Tuning Analog PID Controllers by Multi-Objective Genetic

Algorithms with Fuzzy Aggregation

P. H. G. Coelho, J. F. M. Amaral, Y. C. Bacelar, E. N. Rocha, M. Bentes and T. S. Souza

State Univ. of Rio de Janeiro, FEN/DETEL, R. S. Francisco Xavier,524/Sala 5001E, Maracanã, RJ, 20550-900, Brazil

thaynans.souza@gmail.com

Keywords: PID Tuning, Fuzzy Systems, Genetic Algorithms, Artificial Intelligence Applications.

Abstract: This paper deals with a procedure for adjusting the gains of a Proportional-Integral-Derivative (PID)

controller. Multi-objective genetic algorithms with fuzzy aggregation are used for tuning this controller. To

that end, the component values of a known topology of analog PID controller circuit are evolved by a genetic

algorithm to yield acceptable performance specifications. A fuzzy aggregator allows multi-objective

evaluation for the genetic algorithm. Three objectives regarding the PID reference input signal specifications

were considered: overshoot, rise time and settling time. Minimizing these objectives approximates the PID

controller output to the reference signal and leads the genetic algorithm to find the best controller gains. A

case study is presented to illustrate the procedure.

1 INTRODUCTION

Computational intelligence is a set of computational

methodologies and approaches that seek, through

techniques inspired by nature, the development of

intelligent systems that mimic human aspects, such as

learning, perception, reasoning, evolution and

adaptation. Due to the good results obtained with the

use of different techniques involved in the field of

computational intelligence, the number of research

related has grown even more in recent years.

Furthermore, the area of intelligent systems is quite

broad and covers several applications (Figueiredo et

al., 2014) (Luca et al., 2015) (Ignatiev et al.,2017),

(Ghildiyal et al.,2019).

One of the justifications for research in the area

of intelligent systems, especially in knowledge

discovery, data mining, and machine learning, is the

great complexity of modeling some systems and the

huge volume of digital data existing today that are

many times above the human analysis capability

(Coello Coello, 2013). In this way, the development

of these models can be done automatically through

different approaches such as: artificial neural

networks, Bayesian methods, graphical models and

decision trees, or even through systems with more

symbolic approaches, which, in addition to the ability

to express the knowledge in a more comprehensible

way, they allow the introduction of specialist

knowledge, such as fuzzy systems. Fuzzy systems are

based on fuzzy logic and are widely used, especially

in decision support models and control systems. In

addition, there are several related applications in the

literature, such as, for example, in the area of health

and the study of human locomotion, in speech signal

processing, in the recognition of information and

emotions, in economics and in routing systems (Luca

et al., 2015). Its characteristic of expressing human

inference behavior enables a high level of

understanding, with interpretability being a strong

point of fuzzy systems.

Among the points usually addressed in the area

of intelligent systems, an important point is

optimization, which consists of finding the best

solution for a given problem. At this point,

evolutionary algorithms are a commonly used

computational intelligence technique due to their

great search capability. Optimization in evolutionary

algorithms consists of trying several solutions and

using the information obtained in this process in order

to find increasingly better solutions.

Initially, the great concentration of efforts in the

area of optimization consisted in understanding,

developing and applying methods for the

optimization of a single objective function. However,

most real optimization problems involve multiple

objectives and the idea of optimizing each objective

Coelho, P., Amaral, J., Bacelar, Y., Rocha, E., Bentes, M. and Souza, T.

Tuning Analog PID Controllers by Multi-Objective Genetic Algorithms with Fuzzy Aggregation.

DOI: 10.5220/0011976900003467

In Proceedings of the 25th International Conference on Enterprise Information Systems (ICEIS 2023) - Volume 1, pages 563-571

ISBN: 978-989-758-648-4; ISSN: 2184-4992

 2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)

563

in isolation cannot be applied. Each objective has its

degree of importance and many times the objectives

are conflicting with each other (Ajith et al., 2005). In

everyday situations it is common to find contexts that

have different objectives. For example, the process

for deciding to buy a car takes into account its size,

fuel consumption and price. Another example is the

job search where several points are considered to

make an appropriate choice, such as starting salary,

location and associated opportunities. In an industrial

environment, generally, the aim is to maximize the

quality of a product while minimizing its cost.

