FRACTIONAL ORDER SYSTEM IDENTIFICATION BASED ON GENETIC ALGORITHMS

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1 Jounal of Engineeing Science and Technology Vol. 8, No. 6 (2013) School of Engineeing, Taylo niveity FRACTIONAL ORDER SSTEM IDENTIFICATION BASED ON GENETIC ALGORITHMS MAZIN Z. OTHMAN*, EMAD A. AL-SABAWI Compute and Infomation Engineeing Depatment, College of Electonic Engineeing, niveity of Moul, Iaq *Coeponding Autho: d.mazin@uomoul.edu.iq Abtact Sytem identification deal with etimating the plant paamete unde contol uing input-output meauing data. Mot of pactical plant have factional ode dynamic popetie which ae baed on integation and diffeentiation of nonintege ode. In thi wok the tuctue and the paamete of factional ode unknown tanfe function ae etimated uing input-output data. Intege ode Leat Squae identification i ued fit to confim the tuctue (ode) of the unknown tanfe function. Then, Genetic Algoithm (GA) i followed to find the mot accuate factional ode etimate that epeent the ytem. Illutative example ae peented in which factional ode tanfe function ae identified in a way that faithfully etimate the dynamic of the unknown plant. Keywod: Sytem Identification, Leat Squae, Factional ode, Genetic Algoithm. 1. Intoduction Pio knowledge of the paamete of ytem unde contol i often the fit tep in deigning contolle. The concept of ytem identification focued on pope etimation of the paamete of eal plant. Many tatitical and geometic method uch a leat quae, maximum likelihood and egeion model ae extenively ued fo eal-time paamete etimation [1, 2]. The eal wold plant o pocee ae geneally of factional ode [3].The poblem of paamete identification become moe difficult fo a factional ode ytem compaed to an integal ode one. Howeve, it i ued to appoximate a factional ode poce a an intege ode one. In geneal, thi appoximation can caue ignificant diffeence between a eal ytem and it mathematical model. The idea of uing factional-ode calculu in modelling factional ode contol ytem i welladdeed in [4, 5]. Advantage of uing factional-ode ytem identification have been intoduced in a numbe of publication [6-8]. 713

2 714 M. Z. Othman and E. A. Al-Sabawi Nomenclatue D na nb u w A w B y m y p Factional diffeentiation opeato Poitive intege numbe epeent the numbe of coefficient in the denominato Poitive intege numbe epeent the numbe of coefficient in the nominato Input ignal Low fequency of the bandwidth High fequency of the bandwidth. Model output Plant output Howeve, the Evolutionay Algoithm (EA) i ued by many eeache to find the "optimal" olution elative to the defined poblem. Diffeent appoache wee popoed in thi context to chooe an objective function that eally meet the deied pecification and obey the pactical containt (ee fo example [9] in which Paticle Swam Optimization (PSO) algoithm i ued). In thi wok, the Leat Squae (LS) method i adopted in ode to facilitate the poblem of finding the unknown tanfe function tuctue (i.e., the neaet intege numeato and denominato ode). Then the (LS) eult ae utilized a an initial gue in the GA to obtain the coefficient and the factional ode of the tanfe function. 2. Factional Ode Sytem Modelling Factional ode ytem ae decibed by factional ode diffeential equation. Factional calculu allow the deivative and integal to be abitay numbe. Thee ae eveal definition of factional deivative [10]. Gunwald-Letnikov definition i pehap the bet known one due to it mot uitable fo the ealization of dicete contol algoithm. The th ode factional deivative of continuou function f(t) i given by: D f ( t) d f dt ( t) [ x] j lim h ( 1) f( t jh) h j j (1) 0 0 ( ) ( ) t 1... j whee [x] i a tuncation and x h j. The geneal j! calculu opeato, including factional ode and intege ode i defined a: ad t d / dt LLR( ) 0 1LLLLR( ) 0 t ( dτ ) LLR( ) 0 a (2)

