Discretization Schemes for Fractional-Order Differentiators and Integrators

Transcription

1 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 49, NO. 3, MARCH Discetization Schemes fo Factional-Ode Diffeentiatos and Integatos Yang Quan Chen and Kevin L. Mooe Abstact Fo factional-ode diffeentiato whee is a eal numbe, its discetization is a key step in digital implementation. Two discetization methods ae pesented. The fist scheme is a diect ecusive discetization of the Tustin opeato. The second one is a diect discetization method using the Al-Alaoui opeato via continued faction expansion (CFE). The appoximate discetization is minimum phase and stable. Detailed discetization pocedues and shot MATLAB scipts ae given. Examples ae included fo illustation. Index Tems Al-Alaoui opeato, discetization, factional diffeentiato, factional-ode diffeentiato, factional-ode dynamic systems, ecusive, Tustin opeato. I. INTRODUCTION Factional calculus is a 300-yea-old topic. The theoy of factional-ode deivative was developed mainly in the 19th centuy. Recent books [1] [4] povide a good souce of efeences on factional calculus. Howeve, applying factional-ode calculus to dynamic systems contol is just a ecent focus of inteest [5] [9]. Fo pioneeing wok on this egad, we cite [10] [13]. In theoy, the contol systems can include both the factional ode dynamic system o plant to be contolled and the factional-ode contolle. Howeve, in contol pactice it is moe common to conside the factional-ode contolle. This is due to the fact that the plant model may have aleady been obtained as an intege ode model in the classical sense. In most cases, ou objective is to apply factional-ode contol (FOC) to enhance the system contol pefomance. Fo example, as in the CRONE 1 contol [14], [7], [8], factal obustness is pusued. The desied fequency template leads to factional tansmittance [15], [16] on which the CRONE contolle synthesis is based. In the CRONE contolle, the majo ingedient is the factional-ode deivative s, whee is a eal numbe and s is the Laplacian opeato. Anothe example is the PI D contolle [6], [17], an extension of PID contolle. In geneal fom, the tansfe function of PI D is given by K p + T i s 0 + T d s, whee and ae positive eal numbes; K p is the popotional gain, T i the integation constant and T d the diffeentiation constant. Clealy, taking =1and =1, we obtain a classical PID contolle. If =0(T i =0)we obtain a PD contolle, etc. All these types of contolles ae paticula cases of the PI D contolle. It can be expected that the PI D contolle may enhance the systems contol pefomance due to moe tuning knobs intoduced. Actually, in theoy, PI D itself is an infinite dimensional linea filte due to the factional ode in the diffeentiato o integato. It should be pointed out that a band-limit implementation of FOC is impotant in pactice, i.e., the finite-dimensional appoximation of the FOC should be done Manuscipt eceived Januay 26, 2001; evised Septembe 23, 2001 and Novembe 12, This wok was suppoted in pat by U.S. Amy Automotive and Amaments Command (TACOM) Intelligent Mobility Pogam unde Ageement DAAE This pape was ecommended by Associate Edito P. K. Rajan. The authos ae with the Cente fo Self-Oganizing and Intelligent Systems (CSOIS), Depatment of Electical and Compute Engineeing, College of Engineeing, Utah State Univesity, Logan, UT USA ( yqchen@ieee.og). Publishe Item Identifie S (02) CRONE is a Fench abbeviation fo Contôle Robuste d Ode Non Entie (which means nonintege ode obust contol). in a pope ange of fequencies of pactical inteest [18], [16]. Moeove, the factional ode can be a complex numbe as discussed in [18]. In this pape, we focus on the case when the factional ode is a eal numbe. The key step in digital implementation of a FOC is the numeical evaluation o discetization of the factional-ode diffeentiato. In geneal, thee ae two discetization methods: diect discetization and indiect discetization. In indiect discetization methods [18], two steps ae equied, i.e., fequency domain fitting in continuous time domain fist and then discetizing the fit s-tansfe function. Othe fequency-domain fitting methods can also be used but without guaanteeing the stable minimum-phase discetization. Existing diect discetization methods, e.g., [19], [20], applied the diect powe seies expansion (PSE) of the Eule opeato, continuous factional expansion (CFE) of the Tustin opeato and numeical integation based method. In this bief, we fist focus on the diect discetization method using the well known Tustin opeato which is a staightfowad scheme to discetize the factional-ode deivative [19], [20]. The majo contibution of this bief is to intoduce a ecusive fomula fo discetization with diffeent ode of appoximation which simplifies the pogamming effots. Moeove, the discetized tansfe function is stable and minimum phase. Howeve, as pointed