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Second order system with different controllersTiago Filipe Carvalho Rodrigues; Helena Isabel Pais Ferreira de Vargas

Students of the Faculty of Engineering of Porto University, Department of Electrotecnical Engineering

[email protected]; [email protected]

Abstract - This paper describes the operation controllers appliedto a system of second order, starting with the application ofconventional controllers and passing to the use of fuzzycontrollers. At the end there will be a comparison with dataobtained through the implementation of controllers, using asimulator from MatLab called Simulink.

I. INTRODUCTIONIn industry, controllers are widely used nowadays, and are

responsible for industrial process control, using for that

classical PID controllers, fuzzy controllers or even neural

networks.

This paper addresses the classical PID controllers as well

as its design and calculation of earnings for its

implementation, and also the fuzzy controllers (as well as

your project, the membership functions, inference rules, etc),

both applied to a same system of second order.

Well talk about the operation of each of the controlle rs,

explain how they were design, compare the performance

obtained with each one of the controllers and after that, test

the performance of controllers applied to dynamic systems,

ranging from system reference and parametric variation of the

system.

II. CLASSICAL PIDThe classical PID (Proportional, Integral, Derivative), is the

most common and used controller in the industry control

systems, calculating an error value, making the difference

between a measured process variable and a desired set point.

The PID controller will try to minimize the error by

adjusting the process control inputs, by his mathematic

formulation

(1)

On (1), Kp is the proportional gain, which will act on the

present error, reducing the overshoot of the system response.

On (1), Ki is the integral gain, which will act based on the

accumulation of past errors, accelerating the movement

process for the desired set point, also eliminating steady state

error.

On (1), Kd is the derivative gain, which will act base on the

current rate change of errors, giving the possibility to control

the rising time, that is the time that the system takes to reach

the steady state.

The gains of the PID controller, give the option to select

the weighting of each action in the final value of the

controller in order to obtain the best possible performance of

the system.

III. PROJECTING A PIDCONTROLLERFirst we need to analyze the transfer function of the system

that we want to stabilize with a PID controller.

(2)

As we can see, this system (2) will be unstable, because

despite having a pole in the Left Semi Plan (pole at -1), also

has a pole at the origin, which will confer instability to the

system.

We can observe this instability in the open loop response of

the system.

Fig.1 System Response in open loop

As we can see in Fig. 1, the system response evolves

continuously to infinity, never reaching a set point required

for steady state.

Besides the system response in open-loop, we can also

observe the Root Locus to determine the stability of the

system.


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The poles of this system will be in -0,108; -0,287 and

-5,604, matching with the poles shown in Fig.4, and proving

that the system is stable now with all poles on the semi left

half plan.

To see the impulse response of the system, to check the

stability of the system, we did a simulation of the system with

the help of application Simulinc of the software MatLab.

Fig.5 Impulse response of (4)

We can verify that the system response is stable, having no

steady state error, but with some initial small overshot and

relatively slow in reaching the steady state operation.

This shows how effective it is a classic PID controller, in

the use of process control, along with its relative ease of

design and implementation, explains why the classic PID

controller is one of the more widely used controllers in the

entire industry.

IV. FUZZY CONTROLLERSIn the projection of classical PID controllers, attention is

given to system modeling by differential equations,

representing itself, and construct a mathematical controller

that can put the poles of the most advantageous way possible,

for stability and best performance.

In fuzzy logic controllers the logic is the opposite, attention

is given in the intuitive understanding of the system and how

it well be the best way to control the system.

For example, if one person needs to control the watertemperature of a shower, he intuitively follows rules which

allow him to regulate the water temperature, for example:

If temperature is low then increase temperature;

If temperature is pleasant then keep temperature;

If temperature is high then lower temperature;

These rules are formed with the help of a good

understanding of the functioning of the system, in this case

the system was easy to understand, in some complexes cases,

to build a set of rules that allow us to control a system is

difficult.

