Simulation and Discetization of Factional Ode Systems Mohamad Adnan Depatment of Electical and Compute Engineeing, Ameican Univesity of Beiut, Beiut, Lebanon Abstact - This wok deals with the simulation and discetization of continuous time models with factional ode diffeentiation o integation. It addesses the issue of diect discetization of analog factional ode systems followed by appoximating the discetization esult with an intege ode tansfe function o appoximating the analog factional ode system with an intege analog ode system and then discetizing the esulting intege analog appoximation. It povides a counteexample to the claim that analog intege ode appoximation followed by discetization is supeio to the diect discetization appoach. Keywods: opeato, discetization, factional ode systems, simulation, Tustin tansfom. 1 Intoduction Factional systems ae systems that ae epesented by diffeential equations that allow non intege odes. This is a genealization of the intege ode integation and diffeentiation. The ode can take on any eal value, not necessaily only factional values. Thus the factional designation is not accuate. Howeve the misnome is now the accepted designation. The coesponding tansfe functions fo factional ode systems esulting fom applying continuous o discete time tansfoms will be non-ational. An analytical solution fo the simulation and computation of a factional model s output is often not simple to obtain. Algoithms wee developed to appoximate the outputs of factional systems by using eithe continuous o discete ational models. Thus, thee ae thee majo appoaches to the time domain simulations of the output of a factional system: 1. Obtain an analytical solution of the output. 2. Indiect Discetization Method: Develop a ational continuous-time model appoximation of the factional system, then discetize the esulting ational appoximation. 3. Diect Discetization Method: Develop a ational discete-time model to appoximate the factional system. The method of continued faction expansion (CFE) is one of the ealy methods that was employed in indiect discetization appoaches to obtain ational appoximations to iational functions. The vesatile S. C. Dutta Roy had employed it to obtain ational continuous-time appoximations to factional analog systems [1-2]. It had also been employed ecently by Maione [3] to obtain a ational continuous-time appoximation in a manne simila to the appoach in [2]. The mathematical basis of the CFE ae obtained in the classical woks of Wall [4] and Khovanskii [5]. Othe poposed continuous-time appoximations wee intoduced by Oustaloup et. al.[6], and ecently by Aoun et.al. [7-8] Xue et.al. [9]. The method of CFE was applied by Chen et. al. in a diect discetization appoach that employed opeato[1-11]. used linea intepolation of the discete tansfe functions of taditional numeical integatos to obtain new integatos. He inveted the tansfe functions of the intepolated integatos to obtain new discete diffeentiatos. New s-to-z tansfom wee obtained by equating the Laplace tansfom of the ideal analog diffeentiato, s, o the ideal analog integato, 1/s, to the z- tansfe functions of the new diffeentiatos o integatos espectively [12-18]. opeato is obtained by intepolating the tapezoidal and ectangula integation ules [13, 16, 18]. The Intepolation and invesion pocesses poduced, in some cases, unstable integatos and diffeentiatos which wee stabilized by employing the appoach delineated in [12]. Chen and Mooe s wok gave impetus to eseach employing CFE and opeato [1]. Recently Babosa et. al. [19-22] and Fedi [23-24] intoduced least squaes (LS) appoximations of Pade, Pony, and Shanks, in diect discetization schemes, as altenatives to CFE and obtained bette appoximations than those obtained by using CFE. They have also demonstated that the Pade appoach yielded the same esults as CFE, while Pony and Shanks gave bette appoximations. Pony is usually pefeed because it is computationally less demanding than Shanks while yielding simila pefomance. The appoximations obtained by applying the Pade method can be shown to be identical to those obtained by the CFE method [25]. Howeve, CFE is computationally moe efficient than the Pade method. The above LS methods ae taditionally used to obtain digital filtes appoximating analog filtes [26, 27].
