Discretization of Fractional Order Differentiator and Integrator with Different Fractional Orders

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1 Inelligen Conol and Auomaion, 207, 8, hp:// ISSN Online: ISSN Pin: Disceizaion of Facional Ode Diffeeniao and Inegao wih Diffeen Facional Odes Qi Zhang, Baoye Song *, Huadong Zhao, Jiansheng Zhang College of Elecical Engineeing and Auomaion, Shandong Univesiy of Science and Technology, Qingdao, China How o cie his pape: Zhang, Q., Song, B.Y., Zhao, H.D. and Zhang, J.S. (207) Disceizaion of Facional Ode Diffeeniao and Inegao wih Diffeen Facional Odes. Inelligen Conol and Auomaion, 8, hps://doi.og/0.4236/ica Received: Mach 4, 207 Acceped: May 7, 207 Published: May 0, 207 Copyigh 207 by auhos and Scienific Reseach Publishing Inc. This wok is licensed unde he Ceaive Commons Aibuion Inenaional License (CC BY 4.0). hp://ceaivecommons.og/licenses/by/4.0/ Open Access Absac This pape is concened wih he disceizaion of he facional-ode diffeeniao and inegao, which is he foundaion of he digial ealizaion of facional ode conolle. Fisly, he paameeized Al-Alaoui ansfom is pesened as a geneal geneaing funcion wih one vaiable paamee, which can be adjused o obain he commonly used geneaing funcions (e.g. Eule opeao, Tusin opeao and Al-Alaoui opeao). Howeve, he following simulaion esuls show ha he opimal vaiable paamees ae diffeen fo diffeen facional odes. Then he weighed squae inegal index abou he magiude and phase is defined as he objecive funcions o achieve he opimal vaiable paamee fo diffeen facional odes. Finally, he simulaion esuls demonsae ha hee ae gea diffeences on he opimal vaiable paamee fo diffeenial and inegal opeaos wih diffeen facional odes, which should be aacing moe aenions in he design of digial facional ode conolle. Keywods Facional Ode Conol, Facional Ode Opeao, Al-Alaoui Tansfom, Disceizaion. Inoducion Facional ode calculus has a hisoy of moe han 300 yeas, which exends he ode of he classical calculus fom inege numbe o abiay eal numbe and even complex numbe. Compaed wih inegal ode calculus, he facional ode calculus could descibe he dynamic chaaceisics of he acual sysem moe accuaely. Theefoe, facional ode conol is inceasingly becoming one of he mos impoan opics in conol heoy in ecen yeas []. DOI: /ica May 0, 207

2 The disceizaion of he facional-ode diffeeniao and inegao is he foundaion of he digial ealizaion of facional ode conolle. Geneally, hee ae wo mehods fo he disceizaion of he facional-ode diffeeniao and inegao [2], i.e., diec disceizaion and indiec disceizaion, while he fome is moe pacical in eal applicaions [3]. Seveal algoihms have been poposed fo he diec disceizaion mehod, e.g., he powe seies expansion (PSE) of he Eule opeao, he coninued facional expansion (CFE) of he Tusin opeao, ec. [4]. The disceizaion of he facional-ode diffeeniao is aken fo example. The diec disceizaion mehod could be summaized as he following wo seps. Fisly, some kind of geneaing funcion ω ( z ) is used o disceize he diffeeniao s, i.e., s = ω ( z ), whee is he ode of he facional- ode diffeeniao and ω ( z ) is usually expessed as a funcion of he com- plex vaiable z o he shif opeao z, and hen some kind of expansion mehod is applied o geneae he appoximae digial file of he diffeeniao. Fo example, Chen and Vinage popose an IIR (infinie impulse esponse)-ype digial facional-ode diffeeniao wih weighed sum of Simpson inegaion ule and he apezoidal inegaion ule [3]. Al-Alaoui opeao and CFE ae applied in he disceizaion of facional-ode opeao fo bee disceizaion appoximaion [5]. Zhu and Zou popose an impoved ecusive algoihm fo facional-ode sysem soluion based on PSE and Tusin opeao [6]. Miladinovic and his colleagues use geneic algoihm o minimize he deviaion in magiude and phase esponses beween he oiginal facional ode elemen and he aionalized discee ime file in IIR sucue [7]. The disceizaion mehods fo facional-ode diffeeniao ae compaed based on Tusin opeao and hee diffeen expansion algoihms in [8]. A class of geneaing funcion called paameeized Al-Alaoui ansfom is pesened and he simulaion analysis indicaes ha he vaiable paamee could be adjused o achieve ceain opimal digial file appoximaion fo he facional-ode opeao [9]. Howeve, he vaiable paamee of he opimal digial file is diffeen fo he facional-ode opeao wih diffeen odes, his issue has no been inensively discussed in pevious papes and will be sudied in his pape. The main conibuions of his pape ae oulined as heefold. () The paameeized Al-Alaoui ansfom is analysed hooughly o show he effec of he vaiable paamee. (2) A weighed squae inegal index abou he magiude and phase is defined as he objecive funcions o obain he opimal vaiable paamee fo diffeen facional odes. (3) Some simulaions ae implemened o demonsae he gea diffeences on he opimal vaiable paamee fo diffeenial and inegal opeaos wih diffeen facional odes. The emainde of his pape is oganized as follows. The peliminay of facional calculus is biefly inoduced in Secion II. The disceizaion of facionalode opeao wih diffeen odes is discussed in Secion III. Finally, he conclusions ae given in he las secion. 76

