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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; isj@isep.ipp.pt
Received 28 October 2013; Revised 25 March 2014; Accepted 8 April 2014; Published 7 May 2014
Academic Editor: Jyh-Hong Chou
Copyright ยฉ 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 ๐ผ โ ๐ถ, with thefundamental operator being
๐๐ท๐ผ
๐ก, where ๐ and ๐ก 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 (PI๐ผD๐ฝ controller)involves an integrator of order ๐ผ โ R+ and a differentiatorof order ๐ฝ โ 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 [3โ9]. 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 PI๐D๐controllers. Many other applications can be found in [11โ16].
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 Darwinโs theory ofevolution [17โ19].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 [17โ20]. 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 ๐ domain, we often encounter evaluationproblems in the simulations. Therefore, when analyzingfractional-order systems, we usually adopt continuous ordiscrete integer-order approximations of fractional-orderoperators [26โ28]. 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 (๐ผ > 0):
๐๐ท๐ผ
๐ก๐ (๐ก) =
1
ฮ (๐ โ ๐ผ)
๐๐
๐๐ก๐โซ
๐ก
๐
๐ (๐)
(๐ก โ ๐)๐ผโ๐+1
๐๐,
๐ โ 1 < ๐ผ < ๐,
(1)
where ๐ is integer, ๐(๐ก) is the applied function, and ฮ(๐ฅ) isthe Gamma function of ๐ฅ. Another widely used definition isgiven by the Grunwald-Letnikov approach (๐ผ โ R):
๐๐ท๐ผ
๐ก๐ (๐ก) = lim
โโ0
1
โ๐ผ
[(๐กโ๐)/โ]
โ
๐=0
(โ1)๐
(
๐ผ
๐)๐ (๐ก โ ๐โ) , (2a)
(
๐ผ
๐) =
ฮ (๐ผ + 1)
ฮ (๐ + 1) ฮ (๐ผ โ ๐ + 1)
, (2b)
where โ is the time increment and [๐ฅ] means the integer partof ๐ฅ.
Another definition is given by the Caputo approach:
๐ถ
๐๐ท๐ผ
๐ก๐ (๐ก) =
1
ฮ (๐ โ ๐ผ)
โซ
๐ก
๐
๐๐
(๐)
(๐ก โ ๐)๐ผ+1โ๐
๐๐, ๐ โ 1 < ๐ผ < ๐.
(3)
The โmemoryโ effect of these operators is demonstratedby (1)โ(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)โ(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, ๐น(๐ ) = ๐ฟ{๐(๐ก)}, as
๐ฟ {๐๐ท๐ผ
๐ก๐ (๐ก)} = ๐
๐ผ
๐น (๐ ) , ๐ผ โ R. (4)
The fractional operator can be more easily interpreted inthe frequency domain. In fact, the open-loop Bode diagramsof amplitude and phase of ๐ ๐ผ have correspondingly a slope of20๐ผ dB/dec and a constant phase positioned at ๐ผ๐/2 rad overthe entire frequency domain.
2.2. Approximations of Fractional-Order Operators. In thisstudy we adopt discrete integer-order approximations to thefundamental element ๐ ๐ผ(๐ผ โ R) of a fractional-order control(FOC) strategy. The usual approach for obtaining discreteequivalents of continuous operators of type ๐
๐ผ 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 ๐ค(๐ง
โ1
) =
(1 โ ๐งโ1
)/๐๐and performing a power series expansion of
[๐ค(๐งโ1
)]
๐ผ
= [(1 โ ๐งโ1
)/๐๐]
๐ผ give the discretization formulacorresponding to theGrunwald-LetnikovDefinition (2a) and(2b):
๐ท๐ผ
(๐งโ1
) = (
1 โ ๐งโ1
๐๐
)
๐ผ
=
โ
โ
๐=0
(
1
๐๐
)
๐ผ
(โ1)๐
(
๐ผ
๐) ๐งโ๐
=
โ
โ
๐=0
โ๐ผ
(๐) ๐งโ๐
,
(5)
where ๐๐is the sampling period and โ
๐ผ
(๐) is the impulseresponse sequence.
