research article design of fuzzy fractional pd + i...

Research ArticleDesign of Fuzzy Fractional PD + I Controllers Tuned bya Genetic Algorithm

Isabel S. Jesus and Ramiro S. Barbosa

Department of Electrical Engineering, Institute of Engineering/Polytechnic of Porto (ISEP/IPP), GECAD,Knowledge Engineering and Decision Support Research Center, 4200-072 Porto, Portugal

Correspondence should be addressed to Isabel S. Jesus; [email protected]

Received 28 October 2013; Revised 25 March 2014; Accepted 8 April 2014; Published 7 May 2014

Academic Editor: Jyh-Hong Chou

Copyright © 2014 I. S. Jesus and R. S. Barbosa. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

The fractional-order concepts are a useful tool to describe several physical phenomena, and nowadays they are widely used in thefield of automatic control. A genetic algorithm (GA) is a search process for finding approximate solutions in optimization problems.The GA provides further flexibility and robustness that are unique for signal process. In this paper we consider the development ofan optimal fuzzy fractional PD + I controller in which the parameters are tuned by a GA. The performance of the proposed fuzzyfractional control is illustrated through some application examples.

1. Introduction

Fractional calculus (FC) is a generalization of integrationand differentiation to a noninteger order 𝛼 ∈ 𝐶, with thefundamental operator being

𝑎𝐷𝛼

𝑡, where 𝑎 and 𝑡 are the

limits of the operation [1, 2]. The FC concepts constitutea useful tool to describe several physical phenomena, suchas heat, flow, electricity, magnetism, mechanics, or fluiddynamics. Presently, the FC theory is applied in almostall areas of science and engineering, with its ability beingrecognized in bettering the modelling and control of manydynamical systems. In fact, during the last years FC hasbeen used increasingly to model the constitutive behaviorof materials and physical systems exhibiting hereditary andmemory properties. This is the main advantage of fractional-order derivatives in comparison with classical integer-ordermodels, where these effects are simply neglected.

In this paper we investigate several control strate-gies/structures based on fuzzy fractional-order algorithms.The fractional-order PID controller (PI𝛼D𝛽 controller)involves an integrator of order 𝛼 ∈ R+ and a differentiatorof order 𝛽 ∈ R+. It was demonstrated the good performanceof this type of controller in comparisonwith the conventionalPID algorithms. Recently, there have been a lot of researchesin the application of fuzzy PID control [3–9]. The fuzzymethod offers a systematic procedure to design controllers

for many kinds of systems that often leads to a betterperformance than that of the conventional PID controller. Itis a methodology of intelligent control that mimics humanthinking and reacting by using a multivalent fuzzy logic andelements of artificial intelligence.

It was proved that the use of the fuzzy fractional con-trollers improved the results for many kinds of systems, sinceit gives additional flexibility to the design. In this line ofthought many applications of this type of controllers weredeveloped in the last few years. For example, in [3, 4], theauthors proved the effectiveness of fuzzy fractional PD andPID controllers in terms of their digital implementation androbustness. In [10], an intelligent robust fractional surfacesliding mode control for a nonlinear system is studied. In[6], a fractional-order fuzzy PID controller was proposedand compared to classical PID, fuzzy PID, and even PI𝜆D𝜇controllers. Many other applications can be found in [11–16].

A genetic algorithm (GA) is a search technique based onthe natural selection process. The GA is a particular classof evolutionary algorithms that use techniques inspired byevolutionary biology such as inheritance, mutation, naturalselection, and crossover, established by Darwin’s theory ofevolution [17–19].TheGA is used in the field of robotics, strat-egy planning, nonlinear dynamical systems, data analysis, art,evolving pictures, music, and many others in the real worldapplications [17–20]. The GA provides a unique flexibility

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014, Article ID 676121, 14 pageshttp://dx.doi.org/10.1155/2014/676121

2 Mathematical Problems in Engineering

and robustness for process optimization. Due to this reason,during the last years many control applications used the GAin order to find better results. For example, in [6], the authorsoptimized their system with GA while minimizing severalintegral error indices along with the control signal as theobjective function. Hu et al. [21] developed a methodologyfor the systematic design of fuzzy PID controllers based ongenetic optimization, where they proved that the proposedsystem always provides the best control performance. In[22], the GA was used for tuning the PI controller forload frequency control. Herrera and Lozano showed in [23]the benefits derived from the synergy between evolutionaryalgorithms and fuzzy logic systems.

Bearing these ideas in mind, the paper is organized asfollows. Section 2 gives the fundamentals of fractional-ordercontrol systems. Section 3 presents the control and opti-mization strategies. Section 4 gives some simulations resultsassessing the effectiveness of the proposed methodology.Finally, Section 5 draws the main conclusions.

