Abstract
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.
Similar content being viewed by others
References
Antoulas A. C., Willems J. C. (1993) A behavioral approach to linear exact modeling. IEEE Transactions on Automatic Control 38: 1776–1802
Bose N. K. (1982) Applied multidimensional systems theory. Van Nostrand Reinhold, New York, London
Bose, N. K. (eds) (1985) Multidimensional systems theory. D. Reidel, Dordrecht
Bose N. K. (2003) Multidimensional systems theory and applications (2nd ed.). Kluwer, Dordrecht
Bose N. K. (2007) Two decades of Gröbner bases in multidimensional systems. Radon Series Computational Applied Mathematics 3: 1–22
Charoenlarpnopparut C., Bose N. K. (1999) Multidimensional FIR filter bank design using Gröbner bases. IEEE Transactions on Circuits Systems II 46: 1475–1486
Charoenlarpnopparut C., Bose N. K. (2001) Gröbner bases for problem solving in multidimensional systems. Multidimensional Systems Signal Processing 12: 365–376
Goodearl K. R., Warfield R. B. Jr. (2004) An introduction to noncommutative noetherian rings (2nd ed.). London Mathematical Society, London
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.
Kuijper M., Polderman J. W. (2004) Reed-solomon list decoding from a system-theoretic perspective. IEEE Transactions on Information Theory 50: 259–271
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.
Levandovskyy, V., Schindelar, K., & Zerz, E. (2010). Exact linear modeling using ore algebras. Journal of Symbolic Computation.
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: 445–461
Lin Z., Xu L., Wu Q. (2004) Applications of Gröbner bases to signal and image processing: A survey. Linear Algebra and its Applications 391: 169–202
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.
Schindelar, K. (2010). Algorithmic aspects of algebraic system theory. Ph.D. Thesis, RWTH Aachen University.
Willems J. C. (1986) From time series to linear system. Part II: Exact modelling. Automatica 22: 675–694
Zerz E. (2000) Topics in multidimensional linear systems theory. Lecture notes in control and information sciences. Springer, London
Zerz E. (2005) Characteristic frequencies, polynomial-exponential trajectories, and linear exact modeling with multidimensional behaviors. SIAM Journal on Control Optimization 44: 1148–1163
Zerz, E. (2006). Recursive computation of the multidimensional MPUM. In Proceedings of the 17th international symposium on mathematical theory networks systems (MTNS), Kyoto.
Zerz E. (2008) The discrete multidimensional MPUM. Multidimensional Systems Signal Processing 19: 307–321
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zerz, E., Levandovskyy, V. & Schindelar, K. Exact linear modeling with polynomial coefficients. Multidim Syst Sign Process 22, 55–65 (2011). https://doi.org/10.1007/s11045-010-0125-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11045-010-0125-0