A fuzzy controller operates similarly, contains four main

components, which are the following:

1 A system of rules, which contains the knowledge to

control the system as best as possible

2 An inference engine, that decides which rules control

are relevant to the current state of the system, and then decidewhich input would be best for the system

3A system of fuzzification, which modifies the inputs of

the controller in order so that he could interpret and compare

with the system of rules

4A system of defuzzification, which converts the values,

determined by the inference system in the system outputs to

control

Fig.6 Fuzzy controller architecture

One of the greatest advantage of the fuzzy controllers, are

the membership functions used by the inference engine,

because instead of having only two logical values (true or

false/ 1 or 0), the membership functions in fuzzy controllers

are prepared to deal with partial truths, allowing intermediatevalues, unlike crisp functions.

The membership functions of the example given above

could be something like Fig.6.

Fig.7 Membership Functions

In Fig.7, cold, warm and hot are representative of the

variable temperature, represented by membership functions,

where they say the temperature range that each variable has.

One point in this membership function has real meanings, a

true value for each variable.

The vertical line present in Fig.7 contains three arrows that

indicate the true values for each variable.

Since the red arrow is pointing to zero, it means that the

temperature is not hot. The orange arrow points to a value


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that will be around 0.2 may be interpreted as slightly warm

and the blue arrow that points to a value that will be around

0.8 may be interpreted as quite cold.

This membership function, along with a set of intuitive

rules, such as those shown above, would form a fuzzy

controller capable of controlling the water temperature,

knowing what temperature should be chosen in each moment

of operation of the system.Basically, the fuzzy controllers should be viewed as an

artificial intelligence that makes decisions in real time, on

how to control a system. Gather information about the system

output compares it to a given reference and then decide what

should be the input value of the system to ensure the desired

performance.

V. DESIGN OF A FUZZY CONTROLLERThe fuzzy controller will be design to control (2).

The design of a fuzzy controller is based on three basic

components, which are the following:1Choose the inputs and outputs of the fuzzy controller

2Choose the membership functions, used in the inference

mechanism for the input value, and also the membership

functions used in the construction of the inference of the

controller outputs

3 Choose the fuzzification and defuzzification methods,

the methods for the inference mechanisms and the system

control rules

Our objective is to design a PID fuzzy controller type, so to

the inputs we decide to choose the following inputs:

- error- error change- error accelerationThe error is the difference between the output and the input

therefore is considered to be the proportional input. The

variation of the error is given by the derivative of the error, so

it is considered the derivative input and the acceleration of the

error is given by the integral of the error, being the integral

input of the fuzzy controller.

We will just have one controller output, because the

purpose is only to correct the errors that may exist in the

system in relation to the desired set point for steady state

function.

After choosing the inputs and outputs, is time to choose the

membership functions for the three inputs and also for the

output of the controller.

Fortunately the system (2), that we have to control it

doesnt have such a complexity which requires us to have

very complex membership functions, with many variables.

For that reason we create one type of membership function,

common between the three inputs, relatively simple with only

three variables, which are defined by Negative, Zero and

Positive, being every one of the variables of the triangular

type.

For the output, we decided to create five variables, which

are defined by LargeNegative, SmallNegatve, Zero,

SmallPositive and LargePositive, being every one of the

variables of the singleton type.

Since the system input (reference) is a step of unitary

value, the system error will always be the difference betweenthe value of the step and the output of the system, giving the

possibility to predict what values should be the errors, and

what values should be defined the membership functions and

therefore have been chosen the range of values for the inputs

and output of the system between -2 and 2, already giving

some margin of error for the error get away from the desired

set point.

Fig.8 Membership Functions of the inputs

In Fig.8 we can see membership function, which is equal to

the three inputs to the system previously chosen.

Fig.9 Membership Functions of the output

In Fig.9 we can see the membership function chosen for the

output of the fuzzy controller.

