Abstract
We consider the problem of computing the greatest common divisor of a set of univariate polynomials and present applications of this problem in system theory and signal processing. One application is blind system identification: given the responses of a system to unknown inputs, find the system. Assuming that the unknown system is finite impulse response and at least two experiments are done with inputs that have finite support and their Z-transforms have no common factors, the impulse response of the system can be computed up to a scaling factor as the greatest common divisor of the Z-transforms of the outputs. Other applications of greatest common divisor problem in system theory and signal processing are finding the distance of a system to the set of uncontrollable systems and common dynamics estimation in a multi-channel sum-of-exponentials model.
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Bini, D.A., Boito, P.: A fast algorithm for approximate polynomial GCD based on structured matrix computations. In: Bini, D.A., Mehrmann, V., Olshevsky, V., Tyrtyshnikov, E.E., van Barel, M. (eds.) Numerical Methods for Structured Matrices and Applications. Operator Theory: Advances and Applications, vol. 199, pp. 155–173. Birkhäuser, Basel (2010). https://doi.org/10.1007/978-3-7643-8996-3_6
Eising, R.: Distance between controllable and uncontrollable. Control Lett. 4, 263–264 (1984)
Gu, M., Mengi, E., Overton, M., Xia, J., Zhu, J.: Fast methods for estimating the distance to uncontrollability. SIAM J. Matrix Anal. Appl. 28, 477–502 (2006)
Higham, N.: Matrix nearness problems and applications. In: Gover, M., Barnett, S. (eds.) Applications of Matrix Theory, pp. 1–27. Oxford University Press, Oxford (1989)
Hu, G., Davison, E.: Real controllablity/stabilizability radius of LTI systems. IEEE Trans. Automat. Control 49, 254–258 (2004)
Kaltofen, E., Corless, R.M., Jeffrey, D.J.: Challenges of symbolic computation: my favorite open problems. J. Symbolic Comput. 29(6), 891–919 (2000)
Karow, M., Kressner, D.: On the structured distance to uncontrollability. Control Lett. 58, 128–132 (2009)
Khare, S., Pillai, H., Belur, M.: Computing the radius of controllability for state space systems. Control Lett. 61, 327–333 (2012)
Markovsky, I., Van Huffel, S.: An algorithm for approximate common divisor computation. In: Proceedings of the 17th Symposium on Mathematical Theory of Networks and Systems, Kyoto, Japan, pp. 274–279 (2006)
Paige, C.C.: Properties of numerical algorithms related to computing controllability. IEEE Trans. Automat. Control 26, 130–138 (1981)
Papy, J.M., Lathauwer, L.D., Huffel, S.V.: Common pole estimation in multi-channel exponential data modeling. Signal Process. 86(4), 846–858 (2006)
Polderman, J., Willems, J.C.: Introduction to Mathematical Systems Theory. Springer, New York (1998). https://doi.org/10.1007/978-1-4757-2953-5
Qiu, W., Hua, Y., Abed-Meraim, K.: A subspace method for the computation of the GCD of polynomials. Automatica 33(4), 741–743 (1997)
Rupprecht, D.: An algorithm for computing certified approximate GCD of n univariate polynomials. J. Pure Appl. Algebra 139(1–3), 255–284 (1999)
Usevich, K., Markovsky, I.: Variable projection methods for approximate (greatest) common divisor computations. Theor. Comput. Sci. 681, 176–198 (2017)
Willems, J.C.: Paradigms and puzzles in the theory of dynamical systems. IEEE Trans. Automat. Control 36(3), 259–294 (1991)
Zeng, Z., Dayton, B.: The approximate GCD of inexact polynomials. Part I: a univariate algorithm. In: Proceedings of the International Symposium on Symbolic and Algebraic Computation, pp. 320–327 (2004)
Zhi, L., Yang, Z.: Computing approximate GCD of univariate polynomials by structure total least norm. In: Wang, D., Zhi, L. (eds.) International Workshop on Symbolic-Numeric, Xian, China, pp. 188–201 (2005)
Acknowledgements
The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant 258581 “Structured low-rank approximation: Theory, algorithms, and applications”; Fund for Scientific Research (FWO-Vlaanderen), FWO projects G028015N “Decoupling multivariate polynomials in nonlinear system identification”; G090117N “Block-oriented nonlinear identification using Volterra series”; and FWO/FNRS Excellence of Science project 30468160 “Structured low-rank matrix/tensor approximation: numerical optimization-based algorithms and applications”.
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Markovsky, I., Fazzi, A., Guglielmi, N. (2018). Applications of Polynomial Common Factor Computation in Signal Processing. In: Deville, Y., Gannot, S., Mason, R., Plumbley, M., Ward, D. (eds) Latent Variable Analysis and Signal Separation. LVA/ICA 2018. Lecture Notes in Computer Science(), vol 10891. Springer, Cham. https://doi.org/10.1007/978-3-319-93764-9_10
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DOI: https://doi.org/10.1007/978-3-319-93764-9_10
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