Exact linear modeling with polynomial coefficients

  • Eva Zerz
  • Viktor Levandovskyy
  • Kristina Schindelar


Given a finite set of polynomial, multivariate, and vector-valued functions, we show that their span can be written as the solution set of a linear system of partial differential equations (PDE) with polynomial coefficients. We present two different but equivalent ways to construct a PDE system whose solution set is precisely the span of the given trajectories. One is based on commutative algebra and the other one works directly in the Weyl algebra, thus requiring the use of tools from non-commutative computer algebra. In behavioral systems theory, the resulting model for the data is known as the most powerful unfalsified model (MPUM) within the class of linear systems with kernel representations over the Weyl algebra, i.e., the ring of differential operators with polynomial coefficients.


Multidimensional systems Linear systems with polynomial coefficients Behavioral approach Linear exact modeling Polynomial trajectories Most powerful unfalsified model Gröbner bases 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Antoulas A. C., Willems J. C. (1993) A behavioral approach to linear exact modeling. IEEE Transactions on Automatic Control 38: 1776–1802MATHCrossRefMathSciNetGoogle Scholar
  2. Bose N. K. (1982) Applied multidimensional systems theory. Van Nostrand Reinhold, New York, LondonMATHGoogle Scholar
  3. Bose, N. K. (eds) (1985) Multidimensional systems theory. D. Reidel, DordrechtMATHGoogle Scholar
  4. Bose N. K. (2003) Multidimensional systems theory and applications (2nd ed.). Kluwer, DordrechtMATHGoogle Scholar
  5. Bose N. K. (2007) Two decades of Gröbner bases in multidimensional systems. Radon Series Computational Applied Mathematics 3: 1–22Google Scholar
  6. Charoenlarpnopparut C., Bose N. K. (1999) Multidimensional FIR filter bank design using Gröbner bases. IEEE Transactions on Circuits Systems II 46: 1475–1486MATHCrossRefGoogle Scholar
  7. Charoenlarpnopparut C., Bose N. K. (2001) Gröbner bases for problem solving in multidimensional systems. Multidimensional Systems Signal Processing 12: 365–376MATHCrossRefMathSciNetGoogle Scholar
  8. Goodearl K. R., Warfield R. B. Jr. (2004) An introduction to noncommutative noetherian rings (2nd ed.). London Mathematical Society, LondonMATHCrossRefGoogle Scholar
  9. Greuel, G.-M., Pfister, G., & Schönemann, H. (2009). Singular 3-1-0—a computer algebra system for polynomial computations. http://www.singular.uni-kl.de.
  10. Kuijper M., Polderman J. W. (2004) Reed-solomon list decoding from a system-theoretic perspective. IEEE Transactions on Information Theory 50: 259–271CrossRefMathSciNetGoogle Scholar
  11. Levandovskyy, V. (2006). Plural, a non-commutative extension of Singular. In A. Iglesias, & N. Takayama (Eds.), Mathematical software—ICMS 2006. Lecture Notes in Computer Science Vol. 4151. Springer.Google Scholar
  12. Levandovskyy, V., Schindelar, K., & Zerz, E. (2010). Exact linear modeling using ore algebras. Journal of Symbolic Computation.Google Scholar
  13. Lin Z., Xu L., Bose N. K. (2008) A tutorial on Gröbner bases with applications in signals and systems. IEEE Transactions on Circuits Systems I 55: 445–461CrossRefMathSciNetGoogle Scholar
  14. Lin Z., Xu L., Wu Q. (2004) Applications of Gröbner bases to signal and image processing: A survey. Linear Algebra and its Applications 391: 169–202MATHCrossRefMathSciNetGoogle Scholar
  15. Schindelar, K., Levandovskyy, V., & Zerz, E. (2008). Linear exact modeling with variable coefficients. In Proceedings of the 18th international symposium on mathematical theory networks systems (MTNS), Blacksburg.Google Scholar
  16. Schindelar, K. (2010). Algorithmic aspects of algebraic system theory. Ph.D. Thesis, RWTH Aachen University.Google Scholar
  17. Willems J. C. (1986) From time series to linear system. Part II: Exact modelling. Automatica 22: 675–694MATHCrossRefMathSciNetGoogle Scholar
  18. Zerz E. (2000) Topics in multidimensional linear systems theory. Lecture notes in control and information sciences. Springer, LondonGoogle Scholar
  19. Zerz E. (2005) Characteristic frequencies, polynomial-exponential trajectories, and linear exact modeling with multidimensional behaviors. SIAM Journal on Control Optimization 44: 1148–1163MATHCrossRefMathSciNetGoogle Scholar
  20. Zerz, E. (2006). Recursive computation of the multidimensional MPUM. In Proceedings of the 17th international symposium on mathematical theory networks systems (MTNS), Kyoto.Google Scholar
  21. Zerz E. (2008) The discrete multidimensional MPUM. Multidimensional Systems Signal Processing 19: 307–321MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Eva Zerz
    • 1
  • Viktor Levandovskyy
    • 1
  • Kristina Schindelar
    • 1
  1. 1.Lehrstuhl D für MathematikRWTH Aachen UniversityAachenGermany

Personalised recommendations