Fuzzy Supervisory PI Controller Using Hierarchical Genetic Algorithms

Fuzzy Supervisory PI Controller Using HierarchicalGenetic AlgorithmsKiatkajohn Worapradya and Suvalai PratishthanandaDepartment of Electrical Engineering,Faculty of EngineeringChulalongkorn University,Bangkok10330Thailande-mail:john@control.ee.eng.chula.ac.th,suvalai@ee.eng.chula.ac.thAbstractThis paper presents a fuzzy supervisory PI controller using a Hierarchical Genetic Algorithm(HGA).HGA which is an optimization algorithm is used to define the optimal number and shape of membership functions, and fuzzy rules.The fuzzy supervisory PI controller is designed and simulated on a heat exchanger sys-tem.Sum of square error and sum of absolute error are used as objective functions.Simulation results are compared with the PI controller using˚A str¨o m and H¨a gglund tuning.Improved performances can be noticed.1IntroductionThe general structure of fuzzy systems consists of three principle components:a fuzzifier unit which is a process to transform input signals into fuzzy sets,an inference engine which is a decision process to approx-imate reasoning with fuzzy rule bases,and,finally,a defuzzifier unit which transforms the fuzzy sets into output signals.In order to work efficiently,this struc-ture has to use optimal membership functions and rule bases.However,there are no exact methods to con-struct them.Therefore,the conventional approaches usually base on the operators’skills,experience,and trial-and-error method.To solve this problem,many researchers designed optimal membership functions and rule bases by using different methods such as neural network[2],cluster-ing[3,4],gradient method[5],simulated annealing [6,9],and genetic algorithms[7,8,9,10].Among these methods,the genetic algorithms method is the most in-teresting.For examples,it dose not use derivative in-formation,it searches from a population of points(not a single point),and it is a powerful tool for solving a multi-objectives function problem.Although genetic algorithms can search the optimal shape of membership functions and optimal rules,it cannotfind the proper number of membership func-tions and rules.Therefore,Man,Tang,Kwong and Liu [1]suggested a Hierarchical Genetic Algorithms which canfind the optimal number and shape of membership functions,and rules.In this paper,we propose the design of fuzzy supervisory system which is used for supervising the PI controller.We use the HGA for designing opti-mal membership functions and rules.The obtained fuzzy supervisory PI controller is simulated on the heat exchanger.To demonstrate the better performance,we compare the designed responses with the response of PI controller tuning via˚A str¨o m and H¨a gglund method.2Hierarchical Genetic AlgorithmsHGA is a type of the genetic algorithms,introduced by Man,Tang,Kwong and Liu[1].Its structure is moreflexible than the conventional genetic algorithms (GAs)while it still has important genetic operations. The only difference between HGA and GAs is the chro-mosome form.2.1Hierarchical Chromosome Formulation Concept of hierarchical chromosome is regarded as DNA in a biological chromosome.DNA consists of two types of genes which are structure genes and regulatory genes.The structure genes contain the genetic infor-mation and the regulatory genes control coding of the structural genes.Similarly,a hierarchical chromosome can be classified into two different types,i.e.,para-metric genes and control genes.The parametric genes, which contain parameter values,are analogous to the structure genes.The activation of the parametric genes is governed by the value of the control genes which is analogous to regulatory sequences.In Figure1,a three levels gene structure is illustrated. The parameter genes are governed by thefirst level control genes which are governed by the second level control genes,respectively.The value0or1is used to define the activation of the parametric genes,an integer1is assigned for each control gene that is being ignited,where0is for turning off.When the value2nd levelcontrol gene 1st level control geneparametric geneFigure 1:General form of a hierarchical chromosome of control gene is 1,the associated parametric genescorresponding to that control gene are activated and they are inactivated when the value of control gene is 0.2.2Genetic OperationsGenetic operation of HGA consists of three pro-cesses which are selection,crossover,and mutation.Crossover and mutation are operated independently between control genes and parametric genes.A muta-tion method depends on the coding of a chromosome.3HGA for fuzzy