A rational fraction-type approximation can be obtainedthrough a Pade approximation to the impulse responsesequence โ๐ผ(๐), yielding the discrete transfer function:
๐ป(๐งโ1
) =
๐0+ ๐1๐งโ1
+ โ โ โ + ๐๐๐งโ๐
1 + ๐1๐งโ1
+ โ โ โ + ๐๐๐งโ๐
=
โ
โ
๐=0
โ (๐) ๐งโ๐
, (6)
where ๐ โค ๐ and the coefficients ๐๐and ๐๐are determined
by fitting the first ๐ + ๐ + 1 values of โ๐ผ(๐) into the impulse
Mathematical Problems in Engineering 3
R(s) +
โ
E(s)
ControllerFuzzy
PD๐ฝ + IU(s)
Saturation
๐ฟ๐ฟ 1 N(s)
System
G(s)C(s)
Figure 1: Block diagram of the fuzzy control system.
response โ(๐) of the desired approximation๐ป(๐งโ1
). Thus, weobtain an approximation that matches the desired impulseresponse โ๐ผ(๐) for the first๐+๐+1 values of ๐ [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,๐บ๐(๐ ), has a transfer function of the form [28]
๐บ๐(๐ ) =
๐ (๐ )
๐ธ (๐ )
= ๐พ๐+
๐พ๐
๐ ๐ผ+ ๐พ๐๐ ๐ฝ
, ๐ผ, ๐ฝ > 0, (7)
where ๐ผ and ๐ฝ are the orders of the fractional integra-tor and differentiator, respectively. The parameters ๐พ
๐, ๐พ๐,
and ๐พ๐are correspondingly the proportional, integral, and
derivative gains of the controller. Clearly, taking (๐ผ, ๐ฝ) =
{(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, PD๐ฝ controller, PI๐ผ controller, PID๐ฝcontroller, 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 PI๐ผD๐ฝ is represented by
๐ข (๐ก) = ๐พ๐๐ (๐ก) + ๐พ
๐ 0๐ทโ๐ผ
๐ก๐ (๐ก) + ๐พ
๐ 0๐ท๐ฝ
๐ก๐ (๐ก) . (8)
The fractional-order differential operators in (8) areimplemented using the approximations (5) and (6), yieldingthe discrete transfer function:
๐บ๐(๐ง) =
๐ (๐ง)
๐ธ (๐ง)
= ๐พ๐+ ๐พ๐๐ป๐(๐งโ1
) + ๐พ๐๐ป๐(๐งโ1
) ,
(9)
where๐ป๐(๐งโ1
) and๐ป๐(๐งโ1
) 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 Zadehโs 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 operatorโsactions 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 PD๐ฝ actions and the integral of the error isadded to the output in order to find a fuzzy PD๐ฝ+ 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 ๐,the fractional derivative of ๐, and the integral of ๐. The ๐
represents the controller output. Also, the controller has fourgains to be tuned, ๐พ
๐, ๐พie, ๐พce corresponding to the inputs
and๐พ๐ขto the output.
The control action ๐ is generally a nonlinear function oferror ๐ธ, fractional change of error CE, and integral of errorIE:
๐ (๐) = [๐ (๐ธ,CE) + IE]๐พ๐ข
= [๐ (๐พ๐๐ (๐) , ๐พce๐ท
๐ฝ
๐ (๐)) + ๐พie๐ผ๐ (๐)]๐พ๐ข,(10)
where๐ท๐ฝ 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:
๐ผ (๐งโ1
) =
๐๐
1 โ ๐งโ1
. (11)
In fact, we can adopt an integral action of fractional order,I๐ผ, yielding a fuzzy fractional PD๐ฝ+I๐ผ controller [3].However,in this work we consider only values of ๐ผ = 1.
To further illustrate the performance of the fuzzy PD๐ฝ+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
๐ (๐ข) = {
๐ข, |๐ข| < ๐ฟ,
๐ฟ sign (๐ข) , |๐ข| โฅ ๐ฟ,
(12)
where ๐ข and ๐ are, respectively, the input and the output ofthe saturation block and sign(๐ข) 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].