In this article, Genetic Algorithms with fuzzy

aggregator for multi-objective optimization are

applied as a search/optimization technique for the

three gains associated with the traditional classical

PID controller: Kp (proportional gain), Ki (integral

gain and Kd (differential gain). Due to its widespread

use in industry, the tuning of classic PID controllers

is a topic of current research and several works have

appeared (Pan et al., 2018) (Wang et al., 2021),

including the application of Genetic Algorithms

(Zhang et al., 2021) (Pu et al., 2020). The main

objective of this article is to present a method capable

of performing the tuning of an analog PID controller,

having as a starting point the desired step response for

the global closed-loop system using as objectives

specifications with respect to the controller reference

signal. Procedure details are described in section 2.2.

Even considering that more sophisticated controllers,

based on intelligent techniques, can be used in

industrial controls, a great amount of the industrial

controllers in use today use PID control strategies.

Therefore, it is evident the importance of an approach

that enables a good tuning of the PID controllers.

This paper is organized in four sections. The second

section describes the basics of a PID controller and

the structure of the evolutionary environment used for

tuning the PID controller. Section three discusses an

example and results in connection with the

evolutionary analog circuits. Finally, section four

ends the paper with the conclusions.

2 BASIC FOUNDATIONS

2.1 PID Controllers

Figure 1 shows the block diagram of a closed-loop

system with a PID controller in the direct path, which

is the typical connection. The system's output should

get as closely as manageable to the setpoint, i.e., the

reference signal. The PID controller is specified by

three gains, as shown in Figure 2.

In the frequency domain, the relation between the

PID controller input E, i.e., error signal, and the

output U, which is the input to the plant, can be

described by the following transfer function:

(1)

The closed-loop transfer function G

(s) is given by:

(2)

Figure 1: PID control of a plant.

Figure 2: Structure of a PID controller.

The usual tuning of a PID controller involves

selecting gains K

, K

and K

so that performance

specifications are satisfied. By using Ziegler-

Nichols's method for PID tuning (Ogata, 2010) those

gains are obtained through experiments with the

process under control. The step response and the

value of K

which results in marginal stability are

used as achieving points for obtaining gain values that

guarantee an adequate behaviour. Finer adjustments

to the gains may also be conducted which are not an

easy task. It should be noted that the Ziegler-Nichols

method is not applicable to all plants.

2.2 The Multi-Objective Environment

For tuning the PID controller, a procedure is used to

do the evolution of component values of a known PID

analog electronic circuit topology, based on a genetic

algorithm and using a fuzzy system to evaluate

multiple objectives. The traditional fitness

assessment of genetic algorithms is changed, so that

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a fuzzy system is effectively responsible for the

assessment, thus being able to aggregate the different

objectives of the electronic design and generating a

fitness value for each circuit in the population. The

method was used for generating membership

functions (Coelho et al, 2022).

One of the most important advantages of fuzzy

systems is interpretability. This feature makes it

possible to insert preferences and adapt the system to

different situations using a natural and easy-to-

understand language. In this way, the evolutionary

environment presents a simpler and more

interpretable way of inserting preferences and

specifications, as it uses a fuzzy system. Such

specifications are inserted before the evolution of the

circuit, ensuring that it is guided in the desired

direction, preventing the designer from having to

choose the most appropriate solution at the end of the

process. The possibility of incorporating conflicting

inputs, but resulting in a single output that aims to

meet both, is also a strong point that allows its use in

solving problems with multiple objectives.

An implementation based solely on simulation of

circuit models was selected, providing a flexible

environment for case studies and enabling future

applications. In this way, a method for evaluation

through fuzzy systems has become attractive for the

evolution of electronic circuits. The search capability

of genetic algorithms motivated the choice of this

intelligent technique as a basis for use in this work. A

genetic algorithm was developed capable of obtaining

a solution, that is, the developed circuit, according to

preferences established according to the different

objectives of the problem, and, for this, a fuzzy

aggregation system is used. Comparing the model with

the algorithms that use the Pareto concept, this fact is

of great importance because it prevents several

solutions from being presented for later selection of the

best among them by the designer at the end of the

process.

