Abstract
Given a Laurent polynomial \(F \in \mathbb {C}[z_1^{\pm 1}, \ldots , z_n^{\pm 1}]\), its amœba \(\mathcal A_F\) is the image by \(z=(z_1, \ldots , z_n)\in (\mathbb {C}^*)^n \longmapsto (\log |z_1|, \ldots , \log |z_n|)\in \mathbb {R}^n\) of the algebraic zero set \(V(F)=\{z\in (\mathbb {C}^*)^n\,;\, F(z)=0\}\) of the complex torus \(\mathbb {T}^n :=(\mathbb {C}^*)^n\). We relate here the question of the BIBO stability of a multilinear discrete time invariant system with a regular transfer function \( G(z_1, ..., z_n)/F(z_1, \ldots , z_n)\), where \(F, G\in \mathbb {C}[z_1, ..., z_n]\) are coprime or more precisely structural stability, with the geometrical study of the amœba \(\mathcal A_F\). A criterion for strong and weak structural stability is expressed in terms of the position of \(\varvec{0}=(0, \ldots , 0)\in \mathbb {R}^n\) with respect to the amœba \(\mathcal {A}_{F}\). Then we propose a Monte-Carlo integration based algorithm in order to test the structural stability of a given such system. The proposed algorithm is not limited by the curse of dimensionality, as opposed to the state-of-the-art methods: It can be applied to any number of variables n. Several illustrative examples are presented and discussed.
Similar content being viewed by others
Data availibility statement
Strictly speaking, Data sharing is not applicable: Source data for the figures are reproducible following the description given in the paper. Nonetheless, the Matlab code used to generate the data is available at http://gofile.me/6FJod/OtXuYyz4P. The code will also be made available at the French open archive HAL, as a supplementary material, with the paper after acceptance.
Notes
The Matlab computation of all 5 examples took less than 1.4s on a MacBook Pro-2.3 GHz, Intel Core I5.
References
Anderson, B., & Jury, E. (1974). Stability of multidimensional digital filters. IEEE Transactions on Circuits and Systems, 21(2), 300–304. https://doi.org/10.1109/TCS.1974.1083834
Benidir, M. (1991). Sufficient conditions for the stability of multidimensional recursive digital filters. In: International Conference on Acoustics, Speech, and Signal Processing (ICASSP), 4, 2885–2888
Bistritz, Y. (1984). Zero location with respect to the unit circle of discrete-time linear system polynomials. Proceedings of the IEEE, 72(9), 1131–1142.
Bistritz, Y. (1999). Stability testing of two-dimensional discrete linear system polynomials by a two-dimensional tabular form. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 46(6), 666–676. https://doi.org/10.1109/81.768823
Bogdanov, D. V., Kytmanov, A. A., & Sadykov, T. M. (2016). Algorithmic computation of polynomial amoebas. In V. P. Gerdt, W. Koepf, W. M. Seiler, & E. V. Vorozhtsov (Eds.), Computer Algebra in Scientific Computing (pp. 87–100). Cham: Springer.
Bogdanov, D. V., & Sadykov, T. M. (2020). Hypergeometric polynomials are optimal. Mathematical Z, 296, 373–390.
Borcea, J., & Brändén, P. (2009). The Lee-Yang and Pólya-Schur programs. II. Theory of stable polynomials and applications. Communication on Pure and Applied Mathematics 62(12), 1595–1631
Borcea, J., & Brändén, P. (2008). Applications of stable polynomials to mixed determinants: Johnson’s conjectures, unimodality, and symmetrized Fischer products. Duke Mathematical Journal, 143, 205–223.
Bose, N. (1977). Implementation of a new stability test for two-dimensional filters. IEEE Transactions on Acoustics, Speech, and Signal Processing, 25(2), 117–120. https://doi.org/10.1109/TASSP.1977.1162917
Bouzidi, Y., & Rouillier, F. (2016). Certified Algorithms for Proving the Structural Stability of Two Dimensional Systems Possibly with Parameters. In: 22nd International Symposium on Mathematical Theory of Networks and Systems (MTNS), Minneapolis (USA)
Bouzidi, Y., Quadrat, A., & Rouillier, F. (2015). Computer algebra methods for testing the structural stability of multidimensional systems. In: 9th IEEE International Workshop on Multidimensional (nD) Systems (nDS), 1–6
Bouzidi, Y., Quadrat, A., & Rouillier, F. (2019). Certified non-conservative tests for the structural stability of discrete multidimensional systems. Multidimensional Systems and Signal Processing, 30(3), 1205–1235.
Cooley, J. W., & Tukey, J. W. (1965). An algorithm for the machine calculation of complex Fourier series. Mathematics of Computation, 19, 297–301.
DeCarlo, R., Murray, J., & Saeks, R. (1977). Multivariable Nyquist theory. International Journal of Control, 25(5), 657–675. https://doi.org/10.1080/00207177708922261
Dumitrescu, B. (2006). Stability test of multidimensional discrete-time systems via sum-of-squares decomposition. IEEE Transactions on Circuits and Systems I: Regular Papers, 53(4), 928–936.
Forsberg, M., Passare, M., & Tsikh, A. (2000). Laurent determinants and arrangements of hyperplane amoebas. Advances in Mathematics, 151(1), 45–70. https://doi.org/10.1006/aima.1999.1856
Forsgård, J., Matusevich, L., Mehlhop, N., & De Wolff, T. (2019). Lopsided approximation of amoebas. Mathematics of Computation, 88(315), 485–500.