4 Mathematical Problems in Engineering
eKe
D๐ฝ
I
Kce
Kie
E
CE
IE
๏ฟฝ +
+
Kuu
Fuzzy logiccontroller
Figure 2: Fuzzy PD๐ฝ + 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 PD๐ฝ + I controller illustrated in Figure 2,the rule base can be constructed in the following form (seeTable 1):
if ๐ธ is NM and CE is NS, then V is NL,
where NL, NM, NS, ZR, PS, PM, and PL are linguistic valuesrepresenting โnegative large,โ โnegative medium,โ โnegative
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
โ0.5โ1
Figure 4: Membership functions for ๐ธ, CE, and V.
00.5
10
0.51
0
0.5
1
E
CE
v
โ0.5
โ1
โ0.5
โ1โ0.5
โ1
Figure 5: Control surface.
small,โ โzero,โ โpositive small,โ โpositive medium,โ and โpos-itive large,โ respectively. ๐ธ is the error, CE is the fractionalderivative of error, and V is the output of the fuzzy PD๐ฝcontroller. 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 of๐พ
๐,๐พce,๐พie, and๐พ
๐ข. The fuzzy inference
mechanism operates by using the product to combine the
Mathematical Problems in Engineering 5
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 (๐). Initially,a population is generated randomly. Solutions from onepopulation are taken (parents) and used to form a new๐.Thisismotivated by the hope that the new๐will 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 (๐ถ) and the mutation(๐) operators are the most important parts. The ๐ถ 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 ๐ถ point, copy everything before thispoint from the first parent, and then to copy everything afterthe ๐ถ point from the other parent. There are other ways tomake๐ถ; namely, we can choosemore๐ถ 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 ๐ operation randomlychanges the offspring resulting from ๐ถ. This procedureintended to prevent falling of all solutions in the๐ 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 (๐บ). With this tool, such individuals can be lostif they are not selected to reproduce or if they are destroyedby ๐ถ or ๐. Many researchers have found that ES improvessignificantly the GAโs performance [17โ20].
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 PD๐ฝ + 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]
๐ฝ (๐พ๐, ๐พce, ๐พie, ๐พ๐ข) = โซ
โ
0
๐ก |๐ (๐ก) โ ๐ (๐ก)| ๐๐ก, (13)
PopulationDecoded form
Decoded form
Selection Fitness
Fitness
Replacement Parents Objective function
Genetic operation
Subpopulation
Figure 6: Block diagram of a GA.
where (๐พ๐,๐พce,๐พie,๐พ๐ข) are the PD
๐ฝ
+ 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 PD๐ฝ + 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 ๐ท
๐ฝ is implemented using a 4th-order Pade discrete rational transfer function (๐ = ๐ = 4)
of type (5). It is used as sampling period of ๐๐
= 0.01 s.The PD๐ฝ + I controller is tuned through the optimizationof fitness function corresponding to the minimization ofITAE (13) using a GA. We use ๐ฟ = 15. We establish thefollowing values for the GA parameters: population size ๐ =
20, crossover probability ๐ถ = 0.8, mutation probability๐ = 0.05, and number of generations ๐
๐บ= 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
6 Mathematical Problems in Engineering
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 ๐ฝ. Weset the search space of๐พ
๐, ๐พce, ๐พie, and๐พ
๐ขโ [0, 5].
In the first case, we compare a fuzzy fractional PD๐ฝ + Icontroller (๐ฝ = 0.9), with a fuzzy integer PD + I controller(๐ฝ = 1). Figure 7 shows the unit step responses of bothcontrollers.The plant system๐บ
1(๐ ) used is represented by the
transfer function:
๐บ1(๐ ) =
1
(๐ + 1) (1 + ๐ผ๐ ) (1 + ๐ผ2๐ ) (1 + ๐ผ
3๐ )
with ๐ผ = 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: {๐พ
๐, ๐พce, ๐พie, ๐พ๐ข} โก
{1.0808, 0.3408, 0.3442, 4.2095}, with ๐ฝ = 0.8235, and forthe fuzzy fractional PD๐ฝ + I controller to the followingvalues: {๐พ
๐, ๐พce, ๐พie, ๐พ๐ข} โก {0.7581, 0.3510, 0.3038, 4.3276},
with ๐ฝ = 0.7693. These values lead us to conclude that thefuzzy fractional-order controller produced similar results tointeger one; however, the error ๐ฝ is smaller, as can be seenin Figure 8, where it shows the ITAE error as function of๐ฝ. The graph reveals that for this process we have a lowererror for a fractional value of ๐ฝ = 0.9. We verify that thefractional controller is better (in terms of error ๐ฝ) than theinteger version, only in a narrow region of 0.8 โค ๐ฝ < 1.Figure 9 illustrates the variation of FLC parameters (๐พ
๐, ๐พce,
๐พie,๐พ๐ข) as function of the orderโs derivative๐ฝ, while Figure 10shows the variation of the transient response parameters,namely, the settling time ๐ก
๐ , rise time ๐ก
๐, peak time ๐ก
๐, and
overshoot ๐V(%) versus ๐ฝ, for the closed-loop step response.The variation of FLC parameters and the transient responseparameters reveals a smooth variation with ๐ฝ.