Implementing the digital factional ode diffeentiato coesponding to s, whee is a eal numbe and s is the Laplace tansfom opeato, aims at obtaining ational z- tansfom expessions appoximating s. The indiect discetization appoach stats by appoximating s with a ational s-tansfom expession, then apply an s-to-z tansfom to obtain a ational z-tansfom expession appoximating s. The diect discetization appoach substitutes fo s in diectly an s-to-z tansfom which will esult in a non-ational z-expession. The esulting nonational z-expession is then appoximated by a ational z- expession that appoximates s [19]. Thee is no inheent eason to think that one appoach is supeio to the othe in tems of accuacy. This will depend on the diffeent appoximations that evolve ove time fo both appoaches in addition to the poblem at hand. In this wok we addess a poblem that was consideed by poponents of the fist appoach and show that some of the methods of the second appoach yield bette accuacy, thus poviding a counte example. In this wok the above second and thid appoaches will be discussed. This wok employs the LS methods in the discetization schemes. The indiect discetization appoach is pesented in Section 2. The diect discetization appoach is pesented in Section 3. Section 4 pesents a compaison of the esults and conclusions. Then eq. (2) is expanded using CFE, Taylo expansion, o the method of ecusive poles and zeos to obtain an analog appoximation. The analog appoximation with ecusive poles and zeos is descibed in detail in [4], and it will be used in this wok. It is summaized below: s 1+ d ωa b = s 1+ b ωb d ω ω k ' k lim k= N ' 1 + s / ωk 1 + s / ω N k= N k 2k d 2N + 1 = ωa b + 2k b 2N + 1 = ωb d (4) (5) (3) + (6) 2 k= N ' ds + bsωb s ωk s K d 2 (1 ) s + bs ωb + d k= N s + ωk k= N d ωk K = ωa ' b k= Nω (7) k 2 Indiect Discetization: Discetizing SR with Intege We used b = 1 and d = 9 similaly to [4]. The pocedue fo the appoximation can be biefly summaized in the following: 2.1 The The pupose is to appoximate the tansfe function: Hs () = s (1) Fist, we apply the appoach descibed in [7-8], whee the appoximation follows the steps below: - Detemine a fequency ange [ω A, ω B ] whee we need to appoximate the factional ode tansfe function - Pefom an analog intege-ode appoximation of the factional-ode tansfe function - Discetize the analog appoximation s 1+ ωb( ωa + s) ω B (2) s[ ωa, ωb] n 2 ( ωa) ( s + ωbs + (1 ) ωaωb) s 1 + ω A - Given the fequency ange [ω A, ω B ] and N - Based on the factional ode, calculate ω k and ω k accoding to (4) and (5) - Compute K fom (7) - Obtain the appoximate ational tansfe function fom (6) to eplace s 2.2 Discetizing the In the following s is appoximated by using the bilinea tansfom. 2(1 z ) s (8) T (1 + z ) Note that (2) is equation (22) in [7-8].
O the tansfom: 8 (1 z ) s 7 T (1 + z /7) 2.3 Example 1: =.5 - Discetizing the The following values ae used: (9) w A =.1: lowe fequency in the appoximation ange w B = 1: uppe fequency in the appoximation ange = +.5: Factional exponent N = 3, b = 1, d = 9, T =.1. The esulting ational tansfe function is shown below. 94.87 s 9 + 1.573e4 s 8 + 6.357e5 s 7 + 6.686e6 s 6 + 1.831e7 s 5 + 1.32e7 s 4 + 2.378e6 s 3 + 1.73e5 s 2 + 1 s 4.5 s 9 + 1483 s 8 + 1.18e5 s 7 + 2.436e6 s 6 + 1.39e7 s 5 + 1.828e7 s 4 + 6.67e6 s 3 + 6.64e5 s 2 + 1.314e4 s + 47.43 (1) We eplace s in (1) by (8) and (9) and plot the esulting ational z-tansfe function to obtain the coesponding magnitudes and phases shown in Figue 1. 2.4 Example 2: = -.5 - Discetizing the The following values ae used: w A =.1: lowe fequency in the appoximation ange w B = 1: uppe fequency in the appoximation ange = -.5: Factional exponent N = 3, b = 1, d = 9, T =.1. The following analog ational tansfe function is obtained..8538 s 9 + 186.5 s 8 + 1.221e4 s 7 + 2.366e5 s 6 + 1.249e6 s 5 + 1.732e6 s 4 + 6.192e5 s 3 + 5.475e4 s 2 + 1 s 13.5 s 9 + 1739 s 8 + 6.39e4 s 7 + 6.418e5 s 6 + 1.74e6 s 5 + 1.228e6 s 4 + 2.21e5 s 3 + 9165 s 2 + 49.7 s -.4269 (11) Replacing s in (11) by (8) and (9) to obtain a ational z- tansfe function whose coesponding magnitudes and phases plots shown in Figue 2. -5-1 -15-2 -25 1 1 1 1 2 1 3 3 2 1 1 1 1 1 2 1 3 5 4 3 2 1 1 1 1 1 2 1 3 Figue 1: Magnitude and Phase esponses of the Indiect Discetization of the half powe analog diffeentiato. -1-2 -3-4 -5 1 1 1 1 2 1 3 Figue 2: Magnitude and Phase esponses of the Indiect Discetization of the half powe analog integato. 3 Diect Discetization The diect discetization in this pape applies the LS method of Pony to obtain digital ational appoximations, IIR filtes, to continuous factional ode integatos and diffeentiatos, diffeintegatos, of type s, whee is a eal numbe [21].