2. Fractional-Order Control Systems

Fractional-order control systems are characterized by differ-ential equations that have, in the dynamical system and/orin the control algorithm, an integral and/or a derivative offractional order [24, 25]. Due to the fact that these operatorsare defined by irrational continuous transfer functions, inthe Laplace domain, or infinite dimensional discrete transferfunctions, in the 𝑍 domain, we often encounter evaluationproblems in the simulations. Therefore, when analyzingfractional-order systems, we usually adopt continuous ordiscrete integer-order approximations of fractional-orderoperators [26–28]. The following two subsections provide abackground for the remaining of the paper by giving thefundamental aspects of the FC and the discrete integer-orderapproximations of fractional-order operators used in thisstudy.

2.1. Fundamentals of Fractional Calculus. The mathematicaldefinition of a fractional-order derivative and integral (so-called differintegral) has been the subject of several differentapproaches [1, 2]. One commonly used definition for thefractional-order derivative is given by the Riemann-Liouvilledefinition (𝛼 > 0):

𝑎𝐷𝛼

𝑡𝑓 (𝑡) =

1

Γ (𝑛 − 𝛼)

𝑑𝑛

𝑑𝑡𝑛∫

𝑡

𝑎

𝑓 (𝜏)

(𝑡 − 𝜏)𝛼−𝑛+1

𝑑𝜏,

𝑛 − 1 < 𝛼 < 𝑛,

(1)

where 𝑛 is integer, 𝑓(𝑡) is the applied function, and Γ(𝑥) isthe Gamma function of 𝑥. Another widely used definition isgiven by the Grunwald-Letnikov approach (𝛼 ∈ R):

𝑎𝐷𝛼

𝑡𝑓 (𝑡) = lim

ℎ→0

1

ℎ𝛼

[(𝑡−𝑎)/ℎ]

∑

𝑘=0

(−1)𝑘

(

𝛼

𝑘)𝑓 (𝑡 − 𝑘ℎ) , (2a)

(

𝛼

𝑘) =

Γ (𝛼 + 1)

Γ (𝑘 + 1) Γ (𝛼 − 𝑘 + 1)

, (2b)

where ℎ is the time increment and [𝑥] means the integer partof 𝑥.

Another definition is given by the Caputo approach:

𝐶

𝑎𝐷𝛼

𝑡𝑓 (𝑡) =

1

Γ (𝑛 − 𝛼)

∫

𝑡

𝑎

𝑓𝑛

(𝜏)

(𝑡 − 𝜏)𝛼+1−𝑛

𝑑𝜏, 𝑛 − 1 < 𝛼 < 𝑛.

(3)

The “memory” effect of these operators is demonstratedby (1)–(3), where the convolution integral in (1) and (3)and the infinite series in (2a) and (2b) reveal the unlimitedmemory of these operators, ideal for modelling hereditaryand memory properties in physical systems and materials.

An alternative definition to (1)–(3), which reveals usefulfor the analysis of fractional-order control systems, is given bythe Laplace transformmethod. Considering vanishing initialconditions, the fractional differintegration is defined in theLaplace domain, 𝐹(𝑠) = 𝐿{𝑓(𝑡)}, as

𝐿 {𝑎𝐷𝛼

𝑡𝑓 (𝑡)} = 𝑠

𝛼

𝐹 (𝑠) , 𝛼 ∈ R. (4)

The fractional operator can be more easily interpreted inthe frequency domain. In fact, the open-loop Bode diagramsof amplitude and phase of 𝑠𝛼 have correspondingly a slope of20𝛼 dB/dec and a constant phase positioned at 𝛼𝜋/2 rad overthe entire frequency domain.

2.2. Approximations of Fractional-Order Operators. In thisstudy we adopt discrete integer-order approximations to thefundamental element 𝑠𝛼(𝛼 ∈ R) of a fractional-order control(FOC) strategy. The usual approach for obtaining discreteequivalents of continuous operators of type 𝑠

𝛼 adopts theEuler, Tustin, andAl-Alaoui generating functions [26, 29, 30].

It is well known that rational-type approximations fre-quently converge faster than polynomial-type approxima-tions and have a wider domain of convergence in the complexdomain [29]. Thus, using the Euler operator 𝑤(𝑧

−1

) =

(1 − 𝑧−1

)/𝑇𝑐and performing a power series expansion of

[𝑤(𝑧−1

)]

𝛼

= [(1 − 𝑧−1

)/𝑇𝑐]

𝛼 give the discretization formulacorresponding to theGrunwald-LetnikovDefinition (2a) and(2b):

𝐷𝛼

(𝑧−1

) = (

1 − 𝑧−1

𝑇𝑐

)

𝛼

=

∞

∑

𝑘=0

(

1

𝑇𝑐

)

𝛼

(−1)𝑘

(

𝛼

𝑘) 𝑧−𝑘

=

∞

∑

𝑘=0

ℎ𝛼

(𝑘) 𝑧−𝑘

,

(5)

where 𝑇𝑐is the sampling period and ℎ

𝛼

(𝑘) is the impulseresponse sequence.

