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Al-Alaoui operator and the new transformation polynomials for discretization of analogue systems

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Abstract

The “new transformation polynomials for discretization of analogue systems” was recently introduced. The work proposes that the discretization of 1/s n should be done independently rather than by raising the discrete representation of 1/s to the power n. Several examples are given in to back this idea. In this paper it is shown that the “new transformation polynomials for discretization of analogue systems” is exactly the same as the parameterized Al-Alaoui operator. In the following sections, we will show that the same results could be obtained with the parameterized Al-Alaoui operator.

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References

  1. Enden VD (1989). Discrete-time signal processing. Prentice- Hall, Englewood Cliffs

    Google Scholar 

  2. Franklin FG and Powell JD (1980). Digital control of dynamic systems. Addison-Wesley, Reading

    Google Scholar 

  3. Lam HY-F (1979). Analog and digital filters. Prentice-Hall, Englewood Cliffs

    Google Scholar 

  4. Mitra SK (2006). Digital signal processing, 3rd edn. McGraw-Hill, Boston

    Google Scholar 

  5. Phillips CL and Nagle HT (1984). Digital control system analysis and design. Prentice-Hall, Englewood Cliffs

    Google Scholar 

  6. Rabiner LR and Gold B (1975). Theory and applications of digital signal processing. Prentice-Hall, Englewood Cliffs

    Google Scholar 

  7. Al-Alaoui MA (1993) Novel digital integrator and differentiator. IEE Electr Lett 29(4):376-378, 18 February 1993. (See also ERRATA, IEE Electr Lett 29(10):934, 1993.)

  8. Al-Alaoui MA (1997). Filling the gap between the bilinear and the backward difference transforms: an interactive design approach. Int J Electr Eng Educat 34(4): 331–337

    Google Scholar 

  9. Al-Alaoui MA (2001). Novel stable higher order s-to-z transforms. IEEE Trans Circuits Syst I: Fundam Theory Appli 48(11): 1326–1329

    Article  Google Scholar 

  10. Šekara TB, Stojic MR (2005) Application of the α-approximation for discretization of analogue systems. Facta Universitatis, Ser: Elec. Energ (3):571–586 2005. http://factaee.elfak.ni.ac.yu/fu2k53/Sekara.pdf

  11. Šekara T (2006). New transformation polynomials for discretization of analogue systems. Electr Eng 89(2): 137–147

    Article  Google Scholar 

  12. Al-Alaoui MA (2006). Al-Alaoui operator and the α-approximation for discretization of analog systems. Facta Universitatis (NIS), Ser: Elec Energ 19(1): 143–146

    Google Scholar 

  13. Chen YQ and Moore KL (2002). Discretization for fractional order differentiators and integrators. IEEE Trans Circuits Syst I 49: 363–367

    Article  MathSciNet  Google Scholar 

  14. Al-Alaoui MA (1992). Novel approach to designing digital differentiators. IEEE Electron lett 28(15): 1376–1378

    Article  Google Scholar 

  15. Al-Alaoui MA (1994). Novel IIR differentiator from the simpson integration rule. IEEE Trans Circuits Syst I 42(2): 186–187

    Article  Google Scholar 

  16. Al-Alaoui MA (1996). A class of numerical integration rules with first order derivatives. ACM SIGNUM Newslett 31(2): 25–44

    Article  Google Scholar 

  17. Al-Alaoui MA and Ferzli R (2006). An Enhanced first-order sigma–delta modulator with a controllable signal-to-noise ratio. IEEE Trans Circuits Syst I 53(3): 634–643

    Article  Google Scholar 

  18. Al-Alaoui MA (2007). Linear phase lowpass IIR digital differentiators. IEEE Trans Signal Process 55(2): 697–706

    Article  MathSciNet  Google Scholar 

  19. Al-Alaoui MA (2007). Novel Approach to Analog to Digital Transforms. IEEE Trans Circuits Syst I: Fundam Theory Appl 54(2): 338–350

    Article  MathSciNet  Google Scholar 

  20. Al-Alaoui MA (2007). Using fractional delay to control the magnitudes and phases of integrators and differentiators. IET Signal Process 1(2): 107–119

    Article  MathSciNet  Google Scholar 

  21. Aoun M (2004). Numerical simulations of fractional systems: an overview of existing methods and improvements. Nonlionear Dynam 38(1–2): 117–131

    Article  MATH  Google Scholar 

  22. Barbosa RS, Machado JAT, Ferreira IM (2005) Pole-zero approximations of digital fractional-order integrators and differentiators using signal modeling techniques. In: 16th IFAC world congress, Prague, Czech Republic, July 4–8, 2005

  23. Barbosa RS and Machado JAT (2006). Implementation of discrete-time fractional order controllers based on LS approximations. Acta Polytech Hungarica 3(4): 5–22

    Google Scholar 

  24. Barbosa RS, Machado JAT and Silva MF (2006). Time domain design of fractional differintegrators using least-squares. Signal Process 86(10): 2567–2581

    Article  MATH  Google Scholar 

  25. Barbosa RS, Machado JAT and Galhano AM (2007). Performance of fractional PID algorithms controlling nonlinear systems with saturation and backlash phenomena. J Vib Contr 13(9–10): 1407–1418

