When this is completed, the  is utilized to discover the estimations of .

The following stage when activating the simulation is that the integrator of the initial plant as a it’s underlying condition is fixed to zero. On the other hand, to test the observer’s performance, the integrator of observer its underlying condition is fixed not quite the same as the . Subsequent to the running simulation, we will get four diverse output, the  as well as , , to test the exhibition,  should be compared with the x plants’ output, just as y output designed for both. On the off chance that the observer is very much structured after some time, then the error should move toward zero. This was conducted on Matlab and below is the output diagram.


Fig. 28. Comparison between the State space and the observer outputs (X- and-Y)


 

VII.        FURTHERINVESTIGATION

 

A.     Noise

 


 

Noise is a random factor that cannot be predetermined which provides an additional magnitude input into the system. Noise of varying magnitudes is inputted into the PID control system at which provides (Fig. 36):

It is observed that the PID controller is able to successfully reject noise as the PID controller continues typical PID con- trol behaviour after smoothing down the waveform from the random injected noise. Noise within SIMULINK is simulated via a random number generator.

 

 

B.     Disturbanceof The Theory Behind the Different Controller Approaches


VIII.         CONCLUSIONof The Theory Behind the Different Controller Approaches

In conclusion, this has been a successful investigation, into differentcontrollers,whichhavebeenevaluatedandcompared. Alltheinitialaimsandobjectivesoftheprojecthavebeenmet, and compared. The P,PI and PID controllers have been built and compared, and the parameters for those controllers were calculated using the Ziegler-Nichols tuning method. The PID controller has been evaluated with different disturbance and their effect. Digital PI and PID controllers have been built and compared with the initial PID system response. A state  space model has been built and evaluated. All the controllers have been compared and it was concluded that the PI and PID response would be suitable for the wind turbine, but the state space model looked to be the best option, due to its simplicity andthefactthatithadnosteadystateerrororoscillations.

 

REFERENCESof The Theory Behind the Different Controller Approaches

[1]   How does a PID controller work?, H. (2018). ¿ jjrobots. [online] jjrobots. Available at: https://www.jjrobots.com/pid/ [Accessed 10 Nov.2018].

[2]   Nptel.ac.in.   (2018).      [online]      Available    at: https://nptel.ac.in/courses/108103008/PDF/module1/m1 lec1.pdf [Accessed 12 Nov.2018].

[3]   Pdfs.semanticscholar.org.     (2018).   [online]    Available          at: https://pdfs.semanticscholar.org/d149/c7ad9deba6d0f3a7a615a48fbd11fc16a1a8.pdf [Accessed 10 Nov.2018].

[4]   Anon,       (2018).       [online]       Available      at: https://www.researchgate.net/publication /308325095 SIMPLE DIGITAL PID CONTROLLER FOR SMART SYSTEM [Accessed 15 Nov.2018].

                              

[5]   Ece.gmu.edu. (2019).     [online]     Available     at: https://ece.gmu.edu/                       gbeale/ece  421/compfreqlead.pdf   [Accessed 31 Jan.2019].

[6]   Ece.gmu.edu. (2019).     [online]     Available     at: https://ece.gmu.edu/                       gbeale/ece  421/compfreqlag.pdf     [Accessed 31 Jan.2019].

[7]   Web.boun.edu.tr.              (2019).     [online]      Available          at: http://web.boun.edu.tr/koseemre/me537/classnotes.pdf [Accessed                    31

Jan. 2019].


 

 

 

 

Similar methodology is employed within disturbance testing as a predetermined disturbance signal of varying magnitudes is set into the PID controller. Once again it is clear that the PIDcontrolleroffershighlevelsofdisturbancerejection.


IX.       APPENDIX


15 set(lineobj,’linewidth’, 1.8) ;%change thelinewidthto1.8.


1

2 MatlabcodetocalculatethePIDform2 valuesfromPIDform1values.

3

4  k cr=input(’pleaseenterthevalueof kcr?’);%asktheusertoinputthe g a i nKcr.

5

6  p cr=input(’pleaseenterthevalueof pcr?’);%asktheusertoinputthe p e r i o dpc r.

7%EquationstofindparametersinPID form1andthenform2.

