AALOG REALIZATIOS OF FRACTIOAL-ORDER ITEGRATORS/DIFFERETIATORS A Coparion Guido DEESD, Technical Univerity of Bari, Via de Gaperi, nc, I-7, Taranto, Italy gaione@poliba.it Keyword: Abtract: on-integer-order operator, Fractional-order controller, Rational approxiation, Interlaced ingularitie. on-integer differential or integral operator can be ued to realize fractional-order controller, which provide better perforance than conventional PID controller, epecially if controlled plant are of noninteger-order. In any cae, fractional-order controller are ore flexible than PID and enure robutne for high gain variation. Thi paper copare three different approache to approxiate fractional-order differentiator or integrator. Each approxiation realize a rational tranfer function characterized by a equence of interlaced iniu-phae zero and table pole. The frequency-doain coparion how that bet approxiation have nearly the ae zero-pole location, even if they are obtained tarting fro different point of view. ITRODUCTIO Originally, the invetigation of integral and derivative of any order wa a topic known a fractional calculu. In recent year, however, coniderable attention ha been paid to the concept of non-integer derivative and integral to odel yte in variou field of cience and engineering. In the reearch area of control theory, everal author have provided generalization of claical controller introducing variou type of Fractional- Order Controller (FOC). For exaple, the CROE (French acrony for Coande Robute d Ordre on Entièr ) controller (Outaloup, 99; Outaloup, 995) and Fractional-Order Proportional- Integral-Derivative (FOPID) controller PI λ D μ (Podlubny, 999a; Podlubny, 999b) have been recently conidered. Moreover, FOC have been uccefully applied in rigid robot, both for poition control and for hybrid poition-force-control (Tenreiro Machado and Azenha, 998; Valerio and Sá da Cota, 2). In general, FOC provide better perforance than PID controller, if the controlled plant are of non-integer-order. In other cae, FOC how high flexibility and can enure high robutne for high gain variation. More particularly, in SISO yte, they can ake the phae argin nearly not changing in a wide range around the gain croover frequency, even if high gain variation produce high change in gain croover frequency. Application in echatronic are tetified by everal paper (Canat and Faucher, 25; Li and Hori, 27; Ma and Hori, 2a; Ma and Hori, 2b; Ma and Hori, 27; Melchior et al., 25). The baic eleent of tranfer function of FOPID controller i the fractional differentiator/integrator ν, with ν poitive or negative real nuber. Thi operator i infinite dienional, even if it can be approxiated by finite-dienion tranfer function, whoe coefficient depend on the non-integer exponent ν. A good rational approxiation can be obtained by truncating the continued fraction expanion (CFE) of ν (, 26;, 28). Recently, in (Barboa et al., 26), leat-quare-baed ethod are ued for obtaining Fractional-Order Differential Filter (FODF) approxiating ν. In thi paper, a novel approach i copared to two coonly ued ethod to realize a rational approxiation of fractional-order differentiator or integrator. Thee operator are the baic eleent in fractional-order controller of echatronic yte. Section 2 reviit the three different ethod yteatically. Section copare the in the frequency doain. Section draw the concluion with oe reark. 8