A rational fraction-type approximation can be obtainedthrough a Pade approximation to the impulse responsesequence ℎ𝛼(𝑘), yielding the discrete transfer function:

𝐻(𝑧−1

) =

𝑏0+ 𝑏1𝑧−1

+ ⋅ ⋅ ⋅ + 𝑏𝑚𝑧−𝑚

1 + 𝑎1𝑧−1

+ ⋅ ⋅ ⋅ + 𝑎𝑛𝑧−𝑛

=

∞

∑

𝑘=0

ℎ (𝑘) 𝑧−𝑘

, (6)

where 𝑚 ≤ 𝑛 and the coefficients 𝑎𝑘and 𝑏𝑘are determined

by fitting the first 𝑚 + 𝑛 + 1 values of ℎ𝛼(𝑘) into the impulse

Mathematical Problems in Engineering 3

R(s) +

−

E(s)

ControllerFuzzy

PD𝛽 + IU(s)

Saturation

𝛿𝛿 1 N(s)

System

G(s)C(s)

Figure 1: Block diagram of the fuzzy control system.

response ℎ(𝑘) of the desired approximation𝐻(𝑧−1

). Thus, weobtain an approximation that matches the desired impulseresponse ℎ𝛼(𝑘) for the first𝑚+𝑛+1 values of 𝑘 [26]. Note thatthe above Pade approximation is obtained by considering theEuler operator but the determination process will be exactlythe same for other types of discretization schemes.

3. Control and Optimization Strategies

3.1. Fractional PID Control. The generalized PID controller,𝐺𝑐(𝑠), has a transfer function of the form [28]

𝐺𝑐(𝑠) =

𝑈 (𝑠)

𝐸 (𝑠)

= 𝐾𝑝+

𝐾𝑖

𝑠𝛼+ 𝐾𝑑𝑠𝛽

, 𝛼, 𝛽 > 0, (7)

where 𝛼 and 𝛽 are the orders of the fractional integra-tor and differentiator, respectively. The parameters 𝐾

𝑝, 𝐾𝑖,

and 𝐾𝑑are correspondingly the proportional, integral, and

derivative gains of the controller. Clearly, taking (𝛼, 𝛽) =

{(1, 1), (1, 0), (0, 1), (0, 0)}we get the classical {PID,PI,PD,P}controllers, respectively [24, 31, 32]. Other PID controllersare possible, namely, PD𝛽 controller, PI𝛼 controller, PID𝛽controller, and so on. The fractional-order controller is moreflexible and gives the possibility of adjusting more carefullythe closed-loop system characteristics [2, 33].

In the time domain the PI𝛼D𝛽 is represented by

𝑢 (𝑡) = 𝐾𝑝𝑒 (𝑡) + 𝐾

𝑖 0𝐷−𝛼

𝑡𝑒 (𝑡) + 𝐾

𝑑 0𝐷𝛽

𝑡𝑒 (𝑡) . (8)

The fractional-order differential operators in (8) areimplemented using the approximations (5) and (6), yieldingthe discrete transfer function:

𝐺𝑐(𝑧) =

𝑈 (𝑧)

𝐸 (𝑧)

= 𝐾𝑝+ 𝐾𝑖𝐻𝑖(𝑧−1

) + 𝐾𝑑𝐻𝑑(𝑧−1

) ,

(9)

where𝐻𝑖(𝑧−1

) and𝐻𝑑(𝑧−1

) are the fraction-type approxima-tions to fractional-order integral and derivative, respectively.

3.2. Fuzzy Fractional PD + I Control. Fuzzy control emergedon the foundations of Zadeh’s fuzzy set theory [3, 4, 9]. Thiskind of control is based on the ability of a human being tofind solutions for particular problematic situations. It is wellknown from our experience that humans have the ability tosimultaneously process a large amount of information andmake effective decisions, although neither input informationnor consequent actions is precisely defined. Through multi-valent fuzzy logic, linguistic expressions in antecedent and

consequent parts of IF-THEN rules describing the operator’sactions can be efficaciously converted into a fully structuredcontrol algorithm.

The fuzzy logic controllers are not dependent on accuratemathematical models, which are one of the most importantadvantages in its use, particularly in applications wheresystems are difficult to model or contain significant nonlin-earities.

In the system of Figure 1, we apply a fuzzy logic control(FLC) for the PD𝛽 actions and the integral of the error isadded to the output in order to find a fuzzy PD𝛽+ I controller[3, 34]. This kind of configuration eliminates the steady stateerror due to the integer integrative action.The block diagramof Figure 2 illustrates the configuration of the proposed fuzzycontroller.

In this controller, the control actions are the error 𝑒,the fractional derivative of 𝑒, and the integral of 𝑒. The 𝑈

represents the controller output. Also, the controller has fourgains to be tuned, 𝐾

𝑒, 𝐾ie, 𝐾ce corresponding to the inputs

and𝐾𝑢to the output.

