Rational Approximation in Systems Engineering pp 321-343 | Cite as

# Nested-Feedback-Loops Realization of 2-D Systems

Chapter

## Abstract

Algorithms are derived for representation of double power series by branched continued fractions. Some properties of the branched continued fractions are given. Two new alternative structures for realization of a two-dimensional, zero memory nonlinear transfer characteristic are proposed. These structures permit the design of 2-D systems with relatively low sensitivity of the transfer function to coefficient errors. Generalized equivalence transformations of branched continued fractions are also presented. Illustrative examples are included.

### Keywords

Univer Active Element serA Summing## Preview

Unable to display preview. Download preview PDF.

### References

- 1.A. Bultheel, “Division algorithm for continued fractions and the Padé table,” Journal Comp. Appl. Math., 1980, pp. 259-266.Google Scholar
- 2.J. S. R. Chisholm, “Rational approximants defined from double power series,” Math. Comp., 1973, No 124, pp. 841–848.CrossRefGoogle Scholar
- 3.N. K. Bose, and S. Basu, “Two dimensional matrix Padé approximants,” IEEE Trans. on Automatic Control, vol. AC-25, No 3, June 1980, pp. 509–514.CrossRefGoogle Scholar
- 4.P. I. Bodnarčuk and W. Ja. Skorobogat’ko, “Branched Continued Fractions and Their Applications,” Kiev, Naukowaja Dumka, 1974, (in Ukranian).Google Scholar
- 5.Kh. J. Kutschminskaja, “Corresponding and associated branching continued fractions for the double power series,” Dokl. Akad. Nauk USSR, No 7, ser A, 1978, pp. 614–617, (in Russian).Google Scholar
- 6.J. A. Murphy and M. R. O’Donohoe, “A two-variable generalization of the Stieltjes-type continued fraction,” J. Comp. Appl. Math., No 3, 1978, pp. 181–190.CrossRefGoogle Scholar
- 7.J. A. Murphy and M. R. O’Donohoe, “A class of algorithms for obtaining rational approximants to functions which are defined by power series,” Journ. of Appl. Math. and Physics (ZAMP), No 28, 1977, pp. 1121-1131.Google Scholar
- 8.W. Siemaszko, “Branched continued fraction for double power series,” J. Comp. Appl. Math., No 2, 1980, pp. 121–125.CrossRefGoogle Scholar
- 9.R. E. Kaiman, “On partial realization, transfer functions and canonical forms,” Acta Politechnica Scandinavica, Helsinki, 1979, pp. 9-32.Google Scholar
- 10.W. B. Gragg, “Matrix interpretations and applications of the continued fraction algorithm,” Rocky Mountain J. of Math., No 4, 1974, pp. 213–225.CrossRefGoogle Scholar
- 11.L. Wuytack (Ed.), “Padé Approximation and its Applications,” Springer-Verlag, Berlin 1979 Proceeding of the conference at Antwerp, Belgium, 1979, Lect. Notes in Math. 765.Google Scholar
- 12.C. Brezinski, “Padé-Type Approximation and General Orthogonal Polynomials,” Birkhäuser Verlag, Basel, 1980.Google Scholar
- 13.V. Belevith and Y. Genin, “Implicit interpolation, trigradients and continued fractions,” Philips Res. Repts., vol. 26, 1971, pp. 453–470.Google Scholar
- 14.W. B. Jones and W. J. Thron, “Numerical stability in evaluating continued fractions,” Math. of Computation, vol. 28, No 127, July 1974, pp. 798–810.CrossRefGoogle Scholar
- 15.P. Van der Cruyssen, “Stable evaluation of generalized continued fractions,” Report 80-08, Universiteit Antwerpen, March 1980.Google Scholar
- 16.J. Huertas, “D-T adoptor: Applications to the design of non-linear
*n*-ports,” Int. J. Circuit Theory and Applications, vol. 8, No 3, 1980, pp. 273–290.CrossRefGoogle Scholar - 17.L. O. Chua, “Device modeling via basic nonlinear circuit elements,” IEEE Trans. Circuits and Systems, vol. CAS 27, No 11, 1980, pp. 1014–1044.CrossRefGoogle Scholar
- 18.A. Cichocki, “Synthesis of nonlinear functions using continued fraction techniques,” Electronics Letters, No 11, 1980, pp. 431–433.CrossRefGoogle Scholar
- 19.A. Cichocki, “Modeling of
*n*-dimensional functions using multibranch continued fractions,” Procedings ECCTD’ 80, Warsaw 1980, pp. 331–336.Google Scholar - 20.A. Cichocki, “Generalized continued fraction expansion of multidimensional rational functions and its application in synthesis,” Proc. ECCTD’ 80, Warsaw 1980, pp. 286–291.Google Scholar
- 21.A. Cichocki and S. Osowski, “Matrix continuants, some properties and applications,” AEÜ, Band 31, 1977, pp. 431–435.Google Scholar
- 22.S. K. Mitra and R. Sherwood, “Canonic realizations of digital filters using the continued fraction expansion,” IEEE Trans. Audio and Electoacoustics, No 3, Aug. 1972, pp. 185–194.CrossRefGoogle Scholar
- 23.A. D. Field and D. H. Owens, “A canonical form for the reduction of linear scalar systems,” Proc. IEE, vol. 125, No 4, April 1978, pp. 337–342.Google Scholar
- 24.T. Muir, “A Treatise on the Theory of Determinants,” New York, Dover 1960, pp. 516-565.Google Scholar
- 25.A. Cichocki, “On realizations of orthogonal polynomials and their applications to the synthesis of nonlinear networks,” ECCTD’ 81, the Hague, 1981 pp. 814-821.Google Scholar
- 26.A. Cichocki, “Synthesis of nonlinear networks using operational amplifiers and controlled elements,” (in Polish), Zeszyty Naukowe Elektryka, WPW, Nr 67, 1982.Google Scholar
- 27.F. W. J. Olver, “Bounds for the solutions of second-order linear difference equations,” Journal of Research of National Bureau of Standards, vol. 71B, no 4, 1967, pp. 161–166.CrossRefGoogle Scholar
- 28.O. Perron, “Die Lehre von den Kettenbrüchen,” Band II, B. G. Teubner, Stuttgard, 1957.Google Scholar

## Copyright information

© Springer Science+Business Media New York 1982