COMPARISON OF THE METHODS FOR DISCRETE APPROXIMATION OF THE FRACTIONAL ORDER OPERATOR

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1 COMPARISON OF THE METHODS FOR DISCRETE APPROXIMATION OF THE FRACTIONAL ORDER OPERATOR Ľubomír DORČÁK, Ivo PETRÁŠ, Já TERPÁK ad Marti ZBOROVJAN Departmet of Iformatics ad Process Cotrol, BERG Faculty, Techical Uiversity of Košice, B. Němcovej 3, 4 Košice, Slovak Republic {lubomir.dorcak, ja.terpak, marti.borovja}@tuke.sk Steve s Electroic Services, 37 3 Upper MacLure Rd. Abbotsford, B.C., VT N4, Caada, petras@telus.et Abstract: I this paper we will preset some alterative types of discretiatio methods (discrete approximatio) for the fractioal-order (FO) differetiator ad their applicatio to the FO dyamical system described by the FO differetial equatio (FDE). With aalytical solutio ad umerical solutio by power series expasio (PSE) method are compared two effective methods - the Muir expasio of the Tusti operator ad cotiued fractio expasio method (CFE) with the Tusti operator ad the Al-Alaoui operator. Except detailed mathematical descriptio preseted are also simulatio results. From the Bode plots of the FO differetiator ad FDE ad from the solutio i the time domai we ca see, that the CFE is a more effective method accordig to the PSE method, but there are some restrictios for the choice of the time step. The Muir expasio is almost uusable. Key words: fractioal-order dyamic system, fractioal-order differetiator, discretiatio, power series expasio, cotiued fractio expasio, Tusti operator, Al-Alaoui operator. Itroductio The fractioal-order calculus (FOCA) is about 3-years old topic. The theory of FO derivative was developed maily i the 9th cetury. I the last decades besides theoretical research of the FO derivatives ad itegrals [Oldham, Spaier 974, Samko, Kilbas, Marichev 987, Podlubý 994, 999, ad may others] there are growig umber of applicatios of the FOCA i such differet areas as e.g. log electrical lies, electrochemical processes, dielectric polariatio [Westerlud 994], colored oise, viscoelastic materials, chaos, i cotrol theory [Maabe 96, Oustalup 988, 99, Axtel, Bise 99, Dorčák 994, Podlubý, Dorčák, Koštial 997, etc.] ad i may other areas. 8

2 I works [Maabe 96, Oustalup 988, Axtel ad Bise 99, etc.] the first geeraliatios of aalysis methods for FO cotrol systems were made (s-plae, frequecy respose, etc.). Some of our works were orieted to the methods of FO system parameters idetificatio, methods of FO cotrollers sythesis [Dorčák 994, Podluby et al. 999, Petráš 998, etc.], methods of stability aalysis [Dorčák et al. 998, Petráš et al. 999], etc. The key step i digital implemetatio of the FO differetiator i the PI λ D cotrollers is their discretiatio. Methods of discretiatio of the FO differetiator or itegrator I geeral, there are two discretiatio methods: direct discretiatio ad idirect discretiatio. I idirect methods, two steps are required, i.e., frequecy domai fittig i cotiuous time domai first ad the discretiig the fit s-trasfer fuctio [Oustaloup et al., Viagre et al. ]. I this paper, we will focus o the direct discretiatio methods. There are several approaches to direct discretiatio. We will cosider oly few of them. First is a power series expasio (PSE), secod is cotiued fractio expasio (CFE), third is matrix approach proposed by [Podluby ], fourth is a Taylor/MacLauri series expasio of geeratig fuctio suggested by [Duarte et al., Machado et al. 997] ad last oe is Diethelm s approach based o solvig of FDE i discrete time steps. All approaches are based o evolutio of a geeratig fuctio. As geeratig fuctio we ca use a simple Euler s backward differece, the Tusti rule, the Simpso rule or the Al-Alaoui operator, which is mixed scheme of Euler s rule ad the Tusti trapeoidal rule. The ivestigatio of these methods ad compariso with each other ad also with other methods was studied by several authors. The simplest ad most straightforward method is the direct discretiatio usig fiite memory legth expasio of Euler s rule from the Gruwald- Letikov defiitio of fractioal derivative. This approach was described by [Samko et al. 987, Dorčák 994,, Goreflo 996, Podluby 999, etc.]. Goreflo also suggested a PSE of the Tusti rule. The high order approximatio by usig the quadrature formula was used i [Lubich 986] work. All these ways of approximatio lead to form a FIR filter. It is well kow that, for iterpolatio or evaluatio purposes, ratioal fuctios are sometimes superior to polyomials, roughly speakig, because of their ability to model fuctios with eros ad poles [Viagre et al.,, ] ad therefore form of a IIR filter is much better. O the other had we also do ot eed a large umber of coefficiets. This form of approximatio was described by Viagre ad Che ad compared with PSE ad the others as for example Muir recursio [Che et al. ] or cotiuous methods [Viagre et al. ]. Last but ot least we should metio the approach proposed by Hwag [Hwag et al. ], which is based o B-splies fuctio. Muir recursio applied to the Tusti geeratig fuctio (GF) ad CFE applied to the Tusti or Al-Alaoui GF leads to the followig approximatios of the FO differetiator or itegrator : ± ± ± (, ) (, ) ( ) K A + K P + D lim (, ), () K T A K T Q (, ) ± ± ± ( ) ± P K K p D ( ) CFE /, () K T ( ) + K K T Qq p, q 8

