ELEKTRYKA 2016 Zeszyt 3-4 ( )

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1 ELEKTRYKA 206 Zeszyt 3-4 ( ) Rok LXII Waldemar BAUER, Jerzy BARANOWSKI, Tomasz DZIWIŃSKI, Paweł PIĄTEK, Marta ZAGÓROWSKA AGH Unversty of Sene and Tehnology, Kraków OUSTALOUP PARALLEL APPROXIMATION Summary: Ths artle presents Oustaloup approxmaton method for fratonal dfferental systems, whh has been modfed by the authors. Applaton of ths method by usng numeral experments has been shown and behavor of dsrete system has been analyzed. Keywords: Oustaloup, non-nteger order, dsretzaton, lnear systems APROKSYMACJA RÓWNOLEGŁA OUSTALOUPA Streszzene. W artykule przedstawono zmodyfkowaną przez autorów metodę aproksymaj Oustaloupa dla układów różnzkowyh neałkowtego rzędu. Pokazano jej wykorzystane poprzez eksperyment numeryzny oraz przeanalzowano zahowane systemu dyskretnego. Słowa kluzowe: Oustaloup, systemy neałkowtego rzędu, dyskretyzaja, systemy lnowe. INTRODUCTION Non-nteger ontrollers are a broadly researhed top. Questons of great mportane are the desgn of non-nteger order ontrollers and ther approxmaton allowng dsrete mplementaton (see []). There are some popular methods of onstrutng non-nteger order systems n the form of nteger order transfer funtons. There are however ertan ssues wth ther dsretzaton and subsequent mplementaton. In ths paper the authors propound a new method of approxmaton of non-nteger order systems based on a method proposed by Oustaloup (see [3]).

2 60 W. Bauer, J. Baranowsk, T. Dzwńsk, P. Pątek, M. Zagórowska 2. OUSTALOUP APPROXIMATION Oustaloup flter approxmaton to a fratonal-order dfferentator G ( s) s s wdely used n applatons \te{monje200fratonal}. An Oustaloup flter an be desgned as: where: G ( s) K t ' b N ' s s (2 ) / N u () (2) (3) b ( 2 ) / N u K h (4) h u (5) b Approxmaton s desgned for frequenes, b h and N s the order of the approxmaton. We may observe that ts representaton assumes the form of a produt of a seres of stable frst order lnear systems. Choosng a wde approxmaton band results n large u and hgh order of N results n spang of poles from lose to b. Ths spang s not lnear (there s a groupng near b dsretzaton. h to those very lose to ) and auses problems n 3. OUSTALOUP PARALLEL APPROXIMATION The method proposed n ths paper ams to mprove the spang of poles. Instead of reatng a hgh-order approxmaton (for N>5) over entre, nterval, t s proposed to reate a sum of the two approxmatons: one for lower L (s) and the other for hgher H (s) frequenes. Order of both approxmatons s N / 2 n. b b h L( j) j,, (6) h H ( j) j,, (7) The dvson s loated n the entral frequeny. Both approxmatons should be onneted n a seres wth lowpass and hghpass flters, respetvely (both wth utoff

3 Oustaloup Parallel Approxmaton 6 frequeny of presented n fgure. ). Those seres branhes are then onneted n parallel. Suh desgn s Beause low and hgh bands are approxmated separately, suh parallel onneton s onsstent wth approxmaton over the entre band; however, the poles are dfferently spaed. In ths paper onventonal frst order flters were used. Fg.. Oustaloup parallel approxmaton sheme Rys.. Shemat aproksymaj równoległej Oustaloupa 3. EXAMPLE The system 0.5 G( s) s has been approxmated by onventonal and parallel Oustaloup method. The order of approxmaton was hosen as N = 0 and the band of approxmaton was 0 6, 0 6. Central frequeny was hosen at. Parallel Oustaloup method for ontnuous frequeny responses behaves n a manner very smlar to the onventonal one (see Fg.2). Bggest dfferenes are vsble n phase haraterst at the entral frequeny. However, they are not sgnfant. Man reason for ntrodung Oustaloup parallel approxmaton was the dsretzaton proess. Stable onventonal Oustaloup approxmaton beomes unstable when dsretzed (f. Fg.3). Tustn dsretzaton was used; ths preserves stablty for deal numeral omputatons, but groupng of poles near b ntrodues errors. These errors n bas omputatons lead to dsrete poles that are grouped lose to, but also outsde unt rle. The presented method preserves the stablty better n the dsretzaton proess, as shown n Fg.4.