3 2. Peliminay of Facional Calculus A. Definiion of Facional Calculus Facional ode calculus is a naual genealizaion of he classical inegal ode calculus, which exends he ode of he inegaion and diffeeniaion o he non-inege o facional ode [0]. Two of he commonly used definiions of he facional calculus ae he Günwald-Lenikov (GL fo sho) definiion and he Riemann-Liouville (RL fo sho) definiion. The GL definiion is defined as a a h j D f ( ) = lim h ( ) f ( jh), () h 0 j= 0 j whee ad is he facional-ode calculus opeao, a and ae he limis of he opeao especively, is he ode of he opeao and [ ] means he inege pa. The RL definiion is defined as f ( τ ) ( ) d D d τ, (2) ( n ) d n a f ( ) = Γ n a n τ + whee, n n Γ is he Gamma funcion. Acually, he afoemenioned definiions ae equivalen o each ohe in he eal physical sysems and engineeing applicaions. The Laplace ansfom of facional ode calculus wih zeo iniial condiions fo ode is < < and ( ) whee {} ansfom of funcion f ( ) and { a f ( ) } = s ( s), L D F (3) L denoes he opeaion of Laplace ansfom, ( s) F is he Laplace s denoes he facional-ode opeao []. The disceizaion of facional-ode opeao is o design a digial file o ap poximae he opeao s, whee could be posiive o negaive fo diffeeniao and inegao, especively. B. Paameeized Al-Alaoui Tansfom The disceizaion of he facional-ode diffeeniao and inegao is o design a digial file fo he facional-ode opeao s. Fisly, a geneaing funcion is used o ealize he ansfom of he Laplace opeao fom he coninuous complex fequency domain o he discee complex fequency domain. Accoding o he definiion of Z ansfom, he mapping elaionship beween he wo domains is st z = e, (4) whee T is he sampling peiod. Fuhemoe, he equivalen elaion can be fomulaed ( α ) Ts st s( ( α) T+ αt) e z = e = e =, (5) αts e whee [ ] 0, α. Take powe seies expansion o he numeao and denominao of Equaion (5) and neglec he high-ode ems, Equaion (5) can be 77