AALOG REALIZATIOS OF FRACTIOAL-ORDER ITEGRATORS/DIFFERETIATORS - A Coparion 2 REVISITIG THREE RATIOAL APPROXIMATIOS In thi ection, three ethod are copared. They are hortly reviited, for aking a direct coparion baed on tranfer function putted in the ae for. All the conidered realization are known to be iniu-phae and table, with pole interlacing zero along the negative real half-axi of the -plane. Thi property i enlightened by the for of the three tranfer function, which explicitly how the frequencie correponding to the alternated zero and pole. The interlacing property i iportant for coparion purpoe, becaue the poition of the zero-pole pair deterine the quality of the odel approxiating phae and agnitude of the irrational operator (j) ν. Hence, for coparion purpoe, realization are contrained to have both their zero with iniu odule and their pole with axiu odule approxiately equal. All the approxiating tranfer function are in a factorized for, which put in evidence the break frequencie. Then, the lowet and highet break frequencie of the propoed ethod are taken a reference. 2. The Propoed CFE Approxiation The tarting point i the following continued fraction expanion (CFE): a a2 a j ( + x) ν b + () b + b + b + with b b, a ν x and: 2 a j n (n ν) x, b j 2n (2) a j+ n (n+ν) x, b j+ 2n+ () for j 2n, with n natural nuber (Khovankii, 965). The analog approxiation for the operator ν, with < ν <, i given in (, 28), where x i ued in () to obtain the (2)-th convergent of the reulting CFE a approxiating tranfer function: p ( ν ) + p( ν ) + + p ( ν ) () q ( ν ) + q ( ν ) + + q ( ν ) where j! and C(, j) i the binoial j!( j)! coefficient. Moreover: (ν+j+) (-j) (ν+j+) (ν+j+2) (ν+) (6) (ν ) (j) (ν ) (ν +) (ν +j+) (7) define the Pochaer function with (ν ) () (Spanier and Oldha, 987). A it i eaily noted, in thi ethod the coefficient p j (ν) and q j (ν) are explicitly given in ter of the fractional order ν. Obviouly, the poition of zero and pole in the - plane alo depend on ν. So, can be written in the for: + zi G ( k. (8) i + pi A it i proved in (, 28), zero ( z ) i and pole ( p ) of are all real and i interlace along the negative real half-axi in the -plane, with: <. (9) z p < z < p < < z < 2 2 p 2.2 Outaloup Recurive Approxiation The CROE controller i an integer-order frequency doain approxiation of ν in the for: + zi k. () i + The gain k i adjuted o that ν,) ha the ae croover frequency a the ideal operator ν. The nuber of zero and pole of the approxiating tranfer function i choen in advance. They alternate on the negative real half-axi of the -plane o that the frequencie atify: < < < < < < z p z2 p2 z p. () p i p j (ν) q,-j (ν) ( ) j C(, j) (ν+j+) (-j) (ν ) (j) (5) 85
ICICO 29-6th International Conference on Inforatic in Control, Autoation and Robotic In thi way, zero and pole interlace on the negative real half-axi, leading to a gain which i, approxiately, a linear function of the logarith of frequency. The phae i nearly contant and approxiate ν π / 2. The paraeter zi and pi are deterined by placing zero and pole a follow: ν H α L H ; η L ν ; z L η (2) p α i,..., () i z i z η i p i,...,. + i () The frequencie L and H are appropriately choen a L < z and H > p, o that it hold z z and p p. 2. Matuda Approxiation The Matuda ethod approxiate the operator ν fro it gain ν. The gain i deterined at 2+ frequencie,,, 2, which are taken logarithically paced in the approxiation interval. The interval [, 2 ] i choen o that the lowet break frequency ˆ z and the highet break frequency ˆ p in the odel atify: ˆ z z and ˆ p p, repectively. ote that, uually, an odd value of i ued, o that the reulting approxiation i proper. Then, the following function are defined: ( ) ; ( ) ; 2 k ν ( ) ( ) k 2 k ( ) 2 ( ) ; ( ) ( ) k ( 2 k ( ; ) 2 ) (5) fro which the following et of paraeter are obtained: ( ) ν α (6) k k α k (7) ) ( ) k ( k k k for k, 2,, 2. Uing the k and α k, the CFE can be written a: ν 2 α + (8) α + α + α + whoe convergent provide the rational approxiation to the irrational operator ν. The (2)-th convergent of (8) can be eaily converted into the rational approxiation, a the ratio Gˆ ( of two polynoial with degree. Then, the factorization of thee polynoial lead to: 2 + zi G kˆ ˆ ˆ( ν, ). (9) i + ˆ uerical experient how that, alo in thi cae, it hold: < < < ˆ ˆ. (2) ˆ z ˆ p < ˆ z < ˆ 2 p2 pi < z p A COMPARISO BETWEE THREE METHODS The approache of the previou ection are here copared, by chooing and then. Thee value are choen to ake the order of the FOC realization a low a poible, copatibly with good perforance. Figure, 2, and how the Bode plot of phae and aplitude, for the typical fractional order ν.5. Other value of the integer and of ν, with < ν <, can be conidered. A previouly tated, the approxiation i perfored o that, ν,) and Gˆ ( have their firt zero-frequency and their lat pole-frequency nearly equal. Hence, the zero-frequency z and the polefrequency p or p of are aued a reference. In concluion, it ut nearly hold: z z, ˆ z z, p p, and ˆ p p, when, and z z, ˆ z z, p p, and ˆ p p, when. Firt, the paraeter of are deterined. For ν.5 and, forula (8) give:.52,.66,.9, z z 2 z 86
AALOG REALIZATIOS OF FRACTIOAL-ORDER ITEGRATORS/DIFFERETIATORS - A Coparion p.29, p 2.572, p 9.957, and k.29. Thee value clearly indicate that i iniu-phae, table, with interlacing zero and pole. Figure report the phae Bode diagra of arg [ ν, j)] ( curve). Phae (degree) 5 5 5 25 2 5 Outaloup 5-2 - 2 Frequency (rad/) Matuda Figure : Phae Bode diagra for the approxiation of order a fractional-order differentiator, ν.5. ow, the procedure for deterining the function ν,) i conidered. With reference to (2), the interval [ L, H ] i choen larger than [, z ] p. More preciely, L z λ and H p λ 2, where λ and λ 2 are coefficient to be fixed o that the Outaloup algorith lead to z z and p p. Thee coefficient are choen by a rule of thub. Since z. 52 and p 9. 957, iple coputer experient in MATLAB how that chooing λ.55 and λ 2.8 yield: z.58, z. 2 559, z 5. 86, p.688, p. 2 7972, p 9. 29, k.692. A it i noted, the contraint z z and p p are repected. In Figure, arg[ν, j)] i alo reported (Outaloup curve). Finally, for applying the Matuda ethod, the apling frequencie are logarithically ditributed inide the approxiation interval, o that it ut reult: ˆ z z and ˆ p p, a requeted. Thi reult i achieved by chooing 2 λ z and p / λ. The paraeter λ i fixed by coputer experient to λ 5. aely, the following breaking frequencie of G ˆ( ν, j) reult: ˆ z ˆ z 2 ˆ z.85,.628,.5, ˆ p ˆ p 2 ˆ p.227,.6, 2.627, and ˆk.7. Thee value how that the contraint ˆ z z and ˆ p p are alo atified. A it can be eaily oberved, however, all the reaining frequencie and the gain of the Matuda odel are nearly equal to thoe of the author approxiating tranfer function. Thi fact i confired by the behaviour of arg [ Gˆ( ν, j)] in Figure (Matuda curve). The Bode plot, indeed, i nearly inditinguihable fro the plot of arg [ ν, j)]. In concluion, Figure how that arg [ Gˆ( ν, j)] and arg [ ν, j)] are nearly flat and give a good approxiation of arg [( j) ν ] ν π / 2. The plot of arg [ Gˆ( ν, j)] yield a lightly wort approxiation. Figure 2 confir that the agnitude plot of Gˆ ( and are nearly coincident. They give a better approxiation of ν than ν,), alo in thi cae. Aplitude (db) 2 5 5-5 - -5 Matuda Outaloup -2-2 - 2 Frequency (rad/) Figure 2: Aplitude Bode diagra for the approxiation of order of a fractional-order differentiator, ν.5. ow, let u conider a different approxiation obtained by uing and the ae procedure. For ν.5, forula (8) give: z., z 2., z.2, z 7.586, p.25, p 2.7, p., p 2.6, and k.. Then, G ( ν, ) i iniu-phae, table, with interlacing zero and pole. Figure how the phae Bode diagra of arg [ ν, j)] ( curve). For the Outaloup approxiation, λ.6 and λ 2.6 yield: z., z. 2 226, z.69, z. 927, p. 89, p.69 2, p. 27, p 2. 2, and k.77. The contraint z and z 87