Gelfand, I. M., Kapranov, M. M., & Zelevinsky, A. V. (1994). Discriminants, Resultants, and Multidimensional Determinants. Birkhäuser Boston Inc., Boston, MA. https://doi.org/10.1007/978-0-8176-4771-1
Goodman, D. (1977). Some stability properties of two-dimensional linear shift-invariant digital filters. IEEE Transactions on Circuits and Systems, 24(4), 201–208. https://doi.org/10.1109/TCS.1977.1084322
Grinshpan, A., Kaliuzhnyi-Verbovetskyi, D. S., Vinnikov, V., & Woerdeman, H. J. (2016). Matrix-valued Hermitian positivstellensatz, lurking contractions, and contractive determinantal representations of stable polynomials. In T. Eisner, B. Jacob, A. Ran, & H. Zwart (Eds.), Operator Theory, Function Spaces, and Applications (pp. 123–136). Cham: Springer.
Grinshpan, A., Kaliuzhnyi-Verbovetskyi, D. S., & Woerdeman, H. J. (2013). Norm-constrained determinantal representations of multivariable polynomials. Complex Analysis and Operator Theory, 7(3), 635–654. https://doi.org/10.1007/s11785-012-0262-6
Huang, T. (1972). Stability of two-dimensional recursive filters. IEEE Transactions on Audio and Electroacoustics, 20(2), 158–163. https://doi.org/10.1109/TAU.1972.1162364
Justice, J., & Shanks, J. (1973). Stability criterion for N-dimensional digital filters. IEEE Transactions on Automatic Control, 18(3), 284–286.
Li, L., Xu, L., & Lin, Z. (2013). Stability and stabilisation of linear multidimensional discrete systems in the frequency domain. International Journal of Control, 86(11), 1969–1989.
Lundqvist, J. (2015). An explicit calculation of the Ronkin function. Annales de la Faculté des sciences de Toulouse : Mathématiques - Série 6, 24(2), 227–250
Nisse, M., & Sadykov, T. (2019). Amoeba-shaped polyhedral complex of an algebraic hypersurface. Journal of Geometrical Analysis, 29, 1356–1368.
Ossete Ingoba, W. (2019). A new insight on ronkin functions or currents. Complex Analysis and Operator Theory, 13(2), 525–562. https://doi.org/10.1007/s11785-018-0845-y
Passare, M., & Rullgård, H. (2004). Amoebas, Monge-Ampere measures and triangulations of the Newton polytope. Duke Mathematical Journal, 121, 481–507.
Purbhoo, K. (2008). A nullstellensatz for amoebas. Duke Mathematical Journal, 141(3), 407–445. https://doi.org/10.1215/00127094-2007-001
Ronkin, L. I. (2000). On zeros of almost periodic functions generated by holomorphic functions in a multicircular domain. Complex analysis in modern mathematics (Russian), 243–256
Schussler, H. (1976). A stability theorem for discrete systems. IEEE Transactions on Acoustics, Speech, and Signal Processing, 24(1), 87–89. https://doi.org/10.1109/TASSP.1976.1162762
Serban, I., & Najim, M. (2007a). A multidimensional Schur-Cohn algorithm in tabular forms. In: 2007 European Control Conference (ECC), 3900–3905. https://doi.org/10.23919/ECC.2007.7068694
Serban, I., & Najim, M. (2007b). A new multidimensional Schur-Cohn type stability criterion. In: 2007 American Control Conference, 5533–5538. https://doi.org/10.1109/ACC.2007.4282645
Serban, I., & Najim, M. (2006). Schur coefficients in several variables. Journal of Mathematical Analysis and Applications, 320(1), 293–302. https://doi.org/10.1016/j.jmaa.2005.06.068
Serban, I., & Najim, M. (2007c). Multidimensional systems: BIBO stability test based on functional Schur coefficients. IEEE Transactions on Signal Processing, 55(11), 5277–5285. https://doi.org/10.1109/TSP.2007.896070
Serban, I., Turcu, F., Stitou, Y., & Najim, M. (2005). Multidimensional Schur coefficients and BIBO stability. Communication Information Systems, 5(1), 131–142.
Strintzis, M. (1977). Tests of stability of multidimensional filters. IEEE Transactions on Circuits and Systems, 24(8), 432–437.
Wagner, D. (2011). Multivariate stable polynomials: theory and applications. Bulletin of the American Mathematical Society, 48(1), 53–84.
Walsh, J. (1964). A theorem of Grace on the zeros of polynomials, revisited. Proceedings of the American Mathematical Society, 15(3), 354–360.
Yger, A. (2012). Tropical geometry and amoebas, Université Bordeaux 1, France. Lecture. https://cel.archives-ouvertes.fr/cel-00728880
Funding
The PhD program of the first author is supported by the research grant PAU/ADM/PAUSTI/2016/2 from the African Union Commission through Pan African University Institute of Basic Science, Technology and Innovations.
Author information
Authors and Affiliations
Contributions
All authors have equally contributed in this paper.
Corresponding author
Ethics declarations
Conflict of interest
All authors certify that they have no affiliations with or involvement in any organization or entity with any financial interest or non-financial interest in the subject matter or materials discussed in this manuscript.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Bossoto, B., Mboup, M. & Yger, A. Amœbas and structural stability of multidimensional systems: a test algorithm based on Monte-Carlo integration. Multidim Syst Sign Process 34, 479–502 (2023). https://doi.org/10.1007/s11045-023-00869-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11045-023-00869-9