out in [21], [22], the Tustin opeato based discetization scheme exhibits lage eos in high fequency ange. A new mixed scheme of Eule and Tustin opeatos is poposed which yields the Al-Alaoui opeato [21]. Using the continued faction expansion of the Al-Alaoui opeato, this pape contibutes a new diect discetization scheme with a vey good magnitude fit to that of the oiginal continuous factional diffeentiato. This bief is oganized as follows. In Section II, we pesent a new ecusive fomula fo Tustin discetization of the factional-ode deivative with a symbolic validation and an illustative example. Section III pesents anothe new diect discetization scheme by continued faction expansion of the Al-Alaoui opeato with an illustative example, too. Section IV concludes this pape. II. RECURSIVE TUSTIN DISCRETIZATION OF FRACTIONAL-ORDER DERIVATIVE A. Geneating Function In geneal, the discetization of the factional-ode diffeentiato s ( is a eal numbe) can be expessed by the so-called geneating function s =!(z 01 ). This geneating function and its expansion detemine both the fom of the appoximation and the coefficients [23]. Fo example, when a backwad diffeence ule is used, i.e.,!(z 01 )=(10 z 01 )=T, pefoming the powe seies expansion (PSE) of (10z 01 ) 6 gives the discetization fomula. In this bief, we fist conside the tapezoidal (Tustin) ule as a geneating function!(z 01 ) 6 = 2 T 1+z 01 6 fo obtaining the coefficients and the fom of the appoximation. B. Recusive Tustin Discetization The key point of the Tustin discetization of factional-ode diffeentiato is how to get a ecusive fomula. Hee we intoduce the so-called Mui-ecusion oiginally used in geophysical data pocessing with applications to petoleum pospecting [24]. The Mui-ecusion motivated in computing the vetical plane wave (1) /02$ IEEE

2 364 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 49, NO. 3, MARCH 2002 TABLE I TABLE OF FORMULAS A (z ;) FOR n =1;...; z z z z z z 09 : (5) eflection esponse via the impedance of a stack of n-layeed eath can be used in ecusive discetization of factional-ode diffeentiato of the Tustin geneating function. In the following, without loss of geneality, assume that 2 [01; 1]. Moeove, without complicating the pesentation, we only give the ecusive fomula fo positive We can do the simila Taylo expansion fo (A n(z 01 ;)=A n(z 01 ; 0)). Fistly, efe to the fomulas A n (z 01 ;) fo n = 1;...; 9 listed in Table I. We now examine the case when n = 9. By collecting the tems in each coefficient, we found that Taylo expansion coefficients ae all the same with those given in (5) until z 09. Theefoe, it is veified that the poposed ecusive fomula is as coect as Taylo seies expansion till the ode of appoximation. D. An Illustative Example Using the ecusive method mentioned in this subsection, the discetization of s 0:5 sampled at s. is studied numeically, and the appoximate models ae!(z 01 ) = 2 T = 2 T whee A 0(z 01 ;)=1, and and Theefoe, 1+z 01 lim n!1 An(z01 ;) A n(z01 ; 0) A n (z 01 ;)=A n01(z 01 ;) 0 c n z n A n01(z; ); (3) c n = s =n; n is odd; 0; n is even. 2 T A n(z 01 ;) A n(z01 ; 0) : Symbolically, Table I lists the expessions of A n (z 01 ;) fo n = 1;...; 9. C. Symbolic Validation We now examine the coectness of the ecusive discetization of the factional-ode deivative opeato by using MATLAB Symbolic Toolbox. We can get the symbolic Taylo expansion fo (1) fo the fist 9 tems. The esult is summaized as follows: 1+z (2)z 01 +(2 2 )z z 03 (2) (4) 44:72z 0 22:36 G 1(z) = z +0:5 G 3 (z) = 44:72z3 0 22:36z 2 +3:727z 0 7:454 z 3 +0:5z 2 +0:083 33z +0:1667 G 7(z) = G 9(z) = 44:72z :36z 6 +4:792z 5 0 7:986z 4 +2:795z 3 04:792z 2 +1:597z 0 3:194 z 7 +0:5z 6 +0:1071z 5 +0:1786z 4 +0:0625z 3 +0:1071z 2 +0:0357z +0: :72z :36z 8 +4:969z 7 0 8:075z 6 +3:061z 5 0 4:947z 4 +2:041z 3 0 3:461z 2 +1:242z 0 2:485 z 9 +0:5z 8 +0:1111z 7 +0:1806z 6 +0:068 45z 5 +0:1106z 4 +0:045 63z 3 +0:077 38z 2 +0:027 78z +0:055 56: We pesent fou plots as shown in Fig. 1 to show the effectiveness of the appoximate discetization. It should be pointed out that the diect discetization method intoduced above always gives a Z-tansfe function with stable minimum phase chaacteistics. The convegence of the ecusive scheme [24] is clealy demonstated by Fig. 1. III. DIRECT DISCRETIZATION USING CONTINUED FRACTION EXPANSION (CFE) OF AL-ALAOUI OPERATOR We see fom Fig. 1 that the Bode plot of the discetized model based on the ecusive expansion of the Tustin opeato o tapezoidal ule is with lage eo in the high fequency ange. To have a bette appoximation in the high fequency ange, hee we use the Al-Alaoui opeato [21] which is a weighted sum of the ectangula ule o Eule opeato and the tapezoidal ule. This will give a new staightfowad scheme using continued faction expansion (CFE) to discetize the factional-ode deivative with the aid of a symbolic computation tool.