system3.1Coding of fuzzy systemGenerally,problems solved by GAs,the variables are coded to chromosome form.Similarly,in HGA the parameters of membership functions and fuzzy rules are coded to hierarchical chromosome form which is different between membership chromosomes and fuzzy rule chromosomes.The HGA chromosome consists of the usual two types of genes,the control genes (z c )and parametric genes (z p ).The control genes,in the form of bits,deter-mine the membership function activation,whereas the parametric genes are in the form of real numbers to represent the membership functions.The HGA coding is shown in Figure 2.With the two input fuzzy sets of error signal (e )and error rate (∆e )and the output fuzzy set of ∆u ,we can define the parametric genes of the membership chromosome as z p={αE1a ,αE 1b ,αE 1c ,...,αE ma ,αE mb ,αE mc ,β∆E1a ,β∆E 1b ,β∆E 1c ,...,β∆E na ,β∆E nb ,β∆E nc ,γ∆U1a ,γ∆U 1b ,γ∆U 1c ,...,γ∆U pa ,γ∆U pb ,γ∆U pc }(1)where m ,n ,and p are the maximum allowable numbers of the fuzzy subset of e ,∆e ,and ∆u ,respectively,αEia ,αE ib ,αE ic define the input membership functionof the i th fuzzy subset of e ,β∆Eja ,β∆E jb ,β∆E jc define the input membership function of the j th fuzzy subset1a1cnanbnccontrol genes (z )parameter genes (z ) pnanbnc2a2b2c1b1c1a1bFigure 2:A membership chromosomeof ∆e ,γ∆U ka,γ∆U kb ,γ∆Ukc define the input membership function of the k th fuzzy subset of ∆uTo code a fuzzy rule into a fuzzy rule chromosome,the number of rules depend on the number of fuzzy subset of e and ∆e which is corresponding to the number of actived control genes.Therefore,the fuzzy rule base should be classified,corresponding to the number of actived control genes.In order to make it easy,the fuzzy rule base should be coded in matrix form which is similar to a fuzzy rule table as shown in Table 1.Table 1:The Rule Base in Tabular FormD 1D 2···D j ···D xE 1U 1U 2···U j E 2U 2U 3···U j......E i ···U k ···......E wU i···U yWe code the fuzzy rule table in Table 1into the fuzzy rule chromosome H (w,x,y )which is formulated in the form of an integer matrixH (w,x,y )=⎡⎢⎣h 1,1···h 1,j .........h i,1···h i,j⎤⎥⎦(2)where h i,j ∈[1,y ]and ∀i ≤w,j ≤x and the i −jelement implies the following rule:R ij :If e is E i and ∆e is D j then ∆u is U kwhere E i ,D j and U k are the linguistic names which characterize the fuzzy subsets of e ,∆e ,and ∆u ,re-spectively.3.2Genetic Operations for fuzzy systemDue to the fact that the membership chromosomes con-sist of control genes and parametric genes,each type of gene has to use different genetic operations.For the crossover operation,a one-point crossover is appliedseparately for both the control and parametric genes of membership chromosomes within certain operation rates.The fuzzy rule chromosome has no crossover operation because there is only one fuzzy rule chromo-some per one matrix dimension.Bit mutation is applied for the control gene of the mem-bership chromosome.Each bit of the control gene is changed from1to0or from0to1if a probability test is satisfied(a big probability).As for the para-metric genes,which are real-number represented,ran-dom mutation is applied.The random mutation is presented byg=g+ψ(µ,σ)(3) where g is the real value gene,ψis a random function which may be Gaussian or normally distributed,µand σare the mean and variance related with the random function,respectively.To mutate the fuzzy rule chromosome,a special muta-tion operation is applied.It is called delta operation which alters each element in the fuzzy rule chromosome as follows:h i,j=h i+∆i,j+∆j(4) where∆i and∆j have an equal chance to be1or-1, with a small probability.4Design example:The fuzzysupervisory PI controller ofa heat exchangerFigure3:The fuzzy supervisory PI controllerIn this section,we present an example of the fuzzy supervisory control system design using HGA which is discussed in last section.This fuzzy supervisory system,as a second level controller,is used for super-vising a PI controller of heat exchanger.The fuzzy supervisory control system is shown in Figure3.The membership functionsand fuzzy rules,are defined by HGA and this fuzzy structure is used for K p and K I parameters tuning of the PI controller.The tuning occurs when disturbance is applied or the set point is changed.ShutterTo Ti , Fi InletFigure4:Heat exchanger system4.1Characteristic and mathematical model ofa heat exchangerA heat exchanger is shown in Figure4.A constant velocity blower draws the air into the tube and the air volume is controlled by a shutter.The air which passes through the tube is heated by a heater.The heater power