The methodology used in the present work allows

the evolution of electronic circuits with characteristics

to be optimized, focusing on the adjustment of the

values of the components of pre-defined PID

topologies and whose model is available or can be

built. Basically, an evolutionary algorithm is used to

search for the best circuit that meets the objectives. The

evolutionary algorithm used is a genetic algorithm

based on GAOT (Genetic Algorithm Optimization

Toolbox) (Houck et al., 1996) and implemented in

Matlab. For the simulations, mathematical models of

the circuits were used. The genetic algorithm used in

the work follows the model presented in Figure 3. The

algorithm starts with a population normally generated

randomly, but which can also be generated from a seed

with potentially good solutions obtained from other

methods. The traditional fitness assessment is

performed from a fitness function defined by the

designer.

Figure 3: Basic structure with genetic algorithm and fuzzy

aggregation.

Such a function generates a scalar number for each

evaluated individual, which corresponds to the

individual's aptitude in relation to the objective

established by the defined function. In this paper, the

evaluation is performed by a Fuzzy Inference System

(FIS), called fuzzy aggregation. The fuzzy aggregator

system makes it possible to evaluate all objectives

simultaneously, integrating the user's preferences and

specifications in relation to each objective and each

situation, in a natural way. Figure 4 shows the

proposed evaluation model.

Figure 4: Fitness evaluation model with aggregator system

A general model for aggregating two objectives,

as an example,was developed, which can be used as a

basis for application to any multi-objective problem.

The model has five triangular membership functions

uniformly distributed within the range from 0 to 1 for

the inputs, corresponding to the variation limits of

each input that must be normalized to facilitate and

generalize the application, as shown in Figure 5.

Tuning Analog PID Controllers by Multi-Objective Genetic Algorithms with Fuzzy Aggregation

565

Figure 5: Typical membership functions for inputs.

The defuzzified output of the fuzzy system

represents the general fitness assessment of the

individual being evaluated. For the membership

functions of the output, the format shown in Figure 6

is used as standard, consisting of five membership

functions.

Figure 6: Typical membership functions for the output.

The fuzzy aggregator system is of the Mamdani

type, characterized by being simpler and more

interpretable than TSK-type fuzzy systems and all

rules have the same degree of importance. The rules

of the fuzzy aggregator system are designed to meet

the problem specifications considering each of the

objectives. To exemplify the process of creating rules,

Table 1 shows basic rules for minimizing two

objectives without preference between their

minimization, that is, the minimization of both is

sought equally. Thus, when the entries correspond to

a Very Low value, they generate a Very Good

aptitude assessment. Likewise, entries with a Very

High value have a Very Bad aptitude rating.

Table 1: Base model for minimization rules.

In case where it is desired to prioritize the

minimization of one objective in relation to the other,

the rules must be modified to meet this preference.

Likewise, if the problem involves maximization, the

same rules can be used by inverting only the linguistic

terms of the antecedents, or the designer can create a

new set of rules. The operators used in the system are

the minimum and maximum operators and

defuzzification is performed using the center of

gravity method. After the evaluation of all the

individuals of the population of the current

generation, the genetic algorithm continues the

evolution process in the traditional way, until the

evaluation of the next generation, where the

evaluation process through the fuzzy aggregator

system is executed again for all the individuals, until

the stopping criterion is reached. To carry out the

evolution of circuits with multiple objectives and the

fuzzy aggregator, the project must be carried out in a

simulated environment. Figure 7 shows a block

diagram of the proposal, illustrating in general the

interconnections between the components used.

Figure 7: Fundamental used structure.

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An implementation based purely on simulation of

circuit models was chosen, providing a flexible

environment for case studies and enabling future

applications. Evolutions of analog electronic circuits

in different application areas can be evaluated

through computer simulations.

3 CASE STUDY

Control systems are needed in many fields of activity.

Obtaining a stable process implies more efficient

results, better quality products, reduced reprocessing,

raw material savings, among other highly important

factors, whether in an industry, laboratory or any

environment that demands efficient control. Classic

PID controllers (Proportional Integral Derivative)

have general applicability in most control systems

and correspond to most industrial controllers. In this

way, a fine adjustment of their control parameters is

essential for a stable process.

