Geodesic
A fast numerical test of multivariate polynomial positiveness with applications
Kybernetika, Tome 54 (2018) no. 2, pp. 289-303
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

The paper presents a simple method to check a positiveness of symmetric multivariate polynomials on the unit multi-circle. The method is based on the sampling polynomials using the fast Fourier transform. The algorithm is described and its possible applications are proposed. One of the aims of the paper is to show that presented algorithm is significantly faster than commonly used method based on the semi-definite programming expression.
The paper presents a simple method to check a positiveness of symmetric multivariate polynomials on the unit multi-circle. The method is based on the sampling polynomials using the fast Fourier transform. The algorithm is described and its possible applications are proposed. One of the aims of the paper is to show that presented algorithm is significantly faster than commonly used method based on the semi-definite programming expression.
DOI : 10.14736/kyb-2018-2-0289
Classification : 12D10, 47N70, 65T50, 65Y20
Keywords: multidimensional systems; positive polynomials; fast Fourier transforms; stability; numerical algorithm 
@article{10_14736_kyb_2018_2_0289,
     author = {Augusta, Petr and Augustov\'a, Petra},
     title = {A fast numerical test of multivariate polynomial positiveness with applications},
     journal = {Kybernetika},
     pages = {289--303},
     year = {2018},
     volume = {54},
     number = {2},
     doi = {10.14736/kyb-2018-2-0289},
     mrnumber = {3807716},
     zbl = {06890421},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.14736/kyb-2018-2-0289/}
}
TY  - JOUR
AU  - Augusta, Petr
AU  - Augustová, Petra
TI  - A fast numerical test of multivariate polynomial positiveness with applications
JO  - Kybernetika
PY  - 2018
SP  - 289
EP  - 303
VL  - 54
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.14736/kyb-2018-2-0289/
DO  - 10.14736/kyb-2018-2-0289
LA  - en
ID  - 10_14736_kyb_2018_2_0289
ER  - 
%0 Journal Article
%A Augusta, Petr
%A Augustová, Petra
%T A fast numerical test of multivariate polynomial positiveness with applications
%J Kybernetika
%D 2018
%P 289-303
%V 54
%N 2
%U http://geodesic.mathdoc.fr/articles/10.14736/kyb-2018-2-0289/
%R 10.14736/kyb-2018-2-0289
%G en
%F 10_14736_kyb_2018_2_0289
Augusta, Petr; Augustová, Petra. A fast numerical test of multivariate polynomial positiveness with applications. Kybernetika, Tome 54 (2018) no. 2, pp. 289-303. doi: 10.14736/kyb-2018-2-0289

[1] Ali, A., Ali, M.: Identification of Box-Jenkins model for spatially interconnected systems in closed-loop. Multidimensional Systems Signal Process. 28 (2016), 1-11. | DOI | MR

[2] Augusta, P.: A numerical test of positiveness on the unit circle based on the fast Fourier transform. In: Proc. 21st Mediterranean Conference on Control and Automation, 2013, pp. 1050-1054. | DOI

[3] Augusta, P.: A simple method for stabilisation of (2+1)D systems. In: Proc. 8th International Workshop on Multidimensional Systems, 2013, pp. 163-167.

[4] Augusta, P., Augustová, P.: On stabilisability of 2-D MIMO shift-invariant systems. J. Franklin Inst. 350 (2013), 2949-2966. | DOI | MR

[5] Augusta, P., Cichy, B., Galkowski, K., Rogers, E.: An unconditionally stable finite difference scheme systems described by second order partial differential equations. In: Proc. IEEE 9th International Workshop on Multidimensional Systems, 2015, pp. 134-139. | DOI

[6] Augusta, P., Cichy, B., Galkowski, K., Rogers, E.: An unconditionally stable approximation of a circular flexible plate described by a fourth order partial differential equation. In: Proc. 21st International Conference on Methods and Models in Automation and Robotics, 2016. | DOI

[7] Augusta, P., Hurák, Z.: Distributed stabilisation of spatially invariant systems - positive polynomial approach. Multidimensional Systems Signal Process. 24 (2013), 3-21. | DOI | MR

[8] Bose, N. K.: Multidimensional Systems Theory - Progress, Directions and Open Problems in Multidimensional Systems. D. Riedel Publishing Company, 1985. | DOI | MR

[9] Bose, N. K.: Multidimensional Systems Theory and Applications. Second edition. Kluwer Academic Publishers, 2003. | MR

