Geodesic
The investigation of nonlinear polynomial control systems
Modelirovanie i analiz informacionnyh sistem, Tome 28 (2021) no. 3, pp. 238-249 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The paper considers methods for estimating stability using Lyapunov functions, which are used for nonlinear polynomial control systems. The apparatus of the Gröbner basis method is used to assess the stability of a dynamical system. A description of the Gröbner basis method is given. To apply the method, the canonical relations of the nonlinear system are approximated by polynomials of the components of the state and control vectors. To calculate the Gröbner basis, the Buchberger algorithm is used, which is implemented in symbolic computation programs for solving systems of nonlinear polynomial equations. The use of the Gröbner basis for finding solutions of a nonlinear system of polynomial equations is considered, similar to the application of the Gauss method for solving a system of linear equations. The equilibrium states of a nonlinear polynomial system are determined as solutions of a nonlinear system of polynomial equations. An example of determining the equilibrium states of a nonlinear polynomial system using the Gröbner basis method is given. An example of finding the critical points of a nonlinear polynomial system using the Gröbner basis method and the Wolfram Mathematica application software is given. The Wolfram Mathematica program uses the function of determining the reduced Gröbner basis. The application of the Gröbner basis method for estimating the attraction domain of a nonlinear dynamic system with respect to the equilibrium point is considered. To determine the scalar potential, the vector field of the dynamic system is decomposed into gradient and vortex components. For the gradient component, the scalar potential and the Lyapunov function in polynomial form are determined by applying the homotopy operator. The use of Gröbner bases in the gradient method for finding the Lyapunov function of a nonlinear dynamical system is considered. The coordination of input-output signals of the system based on the construction of Gröbner bases is considered.
Keywords: nonlinear systems, polynomial systems, Lyapunov functions, Gröbner bases. 
@article{MAIS_2021_28_3_a2,
     author = {S. N. Chukanov and I. S. Chukanov},
     title = {The investigation of nonlinear polynomial control systems},
     journal = {Modelirovanie i analiz informacionnyh sistem},
     pages = {238--249},
     year = {2021},
     volume = {28},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/MAIS_2021_28_3_a2/}
}
TY  - JOUR
AU  - S. N. Chukanov
AU  - I. S. Chukanov
TI  - The investigation of nonlinear polynomial control systems
JO  - Modelirovanie i analiz informacionnyh sistem
PY  - 2021
SP  - 238
EP  - 249
VL  - 28
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/MAIS_2021_28_3_a2/
LA  - en
ID  - MAIS_2021_28_3_a2
ER  - 
%0 Journal Article
%A S. N. Chukanov
%A I. S. Chukanov
%T The investigation of nonlinear polynomial control systems
%J Modelirovanie i analiz informacionnyh sistem
%D 2021
%P 238-249
%V 28
%N 3
%U http://geodesic.mathdoc.fr/item/MAIS_2021_28_3_a2/
%G en
%F MAIS_2021_28_3_a2
S. N. Chukanov; I. S. Chukanov. The investigation of nonlinear polynomial control systems. Modelirovanie i analiz informacionnyh sistem, Tome 28 (2021) no. 3, pp. 238-249. http://geodesic.mathdoc.fr/item/MAIS_2021_28_3_a2/

[1] N. N. Krasovsky, Problems of the theory of stability of motion, Mir, M., 1959

[2] A. Papachristodoulou, S. Prajna, “On the construction of Lyapunov functions using the sum of squares decomposition”, Proceedings of the 41st IEEE conference on decision and control (2002), v. 3, IEEE, 2002, 3482–3487 | DOI

[3] K. Forsman, “Construction of Lyapunov functions using Gröbner bases”, 1991 proceedings of the 30th IEEE conference on decision and control, IEEE, 1991, 798–799

[4] S. N. Chukanov, D. V. Ulyanov, “Decomposition of the vector field of control system by constructing a homotopy operator”, Probl. Upr., 6 (2012), 2–6

[5] D. G. Edelen, Applied exterior calculus, John Wiley Sons, Inc., 1985 | Zbl

[6] X. Liao, L. Wang, P. Yu, Stability of dynamical systems, Elsevier, 2007 | Zbl

[7] H. K. Khalil, Nonlinear systems, Prentice hall, 2002 | Zbl

[8] D. Nesic, I. Mareels, T. Glad, M. Jirstrand, “The Gröbner basis method in control design: an overview”, IFAC proceedings volumes, 35:1 (2002), 13-18 | DOI

[9] B. Buchberger, “Ein algorithmisches kriterium für die lösbarkeit eines algebraischen gleichungssystems”, Aequationes Muthematicae, 4:3 (1970), 374–383 | DOI | MR | Zbl

[10] J. L. Awange, é Paláncz, R. H. Lewis, L. Völgyesi, Mathematical geosciences – hybrid symbolic-numeric methods, Springer International Publishers, 2018 | Zbl

[11] S. Wolfram, The mathematica book, Wolfram Media, 2003

[12] M. Demenkov, “A Matlab tool for regions of attraction estimation via numerical algebraic geometry”, 2015 international conference on mechanics-seventh Polyakhov's reading, 2015, 1–5

[13] N. Sidorov, D. Sidorov, Y. Li, “Basins of attraction and stability of nonlinear systems equilibrium points”, Differential Equations and Dynamical Systems, Springer, 2019, 1––10

[14] D. G. Luenberger, Y. Ye, Linear and nonlinear programming, Springer, 2016 | Zbl