The following steps delineate the pocess. a) Substitute fo s in s a chosen s-to-z tansfom to obtain a geneating function H(z). In this pape we will estict ouselves to the Tustin and tansfoms. b) Fom a PSE (Powe Seies Expansion) o Taylo seies expansion ove H(z) to obtain the impulse esponse sequence of the factional discete equivalent h(k). Apply the signal modeling technique of Pony to h(k) to get the desied ational IIR filte appoximation. 3.1 Discetization Diect discetization of s can be expessed by the geneating function obtained fom an s-to z tansfom as s=g(z). Then apply methods used in [21], i.e. eplace s by (8) o (9), then use the least squaes techniques (Pade, Pony, Shanks) to appoximate the obtained digital factional-ode geneating function s =(G(z)). The geneating function expession fo the bilinea (Tustin) ule is 2(1 z ) = ( ( )) = T ( ) = ( ) s G z H z T(1 + z ) (12) The geneating function expession fo opeato is 8(1 z ) = ( ( )) = A( ) = ( ) s G z H z 7 T(1 + z /7) (13) Tansfom (Sampling time:.1) Applying the Tustin tansfom yields the following tansfe function: 1.69 z 9-5.84 z 8 + 99.98 z 7-14.3 z 6 + 61.4 z 5-19.12 z 4 + 2.471 z 3 +.8932 z 2 -.3677 z +.6427 z 9-4.184 z 8 + 7.43 z 7-6.12 z 6 + 2.64 z 5 -.482 z 4 -.1864 z 3 +.1412 z 2 -.4248 z - 5.452e-5 (15) Plotting the esulting ational z-tansfe functions yields the coesponding magnitudes and phases shown in Figue 3. 6 4 2-2 1 1 1 1 2 1 3 6 4 2 1 1 1 1 2 1 3 Figue 3: Magnitude and Phase esponses of the diect discetization of the half powe analog diffeentiato. Example 1: =.5 Applying the methods of [21], ie. discetizing the Analog Expession then applying Pony. We let: m=9; n=9; M = 1: Ode of Taylo Seies Expansion Tustin Tansfom (Sampling time:.1) Applying the Tustin tansfom yields the following tansfe function: 14.14 z 9-3.135 z 8-34.15 z 7 + 6.414 z 6 + 28.19 z 5-4.184 z 4-8.963 z 3 +.944 z 2 +.8223 z -.3276 z 9 +.7783 z 8-2.137 z 7-1.572 z 6 + 1.54 z 5 + 1.9 z 4 -.3823 z 3 -.2133 z 2 +.2446 z +.7461 (14) Example 2: = -.5 Applying the methods of [21], ie. discetizing the Analog Expession then applying Pony: We let: m=9; n=9; M = 1: Ode of Taylo Seies Expansion Tustin (): (Sampling time:.1) Applying the Tustin tansfom yields the following tansfe function:.771 z 9 +.1568 z 8 -.178 z 7 -.327 z 6 +.141 z 5 +.292 z 4 -.4482 z 3 -.4522 z 2 +.4112 z +.1638 z 9 -.7783 z 8-2.137 z 7 + 1.572 z 6 + 1.54 z 5-1.9 z 4 -.3823 z 3 +.2133 z 2 +.2446 z -.7461 (16)
: (Sampling time=.1) Applying tansfom yields the following tansfe function:.9354 z 9 -.2932 z 8 +.263 z 7 +.7978 z 6 -.2662 z 5 +.1571 z 4 -.379 z 3 -.1182 z 2 +.673 z - 4.713e- 6 z 9-3.76 z 8 + 4.493 z 7 -.5399 z 6-3.413 z 5 + 3.159 z 4-1.144 z 3 +.149 z 2 +.3654 z -.19 (17) Absolute Magnituded Eo 2 18 16 14 12 1 8 6 4 2 + + + Pony + Pony The plots of the above esulting ational z-tansfe functions obtain the coesponding magnitudes and phases shown in Figue 4. 