ICICO 29-6th International Conference on Inforatic in Control, Autoation and Robotic p p are repected. In Figure, arg[ν, j)] i alo reported (Outaloup curve). For the Matuda approxiation, λ 9 give: ˆ z., ˆ z 2.27, ˆ z.2, ˆ z 7.572, ˆ p.2, ˆ p 2.75, ˆ p.55, ˆ p 2.2772, and ˆk.9. For, the frequency repone of arg [ Gˆ( ν, j)] i practically inditinguihable fro that of arg [ ν, j)] (Matuda and curve are practically the ae). Phae (degree) 5 5 5 25 2 5 Outaloup Matuda -2-2 Frequency (rad/) Figure : Phae Bode diagra for the approxiation of order of a fractional-order differentiator, ν.5. Aplitude (db) 2 5 5-5 - -5-2 -2-2 Frequency (rad/) Matuda Outaloup Figure : Aplitude Bode diagra for the approxiation of order of a fractional-order differentiator, ν.5. Figure confir that the agnitude plot of ˆ and are nearly the ae and give a better approxiation of ν than ν,), for. COCLUDIG REMARKS Thi paper copared three different ethod to approxiate non-integer-order differential or integral operator in fractional-order controller: thee ethod are the author, the Outaloup, and the Matuda, repectively. All approxiation of the irrational operator ν were realized through analog tranfer function characterized by table pole and iniu-phae zero. In particular, zero and pole were interlaced along the negative real half-axi of the -plane, and the firt and lat ingularitie were contrained to be nearly the ae in all approxiation. The interlacing property allowed u the coparion to find the bet ditribution of ingularitie. aely, a frequency doain analyi of the phae diagra howed that the author and Matuda approxiation outperfored the well-known by Outaloup. ote that all realization were liited to the lowet order that could guarantee good perforance. The better reult achieved by the propoed approxiation are due to a better ditribution of interlaced zero and pole. It i alo intereting to note how the propoed approxiation achieve nearly the ae zero-pole pair of the Matuda approxiation, even if the tarting point of the two ethod are copletely different. REFERECES Barboa, R.S., Tenreiro Machado, J.A., Silva, M.F., 26. Tie doain deign of fractional differintegrator uing leat-quare. Signal Proceing, Vol. 86, o., pp. 2567-258. Canat, S., Faucher, J., 25. Modeling, identification and iulation of induction achine with fractional derivative. In Fractional Differentiation and it Application, Le Mehauté, A., Tenreiro Machado, J.A., Trigeaou, J.C., Sabatier, J. (Ed.), Ubook Verlag Ed., euäß, Vol. 2, pp. 95-26. Khovankii, A.., 965. Continued fraction. In Lyuternik, L.A., Yanpol kii, A.R. (Ed.): Matheatical Analyi - Function, Liit, Serie, Continued Fraction, chap. V, Pergaon Pre. Oxford, International Serie Monograph in Pure and Applied Matheatic (tranl. by D. E. Brown). Li, W., Hori, Y., 27. Vibration uppreion uing ingle neuron-baed PI fuzzy controller and fractional-order diturbance oberver. IEEE Tranaction on Indutrial Electronic, Vol. 5, o., pp. 7-26. Ma, C., Hori, Y., 2a. Backlah vibration uppreion control of torional yte by novel fractional-order PID k controller. Tranaction of IEE Japan on Indutry Application, Vol. 2, o., pp. 2-7. Ma, C., Hori, Y., 2b. Fractional order control and it application of PI α D controller for robut two-inertia peed control. In Proceeding of the th International Power Electronic and Motion Control Conference 88
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