4 appoximaed as ( α ) Ts n! + 0 ( α n ) Ts = z =, k αts k! ( αts) k= 0 and hen he complex vaiable s can be solved o yield n z s = f ( z, α ) =. T + α ( z ) (6) (7) Equaion (7) is defined as he α ansfom fom coninuous complex fequency domain o he discee complex fequency domain in [2] and [3]. The analyses pesened in [5] and [4] pove ha he so-called α ansfom is acually equivalen o he Al-Alaoui ansfom [5] [6] wih vaiable paamee a when paamee α is se o ( a) + 2, i.e., 2( z ) ( + ( + )) z s = =. T + α ( z) T a a z Specially, hee commonly used geneaing funcions, i.e., Tusin ansfom, Al-Alaoui ansfom and Eule ansfom can be fomulaed when a is se o 0, 3/4 and, especively. A digial inegao wih adjusable paamees, which is used in he discei- zaion of facional-ode opeao [7], is pesened in [8]. The expession of he geneaing funcion is z s = βt γ + γ z ( ) whee β and γ ae gain adjusing paamee and phase adjusing paamee, especively. The facional-ode opeao can be adjused moe accuaely wih he adjusing paamee accoding o diffeen eal applicaions. Specially, Equaion (9) is equivalen o Equaion (7) when paamee β is se o. Addiionally, if he paamee α in Equaion (7) is se o ( + δ ), he geneaing funcion is fomulaed + δ z s = T + δ z which is used fo he disceizaion of facional-ode opeao in [9]. Table shows he elaionship of he common geneaing funcions and he vaiable paamees. Fom above analyses, he geneaing funcions in Equaions (7), (8), (9) and (0) ae acually equivalen o each ohe when ceain elaions of he vaiable paamees ae saisfied as shown in Figue. We call hem paameeized Al-Alaoui ansfom in [9], a ceain expession could be used in ceain specific issues. C. Powe Seies Expansion of Geneaing Funcion Two main mehods fo he expansion of geneaing funcion ae coninued facional expansion and powe seies expansion. Geneally, CFE mehod could,, (8) (9) (0) 78

5 Figue. Relaions of paameeized Al-Alaoui ansfom. Table. Relaionship of common geneaing funcions and vaiable paamees. α a γ δ s z Tansfom 0 z s T Eule 7/8 3/4 7/8 /7 /2 0 /2 8 z s + 7T z 7 2 z s T + z Al-Alaoui Tusin obain IIR-ype digial file which is easy fo he digial file design, while PSE mehod could obain FIR (finie impulse esponse)-ype digial file and equies less compuaion cos in compaison wih CFE unde simila accuacy cieion [4]. In his pape, Equaion (0) is used as he geneaing funcion, and he powe seies expansion algoihm pesened in [20] is applied o obain an IIR-ype digial file. The geneal fom of he disceizaion of facional-ode opeao is D ( z ) PSE{ ω ( z )} pq, 2 z PSE T + z PSE 2 = T PSE + P Q pq, {( z ) } ( z ) { } ( z ) ( z ) n p0 + pz + + pz n n 0 n 2 = T q + qz + + qz whee, ( z ) PSE{} indicaes he powe seies expansion, Pp ( z ) and q ( ) p q p q () D is he digial file of he discee facional-ode opeao, Q z ae he 79

6 numeao and denominao polynomials of he digial file, and p and q ae hei odes especively. Wihou loss of genealiy, he appoximae odes of he numeao and denominao polynomials ae se o n (see [20] fo moe deails). 3. Disceizaion of Facional-Ode Opeao wih Diffeen Odes A. Objecive Funcion Al-Alaoui ansfom has bee popeies in he disceizaion of facionalode opeao in compaison wih ohe common geneaing funcions, which have been epoed in seveal papes (see e.g. [4] [7] [2]). In he paameeized Al-Alaoui ansfom, he paamee δ could be adjused o achieve ceain opimal digial file appoximaion of he facional-ode opeao. An objecive funcion is defined in [7], which is o obain he opimal IIR-ype digial file ealizaion by minimizing he weighed sum of he discepancies beween he esponses of he coninuous ime facional ode file and is appoximae digial file ealizaion. Howeve, he papes menioned above usually conside ceain specific facional ode (e.g. facional ode 0.5) and ae no concened wih he disceizaion of facional-ode opeao wih diffeen odes, which will influence he design of he digial file. In his pape, we define he following objecive funcions ωu 2 Jmag = Mc( jω) Md ( jω) dω ω ωl b ωu 2 Jag = Ac( jω) Ad ( jω) dω ω (2) ωl b min J = w J + w J mag ( ) whee Mc ( jω ) and Md ( ) ode diffe-inegao and is discee counepa; Ac ( jω ) and Ad ( ) jω ae he magiude esponses of he facionaljω ae he coesponding phase esponses; ω b is he bandwidh beweeen he lowe and uppe limis ω l and ω u (i.e. ( ωb = ωu ωl) ) wihin a chosen fequency band, e.g. ω 0, ω N wih Nyquis fequency ω N. The nomalized J mag and J ag depend on no only he paamee δ, bu also he diffeen odes. The following simulaion analyses ae o find he opimal paamee δ fo diffeen facional odes, which could achieve he minimal objecive funcion J wih specific weigh w. B. Simulaion Resuls In he simulaion, he weigh w is aken as 0.75 wihou loss of genealiy. The odes of he powe seies expansion ae aken as 5 fo simpliciy, he sampling peiod is aken as 0.00, and he odes of he facional-ode opeao ae ypically aken as 0., 0.5, 0.9 and 0., 0.5, 0.9 fo diffeenial and inegal opeaos especively. Figues 2-4 ae he objecive funcions fo diffeenial opeao wih facional ode 0., 0.5 and 0.9, whee he hoizonal coodinae indicaes he diffeen ag 80

