THE FRACTIONAL INTEGRATOR AS REFERENCE FUNCTION 1. B. M. Vinagre C. A. Monje A. J. Calderón Y. Q. Chen V. Feliu

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1 THE FRACTIONAL INTEGRATOR AS REFERENCE FUNCTION B. M. Vnagre C. A. Monje A. J. Calderón Y. Q. Chen V. Felu Ecuela de Ingenería Indutrale, Unverdad de Extremadura, Badajoz, Span, emal: CSOIS, Utah State Unverty, Logan, Utah, USA, emal: Ecuela Técnca Superor de Ingenero Indutrale, Unverdad de Catlla-La Mancha, Cudad Real, Span, emal: Abtract: The purpoe of th paper, on one hand, to analyze the fractonal ntegrator a reference ytem for control by comparon to the poton ervo n order to nd ther mltude and d erence, and, on the other hand, to revew the approache prevouly made for degn pec caton n every doman when ung the fractonal ntegrator a reference ytem, and to propoe and dcu new approache. Keyword: Fractonal Integrator, Reference Functon, Servo Mechanm, Degn Spec caton. INTRODUCTION Tradtonally the poton ervo ha been ued a reference open loop ytem for automatc control purpoe (ee (Ogata, Özbay 999)). That, a reference ytem for controller degn n order to obtan a cloed loop controlled ytem wth precrbed behavour ung pec caton n the frequency doman (phae margn, croover frequency, etc.), the complex plane (domnant pole locaton) or the tme doman tep repone (re tme, overhoot, peak tme, ettlng tme, etc.), and the correpondng equvalence between thoe doman and wth the charactertc parameter (dampng rato and natural frequency). On the other hand, from the very begnnng of the ue of fractonal calculu n control (ee Th work ha been partally upported by the Spanh Reearch Grant PRA4 (Junta de Extremadura) (Manabe 96, Outaloup 995)) the fractonal ntegrator ha been condered a an alternatve reference ytem for control purpoe n order to obtan cloed loop controlled ytem robut to gan change. In any cae, for ung a ytem a a reference one, t neceary to know t frequency and tme behavour and to de ne clearly t charactertc parameter and t relaton wth ueful pec caton. The purpoe of th paper, on the one hand, to compare the two reference ytem cted above n order to nd ther mltude and d erence, and, on the other hand, to revew the approache prevouly made for degn pec caton n every doman when ung the fractonal ntegrator a reference ytem, and to propoe and dcu new approache. The paper organzed a follow. Secton devoted to a bref revew of the poton ervo a reference ytem and the orgn and meanng

2 y(t) of ome relaton between pec caton n every doman of analy. Secton 3 follow an analogou procedure for the analy of the fractonal ntegrator n order to etablh ome gudelne when ung t a reference ytem for control. Secton 4 gve ome concluon.. THE POSITION SERVO A typcal poton ervo hown n gure, where J and B repreent the load element correpondng to nerta (J ) and vcou frcton (B) e ect. The cloed loop tranfer functon can be expreed a: the rt tme t nal value), and the peak tme, t p ; (the tme at whch the abolute maxmum occur) decreae, and the ettlng tme, t ; (the tme requred for the repone to ettle wthn a mall devaton from t nal value), and the overhoot, M p ; ncreae. It can be alo oberved that the lope of the curve are very cloe near the orgn, and have ncreang value a decreae Unty tep repone of the ervo ( ω n =) δ=! n F () = +! n +! () n where = B=( p JK) repreent the dampng rato, and! n = p K=J the undamped natural frequency δ= tme (ec) Fg.. Step repone of the cloed loop ervo for [; ): Fg.. Typcal ervo ytem. The pole of () are n: ; =! n j! n p () p beng! n t magntude and tan t argument. The unt tep repone of the form: n y(t) = e!nt p (3) p! p! n t + tan whch, for the uual cae of underdamped ytem ( < < ), a damped ocllaton wth attenuaton or decayng factor =! n ; and damped or proper frequency! p =! n p ; beng! n the undamped natural frequency and the dampng rato or attenuaton for unty natural frequency. Th unty tep repone repreented n gure for [; ) and! n = (note that! n 6= re ected a a tme calng). In th gure t can be oberved that a decreae, the re tme, t r ; (tme for the repone to acheve for For the de nton of the charactertc parameter and pec caton of the ytem () n the frequency doman, t mut be taken nto account that the tartng pont of uch a tranfer functon the damped RLC ocllator, n whch the qualty factor, Q; de ned for loy col (R 6= ) a the rato of reactance to retance at frequency! n = = p LC (ee, e.g., (Van Valkenburg 98)). The frequency doman charactertc for the normalzed cae, jf (j)j = ; can be obtaned a follow: Qualty factor: Q = jf (j! n )j = (4) Reonant frequency (frequency for maxmum of the magntude):! r =! n p (=Q ) =! n p (5) Reonance factor (maxmum of the magntude): M r = jf (j! r )j = = p Q p (=4Q ) = (6) The Bode plot of the cloed loop ervo are preented n gure 3 for [; ) and! n = : It can be oberved that aymptotcally (!! ) the lope of the magntude curve 4dB=dec and the phae : Furthermore, there an ncreang wdeband wth the decreang of ; and