In a second experiment, we consider a fuzzy PD๐ฝ + I (๐ฝ =
0.2) controller, for process ๐บ2(๐ ) with a right-half plane zero,
represented by the transfer function:
๐บ2(๐ ) =
1 โ ๐ผ๐
(๐ + 1)3
with ๐ผ = 5.0. (15)
Once again, we consider for comparison the correspond-ing integer version (๐ฝ = 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: {๐พ
๐, ๐พce, ๐พie, ๐พ๐ข} โก {0.5393, 0.4647,
0.3439, 0.2486}, with ๐ฝ = 75.4509, and for the fuzzyfractional controller: {๐พ
๐, ๐พce, ๐พie, ๐พ๐ข} โก {0.6061, 0.0326,
0.3175, 0.2909}, with ๐ฝ = 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 ๐ฝ are smaller. Figure 12 shows the ITAE error asfunction of ๐ฝ. The graph reveals that for this process wehave a lower error for ๐ฝ = 0.2. Also note that the fractionalcontroller is better (in terms of error ๐ฝ) than the integer
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
1.2
Time (s)
c(t)
PIDPID๐ฝ
Figure 7: Step responses of closed-loop system with fuzzy PD + Iand PD๐ฝ + I (๐ฝ = 0.9) controllers.
0 0.2 0.4 0.6 0.8 1
J
101
100
10โ1
๐ฝ
Figure 8: Error ๐ฝ versus ๐ฝ for ๐บ1(๐ ).
one for 0.1 โค ๐ฝ < 1. Figure 13 illustrates the variation of FLCparameters (๐พ
๐, ๐พce, ๐พie, ๐พ
๐ข) as function of the orderโs
derivative ๐ฝ, while Figure 14 shows the variation of ๐ก๐ , ๐ก๐, ๐ก๐
and ๐V(%) versus ๐ฝ, for the closed-loop step response. In thiscase, the ๐พce and the ๐พie have the smaller value near the bestcase (i.e., ๐ฝ = 0.2) and ๐ก
๐ , ๐ก๐, and ๐ก
๐reveal similar values with
the variation of ๐ฝ.In a third study we consider a fuzzy PD๐ฝ + I (๐ฝ =
0.9) controller, for process ๐บ3(๐ ), represented by the transfer
function (16), where the time delay is ๐ = 1 [s]:
๐บ3(๐ ) =
๐โ๐ ๐
(1 + ๐ผ๐ )2
with ๐ผ = 2.0. (16)
For comparison purposes, we consider the correspondinginteger version (๐ฝ = 1). Figure 15 shows the unit stepresponses of both controllers.
Mathematical Problems in Engineering 7
0 0.5 1 0 0.5 1
0 0.5 10 0.5 1
101
100
101
100
10โ1
100
10โ1
100
10โ110โ2
๐ฝ ๐ฝ
๐ฝ๐ฝ
Ke
Ku
Kce
Kie
Figure 9: The PID๐ฝ parameters (๐พ๐, ๐พce, ๐พie, ๐พ๐ข) versus ๐ฝ for ๐บ
1(๐ ).