7 Figue 2. Objecive funcions fo diffeenial opeao wih facional ode 0.. Figue 3. Objecive funcions fo diffeenial opeao wih facional ode 0.5. Figue 4. Objecive funcions fo diffeenial opeao wih facional ode

8 vaiable paamee δ and he longiudinal coodinaes indicae he vaiaion of J mag, J ag and J especively. Figues 5-7 ae he counepas of he inegal opeao. All he figues demonsae ha he vaiaion ends of he objecive funcions wih vaiable paamee δ seem consisen wih each ohe, while he opimal vaiable paamees ae oally diffeen fo facional diffeenial and inegal opeao wih diffeen facional odes unde he seleced objecive funcions. Table 2 shows he opimal vaiable paamees δ fo he specific odes of he diffeenial and inegal opeaos. Figue 5. Objecive funcions fo diffeenial opeao wih facional ode 0.. Figue 6. Objecive funcions fo diffeenial opeao wih facional ode

9 Figue 7. Objecive funcions fo diffeenial opeao wih facional ode 0.9. Table 2. Opimal vaiable paamee δ. Facional Ode Opimal δ fo Opimal δ fo Diffeenial Opeao Inegal Opeao Conclusions This pape is concened wih he disceizaion of he facional-ode diffeeniao and inegao wih diffeen facional odes, which is seldom consideed in pevious lieaues. The paameeized Al-Alaoui ansfom wih one vaiable paamee is pesened as a geneal geneaing funcion, and he objecive funcions ae defined o achieve he opimal vaiable paamee fo he disceizaion. The simulaion esuls demonsae ha hee ae gea diffeences on he opimal vaiable paamees fo he disceizaion of diffeeniao and inegao wih diffeen facional odes. Howeve, he weigh in he simulaion is se as 0.75 wihou loss of genealiy, and i is undoubedly abiay o selec he pope weigh fo specific disceizaion pupose, e.g. selec smalle weigh fo moe accuae phase appoximaion. In he fuue, we will conside he opimal vaiable paamee ino he pacical digial facional ode conolle design o acquie he opimal conol pefomances. 83