3 Phae (degree) Magntude (db) - Servo (Cloed loop, δ=[,), ω n =) Th ytem alo deal n the ene that for t the Bode gan-phae relaton (ee (?, Özbay 999)) are exact relaton. For (8), the equaton: º frequency (rad/) Fg. 3. Bode plot of the cloed loop ervo for [; ): a crong pont of the phae curve at! = wth phae value : Ueful relaton between tme and frequency charactertc can be tablhed. For example, the phae margn can be obtaned a: P M = tan q p (7) 3. THE FRACTIONAL INTEGRATOR 3. Htorcal Overvew Maybe the rt menton of the nteret of conderng a fractonal ntegrod erental operator n a feedback loop, though wthout menton of the term fractonal, wa made by Bode n (Bode ), and next n a more comprehenve way n (Bode 945). A key problem n the degn of a feedback ampl er wa to come up wth a feedback loop o that the performance of the cloed loop were nvarant to change n the ampl er gan. Bode preented an elegant oluton to th robut degn problem, whch he called the deal cuto charactertc, nowaday known a deal loop tranfer functon, whoe Nyqut plot a traght lne through the orgn gvng a phae margn nvarant to gan change. Clearly, th deal ytem, from our pont of vew, a fractonal ntegrator wth tranfer functon:!g G () = (8) beng the gan croover frequency, and the contant phae margn, P M = (9) wth: arg (G (j! )) =! u = ln! Z ; M(u) = d juj W (u) = ln coth gve: M(u)W (u)du () du jg (j! e u )j ;() () arg (G (j! )) = (3) that, the phae at any frequency contant, and o, exactly proportonal to the dervatve of the magntude (lope) on a logarthmc frequency cale: Though the noton ntroduced by Bode n (Bode ) (mnmum phae, phae margn, Bode plot, etc.) became key element of automatc control, Bode deal loop tranfer functon wa not much dcued n the lterature for a long tme, although t can be condered a natural begnnng of robut control. The rt propoal for the ue of the fractonal ntegrator (n fact the Bode deal ytem) wa made n (Tutn et al. 958). In th work, the requrement for the degn of large poton-control ytem lead to contrant on the form of the frequency charactertc, and the deal one reult to be a fractonal order ntegrod erentator ful llng the Bode gan-phae relatonhp. It alo mentoned n that paper the deal ervo mechanm (no frcton) havng a tranfer functon C() = k (4) whch a lmt cae of (8) wth = : The rt explct menton of the fractonal ntegrator n control can be found n (Manabe 96). In that paper the fractonal ntegrator tuded for the rt tme a a reference ytem for control, t frequency and tep repone are analyzed, and expreon for charactertc parameter and uual tme and frequency pec caton are gven. De ntve apportaton for the ue of the fractonal ntegrator a reference ytem for control were made durng the evente n two way. The rt one, recoverng the Bode dea of robutne and extendng t to other parameter varaton, led to the Qualtatve Control Theory of Horowtz (ee, e.g., (Horowtz 993),(Åtröm 999)). The econd one, partcularly nteretng for u, wa the