8

9 p kp= 0 . 5 k c r

10

11 pikp= 0 . 45 k c r

12

13 piti=(1/1.2)pcr

14

15 pid kp= 0 . 6 kc r

16

17 p i dt i= 0 . 5 p c r

18

19 pidtd=0.125pcr

20

 

21

22 p p=p kp

23

24 pi p=pi    kp

25

26 pi  i=(pi   kp) /(pi      ti)

27

28 pid p=pid kp

29

30 p i d i=p id   ti

31

32 pid d=pid kppidtd

 

1

Matlabcode2isusedtoplotgraphs.

3

4      den= [0.5,2.2,4.2055,3.61];

5 num=   7 . 5 8 1 0 ;

6   t e s tk=  1 . 9 6

7 %i sused

−                      

8   p lot(t,PID     out,’K      ’,t,simout2,’             ’)%

9   grid

10 xlabel(’Time(sec.)’,’fontsize’,14)%add xlabel

11 ylabel(’Amplitude’,’fontsize’,14)%add ylabel

12  title(’TunedPIDcontrolleroutputgrpah ’,’fontsize’,14)%addtitle

13

14 lineobj=findobj(’type’,’line’);


16

17 textobj=findobj(’type’,’text’);

18 set(lineobj,’linewidth’, 1.8) ;%change thetexttowidthto1.8.

 

1

Matlabcodei susedt of i n dt h ev a l u eof NumandDenforspecificTinputvalue

oft h ed i g i t a lc o n t r o l l e r.

3

4  %thisscriptconvertsthetransfer function

5     %Gp(s)=1/s(s+1)toadiscrete                  time system

6 %wi t has a m p l i n gp e r i o dofT=1s e c.

7 %

8 num= [ 0 . 8 8 2 88 . 68775 4 8 . 7 7 5 0 9 2 6 3 ] ;den=

[ 1100 0 ] ;s y s c=t f(num,den) ;%The valuesofNumandDencalculatedby substatingthevaluesofPIDform2

i n t o(Eq. 8 )

9      T=  0.05;%steptimeinput.

10 [sy sd]  =c2d(sysc,T,’zoh’) ;

 

1 Leadcom pensat ord e s i g ncode’s

2 %I n i t i a lP l a n tC o n i d i t o n s

3   clearall

4   c lf

5   c lc

6 num= [ 7 . 8 5 1 ] ;

7 den= [ 0 . 5 2 . 2 4 . 2055 3 . 6 1 ] ;

8   plant=tf(num,den);

9

10 %1GpI n d e p e n d a n t

11 figure(1)

12 bode(p l a n t)

13 margin(plant)

14 legend

15 grid

16

17  %2Gpandg a i nKc

18     %ToreduceS.S.E

19 figure(2)

20 kc= 12%Gaini ss e ta sav a l u eof10

21 gainAndPlant=plant         kc

22 bode(g a i n A n d P l a n t)

23 margin(gainAndPlant)

24 [Gm,Pm,Wcg,Wcp]=margin(gainAndPlant)

25 legend

26 grid

27

28 %3DesignofCompensatorC o n t r o l l e r

29 %Tor e c o v e rs t a b l i l i t ymargin

30      a=        Pm+ 10%Allowsphasemargin headroom


31 alpha= (1       sind(a))/(1+sind(a))% Calculatealphavalue

∗ ∗

32  T=s q r t( 1 0 ) /(Wcg 2pi)%CalculateT value

33  %Substitutingvaluesintotransfer functionformat:


 

17

18  %2Gpandg a i nKc

19     %ToreduceS.S.E

20 figure(2)

21 kc= 12%Gaini ss e ta sav a l u eof10

22 gainAndPlant=plant       kc


34 compensator=tf([alpha       Talpha],[alpha       T23bode(gainAndPlant)


1 ] )%TFform

35

36 figure(3)

37 bode(comp ensat or)

38 margin(compensator)

39 legend

40 grid

41

42 %4Gain,c o n t r o l l e randp l a n tcombined

43 %Combinegainkcwithcompensator controller:

44        kcAndCompensator=kc  compensator

45     %Connectingkcwithcompensatortoplant:

46 completeSystem=series(kcAndCompensator, plant)

47

48%GraphicalrepresentationofUncompvs Comp

49 figure(4)

50 c lc

51  t i t l e(’UncompensatedVs.LeadCompensated System’)


24 margin(gainAndPlant)

25 [Gm,Pm,Wcg,Wcp]=margin(gainAndPlant)