2-2 -4-6 1 1 1 1 2 1 3 5 1 15 2 25 3 35 Figue 5: Eo Magnitudes of the discetization of the half powe analog diffeentiato It is shown that in the fequency ange of inteest [.1 1],.5 8(1 z ) applying Pony to yields the lowest eo. T (7 + z ).5 b) s : The eo of the magnitudes of the discetization of the half powe analog diffeentiato ae shown in Figue 6. -2-4 -6 1 1 1 1 2 1 3 Figue 4: Magnitude and Phase esponses of the diect discetization of the half powe analog integato. Absolute Magnituded Eo.6.5.4.3.2 + + + Pony + Pony 4 Eo Compaison and Conclusions Finally, we compae the absolute magnitude eo between the diffeent methods:.5 a) s : The eo of the magnitudes of the discetization of the half powe analog diffeentiato ae shown in Figue 5..1 5 1 15 2 25 3 35 Figue 6: Eo Magnitudes of the discetization of the half powe analog integato. Compaing the absolute magnitude eo between the diffeent methods and the ideal esponse de monstates that in the fequency ange of inteest, [.1 1], applying Al-.5 8(1 z ) Alaoui Pony to yields the lowest eo. T (7 + z ) Thus it is shown that the diect discetization appoach Al- Alaoui opeato and using the Pony LS appoximation yielded the lowest eo, in the fequency ange of inteest,
lowe than the indiect appoaches fo both =.5 and = -.5. This povides a counte example to the claim that the indiect discetization appoaches ae supeio to the diect discetization methods. 5 Acknowledgements This eseach was initiated duing my visit to the Adaptive Systems Laboatoy at UCLA in 27-28. It is indeed a pleasue to acknowledge Pofesso A. H. Sayed fo his invitation and fo poviding the atmosphee conducive to eseach. I am gateful to Cassio Lopes and Qiyue Zou fo thei help duing my stay at UCLA. I am gateful to the highly talented gaduates of the ECE Depatment at the Ameican Univesity of Beiut. It is indeed a pleasue to acknowledge Jimmy Aza, Abbas Slim, Ghassan Deeb, and Elias Yaacoub fo thei invaluable help in the poduction of this wok. This eseach was suppoted, in pat, by the Univesity Reseach Boad, URB, of the Ameican Univesity of Beiut. 6 Refeences [1] S. C. Dutta Roy, On the Realization of Constant- Agument Immittance o Factional Opeato, IEEE Tans. Cicuit Theoy, Vol. CT-14, pp. 264-274, Sept. 1967. [2] S. C. Dutta Roy, A Rational Discete Appoximation of Some Iational Functions Though a Flexible Continued Faction Expansion, Poc. IEEE (Lettes), vol. 7, no. 1, pp.84-85, Jan 1982. [3] G. Maione, A Rational Discete Appoximation to the Opeato, IEEE Signal Pocessing Lettes, vol. 13, no. 3, pp. 141-144, Ma. 26. [4] H. S. Wall: Analytic Theoy of Continued Factions: New Yok, Van Nostand, 1948. [5] A. N. Khovanskii, The Application of Continued Factions and Thei Genealizations to Poblems in Appoximation Theoy, Tansl. Goningen, The Nethelands: Noodhoff, 1963. [6] A. Oustaloup, F. Levon, F. Nanot, and B. Mathieu, Fequency Band Complex NonIintege Diffeentiato: Chaacteization and Synthesis, IEEE Tans. Cicuits Syst. I, vol. 47, pp. 25-4, Januay 2. [7] M. Aoun, R. Malti, F. Levon, A. Oustaloup, Numeical Simulations of Factional Systems, Poceedings of DETC 3, Septembe, 23. [8] M. Aoun, R. Malti, F. Levon, A. Oustaloup, Numeical Simulations of Factional Systems: An Oveview of Existing Methods and Impovements, Nonlinea Dynamics 38, pp. 117-131, 24. [9] D. Xue, C. Zhao, Y. Q. Chen, Factional Ode PID Contol of a DC Moto with Elastic Shaft: A Case Study, Poceedings of the 26 Ameican Contol Confeence, June, 26. [1] Y.Q. Chen