The control action 𝑈 is generally a nonlinear function oferror 𝐸, fractional change of error CE, and integral of errorIE:

𝑈 (𝑘) = [𝑓 (𝐸,CE) + IE]𝐾𝑢

= [𝑓 (𝐾𝑒𝑒 (𝑘) , 𝐾ce𝐷

𝛽

𝑒 (𝑘)) + 𝐾ie𝐼𝑒 (𝑘)]𝐾𝑢,(10)

where𝐷𝛽 is the discrete fractional derivative implemented asrational fraction approximation (6) using the Euler scheme(5); the integral of error is calculated by rectangular integra-tion:

𝐼 (𝑧−1

) =

𝑇𝑐

1 − 𝑧−1

. (11)

In fact, we can adopt an integral action of fractional order,I𝛼, yielding a fuzzy fractional PD𝛽+I𝛼 controller [3].However,in this work we consider only values of 𝛼 = 1.

To further illustrate the performance of the fuzzy PD𝛽+I asaturation nonlinearity is included in the closed-loop systemof Figure 1 and inserted in series with the output of the fuzzycontroller. The saturation element is defined as

𝑛 (𝑢) = {

𝑢, |𝑢| < 𝛿,

𝛿 sign (𝑢) , |𝑢| ≥ 𝛿,

(12)

where 𝑢 and 𝑛 are, respectively, the input and the output ofthe saturation block and sign(𝑢) is the signum function.

Here we give an emphasis of the proposed FLC presentedin Figure 2. The basic structure for FLC is illustrated inFigure 3 [35].


eKe

D𝛽

I

Kce

Kie

E

CE

IE

� +

+

Kuu

Fuzzy logiccontroller

Figure 2: Fuzzy PD𝛽 + I controller.

Input

Fuzzification

Fuzzy rule base Fuzzy inference

Defuzzification

Output

Figure 3: Structure for fuzzy logic controller.

Table 1: Fuzzy control rules.

CE ENL NM NS ZR PS PM PL

NL NL NL NL NL NM NS ZRNM NL NL NL NM NS ZR PSNS NL NL NM NS ZR PS PMZR NL NM NS ZR PS PM PLPS NM NS ZR PS PM PL PLPM NS ZR PS PM PL PL PLPL ZR PS PM PL PL PL PL

The fuzzy rule base, which reflects the collected knowl-edge about how a particular control problemmust be treated,is one of themain components of a fuzzy controller.The otherparts of the controller perform make up the tasks necessaryfor the controller to be efficient.

For the fuzzy PD𝛽 + I controller illustrated in Figure 2,the rule base can be constructed in the following form (seeTable 1):

if 𝐸 is NM and CE is NS, then V is NL,

where NL, NM, NS, ZR, PS, PM, and PL are linguistic valuesrepresenting “negative large,” “negative medium,” “negative

0 0.5 1

0

0.2

0.4

0.6

0.8

1

Mem

bers

hip

(deg

)

NL NM NS ZR PS PM PL

−0.5−1

Figure 4: Membership functions for 𝐸, CE, and V.

00.5

10

0.51

0

0.5

1

E

CE

v

−0.5

−1

−0.5

−1−0.5

−1

Figure 5: Control surface.

small,” “zero,” “positive small,” “positive medium,” and “pos-itive large,” respectively. 𝐸 is the error, CE is the fractionalderivative of error, and V is the output of the fuzzy PD𝛽controller. The membership functions for the premises andconsequents of the rules are shown in Figure 4.

With two inputs and one output the input-output map-ping of the fuzzy logic controller is described by a nonlinearsurface, as presented in Figure 5.

The fuzzy controller will be adjusted by changing theparameter values of𝐾

𝑒,𝐾ce,𝐾ie, and𝐾

𝑢. The fuzzy inference

mechanism operates by using the product to combine the


conjunctions in the premise of the rules and in the represen-tation of the fuzzy implication. For the defuzzification processwe use the centroid method.

3.3. Genetic Optimization. The evolutionary computing wasintroduced in the 60s by I. Rechenberg, and the GA wasinvented by JohnHollandwho published a book in 1975 aboutthis subject.

These algorithms begin with a set of solutions, rep-resented by chromosomes, called population (𝑃). Initially,a population is generated randomly. Solutions from onepopulation are taken (parents) and used to form a new𝑃.Thisismotivated by the hope that the new𝑃will be better than theold one. Individuals are then selected to form new solutionsaccording to their fitness; therefore, the more suitable theyare the more chances they have to reproduce.This is repeateduntil some condition is satisfied. Figure 6 presents the blockdiagram representative of the methodology used in the GA.