4 ntroducton of the Robut Control of Non Integer Order (CRONE control (Outaloup 995)), n whch, takng the fractonal ntegrator a reference ytem for loop hapng (partcularly n the econd generaton CRONE control), ytematc procedure for ytem analy, controller degn, and controller mplementaton have been developed durng two decade, becomng an alternatve paradgm n robut control. In what follow the fractonal ntegrator n cloed loop wll be analyzed n the d erent doman n order to compare the reult wth the obtaned for the ervo. 3. Complex plane analy The fractonal ntegrator (8) n cloed loop, ha a tranfer functon of the form: F () = +!g = (5) + and the cloed loop pole, for < < (equvalent to the underdamped ervo, < < ); are: ; = e j =!g co j n (6) beng the croover frequency,, t magntude and tan t argument (It mut be noted that the condered oluton of the charactertc equaton n 5 are only the correpondng to the man heet of the Remann urface de ned by < arg() < ). By comparng expreon () and (6), expreon for charactertc parameter can be obtaned a follow: Natural frequency a magntude of the pole:! n = j ; j = (7) Attenuaton a real part of the pole: = real( ; ) = co (8a) Dampng rato a attenuaton for! n = : = real( ; )!g= = co ; (9) Damped frequency a magnary part of the pole:! d = Im ( ; ) = n q =! n : () A can be oberved, the charactertc parameter have expreon analogou to the cae of the ervo, wth the dampng rato dependng only on : But n th cae, the natural frequency (whch the ocllaton frequency for no dampng cae, = ) equal to the croover frequency, whch not the cae of the ervo. The former expreon, (7) to (), are the expreon ued n (Outaloup 995). 3.3 Frequency doman analy The expreon for the frequency repone of (5) : F (j!) = +! co + j! n () From () and wth the uual de nton, the followng expreon can be obtaned for the charactertc performance: Qualty factor (magntude for natural frequency): Q a = jf (j )j = = p p + co () Reonant frequency (frequency for maxmum magntude):! r = co (3) Reonance factor (maxmum magntude): M r = jf (j! r )j = n (4) The lat two expreon, (3) and (4), are ued n (Outaloup 995) and (Manabe 96), but n the lat paper an alternatve de nton of dampng rato ntroduced baed on complex planefrequency relaton of the poton ervo (equaton (5) and (6)). By dong o, the expreon for the dampng rato : m = r p + co (5) beng th equvalent dampng rato, m ; the dampng rato of a ervo whoe reonance peak equal to the reonant peak of the fractonal ntegrator. It mut be noted that, n correpondence wth the doman of de nton, for = the dampng :77 (frequency doman, (5)), or (complex plane, (9)). The frequency repone of the fractonal ntegrator n cloed loop hown n gure 4. A can be oberved, there are mportant d erence between thee frequency repone and the correpondng to the ervo. In th cae, there no crong pont n phae, but there a crong pont n magntude n 6dB, that, there a contant 6dB wdeband. The aymptotc value are arg(f (j) = for phae, and the aymptotc lope of the magntude curve are jf (j)j = db=dec (t mut noted that for the ervo cae, thee aymptotc value were

5 ndependent of ): Of coure, thee d erence wll be re ected n the tme doman unty tep repone Fractonal Integrator (Cloed loop, ω n =) - - Fg. 4. Bode plot of the fractonal ntegrator n cloed loop for (; ]. 3.4 Tme doman analy The unty tep repone of the cloed loop ytem (5) of the form (ee (Goren o and Manard 996, Barboa et al. 3)): 3 y (t) = $ 4 5 = + E ( ( t) ) (6) denotng by $ [] the nvere Laplace tranformaton, and beng E () the one-parameter Mttag-Le er functon (Podlubny 999, Goren- o and Manard 996). Followng (Goren o and Manard 996), for the cae of < < ; the functon E ( t ) can be expreed on the form: E ( t ) = f (t) + g (t) (7) beng the functon f (t); Z f (t) = n () + co () + e t d (8) a functon whch vanhe dentcally f an nteger, and negatve for all f < <, and the functon g (t) the correpondng to relevant pole (thoe tuated n the man Remann heet de ned by < arg < ), a vanhng ocllatng functon that can be expreed a: g (t) = h et co(=) co t n -6db (9) So, the unt tep repone, y (t); for the normalzed cae = (t mut be noted that 6=, n the tme doman jut a cale factor), can be formulated a: Z y (t) = + n () + co () + e t d h et co(=) co t n (3) It clear that even expreon (3) not adequate for dervng the tme doman uual charactertc and pec caton. For the moment, t can be etablhed that, nce g (t) the vanhng ocllatng part, the atenuaton of (3) = co (=) ; the dampng rato = co (=) ; the damped frequency! d = n, and the natural frequency!n = : A can be oberved, thee expreon are the ame obtaned n complex plane analy. The problem now to derve ome ueful expreon for tme doman pec caton: overhoot (M p ), re tme (t r ), peak tme (t p ) and ettlng tme (t ). In (Manabe 96) an approxmated expreon gven for the overhoot, obtaned by tep repone mulaton and curve ttng. In (?) other approxmated formulae are gven for overhoot and tme pec caton, baed on curve ttng by polynomal (M p ) or ratonal (t p ; t r ) functon of ; or, n the cae of ettlng tme, by functon analogou to the obtaned for the ervo wth the correpondng expreon of dampng rato (9) a a functon of : Though that approxmated equaton are ueful enough for typcal degn problem, at leat ome of thee parameter can be analytcally obtaned. It mut be noted that n (3) the ocllatng part (9), o an analytcal expreon for t p can be obtaned by d erentatng th ocllatng part and next equatng the reult to zero, that : d h dt et co(=) co t n = t=t p By dong o, and takng nto account the perodcty of the trgonometrc functon, the expreon for the peak tme ( ee gure 5) t p = tan (= tan(=)) + n (=) (3) For obtanng a qua-analytcal expreon for overhoot, the functon (8) can be approxmated (leat quare mnmzaton) by b f a (t) = :9( )e t ; and then the approxmated expreon for the unt tep repone become by (t) = + :9( )e t (3) h et co(=) co t n