26 legend

27 grid

28

29 %3DesignofCompensatorC o n t r o l l e r

30 %Tor e c o v e rs t a b l i l i t ymargin

31      a=        Pm+ 10%Allowsphasemargin headroom

32 alpha= (1       sind(a))/(1+sind(a))% Calculatealphavalue

∗ ∗

33  T=s q r t( 1 0 ) /(Wcg 2pi)%CalculateT value

34  %Substitutingvaluesintotransfer functionformat:

∗                       

35 compensator=tf([alpha         Talpha],[alpha         T 1 ])%TFform

36

37 figure(3)

38 bode(com pensator)

39 margin(compensator)

40 legend


52 Uncompensated=f e e d b a c k(p l a n t, 1 )

 

41

g r i d

53 s t e p(Uncompensated, 40)

 

42

 

54 holdon

 

43

%4Gain,c o n t r o l l e randp l a n tcombined

55 Compensated=f e e d b a c k(complete System,

56 s t e p(Compensated, 40)

57 holdon

1 )

44

 

 

45

%Combineg a i nkcwi t hcomp ensato r c o n t r o l l e r:

kcAndCompensator=kc  compensator

58 legend

 

46

%C o n n e c t i n gkcwi t hcomp ensato rt op l a n t:

59 grid

 

47

complete System=s e r i e s(kc And Compensator,

60

61 figure(5)

 

 

 

48

p l a n t)

62 margin(completeSystem)

63 grid

 

49

 

 

50

%GraphicalrepresentationofUncompvs Comp

f i g u r e( 4 )

1 Lagcomp ensat orcode’s

 

51

c l c

2

3 %I n i t i a lP l a n tC o n i d i t o n s

4  clearall

 

52

 

 

53

t i t l e(’UncompensatedVs.LeadCompensated System’)

Uncompensated=f e e d b a c k(p l a n t, 1 )

5  c lf

 

54

s t e p(Uncompensated, 40 )

6  c lc

 

55

ho l don

7

num=

[ 7 . 8 5 1 ] ;

 

 

56

Compensated=feedback(completeSystem,1)

8

den=

[ 0 . 5 2 . 2

4 . 2055

3 . 6 1 ] ;

57

s t e p(Compensated, 40 )

9  plant=tf(num,den);

58

ho l don

10

59

l e g e n d

11 %1GpI n d e p e n d a n t

60

g r i d

12 figure(1)

61

 

13 bode(p l a n t)

62

f i g u r e( 5 )

14 margin(plant)

63

margin(complete System)

15 legend

16 grid

64

g r i d


1   Statespacecod’s

2

 

3

4 Num= [ 7 . 8 5 1 ]

5 den= [ 0 . 5 2 . 2 4 . 2055 3 . 6 1 ]

6   plant=tf(Num,den)

7   i=s qrt(      −1);

8

9 %Convertingbetweentransferfunctionto statespace

10 [a1b1c1d1] =t f 2 ss(Num,den)

11 polesoftheclosedloop=eig(a1)              1.5

12 k=p l a c e(a1,b1,p o l e s o f t h e c l o s e d l o o p)

13 Youtss= 0.4722

14 thegain=1/Youtss

15

 

16

 

17

−                     

18 observerPoles=eig(a1)             3%Additional       1

19 L=p l a c e(a1’ ,c1’ ,o b s e r v e r P o l e s)

20

21 figure(1);

22

23 plot(out);

24  title(’Outputgrpahofthestatesapce

wi t hag a i nv a l u eof1’,’F o n t S i z e’, 1 4 )

25 xlabel(’Amplitude’,’FontSize’,16)

26 ylabel(’Time(second)’,’FontSize’,16)

27 grid

28

 

29

30 figure(2);

31

32 plot(out);

33

34  title(’Outputgraphofthestatespace

−   

withagainof1andgainofy                        out ss ’,’FontSize’,11)

35 xlabel(’Amplitude’,’FontSize’,16)

36 ylabel(’Time(second)’,’FontSize’,16)

37 holdon

38 plot(out1);

−   

39 legend(’gain=y        outs s’,’g a i n=1’) ;

40 grid

41

 

42

43 figure(3);

44 plot(yOutobserver);

45 holdon

46 plot(obsout);

47 holdon

48 plot(xOut);

49 holdon

50 plot(obsXOut) ;

51

52  title(’StatespacevsObservervalues outputgraph’,’FontSize’,14)

53 xlabel(’Amplitude’,’FontSize’,16)


54 ylabel(’Time(second)’,’FontSize’,16)

55 grid

56 legend

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