3 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 49, NO. 3, MARCH Fig. 1. Recusive Tustin discetization of s at T = 0:001 s. A. Al-Alaoui Opeato Based Discetization The Al-Alaoui opeato is a mixed scheme of Eule and Tustin opeatos. Based on the Al-Alaoui opeato, the geneating function fo discetization is!(z 01 ) 6 = 8 1+z 01 =7 6 : (6) Clealy, (6) is an infinite ode of ational discete-time tansfe function. To appoximate it with a finite ode ational one, continued faction expansion (CFE) is an efficient way. In geneal, any function G(z) can be epesented by continued factions in the fom of b 1 (z) G(z) ' a 0(z)+ b 2 (z) a 1 (z) + b 3(z) a 2 (z)+ a 3(z) +111 whee the coefficients a i and b i ae eithe ational functions of the vaiable z o constants. By tuncation, an appoximate ational function, ^G(z), can be obtained. The esulting discete tansfe function, appoximating factionalode opeatos, can be expessed as: D 6 (z) 8 = CFE P p (z 01 ) Q q (z 01 ) 1+z 01 =7 6 p; q (7) whee CFEfug denotes the continued faction expansion of u; p and q ae the odes of the appoximation and P and Q ae polynomials of degees p and q. Nomally, we can set p = q = n. In MATLAB Symbolic Toolbox, we can easily get the appoximate diect discetization of the factional ode deivative by the following scipt, fo a given n (eplace 14 by 2n): clea all; close all; symsxz %Al 0 Alouoi 0 s scheme x =((10z)=(1 + z=7))^; [RESULT; STATUS] =maple (\with (numtheoy)") %7-th ode; put hee: h7 = maple (\cfac"; x; z; 14); h7n = maple (\nthnume (%%; 14)"); h7d = maple (\nthdenom (%%%; 14)"); h7ns = sym (h7n); h7ds =(h7d); num7 = collect (h7ns; z); den7 = collect (h7ds; z); fn7 = subs (num7; z; 1=z); fd7 = subs (den7; z; 1=z): Remak III.1: The CFE scheme pesented in the above contains two tuning paametes, namely p and q, while in the ecusive Tustin method intoduced in Section II, thee is only one tuning paamete, n. The optimal choice of these paametes is possible based on a quantitative measue. One possibility is the use of the least squaes (LS) eo between the continuous-fequency esponse and discetized fequency esponse. Note that in pactice, p and q can usually be set equal. So, in the discetization methods pesented in this pape, thee is only one tuning knob which will be moe attactive in pactice. Compaed to

4 366 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 49, NO. 3, MARCH 2002 Fig. 2. CFE(Al-Alaoui) discetization of s at T = 0:001 s. the fequency-banded fitting methods, e.g., [18], [16], the diect discetization methods pesented in this pape is staightfowad and no band-pass weighting filte is assumed. B. An Illustative Example The discetization of the half-diffeentiato s 0:5 sampled at s. is studied numeically, and the appoximate models ae 236:6z G 1 (z) = 7z 0 1 G 3 (z) = 1657z z z 0 62:78 49z z 2 +7z +1 2:47e04z 5 05:999e04z 4 +4:941e04z 3 01:512e04z :9z+98:48 G 5(z) = 730:7z z :7z 3 089:48z 2 015:52z+1 G 7 (z) = 3:128e05z 7 01:028e06z 6 +1:283e06z 5 07:433e05z 4 +1:87e05z z z+104:5 9253z 7 02:512e004z 6 +2:436e004z z :7z :7z 2 023:67z01 We pesent fou plots as shown in Fig. 1 to show the effectiveness of the appoximate discetization. We can obseve fom Fig. 2 that the new scheme intoduced in this lette is much bette than the Tustin scheme in magnitude fit to the oiginal s. Afte the linea phase compensation, the maximum phase eo of the Al-Alaoui opeato based discetization scheme is aound 28:25 at 55% of the Nyquist fequency (aound 275 Hz in this example) as shown in Fig. 2. To compensate the : linea phase dop, a half sample phase advance is used which means that we should cascade z 0:5 to the obtained appoximately discetized tansfe function G(z). In this example, the phase compensato is z 0:25 which is noncausal. In implementation, we can simply use z 00:75 =z 01 instead. Note that the linea phase dop compensation does not