3 where T is the sample period, is the order of differetiator or itegrator, K ad K are the costats accordig to the GF (K, K for the Tusti GF, K 8, K 7 for the Al-Alaoui GF), P ad Q are polyomials of degrees p ad q, respectively, i the variable - (pq for Muir recursio). For the umerical computatio we have used pq, 7 ad 9. For e.g. pq the polyomials have the followig coefficiets for the Muir recursio : Q ( - ) / - +/ - 4 +(/3+/ 3 ) -3 +/ , for CFE with the Tusti GF : Q ( - ) ( ) - +( ) -4 +( 3-73) -3 +(4 3 -) , (the polyomials P 9 has the same coefficiets except the odd coefficiets, which have opposite sig) ad for CFE with Al-Alaoui GF : Q ( - ){( ) - + ( ) -4 + ( ) -3 +( ) - +( ) } ad P ( - ){( ) - +( ) -4 + ( ) -3 + ( ) - +( ) Compariso of the discretiatio methods For all three above metioed discretiatio methods we have compared their Bode characteristics first oly for a simple FO differetiator ad secod by their applicatio to the followig FDE : ( ) a y + a y( u( ) ( t, (3) where is order of differetiatio - geerally a real umber, a, a are arbitrary costats. This FDE (3) we have used oly for evaluatio purposes, because the FO PI λ D cotroller, λ which is described by the FDE u( Ke( + Ti Dt e( + Td Dt e(, after their applicatio leads to the similar FDE - oly more complex. For equatio (3) we have compared the Bode characteristic too for their correspodig approximated trasfer fuctio : G( ) a i Q i i i Q i i K + a K T ± i B i i (4) ad we have made compariso of this approximatio methods i the time domai too. We have compared the aalytical solutio of the FDE (3) [Podlubý 994, Dorčák 994] for (, : y( a () () k t E +, t, Eα, β a a k Γ( αk + β ) (ad for spatial cases or the followig classical aalytical solutios of equatio (3) too y(/a (-exp(-a /a ), y(/a (-cos((a /a ) / ) with the umerical solutio of the FDE (3) based o the Gruwald-Letikov defiitio of the FO operator - PSE [Dorčák 994, etc.]: 83

4 y k u a T k at k b y i j k j, k,,...for (,, k,3,...for (,, (6) + a ( ) ( ) ( ) where b j are biomial coefficiets ( ) b, b j ( + ( ± )) / j b, ad with the j umerical solutio of the FDE (3) based o the Muir recursio or CFE with Tusti or Al- Alaoui operator : K y k a Pi y k i a Qi y k i + Qiu. (7) k i K K T i i i a P + aq K T We ca derive aalytical relatios for the Bode characteristic of the FO differetiator F D (s)t D s : π l( F ( iω) ) l( TD ) + l( ω) + i (8) D ad for the FDE () (F(s)/(a s -a )) : where : Im l( F( iω )), l( Re + Im ) i arctg, (9) Re Re Im [ a ω cos( π / ) + a ] + [ a ω si( π / ) ] [ a ω cos( π / ) + a ] + [ a ω si( π / ) ] a ω cos( π / ) + a a ω si( π / ), () Figure. Uit step resposes Figure. Uit step resposes The umerical solutio (PSE) (6) of the FDE (3) is i good agreemet with the aalytical solutio () at >9 for all (,. The agreemet of this solutio is very good for spatial 84

5 case or with the classical aalytical solutios. The accuracy was better by decreasig the time step. The umerical solutio (7) with the Muir recursio was uusable for the time step ear,s, CFE with Tusti or Al-Alaoui GF are better (Fig., ). I these cases we ca improve the accuracy by icreasig of the to 7 or 9, but the accuracy was worse by decreasig of the time step. There is a border,s for the CFE ad,s for the Muir recursio for all, 7 or 9 for FDE (). We ca see from Fig. 3, 4, that the Bode plots of the discretied FO differetiator ad FDE () are with large errors at higher frequecies ad this discretiatio is uusable for smaller time step (ustable poles ad/or eros). Figure 3. Bode plots of the differetiator Figure 4. Bode plots of the FDE This work was supported by grat VEGA /374/3 from the Sl. Grat Agecy for Sciece. Refereces OLDHAM, K. B., SPANIER, J The Fractioal Calculus. Academic Press, New York. SAMKO, S. G., KILBAS, A. A., MARICHEV, O. I Fractioal itegrals ad derivatives ad some of their applicatios. Nauka i techika, Misk, 987. PODLUBNY, I The Laplace Trasform Method for Liear Differetial Equatios of the FO. UEF--94, The Academy of Scie. Ist. of Exp. Phys., Košice, 994, 3 pp. PODLUBNY, I Fractioal Differetial Equatios. Academic Press, Sa Diego. WESTERLUND, S., EKSAM, L Capacitor Theory. I IEEE Tras.o Dielectrics ad Electrical Isulatio, vol, o, 994, pp MANABE, S. 96. The No-Iteger Itegral ad its Applicatio to Cotrol Systems. ETJ of Japa, 6(3-4):83-87,96. OUSTALOUP, A From Fractality to o Iteger Derivatio through Recursivity, a Property Commo to these two Cocepts : A Fudametal Idea from a ew Process Cotrol Strategy. I Proc. of th IMACS World Co., Paris, 988, vol 3, pp OUSTALOUP, A. 99. La Dérivatio o Etiere. HERMES, Paris. (i Frech) AXTELL, M., BISE, E. M. 99. Fractioal Calculus Applicatios i Cotrol Systems. I Proc. of the IEEE Nat. Aerospace ad Electroics Cof., New York, pp DORČÁK, Ľ Numerical Models for Simulatio the Fractioal-Order Cotrol Systems. UEF SAV, The Academy of Scieces, Ist. of Exp. Ph., Košice, Slovak Rep. 8

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