3 Factional Ode Sytem Identification Baed on Genetic Algoithm 715 whee a and t ae the limit elated to opeation of factional diffeentiation and i the calculu ode (0 < < 1). Accodingly, Factional ode ytem can be epeented by the genealized factional diffeential equation given by [11]: na α nb β j y( t) + a i i D y( t) b j D u( t) i 1 j 0 (3) whee α 1 < α 2 <... < α na, β 0 < β 1 < < β nb ae non-intege poitive numbe and i1, 2,, na, j0, 1, 2,, nb. The Laplace tanfom of Eq. (3) can be given by: nb β j b j ( ) j 0 G( ) ( ) na α a i i i 1 Equation (4) can be appoximated uing indiect dicetization method which conit of two tep. In the fit tep a ational analog tanfe function that appoximate the iational tanfe function of the factional ytem i obtained. In the econd tep an analog to digital tanfomation of the ational analog appoximation i done. (4) Analog ational appoximation The appoximation i caied out in the following two tep: Detemine a fequency ange [ω A, ω B ] whee it i needed to appoximate the factional ode tanfe function. Pefom an analog intege-ode appoximation of the factional-ode tanfe function: ωb ( ω A + ) ωb [ ω A ωb ] n 2 ω A + ωb + ω (5) ( ) ( (1 ) AωB ) ω A Equation (5) i expanded uing continued faction expanion CFE, Taylo eie expanion, o the method of ecuive pole and zeo to obtain an analog appoximation. The analog appoximation with ecuive pole and zeo i ummaized below [12]: d ω A b b ω d B k N ' ω lim k N k N ωk (6)

4 716 M. Z. Othman and E. A. Al-Sabawi 2k ' d 2N+ 1 ω k ω A (7) b + 2k b 2N+ 1 ω k ωb (8) d d 2 + bω K B d(1 ) 2 + bωb+ d d k N K ωk ωa b k N ' ωk k N ' + ωk k N + ωk In thi wok b 10, d 9, ω A 0.01 ad/, ω B ad/, and N7 a in [12].Thee paamete ae elected to enue the dynamic ange of mot indutial plant. The pocedue fo the appoximation can be biefly ummaized in the following tep: a) The fequency ange [ω A, ω B ] and N ae given. b) Baed on the factional ode, calculate ω k and ω k accoding to Eq. (7) and (8). c) Compute K fom Eq. (10). d) Obtain the appoximate ational tanfe function fom Eq. (9) to eplace. (9) (10) Dicetizing the analog appoximation To dicetize the analog ational appoximation of Eq. (9) elect uitable -to-z tanfom. The bilinea tanfomation i employed by eplacing in Eq. (9) by the ight hand ide of Eq. (11) 1 2 (1 z ) 1 (11) T ( z ) 3. The Genetic Algoithm [13, 14] The Genetic Algoithm (GA) i a diect andom each technique, which can find the optimal olution in complex multi-dimenional each pace. It i a model of natual evolution opeato which manipulate individual in a population ove eveal geneation to impove thei fitne gadually. Thee individual that, epeent a candidate olution to the poblem unde conideation, often encoded a a bit ting. Thee ae two baic iue in the GA, the fit one i how to code the poblem thi i called coding. The econd iue i how to qualify each individual (ting) which i called the fitne evaluation. Thi tep depend on the poblem natue. It i eithe mathematic equation o a ule-baed pocedue o in ome cae a combination of both. Thee two iue ae poblem dependent.

5 Factional Ode Sytem Identification Baed on Genetic Algoithm 717 The emaining pat of the GA i poblem independent which conit mainly of the GA opeato. Thee opeato ae baically impotant, that ae election (epoduction), coove, and mutation. The election pocedue i to epoduce moe ting whoe fitne function ae highe than thoe whoe fitne function ae low. Thi i impotant to dive the each towad bette and bette olution. Repoducing can be implemented by many method (fo example Roulette Wheel, Tounament etc.).the coove i impotant to ceate new individual fom aleady exit one. The mutation i an opeato that avoid the each poce to be tapped at a local aea in the hype each pace by exchanging bit of a cetain ting. The following algoithm claifie the main tep in GA: Step-1: Ceate an initial population of ting. Step-2: Calculate the fitne of each ting. Step-3: While ( an acceptable olution i not found) OR ( doe not exceed maximum numbe of geneation) Pefom epoduction fo next geneation. Pefom coove between paent to ceate new offping. Apply mutation with cetain pobability. Calculate the fitne of each offping. End while 4. Genetic-Baed Tuning of FO Sytem The undetaken identification poblem i to find the optimal tuning gain fo the FO ytem given in Eq. (4) that enue cetain objective function. The objective function i uggeted hee to foce FO ytem to behave imila to a pedefined model (the plant). Theefoe, each GA olution ha to minimize the following objective function: Fitne y ( t) y ( t) p m (12) T o To i the numbe of collected data ample. Binay coding i ued to epeent the FO ytem tuning paamete a hown in Fig. 1. Fo each exponent (α i o β j ), 9bit epeentation i ued to enue eolution of While fo each coefficient (a i and b j ), 25 bit epeentation i ued: 7 bit fo the intege pat (0 to 127) and the emainde 18 bit fo the factional pat (i.e., with eolution of ). a 1 a 2 a na α 1 α 2 α na b 0 b 1 b nb β 0 β 1 β nb Fig. 1. FO Sytem Sting Coding. The whole block diagam epeenting the popoed deign method i hown in Fig. 2.