[10] Cichy, B., Gałkowski, K., Rogers, E.: Iterative learning control for spatio-temporal dynamics using Crank-Nicolson discretization. Multidimensional Systems Signal Process. 23 (2012), 185-208. | MR

[11] Cichy, B., Gałkowski, K., Rogers, E., Kummert, A.: An approach to iterative learning control for spatio-temporal dynamics using nD discrete linear systems models. Multidimensional Systems Signal Process. 22 (2011), 83-96. | DOI | MR

[12] Cichy, B., Hladowski, L., Gałkowski, K., Rauh, A., Aschemann, H.: Iterative learning control of an electrostatic microbridge actuator with polytopic uncertainty models. IEEE Trans. Control Systems Technol. 23 (2015), 2035-2043. | DOI

[13] Cooley, J. W., Tukey, J. W.: An algorithm for the machine calculation of complex Fourier series. Mathematics of Computation 19 (1965), 297-301. | DOI | MR

[14] Dumitrescu, B.: Sum-of-squares polynomials and the stability of discrete-time systems. In: Proc. Fourth International Workshop on Multidimensional Systems, 2005, pp. 223-228. | DOI

[15] Dumitrescu, B.: Stability test of multidimensional discrete-time systems via sum-of-squares decomposition. IEEE Trans. Circuts Systems 53 (2006), 928-936. | DOI | MR

[16] Dumitrescu, B.: Positive Trigonometric Polynomials and Signal Processing Applications. Springer, 2007. | MR

[17] Dumitrescu, B.: Positivstellensatz for trigonometric polynomials and multidimensional stability tests. IEEE Trans. Circuits Systems 54 (2007), 353-356. | DOI

[18] Dumitrescu, B., Şicleru, B. C., Ştefan, R.: Minimax design of adjustable FIR filters using 2D polynomial methods. In: Proc. IEEE International Conference on Acoustics, Speech and Signal Processing, 2009, pp. 3181-3184. | DOI

[19] Henrion, D., Garulli, A.: Positive Polynomials in Control.

[20] Kaczorek, T.: Positive 1D and 2D Systems. Springer, London 2002. | DOI

[21] Kaczorek, T.: Selected Problems of Fractional Systems Theory. Springer, Berlin 2011. | MR | Zbl

[22] Löfberg, J.: Yalmip - a toolbox for modeling and optimization in MATLAB. In: Proc. CACSD Conference, 2004. | DOI

[23] MATLAB: version 9.2.0.538062 (R2017a). MathWorks, Natick 2017.

[24] Paszke, W.: Analysis and Synthesis of Multidimensional System Classes Using Linear Matrix Inequality Methods. University of Zielona Góra Press, Zielona Góra 2005. | MR

[25] Rabenstein, R., Steffen, P.: Numerical iterative methods and repetitive processes. Multidimensional Systems Signal Process. 23 (2012), 163-183. | DOI | MR

[26] Ramos, J., Mercère, G.: Subspace algorithm for identifying separable in denominator 2-D systems with deterministic-stochastic inputs. Int. J. Control 89 (2016), 2584-2610. | DOI | MR

[27] Rogers, E., Gałkowski, K., Owens, D. H.: Control Systems Theory and Applications for Linear Repetitive Processes. Springer, 2007. | MR

[28] Rogers, E., Gałkowski, K., Paszke, W., Moore, K. L., Bauer, P. H., Hladowski, L., Dabkowski, P.: Multidimensional control systems - case studies in design and evaluation. Multidimensional Systems Signal Process. 26 (2015), 895-939. | DOI | MR

[29] Strintzis, M. G.: Test of stability of multidimensional filters. IEEE Trans. Automat. Control 24 (1977), 432-437. | DOI | MR

[30] Sturm, J. F.: Using SeDuMi 1.02, a Matlab toolbox for optimization over symmetric cones. Optimiz. Methods Software 11 (1999), 625-653. | DOI | MR

[31] Sulikowski, B., Galkowski, K., Kummert, A.: Proportional plus integral control of ladder circuits modeled in the form of two-dimensional (2D) systems. Multidimensional Systems Signal Process. 26 (2015), 267-290. | DOI | MR

[32] Sulikowski, B., Galkowski, K., Kummert, A.: Stability and stabilisation of active ladder circuits modeled in the form of two-dimensional (2D) systems. In: Proc. IEEE 9th International Workshop on Multidimensional Systems, 2015. | DOI | MR

[33] Zhang, F.: The Schur Complement and its Applications. Springer, 2005. | DOI | MR

Cité par Sources :