After the choice of Inputs/Outputs system and their

membership functions, and explained the reasons for their

choices, it becomes necessary to discuss the methods of

fuzzification, defuzzification, rules chosen for the system and

methods of the inference mechanism.

The Model chosen for the fuzzy PID controller was the

Mamdani Model, which the Linguistic prototype is:

IF the error is positive AND the error change is zero AND

the error acceleration is zero THEN the output change is

zero.


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After verifying the model used in the fuzzy controller, it is

important check the rules chosen to control the system. It was

only needed nine rules to control (2). The rules we used were

this:

- If (E is Negative) and (CE is Negative) and (CCE is

Negative) then (u is LargeNegative)

- If (E is Negative) and (CE is Zero) and (CCE is Zero)

then (u is SmallNegative)- If (E is Negative) and (CE is Positive) and (CCE is

Positive) then (u is Zero)

- If (E is Zero) and (CE is Negative) and (CCE is Negative)

then (u is SmallNegative)

- If (E is Zero) and (CE is Zero) and (CCE is Zero) then (u

is Zero)

- If (E is Zero) and (CE is Positive) and (CCE is Positive)

then (u is SmallPositive)

- If (E is Positive) and (CE is Negative) and (CCE is

Negative) then (u is Zero)

- If (E is Positive) and (CE is Zero) and (CCE is Zero) then

(u is SmallPositive)

- If (E is Positive) and (CE is Positive) and (CCE is

Positive) then (u is LargePositive)

The surface resultant of these rules is in Fig.10.

Fig.10 Surface of the rules for PID controller

In this case, with the use of only nine rules was possible to

control (2) with a good performance, even better than

previously obtained with the classical PID controller.If this set of rules couldnt bring stability to the system, or

if it was necessary to increase the system performance, we

could from each existing rule create three new rules, which

would be made with the change of CCE for each of nine

existing rules leaving a total of twenty seven rules. The more

rules a system has, more knowledge will have to make

decisions in the control system.

To evaluate the firing level of each rule, is used the

operator AND ( ), where it is utilized the

product operator.

After the evaluation of the firing level of each rule, it

evaluates the output of each rule, where the value obtained

with the product operator determines the value of the output,

which is going to have the value of the product, because we

are using singletons on the output.

If we werent using singletons, the output would be like

shown in Fig.11.

Fig.11 Operator AND with the product method

After realizing how the rules and the fire level of each rule

is obtained, it is important to discuss the method of

aggregation of the results for multiples rules, to obtain a

single output in the fuzzy controller.

In our case is used only one rule at a time to control the

system, but diffuse controllers allow the use of multiple rules

for a given time, allowing a more complete control of the

system.

To obtain the aggregation result, are used operators such as

sum, maximum or probabilistic sum.

Finally, after the fuzzy set obtained with the aggregation

operation, it is necessary to do the defuzzification of the

fuzzy set, to transform the actual value of the fuzzy set in a

real value which can be applied to the control the system.

Various methods are used to do this, the most common

choice falls on the Centre of Gravity, which was the method

we choose to use on this paper, but there are other methodsthat can be used, such as Centre of Sums, Mean of Maximum,

etc.

Fig.12 Defuzzification with Centre of Gravity method, for

discrete and continues time

After having designed the fuzzy controller of the type PID,

we pass to the simulation of the system with the fuzzy

controller, with the help of application Simulinc of the

software MatLab, so we can see the impulse response of the

system.


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Fig.13 Impulse response of the System with fuzzy

controller

As can be seen in Fig.13, the system response is stable,

without any overshoot, reaches the steady state operation in a

short time and does not have any errors in continuous

operation, and so we can consider that the system has a good

performance.

By comparison with the classic PID controller, we can seethat it reaches the steady state operation almost four times

faster and without any overshoot, in other words, has a much

better performance than the classic PID controller. If we

increase the number of the rules as suggested above, the

performance observed in the impulse response would be even

better.