The controller parameters, corresponding to the min-imization of the ITAE index, lead to the values for thefuzzy integer controller: {๐พ
๐, ๐พce, ๐พie, ๐พ๐ข} โก {1.0496,
0.9300, 0.1757, 2.2949}, with ๐ฝ = 7.2953, and for thefuzzy fractional controller: {๐พ
๐, ๐พce, ๐พie, ๐พ๐ข} โก {0.6112,
1.9142, 0.1731, 1.7278}, with ๐ฝ = 6.8251. These values leadus again to remain the previous conclusions drawn for ๐บ
1(๐ )
and ๐บ2(๐ ), 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 ๐ฝ are smaller. Figure 16 shows the ITAE error asfunction of ๐ฝ. The graph reveals that for this process we havea lower error for ๐ฝ = 0.9. The fractional controller presentssmaller values of ๐ฝ for 0.8 < ๐ฝ < 1. Figure 17 illustrates thevariation of FLC parameters (๐พ
๐, ๐พce, ๐พie, ๐พ๐ข) as function of
the orderโs derivative๐ฝ, while Figure 18 shows the variation of๐ก๐ , ๐ก๐, ๐ก๐, and ๐V(%) versus๐ฝ for the closed-loop step response.
The FLC parameters reveal a slight variation with ๐ฝ, as wellthe transient response parameters.
In conclusion, with the fuzzy fractional PD๐ฝ+ 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 (๐
1) used in the previous section with the structures
of Figure 19 (๐2) and Figure 20 (๐
3) applied to process ๐บ
3(๐ ).
TheGAwas executed 10 times, andwe get the result that leadsto the smaller ๐ฝ.We consider the same fuzzy control rules baseof Table 1 in all the structures (๐
1, ๐2, and ๐
3).
The control action for these new structures is, respec-tively, for (๐
2) and (๐
3):
๐ (๐) = ๐พ๐ขV + ๐พie๐ผV = ๐พ
๐ข[๐ (๐ธ, ๐ถ๐ธ)] + ๐พie๐ผ [๐ (๐ธ, ๐ถ๐ธ)]
= ๐พ๐ข[๐ (๐พ
๐๐ (๐) , ๐พce๐ท
๐ฝ
๐ (๐))]
+ ๐พ๐๐๐ผ [๐ (๐พ
๐๐ (๐) , ๐พce๐ท
๐ฝ
๐ (๐))]
(17)
8 Mathematical Problems in Engineering
0 0.5 1
0 0.5 10 0.5 1
0 0.5 1
102
101
100
101
100
101
100
101
100
10โ1
๐ฝ ๐ฝ
๐ฝ๐ฝ
t s(s
)
t r(s
)
t p(s
)
ov (%
)
Figure 10: Parameters ๐ก๐ , ๐ก๐, ๐ก๐, ๐V(%) versus ๐ฝ of the step responses of the closed-loop system with ๐บ
1(๐ ) and with a fuzzy PD๐ฝ + I controller.
0 10 20 30 40 50
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
c(t)
PIDPID๐ฝ
โ0.2
โ0.4
Figure 11: Step responses of closed-loop system with fuzzy PD + I and PD๐ฝ + I (๐ฝ = 0.2) controllers.
Mathematical Problems in Engineering 9
0 0.2 0.4 0.6 0.8 1
J
102
101
๐ฝ
Figure 12: Error ๐ฝ versus ๐ฝ for ๐บ2(๐ ).
0 0.5 10 0.5 1
0 0.5 1 0 0.5 1
100
10โ1
100
10โ1
100
10โ1
101
100
10โ1
10โ2
๐ฝ
๐ฝ ๐ฝ
๐ฝ
Ke
Ku
Kce
Kie
Figure 13: The PD๐ฝ + I parameters (๐พ๐, ๐พce, ๐พie, ๐พ๐ข) versus ๐ฝ for ๐บ
2(๐ ).
10 Mathematical Problems in Engineering
0 0.5 1
0 0.5 10 0.5 1
0 0.5 1
101102
101 100
102
101
100
102
101
100
๐ฝ ๐ฝ
๐ฝ ๐ฝ
t s(s
)
t r(s
)
t p(s
)
ov (%
)
Figure 14: Parameters ๐ก๐ , ๐ก๐, ๐ก๐, ๐V(%) versus ๐ฝ of the step responses of the closed-loop systemwith๐บ
2(๐ ) and with a fuzzy PD๐ฝ+ I (0 < ๐ฝ โค 1)
controller.