Since the appearance of the first tuning method

for controllers, proposed by Ziegler & Nichols

(Ziegler et al., 1942), several PID tuning techniques

have been proposed in the literature, among them,

intelligent control techniques, such as fuzzy logic,

neural networks and genetic algorithms (Amaral et

al., 2001), (Zhou, 2022), (Ding et al., 2022),

(Lakmesari et al., 2022). Based on this need, a case

study applied to the tuning of analog PID controller

was developed in this work in order to obtain an

adequate control performance. For this, the search

technique of a genetic algorithm is used to find the

best controller gains, that is, the proportional, integral

and derivative gains (Kp, Ki and Kd). Multi-objective

optimization is applied to this problem in order to

obtain the best system according to each project need.

The first step in designing control systems is to obtain

a mathematical model of the system and from that it

is possible to analyze its performance.

In the analysis of the control system, input signals

are used as a reference to allow a performance

comparison based on certain specifications. Among

the main specifications considered in the time

domain, the overshoot or maximum overshoot value

(Mp), the rise time (tr), the settling time (ts) and the

delay time (td) stand out. Figure 8 shows these four

parameters. The overshoot corresponds to the

maximum point obtained beyond the reference signal.

The rise time is the time required for the output signal

to vary from 10 to 90% of the final value. Settling

time corresponds to the time taken for the value to

settle within a range (ess), usually 2% or 5%, of the

final value and the settling time delay is the time taken

for the signal to reach 50% of the final value. In this

work, it was decided to analyze three objectives: the

overshoot, the rise time and the settling time. Thus,

the implemented fuzzy aggregator has three inputs

and one output. The membership functions used for

evaluation were created from the required

specifications. For example, the overshoot is

measured in percentage, so a scale from 0 to 100%

was used as shown in Figure 9. For values above 45%

the overshoot is considered Very High and above

55% is no longer acceptable. Values below 12% are

considered ideal and therefore characterize the

linguistic term Low. Values in intermediate ranges

are considered Medium or High.

Figure 8: Control system step response.

Figure 9: Membership functions of input variable

Overshoot.

For the rise time, it was considered that a time above

1 second is Very High, as well as a time below 0.4

seconds is Low and desirable, as shown in Figure 10.

Tuning Analog PID Controllers by Multi-Objective Genetic Algorithms with Fuzzy Aggregation

567

Figure 10: Membership functions of input variable rise

time.

The settling time was represented by three sets:

Low, Medium and High. The Medium set was

centered on 2 seconds, the Low for values up to 1.5

seconds and High above 2.5 seconds, as shown in

Figure 11.

Figure 11: Membership functions of input variable settling

time.

The membership functions for the output are the

same as those illustrated in figure 6, which are the

typical ones considered in section 2.2. The rules were

created in order to minimize the three objectives. A

High value in any of the objectives is considered Bad

and likewise a Very High value is considered Very

Bad. The 11 rules created for the fuzzy aggregator

applied to control systems are shown in Table 2.

Table 2: Fuzzy aggregator rules for control systems.

rshoot Rise Time Settling

Time

Fitness

Low Low -

Very Good

Low Medium - Good

Medium Low - Good

Medium Medium - Medium

High - - Bad

Very High

- - Very Bad

- High - Bad

Very High

- Very Bad

- - Low

Very Good

- - Medium Good

- - High Very Bad

The electronic implementation for the topology of the

analog PID controllers used in this work can be found

in (Ogata, 2010) and is illustrated in Figure 12.

Figure 12: Base Circuit Topology for Analog PID

Controllers.

The transfer function of this circuit is given by:

𝐸



𝑠

𝐸



𝑠



𝑅



𝑅



𝑅



𝑅



𝑅



𝐶



𝑠  1𝑅



𝐶



𝑠1

𝑅



𝐶



𝑠

(3)

The evolution is performed considering the

chromosome representing the six components used to

calculate the transfer function (3). The chromosome

is illustrated in Figure 13. From these parameters, it

is possible to calculate Kp, Ki and Kd, as shown in

equations 4, 5 and 6.

Figure 13: Chromosome for the Evolution of PID

Controllers.

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𝐾





𝑅



𝑅



𝐶



 𝑅



𝐶





𝑅



𝑅



𝐶



(4)

𝐾





𝑅



𝑅



𝑅



𝐶



(5)

𝐾





𝑅



𝑅



𝐶



𝑅



(6)

The search interval used is between 0 and 100kΩ for

resistors and from 1 kpF to 100 µF for capacitors. The

parameters used to configure the evolution of the

genetic algorithm are in Table 3.