In these algorithms the crossover (𝐶) and the mutation(𝑀) operators are the most important parts. The 𝐶 is arecombination operator that combines subparts of two parentchromosomes to produce offspring that contain some partsof both parents genetic material. The simplest way to do it isto choose some random 𝐶 point, copy everything before thispoint from the first parent, and then to copy everything afterthe 𝐶 point from the other parent. There are other ways tomake𝐶; namely, we can choosemore𝐶 points.Themost usedway of encoding is a binary string; however, there are manyother ways of encoding, such as to encode directly throughreal numbers.

The selection of a better encoding technique depends onthe problem we have to solve. The 𝑀 operation randomlychanges the offspring resulting from 𝐶. This procedureintended to prevent falling of all solutions in the𝑃 into a localoptimum. In case of binary encoding, we can switch a fewrandomly chosen bits from 1 to 0 or from 0 to 1.

Another important concept in GA is the Elitism. TheElitism strategy (ES) was introduced by Kenneth De Jong in1975 and is an addition tomany selectionmethods that forcesthe GA to retain some number of the best individuals at eachgeneration (𝐺). With this tool, such individuals can be lostif they are not selected to reproduce or if they are destroyedby 𝐶 or 𝑀. Many researchers have found that ES improvessignificantly the GA’s performance [17–20].

The advantage of GA is in its parallelism. GA is travellingin a search space using more individuals than other methods.However, GA also has disadvantages, namely, the computa-tional cost, because many times these algorithms are slowerthan other methodologies.

In this work we propose a fuzzy fractional PD𝛽 + I con-troller, where the gains will be tuned through the applicationof a GA, in order to achieve a superior control performanceof the control system of Figure 1. The optimization fitnessfunction corresponds to theminimization of the integral timeabsolute error (ITAE) criterion that measures the responseerror as defined as [20]

𝐽 (𝐾𝑒, 𝐾ce, 𝐾ie, 𝐾𝑢) = ∫

∞

0

𝑡 |𝑟 (𝑡) − 𝑐 (𝑡)| 𝑑𝑡, (13)

PopulationDecoded form

Decoded form

Selection Fitness

Fitness

Replacement Parents Objective function

Genetic operation

Subpopulation

Figure 6: Block diagram of a GA.

where (𝐾𝑒,𝐾ce,𝐾ie,𝐾𝑢) are the PD

𝛽

+ I controller parametersto be optimized.

During the minimization of the fitness function in GA,the chromosomes (potential solutions) that lead to unstableresponses make the error take very high values penalizingthe fitness function. Therefore, these chromosomes are lesslikely to be selected for reproduction or elitism, this way beingdestroyed or lost in the process. This mechanism ensuresthat only the best chromosomes are chosen to produce newpopulation and that the GA converges to an optimal solutionwith a stable closed-loop system.

We can use other integral performance criteria such as theintegral absolute error (IAE), the integral square error (ISE),or the integral time square error (ITSE). In the present studythe ITAE criterion produced good results and is adopted inthe sequel. Furthermore, the ITAE criterion enables us tostudy the influence of time in the error generated by thesystem.

4. Simulation Results

In this section we analyze the closed-loop system of Figure 1with the fuzzy fractional PD𝛽 + I controller of Figure 2. Thesystems used correspond to typical plants [36, 37], namely,a high order process, a high order process with a zero,and a system with a time delay. In all the experiments, thefractional-order derivative 𝐷

𝛽 is implemented using a 4th-order Pade discrete rational transfer function (𝑚 = 𝑛 = 4)

of type (5). It is used as sampling period of 𝑇𝑐

= 0.01 s.The PD𝛽 + I controller is tuned through the optimizationof fitness function corresponding to the minimization ofITAE (13) using a GA. We use 𝛿 = 15. We establish thefollowing values for the GA parameters: population size 𝑃 =

20, crossover probability 𝐶 = 0.8, mutation probability𝑀 = 0.05, and number of generations 𝑁

𝐺= 100. The

gene codification adopts a decimal code. It is important torefer that a reliable execution and analysis of a GA usuallyrequire a large number of simulations to guarantee that


stochastic effects are properly considered [17, 19]. Therefore,the experiments consist in executing the GA several times,for generating a good combination of controller parameters.In this study the GA is repeated 10 times and we get the bestresult, that is, the simulation that leads to the smaller 𝐽. Weset the search space of𝐾

𝑒, 𝐾ce, 𝐾ie, and𝐾

𝑢∈ [0, 5].

In the first case, we compare a fuzzy fractional PD𝛽 + Icontroller (𝛽 = 0.9), with a fuzzy integer PD + I controller(𝛽 = 1). Figure 7 shows the unit step responses of bothcontrollers.The plant system𝐺

1(𝑠) used is represented by the

transfer function:

𝐺1(𝑠) =

1

(𝑠 + 1) (1 + 𝛼𝑠) (1 + 𝛼2𝑠) (1 + 𝛼

3𝑠)

with 𝛼 = 0.5.