10 Acknowledgemens This wok is suppoed by a Pojec of Shandong Povince Highe Educaional Science and Technology Pogam unde Gan J4LN34. Refeences [] Chen, Y., Peáš, I. and Xue, D. (2009) Facional Ode Conol A Tuoial. Poc. Ameican Conol Confeence, S. Louis, MO, 0-2 June 2009, hps://doi.og/0.09/acc [2] Chen, Y. and Mooe, K. (2002) Disceizaion Schemes fo Facional-Ode Diffeeniaos and Inegaos. IEEE Tansacions on Cicuis and Sysems I: Fundamenal Theoy and Applicaions, 49, hps://doi.og/0.09/ [3] Chen, Y., Vinage, B. and Podlubny, I. (2004) Coninued Facion Expansion Appoaches o Disceizing Facional Ode Deivaives An Exposioy Review. Nonlinea Dynamics, 38, hps://doi.og/0.007/s x [4] Song, B., Xu, L. and Lu, X. (204) A Compaaive Sudy on Tusin Rule Based Disceizaion Mehods fo Facional Ode Diffeeniao. Poceedings of Inenaional Confeence on Infomaion Science and Technology, Shenzhen, Apil 204, [5] Al-Alaoui, M.A. (2006) Al-Alaoui Opeao and he α-appoximaion fo Disceizaion of Analogue Sysems. Faca Univesiais, Se.: Elec. and Enge., 9, [6] Zhu, C. and Zou, Y. (2009) Impoved Recusive Algoihm fo Facional-Ode Sysem Soluion based on PSE and Tusin Tansfom. Sysems Engineeing and Eleconics, 3, [7] Das, S., Majumde, B., Pakhia, A., Pan, I., Das, S. and Gupa, A. (202) Opimizing Coninued Facion Expansion Based IIR Realizaion of Facional Ode Diffe- Inegaos wih Geneic Algoihm. 20 Inenaional Confeence on Pocess Auomaion, Conol and Compuing, Coimbaoe, July 20, -6. [8] Song, B., Xu, L. and Lu, X. (205) Discee Appoximaion of Facional-Ode Diffeeniao Based on Tusin Tansfom. Science Technology and Engineeing, 5, [9] Song, B., Xu, L., Lu, X. and Wang, H. (205) Disceizaion Scheme fo Facional- Ode Diffeeniaos and Inegaos Based on Paameeized Al-Alaoui Tansfom. Poceedings of 34h Chinese Conol Confeence, Hangzhou, July 205, [0] Chen, Y., Vinage, B., Xue, D. and Feliu, V. (200) Facional-Ode Sysems and Conols Fundamenals and Applicaions. Spinge, London. [] Vinage, B., Podlubny, I., Henandez, A. and Feliu, V. (2000) Some Appoximaions of Facional Ode Opeaos used in Conol Theoy and Applicaions. Facional Calculus and Applied Analysis, 3, [2] Sekaa, T.B. and Sojic, M.R. (2005) Applicaion of he α-appoximaion fo Disceizaion of Analogue Sysems. Faca Univesiais, Se.: Elec. and Eneg., 8, [3] Sekaa, T.B. (2006) New Tansfomaion Polynomials fo Disceizaion of Analogue Sysems. Elecical Engineeing, 89, hps://doi.og/0.007/s [4] Al-Alaoui, M.A. (2008) Al-Alaoui Opeao and he New Tansfomaion Polynomials fo Disceizaion of Analogue Sysems. Elecical Engineeing, 90, hps://doi.og/0.007/s

11 [5] Al-Alaoui, M.A. (993) Novel Digial Inegao and Diffeeniao. Eleconics Lees, 29, hps://doi.og/0.049/el: [6] Al-Alaoui, M.A. (997) Filling he Gap beween he Bilinea and he Backwad Diffeence Tansfoms: An Ineacive Design Appoach. Inenaional Jounal of Elecical Engineeing Educaion, 34, hps://doi.og/0.77/ [7] Babosa, R.S., Machado, T. and Silva, M.F. (2006) Time Domain Design of Facional Diffeinegaos Using Leas-Squaes. Signal Pocessing, 86, [8] Smih, J.M. (987) Mahemaical Modeling and Digial Simulaion fo Enginees and Scienis. 2nd Ediion, Wiley, New Yok. [9] Caponeo, R., Dongola, G. and Founa, L. (200) Facional Ode Sysems: Modeling and Conol Applicaions. Wold Scienific Publishing, Singapoe. [20] Peas, I. (202) A New Discee Appoximaion of he Facional-Ode Opeao. 3h Inenaional Capahian Conol Confeence (ICCC), High Taas, 28-3 May 202, hps://doi.og/0.09/capahiancc [2] Chen, Y. and Vinage, B. (2003) A New IIR-Type Digial Facional Ode Diffeeniao. Signal Pocessing, 83, Submi o ecommend nex manuscip o SCIRP and we will povide bes sevice fo you: Acceping pe-submission inquiies hough , Facebook, LinkedIn, Twie, ec. A wide selecion of jounals (inclusive of 9 subjecs, moe han 200 jounals) Poviding 24-hou high-qualiy sevice Use-fiendly online submission sysem Fai and swif pee-eview sysem Efficien ypeseing and poofeading pocedue Display of he esul of downloads and visis, as well as he numbe of cied aicles Maximum disseminaion of you eseach wok Submi you manuscip a: hp://papesubmission.scip.og/ O conac ica@scip.og 85

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