affect the magnitude fit. When multi-ate sampling is possible, the phase-dop compensation can be implemented with a smalle eo which deseves futhe investigation. It should also be pointed out that, by checking the pole-zeo map, the diect discetization method intoduced above always gives a Z-tansfe function with stable minimum phase chaacteistics. IV. CONCLUDING REMARKS This bief pesents two discetization methods fo the factionalode diffeentiato s whee is a eal numbe. The discetization is a key step in digital implementation of the factional-ode contolle containing s. The fist scheme is a diect ecusive discetization of the Tustin opeato. The second one is a diect discetization method using the Al-Alaoui opeato via CFE. The appoximate discetization schemes give minimum phase and stable finite dimensional discete tansfe functions. Detailed discetization pocedues and shot MATLAB scipts ae given. Examples ae included fo illustation. ACKNOWLEDGMENT The authos wish to thank the anonymous efeees who caefully eviewed the pape and povided vey useful comments to impove it, in paticula, the Reviewe #5 who had epoduced the esults of this pape fom which the authos could coect some eos in the pape.

5 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 49, NO. 3, MARCH REFERENCES [1] K. B. Oldham and J. Spanie, The Factional Calculus. New Yok: Academic, [2] S. G. Samko, A. A. Kilbas, and O. I. Maichev, Factional integals and deivatives and some of thei applications, in Nauka i Technika, Minsk, U.S.S.R., [3] K. S. Mille and B. Ross, An Intoduction to the Factional Calculus and Factional Diffeential Equations. New Yok: Wiley, [4] I. Podlubny, Factional Diffeential Equations. San Diego, CA: Academic, [5] B. J. Luie, Thee-paamete tunable tilt-integal-deivative (TID) contolle, U.S. Patent , [6] I. Podlubny, Factional-ode systems and PI D -contolles, IEEE Tans. Automat. Cont., vol. 44, no. 1, pp , Jan [7] A. Oustaloup, B. Mathieu, and P. Lanusse, The CRONE contol of esonant plants: Application to a flexible tansmission, Eu. J. Cont., vol. 1, no. 2, [8] A. Oustaloup, X. Moeau, and M. Nouillant, The CRONE suspension, Contol Eng. Pact., vol. 4, no. 8, pp , [9] H.-F. Raynaud and A. ZegaInoh, State-space epesentation fo factional ode contolles, Automatica, vol. 36, no. 7, pp , [10] S. Manabe, The nonintege integal and its application to contol systems, JIEE, vol. 80, no. 860, pp , [11], The system design by the use of nonintege integal and tanspot delay, Jap. Inst. Elect. Eng. J., vol. 81, no. 878, pp , [12] A. Oustaloup, Linea feedback contol systems of factional ode between 1 and 2, in Poc. IEEE Symp. Cicuit and Systems, Chicago, IL, 1981, 4. [13] M. Axtell and E. M. Bise, Factional calculus applications in contol systems, in Poc. IEEE 1990 Nat. Aeospace and Electonics Conf., New Yok, NY, 1990, pp [14] A. Oustaloup, La Déivation non Entièe. Pais, Fance: Hemes, [15] A. Oustaloup, J. Sabatie, and P. Lanusse, Fom factal obustness to CRONE contol, Factionnal Calculus and Applied Analysis, vol. 2, no. 1, pp. 1 30, [16] A. Oustaloup and B. Mathieu, La Commande CRONE: Du Scalaie au Multivaiable. Pais, Fance: Hemes, [17] I. Petáš, The factional-ode contolles: Methods fo thei synthesis and application, J. Elect. Eng., vol. 50, no. 9 10, pp , [18] A. Oustaloup, F. Levon, F. Nanot, and B. Mathieu, Fequency band complex non intege diffeentiato: Chaacteization and synthesis, IEEE Tans. Cicuits Syst. I, vol. 47, pp , Jan [19] J. A. T. Machado, Analysis and design of factional-ode digital contol systems, J. Syst. Anal. Modeling-Simulation, vol. 27, pp , [20] B. M. Vinage, I. Petas, P. Mechant, and L. Docak, Two digital ealization of factional contolles: Application to tempeatue contol of a solid, in Poc. Eu. Contol Conf. (ECC 01), Poto, Potugal, Sept. 2001, pp [21] M. A. Al-Alaoui, Novel digital integato and