0 10 20 30 40 500
0.2
0.4
0.6
0.8
1
1.2
Time (s)
c(t)
PIDPID๐ฝ
Figure 15: Step responses of closed-loop system with fuzzy PD + I and PD๐ฝ + I (๐ฝ = 0.9) controllers.
Mathematical Problems in Engineering 11
0 0.2 0.4 0.6 0.8 1
J
102
101
100
๐ฝ
Figure 16: Error ๐ฝ versus ๐ฝ for ๐บ3(๐ ).
0 0.5 1
0 0.5 1 0 0.5 1
0 0.5 1
101
100
100
10โ1
10โ2
101
100
10โ1
10โ2
101
100
10โ1
10โ2
๐ฝ
๐ฝ
๐ฝ
๐ฝ
Ke
Ku
Kce
Kie
Figure 17: The PD๐ฝ + I parameters (๐พ๐, ๐พce, ๐พie, ๐พ๐ข) versus ๐ฝ for ๐บ
3(๐ ).
12 Mathematical Problems in Engineering
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
10โ1
๐ฝ
๐ฝ
๐ฝ
๐ฝ
t s(s
)
t r(s
)
t p(s
)
ov (%
)
Figure 18: Parameters ๐ก๐ , ๐ก๐, ๐ก๐, ๐V(%) versus ๐ฝ of the step responses of the closed-loop systemwith๐บ
3(๐ ) and with a fuzzy PD๐ฝ + I (0 < ๐ฝ โค 1)
controller.
e
D๐ฝ
Ke
Kce
E
CE
๏ฟฝKu
I KieIE
u
+
+
Fuzzy logiccontroller
Figure 19: Fuzzy fractional PID controller (๐2).
๐ (๐) = ๐พPDV1 + ๐พPI๐ผV2, (18)
where
V1= ๐(๐ธ
1,CE1) = ๐(๐พ
๐1๐(๐), ๐พce1๐ท
๐ฝ
๐(๐)),V2= ๐(๐ธ
2,CE2) = ๐(๐พ
๐2๐(๐), ๐พce2๐ท
๐ฝ
๐(๐)).
Figure 21 shows the unit step responses of fuzzy fractionalcontroller for ๐
1, ๐2, and ๐
3, for the best ๐ฝ. Figure 22 depicts
the corresponding error as function of ๐ฝ. We verify that thelower error occurs for ๐ฝ = 0.9 in ๐
1, ๐2, and ๐
3, leading to
errors ๐ฝ = 6.8251, 5.8000, and 6.2900, respectively.
Analyzing Figure 21 we observe that the step response isless oscillatory for ๐
1, with smaller values of ๐ก
๐and ๐ก๐ than
other structures. Figure 22 shows that all three structurespresent similar results in terms of error ๐ฝ as function of order๐ฝ. Structures ๐
1and ๐
2give the most similar results while
๐3differs mostly on the regions around ๐ฝ = 0.1, 0.5, and
0.6.The different structures analyzed lead to similar results
both in the step responses and error ๐ฝ. In fact, it is interestingto notice that the best ๐ฝ = 0.9 for all the structures whenminimizing the ITAE index.
Mathematical Problems in Engineering 13
eKe1
D๐ฝ Kce1
Ke2
Kce2
E1
CE1
E2
CE2
๏ฟฝ1
๏ฟฝ2
KPD
I KPIIE
+
+
u
Fuzzy logiccontroller 2
Fuzzy logiccontroller 1
Figure 20: Fuzzy fractional PD + PI controller (๐3).
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 ๐
1, ๐2, and ๐
3for ๐บ3(๐ ).
Based on this analysis we verify that the proposed PD๐ฝ+ Icontroller (๐
1) gives comparable and, in some situations, even
better results than other two structures (๐2and ๐3). Moreover,
the adopted structure is simpler and easier to analyze andimplement. These facts justify the choice of the structure ๐
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 PD๐ฝ + 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
๐ฝ
S1S2S3
Figure 22: Error ๐ฝ versus ๐ฝ with the structures ๐1, ๐2, and ๐
3for
๐บ3(๐ ).
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 CompetitividadeโCOMPETE program and by National Funds through FCTFundacao para a Ciencia e a Tecnologia.
14 Mathematical Problems in Engineering
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