is applied by thyrister as the actuator corresponding to control signal.The heated airflows to the end of the tube and is detected by the temperature sensor.The detected temperature is transformed into the voltage by bridge circuit and transmitted to control the system.For linearized model of the heat exchanger,which is a first order plus time delay system,Wang[7]as:G(s)=10e−0.13s0.33s+1(5)where the input is the voltage supplied to heater and the output is the temperature measured by the detector.4.2SimulationsTo simulate,we consider the set point change problem. The temperature at the detecting point is required to be increased to60◦C from an initial value of20◦C at 6th second and then decreased back to20◦C at12th second.The parameters of optimization are as follows:•The parameters of the fuzzy system.–e∈[−50,50]and∆e∈[−5000,5000].–K P∈[0.06,0.3]and K I∈[0.2,0.5].–Minimum inference engine.–Center average defuzzifier.•The parameters of HGA.–The maximum number of the input fuzzysubsets of e and∆e=5and the number ofthe output fuzzy subsets of K P and K I=2(m=n=5and p=q=2).–The number of generations=20.–The other parameters are shown in Table2.Table 2:The parameters of HGAMembership Chromosome Fuzzy Rule Control Parameter ChromosomeGenes Genes Representation Binary RealInteger Population Size 202016No.of Offspring 221Crossover One point One point -Crossover Crossover-Crossover Rate 0.90.9-Mutation Bit Random Delta Mutation Mutation Operation Mutation Rate 0.010.010.01SelectionRoulatte Wheel SelectionBase on the no.of active fuzzy subsetsTo define the fitness functions (F ),the fitness functions are selected as inverse proportion to the objective func-tion (J ).This process will yield a high sentivity string selection,as suggested by Wang [7].In this paper,two different objective functions and fitness functions are used.These objective functions are standard for dis-turbance rejection and set point following problems.•Case I :Sum of squared error objective function is shown as formula (6)and the fitness function is shown as formula (7).J 1=k i =1e 2i(6)F 1=106J(7)•Case II :Sum of absolute error objective func-tion is shown as formula (8)and the fitness func-tion is shown as formula (9).J 2=k i =1|e i |(8)F 2=105J(9)where e is error.4.3Simulation resultsIn Case I,the largest fitness function gives the fuzzy structure as:the optimal number of the input fuzzy subsets of both e and ∆e are 4.The shape of them are shown in Figure 5and Figure 6,respectively.The optimal number of output fuzzy subsets of both K P and K I are 2and the shape of them are shown in Figure 7(a)and 7(b),respectively.The optimal fuzzy rule bases of K P and K I are shown in Table 3and 4,respectively.In case II,The optimal number of the input fuzzy sub-sets of e and ∆e are 3and 2,and the shape of them-500-10-20-30-401020304050E1E2E3E41Figure 5:Membership function of e :Case I0-1000-2000-3000-4000-500010002000300040005000D1D2D3D41Figure 6:Membership function of ∆e :Case I0.30.30.40.20.20.10.06U1U2V1(b)1V2(a)0.5Figure 7:(a)Membership function of K P (b)Member-ship function of K I :Case ITable 3:Optimal rule table for K P :Case I∆e D 1D 2D 3D 4E 1U 2U 1U 1U 1eE 2U 2U 2U 2U 2E 3U 2U 1U 1U 1E 4U 2U 1U 2U 1Table 4:Optimal rule table for K I :Case I∆e D 1D 2D 3D 4E 1V 2V 1V 1V 1eE 2V 2V 2V 2V 2E 3V 2V 1V 1V 1E 4V 2V 2V 2V 2are shown in Figure 8and 9,respectively.The opti-mal number of output subsets of both K P and K I are 2and the shape of them are shown in Figure 10(a)and 10(b),respectively.The corresponding rule tables are shown in Table 5(a)and 5(b),respectively.The best objective values in each generations are ploted in Figure 11.These optimal fuzzy supervisory systems are simulated on the heat exchanger.Simulation results,which are compared with the PI controller using ˚A str¨o m and H¨a gglund tuning,are shown in Figure 12and 13.Case I has the shortest rise time,while Case II has the shortest settling time and the smallest overshoot.-500-10-20-30-401020304050E1E2E3Figure 8:Membership function of e :Case II0-1000-2000-3000-4000-500010002000300040005000D1D21Figure 9:Membership function of ∆e :Case II0.30.40.20.50.20.10.060.3U1U2V1V21(a)(b)Figure 10:(a)Membership function of K P (b)Member-ship function of K I :Case IITable 5:(a)Optimal rule table for K I (b)Optimalrule table for K I :Case II (a)(b)∆e D 1D 2E 1U 1U 2e E 2U 2U 1E 3U 1U 2∆e D 1D 2E 1V 1V 1e E 2V 2V 2E 3V 1V 1Case I uses almost the same control signal as the ˚A str¨o m-H¨a gglund while Case II uses the smallest.5ConclusionsIn this paper,the design of the fuzzy 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