6 718 M. Z. Othman and E. A. Al-Sabawi FO Sytem y m a i, α i, b j, β j GA ( min e ) e - + u Plant y p Fig. 2. The Block Diagam of the Popoed Deign Method. 5. Popoed Pocedue The poblem unde conideation i to find the tuctue and the paamete of a factional ode ytem defined by [4] uing input-output meaued data. Theefoe, tatitical tandad Leat-quae baed linea etimato have been employed with ARMAX (Auto-Regeive Moving Aveage with exogenou input). Such tep give an initial gue of the ytem intege ode tanfe function. In othe wod, it fit the input-output data into the following model: n ( ) b0 + b bn ( ) m a a 1 m Pactically, m > n.now, knowing the ytem intege ode (n and m) and it paamete, the GA will employed uch infomation a an initial gue to each fo a factional veion of [11]. (13) 6. Illutative Example In ode to illutate the above pocedue in identification of factional ode ytem, two example ae peented hee Example-1: Factional ode hypothetical model Conide the following factional ode tanfe function, 0.5 ( ) ( ) Following the pocedue of Sec 2, the imulation of Eq. (14) i accomplihed unde MATLAB envionment. The ampling time i choen to be Seveal tial have been accomplihed uing LS identification method (ax.m in MATLAB identification Toolbox) to dicove the 'bet' intege ode model fom a et of input-output data a hown in Table 1. It i found that the following model ha the leat um abolute eo between the actual and the model output. (14)

7 Factional Ode Sytem Identification Baed on Genetic Algoithm 719 ( ) ( ) LS Table 1. LS Identification of Example-1. Model Ode Sum of abolute eo RMS of eo (15) Knowing the ytem tuctue, the GA i now ued to find the unknown paamete which ae hown in Eq. (16). The population ize i 1200, the maximum numbe of geneation i 150, the numbe of collected data ample i ample, and the coove pobability and the mutation pobability ae choen by tial and eo to be 0.2 and 0.35, epectively (the coove pobability and the mutation pobability value depend on the complexity degee of the hype uface that elate all the paamete to be optimized [14]). The fitne function ued i the um of abolute deviation fom the actual et of output obevation ( ) ( ) (16) The epone of the conideed ytem, the LS identified and the GA identified model ae hown in Fig. 3. The eo of LS identified and the GA FO identified model ae hown in Fig. 4. The um of abolute eo of the GA FO identified model i ( ) (RMS0.0053) Sytem LSI Sytem FOGI Sytem time Fig. 3. The Step Repone of the Conideed Sytem, LS and GA FO Identified Model.

8 720 M. Z. Othman and E. A. Al-Sabawi LSI Sytem Eo FOGI Sytem Eo time Fig. 4. The eo of LS and GA FO Identified Model Example-2: Factional ode Model of DC Moto A pactical gatheed data fom 3KW DC moto i conideed. The data ae ampled at 0.01 ; the excitation of the moto i DC voltage wheea the epone i peed in pm. ing LS identification method, it i found that the bet intege ode model i Eq. (17). ( ) ( ) LS Knowing the ytem tuctue, the GA i now ued to find the unknown paamete which ae hown in (23). The population ize i 1200, the maximum numbe of geneation i 150, the numbe of collected data ample i ample. (17) ( ) ( ) (18) The epone of pactical ytem, the LS identified and the GA FO identified model ae hown in Fig. 5. The eo of LS identified and the GA FO identified model ae hown in Fig. 6. The um of abolute eo of LS identified and the GA FO identified model ae (11655) (RMS ) and (10104) (RMS ) epectively. 7. Concluion In thi pape a pocedue wa uggeted to find the 'bet' tuctue and coefficient of a factional ode contol ytem. Thi pocedue utilize fitly the LS method to find the uitable intege ode tuctue (pole and zeo) that fit the input-output ytem data. Then, GA wa exploited to find the factional ode veion of that obtained uing LS method.

9 Factional Ode Sytem Identification Baed on Genetic Algoithm Sytem LSI Sytem FOGI Sytem time Fig. 5. The Repone of the Pactical Sytem, LS and GA FO Identified Model LSI Sytem Eo FOGI Sytem Eo time Fig. 6. The Eo of LS and GA FO Identified Model. Thi pocedue i illutated uing an example of factional ode model and pactical voltage-peed data of a 3KW DC moto. It i found that the um of the abolute eo between the 'bet' LS tep epone and the GA factional ode epone ae ( ) and ( ) fo the fit example and (11655) and (10104) fo the econd example, epectively. Theefoe, the identified GA factional ode model obtained by the popoed pocedue ae moe uitable to be ued in contol algoithm that fully equie the ytem dynamic.

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