(14)

The controller parameters, corresponding to the min-imization of the ITAE index, lead to the values for thefuzzy integer PD + I controller: {𝐾

𝑒, 𝐾ce, 𝐾ie, 𝐾𝑢} ≡

{1.0808, 0.3408, 0.3442, 4.2095}, with 𝐽 = 0.8235, and forthe fuzzy fractional PD𝛽 + I controller to the followingvalues: {𝐾

𝑒, 𝐾ce, 𝐾ie, 𝐾𝑢} ≡ {0.7581, 0.3510, 0.3038, 4.3276},

with 𝐽 = 0.7693. These values lead us to conclude that thefuzzy fractional-order controller produced similar results tointeger one; however, the error 𝐽 is smaller, as can be seenin Figure 8, where it shows the ITAE error as function of𝛽. The graph reveals that for this process we have a lowererror for a fractional value of 𝛽 = 0.9. We verify that thefractional controller is better (in terms of error 𝐽) than theinteger version, only in a narrow region of 0.8 ≤ 𝛽 < 1.Figure 9 illustrates the variation of FLC parameters (𝐾

𝑒, 𝐾ce,

𝐾ie,𝐾𝑢) as function of the order’s derivative𝛽, while Figure 10shows the variation of the transient response parameters,namely, the settling time 𝑡

𝑠, rise time 𝑡

𝑟, peak time 𝑡

𝑝, and

overshoot 𝑜V(%) versus 𝛽, for the closed-loop step response.The variation of FLC parameters and the transient responseparameters reveals a smooth variation with 𝛽.

In a second experiment, we consider a fuzzy PD𝛽 + I (𝛽 =

0.2) controller, for process 𝐺2(𝑠) with a right-half plane zero,

represented by the transfer function:

𝐺2(𝑠) =

1 − 𝛼𝑠

(𝑠 + 1)3

with 𝛼 = 5.0. (15)

Once again, we consider for comparison the correspond-ing integer version (𝛽 = 1). Figure 11 shows the unit stepresponses of both controllers.

The controller parameters, corresponding to the min-imization of the ITAE index, lead to the values for thefuzzy integer controller: {𝐾

𝑒, 𝐾ce, 𝐾ie, 𝐾𝑢} ≡ {0.5393, 0.4647,

0.3439, 0.2486}, with 𝐽 = 75.4509, and for the fuzzyfractional controller: {𝐾

𝑒, 𝐾ce, 𝐾ie, 𝐾𝑢} ≡ {0.6061, 0.0326,

0.3175, 0.2909}, with 𝐽 = 55.9414. These values lead us toconclude that the fuzzy fractional-order controller producedbetter results than the integer ones, since the transientresponse (in particular the settling time and rise time) andthe error 𝐽 are smaller. Figure 12 shows the ITAE error asfunction of 𝛽. The graph reveals that for this process wehave a lower error for 𝛽 = 0.2. Also note that the fractionalcontroller is better (in terms of error 𝐽) than the integer

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

Time (s)

c(t)

PIDPID𝛽

Figure 7: Step responses of closed-loop system with fuzzy PD + Iand PD𝛽 + I (𝛽 = 0.9) controllers.

0 0.2 0.4 0.6 0.8 1

J

101

100

10−1

𝛽

Figure 8: Error 𝐽 versus 𝛽 for 𝐺1(𝑠).

one for 0.1 ≤ 𝛽 < 1. Figure 13 illustrates the variation of FLCparameters (𝐾

𝑒, 𝐾ce, 𝐾ie, 𝐾

𝑢) as function of the order’s

derivative 𝛽, while Figure 14 shows the variation of 𝑡𝑠, 𝑡𝑟, 𝑡𝑝

and 𝑜V(%) versus 𝛽, for the closed-loop step response. In thiscase, the 𝐾ce and the 𝐾ie have the smaller value near the bestcase (i.e., 𝛽 = 0.2) and 𝑡

𝑠, 𝑡𝑟, and 𝑡

𝑝reveal similar values with

the variation of 𝛽.In a third study we consider a fuzzy PD𝛽 + I (𝛽 =

0.9) controller, for process 𝐺3(𝑠), represented by the transfer

function (16), where the time delay is 𝑇 = 1 [s]:

𝐺3(𝑠) =

𝑒−𝑠𝑇

(1 + 𝛼𝑠)2

with 𝛼 = 2.0. (16)

For comparison purposes, we consider the correspondinginteger version (𝛽 = 1). Figure 15 shows the unit stepresponses of both controllers.


0 0.5 1 0 0.5 1

0 0.5 10 0.5 1

101

100

101

100

10−1

100

10−1

100

10−110−2

𝛽 𝛽

𝛽𝛽

Ke

Ku

Kce

Kie

Figure 9: The PID𝛽 parameters (𝐾𝑒, 𝐾ce, 𝐾ie, 𝐾𝑢) versus 𝛽 for 𝐺

1(𝑠).