diffeentiato, Electon. Lett., vol. 29, no. 4, pp , [22], Filling the gap between the bilinea and the backwad diffeence tansfoms: An inteactive design appoach, Int. J. Elect. Eng. Edu., vol. 34, no. 4, pp , [23] C. H. Lubich, Discetized factional calculus, SIAM J. Math. Anal., vol. 17, no. 3, pp , [24] J. F. Claebout. (1976) Fundamentals of Geophysical Data Pocessing With Applications to Petoleum Pospecting [Online]. Available: Nonlinea Effects of Radio-Fequency Intefeence in Opeational Amplifies Fanco Fioi and Paolo S. Covetti Abstact In this bief, the susceptibility of opeational amplifies to adio fequency intefeence (RFI) is studied by a new analytical model. The poposed model, in paticula, points out the dependence of the RFI induced dc offset voltage shift in opeational amplifies on design paametes and paasitics, giving both a good insight into the nonlinea mechanisms involved in the phenomenon and a suppot to integated cicuit designes in ode to develop high immunity opeational amplifies. The validity of the poposed appoach is discussed compaing model pedictions with the esults of compute simulations and expeimental measuements. Index Tems Electomagnetic compatibility (EMC), electomagnetic intefeence (EMI), hamonic distotion, opeational amplifie. I. INTRODUCTION The inceased level of envionmental electomagnetic pollution makes mandatoy the design of electonic systems which ae immune to adio fequency intefeence (RFI). In fact, metal inteconnections (wies and pinted cicuit boad taces) of electonic systems tanslate electomagnetic intefeence (EMI) into voltages and cuents, which esult supeimposed on nominal system signals. Such an intefeence eaches integated cicuit (IC) teminals and dives distotion phenomena in active nonlinea devices, geneating tempoay o definitive eos in IC and electonic system opeations [1] [3]. Analog cicuits ae paticulaly susceptible to RFI, as they lack the egeneative effect which is typical of digital cicuits. This is one of the main easons why digital signal pocessing is pefeable to analog signal pocessing. Howeve, analog signal pocessing cannot be eplaced in many applications, among which at least the analog-to-digital (A/D) and the digital-to-analog (D/A) convesions must be cited, which ae necessay in digital signal pocessing because of the intinsically analog natue of the infomation in the physical wold. Among analog cicuits, opeational amplifies (opamps) ae extemely susceptible to RF distubances. They demodulate RFI added on nominal input signals and so the nominal output signal is coupted by in-band intefeence. In paticula, the pesence of continuous-wave (CW) intefeence geneates an output offset voltage. This opamp behavio was oiginally obseved in aeonautic electonic systems and subsequently studied pefoming immunity tests on commecial opamps. In paticula, measuements of the output offset voltage induced by RFI conveyed on the input teminals of seveal feedback opamps wee pefomed vaying intefeence fequency and amplitude [4]. These expeimental chaacteization is useful in the design of EMI filtes. Moe ecently, this behavio has been studied by time domain [5] [7] and fequency domain (hamonic balance) [8] compute simulations: seveal effots have been expended on eseach in ode to deive both RFI-oiented numeical models of active devices and opamp macomodels intended to educe compute simulation time. Although these models allow one to pedict efficiently and accuately the RFI-induced upset in opamps, they do not gant a elationship with cicuit paametes and paasitics and so they cannot be diectly applied to deive design citeia /02$ IEEE Manuscipt eceived Mach 31, 2001; evised Octobe 1, This pape was ecommended by Associate Edito G. Palumbo. The authos ae with the Electonic Depatment of the Politecnico di Toino, Tuin 10129, Italy ( fioi@eln.polito.it). Publishe Item Identifie S (02)