Once designed the fuzzy controller of the type PID, we

thought that it would be interesting to try other configurations

with fuzzy controllers such as PD, PD + integrator, PD + PI,

and found that the configuration that the configuration which

best fit to our system to control was the configuration PD +

integrator.

Fig.14 Impulse response of the System with fuzzy

controller

As we can verify, the system is stable, without any error in

steady state operation and it doesnt have overshoot, as we

have seen already in the fuzzy controller of type PID, but has

a better performance, because it can achieve steady state

operation in almost half the time of the PID fuzzy controller

and almost eight times faster than the classic PID controller.

VI. TESTING THE CONTROLLERS WITH REFERENCE ANDPARAMETRIC VARIATION

For this kind of comparison, we could use all controllers

designed in this paper, but this is unnecessary because the

objective of this paper is to compare the performance of the

conventional PID controllers with the fuzzy controllers.

In this case it is sufficient to compare the classic PIDcontroller with the fuzzy controller that has the better

performance with the system under consideration in (2), in

other words, the fuzzy controller PD + integrator.

Fig.15 Impulse response with a Classic PID controller with

reference change

Fig.16 Impulse response with a PD+I fuzzy controller with

reference change

As we can verify through Fig.15 and Fig.16, when the

system is exposed to variations in the reference, the classic

PID controller can reach the new reference given to the

system, but it takes time to reach steady state operation of the

new reference, besides the existent overshoot on the

transition.

In the opposite side we have the fuzzy controller with thePD + integrator configuration, which when exposed to a

change of reference, have a rapid transition to the steady state

operation of the new reference without the occurrence of

overshoot.

In Fig.17 and Fig.18 we can see the behavior of the

controllers when a system changes during the operation, in

other words, there are parametric variations in the model

adopted for designing the controller.


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Fig.17 Impulse response with a Classic PID controller with

parametric change

Fig.18 Impulse response with a PD+I fuzzy controller with

parametric change

As we can see, when the system change during operation,

in other words, when there are parametric variations of the

model used to design the controllers, in this case there isnt an

optimal controller, but may instead be a compromise between

different purposes. In this case we can see that the classic PID

controller can have a lower overshoot when the system

change, but at the same time takes much time to reach again

the steady state operation, while the fuzzy controller has a

response to the parametric variation with a greater overshoot

than the classical controller, but reaches the steady state

operation more quickly.

In summary, in this case the choice of the best controller

would depend on the target, if the target is to have a faster

response to variations of the system, the fuzzy controller

would be best, but if the aim is to have a low overshoot in

response to parametric variations, the best controller would

be the classic.

VII. CONCLUSION

At the end of this paper we make an overview of what we

can learn from the comparison of classical PID controllers

with fuzzy controllers.

The classical PID controller is the most popular and used in

industry, in general achieve good results in practically all

applications and is also easy to implement and tuning.

Besides the classic PID controller turns out to be standardized

by its mathematical equation, which is applicable in all cases,

and for tuning the process controller for each individual

process, simply changing the gain, and can be guide by the

study of the root locus of each process, in other words, it has

a more rigorous design and not dependent of the designer

knowledge about the operation of the system to be controlled.

The fuzzy logic controllers, usual have good performances

in control of systems, but they depend on the level ofknowledge about the operation of the system by the designer,

in a manner that the controller can built a set of rules and

inference mechanism suitable for each system, to obtain the

best performance possible.

We can say that this paper does not fully exploit all the

capabilities allowed by fuzzy controllers, since the infinite

ability to create multiple input variables, as well as the

membership functions that can be created, to the various

options for inference mechanisms, the possibility of using

multiple rules in one given moment, and even different ways

of calculating a real value from a fuzzy set, to be placed at the

input of the control system.

It is also important to say that there are more types of

controllers than those explored in this paper, and in the future

we can explore the possibility to integrate different

controllers to control the same system.

REFERENCES

M.PASSINO,KEVIN;FUZZYCONTROL

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