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Spectral Factorization

Encyclopedia of Systems and Control

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Abstract

For more than half a century, spectral factorization is encountered in various fields of science and engineering. It is a useful tool in robust and optimal control and filtering and many other areas. It is also a nice control-theoretical concept closely related to Riccati equation. As a quadratic equation in polynomials, it is a challenging algebraic task.

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Notes

  1. 1.

    In fact, one can stay with standard one-sided polynomials (either in nonnegative or in nonpositive powers only), if every adjoint \({p}^{{\ast}}(z)\) is multiplied by proper power of z to create a one-sided polynomial \(\bar{p}(z) = {p}^{{\ast}}(z){z}^{n}\).

  2. 2.

    The right and the left spectral factor are sometimes called the factorand the cofactor, respectively, but the terminology is not set at all.

Further Reading

  • Nice tutorial books on polynomials and polynomial matrices in control theory and design are Kučera (1979), Callier et al. (1982), and Kailath (1980)

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  • The concept of spectral factorization was introduced by Wiener (1949), for further information see later original papers Wilson (1972) or Kwakernaak et al. (1994) as well as survey papers Kwakernaak (1991), Sayed et al. (2001) or Kučera (2007).

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  • Nice applications of spectral factorization in control problems can be found e.g. in Green et al. (1990), Henrion et al. (2003) or Zhou et al. (1998). For its use of in other engineering problems see e.g. Sternad et al. (1993).

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Bibliography

  • Callier FM (1985) On polynomial matrix spectral factorization by symmetric extraction. IEEE Trans Autom Control 30:453–464

    Article  MATH  MathSciNet  Google Scholar 

  • Callier FM, Desoer CA (1982) Multivariable feedback systems. Springer, New York

    Book  Google Scholar 

  • Davis MC (1963) Factorising the spectral matrix. IEEE Trans Autom Control 8:296

    Article  Google Scholar 

  • Green M, Glover K, Limebeer DJN, Doyle J (1990) A J-spectral factorization approach to H-infinity control. SIAM J Control Opt 28:1350–1371

    Article  MATH  MathSciNet  Google Scholar 

  • Henrion D, Sebek M (2000) An algorithm for polynomial matrix factor extraction. Int J Control 73(8):686–695

    Article  MATH  MathSciNet  Google Scholar 

  • Henrion D, Šebek M, Kučera V (2003) Positive polynomials and robust stabilization with fixed-order controllers. IEEE Trans Autom Control 48:1178–1186

    Article  Google Scholar 

  • Hromčík M, Šebek M (2007) Numerical algorithms for polynomial Plus/Minus factorization. Int J Robust Nonlinear Control 17(8):786–802

    Article  MATH  Google Scholar 

  • Jakubovič VA (1970) Factorization of symmetric matrix polynomials. Dokl. Akad. Nauk SSSR, 194(3): 532-535

    MathSciNet  Google Scholar 

  • Ježek J, Kučera V (1985) Efficient algorithm for matrix spectral factorization. Automatica 29: 663–669

    Google Scholar 

  • Kailath T (1980) Linear systems. Prentice-Hall, Englewood Cliffs

    MATH  Google Scholar 

  • Kučera V (1979) Discrete linear control: the polynomial equation approach. Wiley, Chichester

    MATH  Google Scholar 

  • Kučera V (2007) Polynomial control: past, present, and future. Int J Robust Nonlinear Control 17:682–705

    Article  MATH  Google Scholar 

  • Kwakernaak H (1991) The polynomial approach to a H-optimal regulation. In: Mosca E, Pandolfi L (eds) H-infinity control theory. Lecture Notes in Maths, vol 1496. Springer, Berlin

    Google Scholar 

  • Kwakernaak H, Šebek M (1994) Polynomial J-spectral factorization. IEEE Trans Autom Control 39:315–328

    Article  MATH  Google Scholar 

  • Oara C, Varga A (2000) Computation of general inner-outer and spectral factorizations. IEEE Trans Autom Control 45:2307–2325

    Article  MATH  MathSciNet  Google Scholar 

  • Sayed AH, Kailath T (2001) A survey of spectral factorization methods. Numer Linear Algebra Appl 8(6–7):467–496

    Article  MATH  MathSciNet  Google Scholar 

  • Sternad M, Ahlén A (1993) Robust filtering and feedforward control based on probabilistic descriptions of model errors. Automatica 29(3):661–679

    Article  MATH  MathSciNet  Google Scholar 

  • Šebek M (1992) J-spectral factorization via Riccati equation. In: Proceedings of the 31st IEEE CDC, Tuscon, pp 3600–3603

    Google Scholar 

  • Vostrý Z (1975). New algorithm for polynomial spectral factorization with quadratic convergence. Kybernetika 11:415, 248

    Google Scholar 

  • Wiener N (1949) Extrapolation, interpolation and smoothing of stationary time series. Wiley, New York

    MATH  Google Scholar 

  • Wilson GT (1972) The factorization of matricial spectral densities. SIAM J Appl Math 23:420

    Article  MATH  MathSciNet  Google Scholar 

  • Youla DC, Kazanjian NN (1978) Bauer-type factorization of positive matrices and the theory of matrix polynomials orthogonal on the unit circle. IEEE Trans Circuits Syst 25:57

    Article  MATH  MathSciNet  Google Scholar 

  • Zhong QC (2005) J-spectral factorization of regular para-Hermitian transfer matrices. Automatica 41:1289–1293

    Article  MATH  Google Scholar 

  • Zhou K, Doyle JC (1998) Essentials of robust control. Prentice-Hall, Upper Saddle River

    Google Scholar 

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Authors and Affiliations

  1. Department of Control Engineering, Faculty of Electrical Engineering, Czech Technical University in Prague, 166 27, Prague 6, Czech Republic

    Professor Michael Sebek

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  1. Professor Michael Sebek
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Correspondence to Professor Michael Sebek .

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Editors and Affiliations

  1. Electrical and Computer Engineering, Boston University, Boston, Massachusetts, USA

    John Baillieul

  2. Automation and Control Solutions, Honeywell, Golden Valley, Minnesota, USA

    Tariq Samad

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Sebek, P. (2014). Spectral Factorization. In: Baillieul, J., Samad, T. (eds) Encyclopedia of Systems and Control. Springer, London. https://doi.org/10.1007/978-1-4471-5102-9_240-1

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  • DOI: https://doi.org/10.1007/978-1-4471-5102-9_240-1

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Chapter history

  1. Latest

    Spectral Factorization
    Published:
    05 May 2015

    DOI: https://doi.org/10.1007/978-1-4471-5102-9_240-2

  2. Original

    Spectral Factorization
    Published:
    25 September 2014

    DOI: https://doi.org/10.1007/978-1-4471-5102-9_240-1

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