The controller parameters, corresponding to the min-imization of the ITAE index, lead to the values for thefuzzy integer controller: {𝐾

𝑒, 𝐾ce, 𝐾ie, 𝐾𝑢} ≡ {1.0496,

0.9300, 0.1757, 2.2949}, with 𝐽 = 7.2953, and for thefuzzy fractional controller: {𝐾

𝑒, 𝐾ce, 𝐾ie, 𝐾𝑢} ≡ {0.6112,

1.9142, 0.1731, 1.7278}, with 𝐽 = 6.8251. These values leadus again to remain the previous conclusions drawn for 𝐺

1(𝑠)

and 𝐺2(𝑠), namely, that the fuzzy fractional-order controller

produced better results than the integer ones, since thetransient response (viz., the overshoot and settling time) andthe error 𝐽 are smaller. Figure 16 shows the ITAE error asfunction of 𝛽. The graph reveals that for this process we havea lower error for 𝛽 = 0.9. The fractional controller presentssmaller values of 𝐽 for 0.8 < 𝛽 < 1. Figure 17 illustrates thevariation of FLC parameters (𝐾

𝑒, 𝐾ce, 𝐾ie, 𝐾𝑢) as function of

the order’s derivative𝛽, while Figure 18 shows the variation of𝑡𝑠, 𝑡𝑟, 𝑡𝑝, and 𝑜V(%) versus𝛽 for the closed-loop step response.

The FLC parameters reveal a slight variation with 𝛽, as wellthe transient response parameters.

In conclusion, with the fuzzy fractional PD𝛽+ I controllerwe get the best controller tuning, superior to the performancerevealed by the integer-order scheme.Moreover, we prove the

effectiveness of this control structure when used in systemswith time delay. In fact, systems with time delay are moredifficult to be controlled with the classical methodologies;however, the proposed algorithm reveals to be very effectivein the control of this type of systems.

4.1. Fuzzy Fractional PID Structures. In this subsection westudy the impact of different fuzzy fractional PID structureson system performance. For that, we compare the structure ofFigure 2 (𝑆

1) used in the previous section with the structures

of Figure 19 (𝑆2) and Figure 20 (𝑆

3) applied to process 𝐺

3(𝑠).

TheGAwas executed 10 times, andwe get the result that leadsto the smaller 𝐽.We consider the same fuzzy control rules baseof Table 1 in all the structures (𝑆

1, 𝑆2, and 𝑆

3).

The control action for these new structures is, respec-tively, for (𝑆

2) and (𝑆

3):

𝑈 (𝑘) = 𝐾𝑢V + 𝐾ie𝐼V = 𝐾

𝑢[𝑓 (𝐸, 𝐶𝐸)] + 𝐾ie𝐼 [𝑓 (𝐸, 𝐶𝐸)]

= 𝐾𝑢[𝑓 (𝐾

𝑒𝑒 (𝑘) , 𝐾ce𝐷

𝛽

𝑒 (𝑘))]

+ 𝐾𝑖𝑒𝐼 [𝑓 (𝐾

𝑒𝑒 (𝑘) , 𝐾ce𝐷

𝛽

𝑒 (𝑘))]

(17)


0 0.5 1

0 0.5 10 0.5 1

0 0.5 1

102

101

100

101

100

101

100

101

100

10−1

𝛽 𝛽

𝛽𝛽

t s(s

)

t r(s

)

t p(s

)

ov (%

)

Figure 10: Parameters 𝑡𝑠, 𝑡𝑟, 𝑡𝑝, 𝑜V(%) versus 𝛽 of the step responses of the closed-loop system with 𝐺

1(𝑠) and with a fuzzy PD𝛽 + I controller.

0 10 20 30 40 50

0

0.2

0.4

0.6

0.8

1

1.2

Time (s)

c(t)

PIDPID𝛽

−0.2

−0.4

Figure 11: Step responses of closed-loop system with fuzzy PD + I and PD𝛽 + I (𝛽 = 0.2) controllers.


0 0.2 0.4 0.6 0.8 1

J

102

101

𝛽


0 0.5 10 0.5 1

0 0.5 1 0 0.5 1

100

10−1

100

10−1

100

10−1

101

100

10−1

10−2

𝛽

𝛽 𝛽

𝛽

Ke

Ku

Kce

Kie

Figure 13: The PD𝛽 + I parameters (𝐾𝑒, 𝐾ce, 𝐾ie, 𝐾𝑢) versus 𝛽 for 𝐺

2(𝑠).


0 0.5 1

0 0.5 10 0.5 1

0 0.5 1

101102

101 100

102

101

100

102

101

100

𝛽 𝛽

𝛽 𝛽

t s(s

)

t r(s

)

t p(s

)

ov (%

)

Figure 14: Parameters 𝑡𝑠, 𝑡𝑟, 𝑡𝑝, 𝑜V(%) versus 𝛽 of the step responses of the closed-loop systemwith𝐺

2(𝑠) and with a fuzzy PD𝛽+ I (0 < 𝛽 ≤ 1)

controller.