6 y α (t)-y~ α (t) y~ α (t) y α (t) t p M p α α Fg. 5. Peak tme, t p, a a functon of : In gure 6 th approxmated tep repone are gven wth the correpondng error, y (t) by (t). A can be oberved, there are everal d erence wth the tep repone of the ervo, manly, we can remark that n th cae there a crong pont near y (t) = :75; and the lope near the orgn are decreang wth ncreang value of (revere for the ervo). Furthermore, the peak tme not unformely decreang a ncreae. Thee fact are n correpondence wth the hgh frequence behavour n gure tm e ( e c.) Fg. 6. Analytcal and approxmated tep repone and correpondng error. By ung th expreon the overhoot can be obtaned by ubttutng (3) nto (3): M p ' by (t p ) (33) In gure 7 the overhoot repreented a a functon of : In order to obtan an expreon for ettlng tme, t mut be noted that n (3) the rt exponental term ha few n uence for long tme, and the envelope of the vanhng ocllaton term are de ned by (=) e t co(=) : So, a tme contant Fg. 7. Overhoot, M p ; a a functon of : can be de ned a & = =( co(=)); and the general expreon for the ettlng tme become: k t = k& = ( co(=)) where k de ne the tolerance margn. 4. CONCLUSIONS (34) In th paper the fractonal ntegrator a reference ytem for control ha been analyzed by comparon to the poton ervo n order to nd ther mltude and d erence, ha been made a revew of the approache prevouly made for degn pec caton n every doman when ung the fractonal ntegrator a reference ytem, and new approache have been propoed. By takng nto account the comment and reult of the former ecton, the followng gudelne can be tablhed for proper ue of the fractonal ntegrator a reference ytem: () The fractonal ntegrator (Bode deal functon) can be ued a an alternatve reference ytem for control, but conderng t own properte, not by extrapolatng the properte of the poton ervo, becaue: (a) relaton between parameter n d erent doman are not the ame; (b) the hape of the frequency and tep repone are qualtatvely d erent though equal for the lmt ocllatng cae, = ; = (For the other lmt cae, = = ; the crtcally damped tep repone of the ervo (y(t) = e!nt ( +! n t)), fater than the one of the fractonal ntegrator (y (t) = e!gt ), whch, n fact, a rt order ytem.) () Expreon for charactertc parameter and pec caton mut be taken from ther own doman of de nton: (a) dampng, attenuaton and natural frequency, from complex plane or, alternatvely, from tme doman; (b) phae margn, croover frequency and reonance charactertc from frequency doman;

7 (c) overhoot and tme pec caton from tme doman. (3) The bac degn parameter de nng the reference model are not dampng rato and natural frequency, but ntegraton order and croover frequency. (4) The bac degn equaton are: (9), (7) to (), () to (4), and (3), (33) and (34). Of coure, analytcal or graphycal relatonhp could be tablhed between the d erent parameter and doman for degn purpoe. REFERENCES Åtröm, K. J. (999). Lmtaton on Control Sytem Performance. Preprnt Avalable at Barboa, R. S., J. A. Tenrero and I. M. Ferrera (3). A Fractonal Calculu Perpectve of PID Tunng. In: Proceedng of the ASME Internatonal 9th Bennal Conference on Mechancal Vbraton and Noe. Chcago, Illno, Ua. Bode, H. W. (945). Network Analy and Feedback Ampl er Degn. Van Notrand. Bode, H. W. (). Relaton Between Attenuaton and Phae n Feedback Ampl er Degn. In: Control Theory: Twenty-Fve Semnal Paper (93-98) (T. Baar, Ed.). pp Goren o, R. and F. Manard (996). Fractonal Ocllaton and Mttag-Le er Functon. Mathematk Sere A, Preprnt No.A- 4/96. Horowtz, I. M. (993). Qualtatve Feedback Degn Theory (QFT). QFT Publcaton. Manabe, S. (96). The Non-Integer Integral and It Applcaton to Control Sytem. ETJ of Japan 6(3/4), Ogata, K. (). Modern Control Engneerng. fourth ed.. Prentce Hall. Outaloup, A. (995). La Dérvaton Non Entère: Théore, Synthèe et Applcaton. Hermè. Özbay, H. (999). Introducton to Feedback Control Theory. CRC Pre. Podlubny, I. (999). Fractonal-Order Sytem and PI D - Controller. IEEE Tranacton on Automatc Control 44(), 8 4. Tutn, A., J. T. Allanon, J. M. Layton and R. J. Jakeway (958). The Degn of Sytem for Autonomou Control of the Poton of Mave Object. In: The Proceedng of the IEEE, Vol. 5, Part C, Sup. no.. pp. 57. Van Valkenburg, M. E. (98). Analog Flter Degn. HRW.