4 62 W. Bauer, J. Baranowsk, T. Dzwńsk, P. Pątek, M. Zagórowska Fg. 2. Oustaloup parallel approxmaton results Rys. 2. Wynk aproksymaj równoległej Oustaloupa Pole Zero Map Imagnary Axs Real Axs Fg. 3. Oustaloup approxmaton: dsrete poles and zeros Rys. 3. Aproksymaja Oustaloupa: beguny zera układu dyskretnego

5 Oustaloup Parallel Approxmaton 63 Pole Zero Map Imagnary Axs Real Axs Fg.4. Oustaloup parallel approxmaton: dsrete poles and zeros Rys.4. Aproksymaja równoległa Oustaloupa: beguny zera układu dyskretnego 4. CONCLUSIONS Proposed method shows promse n preservng stablty. In further researh t s planned to mplement PI α D µ ontrollers wth ths method, nvestgate seleton of dfferent flters to mprove frequeny response near and experment n ontrol of laboratory systems. ACKNOWLEDGMENT Work arred out n the framework of projet enttled Desgn and applaton of nonnteger order subsystems n ontrol systems. Projet was fnaned by Natonal Sene Centre (deson No. DEC-203/09/D/ST7/03960). REFERENCES. Bana P. and Baranowsk J.: Laguerre Polynomal approxmaton of fratonal order lnear systems. In W. Mtkowsk, J. Kaprzyk and J. Baranowsk, edtors, Advanes n the Theory and Applatons of Non-nteger Order Systems: 5 th Conferene on Non-nteger Order and Its Applatons, Craow, Poland, pages 7-82, Sprnger, 203

6 64 W. Bauer, J. Baranowsk, T. Dzwńsk, P. Pątek, M. Zagórowska 2. Monje C. A., Chen Y., Vnagre B. M., Xue D. and Felu V.: Fratonal-order systems and ontrols. Fundamentals and applatons. Advanes n Industral Control. Sprnger-Verlag, London, Oustaloup A., Levron F., Matheu B., and Nanot F.M.: Frequeny-band omplex nonnteger dfferentator: haraterzaton and synthess. Cruts and Systems I: Fundamental Theory and Applatons, IEEE Transatons on, 47():25_39, Mgr nż. Waldemar Bauer AGH Akadema Górnzo-Hutnza Faulty of Eletral Engneerng, Automats Computer Sene and Bomedal Engneerng Al. Adama Mkewza 30, Pawlon B e-mal: bauer@agh.edu.pl Dr nż. Jerzy Baranowsk AGH Unversty of Sene and Tehnology Faulty of Eletral Engneerng, Automats Computer Sene and Bomedal Engneerng Al. Adama Mkewza 30, Pawlon B e-mal: jb@agh.edu.pl Mgr nż. Tomasz Dzwńsk AGH Unversty of Sene and Tehnology Faulty of Eletral Engneerng, Automats Computer Sene and Bomedal Engneerng Al. Adama Mkewza 30, Pawlon B e-mal: tdz@agh.edu.pl Dr nż. Paweł Pątek AGH Unversty of Sene and Tehnology Faulty of Eletral Engneerng, Automats Computer Sene and Bomedal Engneerng Al. Adama Mkewza 30, Pawlon B e-mal: pp@agh.edu.pl Mgr nż. Marta Zagórowska AGH Unversty of Sene and Tehnology Faulty of Eletral Engneerng, Automats Computer Sene and Bomedal Engneerng Al. Adama Mkewza 30, Pawlon B e-mal: zagor@agh.edu.pl