    Article  MATH  Google Scholar 

  26. Backmutsky V, Shenkman A, Vaisman G, Zmudikov V, Kottick D and Blau M (1999). New DSP method for investigating dynamic behavior of power systems. Electric Mach Power Syst 27(4): 399–427

    Article  Google Scholar 

  27. Chen Y, Vinagre BM, Podlubny I (2003) A new discretization method for fractional order differentiators via continued fraction expansion. In: Proceedings of the ASME design engineering technical conference 5 A, pp 761–769

  28. Chen YQ and Vinagre BM (2003). A new IIR-type digital fractional order differentiator. Signal Process 83(11): 2359–2365

    Article  MATH  Google Scholar 

  29. Chen YQ, Vinagre BM and Podlubny I (2004). Continued fraction expansion approaches to discretizing fractional order derivatives—an expository review. Nonlinear Dynam 38(1–2): 155–170

    Article  MATH  MathSciNet  Google Scholar 

  30. Dabroom AM and Khalil HK (2000). Output feedback sampled-data control of nonlinear systems using high-gain observers. IEEE Trans Automatic Control 46(11, 1): 1712–1725

    MathSciNet  Google Scholar 

  31. Ferdi Y, Boucheham B (2004) Recursive filter approximation of digital differentiator and integrator based on Prony’s method. In: Proceedings of the first IFAC workshop on fractional differentiation and its applications (FDA’04), Bordeaux, France, July 19–21, 2004, pp 428–433

  32. Ferdi Y (2006). Computation of fractional order derivative and integral via power series expansion and signal modelling. Nonlinear Dynam 46(1): 1–15

    Article  MathSciNet  MATH  Google Scholar 

  33. Li Y, Dempster A, Li B, Wang J, Rizos C (2005) A low-cost attitude heading reference system by combination of GPS and magnetometers and MEMS inertial sensors for mobile applications. In: The International Symposium on GPS/GNSS

  34. Mrad F, Dandach SH, Azar S, Deeb G (2005) Operator friendly common sense controller with experimental verification using LabVIEW. In: Proceedings of the 2005 IEEE international symposium on, intelligent control Mediterrean conference on control and automation

  35. Ngo NQ (2006). A new approach for the design of wideband digital integrator and differentiator. IEEE Trans Circuits Syst II 53(9): 936–940

    Article  Google Scholar 

  36. Papamarkos N and Chamzas C (1996). A new approach for the design of digital integrators. IEEE Trans Circuits Syst I 43(9): 785–791

    Article  Google Scholar 

  37. Romero M, de Madrid AP, Manoso C, Hernandez R (2006) Discretization of the fractional-order differentiator/integrator by the rational chebyshev approximation. In: CONTROLO 2006, 7th Portuguese conference on automatic control, Instituto Superior Tecnico, Lisboa, Portugal, September 11–13, 2006

  38. Tomov L (2006). Matrix method for generalized discretization of transfer functions through formal replacement with first order operator. J Fundam Sci Appl 13(4): 61–68

    Google Scholar 

  39. Tseng C-C (2006) Digital integrator design using simpson rule and fractional delay filter. IEEE Proc Vis Image Signal Process 153(1)

  40. Tseng C-C (2007). Closed form design of digital IIR integrators using numerical integration rules and fractional sample delays. IEEE Trans Circuits Syst I 54(3): 643–655

    Article  MathSciNet  Google Scholar 

  41. Varshney P, Gupta M and Visweswaran GS (2007). New switched capacitor fractional order integrator. J Active Passive Electron Dev 2: 187–197

    Google Scholar 

  42. Wong CX and Worden K (2007). Generalized NARX shunting neural network modelling of friction. Mech Syst Signal Process 21(1): 553–572

    Article  Google Scholar 

  43. Worden K, Wong CX, Parlitz U, Hornstein A, Engster D, Tjahjowidodo T, Al-Bender F, Rizos DD and Fassois SD (2006). Identification of pre-sliding and sliding friction dynamics: grey box and black-box models. Mech Syst Signal Process 21(1): 514–534, 2007; see also vol 13(4), 2006

    Article  Google Scholar 

  44. Zhang J, Zhang K, Grenfell R, Li Y and Deakin R (2005). Real-time Doppler/Doppler rate derivation for dynamic applications. J Global Position Syst 4(1–2): 95–105

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Electrical and Computer Engineering, American University of Beirut, P.O. Box 11-0236, 179 Bliss Street, Beirut, 1107 2020, Lebanon

    Mohamad Adnan Al-Alaoui

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  1. Mohamad Adnan Al-Alaoui
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Correspondence to Mohamad Adnan Al-Alaoui.

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This research was supported, in part, by the University Research Board of the American University of Beirut.

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Al-Alaoui, M.A. Al-Alaoui operator and the new transformation polynomials for discretization of analogue systems. Electr Eng 90, 455–467 (2008). https://doi.org/10.1007/s00202-007-0092-0

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  • DOI: https://doi.org/10.1007/s00202-007-0092-0

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