Table 3: Fuzzy aggregator rules for control systems.

Parameter Value

Number of Generations 100

Number of individuals

in the population

100

Crossing Rate 70 %

Mutation Rate 1 %

For the analyzed plant, comparisons were made with

the results of a genetic algorithm with traditional

evaluation and with results of traditional techniques

for parameter tuning. A 2nd order plant (7) was used

as a case study (Ogata, 2010).

(7)

Figure 14 illustrates the control system implemented

in Simulink, with the application of a unit step at the

input and it is desirable that the controller obtain a

response as close as possible to this applied input

signal.

Figure 14: Block diagram of the 2nd order control system.

In (Ogata, 2010) an analytical compensator is

developed for this system whose transfer function is:

(8)

In this work, a single objective genetic algorithm was

also implemented in order to minimize the RMSE and

the multi-objective algorithm using the fuzzy

aggregator to minimize the overshoot, the rise time

and the settling time. The analytical compensator

presented in (8) was used for comparison with the

PID controllers obtained by the genetic algorithms.

Table 4 presents the gain values found by the two

genetic algorithms.

Table 4: Comparison of gains for 2nd order plant.

PID Gains

Mono-objective

G.A.

3 Objective

G.A. with

Fuzzy

ato

Kp 100 99.9996

Ki 0.0001 0.2436

4.7514 8.3985

Table 5 presents the values for the overshoot, the rise

time and the settling time that were obtained by the

three analyzed methods.

Table 5: Evaluation parameters for 2nd order plant.

Parameters Analytical

Comp.

Mono-

objective

G.A.

Objective

G.A. with

Fuzzy

ato

Overshoot

21.1612

17.1186

0.4982

Rise Time

(s)

0.3159

0.0806

0.1342

Settling T.

(s)

3.4010

0.4041

0.2123

Figure 15 presents the response obtained as a function

of time in a 15-second simulation carried out in

Simulink with the application of a unit step at the

input. The three analyzed methods are shown in the

figure.

Figure 15: Response to a unit step by the three analyzed

techniques.

Tuning Analog PID Controllers by Multi-Objective Genetic Algorithms with Fuzzy Aggregation

569

Through Figure 15 and the values presented in Table

5, it can be seen that the response obtained by the GA

with fuzzy aggregation obtained the lowest overshoot

and settling time. The rise time achieved by the single

objective GA was the lowest, but the overshoot was

much greater than that obtained by the fuzzy

aggregator, which is not desirable. The analytical

compensator obtained higher values in the three

analyzed parameters. Thus, the obtained results show

that the fuzzy aggregation method was able to

minimize the three parameters adequately and

satisfactorily, obtaining good results compared to the

other controllers. The evolved circuit is shown in

Figure 16.

Figure 16: Evolved analog PID controller.

4 CONCLUSIONS

In this work, an evolutionary model was used for the

development of a PID analog electronic circuit, which

uses a method for evaluation that considers more than

one objective and uses, for that, a process of

aggregation of objectives through a fuzzy system.

This method was called fuzzy aggregator and was

applied in the evaluation process of genetic

algorithms, modifying the traditional method of these

algorithms and including, in this way, the feature of

multi-objective evaluation to such evolutionary

algorithms for obtaining the gains of a PID controller.

The obtained results show that the fuzzy aggregation

method managed to minimize the three parameters

adequately. Compared to the other methods, it

obtained the lowest overshoot and settling time. The

shortest rise time was obtained by the single objective

AG, but it was very close to the time obtained by the

fuzzy aggregator. The analytical compensator method

obtained the highest values for the three analyzed

parameters. In this way, it is concluded that the fuzzy

aggregation method was able to obtain good values

for the gains of a PID controller, generating an

adequate control system.

Future work will include comparisons with other PID

tuning methods, when applicable, and also

investigations with more complex plants.

ACKNOWLEDGEMENTS

This study was financed in part by the Coordenação

de Aperfeiçoamento de Pessoal de Nível Superior –

Brasil (CAPES) – Finance Code 001, and FAPERJ.

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