and K.L. Mooe, Discetization fo Factional Ode Diffeentiatos and Integatos, IEEE Tans. on Cicuits and Systems I, vol. 49, pp 363-367, Mach 22. [11] Chen, Y., Vinage, B.M., Podlubny, I. Continued faction expansion appoaches to discetizing factional ode deivatives-an expositoy eview, Nonlinea Dynamics 38 (1-4), pp. 155-17, 24. [12] M. A., "Novel Appoach to Designing Digital Diffeentiatos"; IEE Electonics lettes, Vol. 28, No. 15, pp. 1376-1378, 16 July, 1992. [13] M. A., "Novel Digital Integato and Diffeentiato"; IEE Electonics Lettes, Vol. 29, No. 4, pp. 376-378, 18 Febuay 1993. (See also ERRATA, IEE Electonics Lettes, Vol. 29, No. 1, pp.934, 1993.) [14] M. A., Novel IIR Diffeentiato fom The Simpson Integation Rule"; IEEE Tansactions on Cicuits and Systems I. Fundamental Theoy and Applications, Vol. 41, No. 2, pp. 186-187, Febuay, 1994 [15] M. A., "A Class of Second Ode Integatos and Lowpass Diffeentiatos"; IEEE Tansactions on Cicuits and Systems I. Fundamental Theoy and Applications, Vol. 42, No. 4, pp. 22-223, Apil, 1995 [16] M. A., " Filling the Gap Between the and the Backwad Diffeence Tansfoms: An Inteactive Design Appoach"; Intenational Jounal of Electical Engineeing Education. Vol. 34, No. 4, pp. 331-337, Octobe 1997. [17] M. A., Novel stable highe ode s-to-z tansfoms; Cicuits and Systems I: Fundamental Theoy and Applications, IEEE Tansactions on, Volume: 48 Issue: 11, pp. 1326-1329, Nov. 21 [18] M. A., Opeato and the α- Appoximation fo Discetization of Analog Systems, Facta Univesitatis (NIS), Se.:Elec. Eneg., Vol. 19, No. 1, pp. 143-146, Apil 26. [19] R.S. Babosa, J.A.T. Machado, I.M. Feeia, Leastsquaes design of digital factional-ode opeatos, in:
Poceedings of the Fist IFAC Wokshop on Factional Diffeentiation and its Applications (FDA'4), Bodeaux, Fance, July 19-21, 24, pp. 434-439. [2] R.S. Babosa, J.A.T. Machado, I.M. Feeia, Pole-zeo appoximations of digital factional-ode integatos and diffeentiatos using signal modeling techniques, in: 16th IFAC Wold Congess, Pague, Czech Republic, July 4-8, 25. [21] R. S. Babosa, and J. A. Teneio Machado, Implementation of Discete-Time Factional Ode Contolles based on LS Appoximations, Acta Polytechnica Hungaica, Vol. 3, No. 4, pp. 5-22, 26. [22] Babosa, R.S., Teneio Machado, J.A., Silva, M.F., Time domain design of factional diffeintegatos using least-squaes, Signal Pocessing 86 (1), pp. 2567-2581, 26. [23] Y. Fedi, B. Boucheham, Recusive filte appoximation of digital diffeentiato and integato based on Pony's method, in: Poceedings of the Fist IFAC Wokshop on Factional Diffeentiation and its Applications (FDA'4), Bodeaux, Fance, July 19-21, 24, pp. 428-433. [24] Fedi, Y., Computation of factional ode deivative and integal via powe seies expansion and signal modeling, Nonlinea Dynamics 46 (1-2), pp. 1-15, 26. [25] L. Loentzen, and H. Waadeland: Continued Factions with Applications, Addison- Wesley, Noth-Holland, Amstedam, 1992. [26] M. H. Hayes: Statistical Digital Signal Pocessing and Modeling, Wiley-Intescience, New Yok, 1996. [27] J. G. Poakis, and D. G. Monolakis: Digital Signal Pocessing: Pinciples, Algoithms, and Applications, Pentice-Hall, 3d Edition, Uppe Saddle Rive, 1996.