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

1.2

Time (s)

c(t)

PIDPID𝛽

Figure 15: Step responses of closed-loop system with fuzzy PD + I and PD𝛽 + I (𝛽 = 0.9) controllers.


0 0.2 0.4 0.6 0.8 1

J

102

101

100

𝛽


0 0.5 1

0 0.5 1 0 0.5 1

0 0.5 1

101

100

100

10−1

10−2

101

100

10−1

10−2

101

100

10−1

10−2

𝛽

𝛽

𝛽

𝛽

Ke

Ku

Kce

Kie

Figure 17: The PD𝛽 + I parameters (𝐾𝑒, 𝐾ce, 𝐾ie, 𝐾𝑢) versus 𝛽 for 𝐺

3(𝑠).


0 0.5 1 0 0.5 1

0 0.5 1 0 0.5 1

102

101

100

101

100

102

101

100

101

100

10−1

𝛽

𝛽

𝛽

𝛽

t s(s

)

t r(s

)

t p(s

)

ov (%

)

Figure 18: Parameters 𝑡𝑠, 𝑡𝑟, 𝑡𝑝, 𝑜V(%) versus 𝛽 of the step responses of the closed-loop systemwith𝐺

3(𝑠) and with a fuzzy PD𝛽 + I (0 < 𝛽 ≤ 1)

controller.

e

D𝛽

Ke

Kce

E

CE

�Ku

I KieIE

u

+

+

Fuzzy logiccontroller

Figure 19: Fuzzy fractional PID controller (𝑆2).

𝑈 (𝑘) = 𝐾PDV1 + 𝐾PI𝐼V2, (18)

where

V1= 𝑓(𝐸

1,CE1) = 𝑓(𝐾

𝑒1𝑒(𝑘), 𝐾ce1𝐷

𝛽

𝑒(𝑘)),V2= 𝑓(𝐸

2,CE2) = 𝑓(𝐾

𝑒2𝑒(𝑘), 𝐾ce2𝐷

𝛽

𝑒(𝑘)).

Figure 21 shows the unit step responses of fuzzy fractionalcontroller for 𝑆

1, 𝑆2, and 𝑆

3, for the best 𝐽. Figure 22 depicts

the corresponding error as function of 𝛽. We verify that thelower error occurs for 𝛽 = 0.9 in 𝑆

1, 𝑆2, and 𝑆

3, leading to

errors 𝐽 = 6.8251, 5.8000, and 6.2900, respectively.

Analyzing Figure 21 we observe that the step response isless oscillatory for 𝑆

1, with smaller values of 𝑡

𝑟and 𝑡𝑠than

other structures. Figure 22 shows that all three structurespresent similar results in terms of error 𝐽 as function of order𝛽. Structures 𝑆

1and 𝑆

2give the most similar results while

𝑆3differs mostly on the regions around 𝛽 = 0.1, 0.5, and

0.6.The different structures analyzed lead to similar results

both in the step responses and error 𝐽. In fact, it is interestingto notice that the best 𝛽 = 0.9 for all the structures whenminimizing the ITAE index.


eKe1

D𝛽 Kce1

Ke2

Kce2

E1

CE1

E2

CE2

�1

�2

KPD

I KPIIE

+

+

u

Fuzzy logiccontroller 2

Fuzzy logiccontroller 1

Figure 20: Fuzzy fractional PD + PI controller (𝑆3).

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

Time (s)

c(t)

PIDS1PIDS2PIDS3

Figure 21: Step responses of closed-loop system with fuzzy frac-tional PID structures 𝑆

1, 𝑆2, and 𝑆

3for 𝐺3(𝑠).

Based on this analysis we verify that the proposed PD𝛽+ Icontroller (𝑆

1) gives comparable and, in some situations, even

better results than other two structures (𝑆2and 𝑆3). Moreover,

the adopted structure is simpler and easier to analyze andimplement. These facts justify the choice of the structure 𝑆

1

used in this study.

5. Conclusion

This paper presented the fundamental aspects of applicationof the FC theory in fuzzy control systems. In this line ofthought, several typical plants were studied. The dynamics ofthe closed-loop systems were analyzed in the perspective ofFC, with the use of a fuzzy PD𝛽 + I controller in which theparameters were tuned through a GA algorithm.

In general, the control strategies presented give betterresults than those obtained with conventional integer controlstructures, showing their effectiveness in the control ofnonlinear systems.

0 0.2 0.4 0.6 0.8 1

J

102

101

100

𝛽

S1S2S3

Figure 22: Error 𝐽 versus 𝛽 with the structures 𝑆1, 𝑆2, and 𝑆

3for

𝐺3(𝑠).

However, much research is still needed to understandtheir effective application in systems with larger time delays,and in the presence of noise and different actuator saturationlevels.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

This work is supported by FEDER Funds through thePrograma Operacional Factores de Competitividade—COMPETE program and by National Funds through FCTFundacao para a Ciencia e a Tecnologia.


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